Wiley Capital Asset Investment Strategy, Tactics

Capital Asset Investment

Strategy, Tactics & Tools

Anthony F. Herbst

JOHN WILEY & SONS, LTD

Copyright C© 2002 John Wiley & Sons Ltd, Baffins Lane, Chichester,West Sussex PO19 1UD, England

Telephone (+44) 1243 779777

Email (for orders and customer service enquiries): [email protected] our Home Page on www.wiley.co.uk or www.wiley.com

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system ortransmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning orotherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms ofa licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, W1P 0LP,UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed tothe Permissions Department, John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex PO191UD, England, or emailed to [email protected], or faxed to (+44) 1243 770571.

This publication is designed to provide accurate and authoritative information in regard to the subjectmatter covered. It is sold on the understanding that the Publisher is not engaged in renderingprofessional services. If professional advice or other expert assistance is required, the services of acompetent professional should be sought.

Other Wiley Editorial Offices

John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA

Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA

Wiley-VCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany

John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia

John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809

John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1

Library of Congress Cataloging-in-Publication Data

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0-470-84511-2

Typeset in 10/12 pt Times by TechBooks, New Delhi, IndiaPrinted and bound in Great Britain by Biddles Ltd, Guildford and King’s LynnThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

Contents

Preface xiii

Acknowledgments xv

1 Introduction: the big picture, environment, terminology, and preview 1Magnitude of capital investment 1General perspective on capital investment 3Capital budgeting 5Cash flows 6Cost of capital 6Risk and uncertainty 7

Risk 7Uncertainty 8

2 The Objective of capital budgeting 9A normative model for capital budgeting 10Basic valuation model 11

Operational adaptation 11The cash flows 12

Cash flows and the public sector 12

3 Estimating basic project characteristics 13Project types 13Project characteristics 14Initial cost 15

Sunk cost 15Components of initial cost 16

Useful life 17Physical deterioration 18Technological obsolescence 19

Cash flows 19Cash inflows (cash receipts) 21Cash outflows 22

vi Contents

Taxes and depreciation 23Depreciation 23Straight-line depreciation 24Double-declining-balance depreciation 24Sum-of-the-years’ digits depreciation 24Comparison of the basic depreciation methods 25Example 3.1 25Example 3.2 26Example 3.3 26ACRS Depreciation 27Canadian depreciation 28Investment tax credit 29Inflation 30

4 Cost of capital 31Introduction 31Cost of capital components 31

Debt 31Preferred stock 32Common stock and retained earnings 33Example 4.1 34

Overall cost of capital 34Optimal capital structure 36Interaction of financing and investment 38Cautionary note 38

5 Traditional methods that ignore time-value of money 39Payback and naive rate of return 39

Payback 39The naive rate of return 41Strong points of payback 41Weak points of payback 41Unrecovered investment 43

Accounting method: alias average return on average investment 43Strong points of accounting method 43Weak points of accounting method 44

Comprehensive example 44

6 Traditional methods that recognize time-value of money: the net present value 47Unequal project size 51

The profitability index 51Unequal project lives 51

Level annuities 53Summary and conclusion 53

Strong points of NPV 53Weak points of NPV 54

Contents vii

7 Traditional methods that recognize time-value of money: the internal rateof return 55Definition of the IRR 55A caution and a rule for IRR 58Payback and IRR relationship 58Mathematical logic for finding IRR 59

Interval bisection 60Newton–Raphson method 61Strong points of IRR 62Weak points of IRR 62

A digression on nominal and effective rates 62Investment–financing relationship 62Nominal rate and effective rate 63Clarification of nominal and effective rates 63IRR with quarterly cash flows 64

8 Reinvestment rate assumptions for NPV and IRR and conflictingrankings 65Reinvestment rate assumptions for NPV and IRR 65Conflicting rankings and fisher’s intersection 68Relationship of IRR and NPV 72Adjusted, or modified, IRR 72Summary and conclusion 73

9 The MAPI method 75The concept of duality 75The MAPI framework 77

Challenger and defender 77Capital cost 77Operating inferiority 78Physical deterioration 79Technological obsolescence 79Two basic assumptions 79Adverse minimum 80First standard assumption 80Second standard assumption 80

Application of the MAPI method 80The problem of capacity disparities 82

Conclusion 83

10 The problem of mixed cash flows: I 85Internal rate of return deficiencies 85

Example 10.1 85Descartes’ rule 85

Example 10.2 86The Teichroew, Robichek, AND Montalbano (TRM)analysis 87

viii Contents

The TRM algorithm 89Example 10.3 89Example 10.4 91Example 10.5 92

The unique, real internal rate of return: caveat emptor! 94A new theorem 96

Theorem 96Proof 96Corollary I 96

11 The problem of mixed cash flows: II 97The Wiar method 97

Example 11.1. An application of the Wiar method 98Sinking fund methods 99

The initial investment method 100The traditional sinking fund method 100Initial investment and traditional sinking fund methods 100Example 11.2 101Example 11.3 102Example 11.4 103Example 11.5 104

The multiple investment sinking fund method 107Strong points of the methods 108Weak points of the methods 109

11A Appendix: the problem of mixed cash flows III — a two-stage methodof analysis 110Relationship to other methods 110The two-stage method 111

Example 11A.1 111Example 11A.2 114

Formal definition and relationship to NPV 114Payback stage 115“Borrowing” rate 116Example 11A.3 116

Conclusion 117A brief digression on uncertainty 117

12 Leasing 119Alleged advantages to leasing 119Analysis of leases 120

Traditional analysis 120Example 12.1 121Alternative analysis 122Analysis 124Implications 125

Contents ix

Practical perspective 126Summary and conclusion 127

Appendix 128

13 Leveraged leases 131Definition and characteristics 131Methods: leveraged lease analysis 132Application of the methods 132

Example 13.1 132Example 13.2 133

Analysis of a typical leveraged lease (Childs and Gridley) 137Example 13.3 137

Conclusion 139

14 Alternative investment measures 141Additional rate of return measures 141

Geometric mean rate of return 141Average discounted rate of return 142Example 14.1 142

Time-related measures in investment analysis 144Boulding’s time spread 145Macaulay’s duration 146Unrecovered investment 147

Summary and conclusion 148

15 Project abandonment analysis 149The Robichek–Van Horne analysis 149An alternate method: a parable 151

Comparison to R–VH 155A dynamic programming approach 156Summary and conclusion 158

16 Multiple project capital budgeting 159Budget and other constraints 159General linear programming approach 159

Mutual exclusivity 161Contingent projects 161

Zero–one integer programming 162Example 16.1 164

Goal programming 165Summary and conclusion 169

16A Appendix to multiple project capital budgeting 170

17 Utility and risk aversion 177The concept of utility 177

x Contents

Attitudes toward risk 178Calculating personal utility 180

Whose utility? 184Measures of risk 185

J. C. T. Mao’s survey results 187Risk of ruin 188Summary 189

18 Single project analysis under risk 191The payback method 191Certainty equivalents: method I 192

Example 18.1 192Certainty equivalents: method II 193Risk-adjusted discount rate 194

Example 18.2 195Computer simulation 195

Example 18.3 196Lewellen–Long criticism 200

19 Multiple project selection under risk: computer simulation and otherapproaches 203Decision trees 203

Example 19.1 205Other risk considerations 207

Example 19.2 A comprehensive simulation example 207Conclusion 214

20 Multiple project selection under risk: portfolio approaches 215Introduction 215

Example 20.1 215Generalizations 216Project independence 217Project indivisibility 217

Multiple project selection 218Finding the efficient set 220The Sharpe modification 222

Relating to investor utility 223Epilogue 223

21 The capital asset pricing model 225Assumptions of the CAPM 225The efficient set of portfolios 225

Portfolio choices 226Enter a risk-free investment 226

The security market line and beta 228The CAPM and valuation 230The CAPM and cost of capital 231

Contents xi

The CAPM and capital budgeting 232Comparison with portfolio approaches 232Some criticisms of the CAPM 233The arbitrage pricing theory (APT) 234

Factors — what are they? 235APT and CAPM 236Development of the APT 237

22 Multiple project selection under risk 239Example 22.1: Al’s Appliance Shop, revisited 240Generalizations 241

Project independence — does it really exist? 242Project indivisibility: a capital investment is not a security 243

Example 22.2: Noah Zark 244Generalization on multiple project selection 246

Securities 247Capital investments 247

Summary and conclusion 248

23 Real options 251Acquiring and disposing: the call and put of it 251More than one way to get there 252Where are real options found? 252Comparison to financial options 253Flashback to PI ratio — a rose by another name . . . 253Types of real options 254Real option solution steps 255Option phase diagrams 256Complex projects 257Estimating the underlying value 258What to expect from real option analysis 258Identifying real options — some examples 259Contingent claim analysis 260The binomial option pricing model 260

Example 23.1 Value of strategic flexibility in a ranch/farm 262A game farm, recreational project 264

Option to switch operation 265Case 1: 100 percent switch 265Case 2: Mix operation 266Option to abandon for salvage value 267The option to expand operation (growth option) 268Interaction among strategic options 269Example 23.2 270Contingent claim valuation: 270The optimal exercise of the growth option and firm value 271The abandon option and firm value 272Current value of the firm 273

xii Contents

The strategic dimension of the real-option analysis 273Conclusion 273

Appendix: Financial mathematics tables and formulas 275A.1 Single payment compound amount. To find F for a given P 276A.2 Single payment present worth factor. To find P for a given F 283A.3 Ordinary annuity compound amount factor. To find F for a given R

received at the end of each period 291A.4 Ordinary annuity present worth factor. To find P for a given R received at

the end of each period 299

Bibliography 307

Index 315

Preface

The aim of this book is to tie together the theory, quantitative methods, and applications ofcapital budgeting. Consequently, its coverage omits few, if any, topics important to capitalinvestment. My intention is to effect a harmonious blend of the old, such as the MAPI methodof capital investment appraisal, with the new, such as the capital asset pricing model (CAPM).I have tried to provide a balanced treatment of the different approaches to capital projectevaluation and have explored both the strengths and weaknesses of various project selectionmethods.

A work on this subject necessarily uses mathematics, but the level of mathematical sophis-tication required here is generally not above basic algebra. Although I have favored clarity andreadability over mathematical pyrotechnics, the book’s level of mathematical rigor should besufficiently high to satisfy most users.

The book’s treatment of risk is deliberately deferred to later chapters. The decision to dothis, rather than treating risk earlier, was based on my belief that readers new to the subjectare less overwhelmed by the added complexities of risk considerations — and better able tocomprehend them — after they have become thoroughly familiar with capital budgeting in anenvironment assumed to be risk-free.

In Chapter 21 I try to present a balanced treatment of the CAPM, including some of theimportant criticisms of its use in capital budgeting. I know that some readers might prefer anearlier introduction of the CAPM, as well as its subsequent use as a unifying theme. I chosenot to employ that structure for three reasons.

First, although the CAPM may be adaptable to capital budgeting decisions involving majorprojects (e.g. the acquisition of a new company division), serious questions exist concerning itsapplicability to more typical projects for which estimation of expected returns alone is difficult,to say nothing of also estimating the project’s beta.

Second, company managers increasingly appear to be placing primary emphasis on thesurvival of the firm rather than on consideration of systematic risk in their capital investmentdecisions, thereby diluting the implications of the CAPM. In other words, top managementdoes indeed care about unsystematic (or company) risk, to which portfolio diversification may,in some cases, give little importance. To the management of a company, such risk may notalways be reduced easily, and, if neglected, may imperil the company.

Third, the CAPM is concerned with risk. For the reasons stated earlier, I felt that the bookwould better serve its audience if it examined capital investment under assumed uncertaintyfirst, without the added complexities that a simultaneous treatment of risk would entail.

xiv Preface

For those who may wish to obtain them I have developed a computer program and spreadsheettemplates for several applications illustrated in the book. Also, for those who adopt the book forteaching, I have a separate book of end-of-chapter questions and problems, and an instructor’ssolutions manual. Please contact the following website http://www.utep.edu/futures

A. F. H.Department of Economics and Finance/CBAThe University of Texas at El PasoEl Paso, Texas 79968-0543

E-mail: [email protected]

Acknowledgments

At this stage of my life, many of those who contributed to the development of the person I am,and the accomplishments that I have achieved, are no longer here to receive my expression ofappreciation. Yet I would be remiss if I did not mention them anyway. First are my parents,who sacrificed that I obtain a university education, and who I think would be pleased withwhat their efforts achieved. Second is Joseph Sadony Jr. who, upon my father’s illness, tookup the task of keeping the teenage me on track to success later in life. And I should mentionHarvey Nussbaum, a professor at Wayne State University in Detroit who nudged me into theacademic life from my career in industry and banking. Then there is my wife Betty, who nolonger types my manuscripts, but is supportive in many other ways that facilitate my work. Itis to these persons, and my children Mya and Geoff that I dedicate this work.

Many persons contributed to the technical, academic content of this work. Professor JamesC. T. Mao merits special mention for inspiring me early in my teaching a research careerwith his work in quantitative analysis of financial decisions to undertake my own work in thatrealm. Numerous of my students and colleagues contributed to this work in various ways, fromencouraging me to do it, to helping with it. Special recognition belongs to Jang-Shee (Barry)Lin for co-authoring Chapter 23, and to Marco Antonio G. Dias of Brazil, for reviewing thatchapter.

1Introduction: the Big Picture,

Environment, Terminology, and Preview

Once one consciously thinks about the nature of capital investment decisions, it becomesapparent that such decisions have been made for millennia, since humans first awakened to theidea that capital1 could improve life. The earliest investment decisions involved matters thattoday would be considered very primitive. But to the early nomadic hunter-forager the firstcapital investment decisions were quite significant. To the extent that time and energy had to bediverted from the immediate quest for food, short-term shelter, and the production of tools forthe hunt, into defensive installations, food storage facilities, and so on, capital investments weremade. Such fundamental capital creation required significant time and effort. And the benefitsthat could have been expected to result were uncertain; it took foresight and determination tobuild capital.

As society evolved, the benefits of capital accumulation gradually became more indirect andcomplex, involving specialization and cooperation not previously envisioned, and the associ-ated commitment of resources more permanent. Additionally, social norms and institutionshad to be developed to facilitate the evolution. For example, the changeover from nomadicto agrarian life required a great increase in capital in the form of land clearing, constructionof granaries, mills, irrigation canals, tools, and fortifications. Fortifications were necessary todeter those outsiders who would use force to seize the benefits achieved by investment. Thechangeover required a commitment that tended to be irreversible, at least in the short run. Andit became more and more irreversible as the changes caused social and economic institutions toadapt or be developed to support it and coordinate the various requisite activities. (C. NorthcoteParkinson writes of such differences between agricultural and nomadic societies in his book[122].)

MAGNITUDE OF CAPITAL INVESTMENT

In the United States in 2001, business capital expenditure on new plant and equipment amountedto more than 20 percent of gross domestic product (GDP) according to data from the FederalReserve Bank of St Louis [47]. In the United Kingdom in 2001 the percentage of GDP ac-counted for by investment was slightly less, between 15 and 20 percent. Comparable statisticsfor Germany and France are 20–22 percent; Canada and Italy about 20 percent, and Japanin a category by itself at more than 25 percent. Figure 1.1 displays saving and investment inthe G7 industrial nations. It is apparent that capital investment is a very important compo-nent of GDP in every one of these countries. The larger share of GDP allocated to savingand investment by Japan may arise from that nation’s relatively lesser spending on militaryhardware and weapons development, and also the relative emphasis on electronics manufacture

1The word “capital” is used in several senses. It may refer to physical plant and equipment (economic capital) or to the ownershipclaims on the tangible capital (financial capital). In this book, unless otherwise indicated in a specific instance, the word shall refer tophysical or economic capital.

2 Capital Asset Investment: Strategy, Tactics & Tools

25Perc

ent o

f G

DP

Uni

ted

Stat

es S

avin

g an

d In

vest

men

tSo

urce

: St

. Lou

is F

eder

al R

eser

ve B

ank,

Int

erna

tiona

l Eco

nom

ic T

rend

s, A

ugus

t, 2

001.

Perc

ent o

f G

DP

Uni

ted

Kin

gdom

Sav

ing

and

Inve

stm

ent

Sour

ce:

St. L

ouis

Fed

eral

Res

erve

Ban

k, I

nter

natio

nal E

cono

mic

Tre

nds,

Aug

ust,

200

1.

20 1584

8586

8788

8990

9192

9394

9596

9798

9900

01

Inve

stm

ent

Savi

ng

25 20 15 1083

8485

8687

8889

9091

9293

9495

9697

9890

00

Inve

stm

ent

Inve

stm

ent

Savi

ng

Savi

ng

35Perc

ent o

f G

DP

Japa

n Sa

ving

and

Inv

estm

ent

Sour

ce:

St. L

ouis

Fed

eral

Res

erve

Ban

k, I

nter

natio

nal E

cono

mic

Tre

nds,

Aug

ust,

200

1.

30 2583

8485

8687

8889

9091

9293

9495

9697

9899

00

Inve

stm

entSa

ving

26Perc

ent o

f G

DP,

ann

ual d

ata

Ger

man

y Sa

ving

and

Inv

estm

ent

Sour

ce:

St. L

ouis

Fed

eral

Res

erve

Ban

k, I

nter

natio

nal E

cono

mic

Tre

nds,

Aug

ust,

200

1.

24 22 20 1884

8586

8788

8990

9192

9394

9596

9798

99

Inve

stm

ent

Savi

ng

25Perc

ent o

f G

DP

Can

ada

Savi

ng a

nd I

nves

tmen

tSo

urce

: St

. Lou

is F

eder

al R

eser

ve B

ank,

Int

erna

tiona

l Eco

nom

ic T

rend

s, A

ugus

t, 2

001.

20 15 1083

8485

8687

8890

8991

9293

9495

9697

9899

00

Inve

stm

ent

Savi

ng

24Perc

ent o

f G

DP

Fra

nce

Savi

ng a

nd I

nves

tmen

tSo

urce

: St

. Lou

is F

eder

al R

eser

ve B

ank,

Int

erna

tiona

l Eco

nom

ic T

rend

s, A

ugus

t, 2

001.

22 20 18 1683

8485

8687

8889

9190

9293

9495

9697

9899

00

Inve

stm

ent

Savi

ng

25Perc

ent o

f G

DP

Ital

y Sa

ving

and

Inv

estm

ent

Sour

ce:

St. L

ouis

Fed

eral

Res

erve

Ban

k, I

nter

natio

nal E

cono

mic

Tre

nds,

Aug

ust,

200

1.

23 21 19 1783

8485

8687

8889

9190

9293

9495

9697

9899

00

Fig

ure

1.1

Savi

ngan

din

vest

men

tfor

G7

indu

stri

alna

tions

Introduction: the Big Picture, Environment, Terminology, and Preview 3

Table 1.1 Saving and investment as a percent of gross domestic product, 1973–97

United West UnitedStates Canada Japan France Germanya Germanyb Kingdom

SavingNet savingc 6.6 7.7 18.5 9.3 10.4 7.9 4.7Personal savingd 6.2 7.6 11.6 7.8 8.1 7.8 4.0Gross saving

(net savingplus cons. offixed capital)e 17.5 19.7 32.5 21.6 22.6 20.9 15.9

InvestmentGross

nonresidentialfixed capitalformation 13.8 15.1 24.0 14.8 14.5 14.9 14.0

Gross fixedcapitalformation 18.3 21.1 30.2 21.0 20.5 21.6 17.6

a The statistics for West Germany refer to western Germany (Federal Republic of Germany before unification).The data cover the years 1973–95.b The statistics for Germany refer to Germany after unification. The data cover the years 1991–97.c The main components of the OECD definition of net saving are: personal saving, business saving (undis-tributed corporate profits), and government saving (or dissaving). The OECD definition of net saving differsfrom that used in the National Income and Product Accounts published by the Department of Commerce,primarily because of the treatment of government capital formation.d Personal saving is comprised of household saving and private unincorporated enterprise.e The main components of the OECD definition of consumption of fixed capital are the capital consumptionallowances (depreciation charges) for both the private and the government sector.Source: Derived from National Accounts, Organization for Economic Cooperation and Development (OECD)Statistical Compendium 2000. Prepared by the American Council for Capital Formation Center for PolicyResearch, Washington, DC, June 2001.

that requires hefty capital equipment investment to produce items with short product lifecycles.

Table 1.1 reinforces the impression of the graphs in Figure 1.1. It contains comparativesaving and investment statistics for six of the major industrialized nations over the span 1973–97. It is apparent that in every case gross nonresidential fixed investment is greater than 1/8of GDP, ranging from a low of 13.8 percent for the United States to a high of 24.0 percent forJapan. This category takes into account only the plant and equipment and other fixed assets2

(i.e. depreciable assets, lasting more than a year). Important investments that are not easilymeasured are excluded, but are nevertheless important, such as “human capital” and technologyresearch and development.

GENERAL PERSPECTIVE ON CAPITAL INVESTMENT

Because resources are scarce (as everyone learns in a first course in economics) and becausecapital investment figures so prominently in the economy, decisions on capital budgets ought

2Housing, a major component of investment, is excluded from this measure but included in the bottom row item, gross fixed capitalinvestment.


to be made on a sound, rational basis. The general irreversibility of capital investments, andtheir legacy for future costs as well as benefits, make such decisions of great importance bothfrom the standpoint of the individual firm and at the level of national policy.

For any given level of technology, considerations arising from unemployment, and fromdemands for increased standards of living, are important. These require that for given levelsof technology the stock of capital goods increases apace with growth in the labor supply or, iftechnological growth tends toward the more capital-intensive side, to exceed it. As long as thegoals are the same, this will hold for every form of government because political philosophycannot change the feasible mix of land, labor, and capital for a given technology.

Few in business and industry would dispute the view that the press and media generally tendto emphasize labor’s employment problems while generally ignoring the problems of under-employment of capital, reduced or negative capital accumulation, and return on investment.Such imbalanced editorial policy, however, is understandable to the extent it exists becausemuch of the public identifies their interests with labor rather than with capital. However, it rein-forces popular notions that, although not clearly incorrect, are at least suspect and thus tend tocontribute further to problems faced by labor and capital alike. The notion that the interests oflabor and capital are mutually exclusive may be useful to rhetoricians at the political extremes,but it fails to address the problems of underemployment and sagging productivity in societiesin which labor and capital are undichotomized. In the industrial nations of North America,Europe, and Asia, for example, it is common for workers to own shares in their own employeror other corporations either directly or through their pension funds. Worker representation oncorporate boards has been accepted in some European nations for many years and the practiceseems to be growing. Thus the interests of labor and capital are not easily separable, if at all,because so many are neither entirely laborer nor entirely capitalist.

A proper blend of the factors of production is necessary to minimize unemployment of laborwithout resorting to government “make work” projects. But it depends on wages that reflectproductivity and product value. Adoption of government policies that ignore capital in orderto aim at direct treatment of labor unemployment, or low minimum wages, is bound to failin a broader sense. Still, such policies are often advocated, whereas the capital investmentnecessary to create a real expansion in jobs is largely ignored. Return on capital investment inthe industrial nations today tends to be generally less than can be earned on the same fundsif put into government bonds, after taking capital consumption into account. This has theeffect of dampening enthusiasm for capital investment and consequently moving funds to lessproductive uses. Since capital must complement labor, fewer productive jobs are created, whichleads to higher labor unemployment, more public policies to treat the symptoms, and still moredisincentive to capital investment. In fact, studies at the University of Chicago concluded thatin the United States, capital was being consumed at a faster rate than it was being created;in other words, industry was paying liquidating dividends. This has serious implications forgovernment economic policies.

Concern that capital will replace labor may be legitimate in the short run, if not longer term.If the relative costs of labor and capital should favor a capital-intensive production mix offactors, this should present no problem provided the distribution of benefits is equitable. Inother words, replacement of some labor by capital may increase the quality of life for workersprovided that the benefits are shared by labor. The dismissal of some workers while othersare put on extended overtime or continue to work a normal number of hours does not furtherequitable distribution. But a reduction in the standard work week for everyone may. If it werenot for capital accumulation and improved technology yielding increased productivity, workers


would not have a 40-hour or less standard week but one of 60 hours or so, as was once thestandard. (Determination of what is equitable and what is not is outside the scope of this book.)

Most of the developing nations are capital-poor, but have abundant labor that is not highlyskilled or educated. For those countries it makes sense to adopt labor-intensive methods of pro-duction, gradually shifting to capital-intensive technologies as their capital stock is increasedand the quality of labor increases. In the developed, relatively capital-rich nations, it is notsensible to adopt labor-intensive methods of production because that would waste productivecapacity. It is ironical that while developing nations strive to accumulate capital, developednations often follow policies that discourage capital growth and may even cause the capitalstock to shrink, relatively if not absolutely. One may well wonder if it would not be morehonest and efficient simply to donate capital to the developing nations to raise their capitalintensity while lowering that of the donor nation.

Capital investment decisions have repercussions that may extend far beyond the immediatetime frame, because they involve long-term commitments that are not readily undone. Thedecision not to invest is a capital investment decision also. And future repercussions may becompounded by the often very long planning periods for capital goods and the fact that it takescapital and labor together to produce more capital goods. The “production process” to create“human capital” — skilled labor — is a lengthy one, and the more sophisticated the capitalgoods to be produced the higher the quality of human capital generally required.

CAPITAL BUDGETING

In the private sector, capital investment and the analysis it requires are generally referred to ascapital budgeting. Capital budgeting focuses on alternative measures of project acceptability.Tangible factors are emphasized. However, to the extent that their effects can be factoredinto the process, intangibles must be considered. Capital investment in the private sector hasperhaps tended to pay less attention to intangibles than its public sector counterpart, cost–benefit analysis. This may well be due to the inherently more qualitative nature of social andpolitical goals and constraints. Some would contend that intangibles in the public sector areoften exaggerated with the result that projects are undertaken that cost more than they yieldin benefits. But to the extent this is a problem the remedy falls under the rubric of politicalprocess rather than investment evaluation.

In private enterprise, profitability provides the principal criterion for the acceptability ofparticular prospective investment projects through the effect they are expected to have onthe market value of the enterprise. Capital budgeting centers on an objective function thatmanagement seeks to maximize, subject to various constraints. Often the primary constraint isimposed by the funds available for investment. When this constraint is binding, it is referred to ascapital rationing. In the public sector, a reasonable alternative is provided by minimizing a costfunction, constrained by specifying minimum levels of services to be provided. The aggregateeffect of private and public support for capital investment is to raise a nation’s standard ofliving, both in tangible benefits and in terms of intangible benefits such as greater security andless social stress. To the extent that the benefits accrue to a few rather than to the many, it isthe social and political institutions that are responsible, not the economic framework. (For alucid presentation of a rationale for aggressive national policy encouraging capital formation,see the classical work by E. A. G. Robinson [132].)

Emphasis throughout this book is on private sector capital budgeting. Yet, some of themethods covered are applicable without modification to public sector investment, and others


are applicable with modification. The following sections set forth some principles that will beused throughout the remainder of the book.

CASH FLOWS

Emphasis is on net, after-tax cash flows. Pre-tax net cash flow is defined as the total cashinflow associated with the capital investment less its net cash outflow. Then, after removingthe portion that will be paid out for income taxes we have the net, after-tax cash flows for thetime period over which we have measured or estimated the flows. Cash flows are estimated foreach period of a capital investment’s life. Some may well be zero. Normally the time perioddivisions will be annual, but they can be quarterly, semiannual, or for some other time periodif desired and sensible to do so. The major focus in this book is on what is to be done with thecash flow estimates once they are obtained, rather than on how to make cash flow estimates,although Chapter 3 does address that subject.

In practice, cash flow estimates will normally represent an amalgamation of experiencedjudgment by persons in such diverse functional areas as production engineering, marketing,and accounting more than the result of objective measurements and data analysis. We canestimate future results a priori, but of course we can only measure them ex post, after theyoccur. Managers who are responsible for capital budgeting must ensure that adequately preciseestimates are obtained and, where possible, objective, unbiased forecasts are prepared. It maybe necessary to emphasize to those participating in the process by which estimates are obtainedthat it is cash flows that are sought, not accrued profits or cost savings. It is cash that may beused to pay dividends, employee wages, and vendor bills and it is cash that may be investedin new plant and equipment. Likewise, it is cash flow on which methods of capital investmentanalysis are based.

An early survey in the US revealed that many firms, even some fairly large ones, failed toinclude all associated cash flows in their analysis. Any such omissions of relevant cash flowscan seriously bias the measures of project acceptability and destroy the usefulness of whatevertechniques are employed. It is vitally important that all relevant cash flows attributable to acapital project be included.

COST OF CAPITAL

The organization’s cost of capital, expressed as a decimal or percentage, is used in two basicways in capital budgeting: as a minimum profitability rate (i.e. hurdle rate) that prospectiveproject returns are required to exceed and as a discount rate applied to cash flows.

The literature dealing with cost of capital is extensive. However, the concept and the mea-surement of cost of capital are still somewhat unsettled. For our purposes, cost of capital willbe assumed to be independently derived, along the lines suggested in Chapters 4 and 21, ex-cept where it is explicitly stated otherwise. The importance of cost of capital should not beignored. An adequate estimate of cost of capital is crucial to properly apply capital-budgetingtechniques, because all but the crudest techniques incorporate it in one way or another.

In the author’s experience the importance of obtaining good estimates of the firm’s costof capital is often overlooked. The result is that capital budgeting may, in practice, becomesomewhat of a burlesque: sophisticated techniques yielding accept/reject decisions based oncrude and incorrect data. One of the largest industrial firms in the United States in the 1970swas using 8 percent as its overall, marginal cost of capital. At the time, by generally accepted


measure, the firm had a cost of capital above 10 percent. In capital investment methods thatemploy discounted cash flows, and these methods are all conceptually better, this was a seriouserror. Paradoxically, the same firm went to considerable effort to obtain finely detailed projectcash flow data from its marketing, engineering, production, and accounting staffs at the sametime. As a result, projects were undoubtedly undertaken that, had an adequate cost of capitalbeen used, would have been rejected. The stock market performance of this firm during the1970s, vis-a-vis comparable firms in its industry, tends to support this view.

RISK AND UNCERTAINTY

For better or worse we live in a world of probabilities, with little more certain than the proverbialdeath and taxes. In general, capital budgeting projects are no exception, although some specificclasses of capital investment, such as we find in leasing, may approach certainty sufficientlyto be treated as if they were risk free.

Risk

Risk is usually defined to prevail in situations in which, although exact outcomes cannot beknown in advance, the probability distributions governing the outcomes are either known ormay be satisfactorily estimated. In a risky environment, probabilities may be associated withthe various results that can occur. Life insurance companies, for example, can predict withina close range of error how many policyholders in any age group will survive to age 65, eventhough they cannot predict accurately which specific individuals within the group will reachthat age. Those poker players who are successful in the sense of being net winners over the longrun understand risk. Winning card combinations have associated probabilities, and successfulplayers must take these into account, at least intuitively, to win on balance. In gambling gamesprobabilities can generally be determined with precision, whether the players know the oddsexplicitly or through a sense gained by experience in playing.

In games of chance the thrill of risk-bearing itself may be more important than the prospectof gain. And in capital investment the taking on of risk for the sake of doing so may hold appealto some individuals in management. However, if management is to serve the interests of theorganization’s owners and creditors, it must manage risk, not be managed by it. Managersmay undertake risky investments (later we show that this can actually be beneficial to theenterprise), but they must strive to commit funds to investments that in the aggregate promise agreater probability of gain than loss. This is not to say that individual projects offering a smallprobability of huge gain in return for a great probability of small loss must never be undertaken.Often such probabilities can be altered by managerial action. Moreover, such projects normallywill constitute a relatively small proportion of the firm’s total capital budget. Managers coulddo worse than be guided by the Machiavellian-like principle that a single loss on a clearlyhigh-risk project may do more harm to their reputation than a string of gains on other projects.When portfolio effects are taken into account, this is an unhappy state of affairs, but one thatthose making capital investment decisions must be aware of.

Later we treat in detail the topic of portfolio risk — risk of the total of the enterprise’s in-vestments. Although it is both informative and a necessary beginning to examine individualinvestment projects by themselves, it is important to recognize such interrelationships of in-vestments that may exist and to analyze the effect on the total enterprise that acceptance ofany particular project is likely to cause.


Uncertainty

The generally accepted distinction between risk and uncertainty is that in the case of uncertaintywe know that the possible outcomes are random variables, but we do not know the probabilitydistribution that governs the outcomes, or its parameters, and cannot estimate them a priori.Because capital-budgeting decisions are usually one of a kind, there is insufficient prior expe-rience with similar situations to grasp the probabilities associated with the possible outcomes.The uniqueness of such capital investments means they are not amenable to Bayesian revisionbecause they will not be repeated. In the more extreme cases even the entire range of outcomesthat could reasonably be expected to occur may be unknown.

In the following chapters we shall first consider risk and uncertainty as if a certainty envi-ronment existed for all project parameters. This has the advantage of allowing us to concentrateon basic principles and thus to gain a solid understanding of them before factoring in the com-plicating matters of risk or uncertainty. Later this assumption of certainty is relaxed in order todeal realistically with risk in individual projects and with the risk relationships between capitalinvestment projects and the enterprise. Prior to that, risk and uncertainty considerations willbe mentioned where appropriate as a complicating factor requiring attention and planning forunforeseen contingencies.

2

The Objective of Capital Budgeting

Unless one restricts attention to the very general social goal of accumulating capital in orderto increase national welfare, it is difficult to define only one objective for capital budgetingto achieve. The classical economics assumption of profit-maximizing entrepreneurs cannotbe considered appropriate for government or not-for-profit, private institutions. Furthermore,that assumption is not operationally feasible, and doubts have been expressed as to whether itrepresents the true motivation of managers in either private enterprise or in government.

Alternatives to the classical assumption of profit-maximizing behavior have been proposed.Among these, the more prominent include the concepts of “satisficing” and organizationaldecision-making. Satisficing owes much of its development to Herbert Simon [145], who ob-served that “Administrative theory is peculiarly the theory of intended and bounded rationalityof the behavior of human beings who satisfice because they have not the wits to maximize.”This view is supported by the observation that managers make decisions without the completeinformation classical economic theory assumes they possess. Managers may intuitively takeinto account the classical concepts of rationality, including marginal analysis and game theory.However, there is no evidence to support the notion that managers attempt to perform the com-plex calculations demanded by classical economic theory in other than rare, specific instances.Even if they wished to do so, managers usually do not have the detailed and exhaustive datathat classical theory would require, nor can they obtain it.

Other behaviorally oriented theories of management decision-making have been developed,including those of Cyert and March [26], which expand and supplement the satisficing concept,and Galbraith [51], who views managements of large, widely held corporations as serving theirown interests above those of the owners, and in some cases above those of the nation. Becauseof the diversity of human behavior and the evolution of attitudes and institutions, it is likely thatadditional behavioral theories will be developed that attempt to elaborate or offer alternativesto those proposed to date.

The behavioral theories, although realistically portraying management in human ratherthan mechanistic terms by incorporating a wider spectrum of behavioral assumptions thanprofit maximization, have contributed to better understanding of organizations. Like the clas-sical model of profit maximization, however, they are not operational. Although profit max-imization provides a normative model of management behavior, the behavioral models aredescriptive. They describe what management does rather than what management should door, more specifically, they fail to specify what objective criteria that management should usein reaching decisions that best serve those who employ them.1 Also, although profit max-imization provides a nomothetic model, the behavioral models to date are ideographic, ornonuniversal.

Our intent is to develop capital budgeting as much as possible within a normative framework.Therefore, we shall leave the behavioral models at this point and proceed to define a normative

1An enlightening critical examination of popular theories of managerial behavior is found in the controversial, now out of print bookby James Lee [84].


model that can be operationalized. The model that is adopted can be applied to both privateand public enterprise.

A NORMATIVE MODEL FOR CAPITAL BUDGETING

A serious deficiency of the classical economics principle of profit maximization, which preventsits adoption as operating policy, is that it does not indicate whether long-run or short-run profitsare to be maximized. A firm’s management, for example, might maximize short-run profits byformulating a policy that would simultaneously alienate customers, employees, and creditorsover a period of a few months. It might attempt charging the highest prices the market will bearwhile lowering quality, paying minimum wages, letting the firm’s financial structure, plant,and equipment deteriorate, and so on. Such behavior would surely injure the firm’s chances tosurvive beyond the short run. Once owners of a firm have decided to liquidate it, they mightadopt such an irresponsible mode of operation, but they could not thus operate and expect thefirm to survive for long.

A less extreme manifestation of short-run profit syndrome is not uncommon. It has takenroot in firms that neglect proper maintenance of plant and equipment and steadfastly refuseto abandon worn or technologically obsolete equipment until a new replacement can “payfor itself” in one or two years. This thinking blossoms into the specious yet appealing notionthat plant and equipment, even though seriously worn and obsolete, should not be abandonedbecause they have been “paid for” years ago. In other words, such assets are considered “free”resources to the firm because they were fully depreciated in the past. In truth, the firm paysincreasingly more in high scrap rates, in higher than necessary labor content, and machinerepair and tooling costs as time goes on. The assets were paid for when acquired, not whendepreciated. Depreciation merely enabled the firm to recover part of the cost of the investmentsthrough tax remission over the useful, economic asset lives.

The opposite of this obsession with short-run profits is preoccupation with the prospect ofprofits in the long-run future. In the extreme this is a much less common pathological conditionthan short-run profit obsession. This is, at least in part, due to the fact that firms that seriouslyneglect short-run profits cannot survive to the long run. A firm may be able to neglect long-runprofits and still survive as an economic cripple; a firm that neglects short-run profits is likelyto be an early fatality.

A tendency toward overly great emphasis on long-term profits may be observed in firms that,although showing poor current performance, spend lavishly on public relations, landscapingand lawn care around factory and office facilities, excessive employee benefits, research anddevelopment, and so on. The key word here is “lavishly.” All firms are expected to spendreasonable amounts on such indirectly beneficial things that make the world more pleasantand that may not be strictly defensible in terms of expected tangible benefits. It is ultimatelyfor the owners and creditors of the enterprise to determine what is reasonable, although inreality management may to a considerable extent be protected by the shield of ignorancesurrounding the enterprise’s operations. It is uncommonly difficult for ordinary stockholdersto obtain the detailed information necessary to successfully challenge mismanagement andlargess by management.

It would appear that management is caught between the horns of a dilemma. Short- andlong-run profit maximization seem to be contradictory. Is there any way to resolve the conflict,the ambiguity? Yes, there is, if we recognize that money, our economic numeraire, has time-value.

The Objective of Capital Budgeting 11

BASIC VALUATION MODEL

The model of managerial behavior we adopt is that of modern financial management, whichis not part of classical economics, although it might be considered a direct descendant:

Maximize wealth, or value, V =∞∑

t=0

Rt

(1 + k)t(2.1)

subject to governmental, economic, and managerial constraints. The terms of this basic val-uation model are as follows: t is the time index, Rt the net cash flow in period t, and k theenterprise’s cost of capital.

This model resolves the ambiguity over whether it is long- or short-run profits that shouldbe maximized. It is the total discounted value of all cash flows that is to be maximized. Thetheoretical value of the enterprise is defined in terms of its profitability over time. One may,of course, adopt the continuous analog of this discrete-time model, although traditionally thishas not been common or especially useful.

The nature of this model is such that its maximization may be powerfully facilitated by aneffective financial management that can raise adequate funds at minimum cost. This becomesclear if one considers that denominator terms containing k are raised to progressively higherpowers. And it underscores the need for good estimates of the enterprise’s cost of capital if itis to yield correct decisions. If the firm will accept only capital investments that have a positivenet present value then the value of the firm will be increased, irrespective of the timing of thenet cash flows.

Contemporary literature recognizes that there is interaction between the cost of capital kand the risk characteristics of the cash flow stream R, over time. In other words, if the firmundergoes changes that alter the variability of its overall cash flows over time, this will have aneffect on its cost of capital. The theory behind this notion has not yet become operational, and,although we shall deal with it in a later chapter, for now it will suffice to adopt the principlethat the cost of capital will not be increased if the enterprise invests its funds in capital projectsof similar risk to the existing assets of the firm. If projects are accepted that are less risky thanthe existing asset base of the firm, a tendency will be for the cost of capital to decrease. Thisprinciple will be useful for now; however, it ignores portfolio effects that imply that certaininvestments that are more risky than the existing asset base can sometimes actually serve toreduce the enterprise’s risk and thus its cost of capital. Portfolio effects are considered in detailin later chapters.

Operational Adaptation

We adopt the simple convention of accepting prospective investments that add to the valueof the enterprise. In an environment in which there is no capital rationing — that is, wherefunds are sufficient to accept all projects contributing to the value of the enterprise — weaccept all profitable projects. In the more common capital-rationing situation in which fundsare scarce compared to the investment cost of the array of acceptable investments, we shouldtry to accept those projects that contribute the maximum amount to the value of the firm.Nothing in this contradicts the behavioral theories of management. Other motivations may deterachievement of maximum value increase. But even satisficing management, or managementmerely interested in remaining in control, should still try to attempt to increase the value of theenterprise, even if not to the fullest possible extent. Management that habitually does otherwise


does so at its peril because the owners may replace the existing management. And increases invalue of the firm may indicate that management is alert and responsive to competitive forcesin the market in other ways as well. Thus, a firm that fails to increase in value may be signalingthat there are problems in the current management of which shareholders should be aware.

The Cash Flows

For individual projects that constitute only one of many items of capital equipment, it isdifficult, if not impossible, to associate net cash flows or even accounting revenues directly.In these cases it may sometimes be possible to treat the package of such individual items ofplant and equipment as a single, large capital-budgeting proposal. This is likely to be the mostuseful when the package alone is to support a new product line, or when it is to replace anentire production facility.

Alternatively, the cash flows associated with the individual capital equipment componentsmay result from cost reduction. Cost reduction may be considered equivalent to positive cashflow because it represents the elimination of an opportunity cost. Such costs are defined asthose attributable to inaction or the result of adopting some alternative to the best availableaction.

CASH FLOWS AND THE PUBLIC SECTOR

Because the equivalent of positive cash flows may be obtained from a reduction in costs, thebasic valuation model can be useful in public sector cost–benefit analysis. Cost reduction freesresources that would otherwise be wasted, so that they may satisfy other public demands.Also, maximization of the value of public enterprise may be considered beneficial, becausesuch institutions belong to all citizens of the community. Maximization of the enterprise valuetherefore serves to maximize the public wealth of society. Value in such cases is drawn fromthe tangible and intangible benefits provided to the citizens and not in the funds accumulatedby the institution, or the wages and salaries of its workers and managers, which should becompetitive with those in corporate industry. Any surplus of money accumulated by publicenterprise should be distributed in the form of social dividends, paid through reduction inrequired external funding (by taxes or government borrowing), or by an increase in the servicesprovided. Unfortunately, the performance of some government institutions may lead one tobelieve that this, in fact, is not done, perhaps because the bureaucrats managing them are notanswerable to the citizens nor held accountable by elected officials. The US Postal Serviceprovides an excellent example.

3

Estimating Basic Project Characteristics

In order to apply any objective, systematic method of capital-budgeting evaluation and projectselection, it is first necessary to obtain estimates of the relevant project characteristics orparameters. The conceptually superior methods of evaluation also require an estimate of thefirm’s cost of capital, which is taken up in the next chapter. The present chapter focuses onthose characteristics of a particular project or aggregation of projects that become manifest incash flows attributable to the project.

PROJECT TYPES

Let us define those capital investment projects as major projects, which in themselves generatenet cash inflows, and as component projects those that do not in themselves directly generate netcash flows. Examples will serve to clarify the distinction. Investment in plant and equipment toproduce a new product line that will generate sales revenues as well as production, marketing,and other costs would be classified as a major project. Investment in a new tool room lathewould be classified as a component project. The lathe will be used to service other capitalequipment in the plant or produce prototype parts. The lathe thus will not produce directlyattributable cash inflows, but will necessitate directly attributable cash outflows for operatorwages and fringe benefits, electric power, and so on. A new spline rolling machine that willreplace several milling machines in a plant producing power transmission shafts and gearsis also a component project. The spline rolling machine in such a production facility wouldnot produce a product that is sold without many other machining, heat treatment, inspection,and assembly operations being performed on it. Thus, product revenue cannot be directlyassociated with this machine except through rather tenuous cost accounting procedures notlikely to properly reflect its revenue contributions.

Major projects have both cash inflows and cash outflows directly associated with themselves.Component projects ordinarily do not. Therefore, although attention to value maximization maybe appropriate for major projects, cost minimization will generally be a more suitable approachto component project analysis. For instance, cost reduction brought about by replacing an older(component project) machine by a newer and more efficient machine contributes to net cashinflow as much as an increase in revenue with costs held constant does. In this case the net cashflow is attributable to eliminating opportunity cost that is associated with inefficient productionequipment for the particular operation.

To further clarify the distinction between project types, assume that, for a given costof capital and risk level, we want to maximize net cash flows by holding costs constantwhile increasing cash revenues. For major projects this may be appropriate. On the otherhand, for component projects we may more easily achieve the same result by minimizingcosts for a given level of cash revenue. An analogy may be drawn using the terminologyof mathematical programming, in which the major project is considered the dual prob-lem of cost minimization. We may expect equivalence between maximizing some objective


function, subject to cost constraints, and minimizing cost subject to some performance con-straints.

Most capital-budgeting techniques have been oriented to selecting projects that will con-tribute toward maximization of some measure of project returns. However, the MAPI methodproposed by George Terborgh departs from these by selecting projects that contribute to min-imization of costs. The classical MAPI method is treated in detail in Chapter 9.

PROJECT CHARACTERISTICS

The quantitative parameters of an investment that are relevant to the decision to accept or rejectthe project are:

� Initial cost project� Useful life� Net cash flows in each time period� Salvage value at end of each time period

These parameters, together with the enterprise’s cost of capital, provide information upon whicha rational decision may be based, using objective criteria. In addition to the above quantitativeparameters, sometimes qualitative considerations will also affect the investment decision. Forexample, one production facility may use a collection of custom-designed machinery, whereasanother employs more or less standard production machines. If the specialized machinerycannot readily be converted to producing other machined products, and if the probability thatthe particular products for which this specialized machinery is acquired will be abandonedprematurely is high, then the decision to adopt standard production machinery may be supe-rior even though it may promise somewhat lower benefits if things go well for the productline.

Qualitative considerations may be crucial, especially in contingency planning. In answeringthe “what if” question of what alternative use may be made of capital investment projects ifthings do not go as they are expected to, management may well prefer the array of projectsthat offers flexibility over the somewhat more efficient, but highly specialized, alternative.For example, a firm that produced automobile fabric convertible tops and decided in 1970to acquire new automated equipment to stitch the seams, predicated on a 15-year useful ma-chine life, would have found in 1976 that original equipment market sales were vanishing.In 1976, Cadillac, the last of the United States automakers to produce fabric-covered con-vertibles, announced it was phasing out such models. Unless the replacement market wouldcontinue to provide sufficient sales into the mid-1980s, the firm would need to find alterna-tive products suitable for production on the specialized equipment. Perhaps fabric camper-trailer tops or similar items could be produced as profitably on the specialized machinery. Ifnot, the firm would find that its decision in 1970 was, with benefit of hindsight, the wrongdecision.

No general rules or procedures have been developed for contingency planning in capitalbudgeting. Each case has its own unique attributes that prevent uniform application of rigorousprinciples. It is in such aspects of decision-making that there is no reasonable alternative tothe judgment of management. For instance, how should a firm that produces barrel tubes forshotguns incorporate in its decisions the possibility that a Congress will be elected that isdisposed to outlaw private firearms ownership or restrict ownership drastically? Given thatsuch a Congress is elected, what is the contingent probability that it will find the motivation

Estimating Basic Project Characteristics 15

and time to act? Will a later change in Congressional composition reverse such legislation?If so, would consumer demand return to its former level? Could alternative uses be found forthe machinery, such as making hydraulic cylinders? Such a firm must incorporate factors suchas these into its capital-budgeting decisions in order to ensure flexibility in using its physicalcapital in case the environment in which it operates suddenly changes. And is there any firmnot subject to environmental changes?

INITIAL COST

Capital-budgeting projects will generally require some initial cash outlay for acquiring theproject and putting it into operation. Such cost may arise from construction outlays, purchasecost or initial lease payment, legal fees, transportation, and installation, and possibly from taxliability on a project that is being replaced by the new one or from penalty costs associated withbreaking a lease on the replaced project. Cash costs attributable to the decision to accept a capitalproject should be included in the initial project cost. Costs that would be incurred regardlessof whether or not the project were accepted are “sunk costs” that should not be included inthe project cost. With some projects, especially large ones involving lengthy design work andconstruction that extends over several years, initial costs should be considered as the negativenet cash flows incurred in each period prior to the one in which net cash flow becomes positive.In such cases there is a series of initial costs instead of a single outlay.

Sunk Cost

Unrecoverable costs associated with previous decisions should not be allocated to a new projectunder consideration.1 For instance, assume that a firm purchased a $400,000 machine 5 yearsago, which had at the time an estimated useful life of 20 years, and that $228,571 has still tobe claimed as depreciation against taxes on income at a rate which yields a tax reduction of$118,857 spread over the next 15 years. Assume now that a technologically improved machineis available to replace it. Should the unrecovered tax reduction be added to the other costs ofthe new machine? The answer is no, in this case. However, the unrecovered tax reductions maybe included as costs in the years in which they would have been realized. Now, what if the$400,000 machine suddenly breaks down and cannot be repaired? In this event the unrecoveredtax reductions from depreciation should not be charged to a replacement machine, either in theinitial cost or in the cash flows over the remaining depreciable life of the broken machine. Fortax purposes the broken machine’s remaining value will be charged to the firm’s operationsas a loss which will have no effect on the replacement decision. Similarly, equipment that isdiscarded because of a change in the firm’s operations should not have any of its cost chargedto new equipment that is subsequently acquired.

The problem of what costs to include and what costs to exclude from a particular capitalinvestment project will be resolved by focusing on cash flows and ignoring accounting costs.Does acceptance of the candidate project increase or decrease cash flow in years zero throughthe end of its anticipated economic life? If acceptance of the project precipitates changes incash flows, then these must be taken into account in evaluating the project. Noncash items,and items such as sunk costs, must not be included in project evaluation, even though specificdollar figures are associated with them.

1Of course, for accounting reasons related to minimizing tax liability, they may be included in the bookkeeping for the project.


What if costly preliminary engineering studies have been completed and research and de-velopment costs incurred? Should these be included in the cost of the project for capital-budgeting evaluation? No, they should not, for they represent sunk costs — water over thedam and subsequent acceptance or rejection of the project will not affect them. The handlingof such sunk costs in the accounting framework may well be a different matter; for tax orcontrol reasons they may be associated with the project. Since in this book our concern iscapital budgeting and not accounting, such matters will be ignored. They are irrelevant tothe decision of whether or not to accept a project, except to the extent that they affect cashflows.

Components of Initial Cost

In addition to the obvious component of initial cost, namely, the basic price of the capitalproject in question, some less obvious costs must be included. Among them are:

� Transportation and insurance charges� Installation costs, including special machine foundations, movement of other equipment to

get the new project to its location in the plant, installation of service facilities such as electric,hydraulic and pneumatic lines, and so on

� License or royalty cost� Required additional working capital investment� Operator training costs

Transportation and insurance costs may be included in the vendor’s price. In many in-stances, however, delivery will be FOB the vendor’s plant. In these cases failure to include thetransportation and insurance costs will understate the project’s initial cost, perhaps seriouslyso for large, heavy equipment that is difficult and expensive to ship.

Installation costs include the full expense of project installation. With industrial machinerythat will be placed in an existing plant, it may be necessary to move intervening equipmentto allow room for the new equipment. In some instances the plant structure itself may haveto be temporarily or permanently modified or the disassembled equipment moved in pieces toits site within the plant. Worker safety may require installation of noise-dampening materials,special ventilation equipment, dust collectors, fire extinguisher systems, and so on. Heavymachines often require a concrete “anchor” foundation to be poured prior to installation,with the foundation sometimes being as much as a meter deep and nearly as large as themachine attached to it. Machinery often requires these foundations to be built so that vibrationsemanating from them are not communicated through the plant floor to other machines, wherethey would affect the quality of the operations they perform. Adequate vibration dampeningcan increase installation cost considerably, but it is required in environments in which precisionmay be affected by this type of unwanted disturbance.

License or royalty cost for use of patented equipment or processes may require an initialpayment as well as the customary periodic payments as production gets underway. Theseshould be included in the project cost.

Required additional working capital investment is an item that is easy to overlook. Usually,this will not be a significant factor with component projects, but will be important for mostmajor projects as defined earlier in this chapter. Additional net working capital required tosupport accounts receivable, inventory, or other current asset increases (net of current liabilityincreases) are included in project cost. Recovery of such net working capital requirements at


project termination may be incorporated in salvage value or the last period’s cash flow, whichare equivalent means of handling this factor.

Operator training costs, if not included by the vendor in the basic equipment cost, mustbe added to the project cost by the purchaser. Such training is to be expected with capitalequipment employing a new technology or capital equipment whose operation by nature iscomplex. For instance, it is generally required when large-scale computer or industrial controlsystems are acquired, that operators and support personnel be trained properly in correct useof the equipment. This is true even though the new machine replaced an older model of thesame vendor, or a smaller machine of the same series with a different operating system orfewer options than the new one.

In addition to these initial cost items, others may be found in particular cases. The rule tofollow in determining whether a cost item should be included in the initial cost of a capitalinvestment project is to answer the two following questions. If the answer to both is in theaffirmative, the cost should be included in the initial cost of the project.

1. Is the cost incurred only if the capital project is undertaken, that is, accepted?2. Is the cost represented by a cash outflow?

If the answer to both is “yes,” then only the cash outflow associated with the cost should beincluded in the initial cost of the project. Noncash costs should be excluded. For this purpose weassume net working capital increases to be cash flows, although “internal” to the firm. Increasedworking capital requirements must be funded, and the money committed will generally not bereleased until the end of the project’s useful life.

USEFUL LIFE

The investment merit of a project will depend on its useful economic life. Useful economic lifeof capital equipment may end long before it becomes physically deteriorated to the point ofinoperability. Economic life, and decline in the value of capital equipment over the economiclife, may mean that project abandonment prior to the end of the originally anticipated projectlife will be of greater benefit to the firm than holding the project to the end. This topic is takenup in detail in Chapter 15.

Terborgh, in his classical method, defined the cumulative effects on decline in capital ser-viceability over the period the equipment is held as operating inferiority [151]. This is a usefulconcept, and it is adopted here. Operating inferiority is determined by two components: phys-ical deterioration and technological obsolescence. Physical deterioration is what is normallyconsidered to be the determinant of project life. However, technological obsolescence will befar more important in determining the economic life of some projects.

Consider an accounting firm that purchased a large number of mechanical calculating ma-chines in 1970, assuming an economic life of 10 years. If the machines were to be kept inservice over the full 10 years, they could be expected to undergo steady physical deteriora-tion. As time went on, more frequent and more serious breakdowns would be expected, andexpenditures for repairs would increase accordingly.

Some rotary, mechanical calculators cost $1000 or more in 1970. At about the same pointin time, due to advances in technology, a variety of electronic calculators came on the market.Not only was the cost substantially less, as little as one-fourth the cost of their mechanicalpredecessors for some of them, but they were superior in several respects. The electronic modelswere substantially faster, immensely quieter, less subject to mechanical problems because


of the dearth of moving parts, and provided number displays that were larger (sometimes),illuminated, and generally easy to read. Thus it was that the mechanical calculator fell victimto technological obsolescence. They simply were inferior in most respects to their moderndescendants, even if still quite serviceable.

Or imagine the firm that bought a large quantity of personal computers in the mid-1990sunder the false assumption that they would be quite adequate for at least a decade. Perhaps theywould remain functional, if maintenance were not required, but clearly a PC even four yearsold presents problems in terms of both physical function and obsolescence. After a few years,hardware and software vendors no longer find it worthwhile to support older machines, andthus keeping them running presents an ever-increasing headache for system administrators.And as newer and better software becomes available it will not run on old machines at all, orat best suboptimally.

In firms whose employees devote a large portion of their working time to calculation, by1975 few mechanical calculators were in use. And by the year 2002 few personal computersof vintage more than three years are in use in companies. Why? Because replacement withthe best technology allows less wasted time while waiting for results, a quieter atmospheremore conducive to productive work, and reduced maintenance and repair costs. If an employeeworks half a day with a calculator or computer, and the new technology is 25 percent faster,the same employee can do the equivalent of one hour’s additional work in the same four hours.Such efficiency gains are easily translated into money terms. An office employing four suchpeople, by equipping each with the new technology, could avoid hiring the fifth person whenthe workload expanded by as much as 25 percent.

Physical Deterioration

For many types of capital, experience with similar facilities in the past may provide usefulguidelines. For instance, we would expect that the physical life of a punch press purchasedtoday would, on the average, be similar to that of a new punch press of 20 years ago.

With capital that embodies new technology, or a new application of existing technology,managers and engineers experienced with production equipment may provide useful estimates.However, capital goods produced by a firm not likely to stay in the business and maintain asupply of replacement parts may negate an otherwise good estimate of useful life. Unavail-ability of a crucial part from a supplier will mean either producing the part in the adoptingfirm’s tool room, contracting to have it custom made, modifying the machine to take a similarstandard part, or abandoning the machine. This last alternative obviously ends the useful life.

Physical deterioration of major projects, as defined earlier, may well be affected by thatof plant and individual equipment components. It is not meaningful to speak of the physicaldeterioration of a major project unless this is taken to be synonymous with deterioration of thebuildings housing the operation, or unless the major project is indeed one major item of capitalequipment that dominates all others. If our major project is a division of the firm composed ofone or more buildings, each housing 100 or more items of capital equipment, what meaningcan we attach to physical deterioration of the project? The answer is none; at least this is true ifcomponent projects are added and replaced as time goes on. On the other hand, if our divisionis based on operation of one dominating item of capital, such as a toll bridge or a carwashor parking garage, it may well be meaningful to refer to physical deterioration of the majorproject.


Technological Obsolescence

Although we may often be able to obtain workable estimates of physical life for capitalequipment and the corresponding physical deterioration over time, it is a very different matterfor technological obsolescence. Technological innovations that contribute to the obsolescenceof existing capital tend to occur randomly and unevenly over time. Sometimes technologicalchanges are implemented rapidly during relatively short intervals of time: The technologicaladvances in computer equipment since the 1950s have been profound, and now may continueat a rapid pace for some time yet. Theoretical developments, such as holographic, laser-directed computer memories, once the engineering obstacles have been overcome, and artificialintelligence software promise yet further waves of innovation in the industry.

Terborgh’s approach to incorporating technological obsolescence into the operating infe-riority of capital equipment is difficult to improve upon. His recommendation, basically, isto assume, in the absence of information to the contrary, that technological obsolescence ofexisting capital will accumulate at a constant rate as time goes on [151, p. 65]. In the absenceof information to the contrary, such extrapolation from the past into the future is reasonable.However, should information be available that implies more rapid or less rapid technologicalchange, this information should be employed, even if it is only qualitatively.

Should the firm delay an investment when technological changes are expected in the nearfuture? The answer to this question may be found in evaluating the merit of investing today withreplacement when the technologically improved capital becomes available, and evaluating themerit of the alternative — that of postponing investment until the improved capital is available.Comparison of the merits of the alternatives will serve to determine the better course of action.Methods for performing such analysis, once the net cash flows are determined, are covered inChapters 9 and 23.

The subject of technological obsolescence in capital budgeting is not limited to the firm’sphysical capital; it may apply also to the human capital of the enterprise, and certainly doesapply to the product lines on which the cash flows of investment projects are predicated. Forexample, the replacement of vacuum tubes with transistors and integrated circuits, and laterwith miniaturized circuits made possible by large-scale integration, caused radios and televi-sions incorporating vacuum tubes to become obsolete. A firm that based its capital investmentdecisions on continued demand for sets with vacuum tubes found that later it had still ser-viceable equipment for a product no longer in demand. Similar examples are to be found inmechanical versus electronic calculators, automatic versus manual automobile transmissions,and piston versus jet propulsion aircraft engines, to name just a few of the more obvious.

CASH FLOWS

In capital budgeting we must base our analysis on the net cash flows of the project underconsideration, not on accounting profits. Only cash can be reinvested. Only cash can be used topay dividends and interest and to repay debt. Only cash can be used to pay suppliers, workersand management, and tax authorities.

Successful application of any method of capital project evaluation requires forecasts of es-timated cash flows. Like all forecasting, this is a “damned if you do, damned if you don’t”proposition — one must forecast, but in doing so is destined to be in error. A successful fore-cast is one wherein the forecast error is minimized. It is far beyond the scope of this bookto delve into the arcane art and science of forecasting. That is left to books that specialize


in the subject. However, those who need to use forecasts, or to make them, should consultcredible references on the subject and become familiar with the merits and pitfalls of thecurrently available methods. (For example, one can find good insights and guidance in bookssuch as that by Spyros Makridakis et al. [96].) Whether one makes forecasts or commis-sions others to do so, the manager should be familiar with the strengths and weaknessesof the methods used to develop forecasts. This book assumes that the capital investmentdecision-maker has the best forecasts possible when analyzing projects. This is somewhatof a fiction, but it will keep us on track here instead of sending us on a detour leading farfrom this book’s main theme, that of capital investment management strategy, tactics, andtools.

Over the long run, both total firm cash flow and total accounting profits provide measuresof management performance. However, in the short run the two will generally not be highlycorrelated. For example, the firm may have a very profitable year as measured by “generallyaccepted accounting principles” and yet have no cash to meet its obligations because the“profits” are not yet realized but are tied up in accounts receivable that are not yet collected,and in inventories.

Determination of net cash flows involves consideration of two basic factors: (1) those thatcontribute to cash inflows or cash receipts and (2) those that contribute to cash outflows or cashcosts. Major projects, as defined earlier, will have both these factors affecting them throughouttheir economic lives. Component projects, after the initial outlay, will be directly involved withcash outflows, but only indirectly with cash inflows.

Estimation of cash inflows for major projects generally requires the joint efforts of specialistsin marketing research, sales management, and design engineering, and perhaps staff economistsand others as well. If the firm does not have the required expertise itself, it will have to hirethe services of appropriate consultants.

For reliable cash outflow estimation, the joint contributions of design, production, industrialengineering, production management, cost accounting, and perhaps others, are required formajor projects. Additionally, staff economists and labor relations personnel can contributeinformation relevant to probable cost increases as time goes on.

Financial management is responsible to top management for analyzing the effects on thefirm’s financial strength if the project is undertaken and for obtaining the funds necessary tofinance the undertaking. Finance personnel will undoubtedly be involved in recommendingwhich major projects should be accepted and which should be rejected based on their analyses,project interactions with the existing assets of the firm, and consequently their profitability.Accounting staff will be concerned with the project’s effect on reported profits and tax liabilitiesand all the attendant details.

Cash inflows for a major project will be determined by (1) the price at which each unitof output is sold; (2) the number of units of output sold; and (3) the collection of accountsreceivable from credit sales. Estimation of these items is not easy, especially for a productdissimilar to product lines with which the firm may already have had experience. Properestimation, as mentioned earlier, involves persons from different functional areas within thefirm, and possibly external to it.

For component projects we will be concerned with cash outflows. If a given task must beperformed by capital equipment, then we shall seek to obtain capital that will do so at low-est cost. Thus, on initial selection of mutually exclusive candidate projects, we will selectthe project that minimizes cost or, alternatively and equivalently, the project for which thesaving in cash opportunity cost is maximized. Of course, we must assume that all compo-nent projects admitted to candidate status are capable of performing the tasks that must be


done within the environment of a major project. Projects that cannot should not be treated ascandidates.

Cash Inflows (Cash Receipts)

Assume that we are examining a combination of several machines that will produce a singleproduct our new firm plans to sell. The principals of our firm are experienced in sales andengineering of such a product; both worked for years for a larger company that producedsimilar products. Some preliminary orders have already been obtained from firms that willpurchase the product we will produce. It is estimated that at a price of $17.38 per unit, our firmcan reasonably expect sales of 100,000 units per year.

If we assume that sales and cash receipts on sales will be uniform throughout each year,and that unit price and sales volume will be constant from year to year, the task of cash inflowestimation is trivial. If we have no uncollectable accounts receivable, our cash inflows eachyear will be unit price times number of units, that is, $1,738,000.

Life is usually not so simple as this, however. Sales probably will not be uniform throughoutthe year, but will have seasonal variations. Sales from year to year will seldom be even nearlyconstant. Possibly sales will grow from year to year along a trend of several years’ duration,or decline for several years. Such variations are very difficult to predict in advance, and are ag-gravated by unforeseen developments in competition, the national economy, and other factors.

In the final analysis, estimation of cash inflows will depend on managerial judgment, con-ditioned by the economic environment and knowledge of the firm’s competition and trends inproduct design and improvement. In many industries revenues may be influenced by advertis-ing expenditures so that firms may in fact influence the demand for their products. Seldom willa firm be in the position of having a mathematical model that provides truly reliable demandand revenue forecasts, especially for periods beyond one year. There are too many qualitative,vague, and intangible factors at work that cannot be quantified given the current state of theart in mathematical modeling. Proper incorporation of these factors requires human judgment,and perhaps not a little luck.

It will often be useful to prepare several forecasts, including a “worst case” forecast inwhich it is assumed that whatever can go wrong for the firm’s sales will go wrong. Of course,unanticipated factors and events may make actual events still worse, but we would assume thatfor a proper “worst case” forecast, actual experience would be no worse, say, 95 times out of100. In other words, a good “worst case” forecast should have a small probability associatedwith it that actual events will turn out worse. “Best case” and “most likely” forecasts maysimilarly be made. The population projections of the United States Bureau of the Census, infact, have been prepared on the basis of high, low, and most likely. Because we can neverexpect an exact forecast, it is extremely useful to be able to bracket the actual outcome, sothat we may say that the probability is p percent that the actual outcome will fall betweenthe best case and worst case estimates, where p is a number close to 1.0. For this approachto be useful, the best case and worst case forecasts cannot be so far apart as to make themmeaningless.

Let us now go back to our example. If $17.38 per unit and 100,000 units are taken to beour most likely forecast for the coming year, we may determine that $14.67 and 40,000 unitsis the worst that is likely and $23.00 and 200,000 the best that is likely to happen. If we canattach (albeit subjective) probability of 0.05 or 5 percent that the actual revenue will fall shortof the worst case, we have useful information. The chances of sales revenues being less thanthe worst case are then only 5 in 100.


Cash Outflows

To some extent estimates of cash outflows may be made with more confidence than thosefor cash inflows. For example, the initial investment outlay is made at the beginning of theproject’s life, and may often be estimated precisely. In fact, the supplier may provide a firmprice for the cost of the investment. However, other cost components may prove to be almostas difficult to forecast as factors affecting cash receipts.

What specific items do we consider as cash outflows? Anything requiring cash to be paidout of the firm, or to be made unavailable for other uses. Suppose that the machines we areconsidering for purchase are to be treated as one project. Assume that the project costs $3 millionfor purchase and installation. For simplicity, assume also that we plan to run production on aconstant level, so that labor, electricity, and materials will be constant for at least two years.In addition, cash will be tied up in raw materials and finished goods inventory and in accountsreceivable. Setting aside the notion of worst and best case forecasts for now, let us deal withthe most likely production and cost forecast, and the most likely annual cash revenue forecast.We obtain the following:

Year 0 Cash outflow Cash inflow

Initial investment $3,000,000Wages 100,000Fringe benefits 50,000FICA, etc. 20,000Raw materials inventory 200,000Finished goods inventory 500,000Accounts receivable investment 400,000Electric and other direct variable costs 10,000

$4,280,000Cash receipts on sales, net of discounts,

and bad debt losses $1,738,000Net cash flow ($2,542,000)

Note that no overhead or sunk cost items have been included in the cash flows. In the secondyear of operations, if no new investment in inventories or accounts receivable is required,and the same amounts of labor, electricity, and so on, at the same rates are employed, weobtain:

Year 1 Cash outflow Cash inflow

Wages $100,000Fringe benefits 50,000FICA, etc. 20,000Electric and other direct variable costs 10,000

$180,000Cash receipts on sales, net of discounts,

and bad debt losses $1,738,000Net cash flow $1,558,000

Wages and fringe benefits include those of direct labor plus the portion of indirect labor thatservices the machinery: machine setup, materials handling, and so on.


Ordinarily, in expositions of the various capital-budgeting techniques, we assume for sim-plicity of explanation that cash flows occur only at the end of a period (usually a year) and arenot distributed throughout the period. Period zero in such treatment includes only the installedcost of the capital equipment. Subsequent periods include net cash flows arising from cashreceipts minus cash disbursements, the latter including whatever additional investment thatmay be required in capital equipment, inventories, and so on. Interest expenses are specificallyexcluded from cash costs because methods of project evaluation that employ discounting al-ready incorporate the interest costs implicitly — they are imbedded in the discount rate andnot in the cash flows.

TAXES AND DEPRECIATION

Since firms generally pay income taxes on earnings and depreciation is deductible as anexpense, it has an effect on cash flow. Cash that does not have to be paid as tax to the governmentserves to increase net cash flow because it is a reduction in cash outflow.

The rationale behind allowing depreciation to be tax deductible is that it represents recoveryof investment rather than profit. And since the benefits of capital investment occur over theeconomic life of the project, it is deemed appropriate to spread recognition of the investmentoutlays, as expenses, over the same period. Various accounting conventions and tax authorityrulings on how the depreciation charges may be calculated, and what depreciation lifetimesmay be used for various asset types, have complicated a basically simple concept. Profitabledisposal of capital equipment may subject the firm to additional taxes on residual salvagevalue. Such details are covered by texts on accounting and on financial management. Herewe are concerned only with basic concepts and the effect of depreciation on cash flow, not onthe details of tax rules. Tax laws change from time to time, not only for various theoreticalreasons, but as part of the occasional economic “fine-tuning” by the federal government aimedat encouraging or discouraging new investment, as the situation of the national economy mayindicate.

Several methods of calculating depreciation for each year of a capital investment’s lifehave been devised. Because cash may be invested, it is generally better to charge as muchdepreciation as possible in the early years of a project’s life, thus deferring taxes to later yearsand simultaneously retaining more cash in the early years. This is especially true during timesof rapid inflation. Unless the firm has tax losses larger than it can use to offset taxable income,it will charge the maximum allowable depreciation in the early years of a capital investment’slife. Given that money has time value, to do otherwise would not be in the best interests ofthe owners of the enterprise. The more rapid the rate of price inflation, the more incumbentit is to charge the maximum depreciation in the early years for tax purposes. However, formanagement control purposes the firm may use the depreciation schedule that is considered tomatch most closely the actual economic deterioration in the capital project from year to year.Yet, with high rates of price inflation in capital goods, the depreciation charged against theoriginal cost if unadjusted may be of little usefulness. Such adjustment, however, is beyondthe scope of this book.

Depreciation

Until 1981 there were two tax depreciation frameworks in the United States: (1) the GeneralGuidelines (GG) and (2) the class life asset depreciation range (ADR) system. The latter could


be considered a precursor to the current modified accelerated cost recovery system (MACRS),which in 1986 replaced the accelerated cost recovery system (ACRS) adopted in 1981.

The GG could be used for any depreciable asset. The ADR system was authorized by theRevenue Act of 1971. Examination of these and the fundamental methods of depreciation willhelp in understanding MACRS depreciation. And, given the propensity of Congress to changethe tax laws, we have not seen the end of changes in allowable methods of depreciation. Byunderstanding the basic methods one can easily grasp what is involved in new procedures.

A firm could select the ADR system in preference to the GG for any depreciable assetacquired after 1970 until ACRS came into being in 1981. A firm elected each year eitherto use or to not use the ADR system for those depreciable assets acquired during the fiscalyear. Under the ADR system, assets corresponded to various classes. For example, one classwas 00.241 — light general purpose trucks. The range for that class was three years at thelower limit, four years “guideline,” and five years upper limit. The firm could choose the lowerlimit to achieve the most rapid depreciation. What the change to ACRS depreciation did wastantamount to mandating that all firms would use ADR depreciation with the lower limit.

It will be helpful to understanding if we now review the fundamental methods of depreciation.Then we shall examine the GG and ADR system before considering MACRS, which derivesfrom ADR.

Straight-line Depreciation

The notion behind straight-line depreciation is that the investment’s residual value declines bya constant dollar amount from year to year uniformly over the useful life. Therefore, the initialinvestment is divided by the number of years of useful life, and the result used as the annualdepreciation charge. This is the least useful method for deferring taxes to the later years of theproject’s life.

Double-declining-balance Depreciation

With declining-balance depreciation it is assumed that the remaining depreciable value of theinvestment at the end of any year is a fixed percentage of the remaining depreciable valueat the end of the previous year. Alternatively, we take as declining-balance depreciation aconstant percentage of the remaining depreciable value at the end of the previous period. InUnited States practice, the method used is that of double-declining balance (DDB), in whichthe percentage value of decline is multiplied by a factor of two. This results in acceleratingthe depreciation charges and deferring larger amounts of taxes to later years. Salvage value isexcluded from the calculations in this method.

Sum-of-the-years’ Digits Depreciation

The method of sum-of-the-years’ digits (SYD) depreciation is implemented by writing theyears in the asset’s lifetime in reverse order, and then dividing each by the sum of the yearsin the useful life. Depreciation for each year is then determined by multiplying the originalasset cost by the factor corresponding to each year. A project lasting five years will thereforehave 5/15 (or 1/3) of the original value charged to depreciation in the first year and 1/15charged in the fifth and last year. This method, like that of double-declining balance, providesfor accelerated asset depreciation.


Table 3.1 Comparison of depreciation methods under generalguidelines, with zero salvage

Double-declining Sum-of-years’Year Straight line balance digits

1 $10,000 $20,000 $18,1822 10,000 16,000 16,3643 10,000 12,800 14,5454 10,000 10,240 12,7275 10,000 8,192 10,9096 10,000 6,554 9,0907 10,000 5,243 7,2738 10,000 4,194 5,4559 10,000 3,355 3,636

10 10,000 13,422a 1,819

a The $13,422 in year 10 is $2684 plus the remaining $10,738 undepreciatedbalance, to arrive at zero salvage value.

Comparison of the Basic Depreciation Methods

Assume we have an asset that cost the firm $C, an estimated salvage value of $S and has adepreciable lifetime2 of N years. Then the annual depreciation charges with the three methodsfor 0 < t ≤ N are:

Straight line:

(C − S)/N

Double-declining balance:

C(1 − 2P)t−12P where P = 1/N

Sum-of-years’ digits:

(C − S)(N − t + 1)/[N (N + 1)/2] sinceN∑

t=1

t = N (N + 1)

2

Example 3.1 For illustration, let us take a project with C = $100,000, N = 10, and calcu-late depreciation at the end of each year with each method. Table 3.1 contains a comparisonof the methods when salvage is assumed to be zero. We assume zero salvage value.

The total depreciation charged with each method is equal to the cost of the project. Totaldepreciation cannot, of course, exceed the investment acquisition cost less estimated salvagevalue. Note that with DDB depreciation, the final year’s depreciation charge is the sum of theDDB amount plus the undepreciated balance.

In practice, firms will usually switch from one method of depreciation to another when itis advantageous to do so, and when the tax authorities will allow the change. For instance,under GG a firm may switch from double-declining-balance depreciation to straight-line de-preciation. Of course, the straight-line depreciation charge will not be based on the original

2The depreciable lifetime of an asset will often be different from the useful economic life. This happens because tax authority rulingsconcerning the lifetime that may be used for depreciation may not properly reflect the useful economic life of such asset in any particularfirm.


Table 3.2 Comparison of depreciation methods under generalguidelines, with salvage

Double-declining Sum-of-years’Year Straight line balance digits

1 $12,000 $26,000 $21,8182 12,000 20,800 19,6363 12,000 16,640 17,4554 12,000 13,312 15,2735 12,000 10,650 13,0916 12,000 8,520 10,9097 12,000 6,816 8,7278 12,000 5,754a 6,5459 12,000 5,754 4,634

10 12,000 5,754 2,181

$120,000 $120,000 $120,000

a Note the straight-line depreciation in years 8, 9, and 10 on the remainingdeclining balance. No more than 100 percent of the asset may be depre-ciated. Salvage is expected to be $10,000 and $102,738 has been chargedby the end of year 7. The straight-line amount is obtained by dividing$(120,000 − 102,738) by 3. This yields $5,754, which is larger than theDDB charge of $5,453 would be, so the switch is advantageous.

cost and lifetime, but on the undepreciated balance and remaining depreciable life at the timeof the switch.

In this example the firm could, under the GG, switch to straight-line depreciation in year 6. Inthis year straight-line depreciation of the remaining balance yields the same dollar depreciationas the double-declining-balance method. However, in years 7, 8, and 9 the amount chargedto depreciation is larger with straight line. In no case may the firm depreciate below salvagevalue. This means that if an asset costs $C, and is expected to have a salvage value of $S, nomore than $(C – S) may be depreciated. Let us now consider an example in which salvagevalue must be taken into account. Note that DDB depreciation ignores salvage, although nomore than $(C – S) may be charged.

Example 3.2 Let us take a project with C = $130,000, N = 10, and S = $10,000. We shallassume the firm will switch to straight-line from DDB depreciation, as allowed by the GG assoon as this is advantageous. Table 3.2 contains a comparison of the methods.

Example 3.3 With the ADR system, salvage is treated the same way as with DDB underthe GG; that is, it is ignored. As always, however, no more than 100 percent of the asset valuemay be depreciated. The following example shows the results of using ADR with the sameasset just considered. In practice, because ADR allows a choice of depreciable asset life, weshould not expect that the asset life will remain 10 years. If a shorter life is allowed, the firmwill take it to maximize accelerated write-off. Tables 3.3 and 3.4 contain a comparison of themethods, including optimal depreciation.

GG allows the firm to switch from DDB to straight line only; ADR allows the firm toswitch to SYD from DDB. This allows for the greatest amount of depreciation in the earlyyears. Switching from DDB to SYD will always be advantageous in the second year of the


Table 3.3 Comparison of depreciation methods underADR system, with salvage

Year Straight line Double declining Sum-of-years

1 $13,000 $26,000 $23,6362 13,000 20,800 21,2733 13,000 16,640 18,9094 13,000 13,312 16,5455 13,000 10,650 14,1826 13,000 8,520 11,8187 13,000 6,816 9,4558 13,000 5,453 4,1829 13,000 4,362 0

10 3,000 3,490 0

$120,000 $116,043 $120,000

Table 3.4 Comparison of optimum depreciation to DDB and SYD,with salvage of $10,000 and cost of $130,000

Optimum Method Double-declining Sum-of-years’Year depreciation used balance digits

1 $26,000 DDB $26,000 $23,6362 20,800 SYD 20,800 21,2733 18,489 SYD 16,640 18,9094 16,178 SYD 13,312 16,5455 13,867 SYD 10,650 14,1826 11,556 SYD 8,520 11,8187 9,245 SYD 6,816 9,4558 3,865 SYD 5,453 4,1829 0 SYD 4,362 0

10 0 SYD 3,490 0

$120,000 $116,043 $120,000

project’s life. Applying this to Example 3.2 yields the following results: actual depreciationin the second year will be the same; SYD will yield greater depreciation in the subsequentyears.

ACRS Depreciation

As a part of the Economic Recovery Tax Act of 1981, new mandatory depreciation rules werepromulgated. The new rules were termed the accelerated cost recovery system (ACRS). Thepurpose in the new depreciation rules was to stimulate investment. Under ACRS depreciation,assets belong to one of several asset life classes. ACRS depreciation is based on the assumptionthat all assets are placed into service at the midpoint of their first year (the half-year assumption)regardless of when during the year they are acquired. For example, after the change in the lawthree-year class assets are depreciated over four years. The reason for this change is that thehalf-year convention was made to apply to the last year of service as well as the first by the1986 Act.


Table 3.5 Depreciation rates for ACRS property other than real propertya

Recovery 3-year 5-year 7-year 10-year 15-year 20-yearyear (200% DDB) (200% DDB) (200% DDB) (200% DB) (150% DDB) (150% DDB)

1 33.33 20.00 14.29 10.00 5.00 3.752 44.45 32.00 24.49 18.00 9.50 7.223 14.81 19.20 17.49 14.40 8.55 6.684 7.41 11.52b 12.49 11.52 7.70 6.185 11.52 8.93b 9.22 6.93 5.716 5.76 8.92 7.37 6.23 5.297 8.93 6.55b 5.90b 4.898 4.46 6.55 5.90 4.529 6.56 5.91 4.46b

10 6.55 5.90 4.4611 3.28 5.91 4.4612 5.90 4.4613 5.91 4.4614 5.90 4.4615 5.90 4.4616 2.95 4.4617 4.4618 4.4619 4.4620 4.4621 2.24

a Assumes the half-year convention applies. Accuracy to two decimal places only. Rates are percentages.b Switchover to straight-line depreciation at optimal time.

After 1985 the ACRS schedule was to have been based on DDB depreciation in the first yearof service, with a switch to SYD depreciation in the second year. However, since the law wasfirst enacted there have been several changes, and doubtless there will be more modificationsto the law. Despite the tendency to tinker with the law every year or two, an understanding ofthe fundamental depreciation methods will enable one to adapt that knowledge to subsequentchanges in the law.

The Tax Reform Act of 1986 changed the system to the Modified ACRS and established sixasset classes in place of the four that had existed. Except for real estate, all depreciable assetsfall within one of the six classes. The law also changed by requiring a switch from DDB tostraight-line in the year for which the straight line amount exceeds the DDB amount.

Table 3.5 contains the MACRS schedule for the 1998 tax year (Form 4562). If history isany guide, by the time you read this the schedule may have changed again, and possibly morethan once. Cost — salvage is always ignored — is multiplied by the percentages. For example,a three-year life asset costing $100,000 would have year 1 depreciation of $33,330.

Canadian Depreciation

Depreciation is generally called “capital consumption allowance” in Canada, or “capitalcost allowance.” Because the Income Tax Act was changed in 1949, only the declining-balance method has been generally allowed. This is not a DDB as discussed earlier, but adeclining balance based on assigned, fixed rates. The rate is applied to the undepreciated


book balance of the asset, and the firm need not charge depreciation in years when it haslosses.

All assets acquired within a tax year qualify for a full year’s depreciation. There are 25asset classes, each assigned a fixed capital cost allowance rate. All assets of a class are pooledtogether. Total capital consumption is calculated by multiplying the book balance of each poolby its corresponding capital cost allowance rate and adding the products together. Capital gainsand losses result only when a given asset pool, not an individual asset, is sold. Capital costallowances due to a specific asset can remain in effect indefinitely if the given asset expireswithout salvage value, if there are other assets in the same asset pool, and if the firm generatesincome from which the capital cost allowance can be deducted.

Summary on Depreciation

1. Under the general guidelines, the only switch that can be made is from DDB to straight-linedepreciation.

2. Under the general guidelines, only the DDB method ignores salvage value.3. With the ADR system the depreciable life is chosen from an IRS guideline range for the

type of asset to be depreciated.4. In the ADR system, salvage value is ignored (as it is with DDB under the general guidelines)

in calculating depreciation with SYD and straight line as well as DDB.5. Under the ADR system, it is permitted to switch from DDB to SYD when this becomes

advantageous to the firm. The switch will be made in the second year in order to maximizethe early write-off of an asset.

6. Although salvage is ignored under ADR (and DDB with the general guidelines), the maxi-mum that may be depreciated is the difference between the asset cost and its salvage value.

It can be shown that under the ADR system, maximization of the tax deferral in the project’searly years will always be achieved by using DDB depreciation in the first year, and SYDdepreciation in the second and subsequent years. Because depreciation rules change from timeto time, it is wise to consult the current tax code.

Investment Tax Credit

From time to time the federal government has provided special tax credit on new asset purchasesin order to encourage aggregate investment in the economy. In recent years the credit has beenincreased from 7 to 10 percent (more in certain special cases) on new investment. The effecton cash flow is much the same as that of depreciation: it facilitates cash recovery during thefirst year of the asset’s life. Because the rules governing application of the investment taxcredit contain some complications, and may change from year to year, the current tax codeshould be checked when estimating the cash flows for an investment project that may qualifyfor the investment tax credit. At the time of this writing the investment tax credit has beenrepealed. However, history shows it is likely to be reinstated, especially when the economygoes into recession, creating a desire to stimulate investment. Therefore, it may be instructiveto examine previous investment tax credit rates. If the asset life is less than three years, nocredit may be claimed. The credit applies to one-third the asset cost if the asset has an economiclife of three years but less than five; and two-thirds if the asset cost is five years but less thanseven. The credit applies to assets described as “qualified investment” under Section 38 of the


Tax Code. It is equal to the amount allowed on new assets under Section 38 plus as much as$100,000 of the cost of newly acquired used assets qualified under Section 38. In no case maythe investment tax credit exceed the firm’s total tax liability for the year.

Unused portions of the investment tax credit are treated in the manner of capital losses:they may be carried back three years and forward five. The firm is restricted in applying thetax credit. If its tax liability is above $25,000, the credit claimed for the year may not exceed$25,000 plus 50 percent of the tax liability exceeding $25,000. If an asset is abandoned priorto the end of its estimated life, a portion of the tax credit claimed may have to be added tothe firm’s tax bill in the year. The amount will equal the difference between the credit actuallyclaimed and the amount that would have been used had the actual asset life been used originallyto calculate the credit.

Inflation

Over a span of time when price levels are fairly constant, depreciation rules and practicesmay be reasonably equitable in allowing the cost of the project to be recovered. A capitalinvestment costing, for example, $150,000 will provide recovery of $150,000, which may beused to purchase a successor project at the end of its useful life. However, if capital equipmentprices were to increase at 12 percent per annum, at the end of only 10 years it would cost$465,870 just to replace the worn machine with an identical new one. Existing depreciationrules do not take this into account, and therefore capital recovery is often inadequate to providefor replacement investment when required.

4

Cost of Capital

The cost of capital is a complex and still unsettled subject. It is discussed in finance texts indetail far beyond what we can devote to it in this text. In this chapter, intended primarily asa review, some of the more important considerations from the theory on cost of capital willbe discussed, and some operational principles illustrated. Cost of capital is treated further inChapter 21, within the context of the capital asset pricing model.

INTRODUCTION

Stated succinctly, the traditional view is that the firm’s cost of capital is the combined costof the debt and equity funds required for acquisition of fixed (that is, permanent) assets usedby the firm. Under this definition even such things as permanent, nonseasonal working cap-ital requirements are acquired with capital funds. Short-term financing with trade credit andbank lines of credit is generally excluded from cost of capital considerations. “Short term” isgenerally understood to be one year or less, in which such balance sheet items as accountspayable and line-of-credit financing are expected to be turned over at least once, or eliminated.Alternatively, the firm’s cost of capital is the rate of return it must earn on an investment sothat the value of the firm is neither reduced nor increased.

In terms of the firm’s balance sheet, cost of capital relates to the long-term liabilities, andcapital section to the firm’s capital structure. Although the specific account titles to be foundfor the various components of capital structure may differ, depending on the nature of thefirm’s business, the preferences of its accountants, and tradition within the industry, certaincommonalities exist. There will usually be long-term debt items in the form of bond issues out-standing or long-term loans from banks or insurance companies. There will always be equitysince a firm cannot be solely debt financed and there must be an ownership account. Equityfor corporations means common stock, retained earnings, and perhaps “surplus”; for propri-etorships and partnerships it may be just an undifferentiated “equity” account. For a varietyof reasons the corporate form of business organization is dominant. However, the principlesfor dealing with corporate organization can be applied straightforwardly to proprietorshipsand partnerships, and thus we will concentrate on the corporate form of organization. Eachcomponent item in the firm’s capital structure has its own specific cost associated with it.

COST OF CAPITAL COMPONENTS

Debt

An important characteristic of debt is that interest payments are tax deductible,1 whereasdividend payments are not; the latter are a distribution of after-tax profits. Thus the effective

1Although interest payments are tax deductible, principal repayments are not. This is a point often overlooked, although not aconceptually difficult one to understand.


after-tax cost of debt is (1 − τ ) times the pre-tax cost, where τ (tau) denotes the firm’s marginaltax rate.

If a firm borrows $1 million for 20 years at annual interest of 9 percent, its before-taxcost is $90,000 annually and, if the firm’s marginal income tax rate is 48 percent, the after-tax cost is $46,800, or 4.68 percent. The cost of debt is defined as the rate of return thatmust be earned on investments financed solely with debt,2 in order that returns available tothe owners be kept unchanged. In this example, investment of the $1 million would need togenerate 9 percent return pre-tax, or equivalently 4.68 percent after-tax, to leave the commonstockholders’ earnings unaffected.

For purposes of calculating the component cost of an item of debt, it is not important whetherthe particular debt component is a long-term loan from a bank or insurance company, whetherit is a mortgage bond or debenture, or whether it was sold to the investing public or privatelyplaced. There is one important exception, however — that of convertible debentures. Suchbonds are convertible at the option of the purchaser into shares of common stock in the firm.Because of this feature, they are hybrid securities, not strictly classifiable as either debt orequity. It is beyond the scope of this book to treat such issues, and the reader is referred tostandard managerial finance textbooks as a starting point in the analysis of those securities.

Preferred Stock

Preferred stock fills an intermediate position between debt and common stock. Ordinary pre-ferred stock has little to distinguish it from debt, except that preferred dividends, in contrast tointerest payments on debt, are not tax deductible. And the firm is under no more legally bindingobligation to pay preferred dividends than it is to pay dividends on common stock. However,preferred stock dividends must be paid before any dividends to common shareholders may bepaid, and unpaid preferred dividends are usually cumulative. This means that if they are notpaid in any period, they are carried forward (without interest) until paid.

The cost of preferred stock may be defined similarly to that of debt. It is the rate of returnthat investments financed solely with preferred stock must yield in order that returns availableto the owners (common stockholders) are kept unchanged. Since preferred issues generallyhave no stated maturity, they may be treated as perpetuities,3 as may securities issues withexceptionally long maturities.4 Therefore, the component cost of a preferred stock that pays adividend Dp and can be sold for a net price to the firm of Pp is given by

kp = Dp

Pp(4.1)

There are many variations of preferred stock, including callable issues, participating, voting,and convertible stocks. Convertible preferreds, like convertible bonds, present problems of clas-sification that are beyond the scope of this book. Principles established for treating convertiblebonds are also applicable to convertible preferreds.

2Note that this view ignores risk and the interactions between cost of capital components. In practice, the firm should evaluate potentialinvestments in terms of its overall cost of capital whether or not the actual financing will be carried out by debt, equity, or somecombination.3A perpetuity is a security that has a perpetual life, such as the British consols issued to finance the Napoleonic Wars and, in thetwentieth century, issues of the Canadian Pacific Railroad and the Canadian central government.4The noncallable 4 percent bonds issued by the West Shore Railroad in 1886 and not redeemable before the year 2361 (475-yearmaturity) could be considered perpetuities. Unfortunately for the bond owners, many firms do not have perpetual life, and thus thereis risk that the firm will fail and the bonds will be rendered worthless.

Cost of Capital 33

Common Stock and Retained Earnings

We will consider equity to exclude preferred stock and to include only common stock andretained earnings. In other words, we take equity to mean only the financial interest of theresidual owners of the firm’s assets: Those that have a claim (proportionate to the shares held)of assets remaining after claims of creditors and preferred shareholders are satisfied in theevent of liquidation.

The cost of equity capital has two basic components: (1) the cost of retained earnings and(2) the cost of new shares issued. In general terms, the cost of equity can be defined as the mini-mum rate of return that an entirely equity-financed investment must yield to keep unchanged thereturns available to the common stockholders, and thus the value of existing common shares.

There are two different but theoretically equivalent approaches to measuring the firm’s costof equity capital. The first is a model premised on the notion that the value of a share ofcommon stock is the present value of all expected cash dividends it will yield out to an infinitetime horizon. We shall refer to this as the dividend capitalization model, or Gordon model.(For detailed development of this and related models, see James C. T. Mao [98, Ch. 10].) It isderived under the assumption that the cash dividend is expected to grow at a constant rate gfrom period to period. The model, for retained earnings, is

kre = D

P+ g (4.2)

where D is the expected annual dividend for the forthcoming year, P the current price pershare, and g the annual growth rate in earnings per share.5

Unfortunately, estimation of the cost of equity capital is not simple and objective asequation (4.2) suggests. Dividends may be quite constant for some firms, but price per shareis subject to substantial volatility, even from day to day. And some firms do not pay dividends;Microsoft has never paid a dividend, at least up to the first quarter of 2002.6 This may requirethat one obtain an average, or normalized price. For corporations whose shares are not activelytraded there may be no recent market price quotation. Growth is affected not only by the in-dividual firm’s performance, but also by the condition of the economy. Therefore, estimationof the firm’s cost of equity capital is not simply employing a formula, but also a substantialamount of human judgment. More complex formulations than equation (4.2) have been de-vised, but still no means for bypassing the need for exercising judgment have been seriouslyproposed. The more complex formulas suffer from the same problems as equation (4.2).

The second approach to estimating the firm’s cost of equity capital is with what has come to beknown as the capital asset pricing model, or CAPM. Although the dividend capitalization modelcould be characterized as inductive, the CAPM might be better characterized as deductive. TheCAPM yields the following equation:

ke = RF + β(RM − RF) (4.3)

where RF is the rate of return on a risk-free security, usually meaning a short-term UnitedStates government security7 such as treasury bills, and RM is the rate of return on the marketportfolio — an efficient portfolio in the sense that a higher return cannot be obtained without

5The cost of new shares is found similarly, except that the share price must be adjusted for flotation costs. Thus, if we denote flotationcosts, as a proportion of the share price, by f , the cost of new equity is given by ke = (D/P(1− f )) + g.6Valuation and cost of capital for nondividend-paying firms presents problems not fully resolved. See for instance Mark Kamstra [78].7Even federal government bonds are not entirely risk free because there is risk of change in their value caused by change in the marketrate of interest. With short-term securities, however, such risk is slight, at least for magnitudes of market changes usually seen.


also accepting higher risk. The beta coefficient relates the returns on the firm’s stock to thereturns on the market portfolio. It is obtained by fitting a least-squares regression of thehistorical returns on the firm’s stock to the historical returns on the market portfolio; it isthe slope coefficient of the regression. β (beta) measures the risk of a company’s shares thatcannot be diversified away, and provides an index that indicates the responsiveness of returnson a particular firm’s shares to returns on the market portfolio. (In calculating betas, capitalappreciation is taken into account explicitly along with dividends in the returns calculations.)

The CAPM thus provides us with a means for estimating the firm’s cost of capital with marketdata and the beta coefficient, which relates the firm to the market. The dividend capitalizationmodel, in contrast, requires only the current market price of our firm’s common shares and itscash dividends. Because of this, the CAPM data requirements are greater. Furthermore, thestability of betas over time for individual firms is not assured.

Example 4.1 If we assume that we have obtained the beta for our firm, which is 1.80, thatthe risk-free interest rate is 6 percent and the return on market portfolio is 9 percent, we obtainas our estimate:

ke = 6% + 1.80(9% − 6%) = 11.4%

The CAPM deals with risk explicitly through the firm’s beta coefficient. The dividend cap-italization model, on the other hand, implicitly assumes risk is fully reflected in the marketprice of the firm’s shares.

OVERALL COST OF CAPITAL

The overall cost of capital is obtained by calculating an average of the individual components,weighted according to the proportion of each in the total. Suppose a firm has the followingcapital structure:

Debt (debentures maturing 2018) $30,000,000Preferred stock 20,000,000Common stock 15,000,000Retained earnings 35,000,000

—————–Total $100,000,000

Assume further that the after-tax costs of the components have been estimated as shown inTable 4.l. Calculation of the weighted average cost of capital is performed as shown.

Note that this average cost of capital is based on historical, balance sheet proportions, andon debt and preferred stock costs that were determined at time of issue of these securities.8

In capital investment project analysis, we are not concerned with average cost of capital. Weare interested in the marginal cost, for that is the cost of funds that will be raised to undertakeprospective capital investments. We cannot raise money at average historical cost, but at today’smarginal rate.

Now, what if the firm must raise an additional $10 million? If the capital structure is judgedto be optimal (more about optimal structure later), funds should be raised in proportion to the

8Once a bond or preferred stock issue is sold, the firm is committed to paying a fixed, periodic return per security. Even though capitalmarket conditions may subsequently dictate higher or lower yields for similar securities of comparable risk if they are to be sold now,the interest and dividend payments on such securities sold in the past do not change.

Cost of Capital 35

Table 4.1 Calculation of weighted average cost of capital (WACC)

(1) (2) (3)$ Amount Proportion % (4)(millions) of total Cost (2) × (3)

Debt 30 0.30 4.16 1.25%Preferred 20 0.20 9.00 1.80Common 15 0.15 15.00 2.25Retained earnings 35 0.35 14.00 4.90

WACC 10.20%

Table 4.2 Calculation of marginal cost of capital (MCC)

(1) (2) (3)$ Amount Proportion % (4)(millions) of total Cost (2) × (3)

Debt 3 0.30 4.68 1.40%Preferred 2 0.20 12.00 2.40Common 1.5 0.15 17.00 2.55Retained earnings 3.5 0.35 15.00 5.25

MCC 11.60%

existing capital structure. If its profits are adequate, the firm may utilize retained earnings ratherthan float new common shares, providing that dividend policy will not be seriously affected.Assume that the $10 million will be raised in amounts and at costs illustrated in Table 4.2.

This 11.60 percent marginal cost of capital suggests that market conditions have changed,and that investors require higher yields than formerly. The result of raising additional fundsat a marginal cost higher than the average historical cost will be to raise the new average costfigure somewhat. The new average cost is given by

10.33% = (100 × 10.20% + 10 × 11.60%)/110

One may be tempted to ask why the entire $10 million should not be raised with debt, thus atthe lowest attainable marginal cost. This question arises naturally, but ignores the interrelatednature of financing decisions. Investors and creditors have notions about the proper mix of debtand equity for firms. Therefore, although today the firm might raise the entire $10 million withdebt, at a later date it could find it has no reserve borrowing capacity, and also cannot borrowon favorable terms or at acceptable cost. In such a situation the firm may find that to raise fundsit must float a new issue of common stock at a time when required yields are much higherthan normal, in a depressed stock market. If that were to occur, it would be a disservice to thecurrent stockholders. The cost of any single capital component by itself cannot be consideredthe true cost of capital for yet another reason. The cost of capital that is associated with anyparticular component applies to that component as a part of the whole firm, within the contextof the firm. Bond purchasers are not merely buying bonds, they are buying bonds of a firmwith a balanced financial structure. Because the overall results of the firm are what mattersto the suppliers of funds, and because funds are not segregated by their origin, it would beinappropriate to use the cost of a component as a substitute for the overall cost.


Analysts have observed that stock and bond prices often move in opposite directions. There-fore, the financial manager has some flexibility in establishing the appropriate financing mixover the short run. Indeed, it is expected that financial management will use its best judgmentin such matters. Sometimes it is better to raise funds with bonds, at other times with commonstock. However, future price and yield trends must be anticipated; investors and creditors willnot willingly tolerate marked deviation in capital structure from established norms over longperiods. And the indenture agreements of prior bond issues or loans may well restrict the firm’slatitude in using more leverage.

OPTIMAL CAPITAL STRUCTURE

In the early 1960s considerable controversy erupted over the theory proposed by Modiglianiand Miller. They contended that the firm’s cost of capital is invariant with respect to its capitalstructure [108], depending only on the risk class to which the firm belongs. The original M andM theory did not take taxes into account, particularly the tax deductibility of interest paymentson debt.9 Subsequent modification of the M and M theory to include tax effects weakenedtheir original conclusions. Since business income taxes are reality in most nations, and interestpayments for businesses a tax-deductible expense, most authorities today agree that there isan optimal capital structure or range of optimal structures for any particular firm. The theoryof M and M is elegantly developed and its repercussions are still affecting financial economicstheory. We shall examine the implications of optimal capital structure now, rather than ventureany further into the M and M arguments that hold only in a world true to their restrictiveassumptions.

Existence of an optimal capital structure for any given firm suggests that financial manage-ment should aim to obtain the optimal, or at least to approach it. Examination of the basicvaluation model introduced early in this book reveals why this is desirable. Attainment of thelowest overall cost of capital will do proportionately more to increase the value of the firmthan will an increase in the net cash flows, because of the compounding of terms containing k,the cost of capital.

We must recognize, however, that world conditions do not remain stationary. The quest foroptimal capital structure requires that one follow a moving target, adjusting and readjustingsights as capital market conditions and investor and creditor attitudes change. Optimal capitalstructure, in practice, is not a once-and-for-all-time achievement. Rather, it requires periodicreview and adjustment. Within a range of leverage, the firm may at times choose to financemore heavily with debt than with equity. Such a process, however, cannot continue indefinitelyor the firm’s leverage will become excessive, and with this the risk of insolvency, so that the costof debt increases and onerous indenture conditions are imposed, not to mention the financialleverage effect on variability in common stockholder returns.

At other times the firm may finance more heavily with equity, or employ preferred shares.An analogy may be drawn to a driver who must use accelerator and brake to adjust to a speedthat is optimal for road conditions — a speed that is safe and yet gets the driver to a destinationin minimum time or with minimum fuel consumption. The driver may use the brakes severaltimes in sequence before using the accelerator, or the accelerator for some time without braking.

9Modigliani and Miller also assumed perfect capital markets, with investors able to borrow and lend at the same interest rate and zerotransactions costs. This created arbitrage opportunities not found in the real world.

Cost of Capital 37

After–Tax %Cost of Capital

20

18

16

14

12

10

8

6

4

2

0 10 20 30 40 50 60Leverage(%:Debt /Assets)

ke

kd

k

Figure 4.1 Cost of capital schedule for a firm

(Unlike debt and equity, however, we may surmise that brake and accelerator will never beused simultaneously.)

The following example assumes constant capital market conditions and investor attitudesfor purposes of illustration. Figure 4.l illustrates the cost of capital schedule for a hypotheticalfirm. For simplicity, it is assumed that debt and equity may be raised in arbitrarily smallincrements, although this abstracts from real world considerations that militate against smallissues of either debt or equity.

Note that the cost of equity capital, ke, rises continuously as leverage increases. This reflectsthe increasing risk to the common shareholders as financial leverage increases: shareholdersrequire a greater return as variability in earnings allocated to them increases. The cost of debt,in contrast, begins at 4.68 percent pre-tax × (1 – 0.48 percent marginal tax rate)] and does notrise until leverage goes beyond 20 percent. Beyond this amount creditors become increasinglysensitive to the risk of firm insolvency as earnings become less and less a multiple of theinterest that must be paid.

Lowest overall cost of capital is reached at 40 percent leverage, even though at this leveragethe component costs of equity and debt are not at their lowest levels. Beyond 40 percentleverage the overall cost of capital rises for additional leverage at a faster rate than it declinedprior to the optimum, thereby reflecting the rapidly rising debt and equity schedules.

If one could find an industry in which the firms were similar except for leverage, a schedulelike that in Figure 4.l could be produced. Each set of observations for a given leverage wouldcorrespond to one firm, or the average of several if multiple observations at the same leveragewere obtained. However, it is difficult to find an industry in which the firms are truly similar,because most firms today are in varying degree diversified in their operations, and are normallyclassified on the basis of their major activity. Furthermore, size disparity between firms would


present problems to the extent that there may be differences in access to capital markets,industry dominance, product brand differentiation, and so on.

INTERACTION OF FINANCING AND INVESTMENT

The foregoing discussion of the firm’s cost of capital assumed that cost of capital was inde-pendent of the firm’s capital investment decisions. In practice this view may be unrealistic.

If the firm consistently follows the practice of investing in capital projects that yield a returnequal to or greater than its existing cost of capital, and do not affect the riskiness of overallreturns to the firm on its investments, then we may assume independence of cost of capital andinvestment. However, if the firm adopts a policy of adopting investments that alter its overallprofitability or variability in earnings, we must recognize that there are likely to be resultantchanges in the cost of capital components and thus the firm’s overall cost of capital.

Changes in the firm’s cost of capital through pursuit of investment policy that alters thecharacteristics of risk (that is, variability in return and probability of insolvency) are notnecessarily adverse. The firm may reduce its cost of capital by reducing risk.10 On the otherhand, the firm may increase its cost of capital if it consistently adopts investments that, althoughoffering high expected returns, at the same time contain commensurately high risk. This willnot necessarily be so, but discussion of this point is deferred to Chapter 22, after portfolioeffects have been considered. Depending upon the correlation between project returns on newinvestments and the existing capital assets of the firm, risk may actually be reduced overall byadopting a very risky new investment.

A simple example with an intuitive interpretation will serve to illustrate this point for now.Assume that the firm undertakes a very risky capital investment project, but has returns that areexpected to be highly negatively correlated with the firm’s existing assets. Acceptance of theproject will reduce the overall riskiness of the firm. If, in a particular year, the existing assetsprovide high cash returns, the new investment will provide low returns. However, if the cashreturns on existing assets were low, then the new investment would provide high returns. Theoverall result will be to smooth out variability in earnings and thus reduce risk.

CAUTIONARY NOTE

Estimation of the cost of capital is fraught with practical problems that militate against assigningsuch work to employees who are not aware of them. The dividend capitalization approach, forinstance, requires that the firm pay dividends and that there be a market-determined price and asense of the market’s expectation for future dividend growth. Other approaches are developing,but present other problems requiring assumptions that may not adequately reflect reality. Noris the capital asset pricing model (CAPM) approach free of practical difficulties. In order touse the CAPM to estimate the cost of equity capital, one must have both market return data andreturn data for the firm and the risk-free rate. For a firm whose shares are not actively tradedin the market it may be a stretch to find a sensible measure of returns. And then there remainquestions of which market index to use, etc. For instance, should a large-capitalization indexlike the Standard & Poor’s 500 be used when calculating cost of capital for a small or mediumsize firm, or an index of small and medium-size firms?

10Some portfolio-approach advocates would argue that, since investors can diversify their portfolios to reduce risk, it is unnecessary,and perhaps even detrimental to stockholder interests, for the firm to diversify its investments.

5Traditional Methods that Ignore

Time-value of Money

This chapter discusses some methods that are often encountered in practice which do not takethe time-value of money into account. In most circumstances these methods should be replacedby better ones that integrate the time-value of money into their rationale.

PAYBACK AND NAIVE RATE OF RETURN

Payback

The payback period criterion has consistently been demonstrated to be the single most popularmeasure of project merit used in practice. Possibly, the payback criterion is the oldest ofcapital-budgeting measures as well.

The payback method is extremely simple to employ and intuitively appealing. To apply themethod to a project costing amount C with uniform cash flows of amount R each period, oneneed only take the ratio of C to R:

Payback ≡ C

R(5.1)

when R is uniform each period. In cases in which cash flows are not expected to be uniform,the method is somewhat more complicated:

Payback ≡ P +C −

P∑t=1

Rt

Rt+1(5.2)

where

C −P∑

t=1

Rt > 0 and C −P+1∑t=1

Rt ≤ 0

Two illustrations will serve to make the procedure clear. We consider first a project designatedas project A (Figure 5.1). This investment requires an outlay today (at t = 0) of $5000, and willyield uniform net cash flows of $2500 each year over its economic, or useful life, of 10 years.

Applying formula (5.1), since the cash flows are uniform, we obtain a payback for projectA of

P(A) = C

R= 5000

2500= 2.0 years

The parenthetical A with P serves to distinguish this from the payback on other projects.Next let us consider project B (Figure 5.2), which, although requiring the same initial outlay

in year t = 0 of $5000, has a nonuniform net cash flow sequence. Cash flows for B in years t = 1through t = 4 are $500, $1000, $2000, and $4000, respectively. In years t = 5 through t = 10,


0 1 2 3 4 5 6 7 8 9 10Year

4

2

0

−2

−4

−6

$ T

hous

and

Figure 5.1 Project A

0 1 2 3 4 5 6 7 8 9 10Year

4

6

8

2

0

−2

−4

−6

$ T

hous

and

Figure 5.2 Project B

the cash flows are uniformly $8000. Since project B has nonuniform cash flows, formula (5.1)is not applicable. We must instead use (5.2). To do this we proceed as follows, taking theabsolute value of C, and successively subtracting the net cash flows:

C = $5000−R(1) = 500

4500 P = 1−R(2) = 1000

3500 P = 2−R(3) = 2000

1500 P = 3

The remaining $1500 is less than the cash flow in year 4. It represents the unrecovered portion ofthe initial outlay. The question now is how far into year 4 we must go to recover this remainingamount. Implicit in the payback method is the assumption that cash flows are uniform overa particular period even if nonuniform from period to period. Therefore, we must go intoyear 4:

$1500

R(4)= $1500

$4000= 3

8or 0.375

Combining results, we obtain the payback period for project B of:

P(B) = P + 0.375 = 3.375

Traditional Methods that Ignore Time-value of Money 41

This procedure for calculating payback is in the form of an algorithm. An algorithm is asystematic, multiple step procedure for obtaining a solution to a problem. Other, more compli-cated algorithms will be introduced in later chapters. Many may be converted into computerprograms in order to reduce human effort and with it the chance of calculation errors.

The Naive Rate of Return

When a manager who relies on the payback criterion speaks of rate of return, this normallyrefers to something other than time-adjusted return on investment. For example, a project thatpromises a two-year payback will be said to offer a 50 percent per year rate of return. This werefer to as naive rate of return (NROR), since it ignores the effects of cash flows beyond thepayback period as well as the effects of compounding from period to period.

NROR ≡ 1/Payback (in years) (5.3)

The criticisms of payback are thus equally applicable to naive rate of return.

Strong Points of Payback

1. It is easily understood.2. It favors projects that offer large immediate cash flows.3. It offers a means of coping with risk due to increasing unreliability of forecasted cash flows

as the time horizon increases.4. It provides a powerful tool for capital rationing when the organization has a critical need to

do so.5. Because it is so simple to understand, it provides a means for decentralizing capital-

budgeting decisions by having non-specialists screen proposals at lower levels in the orga-nization.

Weak Points of Payback

1. It ignores all cash flows beyond the payback period.2. It ignores the time-value of funds.3. It does not distinguish between projects of different size in terms of investment required.4. It can be made shorter by postponing replacement of worn and deteriorating capital until a

later period.5. It emphasizes short-run profitability to the exclusion of long-run profitability.

To illustrate, let us consider along with projects A and B another project, C (Figure 5.3). Thisproject also requires an investment of $5000, and yields net cash flows of $5000, $1000, and$500 in years 1, 2, and 3, respectively. It offers no cash returns beyond year 3. Payback forproject C is easily seen to be 1.0 year.

If we rank the three projects now in terms of payback, we obtain:

Project Payback RankingC 1.0 1A 2.0 2B 3.375 3


0 1 2 3 4 5 6 7 8 9 10Year

4

6

2

0

−2

−4

−6

$ T

hous

and

Figure 5.3 Project C

If we assume that A, B, and C are mutually exclusive, the reliance on payback alone impliesthat project C will be selected as the best project because it has the shortest payback of thethree. In the absence of project C, A would be selected as preferable to B. But which projectwould management prefer to have at the end of four years? Clearly, project B is preferable atthat point, if there is any reliability in the estimates of project characteristics, and if we canassume reasonable certainty of these estimates.

Let us next consider the effects of a large negative salvage value for project C in year 4. Hasthe payback measure for this project been changed? No, it has not, because cash flows beyondpayback are ignored.

Terborgh is [151, p. 207] has provided perhaps the most clever and effective statement ofthe way payback favors delaying replacement of an already deteriorated capital project untilit has deteriorated still further. Payback treats relief from losses caused by undue delay inreplacements as return on the new investment. Consider Terborgh’s analogy:

A corporation has a president 70 years of age who in the judgment of the directors can be retired andreplaced at a net annual advantage to the company of $10,000. Someone points out, however, that if heis kept to age 75, and if he suffers in the interval the increasing decrepitude normally to be expected,the gain from replacing him at that time will be $50,000 a year, while it should be substantially higherstill, say $100,000 at the age of 80. It is urged, therefore, that his retirement should be deferred. Thegenius advancing this proposal is recognized at once as a candidate for the booby hatch, yet it is notdifferent in principle from the rate-of-return requirement.

No one can deny that the advantage of $100,000 a year (if such it is) from retiring the presidentat 80 is a real advantage, given the situation then prevailing. The question is whether this situationshould be deliberately created for the sake of reaping this gain. Similarly, the question is whethera machine should be retained beyond its proper service life in order to get a larger benefit from itsreplacement. The answer in both cases is obvious. The executive who knowingly and wilfully followsthis practice should sleep on a spike bed to enjoy the relief of getting up in the morning.

Considering the strong and weak points of payback all together, is there any merit on balancein using this method? Absolutely not, unless payback is not the sole criterion employed. As asingle criterion, payback may be worse than useless because its implicit assumptions disregardimportant information about events beyond payback. As a tie-breaker to supplement othermethods, payback has considerable merit if we assume that, other things equal, rapid invest-ment recovery in the early years of an investment’s life is preferable to the later years. This viewis reinforced if one considers that the further from the present we attempt to estimate anything,the less reliable our estimates become. In other words, once we introduce risk into our consid-erations, we will ceteris paribus prefer early return of our investment to later recovery of it.


Unrecovered Investment

A concept related to payback, but taking the time-value of money into account, is unrecoveredinvestment [35, pp. 200–219]. Again taking C to be the cost or investment committed (at t = 0)to a capital project, and taking rate k as the firm’s per period opportunity cost of the funds tiedup in the project, unrecovered investment is defined by

U (k, t) = C(1 + k)t −t∑

t=1

Ri (1 + k)t−i (5.4)

Note that if we set k = 0 for values of U ≥ 0, and k = ∞ for U < 0, we find that U = 0defines payback as in the previous section.

If for all periods we take k as the firm’s cost of capital, then it follows that the value of t forwhich U = 0 provides us with a time-adjusted payback period. The value of t for which thisis true will not necessarily be an integer. The t value thus obtained suggests how long it willtake for the firm to recover its investment in the project plus the cost of the funds committedto the project. If k is in fact a precise measure of the enterprise’s opportunity cost, then the tfor which U (k, t) = 0 corresponds to the time at which the firm will be no worse off than if ithad never undertaken the project. Since we assume cash flows occur at the end of each period,this relationship will be approximate but nevertheless useful.

The subject of unrecovered investment is treated further in Chapter 14. The concept of time-adjusted payback is used in the method of analysis discussed in the appendix to Chapter 11.

ACCOUNTING METHOD: ALIAS AVERAGE RETURNON AVERAGE INVESTMENT

The accounting method [116] is based on calculation of an average net cash return on theaverage investment in a project. It is not as popular a method of capital-budgeting projectanalysis as payback, but it is nonetheless still sometimes encountered today. It is easy to apply:it requires only that one:

1. Calculate average accounting profit by(1/N )∑N

t=1 At where A is accounting profit in timet . (Note that this is at odds with our stated use of only net, after-tax cash flows for capitalinvestment analysis.)

2. Calculate average investment by (C + S)/2, where S is estimated salvage value.3. Divide average return by average investment, and express as a percentage.

Thus the average return on average investment (ARAI) is defined as:

ARAI ≡

N∑t=1

At

N

(C + S)/2=

N∑t=1

Rt

N

(C + S)/2(5.5)

Note that if A were uniform, with straight-line depreciation and S = 0, ARAI would be similarto the naive rate of return (NROR). This is because the At terms include depreciation, whereasthe Rt terms include the cash flow dollars shielded from taxes by depreciation.

Strong Points of Accounting Method

1. It is easily understood.2. It does not ignore any periods in the project life.


3. It is, in a sense, more conservative than payback and naive rate of return.4. It explicitly recognizes salvage value.5. Because it is easy to understand, it (like payback) may provide a means for decentralizing

the process of preliminary screening of proposals.

Weak Points of Accounting Method

1. Like payback, it ignores the time-value of funds.2. It assumes that capital recovery is linear over time.3. It does not distinguish between projects of different sizes in terms of the investment required.4. It conveys the impression of greater precision than payback since it requires more calculation

effort, while suffering from faults as serious as that method.5. It does not favor early returns over later returns.6. The method violates the criterion that we consider only net cash flows. We can only pay

out dividends in cash and reinvest cash; not book profits.

COMPREHENSIVE EXAMPLE

Consider the following two projects. They have net after-tax cash flows and depreciationcharges as shown. It is assumed that the class life asset depreciation range system is used, witha switch from DDB to SYD method in year two. Project D has zero salvage, E has salvage of$20,000. The depreciable lifetime for D is five years, for E it is eight years. Note that projectD’s economic life is a year longer than the depreciation life used.

Project D Project E

Initial outlay $100,000 $100,000Cash flow/depreciation for year:1 $25,000/$40,000 $40,000/$25,0002 35,000/24,000 30,000/18,7503 40,000/18,000 20,000/16,0714 40,000/12,000 20,000/13,3935 40,000/6,000 20,000/6,7866 40,000/0 20,000/07 0 20,000/08 0 20,000/0

Payback periods are calculated as follows:

Project D

$100,000−25,000

75,000 P = 1−35,000

40,000 P = 2−40,000

0 P = 3


The payback for project D is exactly 3 years.

Project E

$100,000−40,000

60,000 P = 1−30,000

30,000 P = 2−20,000

10,000 P = 3

and 10,000/20,000 = 1/2, so the payback for project E is 31/2 years.We calculate ARAI on the basis of accounting profits, not cash flows. If we assume that the

timing of accounting profits and cash flows is approximately the same, the accounting profitin any year t , At , will be equal to the net, after-tax cash flow, Rt , less the depreciation chargedin that year: At = Rt − Dt .

Then ARAI for the investment projects is:

Project D

1/6 ($25,000 − $40,000 +35,000 − 24,000 +40,000 − 18,000 +40,000 − 12,000 +

40,000 − 6,000 +40,000 − 0)÷

1/2($100,000 + 0)= $20,000/$50,000= 40.00 percent

Project E

1/8 ($40,000 − $25,000 +30,000 − 18,750 +20,000 − 16,071 +20,000 − 13,393 +

20,000 − 6,786 +20,000 − 0 +20,000 − 0 +

20,000 − 0)÷

1/2($100,000 + 20,000 )= $13,750/$60,000= 22.92%


If we assume the firm has a 10 percent annual cost of capital, the unrecovered investment atthe end of year 3 for each project is:

Project D

$100,000 (1.10)3 −$25,000 (1.10)2

−35,000 (1.10)1

−40,000 (1.10)0

= $133,100 − $108,750= $24,350

Project E

$100,000 (1.10)3 −$40,000 (1.10)2

−30,000 (1.10)1

−20,000 (1.10)0

= $133,100 −$101,400= $31,700

6Traditional Methods that Recognize

Time-value of Money: the Net Present Value

The internal rate of return considered in the next chapter involves finding a unique real rootto a polynomial equation with real coefficients. Prior to modern calculators and computers,that required tedious calculations, particularly for projects with unequal cash flows over manyperiods. And the internal rate of return is fraught with danger for the unaware and careless. Thenet present value (NPV) is considered in the present chapter. The NPV calculation requires themuch simpler task of evaluating that polynomial equation with real coefficients for a given dis-count rate. That rate usually will not be a root to the polynomial. NPV is defined in the equation:

NPV ≡ C +N∑

t=1

Rt

(1 + k)t orN∑

t=0

Rt

(1 + k)t (6.1)

where C is the installed cost, k the enterprise’s cost of capital, Rt the net, after-tax cash flow atthe end of time t, and N the years in the project’s economic life. We can denote C = R0 as theright-hand side of equation (6.1) shows. Thus we have the compact form:

NPV ≡N∑

t=0

Rt (1 + k)t (6.2)

This equation appears to be identical to the basic valuation model considered in Chapter 2.However, there is a subtle difference between them that the notation does not reflect. In thebasic valuation model the Rt are the total or aggregated net, after-tax cash flows in each period.In equation (6.1), on the other hand, the Rt are the net, after-tax cash flows of the particular,individual project under analysis. The same symbols are used in them, but the meaning is thussomewhat different.

The NPV of a particular project provides a measure that is compatible with the valuationmodel of the firm. This much is perhaps obvious because of the mathematical form of theequations. Since we take as given the overarching goal of maximization of the enterprise’s value,we must recognize that acceptance of individual projects with a positive NPV will contributeto the increase of that value. In the absence of capital rationing, in other words, if there is noshortage of money to accept all projects with positive NPVs, the enterprise should do so.

In reference to the calculation of NPV, the conventional approach has been to calculate firstthe gross present value, which is the present value of all the net, after-tax cash inflows, andthen subtract the initial outlay that is assumed to be at present value since it is incurred at t = 0.For projects requiring net outlays beyond the initial period, the outlays are brought to presentvalue in the same manner as the cash receipts.

In the calculation of NPV we take the discount rate, k, as given. The rate used is generally1

the organization’s cost of capital; more particularly, it is the marginal cost of capital. If an

1In the case of risky projects, which we are not considering in this section of the text, a “risk-adjusted discount rate” or “hurdle rate”is sometimes used as an alternative to other methods for dealing with risk.


Table 6.1 The NPV calculations for project A

Year Present worth factor × Cash flow = Present value

1 0.86957 $2,500 $2,173.922 0.75614 2,500 1,890.353 0.65752 2,500 1,643.804 0.57175 2,500 1,429.385 0.49718 2,500 1,242.956 0.43233 2,500 1,080.827 0.37594 2,500 939.858 0.32690 2,500 817.259 0.28426 2,500 710.65

10 0.24718 2,500 617.955.01877 $12,546.92

investment yields a discounted return greater than its discounted cost, it will have NPV > 0.Conversely, if the discounted cost exceeds the discounted expected returns, it will have NPV <

0. Therefore the rule for project adoption under the NPV criterion is

If NPV ≤ 0, reject

> 0, accept

The discounting process employed simply allows cash flows to be judged after they have beenadjusted for the time-value of money. The time reference we use is immaterial: we could justas easily have used net future value by adjusting the cash flows to their compounded (ratherthan discounted) value at t = N rather than at t = 0. However, it is a traditional convention thatNPV rather than net future value be used. And it should be noted that NPV is unambiguouswhile, in contrast, there exists an unlimited spectrum of future dates at which future value canbe calculated.

The NPV method can be further clarified by means of an example. Let us assume that k,the discount rate, is 15 percent per annum and find the NPV for project A, considered in theprevious chapter. This project requires an initial outlay of $5000 and returns $2500 each yearover a 10-year useful life, net after taxes. Table 6.1 illustrates the NPV calculations. Noticethat because the cash flows for this project are a uniform $2500, we could have summed thepresent worth factors and then multiplied the sum once by $2500 to get the NPV. However,this summation has already been done: the results are listed in Appendix Table A.4 for annuitypresent worth factors.2

Now, to obtain net present value, we need to remove only the $5000 project cost that, sinceit occurs at t = 0, is already at present value. (That is, the present value factor for t = 0 is1.00000.) Therefore, the NPV of project A at k = 15 percent is

$7546.92 = $12,546.92 − $5000

NPV > 0, so the project is acceptable for investment.

2Alternatively, if we have a modern calculator, we can find the present worth of annuity factor for N periods and rate k from therelationship

aN k

= 1 − (1 + k)−N

k

The Net Present Value 49

Table 6.2 The NPV calculations for project B

Year Present worth factor ×Cash flow = Present value

1 0.86957 $ 500 $ 434.782 0.75614 1,000 756.143 0.65752 2,000 1,315.044 0.57175 4,000 2,287.00

−2.85498 $4,792.965 0.49718 8,000 $3,977.446 0.43233 8,000 3,458.647 0.37594 8,000 3,007.528 0.32690 8,000 2,615.209 0.28426 8,000 2,274.08

10 0.24718 8,000 1,977.445.018772.16379 × 8,000 = 17,310.32

Gross PV $22,103.28Less C at t = 0 −5,000.00

NPV = $17,103.28

Next, let us find the NPV for project B (also considered in the previous chapter). Thecalculations are shown in Table 6.2. Note that since cash flow in years 5–10 is a uniform $8000,the present value of these flows can be obtained by multiplying by the present worth of annuityfactor for 10 years less the present worth of annuity factor for four years: $8000(5.01877 −2.85498).

If we were considering A and B as mutually exclusive projects, B would clearly be preferredbecause its NPV of $17,103.28 is more than 21/2 times larger, and the required $5000 investmentis the same for both projects.

The internal rate of return (IRR), which the next chapter covers, is a special case of NPV.The IRR is that particular discount rate for which the NPV is equal to zero. In other words, theIRR is that rate of discount for which the present value of net cash inflows equals the presentvalue of net cash outflows.

Figure 6.1 illustrates the NPV functions for projects A and B for various discount rates.Negative rates are for purposes of illustration only, not because of any economically meaningfulinterpretation of such rates as cost of capital. Note that the IRR of each project is at that rate ofdiscount where the NPV of that project intersects the horizontal axis, and that NPV = 0 at thesepoints. It can be shown that for all investments with R0 = C < 0, and Rt ≥ 0 for all 0 < t ≤ N ,that the NPV function is concave from above. For projects having some Rt ≥ 0 for 0 < t ≤ Nthis will not necessarily be true.

Let us consider an investment project that has some negative cash flows after t = 0. We willcall this project AA. The cash flows over a four-year economic life are:

R0 R1 R2 R3 R4

−1000 1200 600 300 −1000

This project has two “IRRs”: −8.14 percent and +42.27 percent. Quotation marks are placed


−5−10

010 20 30 40 50 60

%

IRRA

IRRB

10

20

30

40

50

B

A

NPV ($000)

Figure 6.1 NPV functions for projects A and B

–10 –5 5 10 15 20 25 30 35 555040 60%

–8.14%15

30

45

60

105

120

135

$NPV

42.27%

0

Figure 6.2 NPV curve for project AA

on the IRR because, as we shall see in Chapter 10, neither rate is internal to the project andneither rate measures the return on investment for this project. Figure 6.2 contains the graphof the NPV function for this project. Note that NPV is positive until the discount rate reaches42.27 percent, but that the NPV is concave from below up to that point.

We shall consider such projects in more detail in Chapter 10. For now it will be useful tomake a mental note that they are neither purely investments nor purely financing projects (loansto the enterprise) but a mixture of these. Because many interesting projects are of this nature,


it is important that we have means for their proper analysis, although we shall defer this untilChapter 10.

UNEQUAL PROJECT SIZE

A common difficulty that arises with NPV is that marginally valuable projects may show ahigher NPV than more desirable projects simply because they are larger. For example, considerprojects D and E:

Project D Project E

Cost = R0 $100,000 $10,000Cash flows, R1 through R10 30,000 10,000R11. . . . . . . . . . . . 0 0

Again using k = 15 percent, we see that the NPV for project D is $50,570, whereas that forproject E is only $40,190. Therefore, D would seem preferable to E. However, D costs 10times as much as E. The extra NPV is only 11.5 percent of the additional $90,000 requiredinvestment — an amount less than the 15 percent cost of capital.

This problem is a serious one, especially when investment funds are limited and, as aconsequence, there is capital rationing within the organization. Fortunately, this problem iseasily corrected.

The Profitability Index

The problem of unequal project size with the NPV is easily corrected by using what is calledthe profitability index (PI). The PI may be defined in two ways; these are identical except fora constant of 1.0. The more common way is by the ratio of gross present value to project cost.Under this definition an acceptable project, one with NPV < 0, will have a PI > 1.0.

This author’s preference is to define PI as the ratio of net present value to present value ofproject cost:

PI ≡ NPV

Cor

NPV

R0(6.3)

Thus, if NPV > 0, PI > 0. On the other hand, if NPV ≤ 0, PI < 0. It is immaterial whichdefinition is used, provided that it is used consistently. They are identical except that definition(6.3) yields a PI that is 1.0 less than that obtained under the alternative definition.

Let us look again at projects D and E in terms of their PI:

PI: D = 0.5057 E = 4.019

The results obtained strongly favor project E because the return on the investment is muchhigher proportionately than that on project D.

UNEQUAL PROJECT LIVES

Another problem can exist with NPV and the corresponding PI. This is due to the effect ofunequal project lives. For example, let us assume that a given manufacturing operation is


expected to be required for an indefinite time, and that two mutually exclusive projects, F andG, both have acceptable NPVs and IRRs. However, project F is expected to last for 10 yearswhile project G is expected to last for only five, after which it will have to be replaced. Canwe determine which should be accepted on the basis of PI as defined earlier? No. We musttake into account the effect of the difference in project lives. As a focus for defining how thisshould be done, let us consider the specific projects F and G.

Net cash flows

Year Project F Project G

0 $100,000 $100,0001 30,000 40,0002 30,000 40,0003 30,000 40,0004 30,000 40,0005 30,000 40,0006 30,000 07 30,000 08 30,000 09 30,000 0

10 30,000 0

Letting k = 15 percent, we obtain for the projects:

F G

NPV $50,563 $34,086PI 0.506 0.341

Since project G will have to be replaced at the end of year 5, we need to take this into accountsystematically. (We will assume it can be replaced for $100,000 at that time, ignoring the un-certain effects of inflation.) One way is to assume that if it is adopted, project G will be replacedby an identical project at the end of year 5 and then treat the cash flows over the entire 10-yearperiod explicitly. In other words, we calculate NPV and PI for project G that now has cash flows:

Net cash flows

FirstTime Project G Replacement Combined

0 $−100,000 $−100,0001 40,000 40,0002 40,000 40,0003 40,000 40,0004 40,000 40,0005 40,000 $−100,000 −60,0006 40,000 40,0007 40,000 40,0008 40,000 40,0009 40,000 40,000

10 40,000 40,000


The values obtained now for project G are:

NPV = $51,033

PI = 0.510

so that, contrary to the unadjusted results previously obtained, project G is preferable to projectF. The method just illustrated for adjusting for unequal lives can be tedious to apply, sinceit requires us to calculate NPV over the least common denominator number of years of theproject lives. In this example project F lasts exactly twice as long as project G, so this presentsno problem. However, what if we had one project lasting 11 years and another lasting 13 years(both prime numbers)? In this case we would have to evaluate 143 cash flows for each project!Fortunately, there is a better way.

Level Annuities

A much easier equivalent method is to calculate the time-adjusted annual average (that is, thelevel annuity) for each project. This is done by multiplying the unadjusted NPV by the capitalrecovery factor (also called the annuity with present value of 1.0) for the expected number ofyears of useful life in the project. The capital recovery factor (CRF) is simply the reciprocalof the present value of annuity factor.3

For projects F and G, the time-adjusted annual averages are:

Project F: $50,563 × .019925 (10-year CRF) = $10,075

Project G: $34,086 × .029832 (5-year CRF) = $10,169

It is apparent that project G offers a somewhat higher NPV on an annual basis than does F.Now, to convert these results to adjusted NPVs for the projects over the longer-lived project, inthis case 10 years, we need only multiply by the present value of annuity factor correspondingto this number of years. Thus, we obtain adjusted NPVs of:4

Project F: $10,075 × 5.0188 = $50,564

Project G: $10,169 × 5.0188 = $51,036

And the corresponding profitability indexes are 0.506 and 0.510.

SUMMARY AND CONCLUSION

Now that we have seen how two serious difficulties with NPV may be surmounted with theprofitability index and uniform annual equivalents, what can we say in summary about NPV?

Strong Points of NPV

1. It is conceptually superior to both the payback and accounting methods.2. It does not ignore any periods in the project life nor any cash flows.3. It takes into account the time-value of money.

3The CRF is defined by the equation k/[1 − (1 + k)−N ] for rate k and N periods. The CRF enables us to spread out any given presentvalue over a specified number of years as well as providing a means for finding a time-value-adjusted average present value.4Minor rounding errors make the calculated result different by a small amount.


4. It is consistent with the basic valuation model of modern finance.5. It is easier to apply than the IRR since it involves evaluating a polynomial rather than finding

a polynomial root.6. It favors early cash flows over later ones.

Weak Points of NPV

1. Like the IRR, it requires that we have an estimate of the organization’s cost of capital, k.Also, the given k is embedded in the NPV, whereas with the IRR the internal rate can bejudged by management. Management will determine whether it is reasonable that the IRRis greater or less than k when k is not known with confidence. In other words, IRR may bemore intuitive.

2. It is more difficult to apply NPV than payback or the accounting method, and thus lesssuitable for use by lower levels in the organization without proper training in its application.That may not be feasible.

3. Unless modified by conversion to uniform annual equivalents and converted to a profitabilityindex, NPV will give distorted comparison between projects of unequal size and/or unequaleconomic lives.

4. NPV depends on forecasts, estimates of future cash flows that become increasingly lessreliable and more nebulous the further into the future they are expected to occur.

7Traditional Methods that Recognize Time-value

of Money: the Internal Rate of Return

DEFINITION OF THE IRR

The internal rate of return (IRR) is defined as the rate of interest which exactly equates theNPV of all net cash flows to the required investment outlay. Equivalently, the IRR is the ratefor which the NPV of the cash flows is equal to zero. Thus IRR is the rate of discount r, whichsatisfies the relationship

C = R1

(1 + r )1 + R2

(1 + r )2 + R3

(1 + r )3 + · · · + RN

(1 + r )N (7.1)

If we rename C to R0, this can be restated as

0 = R0

(1 + r )0 + R1

(1 + r )1 + R2

(1 + r )2 + · · · + RN

(1 + r )N (7.2)

or in the more compact equivalent forms

0 =N∑

t=0

Rt

(1 + r )t =N∑

t=0

Rt (1 + r )−t (7.3)

Equations (7.1), (7.2), and (7.3) define a polynomial equation with real coefficients R0,R1, . . . , RN . The IRR is thus the root of a polynomial. Unfortunately, under some circum-stances there may be two or more real roots, or an economically misleading root, and thiscan create some significant problems. The general case of multiple roots will be dealt withlater. For now we will consider only investment projects that have just one sign change inthe coefficients, and for which we are assured by Descartes’ rule of signs, can have but onereal, positive root. Cost C, or as we will normally call it from now on, R0, we assume tobe negative, and R1, R2, R3, . . . , RN to be positive. In later chapters this assumption may berelaxed.

In the past, before the personal computer and spreadsheet software, finding the IRR couldoften be a troublesome task, especially when the cash flows after R0 were nonuniform. We shalldeal here first with a project having uniform expected cash flows. To illustrate the calculationof IRR when cash flows are uniform, and we have only a calculator, let us consider project Afrom Chapter 5 again, and use equation (7.1):

$5000 =$2500

1 + r+ $2500

(1 + r )2 + · · · + $2500

(1 + r )10 (7.4)


which, since the Rt are uniform, can be rewritten as

$5000 = $250010∑

t=1

1

(1 + r )t = $250010∑

t=1

(1 + r )−t (7.5)

so that we have

2.0 = $5000

$2500=

10∑t=1

(1 + r )−t = 1 − (1 + r )−10

r≡ a

10 r(7.6)

Since the last term on the right here is the summation representing the present value of anannuity of 1.0 for 10 periods at r percent,1 we look in Appendix Table A.4.2 Moving acrossthe row that corresponds to the 10 periods, we find the factor 2.003 under 49 percent. Sincethis is very close to 2.0, we might conclude that the IRR for project A is essentially 49 percent(the actual rate is 49.1 percent to the nearest one-tenth percent).

Now let us see what must be done when cash flows are not uniform for all periods beyondt = 0. Finding the IRR for project B is not so easy, because the cash flows are not uniform.To find r we use the Appendix tables for single-amount present worth and for annuity presentworth.

First, let us estimate that r is 60 percent. We then proceed as follows:

Present worth PresentYear factor × Cash flow = value

1 0.62500 $500 $312.502 0.39063 1000 390.633 0.24414 2000 488.284 0.15259 4000 610.365 • •6 • •7 0.2391 } 8000 } 1912.808 (1.6515 − 1.4124) •9 • •

10 • •Project present value at 60% = $3714.57

1The term is derived from the formula for the summation of a geometric progression, which is

Sum = αρN − 1

ρ − 1

where ρ is the common ratio (in the case above, ρ = (1 + r )−1), α the first term in the progression, and N the number of periods. Foran ordinary annuity this becomes

(1 + r )−1 (1 + r )−N − 1

(1 + r )−1 − 1= (1 + r )−1 (1 + r )−N − 1

−r (1 + r )−1 = 1 − (1 + r )−N

r≡ a

N r

2Modern hand-held calculators provide a better approach: calculate the factor for a trial r, and revise the estimate until the result issatisfactorily close. If a table is readily available, it provides a good first guess, of course. Many reasonably priced calculators areavailable with circuitry preprogrammed to provide very precise solutions for r.

The Internal Rate of Return 57

Present worth PresentYear factor × Cash flow = value

1 0.66667 $500 $333.332 0.44444 1000 444.443 0.29630 2000 592.604 0.19753 4000 790.125 • •6 • •7 0.3604 } 8000 } 2883.208 (1.9653 – 1.6049) •9 • •

10 • •Project present value at 50% = $5043.69

Since the cash flows in years 5–10 are a uniform $8000, we need multiply this amount onlyonce by the present worth of annuity factor for 10 years at 60 percent less the present worthof annuity factor for four years at 60 percent. Notice that the latter is equal to the sum of thesingle amount present worth factors for years 1–4. The project present value at 60 percent is$3714.57, which is less than the $5000 investment outlay. Therefore we know that 60 percentis too high a discount rate, and that IRR is less than this. Let us next try 50 percent.

This is just slightly above the $5000 investment cost. We have bracketed the IRR, and nowknow that it is very close to 50 percent. We can now refine our result by interpolation.

60% 3714.57

r50%

] [5000

5043.69

−10

50 − r= 1329.12

43.69

r = 50 + 436.90

1329.12= 50.329% or 50.3%

This interpolation assumes a linear relationship, whereas we have an exponential one, so ourresult is only approximate. However, for capital-budgeting applications it will normally beprecise enough. We should generally give interpolated answers to only one decimal or thenearest percent or tenth of a percent in order not to convey the impression of greater precisionthan we in fact have. The actual rate, correct to the nearest tenth of a percent, is 50.3 percentin this case. In general, the smaller the range over which we bracket the rate, the more preciseour interpolated result will be.

Nowadays one should seldom if ever have to resort to interpolation to find an IRR. Allpopularly used spreadsheet programs provide built-in functions to find IRR. Still, it may beuseful to know how to find a result by interpolation should one be caught without access to acomputer spreadsheet program on rare occasions.

By this point it should be clear that calculation of IRR can be a tedious task when cash flowsare not uniform. In fact, to ease the computational burden, computer spreadsheet programshave built-in functions to calculate IRR, and some hand-held calculators are programmed to


solve for IRR.3 Programs to calculate IRR are not difficult to write; the Newton–Raphson orinterval bisection methods, in conjunction with Horner’s method of polynomial evaluation,both yield good results efficiently.

A CAUTION AND A RULE FOR IRR

Assume we have a project costing $4000 that will last only two years and provides cash flowsof $1000 and $5000 in years 1 and 2. To find the IRR for this project we may use the quadratic

formula −b ±√

b2 − 4ac/2a, since (dropping the $ signs):

4000 = 1000

1 + r+ 5000

(1 + r )2 (7.7)

is equivalent to

−4 + 1

1 + r+ 5

(1 + r )2 = 0 (7.8)

Multiplying by (1 +r)2, we get

−4(1 + r )2 + (1 + r ) + 5 = 0 (7.9)

Letting x = 1 +r , we obtain the quadratic equation

−4x2 + x + 5 = 0 (7.10)

which has roots given by

−1 ± √1 + 80

−8

of x = −1.0 and +1.25. Converting to r, we have IRR of −200 percent and +25 percent!Which do we take as the IRR for this project? The rule to follow in such cases is:

In the case of one positive or zero root and one negative root, choose the negative root only if theproject cost is strictly greater than the undiscounted sum of the cash flows in periods 1 through N .

In this case 4000 < 1000 + 5000, so we choose + 25 percent for IRR. A negative IRR makesno economic sense in cases in which the total cash return is greater than or equal to the projectcost. Conversely, for cases in which the cost exceeds the undiscounted cash flows, a positiveIRR makes no economic sense. A negative IRR < −100 percent makes no economic sensebecause it is not possible to lose more than all of what is lost on a bad investment when allcash flows attributable to the project are included.

PAYBACK AND IRR RELATIONSHIP

In Chapter 5 the naive rate of return (NROR) was defined as the reciprocal of payback. Is thereany relationship between NROR and IRR? Yes. Provided that certain conditions are met, theNROR may approximate the IRR.

3Texas Instruments, Hewlett-Packard, and others make such calculators.


Assuming uniform cash flows, R = R1 = R2 = · · · = RN . Then payback = C/R, NROR =1/payback = R/C , and IRR is the r such that

C =N∑

t=1

R

(1 + r )t = RN∑

t=1

(1 + r )−t (7.11)

C

R=

N∑t=1

(1 + r )−t (7.12)

so that

R

C= 1

N∑t=1

(1 + r )−t

= r (1 + r )N

(1 + r )N − 1= r

1 − (1 + r )−N (7.13)

which is defined as the capital recovery factor for N periods at rate r per period. Now, takingthe limit as N increases, we obtain

R

C= lim

N→∞

[r

1 − (1 + r )−N

]= r (7.14)

which is the same result as the NROR provides.Therefore, under some conditions, the NROR may provide a useful approximation to IRR.

What are the conditions? The first condition is that the project life should be at least twice thepayback period. The second is that the cash flows be at least approximately uniform:

1. N ≥ 2C/R and2. uniform R.

Financial calculators make the task of finding the IRR of a uniform cash flow series easy,as long as there are not a great many cash flow amounts to enter on the keypad. However,for projects with nonuniform cash flows, it is useful to employ a computer, especially if thereare many cash flow periods, or if it is desired to deal with the problem of mixed cash flowsaccording to methods described in Chapter 10 and those following it. Occasionally one mightwish to write a computer program that will be used in large-scale production runs, as at a bank.Thus, it is worthwhile to have a general understanding of ways to efficiently find the IRR.

MATHEMATICAL LOGIC FOR FINDING IRR

For projects that we know will have only one real, positive root, by Descartes’ rule of signs,4

several methods can be easily programmed for computer solution. The most straightforwardare interval bisection and Newton–Raphson. A capital investment with one real positive IRRyields an NPV polynomial function that looks like the one in Figure 7.1, that is, concave fromabove.

Note that the IRR is the discount rate for which the discounted value of the project, includingthe cost C (i.e. R0), is zero. In other words, the IRR is the discount rate for which the projectNPV is zero. Thus, to find the IRR we must find the rate for which the polynomial intersectsthe horizontal axis.

4Descartes’ rule states that the number of unique real roots to a polynomial with real coefficients cannot exceed the number of changesin the sign of the coefficients and that complex roots must be in conjugate pairs. That is, if x + ci is a root, x − ci is also a root.


0r

Rate

+$

−$

Figure 7.1 IRR polynomial

$NPV

0m

r

hRate

Figure 7.2 Interval bisection

Interval Bisection

With the interval bisection method, we select a high value and a low value that we think willbracket the IRR. If we are wrong, we will try higher values the next time. Assume we firstselect 0 as our low value and h as our high value. This is shown in Figure 7.2. We evaluate theNPV polynomial at rate 0 and at rate h. Since the NPV is positive for rate 0 and negative forrate h, we know the function has a root (crosses the horizontal axis) between these rates.

Next we improve our results by bisecting the interval between 0 and h and testing to determinein which subinterval the function crosses. We also test for the possibility that the midpoint ofour earlier range, m, may actually be the IRR, even though unlikely. If we determine that theIRR lies in the interval between m and h, we bisect this range again, and repeat the process.We determine at the outset to stop when the difference between the low and high intervalvalues is less than or equal to an arbitrarily small value. This error tolerance cannot be toosmall, however, because all digital computers have intrinsic round-off errors in computationand different capability for precision calculation.5 In general, however, a tolerance of 0.0001

5This is potentially a very serious problem, especially as the number of cash flows becomes large. Logically correct programs mayyield totally incorrect results for which the program user is unprepared after testing the program with small problems having few cashflows and getting correct results.


0

$NPV

rnr

rn + 1

Rate

Figure 7.3 Newton–Raphson method

or 0.01 percent should cause no problems. If it does, a change to double-precision calculationmay remedy the difficulty. The manual method of IRR solution is similar to this, except thatthe final IRR estimate is made by interpolation rather than continued iteration, once it is knownthat the IRR value has been bracketed.

Newton–Raphson Method

This method is somewhat more sophisticated than the interval bisection approach, and in somecases marginally more efficient.6 Any increase in efficiency it yields will likely result in verysmall savings in computer time, however.

The approach with this method is to modify the original “guess” for the IRR by using theintersection of the tangent line to the NPV curve with the rate axis as the improved “guess.”The process is repeated until the NPV is sufficiently close to zero. This might be expressedas an error less than or equal to some small percentage of the project cost, for example, oralternatively an NPV less than some small money amount.

The Newton–Raphson method requires that the numerical value of the derivative with respectto rate of the NPV be obtained at the current rate estimate. If we refer to rn as the current IRRestimate and rn+1 as the revised estimate, f () as the NPV function, and f ’( ) its derivative, then:

rn+1 = rn − f (rn)

f ′(rn)

The value of rn+1 is our revised estimate of the IRR. Graphically, the Newton–Raphsonapproach may be visualized as shown in Figure 7.3.

For the NPV function at rate i, the polynomial is

NPV = f (i) = R0(1 + i)−0 + R1(1 + i)−1 + · · · + Rn(1 + i)−n

and the derivative:

f ′(i) = −R1(1 + i)−2 − 2R2(1 + i)−3 − · · · − n Rn(1 + i)−(n−1)

6Marginal is used here in the sense that one or two seconds of computer time is not usually significant unless a program is run veryfrequently in a commercial environment.


Computer evaluation of f ′(i) involves modification of f (i) by dropping the first term contain-ing R0, negating all the subsequent cash flow terms, multiplying each by the correspondingexponent, and then decrementing each of the exponent powers by one.

Strong Points of IRR

1. It is conceptually superior to the payback and accounting methods.2. It does not ignore any periods in the project life nor any cash flows.3. It takes into account the time-value of funds.4. It is consistent with the basic valuation model of modern finance.5. It yields a percentage that management can examine and make judgment about when k is

not known with confidence. That is, it yields an intuitive figure to management.6. It favors early cash flows over later ones.

Weak Points of IRR

1. It requires an estimate of the organization’s cost of capital, or at least a range of values inwhich this is likely to be found.

2. It is much more difficult to apply without a computer than the payback or accountingmethods, and when cash flows are nonuniform, much more difficult to apply than the NPVmethod.

3. It does not distinguish between projects of different size and/or different economic lives.However, adjustment for this may be made along lines similar to such adjustments for NPV.

4. It often yields multiple, and thus ambiguous, results when there is more than one signchange in the cash flows.

5. Some have criticized the method on the basis that it implicitly assumes that cash flows maybe reinvested at a return equal to the IRR.

6. Like the NPV, the IRR requires forecast estimates of future cash flows, and thus suffersfrom whatever error to which those forecasts are subject.

A DIGRESSION ON NOMINAL AND EFFECTIVE RATES

Investment–Financing Relationship

The mathematics of the IRR are identical to those required for finding the effective interest rateon a loan or the yield to maturity or call on a bond. To the lender, a loan or a bond purchase isan investment. Symmetry of the loan or bond relationship requires that the effective cost rateto the borrower (bond issuer) be the same as the rate of return to the lender (bond purchaser)on a pre-tax basis. Mathematically, the only difference in a loan or bond from the perspectivesof borrower and lender is that the signs of the cash flows will be reversed.

Consider a bond that has a face value of $1000, a nominal interest yield or coupon rate of7 percent, and a 20-year maturity. It sells when first issued for $935. Interest will be calculatedsemiannually and we assume the bond will not be called prior to its maturity date. To thepurchaser of the bond who buys at issue, it is an investment with cash flows in periods zerothrough 40 (semiannual payments over 20 years) and the IRR equation is, dropping the $ signs,

−935 + 35(1 + r s)−1 + 35(1 + r s)

−2 + · · · + 35(1 + r s)−40 + 1000(1 + r s)

−40 = 0

(7.15)


To the issuer of the bond, the equation to calculate the effective cost is

+935 − 35(1 + r s)−1 − 35(1 + r s)

−2 − · · · − 35(1 + r s)−40 − 1000(1 + r s)

−40 = 0(7.16)

The solution to the real, positive root of equation (7.15) is the same as that to equation (7.16).

Nominal Rate and Effective Rate

Note that the rate in equations (7.15) and (7.16) is denoted by rs. This is to flag it as a semiannualrate. The corresponding nominal annual yield or cost rate is two times rs. The effective annualrate is that rate that would be equivalent if interest payments were made annually at the end ofeach year instead of at the end of each semiannual period.

For the bond in question the (pre-tax) rates are:

Semiannual effective yield rs = 3.81964%Nominal annual yield rn = 2rs = 7.63928%Effective annual yield r = (1 + rs)2 − 1 = 7.78518%

It is usual financial practice in the United States to report yields in terms of nominal annualrates rather than effective annual rates. If the investor has the opportunity to reinvest interestpayments as they are received at the effective per-period rate, however, the period-to-periodcompounding implicit in this should be recognized. The effective annual yield recognizes suchcompounding; the nominal annual rate does not.

Clarification of Nominal and Effective Rates

Let us examine here a problem that may help to clarify the relationship between, and meaningof, the various rates. In the case of a bond, the coupon rate is the rate applied to the face or parvalue of the bond to determine the dollar interest to be paid. The coupon rate is a nominal annualrate. The rate advertised by thrift institutions, which is compounded so many times per year,is also a nominal annual rate. For example, 53/4 percent compounded quarterly is a nominalannual rate. If we divide it by the number of compounding periods, we obtain the effective rateper quarter, which is 1.4375 percent per quarter. Money left on deposit will thus earn 1.4375percent every three months. If $1000 is deposited at the beginning of a year under these terms,it will grow to $1058.75 at year end. This is thus an effective increase of 5.875 percent, not53/4 percent.

In general, nominal annual (coupon) rate, per period effective rate, and effective annual rateare related as follows:

1. Nominal annual rate (rn) ÷ number of compounding periods in a year (M) = effective perperiod rate (rp).

2. The quantity one plus the effective per period rate raised to the power corresponding to thenumber of compounding periods in a year = one plus the effective annual rate.

rn

M= rp

(1 + rp)M − 1 = r

Or, comprehensively in one equation:

r = (1 + rn

M

)M− 1


Figure 7.4 Quarterly versus annual cash flows for IRR calculation

Unless it is perfectly clear from the context in which the term is used, rn should never becalled just the “annual rate,” but should be called the “nominal annual rate” or “annual ratecompounded M times per year.” To do otherwise is imprecise, confusing, and misleading.

IRR with Quarterly Cash Flows

Consider an investment project costing $5000 that will last an estimated five years and providesnet, after-tax cash flows at the end of each quarter of $300 in all but in the final quarter which,with salvage, amounts to $1000.

The quarterly, per-period IRR is rp = 2.701 percent. The nominal annual IRR, rn is 10.804percent compounded quarterly. The effective annual IRR rate, r = 11.249 percent. If we wereto ignore the fact that the cash flows occur quarterly, and instead treat the flows for each yearas if they fell on the last day of the year, we would get an IRR of r = 9.880 percent. Note thatthis is significantly less than if the cash flows were treated as falling at the end of every threemonths, because the cash flows, on average, are pushed into the future some one-and-one-halfquarters or some four-and-one-half months (see Figure 7.4). One may take comfort in the factthat such differences in the IRR will seldom cause a worthwhile investment to be shunned ora poor one to be accepted.

8Reinvestment Rate Assumptions for NPV

and IRR and Conflicting Rankings

This chapter discusses some unsettled conceptual problems that have figured prominently inthe literature of capital budgeting. Warning: the material presented here may be somewhatcontroversial to other writers in the field of capital investments. My intent is that this chapterwill motivate readers who are already familiar with the standard lore of capital budgeting tothink critically of some things that may have been taken for granted, and judge for themselvesthe truth of the matter.

REINVESTMENT RATE ASSUMPTIONS FOR NPV AND IRR

Comparison of equations (7.3) with (6.1) and (6.2) shows that the internal rate of return (IRR)is nothing more nor less than a special case of net present value (NPV). The IRR is defined asthe rate for which NPV is zero. With this in mind, let us examine a general equation combining(7.3) and (6.1):

X = R0

(1 + d)0 + R1

(1 + d)1 + · · · + RN

(1 + d)N (8.1)

or

X =N∑

t=0

Rt

(1 + d)t =N∑

t=0

Rt (1 + d)−t (8.2)

If the discount rate d is such that X = 0, we say that d is the IRR. If X = 0, we say that X isthe NPV for cost of capital k = d. Therefore, by employing equations (8.1) and (8.2) we candeal simultaneously with both IRR and NPV.

So far, all mathematical formulations for IRR and NPV have been in terms of present value.We can easily convert present value to future value by multiplying both sides of the equationby (1 + d)N , where d is either the IRR or cost of capital k, as the case may be. If we do thiswith equations (8.1) and (8.2), we obtain

X (1 + d)N = R0(1 + d)N + R1(1 + d)N−1 + · · · + RN (8.3)

X (1 + d)N =N∑

t=0

Rt

(1 + d)t−N =N∑

t=0

Rt (1 + d)N−t (8.4)

Note that if d is the IRR, then X = 0 and X (1 + d)N is zero. Therefore, IRR could be as easilyobtained from a future value formulation as from the conventional present value formulation.Also, if X were not zero, that is, if X were an NPV for k = d , the effect of multiplying by(1 + d)N simply moves the reference point from t = 0 to t = N .

In other words, looking at equation (8.3), we obtain the same results except for a constantof (1 + d)N times the NPV by assuming the cash flows are reinvested at earning rate d, rather


than being discounted at that rate. The R0, which for most projects will be the cost, or initialoutlay, is “invested” for N periods at rate d compounded each period. For the initial outlaythis may be interpreted as the opportunity cost of committing funds to this project insteadof to an alternative purpose in which rate d could be earned. Or, in the case where d = k,the opportunity cost may arise from the decision to undertake a project requiring funds to beraised, whereas without the project no new funds would need to be raised. The mathematicalsymmetry between these formulations has been a cause for concern among finance theorists.Let us consider the kernel of their concern.

The implication of future value formulation is that the project return, whether measured byIRR or by NPV, will depend on the rate at which cash flows can be reinvested. For a firm ina growth situation, in which profitable investment opportunities abound, the IRR assumptionthat cash flows may be reinvested at a rate of earning equal to the IRR may thus be reasonable.For other firms, and government institutions, some analysts think it is more realistic to assumethat the cash flows can be reinvested at a rate equal to the cost of capital k. This is the usualformulation of the reinvestment rate assumption.

Let us now consider the IRR and reinvestment rate in another light. Consider a loan of$100,000 that is made by a bank to an individual business proprietor for a period of five years.The loan is to be repaid in equal installments of $33,438 (to the nearest whole dollar). Fromthe bank’s viewpoint this is an investment, with cash flows:

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5−$100,000 $33,438 $33,438 $33,438 $33,438 $33,438

and yield (that is, IRR) of 20 percent.From the borrower’s viewpoint the cash flows are identical except that the signs are reversed.

The cash flows of the borrower are precisely the cash flows of the lender multiplied by minusone. Therefore, the borrower has cash flows:

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5+$100,000 −$33,438 −$33,438 −$33,438 −$33,438 −$33,438

and his effective cost is 20 percent on the loan. The borrower must earn at least 20 percent perperiod on the loan just to be able to repay it. The pre-tax return to the lender on the investment(loan) cannot be less than the cost to the borrower. Even if the bank does not reinvest the cashflows as the loan is repaid, its implicit return will still be 20 percent. The return is measuredas a time-adjusted percentage of the principal amount outstanding, and is independent of whatdisposition is made of the cash flows as they are received. This is not to say that the uses to whichthe cash flows are put will have no effect on the organization, for they will. However, althoughthe yield on the funds originally invested may be increased by such uses, it cannot be reduced bylack of such investment opportunities. This is a strong position to take, and requires explanation.

The payments made by the borrower, once given over to the lender, can earn nothing forthe borrower. The borrower must, in the absence of other sources of funds, be able to earn20 percent per period on the remaining loan principal. If the borrower is unable to earn anythingon the remaining loan principal, he or she must still make the required periodical payments.The payments, even if made from other sources of funds, will be the same as those required ifthe loan were to generate funds at 20 percent per period. If funds must be diverted from otherprojects to repay the loan, the opportunity cost to the borrower may be more than 20 percent,if the funds could have earned more than this percentage in other uses. The cost internal to theloan itself, however, is 20 percent.

Reinvestment Rate Assumptions for NPV and IRR 67

Table 8.1 Component breakdown of cash flows (amountsrounded to nearest $)

Beginning Interest on Principalt principal principal repayment

1 100,000 20,000 13,438a

2 86,562 17,312 16,1263 70,437 14,087 19,3514 51,086 10,217 23,2215 27,865 5,573 27,865

a Assumes that end-of-year payment of $33,438 is composed of interestof 20% on the beginning balance plus a repayment of principal (that is,$20,000 + $13,438).

Table 8.1 provides a breakdown of the loan payments into the principal and interest com-ponents implicit in the IRR method of rate calculation. Note that the interest is computed at20 percent per year on the beginning-of-period principal balance. The excess of payment overthis amount is used to reduce the principal.

The following loan (to the borrower) will have an identical cost of 20 percent. However, theprincipal is not amortized but is paid in full at the very end of the loan:

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5+$100,000 −$20,000 −$20,000 −$20,000 −$20,000 −$120,000

The borrower may place the loan principal in a (hypothetical) bank account that pays exactly20 percent annually on the deposit. At the end of each year the borrower withdraws the interestand pays it to the lender. At the termination of the loan the borrower withdraws the principalplus interest and repays the loan. Since the interest is paid to the lender as soon as it is earned,the borrower does not earn interest on interest.

The bank pays the borrower exactly 20 percent annually on the deposit, which he or sheimmediately turns over to the lender. The loan costs are exactly equal to the 20 percent annualinterest the bank pays the borrower for the deposit, so there is no net gain to him or her. (Wehave ignored transactions costs to simplify discussion.)

For the lender the loan also yields exactly 20 percent. However, there is an important dif-ference: the lender may reinvest the interest payments if desired and thus increase the gain.Such increase, significantly, does not depend on the loan itself, but on reinvestment oppor-tunities available for the loan interest when it is received. The 20 percent return is thus aminimum return on the loan, and this minimum is independent of reinvestment opportuni-ties. The reinvestment rate could be zero and still the lender would earn 20 percent on theloan.

The only difference between these two loans is the handling of principal repayment: in thefirst, the principal is amortized over the life of the loan. In the second loan, the entire principalrepayment is made at the loan maturity date. The first loan does provide better reinvestmentopportunities to the lender since larger payments are received in all but the last year, and thelender may be able to reinvest them and so raise the return on the loan. Once again, however,20 percent is the minimum return to be expected, even if the reinvestment opportunity rate wereto be zero. The lender earns exactly 20 percent on the principal amount still in the hands of theborrower. If the lender can earn 20 percent or more on the recovered principal, so much the


Table 8.2 Per period return on remaining principal(constant amortization of $20,000 per period)

Principalt remaining % Return

1 $100,000 13.4382 80,000 16.7983 60,000 22.3974 40,000 33.5955 20,000 67.190

better for the lender. If he or she cannot, the lender nevertheless continues to earn 20 percenton the still unrecovered principal.

Let us assume a zero reinvestment rate. With the first loan, let us consider that the $100,000principal is returned in equal annual installments of $20,000 over the five-year loan maturity.This means that $13,438 over and above the principal repayment is earned on the remainingprincipal. The percentage return on the remaining principal in each year is then as shown inTable 8.2. The geometric mean return is 29.4 percent, the arithmetic mean return 30 percent, andthe median return 22.4 percent. This treatment differs from the IRR formulation in assuming afixed allocation of periodical payments to principal amortization (straight-line amortization ofthe principal) rather than a gradually increasing amortization payment. Therefore, the principalis more quickly reduced, consequently yielding a higher return on that which remains.

These results do not require any reinvestment rate other than zero. They show that thepercentage return on an investment does not depend on the available reinvestment rate. Theactual gain to the investor (or lender) may, of course, be higher than this minimum amountif the available reinvestment rate is greater than zero, but that is a condition external to theinvestment. The IRR is concerned with the internal characteristics only, and therefore providesa measure of the minimum return on the investment.

In summary, the conceptual difficulty with the reinvestment rate assumption arises from fo-cusing on the superficial aspects of the mathematics of the IRR while neglecting the economicinterpretation of the initial investment and the subsequent cash flows. It is important to remem-ber that mathematics is a tool in finance, economics the master. The reinvestment rate problemarises from confusion of this hierarchy — from trying to make the economics conform to themathematics. The IRR might be called more properly return on invested capital to make clearits economic assumptions. However, this term has another specific meaning that is coveredlater, and is thus reserved for that.

Treatment of the reinvestment rate with the NPV, along lines similar to that of the IRR, isleft as an exercise.

CONFLICTING RANKINGS AND FISHER’S INTERSECTION[1]

The rules for project acceptance once again are:

IRR NPVAccept if r > k NPV > 0Reject if r ≤ k NPV ≤ 0

Now it would seem that if we are comparing two acceptable, but mutually exclusive projects weshould get the same preference ranking by NPV as we do by IRR. However, this will not always


10

$(1000)

8

6

4

2

0 10 20 30 40% Rate ofDiscount

H

I

rf

Figure 8.1 Fisher’s rate, rf, and conflicting IRR–NPV project rankings

be the case. Conflicting rankings may arise because one or both projects have nonuniform cashflows.

To begin, let us consider two projects each costing $7000 at t = 0, and each having aneconomic life of five years, in order that we do not need to correct for unequal project size orlife. The projects have the following cash flows (dropping the $ signs):

Project H Project ICost, R0 7,000 7,000

Cash flowt = l 6,000 250t = 2 3,000 500t = 3 1,500 750t = 4 750 4,000t = 5 375 10,000

The internal rates of return are IRRH = 35.120 percent, IRRI = 19.745 percent. As shown inFigure 8.1, the IRR values are fixed. However, the NPV values change, since they depend onthe value of d that is used. For d = 0 (that is, zero discount) they are for projects H and I $4625and $8500, respectively.

Notice that the two NPV curves intersect at a discount rate of approximately 11 percent(11.408 percent), which we denote as rf. For smaller discount rates, project I has an NPVgreater than that of project H. However, IRRH > IRRI always. Therefore, for discount ratesless than rf, there will always be a conflict between the NPV ranking and the IRR ranking, but,for discount rates greater than rf, both IRR and NPV will yield the same ranking.

Let us examine these projects further by treating them each as loans and inspecting theirrepayment schedules, as in Table 8.3. We assume that each net cash flow is composed of twocomponents: an interest payment and a principal repayment. First, we will use as discount rateFisher’s rate of 11.408 percent.

Referring to Table 8.3, the project NPVs at t = 0 can be found by discounting the amountsin column 4. For project H we have

$996.26(1.11408)−2 + 1500(1.11408)−3 + 750(1.11408)−4 + 375(1.11408)−5

equaling $2592.80. For project I we have $4449.93(1.11408)−5 or $2592.80. As we should


Table 8.3 Component treatment of projects H and I at Fisher’s rate of 11.408%(amounts in $)

(1) (2) (3) (4) (5)Beginning Interest on Principal Excess over Total net cash

t principal principal repayment (2) + (3) flow

Project Hl 7,000.00 798.56 5,201.44 0 6,000.002 1,798.56 205.18 1,798.56 996.26 3,000.003 0 0 0 1,500.00 1,500.004 0 0 0 750.00 750.005 0 0 0 375.00 375.00

Project Il 7,000.00 798.56 −548.56 0 250.002 7,548.56 861.14 −361.14 0 500.003 7,909.70 902.34 −152.34 0 750.004 8,062.04 919.72 3,080.28 0 4,000.005 4,981.75 568.32 4,981.75 4,449.93 10,000.00

have expected, at Fisher’s rate the NPVs of projects H and I are equal, for that is how Fisher’srate is defined.

Notice an important difference between the two projects with respect to column 1, thebeginning-of-period principal remaining. The net cash flows for project H were sufficientlyhigh at the end of periods 1 and 2 to pay off the entire principal after paying the requiredinterest. In fact, at the end of period 2 the net cash flow was $996.26 in excess of what wasrequired to pay off the interest and remaining principal. This is the meaning of NPV, and whysome authors choose to call it excess present value. The NPV is the amount, at discount, bywhich the net cash flows of an investment project exceed what is required for payment ofinterest on remaining principal and principal repayment. Note that although project H paid offthe entire principal by the end of period 2, project I did not pay off its principal until the endof period 5.

We know that projects H and I are not loans, but are capital investment projects. The principalrepayments represent recovery of the initial investment, whereas the interest payments representthe opportunity cost of funds committed to the projects and as yet unrecovered. Which projectis preferable? Both have the same NPV of $2592.80 at Fisher’s rate. Does this mean that weshould feel indifferent about the two projects?

Although projects H and I have identical NPVs at Fisher’s rate of 11.408 percent, project His clearly preferable. The reason is that project H provides for faster recovery of the fundsinvested in it than does project I. In fact, project I is incapable of paying the interest onbeginning-of-period principal in any of periods 1–3. Therefore, instead of giving us back ouroriginal investment with interest from the start, project I requires that we wait until the endof period 4 before any reduction in principal plus accumulated interest can be made. On theprinciple that a bird in the hand is worth two in the bush, or rather that a dollar today is betterthan a dollar plus interest tomorrow (if there is any doubt about getting the dollar tomorrow),we prefer project H to project I. The fact that the NPVs of the two projects are identical shouldnot cause us to be indifferent between projects such as H and I, for the NPV is only one criterionby which project merit may be gauged. If the NPVs are equal, we should, of course, prefer theproject that offers the more rapid capital recovery. Invested funds recovered early are subject


Table 8.4 Component treatment of projects H and I at 5%, a rate less than Fisher’srate ($ amounts)

(1) (2) (3) (4) (5)Beginning Interest on Principal Excess over Total net cash

t principal principal repayment (2) + (3) flow

Project Hl 7,000.00 350.00 5,650.00 0 6,000.002 1,350.00 67.50 1,350.00 1,582.50 3,000.003 0 0 0 1,500.00 1,500.004 0 0 0 750.00 750.005 0 0 0 375.00 375.00

Project Il 7,000.00 350.00 −100.00 0 250.002 7,100.00 355.00 145.00 0 500.003 6,955.00 347.75 402.25 0 750.004 6,552.75 327.64 3,672.36 0 4,000.005 2,880.39 144.02 2,880.39 6,975.59 10,000.00

to less uncertainty of receipt than funds to be recovered later, and funds recovered early canbe reinvested for a longer period to enhance the earnings of the firm.

We see that at Fisher’s rate project H is preferable to project I even though the NPVs areidentical. This preference is consistent with what we would have obtained by choosing betweenthe projects on the basis of their IRRs. Now let us examine these same projects at a rate ofdiscount substantially less than rf, Fisher’s rate. For illustration we will use a discount rate of5 percent. Table 8.4 contains the results of our calculations.

We may again determine the NPVs by bringing the column 4 amounts to t = 0 at a discountrate of 5 percent per period. For project H we obtain an NPV of $3641.98 and for project I anNPV of $5465.56. Since NPV of I > NPV of H, which we would expect for a discount rateless than Fisher’s rate, the NPV criterion favors project I. The profitability indexes (PIs) wouldnot suggest a different relationship, because both projects require the same initial investmentoutlay.

At a 5 percent cost of capital we find that the NPV (or PI) criterion favors project I. TheIRR criterion, however, always favors project H. Which criterion should we use? For a costof capital (discount rate) greater than or equal to Fisher’s rate we would have no problem,because both the IRR and the NPV would favor project H. But we have under consideration arate of 5 percent and a conflict in ranking that must be resolved.

To help us decide which project is preferable, let us again examine columns 1 and 4 ofTable 8.4. Notice that project H (column 1) allows us to recover our entire investment plusinterest (opportunity cost on the funds committed) and yields an excess of $1582.50 at theend of the second year (column 4). Project I, in contrast, cannot even compensate for ouropportunity cost in year 1. It does not yield recovery of our investment until the end of thefifth year. In fact, the entire NPV of project I depends on the large cash flow in period 5. Ifunanticipated events in year 3, 4, or 5 caused all cash flows to be zero, project H would stillhave a positive NPV of $1582.50(1.05)−2 = $1435.37, but project I would have a negativeNPV of −$6308.39. Not only does project H have a higher IRR (35.120 percent) than project I(IRR = 19.751 percent), but also it does not depend on the accuracy of our cash flow estimatesbeyond year 2 to be acceptable.


The foregoing analysis illustrates the danger of relying on any single measure of investmentmerit. This applies to the NPV and the PI as it does to other measures. Some have emphasizedthe NPV–PI criterion to an extent tantamount to recommending it as a universal prescriptionfor capital-budgeting analysis. But we have just seen that the NPV alone does not providesufficient information to choose between two projects that have the same required investmentand useful lives. A formal treatment of risk in capital budgeting is deferred until the latterchapters. However, we must realize that risk is our constant companion whether we choose todeal with it formally or to ignore it.

On the premise that point estimates are subject to error, and that the expected error becomesgreater the further from the present the event we estimate will occur, ceteris paribus weprefer an investment that promises early recovery of funds committed, and early receipt offunds above this amount. In the applied world of capital investments, no single measureadequately captures the multifaceted character of capital-budgeting projects. For this reasoncompanies often examine several measures of a capital investment in their decision-makingprocesses.

RELATIONSHIP OF IRR AND NPV

Mao [98, p. 196] in his classical book has explicitly shown the relationship of IRR to NPV inan interesting way. Let us again examine equations (7.3) and (6.2).

0 =N∑

t=0

Rt (1 + r )−t (7.3)

NPV =N∑

t=0

Rt (1 + k)−t (6.2)

If we subtract (7.3) from (6.2), we obtain

NPV =N∑

t=0

Rt [(1 + k)−t − (1 + r )−t ] (8.5)

Now, taking any term beyond t = 0, we assume that k and r are both positive and the Rt

all nonnegative. For NPV to be positive it is necessary only for the relation r > k to be true.Conversely, for NPV to be negative, it is necessary that the relation r < k be true. This showsthe equivalence of the NPV and IRR criteria for projects that have Rt ≥ 0 for t > 0, in termsof the accept/reject decision.

ADJUSTED, OR MODIFIED, IRR

The controversy over whether or not the IRR should be used when there is doubt that theproject’s cash flows can be reinvested at the IRR led to the development of the adjusted IRR.The idea behind the adjusted IRR is to assume that all cash flows after the initial outlay oroutlays are invested to earn at the firm’s cost of capital or some other conservative reinvestmentrate. Thus all cash flows except for the initial outlay are taken out to a future value at the terminalyear of the project’s life, and zeroes are used to replace them in the adjusted series. After this


Using cost of capital/reinvestment rate of 12.00%:

Project H: t � 0 t � 1 t � 2 t � 3 t � 4 t � 5Original flows:

Adjusted flows:

Adjusted IRR � 19.07% � [(16,752.50/7,000)(1/5)� 1] � 100%

Adjusted IRR � 18.73% � [(16,516.64/7,000)(1/5)� 1] � 100%

($7,000)

($7,000)

6,000 3,000 1,500 750 375

840.001,881.604,214.789,441.12

16,752.500 0 0 0

Project I: t � 0 t � 1 t � 2 t � 3 t � 4 t � 5Original flows:

Adjusted flows:

($7,000)

($7,000)

250 500 750 4,000 10,000

4,480.00940.80702.46393.38

16,516.640 0 0 0

Figure 8.2 Comparison using modified IRR

adjustment it is easy to calculate the implicit rate of return that would take the initial investmentto the terminal amount that contains the compounded sum of all cash flows from t = 1 throught = N .

The adjusted, or modified, IRR is analogous to the rate of return on a zero coupon bond.Since there are no cash flows to be reinvested, the rate at which the firm can reinvest cash is nolonger material to the rate of return. The procedure for calculating an adjusted IRR is clearlya type of sinking fund method. Sinking fund methods in general are covered in Chapter 11.

The modified IRR seems to have been more widely accepted by engineering economiststhan by finance writers. However, that situation may be changing.

The application of the adjusted IRR to projects H and I will serve to clarify the procedureinvolved. Figure 8.2 illustrates the method.


In this chapter we examined (1) the reinvestment assumptions implicit in the discounted cashflow methods, IRR and NPV, (2) the reasons for conflicting rankings between IRR and NPV,and (3) Mao’s treatment of the IRR–NPV relationship.

The reinvestment rate assumptions are seen to arise from the mathematical relationshipbetween present value and future value formulations of the discounted cash flow methods. Itis shown that the return on investment, that is, the remaining unrecovered initial investment, isnot dependent on reinvestment opportunities. To do this, the net cash flows are separated intotwo components: a payment of “interest” (return) on remaining “principal” (investment), anda repayment of “principal” (recovery of investment).

A treatment similar to that used to deal with the reinvestment assumptions is employed toanalyze, in conjunction with Fisher’s rate (rf), the reason for contradictory rankings between


IRR and NPV. It is seen that for rates less than rf the NPV alone is inadequate to judge which oftwo projects is better, since the NPV does not distinguish between the timing of cash receiptsto recover the investment and those that provide a net return.

Mao’s mathematical connection of IRR and NPV inside a single equation serves to illustratethe conditions under which the NPV is positive or negative, and to focus more clearly on theIRR–NPV relationship.

9

The MAPI Method

George Terborgh, Research Director of the Machinery and Allied Products Institute1 (MAPI),authored the classical book, Dynamic Equipment Policy. This work still provides perhapsthe most theoretically sound and practical means for analyzing component projects. About adecade after that treatise was published, a simplified and streamlined version of the method-ology was published in Business Investment Policy, many points of which were contained inAn Introduction to Business Investment Analysis, based on an address delivered in 1958 byTerborgh. That these publications are out of print and no longer available from the originalsource can be confirmed by a visit to the Manufacturers’ Alliance/MAPI website, which nolonger contains references to Terborgh or his works.2

In order to appreciate how Terborgh’s method relates to the other methods that employdiscounted cash flow, it will be useful if first the concept of duality is understood. Those familiarwith mathematical programming may skip the following section without loss of continuity.

THE CONCEPT OF DUALITY

One of the important contributions of mathematical programming is that of duality. Simplystated, duality means that, for every maximization problem, there corresponds a minimizationproblem which yields identical solution values. If the problem at hand, called the primalproblem, is one of maximization, the dual problem will always be a minimization problem.The converse is also true: if the primal problem is a minimization formulation, the dual willbe a maximization formulation. Furthermore, the dual of the dual will be the primal.

Let us examine the general linear programming problem:

Maximize p1q1 + p2q2 + · · · + pnqn

Subject to: a11q1 + a12q2 + · · · + a1nqn ≤ b1

a21q1 + a22q2 + · · · + a2nqn ≤ b2

··

am1q1 + am2q2 + · · · + amnqn ≤ bm

and for all iqi ≥ 0

(9.1)

1The name of the association has been changed, though the acronym MAPI still applies. “The Manufacturers Alliance/MAPI is anexecutive education and policy research organization serving the needs of industry. Founded in 1933 by capital goods manufactur-ers and known then as the Machinery and Allied Products Institute (MAPI), the corporate membership of 450 companies has beenbroadened over the years to encompass the full range of manufacturing industries, such as automotive, aerospace, computer, elec-tronics, chemical, machinery, and pharmaceutical, including manufacturers of a wide range of consumer products and businesses thatprovide related services such as telecommunications, power distribution, and software services.” From the Alliance’s web page athttp://www.mapi.net/html/research.cfm2I am sure many readers would agree that it is a sad commentary on today’s world that such remarkable and lucid works are relegatedto oblivion, but will not belabor the point here. The interested reader may find a copy to borrow or useful information on the methodat http://www.mises.org/wardlibrary detail.asp?control=6480 and http://www.albany.edu/∼renshaw/leading/ess05.html


which in the shorthand notation of matrix algebra becomes

Maximize p · q

Subject to A · q ≤ b

qi ≥ 0 for all i

(9.2)

This particular linear programming problem may be considered one of maximizing total firmprofit (price times quantity for each product) subject to constraints on the output of each productassociated with limitations on machine time, labor, and so on, and the relative requirements ofeach product for these limited resources.

If the above maximization problem has a feasible solution, the dual formulation will alsohave a feasible solution, and the dual solution will yield the same amounts of each product tobe produced. The corresponding dual problem is

Minimize b1u1 + b2u2 + · · · + bmum

Subject to: a11u1 + a21u2 + · · · + am1um ≥ p1

a12u1 + a22u2 + · · · + am2um ≥ p2

··

a1nu1 + a2nu2 + · · · + amnum ≥ pn

and for all iui ≥ 0

(9.3)

which in matrix algebra notation corresponding to the primal becomes

Minimize b′u

Subject to A′u ≥ p′

ui ≥ 0 for all i

(9.4)

where b′ is the transpose of the vector of constraint constants in the primal problem, and soon. The u variables are called shadow prices and represent the opportunity costs of unutilizedresources. Thus, if we minimize the opportunity costs of nonoptimal employment of ourmachine, labor, and other resources, we achieve a lowest cost solution.

The important point of all this is that the primal and the dual formulations both lead tothe same allocation of available resources. Therefore, it is not important which of the twoformulations we solve. For reasons of ease of formulation, calculation, or computer efficiency,we may choose to work with either the primal or the dual formulation.

Now, getting back to the MAPI method of capital-budgeting project evaluation, we may saythat the method is analogous to a dual approach to the goal of maximization of the value ofthe firm expressed in equation (2.1). It will be useful to again state equation (2.1) here as

Maximize V =∞∑

t=0

Rt

(1 + k)t(9.5)

If we recognize that the net cash flows Rt are composed of gross cash profits Pt as well ascash costs Ct , then

Rt = Pt − Ct (9.6)

For component projects we will assume that the Pt are fixed: we cannot directly associatecash inflows with the project. Even for major projects we may have to take the cash inflowsas given, since to some extent they are beyond our control, at least as far as finance and

The MAPI Method 77

production are concerned. Marketing staff may influence gross revenues through advertising,salesmanship, and marketing logistics, but still they will be subject to the state of the generaleconomy, the activities of our competition, and “acts of God.”

If we take the Pt as given, or fixed, and is thus independent of the capital equipment usedin production, then we may reformulate our objective as

Minimize cost =∞∑

t=0

Ct

(1 + k)t(9.7)

which may be made operational by accepting the lowest cost projects that can satisfactorilyperform a given task. This is the basic idea behind the MAPI method, which we will nowexamine.

THE MAPI FRAMEWORK

The methodology of the MAPI method involves calculating the time-adjusted annual averagecost of the project or projects under consideration. However, several concepts vital to intelligentapplication of the method must first be understood.

Challenger and Defender

In Terborgh’s colorful terminology, the capital equipment currently in use is referred to asthe defender, and the alternative that may be considered for replacement as the challenger.The MAPI method as originally developed emphasized capital equipment replacement, butis applicable to nonreplacement decisions as well since, in such cases, the status quo may beconsidered the defender. Various potential challengers may be compared against one anotherin a winnowing process, with the project promising the lowest time-adjusted annual averagecost selected as the challenger.

If the challenger is superior to the defender as well as to presently available rivals, it maystill be inferior to future alternatives. The current challenger is the best replacement for thedefender only when there is no future challenger worth delaying for. This requires that aseries of capital equipment not currently in existence be appraised, and this presents somedifficulties:

Now obviously it is impossible as a rule for mere mortals to foresee the form and character of machinesnot yet in existence. In some cases, no doubt, closely impending developments may be more or lessdimly discerned and so may be weighted, after a fashion, in the replacement analysis, but in no casecan the future be penetrated more than a fraction of the distance that is theoretically necessary for anexact, or even a close solution of the problem. What then is the answer? Since the machines of thefuture cannot be foreseen, their character must be assumed [151, pp. 57–58].

The exact nature of the necessary assumptions is dependent on some additional terminology,which will be introduced at this point.

Capital Cost

The mechanics of the MAPI method require that we obtain the time-adjusted annual averagecost of the project under consideration. The two components that determine what the averagecost will be are: (1) capital cost and (2) operating inferiority. Capital cost must not be confused


with the firm’s cost of capital, because they are quite different things despite the similarity inlabels.

Capital cost in the MAPI framework is the uniform annual dollar amount, including theopportunity cost of funds, that are tied up in the project and must be recovered if the projectis retained for either one year, or for two years, or for three years, and so on. To clarifythis concept, assume we are examining a project that costs $10,000, and that our firm has a15 percent annual cost of capital. Our opportunity cost for funds committed to the project is atleast 15 percent annually since, if funds were not committed to it, our cost of funds would belower by this amount. That is, our firm would require $10,000 less, for which it is incurring anannual 15 percent cost.

Now, if we were to accept this project, and then at the end of one year abandon it, whatis the capital cost? It is $10,000 plus the opportunity cost at 15 percent, or $11,500. What ifthe project is abandoned at the end of the second year? In this case the total capital cost is$10,000(1.15)2 or $13,225. However, since the project will be kept for two years, the annualcapital cost will be much less, not even one-half of the $13,225. The reason for this is thatfunds recovered in the first year do not incur opportunity cost during the second year. At a15 percent annual cost of capital, the time-adjusted annual average capital cost in each of thetwo years will be $6151. If the project were retained for three years, the time-adjusted annualaverage cost will be $4380. These amounts are obtained by multiplying the initial investmentby the capital recovery factor corresponding to the annual percentage cost of capital and thenumber of years the project is retained. The capital recovery factor is the reciprocal of theordinary annuity (uniform series) present worth factor.

The longer a project is kept in service, the smaller the amount of investment that must berecovered in each individual year of the project’s life. A project that is retained only one ortwo years must therefore yield a larger cash flow each year to allow recovery of the initialinvestment, plus opportunity cost of the committed funds, than the same project if kept formany years.

In the present value and internal rate of return methods for capital-budgeting project eval-uation, the initial investment is considered only at time period zero. In the MAPI method, theinitial investment is spread over the years of the project’s life. With the NPV method, since thediscount rate is assumed to be known, that is, it is the firm’s cost of capital, it is possible thatthe initial investment could be treated in the same way as in MAPI. However, in practice it isnot treated that way.

Operating Inferiority

Operating inferiority is defined as the deficiency of the defender, the incumbent, existing projector the status quo, relative to the best available alternative for performing the same functions.Operating inferiority is considered to be composed of two components: physical deteriorationand technological obsolescence. In the MAPI method we measure operating inferiority usingthe best capital equipment as a benchmark:

In the firmament of mechanical alternatives there is but one fixed star: the best machine for the job.This is base point and the standard for evaluating all others. What an operator can afford to pay forany rival or competitor of this machine must therefore be derived by a top-down measurement. Butthe process is not reversible. He cannot properly compute what he can afford to pay for the best bymeasurement upward from its inferiors [151, p. 35].

The MAPI Method 79

Physical Deterioration

Physical deterioration of capital can be determined by comparing the equipment in service withthe same equipment when new and undegraded by past operation. Physical deterioration, then,will be the excess of the operating cost of the old machine over its new replica’s operating cost.We normally would expect rapid physical deterioration in the first few years of a project’s life,then for it to accumulate more slowly, perhaps reaching a steady-state equilibrium, with repaircosts at a relatively constant level per period, keeping the quality of service approximatelyconstant. The comparison of the equipment in service with its new replica should be in termsof operating costs, including maintenance and repair, additional direct labor required, extraindirect labor required (such as for quality control inspection), and the cost of higher scrapoutput.

Technological Obsolescence

Although physical deterioration may be considered an internal, age-related aspect of capitaldegradation, technological obsolescence is external to it and not necessarily related to age.Obsolescence consists of the sum of the excess operating cost of the same capital that is newover that of the best alternative now available plus the deficiency in the value of service relativeto the best alternative. Physical deterioration is degradation of the firm’s existing capital relativeto new, identical capital. Technological obsolescence is the inferiority of the existing capitalrelative to the latest generation of capital for doing the job.

Two Basic Assumptions

Unfortunately, although the concept of operating inferiority is not difficult to grasp, imple-mentation poses some problems. Although we may be able to estimate the cost of operatinginferiority for this year, and perhaps the next as well, the task becomes increasingly difficultand tenuous as we attempt to carry the process into the future. Physical deterioration may notoccur uniformly: it may be substantial in the early years, tapering off later, or it may be just theopposite. Technological developments tend to occur randomly over time. Although some maybe anticipated in advance in rapidly developing fields, such as computer technology, predictionof when they will be brought to market is still a somewhat uncertain enterprise.

In addition to the problems inherent in estimating operating inferiority, there is yet anotherobstacle. This is related to the characteristics of future challengers:

It is true that the challenger has eliminated all presently available rivals. But it has not eliminatedfuture rivals. The latter, though at present mere potentialities, are important figures in the contest. Forthe current challenger can make good its claim to succeed the defender only when there is no futurechallenger worth waiting for. It must engage, as it were, in a two-front war, attacking on one side theaged machine it hopes to dislodge and on the other an array of rivals still unborn who also hope todislodge the same aged machine, but later [151, p. 55].

The MAPI analysis emphasizes the importance of future capital equipment:

For since the choice between living machines can be made only by reference to the machines oftomorrow, the latter remain, whether we like it or not, an indispensable element in the calculation. Itmay be said . . . that the appraisal of the ghosts involved is the heart of the . . . analysis. No replacementtheory, no formula, no rule of thumb that fails to take cognizance of these ghosts and to assess theirrole in the play can lay claim to rational justification [151, p. 57].


In order to deal with these problems Terborgh proposed two “standard” assumptions on thebasis that “The best the analyst can do is to start with a set of standard assumptions and shadethe results of their application as his judgment dictates [151, p. 60].” In other words, if we haveinformation we will use it; if not, we will employ reasonable standard assumptions.

Adverse Minimum

The key to the MAPI method is the “adverse minimum” for the capital project. This is definedas the lowest combined time-adjusted average of capital cost and operating inferiority thatcan be obtained by keeping the project in service the number of years necessary to reach thisminimum, and no longer.

First Standard Assumption

Future challengers will have the same adverse minimum as the present one [151, p. 64].

Second Standard Assumption

The present challenger will accumulate operating inferiority at a constant rate over its servicelife [151, p. 65].

The first standard assumption is justified on the basis that there is no alternative that is morereasonable. In the absence of information to the contrary, what compulsion is there for us toassume that future challengers will have either higher or lower adverse minima? If we haveinformation that leads us to believe that future challengers will have different adverse minima,then we may modify the standard assumption. Furthermore, this standard assumption facilitatesdeveloping a simpler replacement formula than would be otherwise possible.

The second standard assumption again is justified on the basis of methodological necessity,since the analyst typically does not have data on a sufficient sample size of similar equipmentto make a more reasonable assumption. In the absence of information to the contrary, the bestwe can do is predict the future by extrapolation from the present and past experiences on thebasis that there are elements of continuity and recurrence that will be repeated into the future.If we were to reject entirely this continuity and recurrence over time, we would be utterlyincapable of dealing with the future in all but those situations in which change is at least dimlyvisible on the horizon.

APPLICATION OF THE MAPI METHOD

In order to apply the MAPI method to a potential challenger, we require an estimate of the firm’scost of capital, the cost of the project, and an estimate of the project’s first-year accumulationof operating inferiority. From this information we derive the adverse minima of potentialchallengers, thereby selecting the project with the lowest adverse minimum as the challenger.The adverse minimum of the defender, if there is existing capital equipment that may bereplaced by the challenger, is determined similarly. In many cases the defender will be foundto have already passed the point in time at which its adverse minimum occurs. In such casesTerborgh has recommended that the next-year total of capital cost and operating inferiority beused as the defender’s adverse minimum.

The MAPI Method 81

Table 9.1 Derivation of the adverse minimum of a challenger costing $100,000 with inferioritygradient of $7000 a year. Assumes no capital additions and no salvage value. Cost of capital is 15%

(4)(2) (3) Present (5)

(1) Present Present worth of CapitalOperating worth worth of operating recovery (6) (7) (8)

Year inferiority factor operating inferiority, factor Operating Cost Bothof for year for year inferiority accumulated for year inferiority (5) × combinedservice indicated indicated (1) × (2) (3) accumulated indicated (4) × (5) $100,000 (6) + (7)

1 $0 0.86957 $0 $0 1.15000 $0 $115,000 $115,0002 7,000 0.75614 5,293 5,293 0.61512 3,256 61,512 64,7683 14,000 0.65752 9,205 14,498 0.43798 6,350 43,798 50,1484 21,000 0.57175 12,007 26,505 0.35027 9,284 35,027 44,3115 28,000 0.49718 13,921 40,426 0.29832 12,060 29,832 41,8926 35,000 0.43233 15,132 55,558 0.26424 14,681 26,424 41,105a

7 42,000 0.37594 15,789 71,346 0.24036 17,149 24,036 41,1858 49,000 0.32690 16,018 87,364 0.22285 19,469 22,285 41,7549 56,000 0.28426 15,919 103,283 0.20957 21,645 20,957 42,602

10 63,000 0.24718 15,572 118,855 0.19925 23,682 19,925 43,60711 70,000 0.21494 15,046 133,901 0.19107 25,584 19,107 44,69112 77,000 0.18691 14,392 148,293 0.18448 27,357 18,448 45,805

a Adverse minimum.

Let us assume the firm’s cost of capital is 15 percent per annum, and that we have anexisting machine (defender) that will have a combined capital cost and operating inferiority of$70,000. There is only one potential challenger. It costs $100,000 and is estimated to accumulateoperating inferiority during the first year of service amounting to $7000. Application of thesecond standard assumption means we assume operating inferiority will be accumulated at$7000 each year the challenger would be in service. We ignore salvage value for now tosimplify the exposition. Table 9.1 illustrates the technique of finding the challenger’s adverseminimum.

Since the adverse minimum of the challenger of $41,105 per year if held six years issubstantially lower than the next-year operating inferiority of the defender, we would replacethe defender this year. In fact, even if the challenger were to be replaced itself at the end of thesecond year of its service by a still better but yet unbuilt new challenger, the decision wouldstill be correct. If kept in service two years, the challenger costs $64,768,whereas the defenderwill cost $70,000 next year and, if it continues to accumulate operating inferiority, still morethe following year. The figures in column 8 of Table 9.1 are average annual costs, adjusted forthe time-value of money. The lowest of these, as stated earlier, is the adverse minimum.

The time-adjusted annual average costs are composed of a capital cost component thatdeclines over time, and an operating inferiority component that increases over time. For theproject analyzed in Table 9.1, capital cost declines rapidly in the early years of the project’s life,whereas operating inferiority rises rapidly. Therefore, this challenger will obtain its adverseminimum in only six years. Some projects may not reach their adverse minima for many years.For instance, management may have a policy of abandoning equipment at the end of, say,15 years regardless of its condition. In such an environment the adverse minimum may notbe reached. However, if the time-adjusted annual average cost declines constantly, we may


Table 9.2 Comparison of alternatives

Adverse Annual productionMachine minimum capability Cost per unit

Defender $50,000 10,000 $5.00Alternative A 500,000 150,000 3.33Alternative B 1,000,000 400,000 2.50

120,000

100,000

80,000

60,000

40,000

20,000

01 2 3 4 5 6 7 8 9 10 11 12

Period of Service (Years)

Dol

lars

Capital Cost Operating Inferiority Both Combined

Adverse Minimum

Figure 9.1 Graphic of MAPI method (using data from Table 9.1)

then take the last year of service’s value as our adverse minimum. Of course, as we go intothe future further and further, the reliability of the two standard assumptions employed in thebeginning begins to wane. Thus, other things being equal, we would prefer a challenger thatreached its adverse minimum within just a few years to one that took 15 or 20 years to reach it.

Figure 9.1 illustrates graphically how the adverse minimum is determined as the minimumpoint on the total cost function, which is the vertical sum of the individual costs of the componentfunctions for capital cost and for operating inferiority. Readers who are familiar with thederivation of the basic economic order quantity model will note some similarity of the graphs.

The Problem of Capacity Disparities

The existence of alternatives that provide for different production capacities requires somemodification to the MAPI method. Assume that the defender, having a next-year operatinginferiority of $50,000, is capable of producing 10,000 units of output annually. Assume furtherthat potential challengers A and B have respective adverse minima of $500,000 and $1 millionand annual production capabilities of 150,000 and 400,000 units.

At first glance it would appear that the defender should not be replaced, for it has the lowestadverse minimum (next-year operating inferiority). However, if we express the adverse minimain terms of the units of annual production, we find that alternative B promises the lowest costper unit (see Table 9.2).

The MAPI Method 83

Now we see that the ordering of the per unit costs is the opposite of the ordering of theadverse minima for the three alternatives. Which should be selected? The answer depends onthe firm’s annual production requirements. For instance, if the firm requires that 35,000 unitsbe produced each year, then the cost per unit will be $14.28 with alternative A and $28.57 withalternative B. The defender alone cannot produce more than 10,000 units annually. However, ifexact duplicates of the defender could be acquired (which will have the same adverse minimumas the defender), then we would need to add three machines. This would raise capacity to 40,000units in total — 5000 more than required. The cost per unit would be $5.71, which is muchless than the per unit costs of machines A and B.

Therefore, adverse minima in themselves are meaningless unless the alternative machineshave the same annual productive capacity. If they do not, it is necessary to make adjustments.Cost should be expressed in terms of cost per unit, since to do otherwise may lead to improperselection of the challenger and to a wrong replacement decision. Furthermore, the cost per unitmust be based on the firm’s requirements, not on the rated machine capacities, which may belesser or greater than the production required by the firm.

CONCLUSION

The Terborgh method of capital equipment analysis provides an alternative means of investmentevaluation that is based on minimization of costs. It is thus, in a sense, a dual formulation ofmethods based on maximization of some measure of investment return such as the discountedcash flow (DCF) measures.

Because the Terborgh method is based on cost minimization, it is suitable for analysis ofprojects where the customary DCF measures are much more difficult, if not impossible, toapply. Such projects are those this author has defined as component projects — they do nothave cash revenues directly attributable to them alone. For these projects the cash inflows maybe assumed invariant with respect to the production equipment employed, whereas cash costswill vary directly with respect to the choice of equipment. The Terborgh method, unlike theDCF methods, requires no estimates of cash inflows. Instead it requires cost estimates thatoften may be provided more easily, and provided by those personnel whose experience inproduction promises they may be the best obtainable estimates. Conversely, revenue estimatesfor component projects are likely to be based on tenuous premises if not pure guesses.

Proper application of the Terborgh method requires that the adverse minima of alternativeprojects be adjusted to reflect production capacity differences or the firm’s production require-ments. Otherwise the per unit cost of one alternative may be greater than that of another eventhough it may have a lower adverse minimum.

10

The Problem of Mixed Cash Flows: I

When the internal rate of return (IRR) was discussed a restriction was placed on the cashflows that assured there would be only one real IRR in the range of −100 percent to +∞. Therestriction was that there would be only one change in the sign of the flows. In this chapterthat restriction is removed, the consequences examined, and a method analyzed that providesa unique measure of return on investment.

INTERNAL RATE OF RETURN DEFICIENCIES

Under certain circumstances the IRR is not unique, and thus we have to decide which, if any,of the IRRs is a correct measure of return on investment. As we shall see, when there are twoor more IRRs, none of them is a true measure of return on investment.

This difficulty with IRR arises because of mixed cash flows. We define mixed cash flowsas a project cash flow series that has more than one change in arithmetic sign. There must, ofcourse, be one change in sign for us to find any IRR. However, when there is more than onechange in sign, the IRR, even if unique, sometimes will not measure the return on investment.

Example 10.1 Let us consider the following capital-budgeting project, for which we havemixed cash flows:

t = 0 t = 1 t = 2

−$100 +$320 −$240

This project has two IRRs: 20 percent and 100 percent. The net present value (NPV) is positivefor any cost of capital greater than 20 percent but less than 100 percent. Figure 10.1 illustratesthe NPV function for this project. The NPV reaches a maximum of $6.25 for a cost of capitalof 60 percent. The profitability index (PI) of 0.0625 (or 1.0625 by the common, alternativedefinition of PI) is not likely to cause much enthusiasm, but let us retain this example forfurther analysis.

DESCARTES’ RULE

Descartes’ rule of signs states that the number of unique, positive, real roots to a polynomialequation with real coefficients (such as the equation for IRR) must be less than or equal to thenumber of sign changes between the coefficients. If less than the number of sign changes, thenumber of positive,1 real roots must be less by an even number, since complex roots come inconjugate pairs, and an nth degree polynomial will have n roots, not necessarily distinct.

1We are interested in positive, real roots because, in solving any given polynomial, we will find the values of x = 1 + r, where r is theIRR value in decimal form. Therefore, after solving for the roots of the IRR polynomial, we must make the transformation r j = x j − 1for each root, j = 1, . . . , n. Since we work with all net cash flows attributable to the particular project, we therefore cannot lose morethan 100 percent of our investment. Thus limiting the r j to ≥ −1.00 restricts the x j to ≥ 0.


k

4

6

8

2

0

−2

−4

−6

−8−10

$NPV

0% 20% 40% 60% 80% 100% 120% 140% 160%

Figure 10.1 Net present value

Example 10.2 The following cash flows yield a third-degree IRR polynomial equation:

t = 0 t = 1 t = 2 t = 3

−$1000 +$3800 −$4730 +$1936

The IRR equation is (dropping $):

−1000 + 3800(1 + r )−1 − 4730(1 + r )−2 + 1936(1 + r )−3 = 0 (10.1)

or, alternatively, by multiplying by (1 + r )3 to put into future value form, then dividing by 1000and letting x = 1 + r:

1x3 − 3.8x2 + 4.73x − 1.936 = 0 (10.2)

This equation has three real roots, one with a multiplicity of two (double root). The rootsare 10 percent, 10 percent, and 60 percent, which we can verify by generating an equationby multiplication and then comparing it with equation (10.2). If r = 10 percent = 0.10, thenx = 1.1 and x − 1.1 is a zero (root) to the equation. Similarly, x = 1.6 is a zero to the equation.Multiplying, we obtain

x − 1.1

x − 1.1

x2 − 2.2x + 1.21

x − 1.6

x3 − 2.2x2 + 1.21x

− 1.6x2 + 3.52x − 1.936

x3 − 3.8x2 + 4.73x − 1.936

which is identical to equation (10.2).

The Problem of Mixed Cash Flows: I 87

We shall leave the matter of multiple roots now, because, when a project has mixed cashflows, even a unique, real IRR is no assurance that the IRR is a correct measure of investmentreturn.

THE TEICHROEW, ROBICHEK, AND MONTALBANO(TRM) ANALYSIS

Teichroew, and later Teichroew, Robichek, and Montalbano (TRM) [148–150] formally ex-plained the existence of multiple IRRs and proposed an algorithm for determining a uniquemeasure of the return on invested capital (RIC). James C. T. Mao was among the first to offera lucid summary of the main points of the TRM work [97, 98], and his work is still a valuablereference on the topic.

The analysis by TRM demonstrated that projects with mixed cash flows may often be neitherclearly investments nor clearly financing projects. For example, if the firm makes a loan thecash flow sequence will be − + + · · · + + if the loan is amortized with periodic payments.This is identical to a capital investment with the same net cash flows. From the viewpoint of theborrower (or the capital asset, if we can attribute a viewpoint to it), the cash flow sequence willbe the negative of our firm’s: + − − · · · − −. Depending on whether we view the cash flowsequence through the firm’s eyes or those of the borrower, we have what is unambiguouslyeither an investment or a financing project. Since there is only one change in sign between thecash flows, we know that the IRR will be unique.

Now, what if we have instead a cash flow sequence of − + + − + or + − − + −+?Can we say a priori that we have an investment or a financing project based only on examina-tion of the signs of the cash flows? The answer is no, we cannot. The TRM analysis recognizesthat some projects with mixed cash flows have attributes of both investment and financingprojects, while others are purely investments. The returns on projects that have characteristicsof both investment and financing projects are not, as the IRR method assumes, independentof the firm’s cost of capital. To understand the TRM analysis, we need to define some terms.

Let

a0, a1, . . . , an

denote the project cash flows. And let

st (r ) =t∑

i=0

ai (1 + r )t−i , 0 ≤ t < n

and

st (r ) = (1 + r )st−1 + at

denote the project balance equations, and

sn(r ) =n∑

i= 0

ai (1 + r )n−i

denote the future value of the project.The minimum rate is rmin for which all the project balance equations are less than or equal

to zero: st (rmin) ≤ 0 for 0 < t < n.


The project balance at the end of period t, at rate r, is interpreted as the future value of (1) theamount the firm has invested in the project or (2) the firm has received from the project fromperiod zero to the end of period t. By using TRM’s classifications, at rate r, the following occur:

1. If st (r ) ≤ 0 for 0 ≤ t < n, we have a pure investment project.2. If st (r ) ≥ 0 for 0 < t < n, we have a pure financing project.3. If st (r ) ≤ 0 for some t, st (r ) > 0 for some t, and 0 ≤ t < n, we have a mixed project.

A simple project is one in which the sign of a0 is different from the sign of ai for all i > 0. Ina mixed project, the firm has money invested in the project during some periods, and “owes”the project money during some other periods.

It can be shown that all simple investments are pure investments. However, the converse isnot true: not all pure investments are simple investments.

Let us follow TRM’s notation in using PFR to denote the project financing rate, the rateapplied for periods in which the project can be viewed as providing funds to the firm; that is,as a net financing source, with positive project balance. We use k, the firm’s cost of capital forPFR, and PIR to denote the project investment rate, r∗ (TRM use r for this), the rate that theproject yields when the project balance is negative. We also refer to the PIR as the RIC, thereturn on invested capital.

To determine the PIR, or RIC, we proceed as follows, first negating all cash flows if a0 > 0:

s0(r, k) = a0

s1(r, k) = (1 + r )s0 + a1 if s0 < 0

= (1 + k)s0 + a1 if s0 ≥ 0

s2(r, k) = (1 + r )s1 + a2 if s1 < 0

= (1 + k)s1 + a2 if s1 ≥ 0···

sn(r, k) = (1 + r )sn−1 + an if sn−1 < 0

= (1 + k)sn−1 + an if sn−1 ≥ 0

In order to find whether, for any j (0 < j ≤ n), s j (r, k) < 0 or ≥ 0, we substitute rmin for r inevaluating it. Since we will use rmin an estimate of the firm’s cost of capital for k, the onlyunknown in sn(r, k) is r. We solve this equation for r, and since the solution is a particularvalue, the RIC, we refer to it as r∗.

Note that because rmin is defined as the smallest real root for which all the project balanceequations, st (rmin), are ≤ 0, with 0 j t < n , it is an which determines whether the project ispure or mixed. If tsn(rmin) ≥ 0, a greater discount rate r, one for which sn(rmin) = 0, willretain the condition that st (r) ≤ 0 for 0 ≤ t < n. In this case, the r for which sn(r) = 0 willbe the r∗ of the project. It will also be the IRR of the project. From this it follows that theIRR is found by assuming that the project financing rate equals the project investment rate, sothat k does not enter the equation for IRR. In general, this will not be a correct assumption.However, it does not affect our results when sn(r, k) = sn(r), a condition we have for all pureinvestments. In other words, for pure investments only, r∗ = IRR, and k does not affect r∗: theIRR is “internal” to the project.


THE TRM ALGORITHM

The foregoing leads to an algorithm for determining the RIC on an investment, either simpleor with mixed cash flows. The steps of the algorithm are:

1. If a0 > 0, negate all cash flows before beginning. Find rmin, the minimum real rate for whichall the project balance equations, st (rmin) are ≤ 0, for 0 ≤ t < n.

2. Evaluate sn(rmin).(i) If sn(rmin) ≥ 0, then st (r, k) = st (r ) and r∗ equals the unique IRR as traditionally

found.(ii) If sn(rmin) < 0, then we proceed to step (3).

3. Let k be the firm’s cost of capital.

s0 = a0

s1 = (1 + r )s0 + a1 if s0 < 0

= (1 + k)s0 + a1 if s0 ≥ 0

s2 = (1 + r )s1 + a2 if s1 < 0

= (1 + k)s1 + a2 if s1 ≥ 0···

sn = (1 + r )sn−1 + an if sn−1 < 0

= (1 + k)sn−1 + an if sn−1 ≥ 0

In every st (r, k), use rmin for r to determine whether the project balance is less than zeroor greater than or equal to zero.

4. Solve sn(r, k) for unique r. Call this r∗ the return on invested capital.

Note that the return on invested capital r∗ may be the IRR, but in general will not be.

Example 10.3 Let us now take up the project discussed earlier, which had cash flows:

t = 0 t = 1 t = 2

−$100 +$320 −$240

and apply the TRM algorithm.

1. Find rmin:

−100(1 + r ) + 320 = 0

r = 3.2 − 1 = 2.2 or 220%

rmin = r

(In this example there is only one project balance equation.)2. Evaluate

sn(rmin) = −100(1 + r )2 + 320(1 + r ) − 240

= −240 and −240 < 0

So we have a mixed investment.


3. Let k be the firm’s cost of capital:

s0 = a0 = −100 < 0

s1 = s0(1 + r ) + a1 since s0 < 0

= 0 when evaluated for r = rmin

s2 = s1(1 + k) + a2 since s1 ≥ 0

= −100(1 + r )(1 + k) + 320(1 + k) − 240

4. Solve for r∗ = r:

1 + r = 320(1 + k) − 240

100(1 + k)

r∗ = r = 2.2 − 2.4

1 + k

Under the IRR assumption of r = k, we find that

r = 2.2 − 2.4

1 + r

r2 − 1.2r + .2 = 0

and, using the quadratic formula,

r = 1.2 ± √1.44 − .8

2= 1.2 ± .8

2= 1.0, 0.20

= 100 percent, 20 percent

What if r = k? Figure 10.2 shows the function for r∗ in terms of k. The figure shows clearlythat for only two values of k will r∗ = k for this project. Under the rule that we accept aproject if r∗ > k and reject if r∗ < k, this project is acceptable for the same values of k thatwe found with the NPV criterion.

k

160.00%

140.00%

120.00%

100.00%

80.00%

60.00%

40.00%

20.00%

0.00%

−20.00%

r *

0.00% 50.00% 100.00% 150.00% 200.00% 250.00%

Figure 10.2 r∗ as function of k for Example 10.3


Example 10.4 For a second application of the TRM algorithm let us take a project consideredearlier that had cash flows:

t = 0 t = 1 t = 2 t = 3

−$1000 +$3800 −$4730 +$1936

1. Find rmin:(i)

−1000 (1 + r ) + 3800 = 0

r1 = 3.8 − 1 = 2.8 or 280%

(ii)

−1000(1 + r )2 + 3800(1 + r ) − 4730 = 0

r2 = −3.8 ±√

(3.8)2 − (4)(4.73)

−2

has complex roots. Therefore, rmin = 2.8 = r1.2. Evaluate sn(rmin):

sn(rmin) = −1000(3.8)3 + 3800(3.8)2 − 4730(3.8) + 1936 = −16,038 < 0

Hence this is a mixed project.3. Let k be the firm’s cost of capital:

s0 = a0 = −1000 < 0

s1 = s0(1 + r ) + a1 = 0

= −1000(1 + r ) + 3800 = 0 at r = rmin

s2 = s1(1 + k) + a2

= −1000(1 + r )(1 + k) + 3800(1 + k) − 4730 < 0 at r = rmin

s3 = −1000(1 + r )2(1 + k) + 3800(1 + r )(1 + k) − 4730(1 + r ) + 1936

4. Solve for r∗ = r .This yields a fairly complicated expression for r in terms of k, although for a specific valueof k, the solution can be easily accomplished with the quadratic formula. The expressionfor k in terms of r is

k = 1.936 − 4.73(1 + r )

(1 + r )2 − 3.8(1 + r )− 1

from which we may generate values of k corresponding to various r∗ = r and plot thefunction (see Figure 10.3).

Note that r∗ is a double-valued function of k. Since we cannot lose more than we haveinvested in the project,2 values of r < −100 percent are not economically meaningful andmay thus be ignored. We cannot lose more than 100 percent of what we invest in the project,because the cash flows reflect the total effect of the project on the firm; all costs and revenuesattributable to the project are incorporated in the cash flows. At 10 percent, a double root to

2We are still assuming project independence in this chapter.


k

150.00%

100.00%

50.00%

0.00%

−50.00%

−100.00%

r *

−150.00% −100.00% −50.00% 0.00% 50.00% 100.00% 150.00%

Figure 10.3 r∗ as function of k for Example 10.4

k

15.00

10.00

5.00

0.00

−5.00

−10.00

−15.00

$NPV

0.00%150.00% 20.00% 40.00% 60.00% 80.00%

Figure 10.4 NPV function for Example 10.4

the IRR equation, it is interesting that the function touches, but does not cross, the r∗ = k-axis. This project is acceptable for 0 ≤ k < 10 percent and 10% < k < 60%, the same asby the NPV criterion. The NPV function is shown in Figure 10.4.

Example 10.5 Let us now solve for the RIC of a project having cash flows in six periods:

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5

−$100 +$600 −$1509 +$2027 +$1436 +$418

This project has real IRR values of 0 percent, 10 percent, and 90 percent. The IRR equationalso has complex roots of 0 + i and 0 − i .

1. Find rmin:(i) −100(1 + r ) + 600 = 0

r1 = r = 5 or 500 percent


(ii)

−100(1 + r )2 + 600(1 + r ) = 1509 = −100(1 + r )2 + 600(1 + r ) − 1509 = 0

r = −600 ±√

(600)2 − (4)(−100)(−1509)

−200

Since r is a complex root, skip r2.(iii) We can approach this in two ways. First, we could solve the equation

−100(1 + r )3 + 600(1 + r )2 − 1509(1 + r ) + 2027

after setting it equal to zero. This is a laborious process, conducive to errors in calcu-lations, unless we use a computer program. Even in this case there may be time lost inaccessing a computer and waiting for results. Once we did find r we would set r3 = r.The second approach may save us this trouble.

Let us substitute the rmin thus far obtained: r1 = 500 percent. If this equation valueis less than zero for r1 , we do not need to solve for the value of r3. Using this approach,we find the equation value is − 7027, so we need not solve for r3.

(iv) As in (iii), we could solve for r in the equation

−100(1 + r )4 + 600(1 + r )3 − 1509(1 + r )2 + 2027(1 + r ) − 1436

after setting it equal to zero. But again, let us first try the second approach. We obtaina value of −43,595; so again we need not solve for r4.

We have found that rmin = r1 = 500 percent, since for this rate, and no lesser rate, allproject balance equations are less than or equal to zero.

2. Evaluate sn(rmin):

sn(rmin) = −100(6)5 + 600(6)4 − 1509(6)3 + 2027(6)2 − 1436(6) + 418

= −261,152 < 0

So this is a mixed project.3. Let k be the firm’s cost of capital:

s0 = a0 = −100

s1 = −100(1 + r ) + 600 = 0 at r = rmin

s2 = −100(1 + r )(1 + k) + 600(1 + k) − 1509 < 0

s3 = −100(1 + r )2(1 + k) + 600(1 + r )(1 + k) − 1509(1 + r ) + 2027 < 0

s4 = −100(1 + r )3(1 + k) + 600(1 + r )2(1 + k) − 1509(1 + r )2 + 2027(1 + r )

−1436 < 0

s5 = −100(1 + r )4(1 + k) + 600(1 + r )3(1 + k) − 1509(1 + r )3 + 2027(1 + r )2

−1436(1 + r ) + 418

4. Solve for r∗ = r .Because this involves solution of a fourth-degree equation, we will present the solutions

for k = 15 percent and k = 25 percent. At k = 15 percent, r∗ = r = 15.08 percent, and theproject is marginally acceptable. At k = 25 percent, r∗ = r = 25.28 percent, and againthe project is marginally acceptable.


THE UNIQUE, REAL INTERNAL RATE OF RETURN:CAVEAT EMPTOR!3

The internal rate of return (IRR), even when unique and real, may nevertheless be an incorrectmeasure of the return on investment. All projects characterized by negative flows occurringonly at the beginning and the end will be mixed investments for which the IRR, whether uniqueand real or not, is not a correct measure of investment return. Several years ago, W. H. Jean[76] proved that for capital budgeting projects in which only the first and last cash flows werenegative that there would be a unique, real, positive internal rate of return or no positive IRR.J. Hirschleifer [69] subsequently showed that if the sum of cash flows beyond the first was lessthan or equal to the first cash flow, then multiple IRRs can exist for the project. This promptedProfessor Jean [77] to extend his treatment, and further specify the conditions for unique IRRfor such cases as Hirschleifer cited. Although Jean’s results are mathematically interesting, theydo not take into account the way in which such projects violate the assumption of independencebetween the IRR and the firm’s cost of capital, which destroys the economic meaning of theresulting IRR. The conditions under which the IRR of a project is not independent of the firm’scost of capital have been widely ignored in the literature, one noteworthy exception beingMao’s out-of-print text. Jean’s article and examples are cited here for purposes of illustration,since his article is mathematically rigorous.

Mathematical uniqueness of a real root to the traditional IRR equation, although a necessarycondition, is not sufficient to insure that one has obtained a rate independent of the firm’s costof capital, and thus a measure of investment return “internal” to the cash flow of the project.In fact, we shall prove that in the case of a project with negative flows only at beginning andend, for which Professor Jean proved a unique, positive IRR can always be found, the IRR willnever (excluding rare or contrived cases in which the firm’s cost of capital is the same as therate rmin, which is discussed later) be independent of the firm’s cost of capital.

A theorem will be proved later that has two corollaries relating to the discussion of professorsJean and Hirschleifer. First, however, the examples provided by Jean and Hirschleifer will beanalyzed within the framework provided by TRM. A cost of capital k = 10 percent will beassumed for all cases.

Case 1

Cash flows are −1, 5, −6.

IRR = −100%, 200%

rmin = 400%

sn(rmin) = −6 < 0

Hence this is a mixed investment.

s0 = a0 = −1 < 0

s1 = s0(1 + r∗) + a1

= −1(1 + r∗) + 5 = 0 (using rmin for r∗)

s2 = s1(1 + k) + a2

= −1(1 + r∗)(1 + k) + 5(1 + k) − 6

3Reprinted with permission of Journal of Financial and Quantitative Analysis, Copyright c© 1978. With corrections.


Therefore r∗ = −145.45%, the rate for which s2 = 0.

Case 2

First m inflows are negative. Cash flows are −5, −1, 2, 2.

IRR = −15.9%

rmin = −45.9%

sn(rmin) = +2 > 0

Hence this is a pure investment and the IRR is a unique, real measure of project return,independent of k.

Case 3

In middle life, m inflows are negative. Cash flows are −1, 2, −4, 2.

IRR = −36.0%

rmin = 100%

sn(rmin) = −6 < 0

Hence this is a mixed project.

s0 = a0 = −1 < 0

s1 = s0(1 + r∗) + a1

= −1(1 + r∗) + 2 = 0 (using rminfor r∗)

s2 = −1(1 + r∗)(1 + k) + 2(1 + k) − 4 < 0

s3 = s2(1 + r∗) + a3

= −1(1 + r∗)2(1 + k) + 2(1 + r∗)(1 + k) − 4(1 + r∗) + 2

So r∗ = −24.09%, the rate for which s3 = 0.Of the three cases considered, only case 2 has a return “internal” to the project. Case 1 has

two IRRs and thus the IRR is not only an incorrect measure of investment return, but alsoambiguous. Case 3 has a unique, real IRR. However, it is not a proper measure of return oninvestment. This is a crucial criticism of the IRR — even though it may be unique and real inthe mathematical sense, this in itself is not a sufficient condition for it to be a correct measureof return on investment.

An example presented by Professor Mao vividly emphasizes this point in case 4.

Case 4

Cash flows are −10, + 40, −40.

IRR = 100%

rmin = 300%

sn(rmin) = −40 < 0


Hence this is a mixed project.

s0 = a0 = −10 < 0

s1 = s0(1 + r∗) + a1

= −10(1 + r∗) + 40 = 0 (using rmin for r∗)

s2 = s1(1 + k) + a2

= −10(1 + r∗)(1 + k) + 40(1 + k) − 40

So r∗ = −63.64%, the rate for which s2 = 0.This is a mixed investment with return on invested capital of minus 63.64 percent, even

though the project has a unique, real, positive IRR of 100 percent. Thus, use of the IRR wouldlead to acceptance of the project for any cost of capital k < 100 percent — a very undesirableconsequence for a firm with normal financial management goals.

A NEW THEOREM

To generalize our findings, we now present a theorem that has significant implications on theclass of investment projects with negative cash flows only at the beginning and the end.

Theorem Given that a0 < 0, an < 0, at > 0 for t = 1, . . . , n − 1 with some at > 0 for aproject, the project will always be a mixed investment (a mixed financing project 4 if a0 > 0,an < 0, and at ≤ 0 for t = 1, . . . , n − l with some at < 0).

Proof Since a0 < 0, and at > 0 for t = 1 to n − 1, if we look only at the cash flows a0 throughan−1, they form, in themselves a simple investment that TRM have proved has a unique, realrate r for which all the st < 0 for t = 1, 2, . . . , n − 1. In particular, sn−1 < 0. Therefore,this r is the rmin of the project, for any larger r would cause all st to be less than zero fort ≤ n − 1.

In evaluating sn at rate r = rmin , we simply add an , which is less than zero, to (1 + rmin)sn−1, which is equal to zero. Then, since sn < 0, we have a mixed investment as defined byTRM, and the project rate r∗ is not independent of the firm’s cost of capital.

Corollary I If

at < 0 for m < n − 1

and

an < 0

with

at ≥ 0 for m ≤ t ≤ n and some at > 0

then the project will be a mixed investment.

4In this case we negate all cash flows before applying the TRM algorithm.

11

The Problem of Mixed Cash Flows: II

In Chapter 10 the problem of mixed cash flows was introduced, and a particular method ofanalysis, that of Teichroew, Robichek, and Montalbano (TRM), was discussed. Although thetheoretical merits of the rigorously developed TRM approach may be superior, it is computa-tionally very demanding. So other methods are more commonly used in practice to deal withprojects having mixed cash flows. The question of which, if any, of the existing methods ofanalysis is universally “best” may be unresolvable. The appropriateness of any of the methodsto a given investment depends on the extent to which the method’s underlying assumptionsmatch the particular situation and the goals of the enterprise’s management. Different circum-stances may require different analytical assumptions, or desired emphasis from investment,and these in turn may imply different methods of analysis.

In this chapter the methods examined are the Wiar method and the sinking fund fam-ily of methods. Because it is fundamentally different from the others, the Wiar method isdiscussed first.

THE WIAR METHOD

Robert Wiar [171] developed this method for analysis of the investment returns on leases,1

for which he asserted it is inappropriate to analyze directly the net cash flows to equity. Hisapproach was to employ an aspect of the Keynesian theory which states that the supply costof funds cannot exceed the imputed income stream yield. In other words, analysis must behandled by simultaneous treatment of three components:

1. positive cash inflows;2. mortgage bond amortization flows; and3. the required equity investment.

This analysis can be stated in two equivalent ways:

E0(1 + re)t =t∑

i=1

Ri(1 + r∗)t−1 −t∑

i=1

Mi (1 + rb)t−1 (11.1)

or

R0(1 + r∗)t = M0(1 + rb)t + E0(1 + re)t (11.2)

where

M0 = the amount financed by bondsMi = the fixed amortization paymentrb = the effective yield to the bond holdersE0 = the equity investment

1Leveraged leases are covered in detail in Chapter 13. The characteristic of leveraged leases of concern to us now is that they usuallyhave mixed net, after-tax cash flows to equity.


re = the return on equity, ignoring bond serviceR0 = the initial outflow — investment — equity plus bond financingr∗ = the overall return on aggregate investmentRi = the future income stream

In the case of capital-budgeting projects it will normally be appropriate to consider the supplycost of funds as the firm’s overall marginal cost of capital k.2 Let us examine equation (11.2),letting M0 = 0, k = re, and R0 = E0. Then

R0(1 + r∗)t = R0(1 + k)t (11.3)

and it is clear that r∗ = k. This means that the imputed income stream yield, r∗, equals thesupply cost of funds k. This is what we would expect, in equilibrium, at the cut-off point,for a firm not constrained by capital rationing, and it is consistent with the IRR criterion orKeynesian marginal efficiency of capital (MEC).3

Example 11.1 An application of the Wiar method4 Consider a project costing $10 millionthat is expected to yield net, after-tax cash flows of $1.5 million at the end of each year of itsuseful lifetime of 10 years. There is expected to be a salvage value of zero.

The firm’s existing capital structure is 25 percent debt, 75 percent equity, and is consideredoptimal. The project, if accepted, will be financed by a private placement of $2.5 million inbonds yielding 10 percent and maturing in 10 years, and the balance by equity.

Assuming the bonds are sold at par value, the payment (assumed to be made at year end)will be $250,000. The entire $2.5 million must be retired in the tenth year, the year of maturityfor the bond.

The overall cash flow stream is composed of two component streams, as shown in Table11.1. It is the return on equity we are interested in. The equity cash flow stream has mixed cashflows. Applying the Wiar method, we obtain from (11.2) the equation to be solved for re:

0.75(1 + re)10 = 1.0(1.0814)10 − 0.25(1.1000)10

and re = 7.45 percent

which is greater than the IRR of 7.03 percent on the equity stream. The ordinary IRR on themixed equity stream is unique in this particular problem, but it need not be so. The previouschapter showed that, when mixed cash flows are considered, even a unique IRR does notmeasure return on investment.

The re obtained is then compared with the required return on equity. If it is greater than therequired rate, the project will be acceptable.

The Wiar method will be discussed again in Chapter 13 that deals with leveraged leases.For now we recognize that for capital budgeting projects as a class, the method reduces to theIRR method already treated in detail, and that it is inadequate for analyzing projects havingmixed cash flows.

2This is because of risk considerations. No individual capital-budgeting project, unless independent of the firm’s existing assets, canbe properly considered in isolation and without regard to its effect on the firm’s risk characteristics that affect the firm’s cost of capital.3The Keynesian term “marginal efficiency of capital” or “MEC” is normally used in macroeconomic discussions concerning theaggregate investment return curve for an entire national economy, whereas IRR is normally used to refer to the return on individualcapital investment projects.4A nice ExcelTMspreadsheet that illustrates the Wiar method and others can be found at the URL http://www.acst.mq.edu.au/unit info/ACST827/levlease.xls at the Department of Actuarial Studies of Macquarie University, Australia.

The Problem of Mixed Cash Flows: II 99

Table 11.1 Cash flows for Wiar method example

Project For bond Net cash flowYear cash flow service to equity

0 − $10,000,000 $2,500,000 −$7,500,0001 1,500,000 −250,000 1,250,0002 1,500,000 −250,000 1,250,0003 1,500,000 −250,000 1,250,0004 1,500,000 −250,000 1,250,0005 1,500,000 −250,000 1,250,0006 1,500,000 −250,000 1,250,0007 1,500,000 −250,000 1,250,0008 1,500,000 −250,000 1,250,0009 1,500,000 −250,000 1,250,000

10 1,500,000 −2,750,000 −1,250,000IRR of cash flow streams = 8.14% 10.00% 7.03%

SINKING FUND METHODS

Sinking fund methods, as a class, are characterized by some adjustment being made to theoriginal cash flow series, aimed at making the adjusted cash flow series amenable to treatmentby IRR analysis. To apply any of them, we first follow some procedure for systematicallymodifying the cash flow series to remove all negative flows except those at the beginning,5

which represent the initial cash outlay or outlays. All negative cash flows beyond the initialoutlay sequence are forced to zero. Next, because the adjusted cash flows have only one signchange, the IRR procedure is applied. Under this definition, the Teichroew, or Teichroew,Robichek, and Montalbano method discussed in the previous chapter can also be considered asinking fund method.

To avoid, or perhaps clear up, some semantic difficulties, let us state here that (at least) threesinking fund methods have been used in practice. One is the initial investment method (IIM),another is the sinking fund method (SFM), the third is the multiple investment sinking fundmethod (MISFM). A possible semantic problem exists because the SFM, although a particularmethod, carries the name of the entire class of methods. In other words, the sinking fund methodis really only one of several methods, all of which can be characterized as sinking fund methods.As a class they include all of these methods as well as the TRM method. To avoid confusion,we shall refer to “the” sinking fund method as the traditional sinking fund method (TSFM).

The sinking fund earnings rate refers to the assumed rate at which funds that are set asidein a (hypothetical) sinking fund will accrue interest earnings. The sinking fund rate, sinkingfund rate of return, or sinking fund return on investment refers to the IRR on the adjusted cashflow sequence. Similarly, the initial investment rate, and so on, refers to the analogous IRR onthe adjusted cash flow series with the initial investment method.

To avoid complicating matters, we rule out the possibility of early project abandonmentin this chapter. That is, we assume all projects will be held until the end of their economiclives. Furthermore, a uniform period-to-period sinking fund earnings rate is assumed in orderto simplify and streamline exposition.

5The multiple investment sinking fund method is an exception to this because not all negative cash flows are removed beyond theinitial negative flow or flows.


The Initial Investment Method

The initial investment method (IIM) is a type of sinking fund method. In the standard, conser-vative IIM it is assumed that an additional amount is invested, at the beginning of the project,specifically for the purpose of accumulating funds exactly sufficient to cover all negative cashflows occurring after the first positive flow.6 Such initial investment is assumed to earn atsome rate of return compounded from period to period. The rate of return is calculated on therevised initial investment and the subsequent positive cash flows. Negative flows are zeroedout because they are assumed to be exactly offset by the additional initial investment growingat a compound rate. Another way of applying the IIM is to assume that the earliest positivecash flows are set aside into a sinking fund earning just sufficient to match exactly the laternegative cash flows.

To gain an understanding of the IIM, it will help to analyze an example. However, in orderthat the two similar methods may be compared together, this will be postponed until after thefollowing discussion of the traditional sinking fund method.

The Traditional Sinking Fund Method

The traditional sinking fund method (TSFM), like the IIM, assumes that positive cash flows canbe invested at some nonnegative rate of return so that later negative cash flows may be exactlycovered. Unlike the IIM, however, the TSFM assumes that the most proximate positive cashflows preceding the negative flow will be put into a sinking fund to the extent required to offsetthe negative flow or flows. The TSFM, therefore, can be considered a much less conservativemethod than the IIM, to the extent it delays investment for what may be a considerable timeand thus does not provide the benefit of having at least some accumulated earnings should theactual available earnings rate decline later in the project life.

Initial Investment and Traditional Sinking Fund Methods

The initial investment and traditional sinking fund methods are based on similar assumptions.Both the IIM and the TSFM are based on a technique that modifies the cash flows of a projecthaving mixed cash flows (and thus often a mixed project in the TRM sense) to produce asimple, pure project that has zeros where the original had negative flows, in all but the initialoutlays (the nonpositive cash flows preceding the first positive cash flow). In fact, the TRMalgorithm of the preceding chapter is closely related, with the earning rate on a “sinking fund”equal to k, the firm’s cost of capital. However, with the TRM method the timing and the amountof investment in a “sinking fund” are perhaps obscured by the nature and the complexity ofthe algorithm, with the cash flows “compressed”7 prior to the solution for return on investedcapital. With the IIM and TSFM, cash flows that had been negative are zeroed prior to solutionfor rate of return. Thus, the order of the polynomial that will be solved for a unique, real rootwill be less for the TRM than for the IIM or TSFM. This will be clarified by using two of thesame examples that were used in the previous chapter with the TRM algorithm, but this timewith the IIM and TSFM.

6A series of m negative cash flows followed by a series of positive flows would cause no problem in determining return, for we coulduse the IRR directly in such cases.7Compressed in the sense that the polynomial that must be solved for the return on invested capital, r∗, is of lesser degree than theIRR polynomial would be for the same project.


Original Investment +$320

t � 0

−$100

Initial Investment Assumption

$198.35(1.10)2

t � 0

−$198.35

Combined

1

1

+$320

t � 0

−$298.35

2

2

2

0Time

Time

Time

−$240

+$240

1

Figure 11.1 Initial investment method applied to Example 11.2 cash flows

Example 11.2 This example is the same as Example 10.1 used in Chapter 10. The cash flow is:

t = 0 t = 1 t = 2−$100 +$320 −$240

As stated in Chapter 10, this project has two IRRs: 20 percent and 100 percent. The NPV reachesa maximum of $6.67 at 50 percent cost of capital. The return on invested capital for this project,for k = 10 percent, is 1.82 percent. Figure 11.1 indicates the procedure used in applying theinitial investment method to the cash flows. The IIM rate of return on investment is 7.26 percentfor this example. Figure 11.2 suggests the procedure followed in applying the traditional sinkingfund method. It is assumed that the applicable sinking fund’s earning rate is 10 percent.

The essence of application of the TSFM to this project is to set aside sufficient cash, atsome assumed earnings rate, to accumulate to an amount exactly sufficient, one period later,to match the negative cash flow.

Thus, for any positive cost of capital above 1.82 percent, the project would be unacceptablefor investment under either method. Like the IRR method or that of TRM, we compare theproject yield to the enterprise’s cost of capital and reject the project if the yield rate is less thanthe cost of capital. For this project, if the earnings rate on the funds set aside8 is 10 percent, andthe cost of capital is k = 10 percent, then rTSFM = 1.82 percent — the same as that obtainedwith the TRM method in Chapter 10. In general, they will not be the same. However, it isindicative of the relationship between the methods.

8In practice, it will be rare to find an investor actually depositing or setting aside cash to meet outflows later in the life of a project.Rather, the firm will employ the positive cash flows in its operations. To be conservative, we could assume that a zero earnings rate isapplicable. However, it would be normally appropriate to assume that funds earn at the firm’s cost of capital.


Original Investment $320

2Time

−$240

+$240

Time

Time

2

$218.18(1.10)

−$218.18

$101.82Combined

−$100

t � 0

t � 0

t � 0

−$100

Sinking Fund Assumption

1 2

0

1

1

Figure 11.2 Traditional sinking fund method applied to Example 11.2 cash flows

$3800

t � 0

x(1 + i)2

2

1

−$4730−($1000 + x)

Time

+$1936

3

Figure 11.3 Initial investment method: application to Example 11.3

Next we examine another example, this the same as Example 10.2.

Example 11.3 Figure 11.3 displays the net cash flows for this project and indicates theapplication of the IIM.

Note that positive cash flows are compounded forward in time. They are not discountedback to an earlier point in time with the various sinking fund methods.9 Otherwise, the $1936

9An exception to this general rule is to be found in what is referred to as the modified sinking fund method (MSFM). We shall notdiscuss this method here, since it is a straightforward variation of the TSFM in which later positive cash flows may be discounted topay off earlier negative flows


$3800−x$3800−x

(1 + i)

2

$1936

3t � 0

−$10001

Time

−$4730

Figure 11.4 Traditional sinking fund method: application to Example 11.3

t � 0

−$100

+$70+$90

−$22

3

21Time

Figure 11.5 Original cash flow series for Example 11.4

positive cash flow at the end of year 3 could have been discounted back one year to offsetpartially the negative flow of $4730 at the end of year 2.

The cash flow sequence that must be solved for rIIM is: −($1000 + x) at t = 0; $3800 att = 1; 0 at t = 2; and $1936 at t = 3. The value of x depends on the assumed earnings rate onthe sinking fund. It is equal to $4730/(1 + i )2 where i is the applicable rate. For i = 10 percent,we obtain x = $3909.10 and rIIM = 10 percent.

Figure 11.4 illustrates application of the TSFM to the same project. The adjusted cash flowsfrom which we find the rTSFM are: −$1000 at t = 0; $3800 − x at t = 1; 0 at t = 2; and $1936at t = 3. The value of x is $4730/(1 + i ), where i is the sinking fund earnings rate. For i = 10percent, we obtain rTSFM = 10 percent.

Example 11.4 The cash flows for this project are illustrated in the time diagram of Figure11.5. This project has two positive cash flows, one at the end of each year preceding the finalcash flow of minus $22 at the end of year 3.

Figure 11.6 shows the procedure involved in applying the TSFM to Example 11.4. At anassumed earnings rate of 10 percent on the sinking fund, a set-aside of $20 (out of the $90

t � 0

−$100

+$70+$90

321Time

+$20(1.10)+$70

+ $22− 22

$0

Figure 11.6 Traditional sinking fund method, adjusted cash flows for Example 11.4 with i = 10 percent


t � 0

+$70

−$100.00− 16.53 (1.10)3

+$90

1 2 3Time

−$116.53

−$22 22

$0

Figure 11.7 Initial investment method: adjusted cash flows for Example 11.4 with i = 10 percent

$70.00−18.18 (1.10)2

$51.82 $90$22−22

t � 0

−$100

1 2 3Time

$0

Figure 11.8 Initial investment method: adjusted cash flows for Example 11.4 with i = 10 percent

cash flow) at the end of year 2 will increase to $22 a year later. The sinking fund amount of $20plus $2 interest will exactly offset the negative cash flow at the end of year 3. After set-aside of$20 at end of year 2, $70 remains for other uses. The yield on the adjusted cash flows is 25.7percent = rTSFM.

Application of the initial investment method to Example 11.4 is illustrated in Figure 11.7.Again, the assumed earnings rate on the funds set aside is 10 percent. With this method wemust invest an extra $16.53 at t = 0 to offset the negative $22 cash flow at t = 3. The rIIM onthe adjusted cash flow series is 22.91 percent.

The IIM, a special case of the sinking fund method, may be considered more conservativethan the TSFM. This difference lies in the manner of selecting the timing and amounts to beset aside.

One might argue that it is too extreme a conservatism to assume that sufficient extra fundsmust be put into a sinking fund at t = 0 to cover the later negative cash flows. The IIM sofar discussed is the limiting case on the conservative end of the spectrum (especially so ifthe assumed earnings rate were to be zero). As an alternative initial investment approach, wecould assume that sufficient funds are set aside from the earliest positive cash flows to offsetlater negative flows. Figure 11.8 illustrates this variation of the IIM. The rIIM in this case is24.25 percent.

Example 11.5 Now let us consider a somewhat more complex project than those in theprevious examples of this chapter. Consider, for instance, a replacement chain of one capitalinvestment following another. Let us assume we have a mine, which will cost $100 initially todevelop and which will provide net, after-tax cash flows of $150 at the end of each year of itsthree-year economic life. Furthermore, assume that undertaking this mining project commitsour organization to a project for secondary mineral recovery costing $350 to initiate andproviding net, after-tax cash flows of $100 at the end of each year of its three-year useful life.Finally, at the end of the secondary recovery, our organization will have to pay out $350 to


$150$100 $100

$150

t � 0

−$100

−$200

1 2

3

4 5

6Time

−$250

Figure 11.9 Original cash flow series for Example 11.5

$NPV

605040302010

−10−20−30−40−50−60

%

10.23%

Figure 11.10 Net present value as a function of cost of capital for Example

close down operations and rehabilitate the land on which the mine is situated to comply withenvironmental legislation. Since we have some positive cash flows from the projects at the endof years 3 and 6, the overall net outlays in those years are not $350, but are $200 and $250,respectively. Figure 11.9 contains a time diagram illustrating the cash flows for this example.

This project has two IRRs: 10.23 percent and 85.47 percent. From Chapter 10 we know thatneither is a correct indication of the return on investment. The NPV function for the projectis plotted in Figure 11.10. Note that under the NPV criterion the project would be consideredacceptable for values of cost of capital k, such that 10.23 percent < k < 85.47 percent.

Since this project is more complex than those considered previously in this chapter, appli-cation of the TSFM and IIM is illustrated in the tables. Table 11.2 shows, for a 10 percenttraditional sinking fund earnings rate, the adjustment to the cash flows that will be made em-ploying the TSFM. The TSFM yield rate can be seen by inspecting the adjusted cash flows tobe 9.27 percent. Figure 11.11 contains a time diagram showing the adjusted cash flows.

Table 11.3 shows, for a 10 percent sinking fund earnings rate, the procedure employed foradjusting the cash flows with the less conservative variation of the IIM. Figure 11.12 contains


Table 11.2 Traditional sinking fund method adjustment to original cash flows forExample 11.5

Time periodoriginal t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

cash flows −$100 $150 $150 −$200 $100 $100 −$250

−100Step 1 ×(1.1)———–→ 110

−100Step 2 ×(1.1)2————————→ 121

−12.98Step 3 ×(1.1)4——————————————————→19

−137.02Step 4 ×(1.1) → 150.72

−40.73Step 5 ×(1.1)2——————→49.28Sinking fund

method cash −100 109.27 0 0 0 0 0flow

Table 11.3 Modified initial investment method adjustment to original cash flows forExample 11.5

Time periodoriginal t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

cash flows −$100 $150 $150 −$200 $100 $100 −$250

−150Step 1 ×(1.1)2—————-→181.50

16.82Step 2 ×(1.1)→ 18.50

−133.18Step 3 ×(1.1)4———————————————→ 194.99

−45.46Step 4 ×(1.1)2————————→ 55.01Initial investment

methodadjusted cashflows −100 0 0 0 54.54 100 0

Time0 0 0 0 0

1 2 3 4 5 6t � 0

−$100

$109.27

Figure 11.11 Traditional sinking fund method


−$100

1 2

00

3

0

4

$54.54

$100

5 6Time

0t � 0

Figure 11.12 Modified initial investment method: adjusted cash flows for Example 11.5 with i = 10percent

100

80

60

40

20

20 40 60 80 100

TRM (2)

TRM (1)SFM

IIM

k%0

−20

r% r � k

Figure 11.13 Sensitivity analysis of Example 11.5 project

the time diagram corresponding to the adjusted cash flows. The IIM yield rate cannot beobtained by inspection (in contrast to the sinking fund yield rate). By calculation we find it tobe 9.84 percent.

Figure 11.13 and Table 11.4 contain sensitivity analyses of the (less conservative) initialinvestment, traditional sinking fund, and TRM rates of return associated with various cost ofcapital percentages (traditional sinking fund earnings rates). Note that for k = 0 there is noreal solution to the TRM return on investment equation.

THE MULTIPLE INVESTMENT SINKING FUND METHOD

The final sinking fund variant we will discuss in this chapter is the multiple investment sinkingfund method (MISFM). The idea underlying the MISFM is to adjust the cash flow sequence toobtain two distinct, nonoverlapping investment sequences having identical IRRs. For example,the project having a five-year useful life can be decomposed as follows into adjusted cash flowsplus sinking fund. We assume the sinking fund earns at 15 percent per period.


Table 11.4 Sensitivity analysis of Example 11.5project

k% TRM(1)% TRM(2)% IIM% TSFM%

0 3.19 50.0010 51.01 10.11 0.61 0.8720 60.74 2.80 23.10 39.0830 67.07 −2.40 35.81 54.8540 71.94 −6.40 47.50 64.2550 75.87 −9.66 58.01 70.8960 79.14 −12.40 67.25 76.0870 81.90 −14.77 75.26 80.2880 84.29 −16.86 82.14 83.7990 86.38 −18.72 88.02 86.76

100 88.21 −20.39 93.02 89.31110 89.85 −21.91 97.29 91.53120 91.31 −23.30 100.94 93.49130 92.63 −24.58 104.06 95.22

The adjusted cash flows can be considered to be two nonoverlapping investments, eachhaving a unique, positive internal rate of return of 50.0 percent. For this particular project therIIM is 28.93 percent, rTSFM is 40.95 percent, and the RIC is 52.24 percent.

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5Original cash flows −$1000 765 2500 −3500 1000 3000Adjusted cash flows −$1000 500 1500 −2000 1000 3000Sinking fund 265 1000 −1500

The MISFM is much more difficult to apply10 than the IIM or TSFM, because it re-quires that the original cash flows be adjusted so that one, two, or several nonoverlap-ping subsequences remain, all of these having the same unique, real IRR, which is therMISFM. The method is used by some major organizations in their leveraged lease analysis.Limited experience with results of the MISFM suggests that the rate is generally close tothe TRM RIC rate. This suggests that the RIC could be used as a first approximation tothe rMISFM.

Strong Points of the Methods

1. Both the TSFM and IIM are based on theoretically defensible assumptions.2. The methods take into account the time-value of money and do not exclude any cash flows.3. The methods are much easier to apply than the Teichroew, Robichek, and Montalbano

algorithm, except for the MISFM.4. Both methods assure a unique, real measure of return on investment.5. Application of the methods is not especially difficult to learn.

10The method is relatively easy to employ today with modern spreadsheet programs, as illustrated in the ExcelTMworkbook available atthis writing from Actuarial Studies at Macquarie University in Australia: http://www.acst.mq.edu.au/unit info/ACST827/levlease.xls


Weak Points of the Methods

1. Both the TSFM and IIM may require, in addition to an estimate of the organization’s cost ofcapital, an estimate of the sinking fund rate at which cash may be invested. For reinvestmentof funds within the firm this rate could be the same as the cost of capital.

2. Both methods are more difficult to apply without a computer than the NPV method.3. The methods do not distinguish between projects of different size and/or different economic

lives. However, adjustment may be made for this.4. In general, the methods may not adequately reflect the interrelated nature of investment and

financing imbedded within mixed projects.

11AAppendix: The Problem of Mixed Cash

Flows III — a Two-stage Method of Analysis

The two preceding chapters present the Teichroew, Robichek, and Montalbano method andother methods classifiable as sinking fund approaches to obtaining a unique, real rate of returnmeasure on those investments having mixed cash flows. It was shown in Chapter 10 that evena mathematically unique, real IRR is no assurance, under the TRM assumptions, of a measureof return on investment independent of the enterprise’s cost of capital. This chapter presentsan alternative, developed by the author, to the various sinking fund methods.

The method of analysis presented here is designed to explicitly separate the analysis ofinvestments with mixed cash flows into two separate but related decision stages. The methodyields two measures by which investments with mixed cash flows, such as leveraged leases,may be evaluated:

1. The time required to recover the initial investment plus the opportunity cost associated withthe funds committed to the project.

2. The implicit “borrowing” rate contained within the cash flows occurring beyond the capitalrecovery time.

Together, the two measures provide a decision rule: if the capital recovery period is acceptableand the implicit borrowing rate less than the rate at which the firm can acquire funds (the firm’scost of capital), or reinvest the second stage cash flows, the project is acceptable.

RELATIONSHIP TO OTHER METHOD

The net present value (NPV) method assumes that the enterprise cost of capital is the appropriaterate at which to discount the cash flows, whether they be positive or negative. The NPV itself is ameasure of monetary return over and above the investment outlay. Although it is implicit in theNPV, the method and the measure provide no information pertaining to the timing of recoveryof the funds invested in the project. The NPV method relates directly to the basic valuationmodel of modern financial management and is generally preferred by academic writers to othercapital budgeting methods. It will be shown in this chapter that the two-stage method is, unlikethe internal rate of return (IRR), perfectly compatible with the NPV. The two-stage methodyields identical accept/reject decisions to those obtained with the NPV, when the acceptabletime span to recover the invested funds is unconstrained.

The two-stage method, although yielding results identical to NPV, provides decision mea-sures in the separate stages that are in terms of time and percentage rate. Thus, the first stage,by providing a time measure, relates to the ubiquitous payback method. As shall be seen,however, it does not suffer from the well-known shortcomings of the payback method. Thesecond stage relates to the IRR, except that the rate found will be a cost rate rather than a returnrate.

The Problem of Mixed Cash Flows III 111

THE TWO-STAGE METHOD

Studies have shown that many managers employ the payback method very heavily, much to thedismay of those academics who have dwelt upon and publicized its shortcomings. The majorshortcomings of the payback method are that it ignores the time-value of money and ignorescash flows beyond the end of the payback period. Or, equivalently, a zero opportunity cost rateis assigned during the payback period, with an infinite rate thereafter.

In stage one of the method proposed here, a payback period for the initial investment isdetermined. However, this payback period takes the time-value of money into account byrequiring not only that the initial investment be recovered but, in addition, the opportunitycost of the unrecovered funds remaining invested in the project. This is related to Durand’sunrecovered investment: at this payback, and not before, unrecovered investment is zero.Unrecovered investment is discussed in Chapter 14. Because the time-value of funds is takeninto account, this payback measure is economically justifiable.1

In stage two, the cash flows remaining beyond the payback are analyzed, in the “negativeinvestment” or “loan” phase of the project. Thus the second major objection to the traditionalpayback method is of no consequence. In the second stage, an imputed rate is determined thatcan be compared to the enterprise’s cost of capital or the investment opportunity rate availableto the firm over the periods remaining in the project life from payback to the end of the projectlife.2 If the “loan” rate is less than the firm’s cost of capital, or the rate the firm expects to beable to earn on the cash flows from the project, it is acceptable on the rate basis. Since thefirm’s cost of capital would be less than or equal to the reinvestment rate offered by futureacceptable projects,3 we shall assume that comparison will be with the cost of capital.

Otherwise, it could be possible to accept a project with an implicit cost greater than theenterprise cost of capital because the reinvestment rate of return is greater.

The two-stage method can perhaps be best explained by detailed treatment of examples.

Example 11A.1 We shall analyze the cash flows to equity on a leveraged lease discussedby Childs and Gridley [22]. Chapter 13 examines leveraged leases in detail, and this sameleasing project is examined further. For now it will suffice to take the net cash flows to equity asgiven and note that our purpose here is to illustrate the two-stage method of project analysisand not to discuss leveraged leases. Table 11A.1 contains the original net cash flows and theadjusted cash flow series. We assume a cost of capital k = 10 percent.

The project (lease) requires an initial equity outlay of $20. With k = 10 percent the enterprisemust recover $20 plus 10 percent of $20, $20(1 + k), at the end of the following year. Theenterprise will be no better off nor worse off if it can recover the capital committed to theproject along with the cost of those funds; its capital will still be preserved. Because the cashflow at t = 1 is only $13.74, with $8.26 of the $22 remaining to be recovered a year later,which with opportunity cost amounts to $9.09. The cash flow of $5.89 at t = 2 cannot quitesuffice, and therefore $3.52 must be recovered at t = 3. Figure 11A.1 illustrates the procedure.

1For the remainder of this chapter “payback” will refer to this time-value-of-money-adjusted payback — “traditional payback” to themethod as it is usually applied.2Or, if abandonment is to be considered, rates can be found corresponding to abandonment at any time prior to the end of the project’seconomic life.3If the “loan” rate is less than the cost of capital, the project is providing funds at a cost less than the combined total of other sources.Such funds can be invested as they are received to earn the opportunity rate available to the firm. In the idealized construct of a worldwithout capital rationing, the enterprise would be expected to invest funds up to the point at which the marginal rate of return equaledthe marginal cost of capital.


Table 11A.1 Assumed per-annum opportunity cost of funds(cost of capital) is 10%

Original Originalinvestment net after-tax Adjusted

plus opportunity cash cash flowt cost flows seriesa

0 0 $−20.00 $−20.00 2.6981 $−22.00 13.74 −8.26 Year2 −9.09 5.89 −3.20 Payback

3 −3.52 5.04 1.524 4.19 4.195 3.32 3.326 2.43 2.437 1.54 1.548 0.63 0.639 −0.51 −0.51

10 −2.68 −2.6811 −2.90 −2.9012 −3.16 −3.1613 −3.41 −3.4114 −3.70 −3.7015 −3.90 −1.7216 +2.40 0

a The payback is in 2.7 years. The $1.52 remaining after payback in year 3 is assumed to occurat the end of the period, just as the original flow of $5.04 is.

Before solving for the implicit loan rate, the last flow of +$2.40 is discounted back at theopportunity cost rate, using a sinking fund approach until absorbed by the negative flows. Thismay be interpreted as assigning, at discount, the last positive flow to the project — in otherwords, using it to prepay a portion of the loan.

Original Cash Flows

t � 0 t � 1 t � 2 t � 3

5.89

5.04

−$20(1 + k) −22.00− 8.26(1 + k) −9.09

−3.20(1 + k) −3.52+1.52

Adjusted Cash Flows

t � 0 t � 1 t � 2 t � 3+1.52

The payback is thus 2 + 3.52/5.04 years, or 2.7 years.

0 0 0

+$13.74

Figure 11A.1 Stage I analysis


Original Cash Flows

t � 15 t � 16

t � 15 t � 160

t � 14

t � 14

+2.18

−3.70

−1.72

−3.70

−3.90

−1.72

Adjusted Cash Flows

+ 2.40 ÷ (1 + k)

Figure 11A.2 Preparation for stage II analysis

By allowing a noninteger value for payback, we violate the assumption that cash flows occuronly at the end of each period. To be consistent with this assumption we would take three yearsas the payback period required to recover the initial investment plus opportunity cost of funds.However, if we recognize that the assumption of end-of-period cash flows is only to facilitatecalculations, and that cash flows do, in fact, occur more or less uniformly over time, we willbe comfortable with the fractional result as calculated.

The residual $1.52 after investment recovery, or capital recovery, is the first nonzero cashflow in the adjusted series. If it were not for the final cash flow of +$2.40 at t = 16 we wouldhave, in the remaining cash flows, the series of a simple financing project with only onesign change in the cash flows. Before applying stage II of the method, we must first get ridof this last cash flow. One approach, a conservative one, would be to ignore the +$2.40, toassume it will not be received. A better approach, in the author’s opinion, is to assume thatfunds can be borrowed at rate k, and the loan proceeds used to offset one or more of theimmediately preceding negative cash flows. The +$2.40 would be used at t = 16 to repay theloan. Equivalently, we may assume the +$2.40 is assigned (at discount of k) to a creditor. Thislatter approach is the one that will be used. Figure 11A.2 illustrates the method.

The enterprise has recovered its initial investment and the associated cost of the unrecoveredfunds remaining committed by the end of the first 2.7 years. The remaining adjusted cash flowsoccurring subsequent to that time are gratuitous to capital recovery of the initial investment.They are characteristic of a loan, and therefore we are interested in the rate implied by thesecash flows. This “loan” rate is determined to be 3.9 percent, much less than our assumed 10percent cost of capital. If we can obtain funds at a rate lower than our cost of capital, weshould do so,4 and the project is acceptable on this basis.

As will become apparent in the formal development of the two-stage method, if a projectis not rejected on the basis of an unacceptable payback, the accept/reject decision obtainedfrom stage II will be identical to that obtained with the NPV method. This is an important

4Even if we recognize that risk considerations may alter the conclusion, the effect of the project itself on k could have been incorporatedinto the rate itself prior to this analysis.


result. Although the stage II results are perfectly compatible with the NPV, the stage Ipayback may mark a project as unacceptable because capital recovery takes longer than isconsidered acceptable.

Before getting into the formal, mathematical treatment of the two-stage method, it will beuseful to analyze one more example. This is the same as Example 10.2 from Chapter 10.

Example 11A.2 This project has cash flows:

t = 0 t = 1 t = 2 t = 3−$1000 +$3800 −$4730 +$1936

Again assuming k = 10%, the $1000 initial investment plus cost of the funds committed (atotal of $1100) is fully absorbed by the large positive cash flow at t = 1. The stage I paybackfor this project is 1100/3800 = 0.289 or 0.3. The adjusted cash flows are:

t = 0 t = 1 t = 2 t = 30 +$2700 −$4730 +$1936

Before finding the stage II rate, we assume the $1936 is assigned to a creditor, at a discountof k percent, and the proceeds received at t = 2. Thus, the adjusted cash flow at t = 2 is−$4730 + ($1936/1.10) = −$2970. To find the stage II rate, rB, we solve for the IRR of thecash flow series:

t = 0 t = 1 t = 2 t = 30 +$2700 −$2970 0

And $2700(1 + rB) = $2970 so that rB = 10 percent. Since rB = k the project is not accept-able. This project, not coincidentally, as will be shown, has a zero NPV for k = 10 percent.Figure 11A.3 contains a plot of the two-stage method results and NPV for various values of k.

FORMAL DEFINITION AND RELATIONSHIP TO NPV

It was stated earlier in this chapter that the two-stage method yields results identical to thoseobtained with the NPV provided that time for full capital recovery is unimportant. To makethe two-stage NPV relationship explicit and at the same time provide a formal definition ofthe two-stage method, we first write the formula for NPV as

NPV =n∑

t=0

Rt (1 + rr)n−t

(1 + k)n(11A.1)

where Rt are the net, after-tax cash flows, and rr the reinvestment rate.

NPV =n∑

t=0

Rt

(1 + k)t=

n∑t=0

Rt (1 + k)−t (11A.2)

also by reduction of equation (11A.1)Similarly, the two-stage method may be written as the following.


$NPV

30

20

10

0

−10

−20

−30

−40

Years

1.50

1.00

.50

0.00−10 10 20 30 40 50 60 70 80 90 100

%k

%rB

NPV � f(k)

70

60

50

40

30

20

10

−10

−20

rB > k rB > krB < k

rB � f (k)

rB � k

Figure 11A.3 NPV and two-stage measures for Example 11A.2

Payback Stage

P∑t=0

Rt (1 + rr)P−t

(1 + k)P= 0 ⇒ P (11A.3)

P∑t=0

Rt

(1 + k)t=

P∑t=0

Rt (1 + k)−t = 0 ⇒ P (11A.4)

Multiplying equation (11A.4) by (1 + k)n yields the equivalent form actually employed in theproblem treated earlier:

P∑t=0

Rt (1 + k)n−t = 0 ⇒ P (11A.5)


“Borrowing” Rate

n∑t=P+1

Rt (1 + rB)n−t

(1 + rB)n−P=

n∑t=P+1

Rt (1 + rB)−t = 0 ⇒ rB (11A.6)

where rB is the implicit cost of funds inherent in the flows remaining after payback.5 Formula(11A.6) assumes the opportunity cost rate equals the implicit cost of funds. If necessary, thelast cash flow at the end of the project can be forced to zero as illustrated earlier.

Formula (11A.6) can be modified to incorporate the firm’s cost of capital and to obtain apresent value formulation:

n∑t=P+1

Rt (1 + rB)n−t

(1 + rB)n−P= present value (11A.7)

Now, adding (11A.3) and (l1A.7), we obtain

n∑t=0

Rt (1 + rr)P−t

(1 + k)P=

n∑t=P+1

Rt (1 + rB)n−t

(1 + k)n−P(11A.8)

which, for reinvestment rate rr = “borrowing” rate rB = r , reduces to

n∑t=0

Rt (1 + r )n−t

(1 + k)n(11A.9)

which is identical to the NPV formulation of (11A.1).In this chapter we have so far considered projects with mixed cash flows. What if we now

apply the two-stage method to a simple investment? Consider the cash flows in the followingexample.

Example 11A.3

t = 0 t = 1 t = 2 t = 3 t = 4−$1000 1000 1000 1000 1000

If we again let k = 10 percent, the stage I payback is 1.11 periods and the revised cash flowseries is

t = 0 t = 1 t = 2 t = 3 t = 40 0 +$890 1000 1000

There is no real rate that satisfies the second-stage rate equation. However, because theremaining cash flows are all positive, they constitute in themselves a “loan” that does nothave to be repaid, or a gift to the firm, and the project is acceptable on this basis. The two-stage method may be used for simple investments as well as those with mixed cash flows. It isgenerally applicable, which was to be expected from what is basically a special formulationof the NPV.

5The rates rr, rB, and k are assumed to be greater than or equal to zero in order that they have a meaningful economic interpretation.


CONCLUSION

The NPV method of analysis has been largely ignored by those decision-makers who haveshown continuing preference for the traditional payback method and to a lesser extent the(internal) rate of return. The two-stage method discussed in this chapter presents the NPV interms decision-makers are accustomed to: payback and percentage rate. The payback, however,takes into account the time-value of funds at the enterprise’s cost of capital; the percentage rate,for nonsimple investments, is a cost rate implicit in the cash flows after payback. If paybackis not constrained, the two-stage method will always yield the same accept/reject decision asthe NPV method.

Because decision-makers have shown long-standing tenacity for the traditional paybackmethod, the two-stage algorithm may find better acceptance by practitioners than the NPVmethod has received. And the two-stage method makes explicit, in the payback measure, thetime required to recover the capital committed to a project. This is something the NPV methoddoes not do, as it is usually stated.

A BRIEF DIGRESSION ON UNCERTAINTY

Up to this point we have considered the environment in which capital investment decisions areto be made one of certainty. If we relax this assumption, as we do in the next section, we arecompelled to admit that, to the extent that cash flow estimates become increasingly tenuousand subject to error the further they occur from the present, a project that returns the initialinvestment early is to be preferred to one that does not, ceteris paribus. This is particularlyso during times of economic, political, and social instability, the combined effects of whichmay cause cumulative exogenous effects to the enterprise that are impossible to predict far inadvance.

Some time ago a distinction was often made between risk and uncertainty. Today it seems thedistinction is often ignored, perhaps because the theory of finance in general, and investmentsin particular, have been developed to their present state by assuming risk rather than the moreintractable uncertainty. The distinction is this: with risk we take as known the probabilitydistributions of the variables; with uncertainty we assume ignorance of the distributions of thevariables.

The two-stage algorithm for investment analysis provides, in its payback measure, a meansof addressing uncertainty. Two investments with identical NPV may have substantially differentcapital recovery payback and, in a world characterized by uncertainty, the investment with theshorter payback is to be preferred. Because the two-stage method provides a capital recoverymeasure, it allows management to determine whether or not the capital recovery is swiftenough. Because capital preservation may be a goal that overrides possible investment returns,the payback should be of interest.

12

Leasing

A lease is a contract under which the user (lessee) receives use of an asset from its owner(lessor) in return for promising to make a series of periodic payments over the life of thelease. A lease separates use from ownership. The two basic types of leases are operating andfinancial. Operating leases have relatively short terms, provide less than full payout,1 and maybe canceled by the lessee. A hotel room, or home telephone, water or electrical service maythus be considered forms of operating lease. In contrast, a financial lease is for a long term,provides for full payout, and cannot be canceled without penalty by the lessee. We shall notbe concerned with operating leases, but instead focus on financial leases.

Financial leases may be separated into two main categories: ordinary and leveraged. Thischapter is concerned with ordinary financial leases. Chapter 13 considers leveraged leases andtheir unique attributes and problems. Both kinds of financial leases have assumed increasingimportance in recent years and we may expect growth in leasing to continue over the nextdecade, barring major changes in tax laws that apply to them.

ALLEGED ADVANTAGES TO LEASING

Many advantages over conventional financing have been attributed to leasing. Although somehave genuine value, other may have advantages only to certain firms in particular circumstances,and still others may have dubious value altogether. Among the claimed advantages are thefollowing:

1. Off-balance-sheet financing This is of dubious value, since the existence of financialleases must be footnoted and analysts will treat a lease as if its capitalized value were alisted liability.

2. Provides 100 percent financing This may be advantageous when other financing is notavailable or available only under unacceptable terms.

3. Longer maturity than debt For a long-lived asset this may be a significant advantage.Financial leases generally run for the life of the asset. Loan terms, on the other hand, aregenerally set by the policy of the lender and maturity may be much shorter than the asset life.

4. Entire lease payment tax deductible This can be advantageous if land is involved sinceit is not depreciable if owned.

5. Level of required authorization Leases may sometimes be authorized by plant managers,whereas purchase of the same asset may require approval higher in the organization.

6. Avoids underwriting and flotation expense Leasing also avoids the public disclosureassociated with sale of securities.

7. Front-end costs reduced Delivery and installation costs are spread over the life of the lease.

1Full payout for a lease requires that the total of payments be sufficient for the lessor to recover, in addition to the capital investment,the cost of funds and profit.


8. Lease payments fixed over time Both the lessee and the lessor know the costs over thelife of the lease.

9. Less restrictive, quicker, more flexible10. May conserve available credit Possibly, but consider comment under (1).11. Lease may “sell” depreciation The lessee, if unable to use depreciation and investment

tax credit directly because of losses, in essence “sells” them to the lessor for more favorableleasing terms and thus gains from what would otherwise be lost.

12. Leased assets provide own collateral The lessee does not have to pledge other assets thatmight have to be pledged to secure debt financing for the same leased equipment. Becausethe lessor owns the leased asset, he can recover it in the event the terms of the lease arebroken.

ANALYSIS OF LEASES

The analysis of ordinary financial leases in the literature has focused almost exclusively onlease evaluation from the viewpoint of the lessee, the user of the equipment. Very little hasbeen written on lease analysis from the lessor’s view until recently. Evaluation by the lessor isin itself a capital-budgeting problem that, depending on the terms of the lease and the qualityof the lessee, may approach a certainty environment in many respects. In this chapter theanalysis of leases will be considered from both the lessee and the lessor viewpoints. First, thetraditional analysis from the lessee’s position will be considered. Then an integrated treatmentof the lessee’s and the lessor’s positions will be discussed.

It should be made clear at the outset that lease analysis itself does not address the questionof whether a particular asset should be acquired or not. Rather, lease analysis starts with thepremise that the asset should be acquired by the lessee.2 The question that lease analysistries to answer is whether the asset in question should be purchased or leased. This is oftenexpressed as “lease or buy” or “lease or borrow.” The traditional analysis, through the lessee’seyes, involves finding the least cost alternative to acquiring an asset: the minimum of the leasecost and the alternative financing cost. The alternative financing is generally assumed to be100 percent debt financing, since leasing commits the lessee to making periodic payments justas a fully amortized bond would do. And contrary to alleged advantages (1) and (10) above, ithas become widely recognized that leases do displace debt.

Traditional Analysis

Many approaches to the valuation of leases have been proposed. The one proposed by Bower[17] is representative of a broad class of net present value (NPV) models, and thus will bediscussed first. The Bower model (in this author’s notation) is:

NAL = C −H∑

t=1

Lt

(1 + r1)t+

H∑t=1

T Lt

(1 + r2)t−

H∑t=1

T Dt

(1 + r3)t

(12.1)−

H∑t=1

T It

(1 + r4)t+

H∑t=1

Ot (1 − T )

(1 + r5)t− SH

(1 + r6)H

2This question has been answered by the capital-budgeting methods generally applied to determine project acceptance.

Leasing 121

where

NAL = net advantage to leasingC = asset cost if purchasedH = life of the leaseLt = periodic lease paymentT = marginal tax rate on ordinary incomeDt = depreciation charged in period tIt = interest portion of loan paymentOt = operating maintenance cost in period tSH = realized after-tax salvage valuert = applicable discount rate

This model allows for discount rates that are different for each of the terms. However, Bowerconcludes that the appropriate discount rate is the firm’s cost of capital. With this in mind wedrop the term containing It since the interest tax shelter is implicitly contained in the cost ofcapital. The model then becomes

NAL = C −H∑

t=1

Lt

(1 + k)t+

H∑t=1

T Lt

(1 + k)t−

H∑t=1

T Dt

(1 + k)t

(12.2)+

H∑t=1

Ot (1 − T )

(1 + k)t− SH

(1 + k)H

or, by combining terms:

NAL = C −H∑

t=1

Lt (1 − T ) + T Dt − Ot (1 − T )

(1 + k)t− SH

(1 + k)H(12.3)

At this point let us consider a numerical example.

Example 12.1 A firm with 12 percent overall marginal cost of capital has decided to acquirean asset that, if purchased, would cost $100. This same asset may be leased for five years at anannual lease payment of $30. Operating maintenance is expected to be $1 a year, and straight-line depreciation to a zero salvage value would be used if the asset were to be purchased. Theprospective lessee is in the 48 percent marginal tax category. Applying equation (12.3) weobtain

NAL = 100 −5∑

t=1

30(1 − 0.48) + 20(0.48) − (1 − 0.48)

(1.12)t

= 100 −5∑

t=1

15.60 + 9.60 − 0.52

(1.12)t

= 100 −5∑

t=1

24.68

(1.12)t

= $11.03


Since $11.03 > 0, the leasing alternative is preferable to purchase of the asset. But, what ifthe net, realized after-tax salvage were estimated to be $20? In this case the NAL would beonly $6.61 and the lease would be less attractive.

Alternative Analysis3

The discussion so far has been limited to the case of the lessee. Also, the impact of the leaseon the lessee’s debt capacity has not yet been considered. Myers, Dill, and Bautista (MDB)developed a model that allowed for the impact of the lease on the lessee’s debt capacity [114].The MDB model assumes that the lessee borrows 100 percent of the tax shields created byinterest payments, lease payments, and depreciation. This debt constraint is used to eliminatethe debt displacement term normally used in the lease valuation equation, assuming that adollar of debt is displaced by a dollar of lease. Myers, Dill, and Bautista generalize theirmodel by removing the constraint that a dollar of lease displaces a dollar of debt. However,they constrain the proportion of debt displaced by a dollar of the lease, λ, to be equal to theproportion of the tax shields the lessee borrows against, γ .

Here the effects of allowing λ to vary from 1.0 are examined. At the same time it is as-sumed that the lessee borrows 100 percent of the tax shields (γ = 1.0) in order to main-tain an optimal capital structure. The generalized model, using MDB’s model as the startingpoint, is:4

V0 = 1 −H∑

t=1

Pt (1 − T ) + T bt

(1 + r − γ rT )t+

H−1∑t=0

H∑τ=t+1

rT Pτ (γ − λ)

(1 + r − γ rT )t+1(1 + r )τ−t(12.4)

where

V0 = value of the lease to the lesseePt = lease payment in period t (normalized by dividing by the purchase price of the asset

leased)bt = normalized depreciation forgone in period t if the asset is leased instead of purchasedr = lessee’s borrowing rateT = lessee’s marginal tax rate on incomeH = life of the leaseλ = proportion of debt displaced by a dollar of the leaseγ = proportion of the tax shields the lessee borrows against

Equation (12.4) follows MDB’s notation except for the inclusion of γ . Salvage value andforgone investment tax credit are assumed to be zero to simplify the model, and operatingmaintenance expenses absorbed by the lessor are also assumed to be zero.

The valuation model in (12.4), once again, is for the lessee. To determine the value of thelease to the lessor may be somewhat more controversial if for no other reason than little workin this area has been published. The claim of MDB is that the lessor’s valuation is the lessee’svaluation model multiplied by −1.0 to reflect the reverse direction of the cash flow. And forthe lessor they claim that λ is the proportion of debt supported by the lease, because the leaseis an investment to the lessor. (Remember that for the lessee, λ represents the proportion of

3This section is based on the extension to the MDB work by Perg and Herbst [124].4The derivation is contained in the appendix to this chapter.

Leasing 123

debt displaced by the lease.) The lessor’s λ will very likely be different from the lessee’s; somay the tax rate, T.

If λ were to be the proportion of the lessor’s debt supported by the lease (in (12.4) multipliedby −1.0 and with λ = γ ), however, then r would be the lessor’s borrowing rate, not the lessee’sborrowing rate. This presents a problem. If the lessor acts as financial intermediary, thenthe lessor’s borrowing rate will generally be less than the lessee’s borrowing rate, becausethe debt obligations of the lessor are less risky. This lower risk is due to the lessor’s equitycushion, the likely more liquid nature of the lessor’s obligations, and the diversification throughholding many different leases. Financial intermediaries also tend to keep the maturity of theirobligations shorter than their assets in order to take advantage of a yield curve, that is, anaverage, upward sloping. Myers, Dill, and Bautista made a valuable contribution to the literatureon leasing. The problem, however, of two different discount rates (the lessee’s and the lessor’s)make the MDB approach to determining the lessor’s valuation of the lease unsuitable. We willnow look at an alternative model for the lessor.

We base our approach to determining the value of a lease to the lessor on the fact that, tothe lessor, the lease is an investment. The NPV of the lease is equal to the present value ofits after-tax cash flows, valued at the after-tax discount rate appropriate for the level of riskassociated with investment in the lease, less the purchase price of the asset to be leased. It isassumed that the lessor is also a lender, a share value maximizer, and financial markets arecompetitive. From this we can say that the lessor will invest in bonds, including those of thelessee, until their after-tax return is equal to the lessor’s after-tax cost of capital appropriate tothe risk associated with holding the bonds.

If the lease is equivalent in risk5 to the lessee’s bonds, then the cost of capital associatedwith investing in the lease is equal to the cost of capital associated with investing in thosebonds: the after-tax borrowing rate of the lessee. If the risk is not equal, then the lessor’s costof capital for the lease equals χ times the after-tax borrowing rate of the lessee, where χ > 1.0if the lease is riskier than the lessee’s bonds, and χ < 1 if the lease is less risky than the bond.Recognizing this we obtain the model for the value of the lease to the lessor:

V0 =H∑

t=1

Pt (1 − T ) + T bt

[1 + Xr (1 − T )]t− 1 (12.5)

Here Pt , bt , and r are the same as in (12.4). But T now represents the lessor’s marginal tax rate,not the lessee’s, and X is a risk adjustment factor. The factor X can be reasonably expectedto be related to the debt displacement factor λ in (12.4). For example, if leases have financialcharacteristics similar to subordinated debt, then the lessor’s cost of capital for the lease willexceed his cost of capital for the bonds (χ > 1), and a dollar of lease will displace less thana dollar of bonds (λ < 1). On the other hand, if leases possess financial characteristics thatmake them senior to the firm’s bonds, we would expect χ < 1 and λ > 1. These possiblerelationships are discussed in the following analysis.

5Strictly speaking, this section deals with a certainty environment. However, in discussing analysis of leases it is necessary to bringrisk into consideration. The awkward alternative would be either to deal with leases under certainty here and bring in risk in a laterchapter or to postpone treatment of leasing until later, and out of this author’s desired sequence of topics. Risk treatment in a formalsense is deferred, however, until later chapters.


Table 12.1 H = 15, r = 0.10, straight-line depreciation γ = 1

Lessee Lessor

Lessee’s Lessee’s Value of leasemarginal break-eventax rate lease payment

to lessorχ = 1

λ = 1 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0 0.0175207 0.0283163T = 0.25 0.1288274 −0.0201286 0 0.0145813T = 0.50 0.1260179 −0.0414979 −0.0185998 0

χ = 1.2

λ = 0.8 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 −0.1045497 −0.0708275 −0.0378043T = 0.25 0.1337616 −0.0889678 −0.0569969 −0.0266946T = 0.50 0.1410994 −0.0389911 −0.0126358 0.0089385

χ = 1.0

λ = 0.8 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0 0.0175207 0.0283163T = 0.25 0.1337616 0.0174014 0.0326661 0.0401897T = 0.50 0.1410994 0.0732132 0.0812453 0.0782708

χ = 0.8

λ = 1.2 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0.1253474 0.1195513 0.1015017T = 0.25 0.1242443 0.0634667 0.0668896 0.0613131T = 0.50 0.1138490 −0.255119 −0.0088307 0.0035225

χ = 1

λ = 1.2 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0 0.0175207 0.0283163T = 0.25 0.1242443 −0.0549879 −0.0303416 −0.0092031T = 0.50 0.1138490 −0.1340555 −0.0991614 −0.0631539

Analysis

The factors that enter into the possible superiority of leasing over conventional financing are(1) the marginal tax rates (T ) of the lessor and the lessee and (2) the relationship betweenλ and χ . In order to explore the effects of the interactions of these variables on the value ofa lease, we compute the break-even lease payment of the lessee for various combinations ofthe lessee’s tax rate and λ. The break-even lease payment is then used to compute the value of thelease to the lessor for various combinations of the lessor’s tax rate and χ . The patterns of theresults are of major interest. They are not affected by varying H, r, or the use of accelerateddepreciation since these factors are common to both the lessee’s and the lessor’s valuationmodels. Therefore, only the results for H = 15, r = 10%, and straight-line depreciation arepresented in Table 12.1. In calculating the values for Table 12.1 the value of γ is kept equal to1.0 because it is thought the results will be more meaningful if the lessee always maintains anoptimal capital structure.

The topmost section of Table 12.1 confirms the conventional results that hold when leasesare financially equivalent to loans. Thus, if a dollar of lease displaces a dollar of debt (that is,

Leasing 125

λ = 1) and leases have the same risk as loans (χ = 1), then leasing is advantageous if, and onlyif, the lessee’s marginal tax rate is less than that of the lessor. The advantage occurs becausethe lease is tantamount to the sale of depreciation tax shields by the lessee to the lessor. If thelessor has a higher tax rate, then the value of the tax shields is greater to the lessor than thecost to the lessee for giving up the tax shields. The lower the lessee’s tax rate, and the higherthe lessor’s, the more mutually advantageous leasing becomes.

Results change dramatically once the assumption of financial equivalency is dropped. Thesecond and third panels of Table 12.1 illustrate what happens if a dollar of lease displaces only80 cents of debt (λ = 0.8). With this value of lambda the lessee’s break-even lease paymentactually increases as the tax rate increases, rather than decreasing as we might have expectedfrom the standard result at the top of the table. To understand this we refer to an importantpoint made by MDB. The lessee’s lease valuation rests upon the well-known Modigliani andMiller assumptions, which imply (among other things) that the only advantage to borrowingis the interest tax shield. If λ = 1.0, then leasing decreases the present value of the availabletax shields because the depreciation tax shield is surrendered. Therefore, as the lessee’s taxrate rises, his break-even lease payment falls. But if λ is sufficiently less than 1 (λ = 0.8 issufficiently less), then leasing actually increases the present value of the tax shields availablesince the increase in debt capacity — and therefore interest tax shields — more than outweighsthe loss of the depreciation tax shield. Because of this the lessee’s break-even lease paymentrises as the tax rate rises.

It follows that if λ is sufficiently less than 1, the value of leasing is a positive function of thetax rates of both the lessor and the lessee. How high the tax rates must be for lessor and lesseeto make leasing advantageous depends on how much riskier these debt-capacity-increasingleases are than the lending/borrowing alternative. If they are no more risky than lending (thatis, χ = 1), leasing is then advantageous for all positive tax rates, regardless of whether thelessee’s or lessor’s tax rate is higher.

The two panels at the bottom of Table 12.1 show the case in which $1 of lease displaces$1.20 of debt. In this case the lessee’s break-even lease payment falls even faster as his or hertax rate rises than it does when λ = 1. The reason this happens is that the lessee gives up thepresent value of reduced interest tax shields as the lessee’s debt capacity falls, in addition to thepresent value of the depreciation tax shield.6 Here the lessee’s marginal tax rate must be as lowas possible if leasing is to be advantageous. However, it does not follow that it is necessarilybest for the lessor’s tax rate to be as high as possible to make the lease most advantageous.When the debt-capacity-reducing effect of the lease (perhaps as a result of its senior claim onthe leased asset or assets) is reflected in lower risk and a lower cost of capital for the lessor (forexample, χ = 0.8), the greatest gain to leasing occurs if both the lessee and the lessor were tohave zero tax rates.

Implications

From the foregoing it is clear that the conventional condition for leasing to be advantageous —that the lessee’s tax rate be less than the lessor’s — applies only if leasing and borrowingare financially equivalent (that is, both λ = 1.0 and χ = 1.0). In the literature it has becometraditional to argue that financial equivalency should hold. However, for such equivalency toprevail would require that product differentiation of financial instruments not influence their

6Note that for T = 0 the lessee’s break-even lease payment is not affected by λ since the value of the tax shields is zero.


sales. Whether or not product differentiation can affect sales may not be settled to everyone’ssatisfaction. But until empirical evidence refuting parallel effects to those among consumergoods are published and confirmed, it would seem reasonable to expect differentiation toapply. If there really were no marketable differences between debentures, mortgage bonds,leases, and so on, one would be hard-pressed to explain why different financial instrumentsand arrangements persist.

If λ and χ may differ from 1.0, we may ask how they may do so. First, let us consider a lesseein sound financial condition; all debt is on an unsecured basis. A lease, however, is a highlysecured form of debt.7 This extra security — actual ownership of the physical asset — makesthe lease senior to the firm’s debt and impairs the security of the other debt.8 It follows thatif the riskiness of this debt is not going to increase (and it cannot if γ , the cost of capital,is to stay constant), total borrowing — including the obligation on the lease — must fall, andthis means λ > 1. The senior claim of the lease makes it somewhat less risky than the lessee’sdebt. To the extent that proceeds from the sale or re-lease of the asset, if it were to be seizedfrom the delinquent lessee, would be insufficient to cover the unpaid rentals and expensescaused by the default, the lessor becomes a general creditor. Altogether, this implies thatχ < 1.

Now, in contrast, let us consider a highly leveraged firm whose debt is virtually all secured.In such a case existing creditors would not perceive themselves as being harmed by leasing,relative to financing the asset with debt, because the new debt would have been secured inany case. But prospective creditors might find themselves more willing to make new leasesthan new secured loans because of their superior position with respect to recovering the assetthey hold title to if the lessee were to encounter financial difficulty. This enhanced recoveryability may cause such prospective creditors to be more willing to make leases rather thanloans (λ < 1.0) even though risk were to be increased (χ > 1.0).

Practical Perspective

It is probably reasonable to assume that much, if not most, leasing is done in situations whereλ = 1.0 and χ = 1.0. For high-quality, low-risk lessees it is likely that λ > 1.0 and χ < 1.0.From the bottom two panels of Table 12.1 we see that a low tax rate for the lessee is of thegreatest importance in this situation. It is also in the lessee’s vital interest to negotiate terms forthe lease that reflect the low risk associated with the lease. Because most standardized leasesare set up to protect the lessor from the higher-risk lessees, this would tend to limit low-riskprospective lessees to large, negotiated, and (usually) leveraged leases. Leveraged leases arediscussed in the following chapter.

In high-risk leases, λ < 1.0 and χ > 1.0. As shown by the middle two panels of Table 12.1,it may make leasing advantageous if the lessee were to have a high tax rate, provided thatλ is sufficiently less than 1. In this case expansion of his or her debt capacity is of primeconcern to the lessee; the foregoing analysis may understate the value of leasing becausethe value of additional debt capacity to the lessee may well exceed the present value ofthe tax shields since the future of the firm as a viable, going concern may well lie in thebalance.

7The lessor holds title to the asset and thus may recover it more easily than if title were held by the lessee.8If the asset were purchased, the title would be held by the firm that otherwise would be the lessee and it would serve to secure all thatfirm’s debt along with all its other assets.

Leasing 127


A financial lease is a financing arrangement in which the lessee purchases the use of an assetowned by the lessor. Asset use is separate from asset ownership. Lease analysis does notdetermine whether or not an asset should be acquired; it starts with the premise that the assetshould be acquired and attempts to determine if leasing provides an attractive alternative forfinancing the asset. The associated question is normally phrased as “lease or buy” or “lease orborrow.”

Traditional analysis of leases has focused almost exclusively on the problem of the lessee.However, to the lessor the lease is even more of a capital-budgeting decision; it is an investment.The lessor’s decision is not simply the mirror image of the lessee’s decision, because of differenttax rates and different risk implications of the same lease to each of them. Traditional analysishas assumed that leases were substitutes for debt, but it did not address the impact of leasingon the lessee’s debt capacity, or the measure of substitutability.

Myers, Dill, and Bautista extended the traditional analysis of leases to take into account theeffect of leasing on the debt capacity of the lessee. Perg and Herbst extended the analysis tothe lessor and integrated the analysis of lessee and lessor.

Appendix

The derivation of the generalized lease valuation equation for the lessee uses MDB’s notationexcept for introducing γ to represent the proportion of the tax shields that the lessee borrowsagainst. The starting point of the derivation is the debt constraint of the lessee. ϒt is the totaldebt of the firm in period t, L is the initial dollar value of the leased asset, Dt is the debt displacedin t per dollar of asset leased (Dt ≡ ∂Yt/∂L), St is the lessee’s total tax shield due to bookdepreciation on all assets owned in t, and Z is optimal borrowing excluding any contributionto debt capacity made by depreciation and interest tax shields.

Yt = λ

H∑τ = t+1

Pτ L

(1 + r )τ−t

(12A.1)= Z + γ

{ ∞∑τ = t+1

Sτ + rT Yτ

(1 + r )τ−t+

H∑τ = t+1

T Pτ L

(1 + r )τ−t

}

Differentiating ϒH−1 with respect to L and solving for DH−1:

DH−1 = −PH (λ − γ T ) − γ T bH

1 + r − γ rT(12A.2)

Substituting DH−1 into the expression for VH−1 and simplifying, yields the expression

VH−1 = −PH (1 − T ) − T bH

1 + r − γ rT+ rT PH (γ − λ)

(1 + r )(1 + r − γ rT )(12A.3)

Going back to the debt constraint, (12A.1) for YH−2, and differentiating with respect to L, andsolving for DH−2 we get

DH−2 = −PH−1(λ − γ T ) − γ T bH−1

1 + r − γ rT+ DH−1

(1 + r − γ rT )(12A.4)

Substituting (12A.2) into (12A.4) and then substituting (12A.4) into the expression for VH−2

and simplifying gives

VH−2 = −PH−1(1 − T ) − T bH−1 + VH−1

1 + r − γ rT+

H∑τ=H−1

rT Pτ (γ − λ)

(1 + r − γ rT )(1 + r )τ−H+2

(12A.5)

Leasing: Appendix 129

Clearly, this reasoning repeats, and in general (except for t = 0):

Vt = −Pt+1(1 − T ) − T bt+1 + Vt+1

1 + r − γ rT+

H∑τ = t+1

rT Pτ (γ − λ)

(1 + r − γ rT )(1 + r )τ−t(12A.6)

For t = 0:

V0 = −P1(1 − T ) − T b1 + V1

1 + r − γ rT+

H∑τ=1

rT Pτ (γ − λ)

(1 + r − γ rT )(1 + r )τ(12A.7)

By successive substitution, V1, V2, . . . , VH−1 are eliminated to obtain

V0 = 1 −H∑

t=1

−Pt (1 − T ) − T bt

(1 + r − γ rT )t+

H−1∑τ=0

H∑τ=t+1

rT Pτ (γ − λ)

(1 + r − γ rT )t+1(1 + r )τ−t(12A.8)

13

Leveraged Leases

DEFINITION AND CHARACTERISTICS

Analytical techniques discussed in previous chapters are applicable to analysis of the leaseversus purchase decision and to lease analysis in general. This chapter is concerned with theanalysis of a particular category of financial lease termed leveraged leases.

Leveraged leases are tax-sheltered financial leases in the sense that some of the returns tothe lessor are attributable to tax legislation intended to encourage capital investment. The USinvestment tax credit, for example, had its raison d’etre in encouraging capital investment.However, as Campanella [20] points out, companies that could benefit from new capital oftencannot benefit from the investment tax credit. Such firms might not be producing taxableincome, for instance, or carryover from previous tax shelter may make the credit useless.These firms find it advantageous to find a lessor who can use the tax benefits, and then arrangea lease in return for a lease cost that is lowered by those tax benefits. Leveraged leasing grewrapidly during the 1960s and is still significant. Changes in the tax laws have not, as somethought they might, negated the advantages of leveraged lease arrangements.

A leveraged lease is typically done through a trust arrangement. The lessor contributes asmall percentage (typically 20–40%) of the capital equipment cost, and the trust then borrowsthe balance from institutional investors on a nonrecourse basis to the owner. The loans tothe trust are secured by a first lien on the equipment, along with an assignment of the leaseand the lease payments. According to Campanella “a leveraged lease is a direct lease whereinthe lessor, through a trust, has borrowed a portion of the equipment cost to help finance thetransaction. Under the ‘true’ lease concept, the lessor retains a material equity and ownershipin the leased property and no option to purchase, other than a fair market value option at theexpiration of the lease term, is given to the lessee.” This is an important point because, withthe similar railroad trust certificate leases, the lessee is treated as the owner for tax purposes,and no tax advantage exists.

The leveraged lease trust is usually administered by a commercial bank, called the ownertrustee. This trustee takes title to the capital equipment and enters into the lease arrangementwith the actual user, the lessee. The lease is a long-term, full-payment “true” lease. With atrue lease the lessor retains a material equity ownership interest in the capital equipment andno option to purchase the equipment is provided to the lessee, at any price other than “fairmarket value.” The tax treatment on which leveraged leases are predicated rests on their beingtrue leases. After first servicing principal and interest on debt, the trustee remits any remainingfunds, pro rata, to the equity investors. It is the profitability of the leasing arrangement to theequity investors that we are primarily concerned with in this chapter.

Although the returns to the creditors of the trust normally may be straightforwardly calcu-lated as the yield on a fully amortized note, and the cost to the lessee may similarly be found,calculating the return to equity is much more troublesome. The reason for this difficulty is inthe mixed cash flows to equity that is characteristic of leveraged leases. A number of methodsfor computing the attractiveness of returns to equity on leveraged leases have been proposed.


Unfortunately, instead of resolving the issue, the various proposals themselves may have addedto the confusion. When one applies the methods to the typical leveraged lease, the results mayseemingly be contradictory. Leveraged leases provide us with an interesting application ofanalytical techniques that have been proposed for dealing with mixed cash flows. Furthermore,the assumption of certainty is not so unreasonable as it may be with most capital investmentprojects, because the returns are governed by the contractual terms of the lease.

METHODS: LEVERAGED LEASE ANALYSIS

The methods of analysis we shall consider are the net present value, the Wiar method, the sink-ing fund methods (including the Teichroew, Robichek, and Montalbano (TRM) approach),and the two-stage method. Since these methods have been introduced in previous chapters,it is assumed the reader is already familiar with the basic application of each. We shalltherefore analyze several example problems to provide a structure for comparing the variousmethods.

APPLICATION OF THE METHODS

Example 13.1 This example was proposed by Bierman [11] to illustrate the superiority ofthe net present value (NPV) method of analysis. The project has cash flows of −$400, $1100,and −$700 at the end of years t = 0, 1, and 2. Figure 13.1 contains plotted results of NPV fordifferent values of k, the cost of capital. The NPV function reaches a maximum at 27.0 percent.The project has IRR at 0 and 75 percent, and between these values the NPV is positive. Thissuggests that for 0 percent < k < 75 percent the project is acceptable.

That this is a mixed project is apparent if we decompose the cash flows into two subsequencesas follows: subprojects (a) and (c) are investments, whereas (b) and (d) are financing projects.In the particular subproject above, the investment returns are exactly offset by the correspond-ing financing costs. At rates between 0 and 75 percent the investment return dominates the

−.40 −.20 .00 .20 .40 .60 .80 1.00 1.20 1.40k–Axis−60

−40

−20

0

40

20

60

NPV Axis

NPV Function

Figure 13.1 Bierman example. Plot of investment characteristics

Leveraged Leases 133

RIC Function

RIC–Axis

k–Axis−.40 −.20

−.20

.00

.20

.40

.60

.80

1

.00 .20 .40 .60 .80 1.00 1.20 1.40

r � k

Figure 13.2 Bierman example. Plot of investment characteristics

financing cost. Consider the decomposition corresponding to 10 percent cost of capital, forexample:

t = 0 t = 1 t = 2(a) −400 +400 Return = 0 percent(b) +700 −700 Cost = 0 percent

or

t = 0 t = 1 t = 2(c) −400 700 Return = 75 percent(d) 400 −700 Cost = 75 percent

t = 0 t = 1 t = 2

(e) −400 463.64(f) 636.36 −700

The “loan” implicit in cash flow sequence (f) is at 10 percent. The NPV of sequence (e) is21.49 and has an IRR of 15.91 percent.

The TRM return on invested capital (RIC = r∗) is plotted against k in Figure 13.2. Ther∗ values for r∗ > k are for 0 percent < k < 75 percent — the same as obtained with theNPV approach. The initial investment method (IIM) yields rIIM values of 0 and 75 percent forcorresponding cost of capital percentages. Furthermore, the traditional sinking fund method(TSFM) yields rTSFM values of 0 and 75 percent. The two-stage method yields results consistentwith these. Next let us consider a project that has two more cash flows.

Example 13.2 This project has cash flows at end of years 0, 1, 2, 3, and 4 of −$1000, $2000,$2000, $2000, and −$5000, respectively. The NPV is positive between 0 and 400 percent. (TheNPV versus k graph in Figure 13.3 extends only to 140 percent.) The corresponding RIC or r ∗

plot is contained in Figure 13.4. Note that the r∗ is a double-valued function of k. In this casethe plot of interest is for positive values of r∗. We reject the negative values on the basis thatthe adjusted cash flow series from which we find r∗ for a given k value sums to greater than


NPV Axis � 101

NPV Function160

120

40

−40

−.40 −.20 .00 .20 .40 .60 .80

80

−80

0

1 1.20 1.40k−Axis

Figure 13.3 NPV–RIC conflict. Plot of investment characteristics

225

200

175

150

125

0

−25

−50

−75

−100

−125

−150

20 40 60 80 100 120 140k%

r � −100%

r%

RIC Function

Figure 13.4 NPV–RIC conflict

zero. In other words, the simple sum of the adjusted cash flows is positive, so we do not admitnegative returns even though mathematically they are acceptable.

Table 13.1 contains an outline of the TRM solution and some selected dual values ofr∗ for corresponding ki . As we have noted before, returns of less than −100 percent haveno economic significance because it is not possible to “lose more than we lose” with aproject.

In contrast to the Bierman project in Example 13.1, for this case the NPV and the RIC arenot totally in agreement. In fact, the RIC conflicts with the other, NPV-compatible methodsalso, as is to be expected. The NPV is greater than zero between the IRR boundary values of0 and 400 percent. The r∗ (RIC) is greater than k over much of the same range but not all


Table 13.1 TRM analysis of Example 13.2

Solution to projectOriginal cash flows balance equation

−1000 100.00000%2000 173.205082000 191.96396 = rmin

2000−5000

Some solution values are:

k r∗ r∗%

−5 −90.93% 125.500 −100.00 130.255 −107.75 135.25

10 −114.55 139.2520 −126.10 145.5030 −135.73 150.2540 −144.00 154.3450 −151.29 157.55

100 −178.96 167.73400 — 183.12

of it. At a k value of 200 percent, for instance, the r∗ is 176.65 percent whereas the NPV isstill positive. One could dismiss this conflict as irrelevant in a world where k cannot be sogreat, since the r∗ and NPV do agree over the range of k values that firms could, in fact, beexpected to have. However, because they are different over part of the range of k, we shouldabandon any complacency generated by the Bierman example and other examples involvingonly two cash flows beyond the initial outlay. The Bierman example underscores the hazardslurking in the shadows for those who would try to generalize from the three-cash-flow case tothe N-cash-flow case.

The conflict between the NPV and the RIC for large values of k is attributable in this caseto the assumption implicit in the NPV that the investment and financing rates are both equalto the cost of capital, whereas the RIC requires only that the financing rate be equal to ki .

Let us next apply the Wiar method (WM), the initial investment method (IIM), traditionalsinking fund method (TSFM), and the two-stage method (TSM) to this example and summarizethe results obtained in Table 13.2.

The various methods are in agreement that for k values of 10, 20, and 30 percent the projectis acceptable. The Wiar method could not properly be applied because it is predicated onoverall debt service and equity-return cash flow streams. Since no debt was assumed for thisexample, the method could not yield results other than the IRR values of 0 and 400 percent.This would not be a proper test of the method.

This is a mixed project, for which, to find r∗ for a given F , we must solve the equation

−(1 + k)(1 + r∗)3 + 2(1 + k)(1 + r∗)2 + 2(1 + k)(1 + r∗) + [2(1 + k) − 5] = 0


Table 13.2 Summary of analytical results for Example 13.2

For k = 10%NPV method $558.64 AcceptTRM RIC 139.25% AcceptWiar method equity return InapplicableIIM return 17.06% AcceptTSFM return 71.45% AcceptTwo-stage measures 0.55 years, 4.58% Accepta

For k = 20%NPV method $801.70 AcceptTRM RIC 145.50% AcceptWiar method equity return InapplicableIIM return 34.57% AcceptTSFM return 109.09% AcceptTwo-stage measures 0.60 years, 9.05% Accept

For k = 30%NPV method $881.59 AcceptTRM RIC 150.25% AcceptWiar method equity return InapplicableIIM return 52.01% AcceptTSFM return 125.70% AcceptTwo-stage measures 0.65 years, 13.47% Accepta

a Assumes management requirement for capital recovery is not less thanthese fractions of a year.

Table 13.3 Childs and Gridley leveraged lease example — a la Wiar’s analysis, borrow $80 at 8% for15 Years, invest $20 equity, tax rate 48%

NetMortgage Effect Effect Effect on Mortgage after-tax

Lease interest on earnings on earnings principal cash flowt Year payment Depreciation payment before taxes taxes after taxes payment (to equity)

0 1 0 0 0 0 0 0 0 −20.001 2 9.64 16.67 6.40 −13.43 −13.45a 0.02 2.95 13.74a

2 3 9.64 15.15 6.16 −11.67 −5.60 −6.07 3.19 5.893 4 9.64 13.64 5.91 −9.91 −4.75 −5.16 3.44 5.044 5 9.64 12.12 5.63 −8.11 −3.90 −4.21 3.72 4.195 6 9.64 10.61 5.34 −6.31 −3.03 −3.28 4.01 3.326 7 9.64 9.09 5.02 −4.47 −2.14 −2.33 4.33 2.437 8 9.64 7.57 4.67 −2.60 −1.25 −1.35 4.68 1.548 9 9.64 6.06 4.29 −0.71 −0.34 −0.37 5.06 0.639 10 9.64 4.09 3.89 1.66 0.80 0.86 5.46 −0.51

10 11 9.64 0 3.45 6.19 2.97 3.22 5.90 −2.6811 12 9.64 0 2.98 6.66 3.19 3.47 6.37 −2.9012 13 9.64 0 2.47 7.17 3.45 3.72 6.88 −3.1613 14 9.64 0 1.92 7.72 3.70 4.02 7.43 −3.4114 15 9.64 0 1.33 8.31 3.99 4.32 8.02 −3.7015 16 9.64 0 0.68 8.96 4.30 4.66 8.56 −3.90

Total 144.60 95.00 60.14 −10.54 −12.06 1.52 80.00 −3.4816 Residual 0 5.00 0 −5.00 −2.40 −2.60 0 2.40

Total 144.60 100.00 60.14 −15.54 −14.46 −1.08 80.00 −1.08

a Includes 7% investment tax credit on the $100 investment.


ANALYSIS OF A TYPICAL LEVERAGED LEASE(CHILDS AND GRIDLEY)

Example 13.3 This example, first proposed by Childs and Gridley [22], has since beendiscussed extensively in the literature. Table 13.3 contains the salient data for the Childsand Gridley leveraged lease. A 7 percent investment tax credit applies to the equipment.Depreciation is calculated using sum-of-the-years’ digits method and an 11-year lifetime. Theequipment is depreciated to a salvage-for-tax amount of 5 percent of the original depreciablevalue. The depreciation for t = 9 (year 10) is shown as $4.09; the calculated amount is $4.55.This difference is due to the requirement that $5 remain for write-off at t = 15 (year 16). Thecolumn of data of most concern to the equity investor is the “Net-after-tax cash flow”column.It is this column on which our analysis will focus, although the Wiar method incorporatesadditional information.

This series of cash flows is a mixed project in the TRM sense. The rmin can be calculated fromthe series by taking the cash flows for t = 0 through t = 8 and solving for the IRR. Verificationthat the result obtained, 28.5038 percent, is, in fact, the project rmin can be obtained byevaluating the project balance equations corresponding to the preceding cash flows at thisrate; they will all be less than or equal to zero.

The Wiar analysis [ ] can be carried out by evaluating the equation

20(1 + rw)16 − 100(1.045220)16 − 80(1.0416)16

to find that rw = 5.80736 percent after tax (or 11.16799 percent pre-tax). The rate of 4.52220percent is the return on the overall cash flow to the leveraged lease, whereas the 4.16 percent

Table 13.4 Analysis of Childs and Gridley leveraged lease example — a la Wiar

NetEffect on Effect Effect on after-tax

Lease earnings on earnings cash flowt Year payment Depreciation before taxes taxes after taxes (overall lease)

0 1 0 0 0 0 0 −100.001 2 9.64 16.67 −7.03 −10.37a −3.34a 20.01a

2 3 9.64 15.15 −5.51 −2.64 −2.87 12.283 4 9.64 13.64 −4.00 −1.92 −2.08 11.564 5 9.64 12.12 −2.48 −1.19 −1.29 10.835 6 9.64 10.61 −0.97 −0.47 −0.50 10.116 7 9.64 9.09 0.55 0.26 0.29 9.387 8 9.64 7.57 2.07 0.99 1.08 8.658 9 9.64 6.06 3.58 1.72 1.86 7.929 10 9.64 4.09 5.55 2.66 2.89 6.98

10 11 9.64 0 9.64 4.63 5.01 5.0111 12 9.64 0 9.64 4.63 5.01 5.0112 13 9.64 0 9.64 4.63 5.01 5.0113 14 9.64 0 9.64 4.63 5.01 5.0114 15 9.64 0 9.64 4.63 5.01 5.0115 16 9.64 0 9.64 4.63 5.01 5.01

Total 144.60 95.00 49.60 16.82 32.78 7.7816 Residual 0 5.00 −5.00 −2.40 2.60 2.40

Total 144.60 100.00 44.60 14.42 35.38 10.18

a Includes 7 percent investment tax credit on the $100 investment.


30

25

20

15

10

0

−5

−10

5.81

Two−stage Payback

0

5

10

Years

k%

TRM r∗

5

10

15 20 25

Wiar Rate

TRM r∗

r% Two−stage “Loan” Rate

r � k

Trad

ition

alSi

nkin

gFu

nd

Meth

od Rate

Initi

alIn

vestm

ent M

ethod

Rate

Figure 13.5 Several measures on the Childs and Gridley leveraged lease returns

rate is the after-tax interest on the debt financing. The overall cash flows to this lease aredisplayed in Table 13.4. Note that the Wiar rate does not depend on the lessor’s cost of capital.The decision to accept the leasing project as an investment or not is made by comparing therw with the lessor’s cut-off or hurdle rate.

The TRM–RIC function for the Childs and Gridley leveraged lease is plotted in Figure13.5, along with the initial investment, traditional sinking fund, and two-stage measures.The functions are plotted on the same graph to illustrate the consistency of the methods: allbut the Wiar equity return would mark the project acceptable for values of k < 25 percent,and unacceptable for k > 25 percent. The apparent symmetry of the “loan” phase mea-sure of the two-stage method with the traditional sinking fund method is interesting. Thelease is acceptable for k < 25 percent with the two-stage method, provided that the pay-back (recovery of investment plus opportunity cost) is acceptable, since the implicit loanembedded in the later cash flows is at a rate less than k in this range. The fact that thetwo-stage payback goes to infinite value after k reaches 28.5 percent is noteworthy, forthat is the value of rmin in the TRM solution. An infinite value of the two-stage paybackmeans that the initial investment and opportunity cost of the funds committed will neverbe fully recovered. Any such payback beyond the project’s economic life may be similarlyinterpreted.


CONCLUSION

Leveraged leases provide perhaps the closest “real-world” approximation to the certainty envi-ronment we have considered to exist for capital-budgeting analysis to this point. Furthermore,growth in the use of leveraged leasing and the magnitude of funds involved make leveragedlease profitability analysis a significant area for investigation.

In this chapter we examined leveraged leases as an interesting and practical application ofanalytical methods designed to deal with mixed cash flows. The results obtained suggest that anaccept/reject decision may be correctly reached by several methods: NPV; two-stage method;initial investment method; traditional sinking fund method; and the TRM approach.

The Wiar method would reject projects acceptable under the other criteria. In fact, the Wiarrate is truly internal to the project — the cost of capital does not enter into its calculation at all.It is solely a function of the IRR to the overall lease cash flows and the interest rate on the debtcomponent of the lease. Because of this the Wiar rate, rw, is plotted on a straight line parallelto the k-axis.

14

Alternative Investment Measures

This chapter introduces several additional measures that may be applied to capital-budgetingprojects. They include the geometric mean rate of return, the average discounted rate of return,Boulding’s time spread, and Macaulay’s duration.

In order to simplify the exposition, in this chapter capital outlays will be restricted to theinitial outlay, and subsequent net cash flows will be assumed to be nonnegative. All cash flowsare assumed to occur at the end of the corresponding time periods. In other words, only simpleinvestments in the sense defined in Chapter 10 will be considered.

ADDITIONAL RATE OF RETURN MEASURES

Geometric Mean Rate of Return

In contrast to the more generally known arithmetic mean, the geometric mean is obtained bytaking the nth root of the product of the n items, rather than dividing their sum by the number n.When considering the average of interest or growth rates over a period of time, the geometricmean is considered more appropriate than the arithmetic mean because it takes into accountthe effects of period-to-period compounding that the arithmetic mean ignores. Because of thetime-value of money, it is, for example, not correct to say that a deposit of $1000 that earns4 percent the first year, 8 percent the second year, and 9 percent the third year has earned anaverage rate of 7 percent over the three-year period. The amount earned is $224.29 on the $1000principal. The uniform, average annual rate for which $1000 will grow to $1224.29 over threeyears is not 7 percent but 6.98 percent. For a three-year period, this is not a great difference, butthe error of the arithmetic mean over the geometric mean becomes greater the longer the timespan covered. For example, if a principal amount can earn 4 percent for 10 periods, 8 percentfor the next 10 periods, and 9 percent for the last 10 periods of a 30-year investment life, thegeometric mean rate is 6.16 percent, whereas the arithmetic mean is 7 percent.

The geometric mean rate of return on an investment, rg, may be defined by

rg =[

n∏t=1

(1 + yt )

]1/n

− 1 (14.1)

where

yt = Et − Et−1 + Rt

Et−1(14.2)

with Rt the cash flow at end of period t and the Et representing the market value of the investmentat end of period t. Note that this formulation measures the period-to-period changes in equityvalue over the life of the investment, even though the overall change in equity value is notrealized until disposition of the asset at the end of its useful life.


Average Discounted Rate of Return

The average discounted rate of return, ra, may be defined as

ra =

n∑t=1

yt (1 + ra)−t

n∑t=1

(1 + ra)−t

(14.3)

where yt is as defined in equation (14.2). Since ra is defined recursively, it would appear difficultand time consuming to compute. However, existing computer programs for finding IRR canbe used. Equation (14.3) is equivalent to

1 − (1 + ra)−n =n∑

t=1

yt (1 + ra)−t

sincen∑

t=1

(1 + ra)−t = 1 − (1 + ra)−n

ra

which is the equation an¬ra , for the present value of an annuity of one for n periods at rate ra.Rearranging the terms, we obtain

1 =n−1∑t=1

yt (1 + ra)−t + (1 + yn)(1 + ra)−n (14.4)

To solve for ra in equation (14.4), we need only recognize that this is the equation for an“investment” of $1, returning y, . . . yn−1, and a final return of 1 + yn at time n, and applyany computer program for finding IRR. This equation is the same as that for finding yield tomaturity for a variable-payment bond that sells for its par value of $1, and at maturity returnsthe $1 with the final payment of interest.

Robert R. Trippi illustrated the application and usefulness of geometric mean return andaverage discounted rate of return for measuring the returns on investments that undergo changesin market value over their useful lives [162]. Trippi employed 12 illustrative examples in hisexposition. Here we will examine several similar examples to demonstrate the techniques andcompare rg with ra and the IRR, which is denoted by rc for “conventional” rate of return inTrippi’s notation.

To illustrate the meaning and calculation of rg and ra, let us consider the following example,which requires a $1000 initial investment and returns the net, after-tax cash flows indicated.In addition, the market value of the asset changes from period to period.

Example 14.1

t = 1 t = 2 t = 3 t = 4 t = 5Cash flow $200 200 300 400 200Market value $1100 1200 1250 1300 1325

Assuming the asset will be disposed of at the end of year 5, the $1325 market value willbe realized as a positive salvage value and the total cash flow increased by this amount to$1525. For simplicity we assume that the market values are the after-tax proceeds that would

Alternative Investment Measures 143

be realized if the asset were to be disposed of at the end of any indicated year. The IRR (heredenoted rc) is 28.5044 percent, on the cash flow series −$1000, $200, $200, $300, $400, and$1525.

To calculate rg we employ equation (14.2) n = 5 times, substituting into equation (14.1).Thus

y1 = E1 − E0 + R1

E0= 1100 − 1000 + 200

1000= 0.3000

y2 = E2 − E1 + R2

E1= 1200 − 1100 + 200

1100= 0.2727

and so on. Applying equation (14.1), we obtain

rg = [(1.3)(1.27273)(1.29166)(1.36)(1.17308)]1/5 − 1 = 27.8024 percent

Now, to find ra we employ equation (14.4). This results in the following series to which weapply the procedure for finding the IRR

−1 0.3 0.27273 0.29167 0.36 1.17308

Note that this is identical to the series for a hypothetical bond that sells for $1, yields thevarious amounts defined by equation (14.2), and returns the original investment in a balloonpayment of one at t = 5. Thus we find that

ra = 28.6193 percent

For this example, ra > rc > rg. This will not always be the case. In fact, several differentcases were identified by Trippi. Table 14.1 displays the relationship between rg, ra, and rc forsome different patterns of cash flow and market value. It is important to note that the IRR, rc,depends only on the pattern of cash flows, whereas both rg and ra depend on the pattern ofchanges in market value in addition to cash flows. Thus many projects having the same IRRwill have different rg and still different ra.

Trippi proposed the average discounted rate of return as an alternative to the geometricmean rate of return for incorporating the change in equity value and its pattern in measuringthe return on investment. This is something that had received more attention in the area ofsecurities analysis and portfolio management than in capital budgeting. However, the conceptis as applicable to capital investment projects as it is to investments in securities.

Shortly after the appearance of Trippi’s paper, Peter Bacon, Robert Haessler, and RichardWilliams [4] found an interesting counterintuitive example, an asset that produces no cashflow prior to its sale, doubles in value in the first year, and then in the second year declinesto its original value. To quote Bacon, Haessler, and Williams: “The marginal return in year 1is 100% and year 2 is −50%. Calculating the geometric mean and the internal rate of returnor just relying on intuition all indicate that the true return is zero. . . . However, when . . . ra

is computed, the result is 36% [4].” As they go on to point out, the problem with Trippi’sra is that it discounts the marginal returns in addition to averaging them. Since percentageincreases are weighted the same as percentage decreases, the problem is not readily apparentwith those investments that only increase or decrease in market value. However, with assetsthat first increase and then decrease, or vice versa, the investment return is misrepresented byra. The fact that early marginal percentage changes are discounted less than those occurringlater further contributes to the problem.


Table 14.1 Examples of rg, ra and rc for various cases, each requiring $1000

Case no. 1 2 3 4 5 rg% ra% rc (IRR)%

1 Cash flow 0 100 200 300 400Market value 1000 1000 1000 1000 1000 19.1596 16.9082 16.9082






7 Cash flow 200 200 200 200 200Market value 1000 900 700 400 0 −10.7682 −10.5832 0.0000






In his reply to Bacon, Haessler, and Williams, Trippi emphasized [163] that his “primaryintent . . . was not to advocate universal acceptance of one measure over the others, but ratherto demonstrate the general difficulties and frequent lack of conformity of each of the mea-sures. . . . Clearly some non-unity marginal rate of substitution of present for future undis-tributed wealth is likely to apply . . . this phenomenon being totally lost with the conventionalmeasures.”

Thus, although caution must be exercised, the geometric mean rate of return and averagediscounted rate of return may be considered adjuncts in the process of investment evaluation,particularly where there is unrealized (monotonic) increase or decrease over the life of theinvestment. The methods yielding rg and ra should perhaps not be used at all for investmentscharacterized by nonmonotonically changing market value.

Chapter 15 takes up the topic of abandonment value in capital budgeting. There it may beseen that an alternative to ra exists. For now, we shall go on to finish this chapter with the topicsof unrecovered investment, duration, and time spread.

TIME-RELATED MEASURES IN INVESTMENT ANALYSIS

Although the literature has generally concentrated on other aspects of the topic of capitalbudgeting, time-related measures are useful and may provide additional insight. Here we shallconsider two time-related measures: Macaulay’s duration and Boulding’s time spread. Thelatter, although identical to the modern actuaries’ equated time, was proposed in 1936.


Boulding’s Time Spread

Kenneth E. Boulding [15] proposed time spread (TS) as a measure of the average time intervalelapsing between sets of capital outlays and returns. For investments having a single initialoutlay at time t = 0, Boulding’s time spread is defined by

N∑t=1

Rt =N∑

t=1

Rt (1 + r )TS−t (14.5)

Since (1 + r )TS is constant for a given r and TS,

N∑t=1

Rt = (1 + r )TSN∑

t=1

Rt (1 + r )−t (14.5a)

so that

(1 + r )TS =

N∑t=1

Rt

N∑t=1

Rt (1 + r )−t

(14.5b)

and

TS = log

N∑t=1

Rt

N∑t=1

Rt (1 + r )−t

÷ log(1 + r ) (14.6)

In the case of r = r∗ (the IRR), it can be shown (as Boulding did) that

TS = log

N∑t=1

Rt

R0

÷ log(1 + r ) (14.7)

where R0 is the initial (and only) capital outlay for the project. The proof of this involves therecognition that, for the IRR, the initial outlay is equal to the sum of the discounted cash flowsat discount rate r∗.

When used with the IRR, time spread shows how long the initial investment remains investedon average at rate r∗ (the IRR). Time spread provides the point in time at which a single amount,equal to the undiscounted sum of cash flows, would be equivalent to the individual cash flowsat intervals over the life of the investment, at a given rate of interest. It is a measure of theaverage time between capital outlays and net cash receipts. In cases for which the only cashoutlay occurs at t = 0, time spread therefore measures the average time elapsed to receive thenet cash flows over the interval t = 1 through t = N . The following example will clarify thisand set the concept.

Consider case 1 in Table 14.1. This project, costing $1000, yields total cash flows of $2000over its life, including $1000 realized on disposition of the asset at t = 5. Time spread for thisproject is TS = 4.4370 years, and the IRR is rc = 16.9082 percent. The individual net cashflows could be replaced by a single cash flow of $2000 (equal to the undiscounted sum of cash


inflows) at t = TS: the equation

R0 =N∑

t=1

Rt (1 + r∗)−TS = (1 + r∗)−TSN∑

t=1

Rt (14.8)

follows directly from equation (14.7). Substituting the parameters of case 1, we obtain

$1000 = ((1.169082)−4.4370)$2000 = $1000

Thus the entire cash flow series beyond the initial outlay may be replaced by a single amountequal to its sum at t = TS. Similarly, rates other than the IRR may be used, with the sameinterpretation, although for different rates different values for TS will be obtained.

Macaulay’s Duration

Following soon after Boulding, Frederick R. Macaulay [94] developed the concept of durationas an alternative to the conventional time measure for bonds — the term to maturity. For thoseinvestments with a single cash outlay at t = 0, duration is a weighted average of repaymenttimes (or dates) with weights equal to the present values of the cash flows at their respectivedates. Equation (14.9) defines duration:

D =

N∑t=1

t Rt (1 + r )−t

N∑t=1

Rt (1 + r )−t

(14.9)

For r = r∗, the IRR, the denominator, by definition of the IRR, is equal to R0, the initial outlay.Therefore, for r∗,

D =

N∑t=1

t Rt (1 + r∗)−t

R0(14.10)

Calculation of duration is straightforward, even if tedious. As Durand has pointed out [35],even though different, D converges to the same value as TS when N is finite and the discountrate approaches zero. It might be added that when the only cash flow is at t = N , D = TS = N .

The history of development and application of duration is very well described by Weil, who,among other things, points out that Hicks’s elasticity of capital with respect to discount factorsis equivalent to duration [168], although apparently developed independently and somewhatlater. Weil also mentions Tjalling C. Koopmans’s (1942, unpublished) paper on matching lifeinsurance assets and liabilities to “immunize” the company against effects of interest ratechanges. At the time he wrote it Koopmans was employed by Penn Mutual Life InsuranceCompany [168, p. 591]. Credit for the seminal idea on immunization is often awarded toRedington [128] for his later contribution, which appears to be the earliest published paper onthe topic although appearing a decade after Koopmans’s paper was written.

Like time spread, duration provides a useful adjunct measure to be used in capital budgetingalthough developed for another purpose. For case 1 of Table 14.1, D = 4.3696. This providesthe “average” time that elapses for a dollar of present value to be received from this project.This is somewhat less than the time spread value. It may be shown that D ≤ TS.


$2,000

1,500

500

–500

1,000

0

0

–1,000

1 2 3 4 5Time

r � 0%

r � 0%

r � 30%

r � 30%

r � r∗

r � r∗

Figure 14.1 Unrecovered investment at end of time period indicated

Unrecovered Investment

The unrecovered investment of a capital-budgeting project is defined by

U = R0(1 + r )T −T∑

t=1

Rt (1 + r )T −t (14.11)

With the IRR it is implicitly assumed that the IRR rate, r∗, is earned on the unrecoveredprincipal as measured at the beginning of each period over the project life. This was made ex-plicit in the treatment of conflicting IRR–NPV rankings in Chapter 8. With r∗, the unrecoveredinvestment will be exactly zero at t = N . For t < N and with r = r∗, U > 0; the unrecov-ered investment will be positive prior to the end of the project life. For t < N and r < r∗, Ubecomes negative prior to t = N . Figure 14.1 shows the graphs for U as a function of t andr . Since it is assumed that cash flows occur only at the end of each period, discontinuitiesoccur at these points. For r = r∗ the unrecovered investment becomes zero after the end ofperiod 4. At a higher rate, r = 30 percent, for example, the cash flows were not even adequatefor paying the “interest” on the unrecovered investment by t = 5, so that the investment isnot fully recovered by that time. In fact, at t = 5 the unrecovered investment is greater thanthe initial investment at t = 0. For rates less than r∗ the investment is fully recovered priorto t = 5.

Table 14.2, which employs the same type of component breakdown used in Chapter 8,illustrates the concept of unrecovered investment. The table displays unrecovered investmentunder the heading “Ending principal” for the beginning-of-period points. Table 14.2 treats theinvestment case 1 of Table 14.1 as though it were a loan, and unrecovered investment is seento be the “loan principal.”


Table 14.2 Component breakdown of cash flows (amounts rounded to nearestcents) for r = r∗

Interest onBeginning beginning Principal Total Endingprincipal principal repayment payment principal

1 $1000.00 $169.08 $0 $0 $1169.082 1169.08 197.67 0 100 1266.753 1266.75 214.18 0 200 1280.934 1280.93 216.58 83.42 300 1197.515 1197.51 202.48 1400.00 1400 0

Note: It is assumed that end-of-period payments are composed of interest at r∗ = 16.9082% onbeginning-of-period principal plus (if there is an excess over the interest) principal repayment.

Unrecovered investment, U , is related to payback period P. The point at which U becomeszero corresponds to the time at which the investment has been fully recovered. For r = 0,the conventionally defined payback period is obtained. (With the conventional calculation ofpayback, the assumption that cash flows occur only at the end of each period is violated; oncethe payback period has been bracketed, the end-of-period cash flow at the further time periodis treated as if it occurred uniformly over the period. The formula for U does not violate theassumption of end-of-period cash receipts, and to this extent there is a difference with paybackcalculation.)

The concept and measure of unrecovered investment are useful as an adjunct to other capital-budgeting measures. It serves to focus attention on the nature of the process of investmentrecovery implicit within other measures.


The additional investment measures presented in this chapter provide useful adjuncts to othercapital-budgeting measures. They illustrate that factors other than cash flow alone may be ofinterest; as, for instance, Trippi’s average rate of return that, to some extent, incorporates theappreciation in asset value that is not realized until the end of the project life. Used alonethey are not especially useful; but used with other measurements of capital-budgeting projectcharacteristics they can provide additional insight, thereby facilitating better decision-making.

Perhaps Durand states the case as well as anyone when he says:

From all this I conclude that we need to take a far broader view of capital budgeting than we have in thepast. We have squandered altogether too much effort on a futile search for that elusive will-o’-the-wispthe one and only index of profitability; and we have lost valuable perspective thereby [35, p. 191].

15

Project Abandonment Analysis

Up to this point it has been implicitly assumed that capital-budgeting projects, if accepted forinvestment, would (for better or for worse) be held tenaciously until t = N . This is undulyrestrictive and unrealistic. It violates the realities of capital-budgeting practice; capital invest-ments are often abandoned prior to termination of their theoretical maximum useful lives. Inthis chapter we formally consider abandonment prior to the end of project life.

THE ROBICHEK–VAN HORNE ANALYSIS

Alexander A. Robichek and James C. Van Horne (R–VH) presented an algorithm for determin-ing if and when a capital investment project should be abandoned prior to the end of its usefullife at t = N [130]. The original procedure was modified somewhat [131] after Edward A.Dyl and Hugh W. Long [38] showed that the original algorithm could, in some circumstances,break down.

The R–VH paper became widely known and cited, perhaps because it was the first paper ina major journal to have dealt with the subject of abandonment value. It provided an importantprod in the process of awakening academics to a problem that in practice has always been afactor considered by practitioners, but for which little mention was to be found in the literature.In order to facilitate their analysis R–VH assumed, that (1) an adequate estimate of the firm’scost of capital exists; (2) there is no capital rationing; and (3) a unique IRR exists for theprojects considered. Assumption (3) may be satisfied by considering only simple investments,in the sense defined in Chapter 10.

The R–VH algorithm (corrected to satisfy the Dyl–Long critique) is stated as:

(A) Compute PVτ ·a for a = n, where

PVτ ·a =a∑

t=τ+1

ECt ·τ(1 + k)(t−τ )

+ AVa·τ(1 + k)(a−τ )

(B) If PVτ ·n > AVτ , continue to hold project and evaluate it again at time τ + 1, based uponexpectation at that time.

(C) If PVτ ·n < AVτ , compute PVτ ·a for a = n − 1.(D) Compare PVτ ·n−i with AVτ as in (B) and (C) above. Continue this procedure until either

the decision to hold is reached or a = τ + 1.(E) If PVτ ·a ≤ AV for all τ + 1 ≤ a ≤ n, then abandon project at time τ . . . .

where

ECt ·τ = expected cash flow in year t as of year τ .AVt = abandonment value in year t.ACt = “actual” simulated cash flow in year t [131, p. 96].


This algorithm as it is stated appears to have been written to deal with the timing of abandonmentfor a project after it had been accepted. However, R–VH suggest that their rule might beextended to ex ante (prior to acceptance) project analysis. This chapter is primarily concernedwith such ex ante capital investment project analysis. This emphasis, however, should not beconstrued to imply that continued project review, such as that suggested in the R–VH algorithm,is any less important.

Step (A) of the R–VH algorithm defines the present value at time reference point τ as thediscounted sum of all cash flows occurring from the period immediately following τ to theend of the project life plus the present value at τ of the expected salvage value to be receivedat the end of the project life.

Step (B) states that we should keep the project if the present value of continuing to do so asdefined in step (A) is greater than the present value of salvage. Note that these present valuesare at time = τ . Present values are normally calculated for t = 0.

Step (C) requires that we perform additional calculation and analysis before abandoningthe project, even though the salvage at time τ is greater than or equal to the present value ofexpected cash flows from time τ + 1 to the end of the project’s maximum useful life at t = N .This step is necessary in order to avoid premature abandonment of the project; the NPV maypossibly be increased by holding the project for one or more additional time periods eventhough it will not be held all the way to t = N .

Step (D) specifies that the analysis in steps (A) through (C) inclusive be repeated until aperiod is found for which, in light of the expected returns as of today, the decision is to holdon to the project, or else we get to a = τ + 1. In the latter case the decision is to hold on forthe current period, then abandon.

Finally, step (E) prescribes that if the salvage value at any point in time τ exceeds the presentvalues that are potentially to be obtained by holding on to the project, it should be abandonedat time τ .

As specified above, the R–VH algorithm seems designed particularly for ongoing, periodicanalysis of capital investments with a view to whether they should be abandoned or kept inservice for another time period. However, although it may not be clear from the wording ofthe procedure, the R–VH approach is suitable for analyzing capital-budgeting projects ex anteas well as the ex post, which the R–VH paper appears to stress. In such cases it could beused to help answer these questions. (1) What is the optimal period to keep the project if it isaccepted? (2) What is the expected present value if the project were to be accepted and heldfor the optimal period and no longer?

The R–VH paper was useful in calling attention to the problem of project abandonment.However, it is equivalent to a “dual” formulation of the Terborgh–MAPI method discussed inChapter 9. Such formulation yields, instead of the MAPI “adverse minimum,” a “propitiousmaximum” NPV, if NPV is the measure of project acceptability or desirability employed.Associated with this maximum is the optimum number of years over which the project, ifaccepted, would be held. To develop the methodology, we need, in addition to the R–VHassumptions, the assumption that we have or can obtain reliable estimates of salvage valuesfor time periods between the adoption of the project and the end of its useful life at t = N .For those fairly standard types of equipment for which there is a well-developed secondarymarket, this should be a reasonable assumption.1 On the other hand, for plant and for custom-made equipment this assumption will in general lack the reliability associated with the former

1For example, we could develop estimates of salvage value deterioration gradients by careful, systematic analysis of trade publicationscarrying advertisements for used equipment and by consulting with dealers specializing in such equipment.

Project Abandonment Analysis 151

category. For a thorough treatment of the problems associated with extraction of such estimates,the reader is referred to Terborgh’s Dynamic Equipment Policy [151].

The following two sections illustrate application of the modified MAPI procedure to threecapital-budgeting projects and compare results with the R–VH method.

AN ALTERNATIVE METHOD: A PARABLE2

To highlight the points presented above an example will be employed. The capital projectcommittee of Typical Manufacturing Company (TMC) is considering which of three mutuallyexclusive production machines it should purchase to perform certain operations on a newproduct that the company has decided to add to its line. The machines are all of standard design,and hundreds of various vintages are in use across the nation. The following information hasbeen presented to the committee:

Purchase For the year indicatedPrice net after-tax cash benefits

Machine A $2000 $600 $600 $600 $600 $100 $100 $100Machine B 2000 700 600 500 400 300 200 100Machine C 1000 100 200 300 400 300 200 100Year 1 2 3 4 5 6 7

All three machines have useful lives of seven years. Machine A has an estimated salvagevalue of $419.43, B $419.43, and C $478.30 at the end of seven years.

The recommendation provided to the committee is that machine C be purchased because, atthe firm’s 10 percent cost of capital, it has an NPV of $350.69, while A and B have respectivelyNPVs of $286.99 and $346.75. Since most members of the committee are well versed in thetraditional finance literature concerned with capital budgeting, C is chosen for purchase withlittle discussion. Of course, there is some argument over the significance of the slight edge inNPV that C has over B, but A is out of the running from the beginning.

Has the committee selected wisely? No! “But,” the reply will be, “by selecting the projectwith the highest net present value we are assuring the maximum increase in the value of equity.”However, there is more to the story. In the approach to project selection that was followed,no attention was paid to salvage value prior to the end of each machine’s useful life. In theexample presented here the salvage values for A and B represent 20 percent per annum declinesfrom the prior year’s value, beginning with the purchase price paid for each, while that for Crepresents a 10 percent per annum decline.

Tables 15.1(a)–(c) present alternative calculations that might have been performed for ma-chines A, B, and C. Readers familiar with the MAPI method for replacement evaluationpresented by Terborgh will note some similarity in that his method also considers interme-diate salvage values. However, the MAPI method was developed primarily for replacementdecisions, and is based on minimum cost (adverse minima) rather than maximum benefit con-siderations. The method presented here might be considered the dual to Terborgh’s method.Like the MAPI method, that shown in Tables 15.1(a)–(c) uses the concept of time-adjustedannual averages, or level annuities. However, instead of finding adverse minima, we insteadfind what might be called “propitious maxima.” In this instance these are employed to present

2The following pages are reprinted with permission from The Engineering Economist, Vol. 22, No. 1 (Fall 1976). Copyright c©American Institute of Industrial Engineers, Inc., 25 Technology Park/Atlanta, Norcross, GA. 30092.


Tabl

e15

.1(a

)C

apita

l-bu

dget

ing

proj

ectw

ithsa

lvag

eva

lue

aten

dof

each

year

(ini

tialo

utla

y=

$200

0;es

timat

edus

eful

life

=7

year

s;de

clin

ein

salv

age

valu

efr

ombe

ginn

ing

ofpe

riod

valu

e=

20%

year

)

23

45

67

Pres

ent

PVof

Acc

umul

ated

Cap

ital

Lev

elL

evel

1va

lue

retu

rn=

PV=

reco

very

annu

ity=

annu

ity=

Yea

rR

etur

nfa

ctor

—10

%(1

)×

(2)

(3)

Acc

umul

ated

fact

or—

10%

(4)×

(5)

cost

$200

0×

(5)

1$6

000.

9091

545.

4654

5.46

1.10

0060

0.00

−220

0.00

260

00.

8264

495.

8410

41.3

00.

5762

600.

00−1

152.

403

600

0.75

1345

0.78

1492

.08

0.40

2160

0.00

−804

.20

460

00.

6830

409.

8019

01.8

80.

3155

600.

00−6

31.0

05

100

0.62

0962

.09

1963

.97

0.26

3851

8.10

−527

.60

610

00.

5645

56.4

520

20.4

20.

2296

463.

89−4

59.2

07

100

0.51

3251

.32

2071

.74

0.20

5442

5.54

−410

.80

89

1012

13a

14sa

lvag

ePV

ofL

evel

11PV

ofN

PVIn

tern

alva

lue

atsa

lvag

e=

annu

ity=

(6)+

(7)

annu

ity—

of(1

1)=

rate

ofY

ear

year

end

(2)×

(8)

(5)×

(9)

+(1

0)10

%(1

1)×

(12)

retu

rn

116

00.0

014

54.5

616

00.0

00.

00.

909

0.0

0.10

002

1280

.00

1057

.79

609.

5057

.10

1.73

699

.13

0.13

113

1024

.00

769.

3330

9.35

105.

152.

487

261.

510.

1609

481

9.20

559.

5117

6.53

145.

533.

170

461.

330.

1881

565

5.36

406.

9110

7.34

97.8

43.

791

370.

890.

1680

652

4.28

295.

9667

.95

72.6

44.

355

316.

350.

1565

741

9.43

215.

2544

.21

58.9

54.

868

286.

970.

1502

aC

olum

n13

will

gene

rally

beve

rysl

ight

lydi

ffer

entt

han

ifN

PVw

ere

calc

ulat

eddi

rect

ly,d

ueto

roun

ding

.


Tabl

e15

.1(b

)C

apita

l-bu

dget

ing

proj

ectw

ithsa

lvag

eva

lue

aten

dof

each

year

(ini

tialo

utla

y=

$200

0;es

timat

edus

eful

life

=7

year

s;de

clin

ein

salv

age

valu

efr

ombe

ginn

ing

ofpe

riod

=20

%ye

ar)

23

45

67

Pres

ent

PVof

Acc

umul

ated

Cap

ital

Lev

elL

evel

1va

lue

retu

rn=

PV=

reco

very

annu

ity=

annu

ity=

Yea

rR

etur

nfa

ctor

—10

%(1

)×

(2)

(3)

Acc

umul

ated

fact

or—

10%

(4)×

(5)

cost

$200

0×

(5)

1$7

000.

9091

636.

3763

6.37

1.10

0070

0.00

−220

0.00

260

00.

8264

495.

8411

32.2

10.

5762

652.

38−1

152.

403

500

0.75

1337

5.65

1507

.86

0.40

2160

6.31

−804

.20

440

00.

6830

273.

2017

81.0

60.

3155

561.

92−6

31.0

05

300

0.62

0918

6.27

1967

.33

0.26

3851

8.98

−527

.60

620

00.

5645

112.

9020

80.2

30.

2296

477.

62−4

59.2

07

100

0.51

3251

.32

2131

.55

0.20

5443

7.82

−410

.80

89

1012

13a

14sa

lvag

ePV

ofL

evel

11PV

ofN

PVIn

tern

alva

lue

atsa

lvag

e=

annu

ity=

(6)+

(7)

annu

ityof

(11)

=ra

teof

Yea

rye

aren

d(2

)×

(8)

(5)×

(9)

+(1

0)fa

ctor

—10

%(1

1)×

(12)

retu

rn

116

00.0

014

54.5

616

00.0

010

0.00

0.90

990

.90

0.15

002

1280

.00

1057

.79

609.

5010

9.48

1.73

619

0.06

0.16

023

1024

.00

769.

3330

9.35

111.

462.

487

277.

200.

1668

481

9.20

559.

5117

6.53

107.

453.

170

340.

620.

1699

565

5.36

406.

9110

7.34

98.7

23.

791

374.

250.

1696

652

4.28

295.

9667

.95

86.3

74.

355

376.

140.

1662

741

9.43

215.

2544

.21

71.2

34.

868

346.

750.

1599

aC

olum

n13

will

gene

rally

beve

rysl

ight

lydi

ffer

entt

han

ifN

PVw

ere

calc

ulat

eddi

rect

lydu

eto

roun

ding

.


Tabl

e15

.1(c

)C

apita

l-bu

dget

ing

proj

ectw

ithsa

lvag

eva

lue

aten

dof

each

year

(ini

tialo

utla

y=

$200

0;es

timat

edus

eful

life

=7

year

s;de

clin

ein

salv

age

valu

efr

ombe

ginn

ing

ofpe

riod

=20

%Y

ear)

23

45

67

Pres

ent

PVof

Acc

umul

ated

Cap

ital

Lev

elL

evel

1va

lue

retu

rn=

PV=

reco

very

annu

ity=

annu

ity=

Yea

rR

etur

nfa

ctor

—10

%(1

)×

(2)

(3)

Acc

umul

ated

fact

or—

10%

(4)×

(5)

cost

$200

0×

(5)

1$1

000.

9091

90.9

190

.91

1.10

0010

0.00

−110

0.00

220

00.

8264

165.

2825

6.19

0.57

6214

7.62

−576

.20

330

00.

7513

225.

3948

1.58

0.40

2119

3.64

−402

.10

440

00.

6830

273.

2075

4.78

0.31

5523

8.13

−315

.50

530

00.

6209

186.

2794

1.05

0.26

3824

8.25

−263

.80

620

00.

5645

112.

9010

53.9

50.

2296

241.

99−2

29.6

07

100

0.51

3251

.32

1105

.27

0.20

5422

7.02

−205

.40

89

1012

13a

14sa

lvag

ePV

ofL

evel

11PV

ofN

PVIn

tern

alva

lue

atsa

lvag

e=

annu

ity=

(6)+

(7)

annu

ityof

(11)

=ra

teof

Yea

rye

aren

d(2

)×

(8)

(5)×

(9)

+(1

0)fa

ctor

—10

%(1

1)×

(12)

retu

rn

190

0.00

818.

1990

0.00

−100

.00

0.90

9−9

0.90

0.00

002

810.

0066

9.38

385.

70−4

2.88

1.73

6−7

4.44

0.05

623

729.

0054

7.70

220.

2311

.77

2.48

729

.27

0.11

204

656.

1044

8.12

141.

3864

.01

3.17

020

2.91

0.16

375

590.

4936

6.64

96.7

255

.76

3.79

121

1.39

0.18

216

531.

4430

0.00

68.8

881

.27

4.35

535

3.93

0.18

587

478.

3024

5.46

50.4

272

.04

4.86

835

0.69

0.18

12

aC

olum

n13

will

gene

rally

beve

rysl

ight

lydi

ffer

entt

han

ifN

PVw

ere

calc

ulat

eddi

rect

ly,d

ueto

roun

ding

.


matrices representing the continuum of NPV opportunities of each project, assuming that theprojects may be abandoned at the end of years 1, 2, 3, . . . , n where n represents the last yearin the useful life of the project.

Column 6 in the tables gives the level annuity having the same accumulated value at theend of the indicated year as the values in column 4. Column 7 gives the level annuities for thenumber of years indicated that are equivalent to the initial outlay. Column 10 gives the uniformannual equivalent to the salvage value in each indicated year. By summing across columns 6,7, and 10 and then multiplying by the appropriate factors for the present value of an annuity,the NPVs of column 13 are obtained. The figures in column 13 are the NPVs of the projects ifthey are abandoned and sold for salvage at the end of the year indicated. Column 13 could havebeen calculated directly, of course, and to a somewhat greater precision in the trailing digits.

Armed with the information contained in Tables 15.1(a)–(c) the committee probably wouldhave selected project A and not project C. If project A were selected, and then abandoned atthe end of the fourth year of service, it would provide an NPV of $461.33. This is higher thanthat of B, which reaches a peak of $376.14 in the sixth year, and C, which must also be keptfor six years if its peak NPV of $353.93 is to be realized.

Since the projects reach maximum NPV with different timing, we have a situation tantamountto that of projects with unequal economic lives. Thus, it may appear necessary to adjust theTable 15.1 figures to reflect the different timing of optimal abandonment for each project.Ordinarily, the easiest means for adjusting for different economic lives is to find the uniformannual equivalent of the NPVs by multiplying them by the corresponding capital recoveryfactor. However, we already have these results in column 11.

Over a 12-year time horizon, the least common denominator of four- and six-year lives,projects A, B, and C have NPVs of $991.78, $588.46, and $553.71, respectively. These figuresmay be obtained by multiplying the uniform annual equivalents for optimal abandonment bythe present value of annuity factor for 12 periods. Note that since the same present value ofannuity factor is used, the comparison could have been directly between the uniform annualequivalents.

Comparison to R–VH3

This section is concerned with comparison of the method illustrated in the foregoing sectionwith the revised Robichek–Van Horne algorithm.

Let us begin by applying the R–VH rule to the machines that were considered by TMC.Employing the R–VH procedure with fixed, point estimates of cash returns in each periodresults in the values displayed in Table 15.2. The values in row 1 for projects A, B, and Care identical (except for rounding errors in the trailing digits) to the values in column 13 ofTable 15.1. In row 5 of Table 15.2(a) we see that the first figure is negative. The interpretationof this value is that, if at the end of year 4 project A is not abandoned, the company will incuran opportunity cost with present value as of the end of year 4, of $132.51 during year 5.

The figure of −$212.35 that follows in row 5 is the present value as of the end of year 4, of thecumulative opportunity cost if project A is held through years 5 and 6. The figure of −$255.39in the last column in row 5 is the present value, as of the end of year 4, of the cumulativeopportunity cost that will be suffered if project A is held from the end of year 4 through theend of year 7.

3Ibid., pp. 63–71.


Table 15.2 Present value in row year I, of salvage in column year J, plus cumulative returns throughcolumn year J, less salvage value at start of column year J

1 2 3 4 5 6 7

(A)1 0.00 99.17 261.46 461.45 370.94 316.41 287.012 109.09 287.60 507.59 408.03 348.05 315.713 196.36 438.35 328.84 262.86 227.284 266.18 145.72 73.14 34.015 −132.51 −212.35 −255.396 −87.82 −135.177 −52.08

(B)1 90.91 190.08 277.24 340.62 374.30 376.22 346.822 109.09 204.96 274.68 311.73 313.84 281.503 105.46 182.15 222.00 225.22 189.654 84.36 129.19 131.74 92.625 49.31 52.12 9.086 3.09 −44.267 −52.08

(C)1 −90.91 −74.38 29.30 202.92 307.72 353.96 350.732 18.18 132.23 323.22 438.50 489.35 485.803 125.45 335.54 462.34 518.29 514.384 231.09 370.58 432.11 427.825 153.44 221.12 216.406 74.46 69.267 −5.72

Thus, in terms of the figures of Table 15.2, each project should be abandoned at the end ofthe year prior to that corresponding to the row in which the figures become negative. In termsof opportunity cost, the interpretation is that a project should be abandoned when continuedretention results in an opportunity cost, from loss in salvage value, greater than revenues insubsequent periods.

The figures in Table 15.2 could have been generated entirely by the procedure implicit inTable 15.1, simply by shifting the time reference point forward one period for each new rowof figures generated. However, in Table 15.2, for project A the values in column 4 are greaterthan any values in their corresponding rows. Therefore, at t = 0 there is no need to generateany more than the first row of values for each project. However, once a particular project hasbeen selected, it may be useful to reevaluate it at the end of subsequent periods to determineif the optimal time of abandonment has shifted under changing estimates of cash flow andabandonment value. Such a procedure is equivalent to the R–VH approach.

A DYNAMIC PROGRAMMING APPROACH4

The solutions shown in Table 15.3 were obtained by using a dynamic programming approach,which provides a useful alternative to that described in the preceding section and also to theR–VH algorithm. Note that this dynamic programming formulation and solution employ the

4The solutions to the preceding examples, which are shown in Table 15.1, were offered by an anonymous referee, who reviewed thepaper for The Engineering Economist.


Table 15.3 Calculation of optimal abandonment decision and present value for three examplemachines (dynamic programming approach)

End of Return Abandonment Discounted return (DRt ) =year t Rt value (AVt ) max{AVt ; (0.9091)DRt+1} + Rt Decision

Project A (Machine A)7 $100 $419.43 $519.43 –6 100 524.28 624.28 Abandon5 100 655.36 755.36 Abandon4 600 819.20 1419.20 Abandon3 600 1024.00 1890.19 Keep2 600 1280.00 2318.37 Keep1 600 1600.00 2707.63 Keep0 – – 2461.51 –

Project B (Machine B)7 $100 $419.43 $519.43 –6 200 524.28 724.28 Abandon5 300 655.36 958.44 Keep4 400 819.20 1271.32 Keep3 500 1024.00 1655.76 Keep2 600 1280.00 2105.25 Keep1 700 1600.00 2613.88 Keep0 – – 2376.28 Keep

Project C (Machine C)7 $100 $478.30 $578.30 –6 200 531.44 731.44 Abandon5 300 590.49 964.95 Keep4 400 656.10 1277.24 Keep3 300 729.00 1461.14 Keep2 200 810.00 1548.32 Keep1 100 900.00 1489.40 Keep0 – – 1354.00 –

“backward searching” algorithm. (This is described in the chapter on dynamic programming byDaniel Teichroew [148, pp. 610–621].) This is what James L. Pappas used in his contributionon project abandonment [120], which was published simultaneously with the paper from whichthe preceding section was extracted.

Application of dynamic programming to equipment repair and replacement problems is cov-ered in most texts on management science/operations research. The problem of abandonmentvalue reduces to a special case of replacement, one in which an existing asset may be replacedby a hypothetical asset that does not exist, and therefore has a value of zero for the parametersof cost, cash flows, and so on, associated with it.

Since the methodology may not be familiar to many readers, a few words about Table 15.3are in order. Starting with the year 7 values for project A, the value $519.43 is the sum ofthat year’s return and salvage value. Subsequent returns do not have to be considered sincethe machine lasts only seven years. For year 6, the value $624.28 is obtained by addingthe $100 return in year 6 to $524.28, which is the year 6’s abandonment value. Since theabandonment value is $524.28, which is a greater amount than the discounted future returns($519.43 × 0.9091 = $472.21), the decision is to abandon. The decision to abandon holds untilyear 3, where the $1024 abandonment value is less than $1290.19 (= $1419.20 × 0.9091).



It has been shown that consideration of abandonment values can change the selection fromamong alternatives that would otherwise be made if only final salvage values were considered.

The possibility of abandoning a capital investment at a point in time prior to the estimateduseful or economic life has important implications for capital budgeting. Although we have notyet considered the effects of risk and uncertainty, the possibility of abandonment expands theoptions available to management and subsequently reduces the risk associated with decisionsbased on holding assets to the end of their lives. We must recognize that in a world cloudedwith great economic and political uncertainties, abandonment analysis synchronizes with thearray of techniques that fall under the topic broadly termed contingency planning. To neglectthe meaning and impact of abandonment and intermediate salvage values would be to refuse amost valuable instrument for gaining additional insight into the process of capital investmentevaluation.

The origins of abandonment analysis are implicit in writings going back at least as early asTerborgh’s Dynamic Equipment Policy. Actually, the adverse minimum of the Terborgh–MAPImethod does identify the optimal project life. The methods illustrated in this chapter can nodoubt be supplemented, modified, argued, and discussed much further. Some may prefer theR–VH algorithm, some the tabularized procedure, others the dynamic programming technique.Since the methods presented in this chapter yield equivalent results, the question of which oneshould be employed is largely a matter of personal preference.

16

Multiple Project Capital Budgeting

Preceding chapters considered various means for measuring the acceptability of individualcapital-budgeting projects under conditions of certainty and no risk. Ranking of projects waslimited to the problem of choosing which one project from a set of mutually exclusive projectsshould be selected when all the candidate invesments meet at least the minimum criteria foradoption. The problem of capital rationing was not previously considered, although actually nofirm has unlimited capital, and most have funds limitations, at least periodically, that precludeinvestment in the entire set of projects meeting their minimum criteria. Neither, to this point,were the effects of other constraints considered, whether economic, technical, or managerialpolicy.

BUDGET AND OTHER CONSTRAINTS

In this chapter we will continue to assume a world of certainty in order not to let considera-tions relating to risk and uncertainty obscure exposition of basic principles and methodologies.However, we shall deal explicitly with the implications of those factors that constrain man-agement to choose a subset of the total array of projects that would individually be acceptablein the absence of restrictions.

To simplify exposition, a single measure of investment worth will be employed throughoutthis chapter. The net present value (NPV) will be used provide the measure of individual projectdesirability. The NPV chosen as the single measure to be used primarily because it relates moredirectly and unambiguously to the basic valuation model of financial management, which wasintroduced at the outset of this book, than other measures do. Alternatively, if preferred, theprofitability index, internal rate of return, payback, or a composite function of measures canbe used. (We would hope, however, that the payback measure would not be adopted as the solecriterion by anyone who has read this far.)

Consider the three capital investment projects in Table 16.1. If the projects are not mutuallyexclusive and there are no limits on funds that may be invested, all three projects will beundertaken by the firm. However, once we begin to consider capital rationing it becomes clearthat a method is needed for selecting a subset from among candidate projects. For example,various budget limitations yield differing selections and total NPV for the capital budget(see Table 16.2). Here we have only three projects, and the only constraint is the one onfunds available for investments — capital rationing. Other constraints are common and furthercomplicate the selection process.

GENERAL LINEAR PROGRAMMING APPROACH

Additional constraints may take several forms. For example, suppose that one project is toconstruct a new assembly facility a short distance from our existing plant, and another projectis to build an overhead conveyor from our existing plant to the new facility. Obviously, we should


Table 16.1

Project Cost NPV

A $60,000 $30,000B 30,000 20,000C 40,000 25,000

Table 16.2

Budget Accepted Total NPV

≥$130,000 A, B, C $75,000100,000 A, C 55,000

90,000 A, B 50,00070,000 B, C 45,00060,000 A 30,000

not even consider building the conveyor unless the construction project has first been accepted.Another type of constraint is that of mutual exclusivity. For example the two projects: (1) repairthe existing facility now, and the mutually exclusive alternative (2) destroy the existing facilitynow and replace it with something new. Still another type of constraint is the requirement that iftwo projects are both accepted, a third project will also be accepted. Depending upon whetherour objective is to maximize a value (such as NPV) or minimize a value (such as cost), thegeneral linear programming problem may be specified as:

Maximize p1x1 + p2x2 + · · · + pn xn

Subject to a11x1 + a12x2 + · · · + a1n xn ≤ b1

a21x1 + a22x2 + · · · + a2n xn ≤ b2

...am1x1 + am2x2 + · · · + amn xn ≤ bm

and for all i, xi ≥ 0

or (16.1)

Minimize b1u1 + b2u2 + · · · + bmum

Subject to a11u1 + a21u2 + · · · + amum ≥ p1

a12u1 + a22u2 + · · · + am2um ≥ p2...a1nu1 + am2u2 + · · · + amnum ≥ pn

and for all i, ui ≥ 0

which, in matrix algebra notation, becomes

Maximize p · x Minimize b′uSubject to A · x ≤ b or Subject to A′u ≥ p′

xi ≥ 0 for all i ui ≥ 0 for all i

(16.2)

Multiple Project Capital Budgeting 161

By adding the requirement that xi be integer-valued for all i, we have made this into a linearinteger programming problem. Further restriction on the xi , specifically the requirement thatthey take on only the values zero or one, produces a zero–one integer programming problem.We formally specify the following two frequently encountered and important constraints.

Mutual Exclusivity

A set of n projects, from which at most one may be selected, yields the constraintn∑

i=1

xi ≤ 1 (16.3)

Since we already have a nonnegativity constraint on all the xi , this means that only one of thexi may have a nonzero value, namely a value of one. However, the constraint allows for nonebeing accepted, since a zero value for every one of the project xi satisfies the constraint.

Contingent Projects

Project B is said to be contingent on project A if it can be accepted only if A is accepted. Thisyields the constraint

xb ≤ xa (16.4)

which is equivalent to

xb − xa ≤ 0 (16.5)

or

xa − xb ≥ 0

This last constraint form allows project A to be accepted, yet does not force its acceptance.However, an attempt to accept B with A not already accepted produces a value of −1, whichis less than zero and violates the constraint. Therefore this constraint accomplishes what wewant and no more.

As the number of projects increases, the difficulty of selecting a subset that is in some sense“best” increases. When constraints in addition to budget limitations apply to the selectionproblem, things becomes unmanageable without a systematic procedure for carrying out theselection process.

We have used the method of linear programming to illustrate how a subset of projects maybe selected. Linear programming facilitates the handling of constraints, but its use in capitalbudgeting is limited. Capital investments are not finely divisible. We either accept a projectcompletely or reject it; for example, we do not choose to invest in 0.763 or 1.917 of a project.This means that we invest in 0, 1, 2, or some other integer number of projects of the particulartype. In fact, it often will be the case that we have a unique project, so that the relevant values are0 (reject the project) or 1 (accept the project). For such projects there is no option of acceptinga second, a third, and so on, because they do not exist. When there are multiples of a particularproject, for example, construction of one warehouse, construction of a second warehouse, andso on, then each may be considered to be a separate project, identical to the others. Linearprogramming may be made appropriate to capital-budgeting applications by modifying, for


example, the well-known simplex method,1 to incorporate Gomory’s cutting-plane approach[55]. The result of such modification is an integer linear programming algorithm. However,because most capital-budgeting proposals involve one or a few projects of a particular type,and because the problem setup and constraint system are similar, the approach that will betaken here is that of 0–1 (zero–one) integer programming. The 0–1 terminology is useful sincea project is either selected (1) or rejected (0) and thus can be assigned a numerical value ofeither 0 or 1 as the algorithm proceeds.

ZERO–ONE INTEGER PROGRAMMING2

Two similar approaches to 0–1 integer programming are those of Balas’ [5] and of Lawler andBell [83]. Both rest on the concept of vector partial ordering as a heuristic for systematicallysearching out the optimal solution (if one exists) to an array of n projects in m constraints,without the necessity of evaluating each and every one of the 2n possible combinations ofprojects.3 The discussion that follows is patterned on that of Lawler and Bell. Developmentof new zero–one solution algorithms and refinements on existing methods continue becausemethods developed so far have their individual idiosyncracies and none is clearly superior to theothers for all problems. The Lawler and Bell algorithm, for purposes of illustration, is as usefulas any. However, in terms of computational efficiency, there seem to be marked differencesbetween alternative solution algorithms. For instance, Pettway examined the efficiency ofseveral, and found wide differences among them [127]. So has Horvath [71], who developeda computer program for using the algorithm that he found the most efficient and reliable. Avector x is said to be a binary vector if each element is either 0 or 1. The vector may then belooked upon as a binary number. For two binary vectors, x and y, vector partial ordering meansthat x ≤ y if, and only if, xi ≤ yi for all i. For example, x ≤ y for x = (0 0 1 0 1) and y =(0 0 1 1 1). For a particular vector x, there may or may not exist a vector or vectors x ′ suchthat x < x ′.

Lawler and Bell denote by x ∗ the vector following x in numerical (binary number) orderingfor which x = x∗. The vector x ∗ can be calculated by treating the vector x as a binary number.There are three steps involved. First, subtract 1 from x. Next, apply the logical “or” operation4

to x and x −1 to obtain x∗ − 1. Finally, add 1 to x∗ = 1. The alert reader will note that thesethree steps are equivalent to binary addition of “1” to the rightmost “1” in x. Applying thenotion of vector partial ordering to the problem of:

Minimize g0(x)

Subject to −g1(x) ≥ 0

g2(x) ≥ 0

where x = (x1, x2, . . . , xn) and x1 = 0 or 1, Lawler and Bell developed an optimizationalgorithm containing only three decision rules. The vector i denotes the best solution so farobtained.

1For applications of the simplex method and a nicely done FORTRAN implementation, see Hans G. Daellenbach and Earl J. Bell [27].2This section is provided for historical perspective and to provide a look “at the engine room.” It may be skipped by those not concernedwith how the 0–1 algorithms work.3The number 2n rapidly becomes very large as n increases. For example, 225 = 33,554,432, and 235 = 34,359,738,368.4The logical “or” means here that if both the jth element of x and the jth element of x −1 are 0, then the jth element of x∗ −1 is set to0, or else it is set to 1.


Figure 16.1 Flow chart for Lawler and Bell 0–1 integer algorithm

Rule 1. If g0(x) ≥ g0(i), replace x by x∗.Rule 2. If x is feasible, replace x by x∗. Feasibility means that −g1(x) ≥ 0 and g2(x) ≥ 0, orthat gi1(x) − gi2(x) < 0.Rule 3. If for any i, gi1(x∗ − 1) = gi2(x) = 0, replace x by x∗.

If no rule is applicable, replace x by x + 1 and continue. (It is strongly recommended thatthe interested reader refer to the original article by Lawler and Bell for a much more detailedexplanation of the algorithm, and for examples of how problems involving nonbinary integercoefficients and quadratic objective functions may be handled.)

A flow chart for the algorithm with a sample problem and solution is shown in Figure 16.1.In using the algorithm to solve problems, it is suggested, based on the experience of Lawlerand Bell, that those variables that a priori would seem to be least significant be placed so asto occupy the rightmost positions in the solution vector. During so will reduce the number ofiterations required, and hence the requisite solution time.

This algorithm is minimizing, and is predicated on monotonically nondecreasing functionsfor the objective equation and the constraints. However, maximization problems can be handledby negating the objective function and the constraint equations and objective function made


monotonically nondecreasing by substituting x ′ = 1 − x in the original problem. After theoptimal solution to the minimization problem has been obtained, reverse substitution for thex ′ will provide the optimal array of projects for the maximum problem.

Since zero–one programming problems of practical scale do not lend themselves to manualcalculation, and since computer programs are available for application of zero–one integerprogramming algorithms, the steps involved in solving a specific problem will not be illustratedhere. Even small problems, although easily solvable on a computer, involve too many iterationsfor the methods to be considered suitable for manual application. The important thing, sinceproblem setup is similar or identical with all the methods, is an understanding and mastery ofthe problem setup and constraint specification. This is vital, because an incorrectly specifiedconstraint will often cause an incorrect problem solution. In practice, it may be useful to havethe same problem set up independently by two or more persons or teams to provide a checkon results.

Example 16.1 To illustrate the type of problem amenable to solution by zero–one integeralgorithms and the formulation of the constraints, consider the following problem faced bythe management of Tangent Manufacturing Company. Tangent is a small firm engaged in lightmanufacturing. The company has recently experienced a substantial increase in demand that isexpected to continue for several years. The company currently owns a dilapidated warehouse,and a small building for assembling its products. The condition of the existing warehouse issuch that it cannot be used much longer in its current condition.

The executive committee consider it likely that the company will require at least 3000 squarefeet of new warehouse floor space next year and, in the three following years, 7000, 8000, and11,000 square feet of new space. In none of these years, however, do they want more than20,000 square feet of warehouse space. (The old warehouse should be ignored in determininghow much space the firm has.)

The company also wants to expand its assembly operations by either renovating its currentplant or by constructing another building for its assembly operations. Outlays for capitalimprovements are to be limited to $55,000 the first year, and to $45,000, $35,000, and $20,000in the following three years. The treasurer of the company has developed information on thepresent value of returns and on the required outlays for alternatives available to the firm. Thisinformation is presented in Table 16.3.

Management will not allow both project 1 and project 2 to be undertaken. Furthermore,adoption of project 7 is dependent on the prior adoption of project 3; project 8 is dependent onthe prior adoption of project 4. Finally, management requires that there be some expansion inassembly capacity, so either project 5, or project 6, or both must be adopted. Table 16.4 containsthe system of constraints. Notice that the last constraint may be stated in two alternative ways.Also, constraints 5, 6, 9, and 10 are redundant in the presence of constraints 7, 8, 11, and 12,respectively, and can be eliminated.

Since the Lawler and Bell algorithm is predicated on monotonically decreasing func-tions, we must substitute x ′ = 1 − x for in the objective function and in the constraints asfollows:

Case Original constraint Modified constraintI. ai xi ≤ b ai x ′

i ≥ + ai − bII. ai xi ≥ b −ai x ′

i ≥ − ai + b


Table 16.3

$ Thousands required outlays

ProjectPresentvalue Year 1 Year 2 Year 3 Year 4

1 Construct new warehouse (10,000 $50 $25 $7 $0 $0sq. ft) in year 1

2 Renovate existing warehouse 27 10 5 5 5(7000 sq. ft) in year 1

3 Lease warehouse for 2 years (3000 15 6 6 0 0sq. ft) in year 1

4 Lease second warehouse for 2 15 6 6 0 0years (3000 sq. ft) in year 1

5 Construct new assembly plant in 35 20 5 0 0year 1 (3000 unit capacity)

6 Renovate existing plant for 25 12 3 0 0expanded production (1500 unitincrease in capacity

7 Lease warehouse for 2 years (3000 13 0 0 6 6sq. ft) in year 3

8 Lease second warehouse for 2 13 0 0 6 6years (3000 sq. ft) in year 3

9 Construct new warehouse (10,000 40 0 0 25 7sq. ft) in year 3Outlay constraints ≤ 55 ≤ 45 ≤ 35 ≤ 20Warehouse space constraints ≤ 20 ≤ 20 ≤ 20 ≤ 20(thousands of sq. ft) > 3 > 7 > 8 > 11

After substitution, the modified constraint coefficients are ready to be submitted for solution bythe program. The modified constraint system is contained in Table 16.5. Note that all constraintsare now in the “≥” form.

Acceptance of all but projects 1, 7, and 8 yields an objective function NPV value of of $157.All problem constraints are satisfied.

The appendix to this chapter contains computer solutions achieved with ExcelTM and QuattroProTM. For problems that are not truly large these spreadsheet programs are very easy to use andyield quick, accurate solutions. On modern personal computers these spreadsheet programsare capable of providing results that formerly would have required specialized programs onmainframe computers.

GOAL PROGRAMMING

Early on, this book specified maximization of shareholder wealth as the goal of modernfinancial management. Using NPV as the criterion for project selection serves to move the firmtoward this goal when capital markets are approximately perfect and there is certainty withrespect to project parameters. These, however, are abstractions from the reality that typicallyprevails; capital markets are less than perfect and uncertainty does prevail. Situations are oftenencountered in which management has more than one objective. In fact, this is probably thenorm rather than an exception. Management recognizes that the market takes into account more


Table 16.4

Constraint x1 x2 x3 x4 x5 x6 x7 x8 x9 b

Financial constraints 1 25 10 6 6 20 12 0 0 0 ≤ 552 7 5 6 6 5 3 0 0 0 ≤ 453 0 5 0 0 0 0 6 6 25 ≤ 354 0 5 0 0 0 0 6 6 7 ≤ 20

Warehouse spaceconstraints

5 10 7 3 3 0 0 0 0 0 ≤ 20Yr 1

{6 10 7 3 3 0 0 0 0 0 > 3

7 10 7 3 3 0 0 0 0 0 ≤ 20Yr 2

{8 10 7 3 3 0 0 0 0 0 > 7

9 10 7 0 0 0 0 3 3 10 ≤ 20Yr 3

{10 10 7 0 0 0 0 3 3 10 > 8

11 10 7 0 0 0 0 3 3 10 ≤ 20Yr 4

{12 10 10 0 0 0 0 3 3 10 > 11

Projects 1 and 2 13 1 1 0 0 0 0 0 0 0 ≤ 1mutually exclusive

Project 7 dependent 14 0 0 1 0 0 0 −1 0 0 ≥ 0on prior adoptionof project 3

Project 8 dependent 15 0 0 0 1 0 0 0 −1 0 ≥ 0on prior adoptionof project 4

Assembly facilities 16 0 0 0 0 3 1.5 0 0 0 ≥ 1.5must be expanded or

16′ 0 0 0 0 1 1 0 0 0 ≥ 1

Table 16.5 Constraints for Lawler and Bell algorithm solution

Constraint x1 x2 x3 x4 x5 x6 x7 x8 x9 ≥ b

1 25 10 6 6 20 12 0 0 0 242 7 5 6 6 5 3 0 0 0 −133 0 5 0 0 0 0 6 6 25 74 0 5 0 0 0 0 6 6 7 45 10 7 3 3 0 0 0 0 0 36 −10 −7 −3 −3 0 0 0 0 0 −167 10 7 0 0 0 0 3 3 10 138 −10 −7 0 0 0 0 −3 −3 −10 −259 1 1 0 0 0 0 0 0 0 1

10 0 0 −1 0 0 0 1 0 0 011 0 0 0 −1 0 0 0 1 0 012 0 0 0 0 −1 −1 0 0 0 −1

information than the NPV of accepted capital investment projects could provide. Accountingprofits, earnings, and dividend stability and growth, market share, total assets, and so on, affectthe value of shareholder wealth.

Management may have compatible goals, goals for which the achievement of one does notprevent achievement of the others. On the other hand, goals may be incompatible; steps to


reach one goal may require moving further from other goals. Ordinary linear programmingcan lead to results that are less satisfactory than can be obtained with what is known as goalprogramming. And in cases for which there is no solution when target values are treated asconstraints in ordinary linear programming, goal programming can provide feasible solutions.

Goal programming is an extension of ordinary linear programming. In fact, the basic simplexalgorithm and its computerized implementations are suitable in those instances in which integersolutions are not required. The general goal programming problem may be specified as

Minimize f = (M1 y+1 + N1 y−

1 ) + (M1 y+2 + N2 y−

2 ) + · · ·+ (Mn y+

n + Nn y−n )

Subject to a11x1 + a12x2 + · · · + a1n xn − y+1 + y−

1 = b1

a21x1 + a22x2 + · · · + a2n xn − y+2 + y−

2 = b2...am1x1 + am2x2 + · · · + amn xn − y+

m + y−m = bm

and for all i, xi , y+i , y−

i ≥ 0

(16.6)

In the more compact form obtained by using matrix notation for the constraint system thisbecomes

Minimize f =m∑

t=1

(Mi y+i + Ni y−

i )

Subject to A · x − y+ + y− = b

(16.7)

Several things should be noted about the goal programming model. First, many of the y+

and y− will be unimportant; in this case they will not appear in the objective function and maybe ignored. Second, the normal linear programming constraints are present. These representtechnological, economic, legal, or other requirements that must not be violated. Third, sincethe y+

i and y−i represent underachieving or overachieving the same goal, one of them must be

zero. The Mi and Ni provide for different weights to be assigned to under- or overachieving agoal. If either is zero it means that no importance is attached to that particular deviation fromgoal. The Mi and Ni also allow priority levels to be established. For instance, if goal i must beachieved before goal j can be considered, this may be specified by defining the priority levelcoefficients. The relationship

Mi > > > M j

denotes Mi to be a very large value in comparison with M j , so large that goal i will be givenabsolute preference to goal j. If Mi is to take absolute preference over M j , but M j is onlytwice as important as Mk , this can be stated as: Mi >>> M j = 2Mk . Thus, we can (1) definea hierarchical structure of goals, in which each level is fixed in relation to the others and(2) define trade-off functions between goals within a particular hierarchical stratum. When westate that goal j is twice as important as goal k, we are defining a trade-off function. Absolutepriority is accomplished by making the trade-off too costly to be considered.

A goal programming problem formulation requires three main items:

1. An objective function for which the weighted deviations from the target or goal levels areminimized according to specified priority rankings. This is in contrast to the ordinary linear


Table 16.6 Goal specification

Reach at least a specified minimum level: Minimize y−

Do not exceed a specified maximum level: Minimize y+

Approach specified target as closely as possible: Minimize (y+ + y−)Achieve specified minimum level, then move as

far above as possible: Minimize (−y+ + y−)Achieve specified maximum level, then move as

far below as possible: Minimize (y+ −y−)

programming formulation, in which an aggregate value objective function is maximized or,equivalently, the opportunity costs (shadow prices) are minimized.

2. The normal linear programming constraints reflecting economic, technological, legal, andother constraints.

3. The goal constraints that incorporate one or both of y+i and y−

i , the deviational variables.

The objective function must specify (1) the priority level of each goal, (2) the relativeweight of each goal, and (3) the appropriate deviational variables. The deviational variables areviewed as penalty costs associated with under- or overachieving a particular target, or goal. Thespecification of deviational variables in the objective function determines whether a particulargoal is to be reached as exactly as possible, whether either under- or overachievement is to beavoided, or whether it is desired to move as far from some target level as possible. Specificationof these goals is summarized in Table 16.6. A great advantage of goal programming is that itcan handle both complementary and conflicting goals as long as a trade-off function is specifiedthat links the conflicting goals. Since management must make decisions in a world of risk andimperfect capital markets, project characteristics beyond NPV may have to be factored intothe decision process. Interactions between the value of the firm and capital investments mayhave to be recognized. For example, large capital projects may require borrowing that couldaffect the firm’s capital structure and its cost of capital for several years. Also, accountingprofits may be important to the extent they influence the markets for the firm’s shares anddebt instruments. Goal programming is particularly useful, because it allows managementperceptions and policies, and some interrelationships between the firm and the prospectivecapital investments to be handled simultaneously. It enables management to obtain insight intothe implicit costs of its goals and trade-off functions if sensitivity analysis is carried out toshow the effects of changing goal and trade-off parameters.

Goal programming allows the objective of maximizing shareholder wealth — enterprisevalue — to be approached by setting lesser goals that, if reached, contribute to the major ob-jective. In other words, goal programming provides a means for disaggregating a strategicobjective into a series of tactical goals that, taken together, move the firm toward that objec-tive. And the tactical goals themselves may be interrelated by trade-off relationships. Threedifficulties affect the use of goal programming, particularly in capital budgeting:

1. In capital-budgeting projects indivisibility is the rule rather than the exception. This meansthat ordinary linear programming computer codes are generally not suitable and one mustresort instead to integer programming algorithms that are not as generally available.

2. Specification of goals is often based on conjecture or “hunches” about empirical questions.Thus, different managers will generally make different subjective assessments of reality,based on individual experience, perception, and bias. This means that solutions obtained


will depend on whose goals are achieved in the goal programming formulation. Thereis nothing necessarily wrong with this. In fact, since goals must be clearly specified, anadded benefit may result when managers must articulate goals in a form amenable toprogramming solution. What goal programming can provide is an objective procedurefor systematically and accurately reaching goals — goals that themselves may have beensubjectively determined.

3. Conflicting goals require that trade-off relationships be defined. This often requires thatnoncommensurable items be compared, that exchange rates between “apples and oranges”be defined even though the trade-off function may involve less tangible factors than these. Afurther difficulty in this vein is that trade-off functions may not be linear, but may change overa range of values. This third difficulty means that one of the strengths of goal programming,its ability to deal with conflicting goals, is also potentially one of its greatest weaknesses ifnot approached with care.


Selection of which (if any) capital investments out of an array of candidates should be acceptedrequires a systematic approach. The indivisible nature of capital investment projects means thatordinary linear programming may not produce correct results. Integer linear programming maybe used, but requires constraints that reflect the number of each project available for adoption.Zero–one integer programming provides a very useful means of selection, and requires fewerexplicit constraints.

Several methods of zero-one integer programming have been developed, and there willlikely be further refinements and new developments. Experimentation has suggested that somemethods are more generally useful than others. However, since problem specification will beidentical, or at least very similar, for all existing zero–one integer programming algorithms, andprobably for new developments as well, we have concentrated on specification of the problemand constraints.

A programming solution to multiple project selection is particularly useful because manyconstraints beyond that imposed by capital rationing may be handled easily. Great care mustbe exercised, however, in specifying all constraints. A seemingly minor error in one constraintmay cause an entirely incorrect solution to be obtained.

The evolution of add-in programs in the ubiquitous spreadsheet programs now found onvirtually every personal computer, along with abundant computer memory and high-speedcentral processors, makes it not only possible, but desirable to solve many optimization prob-lems without resort to specialized programs. Only the larger problems require that one resortto those formerly necessary remedies. The appendix to this chapter illustrates this.

16AAppendix to Multiple Project

Capital Budgeting

Although methods developed for mainframe computers are still useful, especially for large-scale problems with hundreds of variables and constraints, for many problems personal com-puters and popular spreadsheet programs can be used. One must be careful to ensure that theproblems are accurately specified, and that constraints are correct, or spurious answers willlikely be found. An advantage of using spreadsheet programs instead of mainframe computerprograms is that one may easily examine the effects of changing constraints or variable costsand their payoffs.

Although ExcelTM and Quattro ProTM are used for illustration here, other programs will yieldsimilar results and are used similarly. So users of LotusTM, ApplixwareTM, or Star OfficeTM

need not be at a disadvantage.1 The setup for ExcelTM is contained in Figures 16A.1, 16A.2.A rose by another name is still a rose — Figures 16A.3 and 16A.4 contain the Quattro ProTM

optimizer setup and options. It is obvious that there is no significant difference between them,indicating that either program can be used for this purpose.

Figure 16A.5 contains the ExcelTM spreadsheet for solving the zero–one integer problemof Example 16.1. It should be noted that the lower set of cells contains values obtained bymultiplying the contents in the set above by the 0 and 1 x solution values in cells Q6 throughQ14. The sums of the column contents in the lower set then serve for most of the constraintsin the problem. The Quattro ProTM spreadsheet is so similar in appearance to the ExcelTM thatit is not shown separately.

Figure 16A.1 ExcelTM solver parameters

1Some of these other programs at this writing do not have the add-ins required for these programming applications. However, it is onlya matter of time before they too will include them integrally, or as third-party add-in applications.

Appendix to Multiple Project Capital Budgeting 171

Figure 16A.2 ExcelTM solver options

Figure 16A.3 Quattro ProTM optimizer setup

Figure 16A.4 Quattro ProTM optimizer options

Example 16-1Excel 97 Setup

ProjectConstruct New Warehouse in Year 1Renovate Existing Warehouse in Year 1Lease Warehouse for 2 Years in Year 2Lease 2nd Warehouse for 2 Years in Year 2Construct New Assembly Plant in Year 1Renovate Existing PlantLease Warehouse for 2 Years in Year 3Lease 2nd Warehouse for 2 Years in Year 3Construct New Warehouse in Year 3

Sums =

x1x2x3x4x5x6x7x8x9

Projects 1 and 2 mutually exclusiveProject 7 depends on prior adoption of 3Project 8 depends on prior adoption of 4Assembly facilities must be expanded

(i.e, project 5 or 6 must be accepted)

Sol’nValue

1-1-12

from

NPV$157

SqFt SqFt SqFt SqFt Units Units Units Units

x1 + x2 <= 1x7 - x3 <= 0x8 - x3 <= 0x5 + x6 >= 1

13 13 17 17 4.5 4.5 4.5 4.5 $54 $25 $30 $12

4000

2535151527

0 0 0

0000

10

0

00

7

00000

10

0

07

00000

0733

00000

07

33

7

0

000000

50

000000

5

25

0566

35

000

66

010

2012

000

000

00003

000

00003

1.5 1.5

000

00003

1.5

000

00003

1.5

Capacity Increases Available in Each YearYear 1 Year 1Year2 Year 2 Year 3 Year 4Year3Year4 Investment Outlays

Year 1 Year 2 Year 3 Year 4NPV SqFt SqFt SqFt SqFt Units Units Units Units

$233 23 23 33 33 4.5 4.5 4.5 4.5 79 32 42 24

3 333

3 33 3

NPVs$50

2715153525131340 10 10

7 7 7 710 10 10 10

1.5 1.5 1.5 1.53 3 3 3

SqFt SqFt SqFt SqFt Units Units Units Units

Capacity Increases Available in Each Year

00

000

0000

0000

0000

2510

62012 3

56665 5 57

6 666

25 70

0

001

1

11

11

x1x2x3x4x5x6x7x8x9

Sol’nXjValue ProjectYear 1 Year 2Year 3 Year 4

Req’d OutlaysYear 1 Year2 Year3Year4 Year 1 Year 2 Year 3 Year 4

Figure 16A.5 ExcelTM spreadsheet for Example 16.1.


The solution model from the ExcelTM setup as:

Not using “= binary” Model using x 1 . . . X 9 = binary

=MAX($D$33) =MAX($D$33)

=COUNT($Q$6:$Q$14) =COUNT($Q$6:$Q$14)

=$D$38<=1 =$D$38<=1

=$D$39<=0 =$D$39<=0

=$D$40<=0 =$D$40<=0

=$D$41>=1 =$D$41>=1

=$E$33<=20 =$E$33<=20

=$F$33<=20 =$F$33<=20

=$F$33>=7 =$F$33>=7

=$H$33<=20 =$H$33<=20

=$H$33>=11 =$H$33>=11

=$M$33<=55 =$M$33<=55

=$N$33<=45 =$N$33<=45

=$O$33<=35 =$O$33<=35

=$P$33<=20 =$P$33<=20

=$Q$6<=1 =($Q$8=0)+($Q$8=1)=1

=$Q$6=INT($Q$6) =($Q$7=0)+($Q$7=1)=1

=$Q$6>=0 =($Q$6=0)+($Q$6=1)=1

=$Q$7<=1 =($Q$10=0)+($Q$10=1)=1

=$Q$7=INT($Q$7) =($Q$13=0)+($Q$13=1)=1

=$Q$7>=0 =($Q$12=0)+($Q$12=1)=1

=$Q$8<=1 =($Q$11=0)+($Q$11=1)=1

=$Q$8=INT($Q$8) =($Q$14=0)+($Q$14=1)=1

=$Q$8>=0 =($Q$9=0)+($Q$9=1)=1

=$Q$9<=1 ={100,1500,0.000001,0.01,TRUE,FALSE,FALSE,2,2,2,0.001,FALSE}=$Q$9 = INT($Q$9)

=$Q$10<=1 Note how much more compact this alternative model specification is.=$Q$10=INT($Q$10)

=$Q$10>=0

=$Q$11<=1

=$Q$11=INT($Q$11)

=$Q$11>=0

=$Q$12<=1

=$Q$12=INT($Q$12)

=$Q$12>=0

=$Q$13<=1

=$Q$13=INT($Q$13)

=$Q$13>=0

=$Q$14<=1

=$Q$14=INT($Q$14)

=$Q$14>=0

= {100,1000,0.000001,0.01,True,False,False,1,1,1,0.0001,False}


It should be mentioned that cells Q6 through Q14 are named x 1 through x 9 in order tofacilitate entering the constraints in the model. Thus x 1 can be substituted for Q6, x 2 for Q7,etc., in any of the formulae.

The solution from the ExcelTM solver is given as:

Microsoft Excel Answer ReportWorksheet: [Example 16-1.xls]16-1 SetupReport Created: 3/21/2002 10:20:27 PM

Target Cell (Max)

Cell Name Original Value Final Value

$D$33 xj*NPV $0 $157

Adjustable Cells

Cell Name Original Value Final Value

$Q$6 x 1 0 0

$Q$7 x 2 0 1

$Q$8 x 3 0 1

$Q$9 x 4 0 1

$Q$10 x 5 0 1

$Q$11 x 6 0 1

$Q$12 x 7 0 0

$Q$13 x 8 0 0

$Q$14 x 9 0 1


And the system of constraints after optimization is given by

Constraints

Cell Name Cell Value Formula Status Slack

$D$38 Projects 1 and 2 mutually exclusive Value 1 $D$38<=1 Binding 0

$D$39 Project 7 depends on prior adoption of 3 Value -1 $D$39<=0 Not Binding 1

$D$40 Project 8 depends on prior adoption of 4 Value -1 $D$40<=0 Not Binding 1

$D$41 Assembly facilities must be expanded Value 2 $D$41>=1 Not Binding 1

$E$33 SqFt 13 $E$33<=20 Not Binding 7

$F$33 SqFt 13 $F$33<=20 Not Binding 7

$F$33 SqFt 13 $F$33>=7 Not Binding 6

$H$33 SqFt 17 $H$33<=20 Not Binding 3

$H$33 SqFt 17 $H$33>=11 Not Binding 6

$M$33 Year 1 $54 $M$33<=55 Not Binding 1

$N$33 Year 2 $25 $N$33<=45 Not Binding 20

$O$33 Year 3 $30 $O$33<=35 Not Binding 5

$P$33 Year 4 $12 $P$33<=20 Not Binding 8

$Q$6 x 1 0 $Q$6<=1 Not Binding 1

$Q$6 x 1 0 $Q$6=integer Binding 0

$Q$6 x 1 0 $Q$6>=0 Binding 0

$Q$7 x 2 1 $Q$7<=1 Binding 0


$Q$7 x 2 1 $Q$7=0 Not Binding 1

$Q$8 x 3 1 $Q$8<=1 Binding 0

$Q$8 x 3 1 $Q$8<=integer Binding 0

$Q$8 x 3 1 $Q$8>=0 Not Binding 1

$Q$9 x 4 1 $Q$9<=1 Binding 0


$Q$10 x 5 1 $Q$10<=1 Binding 0


$Q$10 x 5 1 $Q$10>=0 Not Binding 1

$Q$11 x 6 1 $Q$11<=1 Binding 0


$Q$11 x 6 1 $Q$11>=0 Not Binding 1

$Q$12 x 7 0 $Q$12<=1 Not Binding 1


$Q$12 x 7 0 $Q$12>=0 Binding 0

$Q$13 x 8 0 $Q$13<=1 Not Binding 1


$Q$13 x 8 0 $Q$13>=0 Binding 0

$Q$14 x 9 1 $Q$14<=1 Binding 0


$Q$14 x 9 1 $Q$14>=0 Not Binding 1


The alternate solution, using x 1 through x 9 = binary, is given in

Microsoft Excel 10.0 Answer Report

Worksheet: [Example 16-1-Binary.xls]16-1 SetupReport Created: 4/19/2002 12:32:10 PM

Target Cell (Max)Cell Name Original Value Final Value

$D$33 xj*NPV $0 $157

Adjustable CellsCell Name Original Value Final Value

$Q$6 x 1 0 0$Q$7 x 2 0 1$Q$8 x 3 0 1$Q$9 x 4 0 1$Q$10 x 5 0 1$Q$11 x 6 0 1$Q$12 x 7 0 0$Q$13 x 8 0 0$Q$14 9 0 1

ConstraintsCell Name Cell Value Formula Status Slack

$D$38 Projects 1 and 2 mutually exclusive Value 1 $D$38<=1 Binding 0$D$39 Project 7 depends on prior adoption of 3 Value −1 $D$39<=0 Not Binding 1$D$40 Project 8 depends on prior adoption of 4 Value −1 $D$40<=0 Not Binding 1$D$41 Assembly facilities must be expanded Value 2 $D$41>=1 Not Binding 1$E$33 SqFt 13 $E$33<=20 Not Binding 7$F$33 SqFt 13 $F$33<=20 Not Binding 7$F$33 SqFt 13 $F$33>=7 Not Binding 6$H$33 SqFt 17 $H$33<=20 Not Binding 3$H$33 SqFt 17 $H$33>=11 Not Binding 6$M$33 Year 1 $54 $M$33<=55 Not Binding 1$N$33 Year 2 $25 $N$33<=45 Not Binding 20$O$33 Year 3 $30 $O$33<=35 Not Binding 5$P$33 Year 4 $12 $P$33<=20 Not Binding 8$Q$8 x 3 1 $Q$8=binary Binding 0$Q$7 x 2 1 $Q$7=binary Binding 0$Q$6 x 1 0 $Q$6=binary Binding 0$Q$10 x 5 1 $Q$10=binary Binding 0$Q$13 x 8 0 $Q$13=binary Binding 0$Q$12 x 7 0 $Q$12=binary Binding 0$Q$11 x 6 1 $Q$11=binary Binding 0$Q$14 x 9 1 $Q$14=binary Binding 0$Q$9 x 4 1 $Q$9=binary Binding 0

x

17

Utility and Risk Aversion

In this chapter, the first one to explicitly consider risk, we examine the concept of utilityand how a utility function may be calculated, various ways of estimating risk, and problemsassociated with the probability of ruin. This last topic is of some importance to the enterprise(perhaps a small firm) for which candidate capital investment projects are large in relation toits financial resources.

The concept of utility is essential if we are to develop a rationale by which a decision may bereached for a capital investment that has a range of possible outcomes, each with an associatedprobability. Some of the possible outcomes may be large losses, whereas others are large gains.

In a world of certainty we would be able to say whether or not a project is acceptable basedon objective criteria, such as the profitability index. Once risk is introduced, however, we musttake into account the decision-maker’s1 attitude toward risk in order to reach an economicallyrational decision. Two different persons are likely to disagree about accepting risky projectsif they have different attitudes toward risk. One may take risk in stride, whereas the other isconsiderably troubled by it.

THE CONCEPT OF UTILITY

Economists developed the concept of utility long ago, and it is the foundation upon whichmicroeconomic analysis rests. Utility is a reflection of personal satisfaction. Something thatprovides more feeling of pleasure than something else (or, equivalently, less pain) is said tohave greater utility.

In this work we measure investment project traits in units of currency — dollars. Becauseof this our task is much easier than it would be otherwise. We do not have to compare therespective utility of oranges and that of apples — only that of more dollars versus fewer.

The utility of more dollars is assumed to be greater than the utility of fewer dollars, sothat the utility function we will be working with is monotonically nondecreasing in dollars,at least over the range of values we will be considering. A second dollar may have the sameutility value as the first (constant marginal utility), greater value (increasing marginal utility),or lesser value (decreasing marginal utility).

When risk is introduced, we recognize that there is a trade-off between possible dollar gain(and thus utility) and risk. Projects promising the greatest return are also normally the riskiestprojects. If it were otherwise, they would have been snapped up before now.

The classic article on risk as it relates to utility is that of Tobin [159]. The following sectionfollows his terminology.

1We shall avoid the problem of whose attitude specifically is to be taken into account by using the term “decision-maker” or“management.”


Figure 17.1 Relationship of money gain to utility for a risk averter

Attitudes toward Risk

Individual attitudes toward risk are revealed in the curvature of utility functions. Before goinginto the determination of an individual utility function, it will be useful to examine theseattitudes toward risk. First, in Figure 17.1 we have a utility function corresponding to a risk-averse individual, a risk averter. Here the utility function is concave from below. This meansthat each additional monetary gain, while contributing to increased utility, contributes lessthan the same amount when the individual has less to begin with. In other words, the functiondepicts diminishing marginal utility. An extra dollar offers less additional utility to given risk-averse individuals when they have $100,000 than it does when they have $10,000. Diminishingmarginal utility means that the loss of a dollar causes more disutility (negative utility, or lossof utility) than the gain of a dollar offers. For example, consider a person with $10,000 in cashsavings, which may have taken many years to accumulate. An extra $1000 would increaseutility, but not as much as a loss of $1000 would decrease it. If this person were offered a “fairbet,” one with zero expected value based on equal probabilities of gaining or losing a largeamount, the bet would be rejected. It would be rejected because the expected utility is negative,even though the expected monetary gain is zero.

For small amounts of monetary gain or loss, a risk-averse individual may take “fair bets” oreven bets where the expected monetary gain is negative. This may be explained on the basis thatthe negative utility of the small loss is dominated by the utility provided by the entertainmentprovided, or the utility provided by the hope, however remote, of a large monetary gain (as ina lottery).

Figure 17.2 corresponds to a risk-neutral, that is, a risk-indifferent individual. Each ad-ditional dollar, no matter at what point on the function, offers the same utility. The utilityfunction of a risk-indifferent individual is a straight line. The slope is constant and thereforethe marginal utility of monetary gain does not change over the curve. A large loss would reducethis person’s utility by the same amount of utility as would be gained by winning an identicaldollar amount. A risk-indifferent individual could be expected to take fair bets, even thoseinvolving large amounts of possible gain or loss.

The final utility curve is that of a risk lover. This is illustrated in Figure 17.3. Tobin concludedthat there are no 100 percent risk lovers. Such individuals, on reflection, would have to have

Utility and Risk Aversion 179

Figure 17.2 Relationship of money gain to utility for a risk-neutral person

Figure 17.3 Relationship of money gain to utility for a risk lover

Figure 17.4 Overall utility of money for an individual

a self-destructive compulsion to prefer the riskier of any two propositions. The utility curvehas the property of increasing marginal utility, suggesting greed. The important thing aboutsuch utility function is that most individuals have utility functions that exhibit properties ofrisk aversion, risk neutrality, or risk seeking for different ranges of monetary values. This isillustrated in Figure 17.4.


For the individual whose utility function is shown in Figure 17.4 we can observe that theperson is risk averse for amounts less than m1 and greater than m3. This individual is essentiallyrisk neutral for amounts between m2 and m3, and is a risk lover or risk seeker for values betweenm1 and m2. The next section will show you how to estimate your own utility function or thatof someone else. Do not be surprised if it is similar to the one in Figure 17.4, nor shouldyou be bothered if it is not. The benefit you derive from plotting points along your utilityfunction is that you gain a better understanding of how you react to risk. Your attitude towardrisk influences your performance as a decision-maker evaluating capital investments. And itis likely that your attitude toward risk changes as the dollar magnitude of the potential gain orloss changes.

CALCULATING PERSONAL UTILITY

This section presents a method for estimating an individual’s utility function. The methodcould also be used to try to determine a composite utility curve for a group.

The treatment here parallels the presentation contained in Teweles, Harlow, and Stone [158].Their purpose was to illustrate how the individual commodity futures trader could obtain his orher personal utility function for the risks found in such activity. The principle is also applicablein the present context. Consider that you are asked to put yourself in the role of managerhere, and not one who is risking personal funds. That this is a distinction that a sole proprietorcannot make will not be argued. Because most managers in various enterprises do have differentattitudes about their own money vis-a-vis that of their employers, try to make this distinctionto the extent it is possible.

After assuming the “managerial frame of mind,” the next step is to determine the largestdollar gains your capital investment decisions have regularly made and the largest dollar lossesthat have similarly resulted. This may be impossible for many readers. Thus students andmanagers who have not made such decisions will have to imagine what the amounts would be,and try to be perfectly honest about it. This is not a test, there are no right or wrong answers.It is not necessary to do any calculations in reaching your answers, although you may if youprefer. Calculation of the reader’s personal utility function for monetary gains may be done asan exercise. For now we shall only illustrate the procedure.

Let us begin by developing the utility function for an entrepreneur who tells us that he(individually) has regularly made decisions that have resulted in gains of as much as $2 millionand losses of as much as $500,000. We shall not concern ourselves here with whether these aregains and losses over the entire project life, or with other details of the timing of the amounts.The procedure would not be materially different anyway. Having established the largest regulargains and losses, we write them down as shown in Table 17.1, column 4. The gain of $2 millionis associated with utility of 1.0 and the loss of $500,000 with utility 0.0. Other values couldhave been used for the utilities, such as +1.0 and −1.0, but the scaling is unimportant and byassigning zero utility to the worst outcome the calculations are a little easier.

Having established utilities of 1.0 for a gain of $2 million with probability 1.0, and 0.0for a loss of $500,000 with probability 1.0,we now need to find intermediate values. Supposenow we ask our entrepreneur to tell us if he would accept an investment offering a gain of$2 million with probability 0.9 and loss of $500,000 with probability 0.1; in other words,9 chances in 10 of gaining $2 million and one chance in 10 of losing $500,000. He answers“yes,” certainly he would accept such an investment. Now we ask if he would pay us $1 millionfor the opportunity to make such an investment. Yes, he would. $1.5 million? No, not that much.


Table 17.1 Computation of decision-maker’s utilities for variousmonetary gains

Probability of:

(1) (2) (3) (4)Best Worst Computed Dollar cashresult result utilitya equivalent

1.0 0.0 1.0 $2,000,000 Best result0.9 0.1 0.9 1,200,0000.8 0.2 0.8 950,0000.7 0.3 0.7 750,0000.6 0.4 0.6 550,0000.5 0.5 0.5 400,0000.4 0.6 0.4 −100,0000.3 0.7 0.3 −300,0000.2 0.8 0.2 −400,0000.1 0.9 0.1 −450,0000.0 1.0 0.0 −500,000 Worst result

a 1.0 × column 1 + 0.0 × column 2. The utility of the best result is arbitrarilyassigned a value of 1.0, the worst result, a value of 0.0. Other values could beused if desired.

Figure 17.5 Plotted utility function of entrepreneur. From Table 17.1 dollar equivalents

How about $1.2 million? Maybe. At $1.2 million he is not sure. For $50,000 more he will nottake the investment, for $50,000 less he will. Thus in Table 17.1 we write in the second row,rightmost column, the amount $1,200,000.

We repeat the process for gain of $2 million with probability 0.8 and loss of $500,000with probability 0.2. Our entrepreneur will pay up to $950,000 but no more than that foran investment offering these prospects. For each missing value we repeat the process untilcolumn 4 is completed. Then we can plot the results obtained, as shown in Figure 17.5, and fitan approximate curve to the points.


Table 17.2 Computation of refined utility of decision-maker’smonetary gain

Best outcome Worst outcome

(1) (2) (3) (4)Utility Probability Utility Probability

(5)Computed

utilitya

(6)Dollarcash

equivalent

0.5 1.0 0.4 0.0 0.50 $400,0000.5 0.9 0.4 0.1 0.49 360,0000.5 0.8 0.4 0.2 0.48 350,0000.5 0.7 0.4 0.3 0.47 325,0000.5 0.6 0.4 0.4 0.46 280,0000.5 0.5 0.4 0.5 0.45 250,0000.5 0.4 0.4 0.6 0.44 180,0000.5 0.3 0.4 0.7 0.43 160,0000.5 0.2 0.4 0.8 0.42 50,0000.5 0.1 0.4 0.9 0.41 00.5 0.0 0.4 1.0 0.40 −100,000

a Computed utility is obtained as best outcome times best outcome probability plusworst outcome times worst outcome probability.

In constructing his or her own utility curve for monetary gains and losses, the reader canperform a self-interview or work with someone who will perform the function of interviewer.Further insight into the process is found in Teweles et al., cited earlier [158].

The curve obtained and plotted in the graph of Figure 17.5 is reasonably satisfactory, exceptthat we do not have enough data points between −$100,000 and +$400,000 to be confidentin the shape of the curve in that range. We can obtain more information by continuing theinterview process. Let us begin by constructing Table 17.2 with the best result now $400,000and the worst result −$100,000. The associated utilities are, respectively, 0.5 and 0.4.

Values between $400,000 gain and $100,000 loss are obtained as before. The decision-makeris asked if he would accept an investment project offering $400,000 gain with 0.9 probabilityand $100,000 loss with 0.1 probability. Yes. Would he pay $380,000? No. $370,000? No.$360,000? Maybe. Don’t know. $350,000? Yes. We write $360,000 in column 6 of Table 17.2.The process is repeated until column 6 is filled and then the results are plotted. Figure 17.6contains the graph for the section of utility curve between −$100,000 and +$400,000. Thescale is enlarged from that used in Figure 17.5. After combining the information contained inTable 17.2 with that of Table 17.1, and plotting the results, we obtain Figure 17.7. The additionaldetail for utility between −$100,000 and +$400,000 enables a more refined approximation.The tentative judgment that our entrepreneur is a risk seeker for monetary gains seems tobe vindicated by additional information over the range of about $0 to $400,000. Because ofthe rather wide spread between utility values for dollar amounts in the ranges −$300,000 to−$100,000 and $1.2 million to $2 million, we might want to repeat the procedure for obtainingdetail over these ranges, if our entrepreneur’s patience has not been exhausted. We shall notdo this here, however, since the procedure has been illustrated.

The plotted results obtained from our entrepreneur do not deviate very widely from thefitted curve. In fact, they are very close to it, thus indicating a high degree of consistency inevaluating alternatives over the range of values regularly experienced by the decision-maker.We should not have been surprised if the points were much more scattered about the fitted


Figure 17.6 Plot of refined utility of decision-maker’s monetary gains from Table 17.2

Figure 17.7 Revised decision-maker’s utility of monetary gain

curve. What happens if we now attempt to obtain utility values for dollar gains and lossesbeyond the range of the decision-maker’s regular experience? If we attempt to do this, we willmost likely find that the decision-maker becomes increasingly inconsistent as the monetaryvalues become further and further removed from the domain of his experience.

To compute the utility of larger monetary gains, the formula used is

U (Gain) × Pr (Gain) + U (Loss) × Pr (Loss) = U (Cash equivalent) (17.1)

which is equivalent to

U (G)p + U (L)(1 − p) = U (C) (17.2)


By rearranging the terms, we obtain formulas for computing the utilities of gains and lossesoutside the range of the decision-maker’s experience:

U (G) = U (C) − U (L)(1 − p)

p(17.3)

and

U (L) = U (C) − U (G)p

1 − p(17.4)

In calculating the utility of gains and losses for extended amounts for commodity traders,Teweles, Harlow, and Stone suggest computing them three times. The reason behind repeatingthe procedure is to find out how consistent the decision-maker is in making decisions outsidethe range of his or her experience. The reader is referred to the excellent discussion by Teweleset al. for a detailed treatment [158].

The main reason for concern about the decision-maker’s consistency for larger gains andlosses than he or she has regularly experienced is this: if judgments do become increasinglyinconsistent for larger gains and losses, then the decision-maker should either exercise greatercaution (something the person will probably do anyway) in evaluating such prospects, or elseavoid them entirely if possible. Otherwise it is likely that decisions will be made for whichthe perceived a priori and a posteriori utilities are different and the decision-maker regrets thedecision after making it, even before the results are in.

In the next chapter we shall examine capital investments with stochastically determinedcharacteristics. The returns from such projects cannot be known ahead of time with certainty.It will be shown, however, that the probable outcomes can be estimated, and such estimatesevaluated by comparison with the decision-maker’s utility function.

Whose Utility?

In the foregoing discussion of utility it was assumed that the utility function of interest is that ofan individual decision-maker. An entrepreneur with sole responsibility for capital investmentdecisions in a proprietorship provides a clear example. For the entrepreneur it is reasonable toassume that the utility function for monetary gains and losses to the enterprise is none otherthan that of the entrepreneur.

But what is to be done for a partnership, in which each partner shares personally in the gainsand the losses? Can we still obtain a meaningful estimate of the utility of monetary gain? Wecan certainly repeat the procedures illustrated earlier, and in so doing obtain a curve relatingcash equivalents to risky alternatives. This author chooses to dodge the question of whether thecurve obtained is really a utility curve or whether an aggregate utility function exists. Instead,let us refer to the curve obtained for all partners in a partnership as the enterprise’s investmentcurve.

More complex conceptual considerations enter the scene when we begin to examine thesituation for enterprises in which ownership is largely separated from management. For suchenterprises (in which we would include public sector undertakings), the utilities of money gainand loss on investments may cover a wide spectrum just among the owners. Those who makethe capital investment decisions may (and most likely do) have personal utility curves that aresubstantially different from what their decisions as managers would indicate. What managersmust do is surmise the enterprise’s investment curve, the composite of the aggregated owners’


preferences. This may be linked to the personal utilities of individual high-ranking managersinasmuch as their job security, bonuses, reputations, and the like may reflect how well or poorlythey serve the owners. Thus, monetary gains and losses to the enterprise may be associatedwith a manager’s personal utility.

To conclude this section, we adopt the operational guideline that the required utility func-tion or investment curve is that of the person or persons who, individually or collectively,are responsible for making investment decisions of the magnitude with which we are con-cerned. In corporations this may mean the board of directors itself for large projects. Oncethe individual or the group responsible for decisions on capital investments has been iden-tified it is theoretically possible to derive an investment curve along the lines illustratedfor an individual. However, in practice there would likely be considerable difficulty in get-ting a group to cooperate fully over a long enough time to obtain enough data points to beuseful.

MEASURES OF RISK

As mentioned earlier, in a risk environment variables are not known with certainty but followprobability distributions. With risk, the outcome of a capital investment decision is a particularvalue, but the value cannot be known a priori. When risk is present, we may be able to predictwith statistical confidence the range of values within which the outcome may be expected tooccur.

For experiments that may be repeated a number of times we can generally estimate theempirical risk parameters. However, capital investment projects are usually one-of-a-kindundertakings; we will not have the opportunity to repeat them. Therefore, empirical estimationof the parameters is not possible and we must instead make subjective judgments about thegoverning probability distributions. Past experience with similar projects may be helpful. Forexample, we may have determined that the useful, economic life of capital investments is, ingeneral, Poisson distributed, or that it is negative exponentially distributed.

When “risk” is discussed in relation to the overall return on an investment, the word hascome to mean the potential variability of the actual return from its expected value. Other thingsbeing equal, the investment with the greatest range of possible results is said to be the riskiest.The most widely used measure for risk is the variance. Figure 17.8 illustrates the concept.Projects RA and RB are both risky; both have a range of possible outcomes associated withthem. The possible outcomes are distributed around the most likely values V A and V B, whichare defined as the expected values of returns on the projects. Project RB is the riskier projectbecause the range of possible outcomes is much wider than for RA. If we were able to say with95 percent (statistical) confidence that project RA would have a value in the range RAL to RAH

Figure 17.8 Comparative riskiness


and project RB a value in the range RBL to RBH, then RAH − RAL < RBH − RBL. Therefore,RB is riskier. Since the range of values, for any particular statistical level of confidence, isa function of the distribution’s variance (or standard deviation),we can say that for projectswhose outcomes follow the same type of probability distribution, the project having the largervariance of outcome is the riskier.

There is a problem with using the variance alone as risk measure, however, although it iseasily adjusted for: the variance does not distinguish between projects of different size. A largeinvestment undertaking may have a larger variance than a smaller undertaking simply becauseit is larger. This problem is readily resolved by substituting for the variance the coefficientof variation. The coefficient of variation is a “risk relative” measure defined as the ratio ofstandard deviation to expected value. The investment having coefficient of variation 2.3 isrelatively riskier than the investment with coefficient 1.9.

Occasionally, we may want to analyze the pattern of net, after-tax cash flows for pastprojects in order to estimate the cash flows for similar projects under consideration now. Suchan endeavor is predicated on a stable relationship of the cash flows to one or more variables thatpresumably can be accurately forecasted. If we fit a regression for cash flows to the independentvariables, one of the pieces of information obtained is the standard error of the estimate. Thismeasures the variability of actual values from predicted values of the regression equation.In other words, the standard error of estimate is analogous to the standard deviation, but iscalculated from deviations of actual values around the computed regression line. Estimationof values from a regression equation with large standard error is riskier than from a regressionwith small standard error.

The question has been posed as to why we use variance to measure risk when people do notfeel the same about bad results as they do about better-than-expected results. This asymmetryof attitude toward deviation above and below expected values can be explained in terms ofdiminishing marginal utility; the increase in utility from a gain above the expected value is lessthan the utility loss from an outcome with the same dollar amount below the expected value.Consider Figure 17.9.

In Figure 17.9 outcomes less than the value V T are considered undesirable. On the other hand,outcomes above V T are considered fortuitous. Yet the variance does not distinguish betweendeviations above the expected value and those below it. Mao [99] and others have suggestedthat decision-makers are sensitive to, and influenced much more by, outcomes with values lessthan the expected value. In other words, when decision-makers speak of risk, they mean thedistribution of outcomes that are worse than anticipated. This is not measured by the variancebut by the semivariance. The semivariance is calculated like the variance, but only for values

Figure 17.9 Risk perception for results better and worse than expected value


less than the targeted value.2 For an empirical distribution this means summing the squares ofthe deviations equal to or below the target and dividing by the number of observations. For atheoretical distribution it means integrating or summing the density function up to the targetvalue. Above this value deviations are ignored.3

A conceptual problem with the semivariance is that fortuitous outcomes are usually ignoredin risk considerations (although they may nevertheless enter determination of the target value).We might expect that instead of ignoring better-than-expected results, they should to someextent counteract the worse-than-average outcomes that are possible. If we were to use util-ity weights instead of dollar NPV or other value weights, this problem might be alleviated.Although the variance may leave much to be desired as a risk measure, the semivariance offersa debatable advantage to it, and a generalized utility-based distribution may not be obtainable.

In subsequent chapters the variances and covariances of returns are used to illustrate port-folio selection approaches to investment and the capital asset pricing model. Although thesemivariance could, in principle, be substituted, it is not. The reason for this is that the mathe-matical calculations are much less tractable for the semivariance. This suggests a reason whydevelopment of the literature on investments has employed the variance or coefficient of vari-ation almost exclusively. To the extent efforts have been made to use the semivariance, it hasbeen by those who have attempted to modify results obtained originally with variance as therisk measure for a specific application. General substitution of the semivariance has not beensuccessful.

J. C. T. Mao’s Survey Results

Mao’s survey was based on personal interviews with managers of eight medium and largecompanies in electronics, aerospace, petroleum, household equipment, and office equipmentfirms. He found that decision-makers do indeed think of risk as being related to outcomesthat are worse than expected, and that the possibility of a bad outcome significantly influencesthe capital investment decision. To verify the relevance of semivariance he designed a testin which the decision-makers were asked to choose between two (hypothetical) mutuallyexclusive investments, each costing the same amount and returning after one year the cost plusprofit or loss. The probability distributions of the returns were as shown in Figure 17.10.

The reader may readily verify an interesting property of the distributions of projects A andB: they both have expected value of 3 and variance of 4. Since they require the same investmentoutlay, their coefficients of variation are identical, and therefore their risk as conventionallymeasured. But are they equally risky? Mao’s respondents did not think so. Their answers wereconsistent with the semivariances (using zero in each as the critical value) that were 0.2 forproject A and 0 for project B. Note also that there is a nonzero probability that project A couldactually lose money.

To summarize Mao’s findings, they are that: executives were more likely to choose A if theirbusinesses accustomed them to that degree of risk, if they personally preferred risky ventures,and if they could control the loss possibility in A through diversification. When these conditions

21t is necessary to say “targeted value” rather than “expected value” because an arbitrary value may be used in calculating the semi-variance. The arbitrary value could be the expected value of the distribution of outcomes, as is the case with the variance, but it mayjust as well be some critical value established by management policy.3Mao [99, p. 353] describes the semi-variance as follows: consider investment return R, as a random variable with known probabilitydistribution. If h stands for a critical value against which the actual values of R are compared, and (R − h)− stands for (R − h) if(R − h) ≤ 0 and for zero if (R − h) > 0, then Sh (semi-variance with h the reference point) is given by the formula: Sh = E[(R − h)−]2

where E is the expectation operator.


Figure 17.10 Mao’s test distributions

are absent, the executive is more likely to pick B because the absence of loss possibility makesit a more secure investment.

The previous analysis was concerned with a choice between two individual investments. Itis equally important to examine the risk concept from the portfolio viewpoint. The executiveswere asked to imagine the same alternatives on a larger scale, where X represented the totalcompany investment, and questioned as to which portfolio they would select. The answerwas unanimously B, and their reasons were most succinctly voiced by the statement of oneexecutive:

The key is survival. We take a chance on evaluating individual projects rather optimistically, but wewill not take a chance on the main company. One of our obligations is to sustain this company in lifeand every time we put it in a minus position, it dies a little bit if not in total. This evidence is consistentwith semi-variance as a concept of risk. . . . [99, p. 355.]

RISK OF RUIN

If all capital investments were small relative to the enterprise’s financial resources, this sectionwould not be necessary. However, capital investments are often large relative to the enterprise’ssize as well as absolutely. Because of this an adverse investment result may impair the firm’scapital; several such adverse investments with one right after another might destroy the firm.Therefore, in addition to analysis of individual projects, it will be useful to examine therelationship of project size relative to firm size within the context of the risk affecting theprobabilities of return on investment.

Risk of ruin, like many useful statistical concepts, has its origins in gambling. It is applicableto other situations in which a series of losses (adverse outcomes) can result in ruin (loss ofequity). Teweles, Harlow, and Stone illustrated application of the concept to an individual whoundertakes the trading of commodity futures. This can be a very risky venture because of thesmall margin requirements, usually around 5–10 percent of the commodity contract value.Their discussion is highly recommended to readers who wish to reinforce their understandingof the concept and its application. Here we shall focus attention on risk of ruin as it applies tocapital investment decisions.

The probability of eventual ruin (loss of equity, bankruptcy) is given by

R =(

1 − A

1 + A

)C

(17.5)

where A is the investor’s advantage in decimal form and C is the number of investment units withwhich the investor starts [48]. For capital investments we may define the investor’s advantageas the probability of a favorable outcome minus the probability of an unfavorable outcome


resulting in monetary loss.4 If P is the probability of a favorable outcome, this means that theadvantage is P − (1 − P) or 2P − 1. To obtain C we divide the required investment outlayinto the enterprise’s total equity. (If desired, a lesser amount may be used.)

To illustrate, let us take the case of a company considering an investment of $40 million in aproject that is thought to offer 0.60 chance of returning 100 percent and a 0.40 chance of lossof the entire investment. The firm has equity of $160 million. Translating these numbers intorisk of ruin, we get

R =(

1 − 0.2

1 + 0.2

)4

= 0.198 (17.6)

or about one chance in five of eventual ruin.This one project cannot by itself ruin the firm. However, there is 0.40 chance that the entire

investment will be lost. The significance of the risk of ruin is this: if the firm makes a habit ofundertaking investments of similar risk, cost, and return, there is one chance in five that the firmwill go bankrupt as a result. Thus, risk of ruin relates to the firm’s ongoing policy with respectto investments. Should the firm regularly undertake investments for which the probability ofeventual ruin is one-fifth? Is this an exceptional case? If so, how does it relate to the firm’sinvestment policy? Examination of the formula for risk of ruin yields some principles thatmay be useful in policy formulation. Of major significance is the fact that by reducing the sizeof investments that may be considered, the risk of ruin may be greatly reduced, other thingsbeing constant. In the example just considered, reducing the size of the investment to $20million reduces risk of eventual ruin to 0.039 or about four chances in 100. Halving the sizeof investment outlay in this case reduced the risk of ruin by 80 percent. Although one chancein five would be considered an unacceptable risk of ruin by many executives, four chances in100 might well be considered quite good odds.

Risk of ruin as presented above is somewhat an abstraction. Firms do not have continuingopportunity to undertake a series of identical capital investments over the time continuum.Furthermore, enterprise policy is not immutable; it tends to change to adjust to circumstances.If a firm has a run of losses on its investments, sooner or later it will review its investmentpolicies, objectives, and procedures in order to correct its practices.

Nevertheless, risk of eventual ruin provides a gauge for measuring the enterprise’s approachto risky investments. It indicates management’s attitude toward the ultimate investment risk —that of financial ruin, bankruptcy. The measure itself could be incorporated into the decisionprocess as a constraint. For instance, a policy might be adopted constraining investments torisk of ruin of X chances in 100 (based on the calculation above, which assumes the sameinvestment outlay and success/failure probabilities to be repeated over time).

SUMMARY

Risk, in the context of this chapter, was considered without regard to portfolio effects. Thischapter introduced the subject of risk and the notion that risk cannot be adequately handledwithout reference to utility. We saw that the utility function governs the investor’s attitudetoward risky investments, and that diminishing marginal utility of monetary gain leads to riskaversion.

4Ruin, in the sense we are using the word here, rests upon the probability of actual out-of-pocket money losses, if ruin would beconsidered to result from its loss.


Risk is normally measured by the variance or coefficient of variation of the outcome variable.Professor Mao’s interviews with executives who make capital investment decisions yielded at-titudes congruent with semivariance as a measure of risk actually representative of managementthought.

Even though capital investments are typically one of a kind, the concept of risk of eventualruin was introduced. By calculating the probability of ruin it is possible to assess more fullythan otherwise managements’ attitudes toward risky investments. Risk of ruin also ties in wellwith one of Mao’s survey respondents who put survival of the enterprise ahead of any profitaspirations.

18

Single Project Analysis under Risk

This chapter considers methods that are used to analyze capital investment projects when oneor more project characteristics are random variables. Independence of projects (with respectto the existing assets of the firm as well as one another) is assumed throughout this chapterin order to concentrate on risk considerations without the added complexities introduced byportfolio effects. Project analysis with diversification and project interdependence is coveredin Chapter, 20.

Chapter 17 considered those factors that influence our attitudes toward risk and introducedthe concept of risk of ruin. In this chapter we examine means for measuring risk and dealingwith it. Among the matters to be covered here are payback as a risk-coping method, the certaintyequivalent approach, the risk-adjusted discount rate approach, and computer simulation.

THE PAYBACK METHOD

Payback has been rationalized as a method for coping with risk. If we recognize that con-siderable importance may be attached to capital preservation, it is understandable that somemeasure of solace may be gained from the knowledge that a particular investment projectpromises a quick return of the funds invested in it. The longer the time required to recover aproject’s investment cost, the more uncertain we are about the recovery. The more time thatpasses between the present and an anticipated event, the more the unanticipated influences thatcan spring up to render our forecast results worthless.

The shortcomings of the payback criterion are well known; Chapter 5 discusses them. As ameans of dealing with risk or uncertainty, payback does favor those projects that promise earlyrecovery of investment. It does not make much sense to use payback as the sole criterion or evenmajor criterion, but, used as an adjunct to the discounted cash flows, it provides an unequivocal,if crude, means of screening projects to eliminate those that take what management considerstoo long to recover their investment and, as a consequence, are all too likely to fail to helppreserve the enterprise’s capital.

To an extent, reliance on payback may be tantamount to an admission of forecasting impo-tence. That is, if we feel it is not at all possible to make usefully reliable forecasts, then wemay adopt the project that exposes our capital to the ravages of time the least. After havinggot back our investment, if the project then offers further returns, that is all to the good; ifnot, we still have our funds to invest in something else. In relatively stable economic timespayback may have little to recommend it. In times like the present, with forecasting madeall the more difficult by one crisis after another communicated instantly around the world,compounded with periodic surges of inflation, preservation of capital is not a trivial matter, ifit ever was. It is understandable that payback holds a prominent position among the methodsof capital asset selection used by those who must make decisions with their firm’s resourcesand be held accountable for results. And if a project selected on the basis of payback does notperform according to expectations, it will be recognized early, within the payback period, and


appropriate measures taken — no need to wait years for a return only to find it is a will-o-the-wisp that vanishes as approached.

CERTAINTY EQUIVALENTS: METHOD I

In the previous chapter it was shown how a utility curve can be estimated by determining the(certain) cash amount that is considered equivalent to two different risky amounts, each witha probability attached to it. The cash was considered equivalent to the risky investment, whichcould yield either more or less than the cash equivalent depending on the outcome. This isthe essence of the certainty equivalent approach to capital investment analysis. If amount Amay be received with probability P(A) and amount B with probability P(B) = 1 − P(A),and the events are mutually exclusive,1 then amount C is said to be a certainty equivalent tothe investment that could result in outcome A or outcome B, and P(C) = 1.0. This can begeneralized to investments with more than two outcome possibilities.

For investments with more than two outcome possibilities, the certainty equivalent, C ,may be defined as the (certain) cash amount for which the investor is indifferent to the riskyinvestment for which the dollar returns are described by the probability distribution P(I ),where I is a vector. It is important to note that C is not the expected value of the investment,E[I ]. Rather, C is the certain cash amount with utility that is equivalent to the investment. Inother words, if U denotes utility, then

U (C) = E[U (I )] (18.1)

Because it is the utility of the returns vector I that determines investor attitudes toward theinvestment, it is unimportant in determining certainty equivalents whether I is the vector of netpresent values (NPVs), internal rates of return (IRR), or other measure of investment returns.In this chapter we use the NPV as a matter of convenience only, noting that the certaintyequivalent may be based on any measure for which the investor can make consistent choices.

Although utility itself is a personal matter, we may, as suggested in the previous chapter,estimate a composite investment curve for the decision-makers of a particular organization.2

To illustrate the use of certainty equivalents, in Figure 18.1 a risk–return trade-off curve fora given enterprise is illustrated. All points of the curve have equal utility, so we could referto it as an iso-utility curve. Different organizations may be expected to have different curves.Figure 18.1 is consistent with risk aversion because expected returns must rise more rapidlythan risk (however risk is measured) for the utility to remain constant.

Example 18.1 The Ajax Company is evaluating a production mixer that costs $25,000. Themixer is expected to produce cash flows of $10,000, $15,000, $20,000, and $20,000 at the endof each year of its four-year economic life. The firm’s management believes that the cash flowsbecome riskier the further into the future they are expected to occur. Certainty equivalentsare calculated at 0.982, 0.930, 0.694, and 0.422 of the expected cash flows. Since certaintyequivalents are used, we discount at the risk-free rate,3 which is assumed to be 8 percent. The

1Mutually exclusive may be defined in terms of the conditional probabilities as P(A|B) = 0 and P(B|A) = 0.2Some authors have suggested that the equal market valuation curve be employed in calculating certainty equivalents. Although suchan approach may be useful in securities investment analysis, it is questionable within the context of capital investment analysis for anindividual firm.3Here risk free means free of default risk. The rate is generally taken to be the rate on US government securities of the same futurematurity as the cash flow’s receipt.

Single Project Analysis under Risk 193

Figure 18.1 Risk–return trade-off curve for the enterprise

NPV of the mixer is

NPV = $ − 25,000 + (0.982)($10,000)

1.08+ (0.930)($15,000)

(1.08)2

+ (0.694)($20,000)

(1.08)3+ (0.422)($20,000)

(1.08)4

= $ − 25,000 + $9092.59 + $11,599.88 + $11,018.39 + $6203.65

= $13,274.51

which is positive; thus the project is acceptable under the NPV criterion.

CERTAINTY EQUIVALENTS: METHOD II

The previous section shows one way in which certainty equivalents may be found and usedto calculate a project’s certainty equivalent NPV. Another method that is sometimes used isfirst to calculate the project’s NPV at the risk-free interest rate. The next step is to convertto a “certainty” equivalent by subtracting an allowance for risk, usually taken to be V [NPV],the variance of NPV. To illustrate, assume the variance of the NPVs of the cash flows inExample 18.1 is V [NPV] = $14,422.07. First calculate the NPV of the expected cash flows atthe risk-free rate (again assumed to be 8 percent):

NPV = $ − 25,000 + $10,000

(1.08)1+ $15,000

(1.08)2+ $20,000

(1.08)3+ $20,000

(1.08)4

= $ − 25,000 + $9259.26 + $12,860.08 + $15,876.64 + $14,700.60

= $27,696.58

Next, subtract the variance to obtain the certainty equivalent:

NPV = $27,696.58

−V (NPV) = −14,422.07

$ 13,274.51 certainty equivalent


In this case the certainty equivalent is identical to that obtained with method I; this is becausethe variance was chosen to make the results identical.

In general, it is unlikely that this method will yield results close to those obtained withmethod I. Although this is empirically questionable, we can say a priori that for this methodto yield the same results as method I, it is necessary that the variance measure risk in thesame way that calculation of certainty equivalents for the individual period cash flows does.Considering that the certainty equivalents of the individual cash flows are based on utility,and the variance is based on dollars alone, the two methods will not yield comparable resultsexcept for investment curves of a most particular type.4

Despite the conceptual problems imbedded in method II, it is sometimes used, and it isthen necessary to have equations for calculating the expected value of NPV, E[NPV], and thevariance of NPV, V [NPV]. The expected value of NPV is found from

E[NPV] =N∑

t=0

E[Rt ]

(1 + i)t(18.2)

where R0 is the required investment outlay. The variance is found from the relationship

V [NPV] = σ 20 + σ 2

1

(1 + i)2+ σ 2

2

(1 + i)4+ · · · + σ 2

N

(1 + i)2N(18.3)

if the cash flows are independent of one another (uncorrelated). If the cash flows are perfectlycorrelated, the variance is defined by

V [NPV] = σ 20 + σ 2

1

(1 + i)2+ σ 2

2

(1 + i)4+ 2

{Cov(0, 1)

(1 + i)1+ Cov(0, 2)

(1 + i)2+ Cov(1, 2)

(1 + i)3· · ·

}(18.4)

For the general case with three cash flows that are random variables, X0, X1, X2 with weightsa, b, c, the variance is given by

V [NPV] = V [aX0 + bX1 + cX2]

= a2V [X0] + b2V [X1] + c2V [X2](18.5)+ 2ab Cov[X0, X1] + 2ac Cov[X0, X2]

+ 2bc Cov[X1, X2]

where a = (1 + i)–0 = 1, b = (1 + i)–1, c = (1 + i)–2. The complexity of this expressionsuggests why we consider the special cases of zero correlation and perfect correlation, forwhich the considerably simpler equations apply.

RISK-ADJUSTED DISCOUNT RATE

Another method for dealing with risky cash flows is that of the risk-adjusted discount rate.Although the certainty equivalent method essentially adjusts the cash flows or the NPV of the

4The assumption that the distribution of returns is normally distributed means that the distribution is fully described by two parameters:mean and variance. This implies that utility may be maximized by appropriate selection of assets according to a function of meanand variance, provided we know the equation for the investor’s utility in terms of these parameters. Certainty equivalent method II,by subtracting the unweighted variance of returns, assumes a very special type of utility function in addition to normally distributedreturns.


unadjusted flows, the risk-adjusted discount rate makes the adjustment to the discount rate foreach cash flow period prior to calculating the NPV.

In Chapter 6 the NPV method was discussed in the context of a certainty environment.There it was assumed that the enterprise’s cost of capital was the appropriate rate of discount,5

and that this was constant from period to period over the asset’s economic lifetime. Now,assuming a risky environment in which the cash flows differ both in futurity and in uncertainty,it is no longer reasonable to assume that a constant discount rate is appropriate. Ceteris paribus,the riskier an investment return occurring sometime in the future, the greater the required rateof return — discount rate — that investors will apply to it.

Example 18.2 To illustrate the method of risk-adjusted discount rate let us again considerthe Ajax Company mixer from Example 18.1. The cost of the mixer is $25,000 and the cashflows expected at the end of each year of its four-year economic life are $10,000, $15,000,$20,000, and $20,000.

The NPV, using the risk-adjusted discount rate approach with rates k = 9.98 percent, k2 =11.99 percent, k3 = 21.98 percent, and k4 = 34.00 percent, is

NPV = $ − 25,000 + $10,000

(1.0998)1+ $15,000

(1.1199)2+ $20,000

(1.2198)3+ $20,000

(1.3400)4

= $ − 25,000 + $9092.56 + $11,960.04 + $11,019.56 + $6203.68

= $13,275.84

which is within 0.01 percent of the certainty equivalent approach result. The difference is dueto rounding errors in this case because the risk-adjusted discount rates were chosen to yieldidentical results.

If α is the constant that relates the certainty equivalent to its expected value of cash flow,and i the risk-free rate, then the relationship between these variables, as Mao has shown, is

Certainty equivalentαE[Rt ]

(1 + i)t

Risk-adjusted discount rateE[Rt ]

(1 + kt )t

(18.6)

which, after rearranging terms, yields

kt = 1 + i

α1/t− 1 (18.7)

An investor who is perfectly consistent in applying this method and the certainty equivalentmethod should obtain identical results (except for rounding errors). Whether or not decision-makers are consistent in applying the methods is an empirical question, as is the question asto which method is preferred (if either is) by the same decision-makers.

COMPUTER SIMULATION

Although the foregoing methods for dealing with risk can be useful, they have some short-comings. Neither the certainty equivalent nor the risk-adjusted discount rate provides more

5Some would argue that the cost of common equity is the appropriate rate since the net, after-tax cash flows accrue to the commonshareholders. See, for example, J. C. T. Mao [100, p. 138].


than a single, point estimate of the returns on a capital investment. Neither is suited to riskyinvestments for which characteristics other than the cash flows themselves may be randomvariables. For example, the project life itself may be a random variable, or at least we maywish to examine the implications if it is treated as one. Circumstances may exist that compelus to treat the discount rate itself as a random variable, and we may have reason to want totreat the related cash flows according to a more complex relationship than simple correlationallows. And, we may want to examine the effect of having a particular policy that requires theasset to be abandoned if certain conditions are met, or we may wish to examine the effects ofrandom shocks, such as a sudden increase in petroleum price.

Computer simulation (also referred to as Monte Carlo experimentation) enables us to an-alyze complex investments for which direct mathematical methods either do not exist or areinadequate for our purpose. Because of the speed with which modern digital computers canperform complex, repetitive calculations, we can examine the simulated outcome of thousandsof iterations that, taken together, provide a profile of the investment. Through analysis of thesimulated profile, within the context of management’s investment curve (a notion developedin the previous chapter), a rational decision may be reached to accept or reject the project.

The topic of computer simulation is covered thoroughly in texts devoted exclusively tothe subject. For those who desire a detailed development we recommend those works beconsulted. It is not our purpose here, however, to become sidetracked into a detailed discussionof computer simulation. For our purposes it should suffice to state that three basic things arerequired: (1) a computer; (2) a (pseudo-) random number generator; and (3) a mathematicalmodel of the thing we wish to simulate. Since serviceable random number generators are anintegral part of much computer software nowadays, to perform a simulation analysis we needbe mainly concerned with the task of specifying our model carefully, then translating it intosuitable form for the computer.

It is often recommended that one should use the risk-free rate (i.e. the yield on 90-day USTreasury bills) in computer simulations because the simulated distribution itself reveals theproject’s risk. However, the NPV profile obtained with the risk-free rate is difficult to interpretand relate to the NPVs of other projects. Therefore, simulations are in practice often performedusing the organization’s cost of capital. The following example was analyzed with KAPSIM,a program written by this author to simulate single capital investment projects. It can be donewith other simulation packages and add-ins for popular spreadsheet packages or even withina spreadsheet itself with contemporary spreadsheet programs with a little programming work.

Example 18.3 An investment costs $2.5 million and is expected to last nine years. The projectlife is thought to be Poisson distributed, so the variance is also nine. Management policy is todispose of such assets after 20 years; thus if the project should last that long, it would then beabandoned.

The variance of the net, after-tax cash flows is estimated to be ($2 million)2 every year. Thefirst-year cash flow has expected value of −$5 million; that of the second year is +$5 million;from the third through the ninth year the returns are expected to grow by $1 million each year,starting at $6 million at the end of the third year. The returns in years 10 and beyond are zerobecause the project is expected to last only nine years.

A further complication is that, because of structural changes in the economy, the discountrate is expected to increase over time. The expected rate is 10.1 percent for the first year’s cashflow, with variance (1.01 percent)2. Each subsequent year is expected to be 0.1 percent higher,and the standard deviation to be 10 percent of expected value.


Table 18.1 Example 18.3 problem description (Part 1)

Item Distribution Parameter 1 Parameter 2 Parameter 3

Initial outlay Constant 2.50000E + 06 0 −0.0Period 1 return Normal −5.00000E + 06 2.00000E + 06 −0.0Period 2 return Normal 5.00000E + 06 2.00000E + 06 −0.0Period 3 return Normal 6.00000E + 06 2.00000E + 06 −0.0Period 4 return Normal 7.00000E + 06 2.00000E + 06 −0.0Period 5 return Normal 8.00000E + 06 2.00000E + 06 −0.0Period 6 return Normal 9.00000E + 06 2.00000E + 06 −0.0Period 7 return Normal 1.00000E + 07 2.00000E + 06 −0.0Period 8 return Normal 1.10000E + 07 2.00000E + 06 −0.0Period 9 return Normal 1.20000E + 07 2.00000E + 06 −0.0Period 10 return Normal 0 2.00000E + 06 −0.0Period 11 return Normal 0 2.00000E + 06 −0.0Period 12 return Normal 0 2.00000E + 06 −0.0Period 13 return Normal 0 2.00000E + 06 −0.0Period 14 return Normal 0 2.00000E + 06 −0.0Period 15 return Normal 0 2.00000E + 06 −0.0Period 16 return Normal 0 2.00000E + 06 −0.0Period 17 return Normal 0 2.00000E + 06 −0.0Period 18 return Normal 0 2.00000E + 06 −0.0Period 19 return Normal 0 2.00000E + 06 −0.0

Note: Number of iterations equals 1000; expected value of project life is 9 with variance of 9;distribution Poisson; maximum project life is 20.

Finally, because the project may be abandoned prior to the end of its economic lifetime, itis necessary to estimate salvage at the end of each year. Salvage value is assumed to declinefrom the original investment by 20 percent each year. However, the rate of decline is estimatedto be Poisson distributed.

This information is summarized in Tables 18.1 and 18.2, which contain the provided infor-mation as part of the computer-printed simulation results. One thousand iterations are run toobtain a sufficient sample of results, which are measured by the NPV. If too few iterations areused, we cannot draw conclusions from the scattered results, whereas too many iterations wouldoffer little or no additional useful information. The cost of computation has come down greatly(thanks to constantly improving technology). So, it may be best to err somewhat on the side oftoo many iterations rather than too few if there is any doubt about how many should be used.

Tables 18.1 and 18.2 provide a record of the information we put into the computer simulationof the investment project. All simulations should be accompanied by such “echo” output for tworeasons: (1) in order to find mistakes that may have been made in specification of the problemsor preparation of the data and (2) in order to provide an organized record to accompanythe simulated results. Simulation output for one case may look about the same as output foranother. If the input data are attached to the computer printout, one can avoid the frustrationand difficulty of trying to reconstruct the assumptions that produced a particular computerrun’s output.

Table 18.3 contains the frequency interval bounds, and frequency distribution of the simu-lated NPV results for Example 18.3. The frequency count for an interval is determined by thenumber of results that are greater than or equal to the lower bound and strictly less than theupper bound. Figure 18.2 contains a graph corresponding to the values of Table 18.3.


Table 18.2 Example 18.3 problem description (continued)

Item Distribution Parm 1 Parm 2 Parm 3

Period 1 capital cost Normal 0.101000 0.010100 000000Period 2 capital cost Normal 0.102000 0.010200 000000Period 3 capital cost Normal 0.103000 0.010300 000000Period 4 capital cost Normal 0.104000 0.010400 000000Period 5 capital cost Normal 0.105000 0.010500 000000Period 6 capital cost Normal 0.106000 0.010600 000000Period 7 capital cost Normal 0.107000 0.010700 000000Period 8 capital cost Normal 0.108000 0.010800 000000Period 9 capital cost Normal 0.109000 0.010900 000000Period 10 capital cost Normal 0.110000 0.011000 000000Period 11 capital cost Normal 0.111000 0.011100 000000Period 12 capital cost Normal 0.112000 0.011200 000000Period 13 capital cost Normal 0.113000 0.011300 000000Period 14 capital cost Normal 0.114000 0.011400 000000Period 15 capital cost Normal 0.115000 0.011500 000000Period 16 capital cost Normal 0.116000 0.011600 000000Period 17 capital cost Normal 0.117000 0.011700 000000Period 18 capital cost Normal 0.118000 0.011800 000000Period 19 capital cost Normal 0.119000 0.011900 000000Period 20 capital cost Normal 0.120000 0.012000 000000

Note: Salvage assumed to decline from initial outlay by percent per year of: Poisson2.00000E − 01, 2.00000E − 01, 0.

Table 18.3 Example 18.3 NPV frequency distribution

FrequencyInterval Lower bound Upper bound ≥ = LB, < UB

1 −0.308229000E + 08 −0.284193447E + 08 12 −0.284193447E + 08 −0.260157834E + 08 43 −0.260157834E + 08 −0.236122341E + 08 24 −0.236122341E + 08 −0.212086788E + 08 65 −0.212086788E + 08 −0.188051235E + 08 76 −0.188051235E + 08 −0.164015682E + 08 207 −0.164015682E + 08 −0.139980129E + 08 278 −0.139980129E + 08 −0.115944576E + 08 209 −0.115344576E + 08 −0.919090230E + 07 35

10 −0.919090230E + 07 −0.678734700E + 07 4311 −0.678734700E + 07 −0.438379170E + 07 3712 −0.438379170E + 07 −0.198023640E + 07 7413 −0.198023640E + 07 0.423318900E + 06 10514 0.423318900E + 06 0.282687420E + 07 15615 0.282887420E + 07 0.523042950E + 07 17616 0.523042350E + 07 0.763398480E + 07 14017 0.763398480E + 07 0.100375401E + 08 10118 0.100375401E + 08 0.124410354E + 08 3519 0.124410954E + 08 0.148446507E + 08 920 0.148446507E + 08 0.172482060E + 08 1

Note: Avg. NPV = 0.631806495E + 07; std. dev. of NPV = 0.762450328E + 07; Low =−0.308229000E + 08; high = 0.172482060E + 08; range = 0.480711060E + 08.


Figure 18.2 Frequency histogram for Example 18.3 computer simulation

The decision to accept or reject the investment may be made from the frequency distribution,which is clearly not normally distributed. The first 12 intervals contain only negative NPVresults, whereas the thirteenth contains mixed results. If we may assume the distribution withinthe thirteenth interval to be uniformly distribu ed over that interval −0.198023640E + 07 to+0.423318900E + 06, a range of 0.2403553E + 07, then (198/240) × 105, or 87 of the 105simulated observations in that interval are negative, and 18 are positive. Therefore, there isprobability of (276 + 87)/1000 or 363/1000 = 0.363 that this project, if accepted, wouldyield a negative NPV, or better than one chance in three.

Let us say that management will not accept any project for which the odds of losing morethan the initial investment (in this case $2.5 million) exceed 1/10. If we may again assumeresults to be proportionately split over the twelfth interval, 58 of them in the interval are −$2.5million or less. This means the odds are 260/1000, or 0.260 of losing more than the initial


investment (a little less if we were to consider the negative cash flow at the end of the first yearas part of the initial investment). This project would be unacceptable to management underthis policy, even though the mean NPV is $6,318,065.

A virtually limitless number of management policies may be applied to the frequency dis-tribution. The particular ones applied will depend on the enterprise’s investment curve, whichreflects the risk attitudes of its management. For instance, management might consider theproject acceptable if the odds for an NPV exceeding $10 million were greater than the oddsof NPV loss of more than $20 million (on the assumption that management could, and would,intervene to prevent such loss actually accumulating; that is, changing the odds during thegame). Alternatives are limited only by one’s imagination.

This example was not designed to represent necessarily a realistic project, but rather toillustrate how simulation analysis can facilitate the decision-making process for a complexproject. No doubt the example may be made more realistic simply by changing some of theparameters. As it stands, this hypothetical project would probably not be very attractive toa risk-averse, capital preservation-oriented management because there are fairly large oddsthat losses with NPV as large as −$30 million could result from adopting this project. Unlessmanagement believes that the odds can be significantly improved by intervention as time passes,this project will be passed over by most executives if we may extrapolate from Mao’s surveyfindings cited in the previous chapter.

LEWELLEN–LONG CRITICISM

Wilbur G. Lewellen and Michael S. Long, in an insightful comparison of simulation resultsto those obtained from point estimates, draw some thought-provoking conclusions. Theseshould be considered carefully, because of implications that may limit the classes of invest-ment projects that are suitable for Monte Carlo treatment. In summary, the Lewellen–Longconclusions are:

1. If the IRR is used in simulating a capital investment, the relationship between the cash flowsand the IRR will cause the mean IRR to be lower than the IRR obtained by discounting themeans of the respective cash flow distributions. This holds even if the cash flow distributionsare symmetrical about their respective means.

2. By using NPV rather than IRR as the measure of a simulated investment, the bias is elimi-nated because the present value of a future cash flow is a linear function of the size of thecash flow, whereas the IRR is not.

3. The discount rate that should be used in present value simulations is the risk-free rate (freeof default risk, such as federal government securities).

4. Even with the NPV, problems arise when the project lifetime is uncertain. Variable projectlife will, ceteris paribus, cause the mean of the resulting simulated NPV distribution to beslightly less than the NPV obtained by discounting the cash flows over the mean project life.

5. Further disparities can be caused by nonsymmetrical cash flow distributions. This problemis likely to arise when a “most likely,” or modal value, is used in place of the mean inspecifying the simulation.

6. Since, because of the cost and time required to properly perform a computer simulation,it will be limited to the larger and more important projects, most will be evaluated by othermethods, using a point estimate of each cash flow. It is therefore important that expectedvalues rather than modal values be employed.


7. If any elements of the cash flows depend on multiplicative combinations of stochasticvariables, it is important to use the cash flows directly in obtaining the expected values.Only if the multiplicative components are completely independent can E[p · q] be treatedas equal to E[p] · E[q]; otherwise E[p · q] = E[p] · E[q] + σpq where σpq is the co-variance between p and q.

For example, if p represents price and q quantity sold, we should recognize that in allbut perfectly elastic or inelastic demand, price and quantity are correlated. Since price andquantity demanded are inversely related, indirect estimation obtained by multiplying E[p]by E[q] will overstate E[p · q]. Because the correlation coefficient ρ is defined by

ρ = σpq

σpσq

the relationships are

E[p · q] > E[p] · E[q] if ρ > 0

E[p · q] < E[p] · E[q] if ρ < 0

8. All the above problems are either artificial or avoidable, with the exception of the variableproject life. Therefore, as an index of investment project worth the point-estimate analysisshould serve quite well. While not incorrect in determining a proposal’s expected return,simulation is unnecessary.

9. The claim that a major benefit from simulation analysis is that it provides an improvedappreciation of the risk the enterprise would assume may be misleading and evendangerous. It is not the riskiness of the individual investment (its “own” risk) which isimportant, but rather its risk within the context of a collection of assets to which its returnsmay be related. To quote Lewellen and Long [89]:

. . . the one irrelevant feature of an asset’s prospective returns is its “own risk” — the outcome uncer-tainties unique to the asset itself. These can, and will, be diversified away by individual investors andby institutions in their securities portfolios, leaving only the degree of correlation between the asset’sreturns and those of the so-called “market portfolio” as relevant to value, since this connection andits implied risks cannot be extinguished via diversification.

And they conclude by stating that [89, p. 31]:

Simulated profiles of capital expenditure proposal outcomes are, perforce, chiefly descriptions of “ownrisk.” They do not reveal a project’s addition to . . . total risk. Having simulated, therefore, the analyststill cannot legitimately compare and choose among alternatives, because nowhere in the informationhe obtains is the single most important criterion of investment worth — portfolio impact — addressed.However imperfectly, the cost-of-capital/single-point-expected value cash flow forecast approach doesget at the issue of portfolio context. For that reason, . . . not only is such an approach less onerous inexecution but, paradoxically, also has a higher potential for relevant risk recognition. . . .

Although portfolio effects and diversification are topics that are addressed in the followingchapters, the Lewellen–Long criticism of simulation should cause us to reflect carefully onthe facts and circumstances surrounding an investment proposal before boldly embarking ona simulation analysis. And there are circumstances under which a simulation analysis may bepreferable to using the simplified analysis based on point estimates. These might include thefollowing:


1. The proposal is under consideration by an entrepreneur who will be putting all his or herassets into the project if it is accepted.

2. It is not possible to obtain meaningful estimates of the proposal’s correlation with theexisting assets of the firm, or the project being considered is estimated to be uncorrelatedwith them.

3. The project is large in relation to the firm and has a potential for large loss that could threatenthe firm’s survival.

4. Possible pressure on executives to avoid investments in risky assets that would reduce orleave unchanged the organization’s overall risk. If such pressures have been responsiblefor portfolio managers generally avoiding “second-tier” securities, gold, and commodityfutures then we may surmise that similar peer group or external pressures may act toconstrain decisions to “own risk” more than portfolio considerations.

5. The cash flows may be governed by complex relationships which preclude analysis ofpoint estimates. (The Lewellen–Long article did not deal with the problem of simultaneousvariability in cash flows and project life or contingent cash flows, for example.) Althoughsome fairly complex proposals may be analyzed with point estimates, considerable timemay pass before the analyst develops the insight and inspiration that may be necessary.Contingent asset relationships of the type treated in the following chapters, for example,may be impossible to deal with on a point-estimate basis.

6. The matter of risk in the public sector may require a different approach even if the CAPMwere unequivocally and universally accepted for firms in the private sector.

7. The CAPM rests upon a foundation of restrictive assumptions that may not always apply,particularly for smaller firms. For additional insight into the pros and cons of simulationfor the analysis of risky investments, an article by Stewart Myers [113] is recommended.The matter remains controversial and is likely to remain so, at least until some additionalquestions about the CAPM can be unequivocally answered.

With the conclusion of this chapter we shall leave the domain of risk without diversificationand enter that of capital investment in which project proposals are not independent and portfolioaspects must be considered.

19Multiple Project Selection under Risk:

Computer Simulation and Other Approaches

Previously we primarily considered capital investment selection under risk for individual can-didate projects. To simplify exposition we assumed they and existing assets were independent.But projects often are interrelated, and therefore cannot be evaluated in isolation. Up to now itwas useful to assume that all relevant asset characteristics were contained in expected returns,variances and covariances. That enabled selecting efficient combinations of investments: thosewith maximum expected return for a given level of risk or, alternatively, those with minimumrisk for a given level of return.

In this chapter we examine further the problem of capital investment selection. Now, however,we shall not require that returns be normally distributed or that the relationship between projectreturns and existing assets be constant intertemporally. Additionally, we shall consider theproblem introduced by sequential, event-contingent decisions over time. Specifically, mattersrelating to parameters other than the mean and the variance–covariance are considered, as aresimultaneous variability of project characteristics.

DECISION TREES

The decision tree approach is a widely employed analytical device for addressing investmentprojects that contain sequential, event-contingent decisions. Such projects, in other words, havecertain chance-determined outcomes during their lifetimes and these outcomes may influencedecisions made during the project lifetime and interact with them.

Magee’s paper [95] is now a classic on the topic of decision trees, and was responsiblefor focusing attention on this important technique, which others subsequently built upon.Two years prior to Magee, Masse [102] criticized the view of investments as single-perioddecisions isolated from future events. He noted that future alternatives are conditioned bypresent choices. Because of this, investment evaluation cannot be reduced to a single decision,but must be viewed as a sequence of decisions over time.

Decision trees are similar in some respects to Markov chains. They differ in that they includeimbedded decisions and not solely probabilistic event nodes. A Markov chain has the propertiesthat (1) the possible outcome set is finite; (2) the probability of any outcome depends only onthe immediately preceding outcome,1 and (3) the outcome probabilities are constant over time.Markov chains are suitable for many types of problems and the mathematics for their analysisis well developed.

Figure 19.1 contains the structure of a generalized decision tree. The letter D denotes adecision and C a chance event. At D1,1 the decision-maker is faced with two choices.2 The

1This is true only for first-order Markov chains. If the chain is second-order, then an outcome may depend on the two prior events, andso on.2In general, there will be n decision choices, where n is a finite number. Our space limitations dictate that for illustrative purposes onlytwo choices be used.


Figure 19.1 General structure of a decision tree

first subscript number refers to the point in time at which the decision or chance event occurs.Time need not be cardinal. In other words, the actual time between t = n and t = n + 1 maybe either more or less than the time from t = n − 1 to t = n. The second subscript refers tothe position of the event or decision from top to bottom in the diagram.

The problem illustrated in Figure 19.1 requires two decisions, one at the outset and one aftera chance event has occurred. Although they are shown as possibly different events, C2,1 andC2,2 may refer to the same chance event. The effect on the enterprise, however, is differentbecause of the decision made at D1,1. Similarly, D3,1, D3,2, and so on, could refer to thesame decision, tempered by different precedents. Alternatively, the decisions could be quitedifferent.

To find the optimal decisions we examine the payoff utilities associated with a particularsequence of decisions and chance events. The final result, denoted by x and y, is determinedby conditional probability of preceding events. The optimal decisions are found by backward

Multiple Project Selection under Risk: Computer Simulation 205

induction, by looking at the most distant result, and tracing back from there to the very beginningat D1,1. As an alternative to working with the utilities of each separate terminal point, itis possible to find the final monetary outcomes and their associated probabilities for eachbranch from the final decision nodes, then take the certainty equivalents of the uncertainoutcomes.

An example will help clarify how decision trees may be applied to capital investmentdecisions.

Example 19.1 The Northern Lites Manufacturing Company has successfully produced andmarketed home heating units for 30 years. The units are designed to use either gas or fuel oil, butnot both. The engineers of the firm have now designed a new unit that can use interchangeablyeither gas, fuel oil, coal, or cord wood and, with an optional adapter, peanut shells andother waste products. The marketing staff is enthusiastic about the sales prospects for the newmultiple-fuel furnace that they are calling the “Hades Hearth Home Heater.”

The new heating unit could be produced in the existing factory, but changeover would takeat least a year, perhaps longer, and an inventory of replacement parts would have to be builtup to meet the current owners’ needs for repairs of the present models. A new plant could beconstructed to be in operation within a year, and a nearby facility could be leased and readiedfor production in the same time span.

Because of present shortages with rising fuel oil and natural gas prices, sales are expectedto be brisk. However, there is a 30 percent chance of an economic depression in a year thatcould be so severe that even though they might like to buy the new furnace, consumers willnot have the money. If there is such a depression, the firm will likely lose $1000 with a newplant facility, $300 with the leased facility, and $100 if it delays its production by deciding touse the existing plant. A deep depression would mean a halt to production until the economyimproved.

A new plant will allow maximum production. Delaying production to use the existing plantwill allow competitors to take the lead and sales will therefore be less. In two years theengineering staff feels it will have a much improved design that may be required to keep salesup. There is some risk, however, that the improved design may not be as popular as anticipatedbecause of competitive developments and the possibility of meaningful government incentivesfor fuel conservation.

Figure 19.2 illustrates the tree diagram for this problem. If a new plant is built, if there isno depression, and the design is changed and successfully received in the market, the presentvalue to the firm is $900. If not well received, the present value is $360. The remainingresults are similarly interpreted. Note that the expected values shown in Figure 19.2 areexpected monetary values, not expected utilities or certainty equivalents. Table 19.1 showsthe calculations corresponding to the expected monetary values of Figure 19.2 for finding theexpected utilities.

Implicit in the approach illustrated in Table 19.1 is the assumption that the expected utilityis the proper measure by which the optimum course may be determined. However, we shouldnote that this approach does not take into account the distributions of the outcome utilities butonly their expected values. It may well be the case that certainty equivalents correspondingto the monetary outcomes would yield a different ranking. Therefore, it may be important inpractice to have management decide on the basis of the certainty equivalents that it assigns tothe various choices.


Figure 19.2 Decision tree for Example 19.1

As matters stand, the expected utilities in Table 19.1 favor leasing a plant for productionof the multiple fuel furnace instead of either building a new plant or delaying and using theexisting plant. Note that the spread of both dollar and utility outcomes is greater for leasingthan for delaying until the current plant can be used, thereby making the lease arrangementmore risky. By the same reasoning, the building of a new plant would be both more risky andoffer a lower expected utility than leasing; it is thus dominated by the leasing alternative.

An alternative to either choosing the path offering the highest expected utility or choosingthe path offering the highest certainty equivalent is found in the next chapter. There the capitalasset pricing model (CAPM) is discussed and it is shown that the optimal choice might bemade by employing the risk/return trade-offs prevailing in the market.


Table 19.1 Calculation of payoff utilities forExample 19.1

Build new plant0.7 × 0.8 ×U ( $900) = 0.56 × 0.80 = 0.4480.7 × 0.2 ×U ( 360) = 0.14 × 0.50 = 0.0700.7 × 0.6 ×U ( 720) = 0.42 × 0.73 = 0.3070.7 × 0.4 ×U ( 180) = 0.28 × 0.30 = 0.0840.3 × 1.0 ×U (−1000) = 0.30 × −0.90 = −0.270

0.639Lease plant0.7 × 0.8 ×U ( $630) = 0.56 × 0.68 = 0.3810.7 × 0.2 ×U ( 270) = 0.14 × 0.40 = 0.0560.7 × 0.6 ×U ( 540) = 0.42 × 0.63 = 0.2650.7 × 0.4 ×U ( 225) = 0.28 × 0.35 = 0.0980.3 × 1.0 ×U ( −300) = 0.30 × −0.32 = −0.096

0.703Delay production0.7 × 0.7 ×U ( $450) = 0.49 × 0.57 = 0.2790.7 × 0.3 ×U ( 135) = 0.21 × 0.24 = 0.0480.7 × 0.5 ×U ( 360) = 0.35 × 0.49 = 0.1720.7 × 0.5 ×U ( 90) = 0.35 × 0.18 = 0.0630.3 × 1.0 ×U ( −100) = 0.30 × −0.14 = −0.042

0.520

OTHER RISK CONSIDERATIONS

In addition to the types of risk considerations introduced by sequential, event-contingent deci-sions and intervening stochastic events, there are other situations in which portfolio approachesmay not be employed to best advantage. Since portfolio approaches developed by Markowitzand Sharpe assume stabile (or at least not suddenly changing) relationships between projectsand the enterprise’s existing assets, problems in which the relationships may change in re-sponse to some future chance event or events do not lend themselves to portfolio selectionapproaches as they currently exist.

Recognizing the shortcomings pointed out in Chapter 18 for computer simulation of singleinvestment proposals, computer simulation may nevertheless offer the most suitable approachto the analysis of certain multiple-project problems. One such application arises in the caseof situations involving assets with useful lives governed not only by their own unique char-acteristics but also by the life of the aggregate. An example will serve to illustrate this typeof problem. The following example is rather lengthy because of the nature of the problemand because it illustrates a way of analyzing multiple-project problems by means of computersimulation.

Example 19.2 A comprehensive simulation example3 In oilfield primary recovery oper-ations, equipment deterioration and breakdown create problems over the lifetime of the field.These problems are of a reinvestment/abandonment type for which little theoretical guidanceis to be found in texts on capital budgeting, engineering economy, or elsewhere. For instance:

3Extracted from a paper by Anthony F. Herbst and Sayeed Hasan [64]. By permission of the American Institute for Decision Sciences.


(1) When should a particular well be rehabilitated rather than abandoned? And (2) whenshould the entire field be abandoned for continuing primary recovery and secondary recoveryoperations begun?

Discussion is limited in scope to consideration of decision algorithms for determining:

1. Whether a particular well should be repaired after equipment breakdown, or should insteadbe shut down.

2. Under what circumstances the entire field should be abandoned as a primary recovery field.

We consider the related problem of replacement of primary recovery with secondary recovery(by means of gas or water injection) before primary recovery in itself becomes economicallynonviable, though our emphasis is on the above two problems.

For purposes of classification we distinguish between two cases of primary recovery fieldabandonment. First, the weak case, in which the primary recovery operation is in itself stilleconomically viable, but inferior to and economically dominated by a change in technology tosecondary recovery methods. Second, the strong case, in which primary recovery by itself iseconomically nonviable regardless of secondary recovery operations.

A difficult but interesting feature of the problems considered in this section is interdepen-dence between the maximum expected economic life of each well and that of the entire field.Abandonment of any given well or set of wells may result in the entire field becoming unprof-itable with primary recovery. Adoption of the decision to abandon the entire field, of course,implies abandonment of all individual wells whether or not they are operating profitably at thetime.

The decision to abandon the entire field for primary recovery (strong case) is relativelystraightforward. The field should be shut down when the variable cash revenue contributionof the field no longer exceeds variable cash cost. In other words, in the strong case the fieldshould be abandoned when net cash contribution to fixed cost coverage and profits becomeszero.

The weak case decision to abandon primary recovery and replace it with secondary recov-ery technology should be made when the expected net present value (NPV) of investment insecondary recovery by gas or water injection exceeds the expected NPV of the remaining cashflows with primary recovery. We will not at this time get into the problem of determining theNPV of secondary recovery investment, as this appears to be amenable to the more standardcapital budgeting–engineering economy analysis. We do, however, address the problem ofestimating the NPV of the remaining cash flows with primary recovery.

The more difficult question than field abandonment for primary recovery is that of whenany particular stripper well should be abandoned. At least two circumstances must be consid-ered. First, under what circumstances should a given well be abandoned prior to equipmentfailure? Second, given an equipment failure of a particular type, when should repair or re-placement be undertaken, and when should the well be shut down and abandoned for primaryrecovery?

Characteristics of the Problem

The Oilfield

Type of Recovery Process We adopt the terminology that primary oil and gas recovery isthat undertaken by use of pumps and secondary recovery is that undertaken by use of water or


gas injection.4 When pumps are employed to raise the oil to the surface, the wells are calledstripper wells. Each stripper well will have its own pump, and therefore there will be a closecorrespondence between the number of wells and the amount of capital equipment required.

Secondary recovery with water or gas injection is accomplished by drilling wells aroundthe field through which water or gas may be injected to drive out the oil ahead of it. Unlikeprimary recovery, the number and location of injection wells are not strictly determined by thenumber of oil wells and there is no one-to-one correspondence between injection sites and oilwells as there is between pumps and stripper wells.

Breakdown of Stripper Wells Stripper wells are subject to a variety of breakdowns whichrequire the decision be made to either repair the problem or else to abandon that well duringprimary recovery. Major types of breakdown include pump overhaul, need for condensatewash, broken pump rod, and rupture in production tubing. In addition to the normal costs ofrepair, a condensate wash involves several days’ lost production while it is being carried out.Periodic, scheduled maintenance may entail pump overhaul and condensate wash to preventunscheduled breakdowns which may be more costly than planned preventive maintenance.

The probabilities associated over time with different major breakdowns may not be inde-pendent. For instance, a broken pump rod may do other damage to other well equipment, anda condensate wash may sometimes be required after another type of breakdown.

It is not our purpose to become involved in the technical details of the recovery process, butrather to deal with the methodology for determining the optimal reinvestment policy for the oilcompany. Therefore, after this brief introduction to the technology of recovery, we will moveon to the financial, economic, and methodological consideration of our work.

Decision Required

There are three basic decisions which must be made continually over the life of the field:

1. Given a well breakdown, whether to repair or shut down (abandon) the well.2. Whether to replace all well equipment with new equipment of the same type or switch to

another extraction technology and equipment.3. When to shut down the entire field to recovery with the currently employed technology.

The second decision, we propose, is amenable to more or less familiar approaches, such asthe MAPI (Machinery and Allied Products Institute) method of replacement analysis.

Financial Measures of Investment Return

Modern financial management heavily favors the use of discounted cash flow (DCF) methodsof evaluation, and favors the NPV approach over IRR. While the DCF approach has beenfavored, the payback criterion has simultaneously been attacked for use in investment decisions.However, DCF methods require certain information and conditions which may not exist.

In the problem we consider, the economic lives of the oilfield and the individual wells areinterdependent. We do not know the useful life of any individual well, as that life depends on the

4Another classification scheme would relegate to primary recovery the removal of oil by employing the natural subsurface pressure thatmay exist, secondary recovery to the use of pumps, and tertiary recovery to the water or gas injection methods. Under this alternativeclassification a field may be taken from primary to tertiary recovery by omitting the use of pumps entirely.


nature and frequency of required repairs and especially on the life of the entire field. Similarly,the life of the field depends on the lives of the individual wells. When all wells are shut down,obviously the field life, with the particular technology, has ended. When it has been decided toclose the field all wells in it will be closed. It is not known a priori how long the useful lives ofeither the individual wells or the field will be.

Additionally, the DCF methods require that the cash flows for each period be known orestimated. However, in the case of well breakdown the cash flows will be affected by thepattern and types of breakdown experienced, which are stochastically determined. The DCFmeasures are known to be affected by the pattern of cash flows as well as their magnitudes,and although the expected value of the cash flows could be determined the pattern cannot beon an a priori basis.

The absence of a priori knowledge of useful lives and cash flow patterns, coupled with theinterdependence of the well and field characteristics, negates the usefulness of ordinary DCFmethods to the problem we consider.

Methodology

We are interested in determining a policy for repairs that will yield the maximum expected netdiscounted return, and in the absence of a priori information, use the approach of simulationexperience with the field under alternative policies. The results of the simulation then provideinformation that can be converted to the DCF present value measure. In other words, weapproach the problem by simulating the cash flows of the field under different repair policies.The cash flows are carried forward with compounding at the firm’s cost of capital as futurevalues. At abandonment or shutdown of the field the future values are then converted to presentvalue equivalents. Choice of the proper policy will yield results that would be obtainable inordinary circumstances by a direct DCF approach.

Since the payback criterion is already familiar and widely used in industry we adopt it asour policy variable for repair decisions.5 If the net cash flow over n periods, where n is thepayback policy, is insufficient to recover the repair cost, we shut down the well. If the cash flowis greater than the repair cost, we repair and continue to operate the well. After a representativenumber of simulations, employing different seed values for the generation of (pseudo-) randomnumbers, we propose adopting that payback policy that promises the highest NPV for the field.

We also consider as a secondary policy the shutdown of the field when the rate of change inthe cash revenues becomes less than the rate of change in cash costs (including repair costs)over m periods. This provides a means for determining when to shut down the entire fieldbefore all wells are shut down and before net cash flow becomes negative. We define shutdownunder this secondary policy variable as “weak case” shutdown.

The field will be shut down to primary recovery whenever the net cash flow becomes negative.That is, when cash revenues are inadequate to cover cash costs, we will shut down the field.We define shutdown of the field under this condition as “strong case” shutdown.

The Model

I. Net per-period cash revenue for the field in period j:

NR = Total cash revenue − Variable cash cost − Fixed Cash Cost:6

5We do, however, adjust the payback criterion to include the time-value of funds.6Cash costs that would not exist were the field to be closed.


NR = PCFOILN W E L L S∑

i=1

OILBBLi, j

+ PCFGASN W E L L S∑

i=1

GASMCFi, j

− (ROALTO + PROCO)N W E L L S∑

i=1

OILBBLi, j

− (ROALTG + PROCG)N W E L L S∑

i=1

GASMCFi, j

− CASHFC −N W E L L S∑

i=1

REPRCi, j

Strong case decision:

If NR ≤ 0, shut down the entire field where i denotes individual wells.

PCFOIL and PCFGAS = the price per barrel of oil and MCF7 of gas, respectivelyOILBBL and GASMCF = the barrels of oil output and MCF gas output for the ith well,

j th periodROALTO and ROALTG = the royalty charges levied per barrel of oil and MCF of gasPROCO and PROCG = processing costs per barrel of oil and MCF of gasCASHFC = fixed cash expense per period while the field is openREPRC = repair cost for the i th well and jth period if a breakdown

occurs and is repaired

II. For a given well i :Let bk, j be the number of breakdowns of type k in period j . Then the reinvestment costfor repairing the breakdown is Ci, j . Let Pk be the cost of repair for type k breakdown. Werequire recovery of the cost with a payback period of n time periods.

If Ci, j <

j+n∑l= j+1

OILBBLi,l(PCFOIL − ROALTO − PROCO)/(1 + RATE)l− j

+j+n∑

l= j+1

GASMCFi,l(PCFGAS − ROALTG − PROCG)/(1 + RATE)l− j

where RATE is the firm’s per period cost of capital, we repair the well and continueoperation. Otherwise we shut down the well.

III. Breakdowns are assumed to be:

1. Poisson distributed over time, and be2. Independent of previous breakdown history of the well.

Figures 19.3(a) and (b) contain a flow chart of the model.8

7MCF is defined as 1000 cubic feet.8The paper published in the Proceedings did not contain this flow chart because of space limitations.


Figure 19.3(a) Flow chart of the model

Estimating NPV of Remaining Cash Flows with Primary Recovery

Empirical experience with our model, which provides for application of production deterio-ration gradients and changes in price, processing costs, and so on, suggests how remainingfield NPV may be estimated to any arbitrary point in time. Since the accumulated net futurevalue is calculated for each time period prior to shutdown, the NPV of cash flows over anytime span prior to shutdown may be calculated. Thus, if shutdown should occur in period 85,


Figure 19.3(b) Continuation of flow chart of model

and we are interested in the NPV of flows between period 40 and 85, we can determine thiseasily with customary financial mathematics.

Analysis of the remaining value of net cash revenues over time will provide informationuseful in the decision of whether, and when, to switch to other technology for recovery.

Empirical Results

The model was tested with empirical data provided to one of the authors, for a stripper fieldin western Canada. Results of a typical simulation run are summarized in Table 19.2. Thecondition of this field is such that the maximum NPV is obtained with a payback policy of fourperiods (one week equals one period in this run). Such a short payback meant that after amajor breakdown of any type the well involved would be abandoned. This may be attributedto the low oil and gas yields of the wells in a declining old field in relation to the cost ofrepairs.

Breakdown probabilities were assumed to be independent of prior breakdown history ofany well. This assumption may not be warranted, but the authors had no information tothe contrary which would enable them to modify the breakdown probabilities for furtherbreakdowns once a well had been repaired. Because of this, breakdowns of the same type couldoccur within the payback period. This meant that a repair could be carried out only to havethe same type of breakdown result in abandonment before the cost of the first repair had beenrecovered.


Table 19.2 Simulation results

All wells NPV by Strong caseshut down waiting until shutdown NPV within period all wells are in period strong case

NPAOUT No. shut down no. shutdown

4 15 $10,888.50 15 $10,888.505 18 −1,756.10 9 8,312.776 18 −1,756.10 9 8,312.777 18 −2,592.75 5 5,263.988 18 −2,592.75 5 5,263.989 18 −2,592.75 5 5,263.98

10 26 −18,184.30 5 5,263.9811 26 −18,184.30 5 5,263.9812 26 −18,184.30 5 5,263.9813 26 −18,184.30 5 5,263.98

CONCLUSION

The aim of this chapter was to illustrate two approaches to capital investment decision-makingunder risk that may provide useful results when portfolio approaches are not applicable. Thefirst approach shown was that of decision tree analysis. The second was computer simulationof multiple project problems.

In decision tree analysis the problem is characterized by a sequential decision–event–decision–event . . . chain over time. Because decisions are contingent on chance events, it maynot be possible to specify a priori which decision beyond the first will be made. The goalof decision tree analysis is to identify the mixed decision–event sequence that promises themaximum expected utility. By doing so it is possible to identify the superior initial decisionbecause it is determined by all that follows it. Decisions reached through application of thedecision tree approach will often be different from those reached through viewing the problemas an isolated, period decision because this latter approach ignores future choices.

Computer simulation of multiple project capital investment problems can be used to reachoptimal decisions where the interrelationships between projects are complex. It is difficult togeneralize about the type of problem warranting the time and expense of computer simulation.However, problems involving many projects, where their relationships cannot be describedby variance–covariance or correlation, will be candidates for simulation, especially if theparticular problem represents large resource value to the enterprise. Computer simulationshould be viewed not as a replacement for other methods of analysis in normal situations, butrather as a last line of defense when other approaches fail, and a powerful last defense it is. Innormal circumstances the extra time, effort, and expense of computer simulation are difficultto rationalize when other, less arcane methods yield correct decisions.

20Multiple Project Selection under Risk:

Portfolio Approaches

In Chapter 16 we discussed the matter of investment selection from an array of individu-ally acceptable independent1 projects. The selection was constrained by conditions of capitalrationing (that is, budget constraints), management policy, and technical considerations.Chapter 18 considered the analysis of individual risky projects within a framework of completeproject independence. This chapter takes up capital investment analysis when we recognizestochastic dependence between project proposals themselves and between project proposalsand the existing assets of the firm.

INTRODUCTION

When investment projects are statistically related to one another and/or to the currently existingassets of the firm, it is no longer appropriate to treat them as independent for purposes ofanalysis. When there is dependence, it is to be expected that we may find that project A ispreferable to project B, even though A has a greater variance and coefficient of variation inits returns than B, and it is therefore more risky as an individual project. The reason for suchresults as these is that project A’s covariance makes it more favorable for selection than doesproject B’s covariance. Up to this point we assumed (or pretended) that all project covariancesto other projects and to the enterprise’s existing assets to be zero, i.e. complete independence.We shall now drop the assumption of zero covariances. And, because the covariance termsare no longer zero, we shall have to take them explicitly into account; we can no longer basedecisions solely on a project’s individual characteristics of risk and return as isolated factors.

To illustrate the effect of nonindependence between a project proposal and the firm’s existingassets, let us examine the following example.

Example 20.1 The owners of Al’s Appliance Store are considering a major expansion that,if undertaken, would mean construction of an attached laundromat and an equipment rentalshop. In that case, the existing building would then become one part of a triplex. The ownershave carefully considered the disparate natures of the businesses and have determined the datain Table 20.1 to be representative.

Although the return on investment is higher, the riskiness of the proposed expansion isgreater in both absolute and relative terms than those of the firm’s existing assets. If we wereto treat the proposal as independent of the appliance store, it might well appear to be more riskythan the business to which the owners are accustomed. Analysis of the proposed expansion inisolation could be carried out with techniques suggested in previous chapters. Here we are

1We assume here that contingent project relationships do not necessarily reflect stochastic dependence. We shall reserve the termstochastic dependence for dependence between the cash flows of projects and not dependencies seen only at the initial accept–rejectdecision. Contingent relationships are often asymmetrical (“accept B only if A has been accepted,” not vice versa), whereas stochasticdependence is generally taken to be symmetrical.


Table 20.1 Data for Example 20.1

Proposed attachedAppliance store laundromat and(existing assets) equipment rental shop

Market value/cost $1,000,000 $500,000Expected annual return E[R0] = $100,000 E[R1] = $75,000Variance of annual return σ 2

0 = ($10,000)2 σ 21 = ($15,000)2

Coefficient of variation σ0/E[R0] = 0.100 σ1/E[R1] = 0.200Covariance σ 2

0,1 = −($12,000)2

Table 20.2 Combined asset characteristics,Example 20.1

Value of combined assets $1,500,000Expected combined annual returns E[Rc] = $175,000Variance of combined returns σ 2

c = ($2,333)2

Coefficient of variation σc/E[Rc] = 0.013

not interested in the proposal as an entire, stand-alone investment, but as an addition to aninvestment portfolio that already contains assets. Therefore, it is not appropriate to analyzethe expansion without reference to the combined, portfolio effect that its acceptance wouldbring about.

Individually, the proposal is relatively twice as risky as the appliance store. However, thecovariance between them is negative because it is expected that the proposal will responddifferently to changing status of the economy than will the existing business; in fact, it willrespond oppositely. Table 20.2 contains data for the firm assuming the expansion has beenundertaken.

Note that the relative riskiness as measured by the coefficient of variation has dropped farbelow what it was for the appliance store alone, which individually was less risky than theproposal. The variance of returns on the combined assets2 is less than on the proposal alone oron the existing assets, so the absolute risk is also less than it would be without the expansion.This highlights the benefits to be gained by diversification among assets whose returns tend tomove in opposite directions. The correlation in this case between the appliance store and thelaundromat-cum-rental shop is ρ = −0.96, an almost perfect negative correlation. So perfectis the relationship that 92 percent of the variation in the return on one may be statisticallyexplained by variation in the other (ρ2 = 0.92).

Generalizations

In this example the high negative correlation yields dramatic results. But what if the correlationis not so strongly negative? And, what if, instead of negative correlation, there is positivecorrelation? Table 20.3 contains comparative figures.

2The variance of a linear combination of variables, with weights a, b, c, and so on, is given by

Var (ax + by + cz + · · ·) = a2Var (x) + b2Var (y) + c2Var (z) + · · · + 2ab Cov (x, y) + 2acCov (x, z) + · · · + 2bc Cov (y, z) + · · ·

Multiple Project Selection under Risk: Portfolio Approaches 217

Table 20.3 Risk–return relationships fordifferent return correlations between existingfirm and proposed expansion

ρ Cov(0, 1) σ 2c σc/E[Rc]

−1.0 −($12,247)2 (1,667)2 0.010−0.5 −($8,660)2 (6,009)2 0.034

0.0 0 (8,333)2 0.048+0.5 ($8,660)2 (10,138)2 0.058+1.0 ($12,247)2 (11,667)2 0.067

Comparison of the coefficients of variation for the combined assets with the existing assetsreveals the following:

1. As the correlation coefficient changes from perfect negative correlation to perfect positive,the riskiness of the combined enterprise increases.

2. For all correlations, acceptance of the proposal will reduce relative risk for the combinedenterprise to a lower level than that of the original assets (in this example, the appliancestore).

3. The case of ρ = 0.0 corresponds to a covariance of zero and independence of the proposaland the existing firm.

4. Even with perfect positive correlation (ρ = +1.0) the relative risk for the combined assetsis less than that of the proposed expansion alone.

Project Independence

An important implication of point (3) above is: even if the proposed business expansion wereindependent of the existing assets of the firm, proper analysis cannot be done without referenceto them. The coefficient of variation for the proposal is 0.200. Yet, the coefficient for the firmafter undertaking the investment is 0.013,when ρ = 0.0. Thus, although the expansion itselfis per se more risky than the existing firm, its acceptance would actually decrease the riskof the firm. The point is that if the expansion is viewed separately, it may be rejected becauseit is relatively twice as risky as the firm as it exists, without the investment. If, on the otherhand, the accept/reject decision is based on the riskiness of the firm before the investment tothe firm after the investment it is apparent that relative risk has decreased measurably. Therelevant comparison is not in the terms of the “own” risk of the proposal vis-a-vis the firmwithout the proposal, but rather the firm without the proposal to the firm with the proposal. Thisis not to say that techniques of analysis such as computer simulation of individual projectsshould be rejected, but that their use must be appropriately modified to take the foregoinginto account even when the project is statistically independent of the existing assets of theenterprise.

Project Indivisibility

Portfolio selection theory as it is applied to securities, generally assumes that the investmentin any particular asset is finely divisible. This means that the percentage of the portfolio


committed to a particular asset can be any amount between 0 and 100 percent. In the caseof securities such an assumption is reasonable because discontinuities are relatively small,amounting to no more than the smallest investment unit. With capital investments, however,the assumption of finely divisible investments is seldom, if ever, justified. In the foregoingexample the question was not what proportion of the proposal should be invested in between0 and 100 percent inclusive, but whether the investment would be rejected (0 percent) oraccepted (100 percent). Only the extremes were considered because the project was consideredindivisible.

Project indivisibilities are much more troublesome in capital budgeting than in (stock andbond) portfolio selection. Therefore, techniques that were originally developed for securitiesinvestment portfolio selection must be judiciously modified if correct decisions are to be made.Choices of percentage other than 0 percent (rejection) and 100 percent (acceptance) aregenerally not possible.

MULTIPLE PROJECT SELECTION

The formal model for optimal choice of risky assets for an investment portfolio is generallycredited to Markowitz [101]. What is optimal for one investor will generally be different fromwhat is optimal for another because of different attitudes toward risk-to-return relationships.The basic nature of risk and return within the framework of utility was covered in Chapter 17for individual asset choice decisions. Here we extend the treatment of risk–return trade-off tothe multiple asset case.

Let us denote the overall return from a portfolio, R, as a weighted average of the returns,Ri , with weights xi , as

R = x1 R1 + x2 R2 + · · · xn Rn (20.1)

Each xi is the proportion of the total invested in asset i , and Ri is the expected return fromthat asset. The Ri are random variables and therefore R, the portfolio return, is also a randomvariable. If we denote the expected values of the Ri by µi , the variance of asset i by σi i , andthe covariance of asset i and asset j as σi j , we obtain for the portfolio

E[R] = x1µ1 + x2µ2 + · · · + xnµn (20.2)

Var[R] = x21σ11 + x2

2σ22 + · · · + x2nσnn

+ 2x1x2σ12 + 2x1x3σ13 + · · · + 2x1xnσ1n(20.3)

+ 2x2x3σ23 + · · · + 2x2xnσ2n, · · ·=

n∑i=1

n∑j=1

xi x jσi j

Although the optimum portfolio for any particular individual depends on one’s utility func-tion, we can nevertheless narrow considerably the field of choice. For instance, we can eliminatefrom further consideration those portfolios that should not be selected by any rational personregardless of utility function. For this we use the concept of efficient portfolio.

An efficient portfolio is one for which a higher return cannot be had without incurring higherrisk or, equivalently, a portfolio for which one cannot reduce risk without a corresponding lossof return. A portfolio is inefficient if, by changing the proportions of the assets held, wecan obtain a higher return with no more risk, or reduce risk without sacrificing some return.


Figure 20.1 Selection of optimal portfolio of risky assets

Alternatively, an efficient portfolio is the minimum variance portfolio among all portfolioswith the same expected return and it is the portfolio with the maximum expected return amongall with the same variance of return.

Markowitz suggested two steps in finding an investor’s optimum portfolio. First, identify theset of efficient portfolios. Second, select that efficient portfolio that best matches the investor’srisk–return attitude, which maximizes the investor’s utility. The exact portfolio will vary frominvestor to investor, but all of them must be efficient portfolios. Figure 20.1 summarizes theprocedure.

Line AB is the efficient set of portfolios. The Ui represents a particular investor’s set of utilitycurves relating risk to returns.3 The shaded area represents the set of all other portfolios thatcould be formed from the same assets (assuming infinite divisibility). The point of tangencybetween the highest indifference curve (U0) and the efficient set corresponds to the optimumportfolio for this particular investor.

Markowitz considered several different possible forms of utility functions. It was thequadratic form that Farrar [46] employed to determine the coefficients of risk aversion formutual funds; also the quadratic form of utility functions has been widely employed by manyother writers, in no small measure due to its consistency with diminishing marginal utility andother generally accepted aspects of preference theory. The discontinuous utility function usedby Roy [135] provides a linkage to the notion that survival of the enterprise is an importantinvestment objective.

Roy’s utility function was of the form

U [R] = 1 for R ≥ a

= 0 for R < a

which yields expected utility of

E[U ] = 1 · Pr(R ≥ a) + 0 · Pr(R < a)

= 1 − Pr(R < a)

3For investments of the size represented by this portfolio. Chapter 17 suggests that the utility function is related to the size of therequired investment and to risk of ruin as well.


for which maximization of utility means minimization of the probability that the portfolio returnwill fall below amount a. Graphically, this corresponds to selecting the portfolio determinedby the tangency of a line through point a with the efficient set in Figure 20.1. Roy’s modelis consistent with the responses elicited in Mao’s survey of executives cited in Chapter 17.Together, they suggest that where the survival of the enterprise itself is a factor in the decisionprocess, the decision-maker may apply a utility function distinct from that which is used incircumstances in which the enterprise’s survival is not in question.

Finding the Efficient Set

Since the first step in the procedure for finding the optimal investment portfolio is to findthe set of efficient portfolios, we require a systematic method. This may be formulated as amathematical programming problem:

Maximize � = E[R] − λ · Var[R] (20.4)

= xµ − λσ 2 (in vector notation)

subject to∑

i

xi = 1.0

or xi ≥ 0 (20.5)

xi = 0 or 1

Lambda (λ) is the coefficient of risk aversion, which we restrict to the range 0 to + ∞.To obtain the efficient set we must solve the above problem in equation (20.4) for differentvalues of λ until we have enough points on the efficient set to define its curve. It is importantto recognize that this model assumes that the investor’s risk attitude is adequately reflectedby the variance–covariance matrix of all candidate investments. The final constraint restrictsus to either accepting a project entirely (x = 1) or else rejecting it entirely. This constraintis not necessary (under most circumstances) when we are considering securities portfolioinvestments. However, most capital investment projects are indivisible; we either accept orreject a particular project; we do not have the option of owning a partial share of it. In securitiesportfolio selection this constraint is replaced by the restriction that the xi (the weights) sum toa value of 1.0 (100 percent).

If we measure the return on an investment by its net present value (NPV), then we canformulate the problem of finding a point in the efficient set as

Maximize F = p1x1 + p2x2 + · · · + pn xn

−λ(x2

1σ11 + x1x2σ12 + · · · + x1xnσ1n

x2x1σ21 + x22σ22 + · · · + x2xnσ2n

...xn x1σn1 + xn x2σn2 + · · · + x2

nσnn)

(20.6)


subject to a11x1 + a12x2 + · · · + a1n xn ≤ b1 (20.7)

a21x1 + a22x2 + · · · + a2n xn ≤ b2

...

am1x1 + am2x2 + · · · + amn xn ≤ bm

and for all i, xi = 0 or xi = 1

which, in matrix algebra notation, becomes

Maximize F = p · x − λ(x · V · x) (20.8)

subject to A · x ≤ bxi = 0 or 1

(20.9)

where V is the variance–covariance matrix of returns.A point regarding problem setup that seems to be often ignored in attempts to transfer this

model to capital investments is that existing assets of the enterprise need to be included inthe formulation. In other words, we cannot look solely at the proposals under considerationand their interrelationships, but need to include their relationships to the firm as it is now. Inthe programming framework above, this may be done by treating the existing enterprise as a“project” with zero required investment outlay. If divestiture is not to be considered, it willalso be necessary to include a constraint that requires the existing enterprise to be included inthe optimal solution. It may be interesting, however, to solve again without the constraint todetermine if divestiture would be worthwhile.

This formulation may be compared to that of the problem of project selection under certaintyin Chapter 16. In fact, certainty means that the elements of the variance–covariance matrixare all zero, and the problem specified in equations (20.6) through (20.9) reduces to equations(16.1) and (16.2).

Mao [98] shows how the Lawler and Bell zero–one integer programming algorithm (dis-cussed in Chapter 16) may be modified to solve problems with objective functions of theform in equations (20.6) and (20.8). However, Baum, Carlson, and Jucker showed that theMarkowitz approach, when applied to problems with indivisible projects, “may result in either(1) a solution set that does not contain all solutions of interest to the decision maker or (2) arequirement that the (implicit) utility function describing the decision maker’s preferences bea linear function of the mean and variance of return” [7]. By examining all feasible solutions toa problem solved by Mao (pp. 295–296), Baum, Carlson, and Jucker show that no matter whatthe value of λ, the coefficient of risk aversion, an efficient point is missed by the Markowitz ap-proach. Furthermore, the Markowitz approach to Mao’s problem, although missing an efficientpoint, produced a dominated point:4 “More generally, any efficient point which is not locatedat a corner point of the upper boundary of the convex hull of the complete set of efficient pointswould also be missed” [7, p. 338].

The Baum, Carlson, and Jucker article is significant in that it demonstrates that incorrectresults may be obtained when the Markowitz approach is employed with problems involvingselection of indivisible assets. This is a serious problem requiring suitable modifications to

4A dominated point is one that is inferior to other asset combinations for all values of X , the coefficient of risk aversion.


the solution algorithm or alternative methods of solution that find only dominant points, forexample.

Although problems involving complex constraints of the type illustrated in Chapter 16 donot lend themselves to solution by complete enumerations, an enterprise which has only a fewfairly large investment proposals may be able to analyze them quite successfully without relyingon the Markowitz approach per se. Baum et al. found the error in the Mao problem solution byevaluating all 32 possible selections of the 5 candidate projects.5 Full evaluation can get out ofhand quickly, however, as the number of projects considered increases. Nevertheless, with upto 10 projects or so (210 = 1024) evaluation of all combinations of proposals by modern digitalcomputers is not only possible but not particularly costly. After about 10 projects, however,the number of combinations becomes rapidly so great that full enumeration is not possible.In all but the largest enterprises this should not be an insurmountable problem, particularly ifminor projects are aggregated together.

Another possibly rewarding substitute approach might be found in using a modified zero–oneprogramming algorithm, such as that discussed in Chapter 16 to obtain the partial enumerationcombinations that will need to be evaluated in terms of risk in a second step. In other words, if wecan eliminate infeasible combinations before looking at risk, then the number of combinationsto be examined will be reduced, often substantially, particularly where several proposals aremutually exclusive.

The Sharpe Modification

William Sharpe is credited with a modified version of the Markowitz model that has far-reachingimplications for the valuation of assets [142]. Sharpe’s model provides the foundation of thecapital asset pricing model (CAPM) discussed in Chapter 21. The Sharpe model is oftenreferred to as the “diagonal model” of portfolio selection. His model assumes that returns onassets are related only through their correlations with some index. Returns are defined as

Ri = Ai + bi I + µi (20.10)

where I is a random variable denoting an index, and Ai and bi are constants for asset i . Thevariable I is assumed to have a finite mean and finite variance. The µi are random errors due toindependent, external causes, and satisfy the usual least-squares assumptions of zero expectedvalue, finite variance, and independence (zero covariances).

With these specifications any investment can be split into two parts: (1) an investment inAi and µi , the asset’s basic, or unique characteristics and (2) an investment in I , the externalindex. This means that the variance–covariance matrix for n securities plus the external index(n + 1 in total) has zero elements except on the diagonal containing the covariances betweeneach of the n securities and the external index.

Elimination of the covariances between individual assets makes the Sharpe model muchmore amenable to computer solution than the original Markowitz formulation, because thecomputational burden is substantially lessened. It has yet to be determined whether or not theSharpe model suffers from similar shortcomings to those Baum et al. found for the Markowitzformulation when assets are indivisible.

5The number of possible selections is given by 2n , where there are n candidate proposals.


Figure 20.2 Selection of optimal portfolio of risky indivisible assets

RELATING TO INVESTOR UTILITY

After determining the set of efficient assets for various6 coefficients of risk aversion (λ), theactual combination of projects selected will depend on the decision-maker’s utility function.Graphically, the optimum project selection is that which just touches the indifference curve(that is, constant utility curve) corresponding to the highest utility.

In the case of infinitely divisible assets as depicted in Figure 20.1, the point of tangencyequates the marginal rates of substitution of the efficient set curve7 and the tangent indiffer-ence curve. For indivisible assets, the optimum efficient point is the one lying on the highestindifference curve. Figure 20.2 illustrates this.

In practice, it may not be necessary to estimate the family of indifference curves; in fact,most managements will insist on making the final decision on which efficient set of capitalinvestments will be undertaken. There is a marked tendency for executives to resist whatthey interpret as arrogation of their authority and responsibility by technicians armed withcomputers. It may suffice to submit to those who have the decision authority only the fewefficient alternative combinations that are not dominated by others. This will generally meanconsidering which one, two, or few projects will be either removed or entered into the finalselection, because alternative efficient collections are often similar over a rather wide range ofλ values, differing by only few individual included or excluded projects.

EPILOGUE

Subject to qualifications resulting from the article by Baum, Carlson, and Jucker, the quadraticprogramming approaches to asset selection used by Markowitz and by Sharpe yield usefulinsights into the process by which risk-averse investors select investment portfolios. Extensionfrom the domain of securities analysis, where individual investments may be considered tobe finely, if not infinitely, divisible to the domain of indivisible capital project investments,can yield incorrect results. This appears, however, to be more a mechanical problem withimplementation of the procedure than it is a problem with the theoretical concepts. As such itis a flaw that should be amenable to correction by modification of the algorithm.

6If we knew λ in advance, we could solve directly for the optimal portfolio.7The curve defined by all efficient points is called the “efficient frontier.”


Of a more fundamental nature are questions arising from concerns relating to the choiceof measure of return that is used as well as questions about the adequacy of the variance–covariance relationship to reflect all relevant risk–return attitudes, although numerous authorshave defended this latter factor. Still further unresolved concerns stem from the nature of thetrade-off function relating uncertainty and futurity, from considerations of abandonment priorto final project life maturity, and from complex intertemporal contingencies between today’sprojects and future candidate projects.

Estimation of project variances and covariances is a difficult problem. And the question ofsolution sensitivity to misspecification or misestimation of the variance–covariance structureand the stability of these parameters over time has not been resolved to everyone’s satisfactionfor securities investments, not to mention capital expenditure proposals.

These concerns and cautions have not been raised to disparage application of portfolioselection techniques to capital investments, but rather to alert the reader that there are unresolvedand controversial matters that tend to be ignored by the more zealous advocates of portfolioselection techniques to capital investment expenditure analysis.

21

The Capital Asset Pricing Model

Early in this book, the capital asset pricing model (CAPM) was mentioned in relation to thefirm’s cost of equity capital. In Chapter 18 it was mentioned again in discussion of the criticismof computer simulation of individual capital investment proposals. In this chapter we examinethe CAPM more closely, to see how it may enable the firm to rationalize the risk–return trade-off and how it may assist in estimating the firm’s cost of equity capital and in assessing theinteraction between capital investment and cost of capital.

The purpose of this chapter is to present the CAPM as a means for dealing with risk incapital investment. In keeping with this purpose, the model is examined critically in terms ofhow well it facilitates making better decisions, as well as problems that affect application ofthe model to capital investment projects.

ASSUMPTIONS OF THE CAPM

The CAPM is based on several of the following assumptions, some of which do not conformwell to the reality of capital investment projects:

1. Investors are risk-averse maximizers of the expected utility based on wealth at end-of-holding period.

2. Expectations about asset returns are homogeneous: everyone agrees on the probabilitydistributions governing returns.

3. Assets are fixed in number, marketable, and infinitely divisible.4. A risk-free asset exists and investors may borrow or lend without limit at the rate of return

paid by the risk-free asset.5. The market is perfect: no taxes, no transactions costs, no restrictions.6. All investors have the same planning horizon and holding period.7. Investors do not distinguish between sources of returns. They are indifferent between divi-

dends and interest of equal dollar amount.

THE EFFICIENT SET OF PORTFOLIOS

In Chapter 20 portfolio selection techniques were examined. They require that we first findthe set of efficient portfolios, those that cannot be improved upon in this sense: an efficientportfolio is one that has maximum expected return for a given level of risk or, alternatively,has minimum risk for a given expected rate of return. In other words, if a portfolio is efficient,we cannot improve upon it by finding another portfolio that has the same (or greater) returnand lower risk or the same (or less) risk, and greater return.

Since the investor is assumed to be risk averse and utility maximizing, and since greaterutility is associated with greater returns, the only portfolios he or she is interested in are efficientportfolios. In Figure 21.1 the section of curve II′ above point 0 is the efficient frontier. Points


Figure 21.1 The efficient set and investor portfolio choices

on II′ below point 0 represent nonefficient portfolios because for the same risk each has analternate providing higher expected return on the segment above.

Portfolio Choices

The risk-averse, utility-maximizing investor that we have assumed can be interested onlyin efficient portfolios. It remains for us to determine exactly which efficient portfolio isoptimum for that investor. In Chapter 20 this was found to be the point of tangency betweenthe efficiency frontier and the investor’s highest indifference curve. These indifference curvescontain risk–return combinations which, to the particular investor, offer constant utilityalong a given curve. In Figure 21.1 the portfolio choices of three investors (A, B, and C) aredepicted by points 1, 2, and 3. Investor C may be considered the most aggressive of the threebecause point 3, although corresponding to higher returns than the others, is also associatedwith proportionately greater risk. By similar reasoning, investor A may be considered themost conservative of the three. Minimum risk is at point 0, but the return of the portfoliocorresponding to that point is also at a minimum.

So far, we have not admitted the existence of a risk-free asset into our portfolio considerations.At this point we shall assume such an investment exists, and examine implications.

Enter a Risk-free Investment

The foregoing assumed that the only assets that existed were risky. Now we assume thereis a risk-free asset, paying a certain rate of return that is denoted by R f . Because federalgovernment securities are risk free in this sense, and they are financial instruments, we mightalternatively use i to denote the fixed, certain rate of return. Those who are totally averse torisk would be expected to purchase only the risk-free asset for its certain return. Others wouldpurchase combinations of the risk-free asset and the market portfolio.

The risk-free assumption, it should be noted, requires several subsidiary assumptions.Although the payment of a fixed, periodic money interest on federal government bondsis reasonable (except possibly in time of war or revolution that may bring to power newgovernment that will repudiate the debt), it raises questions. Is the rate of return in nominalterms or in real terms? What about capital gains or losses brought about by changes in the

The Capital Asset Pricing Model 227

Figure 21.2 Risk-free asset, market portfolio, and capital market line (CML)

market rate of interest? If there can be capital gains, then the rate of return is not fixed andcertain.

In order for the risk-free return to be in fact risk free, or certain, we need to require either that(1) we measure nominal returns, unadjusted for inflation or (2) we have constant inflation orknow the rate of inflation over the holding period, or (3) the risk-free security is a floating ratebond that pays a certain real return, or (4) the holding period is short enough that the effects ofinflation can be ignored. If both the risk-free return and the market portfolio return are equallyaffected by inflation, the problem diminishes and it is not necessary to do more than insurethat returns are measured on the same basis in either nominal or real terms.

The problem posed by capital gains may be resolved in either of two ways. We can addthe assumption of costless information, which is sometimes listed as one of the requiredassumptions of the CAPM, or we may instead assume a constant interest rate over the holdingperiod or else a short holding period over which interest-induced capital gains and losses wouldbe minimal, such as with three-month Treasury bills, for example.

For our purposes we need not be overly concerned about these questions about the risk-freeasset. However, because such questions tend to arise from students of the CAPM, this seemsan appropriate place to bring them into the open. We shall not dwell on this matter further, butcontinue, assuming that the risk-free rate is indeed risk free.

A risk-free asset in addition to the efficient set means that now every investor can have aportfolio composed of the risk-free asset plus an efficient portfolio of risky assets. Figure 21.2illustrates this. Any point on the straight line may be obtained by the appropriate linear com-bination of the risk-free asset and the efficient portfolio at the point of tangency. If q is theproportion of the investor’s wealth invested in the risk-free asset, then 1 − q is the proportioninvested in the tangent portfolio, and 0 ≤ q ≤ 1.

The line from R f tangent to the efficient set is called the capital market line (CML). Nolonger are investors content to hold only risky assets in efficient portfolios. Now they want tohold some amount of the risk-free asset in combination with an efficient portfolio. And theone efficient portfolio they will hold in such combination is the market portfolio. Every otherportfolio is inferior, because now investors can achieve higher returns for a given level of risk.In terms of indifference curves this means that such curves will not be tangent to the efficientset, but rather will be tangent to line R f R′

f that lies above the efficient set at every point exceptM, the point of tangency corresponding to the market portfolio.


The market portfolio must include every risky asset; otherwise the prices of some wouldrise and others fall until all were included, and all assets must be owned by someone. A changein the risk-free return vis-a-vis the return on the market portfolio means that a new pointof tangency will exist and risky asset prices will adjust until a new equilibrium is reached,corresponding to a new market portfolio.

The line segment of R f R′f above point M corresponds to margin purchases of the market

portfolio. In other words, investors who seek a higher return, albeit with higher risk, can borrowat the risk-free rate (by assumption) and invest the proceeds in the market portfolio. Therefore,linear combinations of the risk-free asset and the market portfolio are all that investors willwant to hold above point M as well as below it. Again, above point M as well as below it,investors will want to stay on line R f R′

f because it offers greater utility.The equation of the capital market line R f R′

f is

E[RM ] = R f + λσM (21.1)

or if we can denote E[R] byR:

RM = R f + λσM (21.2)

The line slope, λ, is the “price of risk.” The risk-free rate and market expectations may changeover time and, accordingly, the price of risk may change as a new equilibrium is reached.Equilibrium is assumed to exist now and to reflect expectations over one period into the future.Lambda (λ) is defined by

λ = RM − R f

σM(21.3)

Thus the price of risk is not simply the difference between the price of the risk-free asset andthe expected return on the market portfolio, but must be adjusted for the standard deviation ofthe portfolio return.

THE SECURITY MARKET LINE AND BETA

The equilibrium conditions for efficient holdings of a risk-free asset and the market portfolioare given by the capital market line. But we are also interested in the return on inefficientholdings, whether portfolios or individual assets. Every asset is held in the market portfolio, aswe have previously stated; otherwise its price would change until it was included. The capitalmarket line provides a measure of the price of risk for the overall market portfolio. But whatis the risk–return trade-off for a given asset or inefficient asset portfolio?

By equating the “price of risk” with the slope of the efficient frontier at point of tangencyM (Figure 21.2), we obtain

RM − R f

σm= E[R j ] − RM(

σ j M − σ 2M

)/σM

(21.4)

from which can be obtained the relationship usually called the capital asset pricing model(CAPM):

E[R j ] = R f + σ j M

σ 2M

[RM − R f ] (21.5)


which is usually written in the form

R j = R f + β j [RM − R f ] (21.6)

where beta (β) is defined as

β j = Cov(R j , RM )

Var(RM )= σ j M

σ 2M

(21.7)

Beta can be thought of as the relative volatility of the jth asset and may be estimated (if dataare available) by fitting an ordinary least-squares regression to the equation:

R j = α j + β j RM (21.8)

Several investment services publish calculated betas for US common stocks. However, becausetrue market returns would have to be based on a market portfolio including every asset weightedaccording to its importance, and because this would be impossible to achieve and prohibitivelycostly to try to approach, the “market” portfolio is in fact a portfolio of New York Stock Ex-change listed common stocks or some more restricted list such as the Standard and Poor’s 500.

If we define total risk of any asset as systematic risk plus unsystematic risk, then betarepresents the systematic risk. The actual return on any asset j is given by

R j = α j + β j RM + µ j (21.9)

with variance

σ 2j = β2

j σ2M + σ 2

µ (21.10)

where µ represents an independent random error term, and R j and RM are random variables.Since in (21.10) the variance is the total risk, the right-hand side contains the two component

risks. The unsystematic risk is σ 2µ, and this can be eliminated by diversification. Because un-

systematic risk can thus be eliminated, the market will not pay to avoid it. However, systematicrisk is another matter entirely. Investors will pay to avoid systematic risk, and in equilibriumevery asset must fall on the security market line. Figure 21.3 illustrates the security marketline. The market portfolio itself has β = 1, as indicated.

Figure 21.3 The security market line (SML)


Figure 21.4 Empirical security market line

Although the capital market line relates a risk-free asset to the market portfolio, the securitymarket line relates risk and return for individual securities. These same individual assets areincluded in the market portfolio. Each has its own systematic, nondiversifiable risk associatedwith it and therefore its own rate of return that the marketplace will require of it. If the returnis higher than necessary, investors will want to add more of it to their holdings, bidding up theprice until, in equilibrium, it lies on the security market line. On the other hand, if the returnis lower than necessary, holders will want to sell and the price will decline until it rests on thesecurity market line.

If we plot data points for risk (that is, beta) and return on securities, we will obtain ascattergram such as that shown in Figure 21.4, to which an empirical security market linemay be fitted by least-squares regression. We shall not address here the question of how wellhistorical data reflect expectations about future performance, the stability of β, and so on.These significant questions have been considered by many researchers, and discussion herewould put us on a lengthy detour from the main track.

An important property of the CAPM stated here without proof is: the β of a portfolio is alinearly weighted average of the individual constituent assets. If a portfolio is composed of xpercent asset A, y percent asset B, and z percent asset C, then the portfolio β is

β = xβA + yβB + zβC (21.11)

This is a very important result because it means that the risk of a portfolio may be foundwithout resort to quadratic programming to determine the efficient set.

THE CAPM AND VALUATION

The CAPM (through the security market line) provides the relationship between risk andrequired return imposed by the interactions of all investors in the market. Expected returnis determined by the price paid for an investment. The return is determined as the ratio ofearnings return (dividend or interest) plus (or minus) the price change in the asset over theholding period to the price paid for the asset. If we assume a single-period holding, the return


is then

R1 = D + (P1 − P0)

P0(21.12)

where D is the income return component and (P1 − P0) the capital gain or loss. Combining Dwith P1 and calling it P ′

1, this becomes

R1 = P ′1 − P0

P0(21.13)

Equating this to the CAPM of equation (21.5) and taking expectations of both sides, we get

P ′1 − P0

P0= R f + σ j M

σ 2M

[RM − R f ] (21.14)

This may be rearranged to obtain P0:

P0 = P ′1

1 + R f + (σ j M/σ 2

M

)[RM − R f ]

(21.15)

or

P0 = P ′1

1 + R f + (λ/σM )σ j M(21.16)

where λ was defined in equation (21.3) as the price of risk. Note that this corresponds to therisk-adjusted discount rate discussed in Chapter 18. The certainty equivalent formulation maybe obtained from (21.16) by substituting for σ j M the equivalent (1/P0)σP1 RM :

P0 = P ′1 − (λ/σM )σP1 RM

1 + R f(21.17)

THE CAPM AND COST OF CAPITAL

Early on we stated that the cost of capital to the firm is the required rate of return. The costof equity is the rate required by common shareholders, the cost of debt is the rate required bycreditors, and so on. Therefore, we can write k for E[R j ], where k is the required rate of return.Then equation (21.6) becomes

ke = R f + βe[RM − R f ] (21.18)

for the cost of equity capital, ke, where βe is the systematic risk of equity in the particular firm.The weighted average cost of capital may be found as illustrated in Chapter 4 once the

component costs have been obtained. Dropping the assumptions used to develop the CAPM,which mean that debt is debt, and therefore risk free, how can the cost of debt be found?In principle the CAPM could be used, but how does one estimate beta for a firm’s bonds,particularly for a firm that has never defaulted? The cost of debt may be observed directly inthe market for publicly traded bonds or estimated from market data on those that are. Companydebt will always be required to yield more than federal government debt because the formeris not risk free, whereas federal government debt is as close as we can come to approximatinga risk-free security.


Figure 21.5 The SML and capital investment

THE CAPM AND CAPITAL BUDGETING

One of the results yielded by the CAPM is that once we determine an asset’s risk we willknow the required rate of return it must yield. This being the case, if we can estimate thesystematic risk of the capital investment proposal, we can apply the CAPM. Estimation of betafor a capital investment is a formidable task because we normally do not have a foundationof historical data on this or similar projects from which we might comfortably estimate theproject’s beta. Assuming we have obtained a beta, we could apply the risk-adjusted discountrate or the certainty equivalent method to find the NPV. Or, as illustrated in Figure 21.5, wemay employ the security market line to determine whether or not individual capital investmentprojects should be undertaken.

When we evaluate a capital investment proposal that has the same risk as the firm as a whole,we can correctly use the firm’s weighted average cost of capital. However, for projects whoserisk is either greater or less than that of the firm, it is not appropriate to use the weightedaverage marginal cost of capital.

In Figure 21.5 proposal B has a higher expected return than the overall firm. Should theproject be accepted? No. The return is greater but the risk is greater yet. In fact, because pointB lies below the security market line, we can say that its risk is too great in relation to itsexpected return, or equivalently that its return is too low in relation to its risk. Proposal A hasa expected return less than the firm. But its risk is also less. In fact, because point A lies abovethe security market line, its return is greater than it need be for its level of risk or its risk lessthan it need be for the expected return. For projects having different systematic risk than thefirm as a whole, the CAPM provides a rationale for accepting or rejecting them.

COMPARISON WITH PORTFOLIO APPROACHES

Two major theoretical approaches to evaluation of risky, interrelated projects are those ofportfolio diversification along the lines pioneered by Markowitz (1962) and the approachimplied by the CAPM.


The Markowitz approach requires that we solve a programming problem with a quadraticobjective function. To do so we need the matrix of the variances and covariances of all assetsthat may be included in the portfolio. Then, once the quadratic programming problem has beensolved, we must either solve again repeatedly until we have the efficient set or efficient frontierand then find a point of tangency with the investor’s utility function or, alternatively, we mustfind the investor’s coefficient of risk aversion at the outset, using this in the quadratic programto solve once for the optimum portfolio.

With the Markowitz portfolio approach we must do considerable computation after obtaininga burdensome amount of data. There may be serious questions about the accuracy of variance-covariance data and utility function estimates as well. Assuming that we have obtained accuratedata and solved the quadratic programming problem, the result is a selection of proposals thatmay be unique to the utility function used or the coefficient of risk aversion. Different utilityfunctions may be expected to yield different optimal portfolios.

In contrast, the CAPM requires only that we know the security market line and the ex-pected return and the beta of any given investment. Since beta measures the systematic ornondiversifiable risk of an asset, other risk is assumed to be unimportant because (in theoryat least) it can be eliminated by suitable diversification. In the case of the CAPM, the marketdetermines, through the security market line, what is an acceptable investment and what is not.In the Markowitz approach, the question of whether or not a particular project is acceptableis internal to the enterprise. It depends on the existing firm, the array of candidate projects,and a utility or investment function. The Markowitz approach assumes there are benefits to begained through active diversification. The CAPM suggests, however, there is no advantage todiversification by the firm, although it implicitly requires that diversifiable risk is eliminatedthrough appropriate portfolio construction.

SOME CRITICISMS OF THE CAPM

The question we need to be concerned with in a capital investment text is how appropriatea model or technique is to capital investments. The CAPM was developed, and its validitytested, for common stocks. Can it be transferred to capital-budgeting applications without lossof validity?

There are some major differences between publicly traded securities and capital investments.Such securities are highly liquid, whereas capital investment projects often do not have asecondary market other than for scrap. Securities approach the CAPM assumption of infinitedivisibility much more closely than most capital investment proposals. There is usually littleinformation available to the public about capital investment projects, much less a consensusabout expected return and risk.

Commodity markets meet the conditions assumed by the CAPM much more closely thancapital investments. Yet, in an empirical test of the CAPM, Holthausen and Hughes concludedthat the CAPM may not fit commodity markets as well as it does security markets. Theyalso observed that returns on commodities do not seem closely related to the measures ofnondiversifiable risk they employed (that is, the betas) [70].

Rendleman points out that although the CAPM provides a basis for valuing securities ina perfect and efficient capital market, there are problems in attempting to use it in capitalbudgeting. He examines the application of the CAPM under capital rationing, concluding thatit is not appropriate “for a firm to use its own beta when computing the expected excess returnof a project with risk characteristics identical to the firm itself” [129, p. 42].


Myers and Turnbull showed that under certain circumstances a capital project’s beta willbe a function of the growth rate of the cash flows and the project life. They suggest it maynot be possible to obtain betas that truly represent the systematic risk for the firm’s cash flows[115].

Levy was concerned with the use of the CAPM in public utility regulation. Although hisremarks are addressed to utility company securities, they apply also, perhaps more strongly,to use of the CAPM for capital budgeting. His comments are:

1. If a public market for the utility’s stock does not exist, the beta cannot be computed.2. Similarly, if a list of diversified companies were determined (on qualitative grounds) to be

equivalent in risk to a particular nontraded utility, the average standard deviation of earningsof firms in the list could probably be used as an estimate of the standard deviation for theutility; but the same inference might not be appropriate regarding the beta.

3. At least two studies . . . find that investors receive some incremental return for incurringdiversifiable risk.

4. Unless the independent variable in the regression equation is fully diversified to includebonds, real estate and other investments, the beta coefficient will not properly distinguishbetween diversifiable and nondiversifiable risk.

5. Large disparities can exist in the beta coefficients for individual stocks when differentcomputational methods are employed [such as time span].

6. Independent studies reveal marked instability of [calculated] betas over time.7. [Studies] indicate that returns on high beta stocks were lower than would be expected and

returns on low beta stocks higher . . . the expected relationship between betas and individualstock returns prevailed less than half the time.

8. . . . tests will show, almost invariably, that the beta coefficients for individual stocks are notsignificantly different in a statistical sense from the general market beta of 1.00 [88].

Fama demonstrated that the future riskless rate, the market price of risk, or the elasticityof a capital project’s expected cash flows with respect to the market return must be certain.Otherwise the CAPM cannot be properly employed. The only parameter that can be uncertainthrough time is that of the cash flows themselves [44].

THE ARBITRAGE PRICING THEORY (APT)

In recent years the CAPM has yielded to the notion of the arbitrage pricing theory. The CAPMprovides a theoretically simple model for measuring and matching risks to expected returns.It affords a way of determining the return that in the view of the overall market is appropriateto the level of risk, and vice versa.

The CAPM postulates that the relevant, or nondiversifiable risk of an asset is fully measuredby its sensitivity to the risk premium on the market portfolio; that is, to (RM − R f ). Thatsensitivity is measured by the asset’s beta (β). While the CAPM is useful, it fails to explainwhy there appear to be persistent differences in common stock returns from influences arisingfrom the industry a firm is in, its size, the term structure of interest rates, and so on. And, ithas been suggested by Roll [133] that the CAPM cannot be adequately tested empirically.

The fundamental idea behind the APT is that there are several influences that togetherdetermine the return on a security or other investment, and also the risk. These influences arecalled factors and have a particular statistical meaning and interpretation.


The general awakening of interest in a multiple factor model by financial economistsfollowed Ross [134], who provided a theoretical foundation that earlier multiple factor modelslacked.

Factors — What Are They?

In multivariate statistics there are two closely related techniques called principal componentsanalysis and factor analysis. Both are based on the notion that observations on real, observable,economic, or other variables can be viewed as being a weighted sum of unobservable variablescalled principal components, or factors. Alternatively, the principal components or factors maybe viewed as being a weighted sum of the observable variables.

In principal components analysis, if there are j variables then it will require j principalcomponents to account for the total variance [9]. The jth principal component accounts forthe jth greatest variance. A small number of principal components may account for most ofthe original variance, but all j of the principal components are required to fully account forthe variance. The principal components are linear combinations of the original, observablevariables.

In common factor analysis the model is based on the premise that there are fewer factors thanthe original j variables. However, the factors may not account for as much of the variance asthe same number of principal components, since the common factors are linear combinationssolely of the common parts of the variables.

Both techniques have been applied to the APT. However, factor analysis is preferred. Factoranalysis is directed toward the correlation, or covariance between the original variables. Prin-cipal components analysis is directed toward the variance of the variables, not the commoninfluence between variables. Since the APT postulates influences common to many assets in themarket, researchers thus normally should use the technique that better probes for such commoninfluences in preference to one that looks for something that might be unique to each asset.

The idea behind the techniques is perhaps best grasped by seeing how they apply graphically.Consider Figure 21.6, which shows a scatter of points in an axis system with returns to assetA on the vertical and returns to asset B on the horizontal. The two assets have returns that

Figure 21.6 Two assets, two factors


are related, and that is why the scatter of points is elliptical and not circular. Perfect positivecorrelation would, of course, appear as a straight line rising at a 45-degree angle from theorigin; perfect negative correlation would appear as a straight line falling from left to right ata 45-degree angle.

The longer axis fitted to the ellipse containing the returns accounts for the greater proportionof the common variance in returns between the two assets. The shorter axis accounts for the restof the common variance since there are only two assets. Points on the two axes that correspondto the original pairs of returns, such as pair x, coincide with an “observation” on a factor orprincipal component. Point q is the value of factor 1, point r the value of factor 2. Because theprojection of the first (i.e. principal) axis on the RB axis is longer than that on the RA axis, weobserve that it is more closely related to asset B’s returns. Factor 1 is said to load more heavilyon asset B.

This graphical model may be extended to three dimensions. Beyond three dimensions themathematical techniques can deal with the calculations even though there may be no meaningfulgraphical exposition of the hyperellipsoids to which an axis system is being fitted.

Multiple correlations between the original observations and the points on the axes providewhat are termed factor loadings. The axis system may be rotated to obtain factor loadings withthe original variables that may facilitate interpretation of the factors. For example, if one wereusing data for interest rates, a high loading of factor n with the yield on US Treasury bonds,corporate AAA-rated bonds, and mortgages, a promising interpretation would be that factor nis a long-term interest rate factor. When a factor loading is very high on an observable variable,we then may use the observable variable in place of the factor, as a surrogate. In searchingfor macroeconomic variables that may be called “factors” such high loadings are what onelooks for.

Rotation may be orthogonal or nonorthogonal. That is, rotation may be done with the axesperpendicular to one another, or nonperpendicular. Principal components are always donewith orthogonal axes. Factor analysis may be done either way, but the nonorthogonal rotationschemes, despite some subjectivity necessary in their use, provide a richer domain of possibleinterpretations.

In comparison to other statistical techniques, factor analysis has not had a large variety oftests of significance developed. A traditional rule is that factors (or principal components)having eigenvalues (that is, factor or component variances) greater than 1.0 are likely to besignificant. The intuitive interpretation of this is that those factors account for more of thecovariance or correlation than any single one of the original variables. The lack of powerfulstatistical tests for determining which, or how many, factors are significant (that is, “priced” inthe market) has probably been central to the controversy that continues over how many factorsaffect the returns of assets, and what interpretations may reasonably be placed upon them.In examining the history of the APT this knowledge will provide a necessary perspective forunderstanding its evolution.

APT and CAPM

In applying the CAPM we fit a regression equation with return on the jth asset as the dependentvariable to one independent variable, return on the market portfolio or index. The APT takesthe process a step further, by adding to the market return the macroeconomic variables that areconsidered to be important in determining the returns on assets. Thus, APT is a multivariateanalog to the CAPM and includes the CAPM as a special case.


Development of the APT

APT is considered to offer a testable alternative to the CAPM introduced by Sharpe [143].Ross argues that the APT is an appropriate alternative because it agrees with the intuitionbehind the CAPM since it is based on a linear return-generating process in which risk maybe separated into diversifiable and nondiversifiable portions. Unlike the CAPM, however, theAPT is claimed to hold in both multi-period and single period instances, and the “market”portfolio plays no special role in APT.

Major differences between APT and Sharpe’s model are that (1) APT allows for more thanone return-generating factor and (2) because no arbitrage profits are possible in a market atequilibrium, every equilibrium may be characterized by a linear relationship between eachasset’s factors. The easily accepted assumption that no riskless arbitrage profits can exist inequilibrium, given the factor-generating model, leads directly to APT.

Bower, Bower, and Logue (BBL) provide an example application of the APT to a sampleof companies [16]. Though they had worked with a larger sample, the one for which resultswere presented in the article cited here was very heavily composed of firms in the broadcastingindustry. Nevertheless, the results obtained are worth looking at for the general lesson theycontain. The authors used four alleged factors (industrial production, inflation, interest rateterm structure, and the spread between low and high grade bonds) in regressions, along withthe market index return as a fifth macroeconomic variable. They compared the results withthose from regressing the stocks’ returns on the market index return, the CAPM model.

BBL found that the R2 for the APT was 0.36 compared to 0.32 for the CAPM. Taking theregression coefficients they had obtained, they applied them to estimate the required rate ofreturn on each firm’s stock. For the sample they found the APT yielded a lower required rate ofreturn than the CAPM, an average of 18.8 percent versus 23.0 percent. However, the standarddeviation of the required returns was 6.05 percent with the APT, compared to 4.43 percent forthe CAPM.

Since BBL do not state whether or not they used the R2 adjusted for degrees of freedom, it isnot possible to determine from the article if the four additional macroeconomic variables madea genuine contribution or not. For each additional independent variable used in a regressionthere is generally an increase in the R2 due to loss of degrees of freedom. Use of R2 (called“r-bar squared”) takes this into account.

Close examination of BBL’s regression coefficients reveals that with the consistent exceptionof the return on the market index, most of them are very close to zero, even though some of themare statistically significant. The reason for mentioning this is to focus attention on the differencebetween statistical significance and economic, or practical significance. For example, for CookInternational, BBL calculate a coefficient of 0.05 (one of the larger such coefficients), witha t-value of 3.32 indicating high significance, for the third macroeconomic variable (interestrate term structure). The coefficient for the market return in contrast is 0.90 with a t-value of3.12. Thus, the statistical significance of interest rate term structure is greater than that of themarket, but its impact on the required return only 1/18th as great!

If indeed there are only four or five factors, one might be forgiven if he or she looks forsomewhat greater economic significance from each.

22

Multiple Project Selection under Risk

Previously we discussed how to select individual risky investments when they are independentof the existing assets of the firm or were very large relative to the firm’s existing assets. Thecapital asset pricing model (CAPM) considered how to select individual risky assets when theycould be considered related to the existing assets of the firm only through an index of returnson a “market” portfolio. This chapter focuses on how to select risky projects that are relateddirectly to the existing assets of the firm, and not solely through a market index.

Thus, we are going to consider an alternative to the CAPM approach for selecting risky assets.This is appropriate for many capital investments when we do not have a reliable estimate of theirbetas, or have reason to question the appropriateness of the CAPM for them. The CAPM worksbest for large projects, those on the scale of acquisition of other, publicly traded companies.The approach illustrated in this chapter can offer advantages for other projects.

Recognition that risky investment projects are statistically related to one another or to thecurrently existing assets of the firm can facilitate better investment decisions. When risky assetshave nonzero correlation between their cash flows we say the project proposals are (statistically)dependent. When different project proposals have different degrees of dependence in relationto the existing assets of the firm we often find that a project that is individually more risky ispreferable to one that is individually less risky.

The correlation coefficient (ρ, the Greek letter rho) between two sets of cash flows (forprojects A and B) is defined as the ratio of the covariance between the two divided by theproduct of their standard deviations:

ρ = σA,B/(σAσB)

The covariance between the cash flows of A and B is analogous to the variance of the cashflows of A, or of B. If we denote an individual cash flow of project A by X, and the average,or expected cash flow by µX (Greek letter mu), then the variance is calculated as the expectedvalue of (X − µX )2; that is, the average or expected value of the quantity X less its expectedvalue squared. If we denote an individual cash flow of project B by Y, then the covariance isdefined as the expected value of (X − µX )(Y − µY ). The value of ρ2 (rho-square) measuresthe proportion of the variability in A that is statistically accounted for by the variability in B.And, since we measure risk as variability, a ρ2 of 1.00 tells us that the cash flows are perfectlyrelated, while a ρ2 of −1.00 tells they are perfectly but oppositely related.

Up to this point we have assumed that covariances, and thus the correlations, betweenprojects and the enterprise’s existing assets were zero, that is, they were completely indepen-dent. Now we will drop the assumption of independence and take explicitly into account thecovariance terms. We will no longer look at decisions based solely on a project’s individualcharacteristics of risk and return as isolated factors, or solely on its relationship with a marketindex.

To illustrate the effect of nonindependence between a project proposal and the firm’s existingassets the following example will be useful.


Table 22.1 Al’s appliance store, etc.

Existing appliance store Proposed laundromat, etc.

Market value (or cost) $1,000,000 $500,000Expected annual return 100,000 75,000Variance of annual return (10,000)2 (15,000)2

Coefficient of variation 0.100 0.200Covariance –($12,000)2

Table 22.2 Combined asset characteristics, Example 22.1

Value of combined assets $1,500,000Expected combined annual returns $175,000 = E[Rc]Variance of combined returns ($2,333)2 = 5.443E + 06 ($2)Coefficient of variation 0.013Correlation coefficient −0.960

Example 22.1: Al’s Appliance Shop, Revisited In Chapter 21 we considered a problemfaced by the managers of Al’s Appliance Arcades, Inc. That chapter used the CAPM framework.Here we will look at it again, but this time in light of the variance, covariance characteristicsof the projects and the firm. As we return to the scene, the owners of Al’s Appliance Arcades,Inc., are again considering a major expansion that would mean construction of an attachedlaundromat and an equipment rental shop. The existing building would become one part ofa triplex. Management has investigated the natures of the businesses and has determined thedata in the following table to be representative. The statistical measures were calculated fromthe historical cash flows of the existing business and data for a laundromat similar to the onethat is being considered for investment. Table 22.1 summarizes the data.

Although the return on investment is higher, the riskiness of the proposed expansion isgreater in both absolute and relative terms than those of the firm’s existing assets.

Analysis of the proposed expansion in isolation could be carried out with techniques sug-gested in previous chapters. That would be appropriate for an investor who did not have otherbusinesses besides the proposal. But here we are not interested in the proposal as a solitaryinvestment holding. Rather, it would be an addition to an investment portfolio that alreadycontains assets.

Individually, the proposal is relatively twice as risky as the appliance store, as indicated bythe respective coefficients of variation, which are defined as the standard deviations dividedby the respective expected returns. However, the covariance between them is negative. Thisindicates that the proposal will tend to respond not only differently, but oppositely, to changesin the economy than the existing business. Table 22.2 contains data for the firm assuming theexpansion has been undertaken.

Note that the relative riskiness as measured by the coefficient of variation has dropped farbelow what it was for the appliance store alone, which was less risky than the proposal. Thevariance of returns on the combined assets1 is less than that of either the proposal alone, or

1When we measure returns and variances in dollars, the variance of a combination of assets is given by

Var(x + y + z + . . .) = Var(x) + Var(y) + Var(z) + . . . + 2 Cov(x, y) + 2 Cov(x, z) + . . . 2 Cov(y, z) + . . .

Multiple Project Selection under Risk 241

Table 22.3 Risk–return relationships for different return correlationsa

between existing firm and proposed expansion

Correlation Cov(0,1) Var(combination) Coefficient of variation

−1.00 −1.5000E + 08 2.5000E + 07 0.029−0.90 −1.3500E + 08 5.5000E + 07 0.042−0.80 −1.2000E + 08 8.5000E + 07 0.053−0.70 −1.0500E + 08 1.1500E + 08 0.061−0.60 −9.0000E + 07 1.4500E + 08 0.069−0.50 −7.5000E + 07 1.7500E + 08 0.076−0.40 −6.0000E + 07 2.0500E + 08 0.082−0.30 −4.5000E + 07 2.3500E + 08 0.088−0.20 −3.0000E + 07 2.6500E + 08 0.093−0.10 −1.5000E + 07 2.9500E + 08 0.098

0.00 0.0000E + 00 3.2500E + 08 0.1030.10 1.5000E + 07 3.5500E + 08 0.1080.20 3.0000E + 07 3.8500E + 08 0.1120.30 4.5000E + 07 4.1500E + 08 0.1160.40 6.0000E + 07 4.4500E + 08 0.1210.50 7.5000E + 07 4.7500E + 08 0.1250.60 9.0000E + 07 5.0500E + 08 0.1280.70 1.0500E + 08 5.3500E + 08 0.1320.80 1.2000E + 08 5.6500E + 08 0.1360.90 1.3500E + 08 5.9500E + 08 0.1391.00 1.5000E + 08 6.2500E + 08 0.143

aNote that the covariance between variable 0 and variable 1 may be found from thecorrelation (R) and the standard deviations as Cov(0, 1) = ρσ0σ1.

that of the existing assets. Therefore, the absolute risk is also less than it would be withoutthe expansion. This highlights the benefits to be gained from diversification. The results ofthe combination are particularly attractive for assets whose returns tend to move in oppositedirections; the correlation in this case between the appliance store and the laundromat-cum-rental shop is ρ = −0.96, an almost perfect negative correlation. Thus, ρ2 = 0.92 and thismeans that 92 percent of the variation in the return on one is statistically accounted for byvariation in the other. In this example, the correlation is computed from

(−12,000)2/(10,000∗15,000) = −(144,000,000)/(150,000,000) = −0.96

Generalizations

In this example the high negative correlation yields dramatic results. But what if the cor-relation were not strongly negative? What if, instead of negative correlation, there is posi-tive correlation? Table 22.3 contains a range of comparative figures. Figure 22.1 illustratesthe relationship between the correlation coefficient and the coefficient of variation betweenthe existing appliance store and the proposed laundromat. From this we can make severalgeneralizations.

Note: If we were working with percentage returns we would have to use a formula that includes the weights (relative proportions) ofthe assets in the portfolio. Thus, the variance of a combination of assets, with weights a, b, c, and so on, is given by

Var(ax + by + cz + . . .) = a2Var(x) + b2Var(y) + c2Var(z) + . . . + 2ab Cov(x, y) + 2ac Cov(x, z) + . . . + 2bc Cov(y, z) + . . .


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Coe

ffic

ient

of

Var

iatio

n

−1 −0.5 0 0.5 1Correlation

Figure 22.1 Al’s Appliance Store correlation vs coefficient of variation

1. As the correlation coefficient changes from perfect negative correlation to perfect positive,the riskiness of the combined enterprise increases. However, it is less than that for eitherthe existing business or the proposal alone for all negative correlations.

2. The greatest reduction in relative risk occurs from investing in the proposed expansion ifthere is perfect negative correlation between the existing assets and the proposed expansion.

3. The case of ρ = 0.0 corresponds to covariance of zero and consequently statistical inde-pendence of returns from the proposal and those from the existing firm.

The combination of existing assets with the proposed expansion project reduces the risk butnot the expected return. The returns are linearly additive (no terms are raised to an exponentother than 1.0). The risk, as measured by the variances or standard deviations, is not linearlyadditive.

PROJECT INDEPENDENCE — DOES IT REALLY EXIST?

Unless the proposed business expansion is independent of the existing assets of the firm, properanalysis cannot be done without reference to them.2 The coefficient of variation for the proposalis 0.200. A ρ = 0.0 means that the projects are statistically independent. Yet, when ρ = 0.0,the coefficient for the firm after undertaking the investment is 0.103, which is greater thanthe 0.100 it would be without the project. Thus, although the expansion itself is per se twiceas risky as the existing firm, and its returns statistically independent, its acceptance wouldactually increase the relative risk of the combined enterprise, the overall firm, very little. Thepoint is that if the expansion is viewed separately, it would probably be rejected because it isrelatively twice as risky as the firm as it exists now, without the investment. If, on the otherhand, the accept/reject decision is based on the riskiness of the firm before the investment tothe firm after the investment it is apparent that relative risk has increased insignificantly. Therelevant comparison is thus not in the terms of the “own” risk of the proposal vis-a-vis the firmwithout the proposal, but rather the firm without the proposal to the firm with the proposal.

2Therefore, certainty equivalents and risk-adjusted discount rates should be adjusted to reflect this fact.


PROJECT INDIVISIBILITY: A CAPITAL INVESTMENTIS NOT A SECURITY

In portfolio selection as it is applied to securities, it is generally assumed that the investment inany particular asset is infinitely divisible and, excluding short sales, that the amount committedto any asset lies between 0 and 100 percent. In the case of securities such an assumption isreasonable because discontinuities are relatively small, amounting to no more than the small-est investment unit. With capital investments, however, the assumption of infinitely divisibleinvestment units is seldom justified. In the foregoing example the question was not what pro-portion of the proposal should be invested, between 0 and 100 percent inclusive, but whetherthe investment would be rejected (0 percent) or accepted (100 percent). Only the extremeswere considered because the project was considered indivisible.

Project indivisibilities are much more troublesome in capital budgeting than in (stock andbond) portfolio selection. Therefore, techniques that were originally developed for securitiesinvestment portfolio selection must be judiciously modified if correct decisions are to result.Choices of percentage other than 0 percent (rejection) and 100 percent (acceptance) are gener-ally not possible. Thus, in adapting portfolio theory, which was developed for common stockinvestments, we cannot assume that for capital investments there is the same (relatively) smoothset of risk–return combinations. With two assets, A and B, we either have 0 percent of each,100 percent of each, or 0 percent of one and 100 percent of the other. We do not have the optionof considering investment of x percent of the firm’s assets in A and (100 −x) percent of thefirm’s assets in B.

Figure 22.2 graphs the risk–return combinations available for the two assets we have beenexamining. Clearly the proposed investment, in combination with the existing firm, can offeran attractive result for some possible correlations with the existing firm. The expected returnequals the sum of those for the individual projects. But the risk, as measured by the standarddeviation, would be greatly reduced with negative correlation, not only from the proposedinvestment’s own rather high level, but from that of the firm without the proposal.

In the case of a corporation with widely held ownership, the shareholders may prefer that theCAPM criterion be used. If only a small percentage of their wealth is held in the form of shares

6.00E+04

8.00E+04

1.00E+05

1.20E+05

1.40E+05

1.60E+05

1.80E+05

Mar

ket V

alue

0.00E+00 5.00E+07 1.00E+08 1.50E+08 2.00E+08 2.50E+08

Variance

Combination

Existing

Proposed

Figure 22.2 Risk–return set Al’s Appliance Store, etc.


Table 22.4 Variances and correlations

Project 1 Project 2 Noah Zark

Variance–covariance matrixProject 1 0.00004119 0.00000702 0.00000595Project 2 0.00000702 0.00001155 0.00000999Noah Zark 0.00000595 0.00000999 0.00000918

Correlation matrixProject 1 1.0000000 0.3218166 0.3059307Project 2 0.3218166 1.0000000 0.9699338Noah Zark 0.3059307 0.9699338 1.0000000

in the original business, risk reduction for this one firm is not likely to figure as prominentlyin their concerns. They can achieve their own, individual, preferred risk–return combinationsimply by buying or selling stock shares in this and other businesses. But, if the firm is ownedby one individual (or a few persons) the proposal could be accepted based on the risk–returncombination of the merged capital assets.

Example 22.2 Noah Zark In Chapter 21 we illustrated the use of betas to determinewhether an investment should be undertaken. This information was then used to decide whetheror not to accept either or both of two proposed investments.

Noah Zark, Inc. is a major manufacturer of pet foods whose management feels it is timeto significantly expand its operations. Two capital investment proposals are being considered,each with very different characteristics. One project is a fish farm that has been operatingfor several years. The other is a factory that can synthesize protein from the carbohydratesin corn. Management requires a recommendation separate from that based on betas aboutwhether either or both of the projects should be accepted for investment. Table 22.4 con-tains the variance–covariance matrix and correlation matrix for Noah Zark and the proposedprojects.

Note that in Table 22.5 the variance is calculated from the relationship

(9.18E − 6)(10/11)2 + (4.119E − 5)(1/11)2 + 2(5.95E − 6)(10/11)(1/11) = 8.911E − 6

But this is not in dollars. To convert it to dollars we need to multiply by the combined assetvalue squared ($1100)2, to obtain ($$)10.78. Then the standard deviation, the square root ofthe variance, is 0.2985073 percent, or $3.28. This dollar measure is obtainable either as thesquare root of ($$)10.78 or from 0.002985073 times $1100. The expected return is

(4.353%)($1000) + (4.500%)($100) = $48.03

The expected returns and variances for the other combinations are obtained in like manner.The results in Table 22.5 can be used to decide whether or not projects 1 or 2 should

be accepted. Figure 22.3 graphs the expected returns against the coefficients of variationfor combinations of Noah Zark with the proposed capital investments. Comparison of thecoefficient of variation for the Noah Zark firm alone with (1) the portfolios composed of theexisting Noah Zark firm with project 1, (2) the Noah Zark firm with project 2, and (3) the NoahZark firm with both projects 1 and 2, reveals several facts.


Table 22.5 Noah Zark with projects 1 and 2—risk : return characteristics

Costs Market value

Project 1 Project 2 Noah Zark

Value weights $100 $200 $1,000Dollar returns $4.50 $6.60 $43.53Dollar (squared) $$0.42 $$0.46 $$9.18

variances

Firm with Firm with Firm withCombinations Firm alone project 1 project 2 projects 1 and 2

Variance ($$) $$9.18 $$10.78 $$13.64 $$15.52Standard deviation $3.03 $3.28 $3.69 $3.94

E[$R] $43.53 $48.03 $50.13 $54.63Coefficient of 0.0696 0.0684 0.0737 0.0721

variation

Weights Combinations Values Weights Squared weights

Firm $1,000 0.90909 0.82645Project 1 100 0.09091 0.00826Total $1,100 1.00000Firm $1,000 0.83333 0.69444Project 2 200 0.16667 0.02778Total $1,200 1.00000Firm $1,000 0.76923 0.59172Project 1 100 0.07692 0.00592Project 2 200 0.15385 0.02367Total $1,300 1.00000

4.1500%

4.2000%

4.2500%

4.3000%

4.3500%

4.4000%

3.00 3.20 3.40 3.60 3.80 4.00

Standard Deviation

Exp

ecte

d R

etur

n

Firm with Project 2

Firm with Both 1 and 2

Firm with Project 1

Firm Alone

Figure 22.3 Risk–return combinations: Noah Zark and its investments

First, acceptance of project 1 increases the total return on investment to 4.37 percent perquarter ($48.03/$1100), or a compound annual rate of 18.64 percent (1.04374 − 1) from4.35 percent per quarter and 18.57 percent per year. Second, along with the increased re-turn there is a decline in the relative risk. The coefficient of variation drops from 0.0696


for the firm alone to 0.0683 for the firm combined with project 1. Thus, project 1 should beaccepted.

In comparison, project 2 if accepted would decrease the expected return on investment to4.18 percent per quarter ($50.13/$1200) or 17.8 percent per year, and at the same time raisethe relative riskiness, as measured by the coefficient of variation, to 0.0737. There would haveto be some compelling intangibles associated with project 2 for management to consider itfurther. It should not be accepted.

If both projects 1 and 2 were to be accepted, the result would be an expected return of 4.20percent per quarter ($54.63/$1300), or 17.7 percent per year. And the coefficient of variationwould be 0.0721. Since this is inferior to the firm alone, or the firm combined with project 1,project 2 will not be accepted.

The acceptance of both projects was indicated when the CAPM was applied to these sameassets in Chapter 21. However, here we look at the firm and the proposed projects, and all theirinterrelationships, but not at the market. With the CAPM we considered only their relationshipsas manifested through the market, and ignored their covariances with one another. The CAPMis appropriate for companies whose shareholders have well-diversified portfolios, such thatthe market for securities governs their risk–return preferences. However, for companies thathave concentrated ownership, and whose owners are not (and perhaps cannot be) so welldiversified, the approach illustrated in this chapter may be more useful.

This example illustrates that accept/reject decisions for capital budgeting proposals canbe obtained which take risk into account even when suitable estimates of their return char-acteristics vis-a-vis the “market” are not available. The decision may be based on the vari-ance/covariance relationships between the firm and the projects calculated from the returnson each of them alone. Thus we have a useful alternative to the CAPM in situations whereeither the CAPM is considered inappropriate or else a second measure is desired to augmentthe CAPM-based recommendation.

It must be mentioned that a project proposal may have both higher expected return andhigher risk than the existing firm. If the correlation is not such that the combination of theproject with the firm produces a coefficient of variation that is not greater than that of the firm,that will create some indeterminacy. Should the project be accepted if its combination with thefirm produces both a higher return and higher relative risk? This is a question that managementmust answer. Without reference to what the market judges to be the appropriate relationship ofrisk to return, management must apply its collective utility function to the question and decideif the return is sufficiently high to justify the greater associated risk. In the case of the CAPMthe decision could be made without reference to management judgment about the risk–returntrade-off. With the present method management must decide, it cannot delegate the decisionto the market.

GENERALIZATION ON MULTIPLE PROJECT SELECTION

The formal model for optimal choice of risky assets for an investment portfolio (generally calledthe Markowitz (1962) model) suggests that what is optimal for one investor will generally bedifferent from what is optimal for another because of different attitudes toward risk–returnrelationships. The basic nature of risk and return within the framework of utility is coveredin Chapter 14. Here we shall adapt the treatment of risk–return trade-off to the multiple assetcase.


The overall return from a portfolio,3 R, is a weighted average of the returns, Ri , with weightsxi , as

R = x1 R1 + x2 R2 + · · · + xn Rn (22.1)

Each xi is the proportion of the total invested in asset i, and Ri is the expected return fromthat asset. The Ri are random variables and therefore R, the portfolio return, is also a randomvariable. If we denote the expected values of the Ri by µi , the variance of asset i by σi i , andthe covariance of asset i and asset j as σi j , we obtain for the portfolio:

E[R] = x1µ1 + x2µ2 + · · · + xnµn (22.2)

Var[R] = x21σ11 + x2

2σ22 + · · · + x2nσnn (22.3)

+2x1x2σ12 + 2x1x3σ13 + · · · + 2x1xnσ1n

+2x2x3σ23 + · · · + 2x2xnσ2n, . . .

=n∑

i=1

n∑j=1

xi x jσi j

Securities

When working with securities — which were assumed to be infinitely divisible — varyingthe weights of the individual assets allows us to obtain an unlimited number of portfolios.However, we want to ignore all portfolios other than those that have expected returns superiorto all others with the same risk4. In other words, we want to consider only those risk–returnpoints which constitute the efficient set, that when plotted on a graph define the efficient frontier.For each point on the efficient set there is no other point having lower risk for that expectedreturn or higher expected return for that same risk.

Capital Investments

Unlike securities, capital investment projects are not usually divisible, and often they are one-of-a-kind investments. What this means is that the xi weights above must take on values of 0 or1 only5. The practical consequence of this is to narrow immensely the possible combinationsof projects with the existing firm.

In the case of a firm that is considering only one large capital investment, the expected returnand variance of the combination reduces to

E[R] = x1µ1 + x2µ2

Var[R] = x21σ11 + x2

2σ22 + 2x1x2σ12

where x1 and x2 are either 0 or 1.

3A portfolio is nothing more nor less than a set, or collection, of assets. Thus it may be the stocks and bonds owned by an individualor a pension fund or insurance company. But it may just as correctly be considered as the individual building, fixtures, and so forthof one retailing store in a national retailing chain, or the collection of all the individual stores owned by the chain. Thus it may be atwhatever level of aggregation we wish to consider.4An equivalent way of stating this is that only portfolios having the lowest risk among those having the same expected return are found.5If there were two projects that were virtually identical then they could either be combined and treated as one project, or else treatedas separate projects that would either be accepted or rejected on their own merits.


Clearly, if asset 1 is the existing firm, then its weight must be 1 unless we are consideringselling the business. Thus, as a practical matter we are considering the result of giving a weightof 1 to the proposed project (accepting it) versus continuing the existing business without it.We are interested in what would happen to the risk–return characteristics of the firm if theproject were to be accepted.

Extension to two proposed projects plus the firm is straightforward:

E[R] = x1µ1 + x2µ2 + x3µ3

Var[R] = x21σ11 + x2

2σ22 + x23σ33 + 2x1x2σ12 + 2x1x3σ13 + 2x2x3σ23

where x1, x2, and x3 are either 0 or 1.A company that has only a few fairly large investment proposals may be able to analyze them

without relying on sophisticated and technical computer programs. For example, Baum et al.[7] evaluated all 32 possible selections of 5 candidate projects. However, full evaluation canget out of hand quickly as the number of projects considered increases; with N projects thereare 2N unique combinations. After about five projects the number of combinations rapidlybecomes so great that full enumeration is unduly burdensome if not impossible. However, inall but the very largest enterprises this should not be a significant problem, especially if minorprojects are aggregated together and treated as a single project.

Most managers are wise to insist on making the final decision on which set of capital in-vestments will be undertaken. In the author’s experience executives tend to resist arrogationof their authority and responsibility by technicians armed with computers. And they are gen-erally correct in this view, because there is as yet no adequate substitute for the judgment ofexperienced managers. And when all is said and done, it is the executive who decides thatis responsible for the consequences. Those performing the detailed project analysis shouldnormally present the executives who have the decision responsibility with a summary of thecharacteristics of the efficient alternative combinations. In the best capital investment analysisthere will be dialogue between those responsible for the final decision and those doing thedetailed analysis as the analysis proceeds. Few things could be expected to result in worsecapital-budgeting decisions that to have an assignment given to the analyst who will have nofurther contact with the decision-makers until he or she presents a recommendation.


This chapter presented an alternative to the CAPM for determining the acceptability of proposedcapital investment projects. It was shown that the combination of a proposed project with thefirm can lower or increase the enterprise’s risk depending upon the correlation of its cash flowswith those of the existing firm.

Projects that decrease the firm’s risk while not decreasing its expected return on investmentare accepted. Projects that increase the risk may be acceptable if management judges that theyincrease the expected return enough to compensate for the greater risk. Thus this method,in contrast to that of the CAPM, requires that management decide whether the risk-to-returntrade-off is appropriate. This is both a strength and a weakness of the method. It is a weaknessif judged by the extent to which decisions may be automated or delegated to lower levels ofmanagement. It is a strength if judged on the basis of management having to commit to adecision based on its collective judgment.


The method illustrated in this chapter provides an alternative to the CAPM. In cases wherethe CAPM is judged to be inappropriate or inapplicable it will allow rational decisions to bereached on the basis of risk-to-return trade-offs. In cases where the CAPM is applicable it canprovide another angle of view by which a proposed project may be examined.

A BETTER WAY TO CALCULATE RISK-ADJUSTED DISCOUNT RATESRecall that the capital asset pricing model (CAPM) can be written as the equation

k = Rf + β(RM − Rf)

Thus if we have the beta for an asset we can determine the appropriate discount rate touse in calculating its net present value (NPV). This suggests that there is a superior wayby which we can calculate risk-adjusted discount rates. To apply the method we need theproject proposal’s beta (β), the risk-free rate of interest (Rf), and the expected return on themarket (RM). For the risk-free rate we normally would use the yield on three-month USTreasury bills. The expected return on the market is the most troublesome item to estimate.Normally we would use the historical rate of return as the estimate of the expected rate.

For the Noah Zark company the risk-adjusted rates of discount we would use for calcu-lating the NPVs of projects 1 and 2 would be calculated as follows, using the data from theexample:

Project 1:k1 = 1.750% + 0.689(3.058% − 1.750%) = 2.651% per quarter which on a compound

annual basis is (100(1.02651)4 − 1) = 11.034%.

Project 2:k2 = 1.750% + 1.193(3.058% − 1.750%) = 3.310% per quarter which on a compound

annual basis is (100(1.0331)4 − 1) = 13.914%.

Thus we see that the more risky project is discounted at a higher rate than the less riskyproject. Not only that, but the discount rate reflects the market’s consensus judgment aboutthe “price of risk” and is therefore an objective alternative to other ways of calculatingrisk-adjusted discount rates.

23

Real Options∗

The term “real options” might better have been called “options on tangible or physical assets”but the term is solidly entrenched now. Real options in the realm of capital investment are thosecontaining choices about capital assets. These choices include whether to invest now, or investlater, whether to divest now or divest later, whether to expand or contract scale of operations,whether to switch use (input, output, or the application of a real asset) and combinations thatinvolve choices of actions over time about capital assets. Stewart Myers coined the term “realoptions” in a 1984 article to contrast them with financial options.

Why should we use real options? The main reason is that the traditional methods of capitalinvestment analysis fail to recognize that effective managers do not stand by passively, but areactive, and adapt to changes in markets and to opportunities for capturing strategic value. “Anoptions approach to capital budgeting has the potential to conceptualize and quantify the valueof options from active management and strategic interactions” [161, p. 4].

ACQUIRING AND DISPOSING: THE CALL AND PUT OF IT

Call options provide their owners with the right to buy something at a specific price, termedthe strike price or exercise price, until (American-style call options) or at (European-style calloption) some specified expiration date. Put options endow their owners with the right to sellsomething at a strike price at, or until, expiration. As a practical matter, there is little differencebetween the theoretical prices of an option as American or European. This is especially sowhen one considers that in the case of capital assets one usually is working with forecasts,approximation, insights, and managerial intuition about the relevant parameters. Other factorsare typically more important in determining the value of a real option on a capital asset.

The theoretical value of an option, which we assume is the market price if it is traded publicly,is determined by the difference between the exercise price and market price of the underlyingasset, the time to expiration, the price volatility of the underlying asset, the risk-free interestrate, and the cash flow from the underlying asset (interest or dividend or cost of storage if aphysical commodity). Most finance texts do not treat options on physical commodities, insteadconcentrating on options on common stock, and sometimes including those on futures andcurrencies. However, there are important differences between options on stocks, futures con-tracts, and physical commodities.1 In the case of real options the valuation models for financialinstruments and commodities may sometimes work, but often other approaches must be used.

The fundamental concepts of put and call options can be viewed in the real option schemeof things as abandon or invest, stop or proceed. The time aspect of a real option is readily

∗The author is indebted to J. Barry Lin in for his advice and co-authorship on this chapter. We both appreciate the critical insights and

valuable advice on this chapter provided by Marco Antonio G. Dias.1Users of this book may download a free ExcelTMspreadsheet, developed by the author, that is designed to calculate option prices andother parameters (the “Greeks”) and provide sensitivity analysis graphics. The spreadsheet is based on an article by James F. Meisnerand John W. Labuszewski [103].


seen to correspond to decide now, or decide up to some future date. Transfer of option pricingmethods from financial instruments and commodities to real options presents difficulties inmost cases because valid estimates of the critical parameters are not available. One needs tohave a sense of the market price of the underlying asset, for instance, and that might not beavailable. Similarly, the volatility of the asset price may present estimation obstacles. On topof that, the time to expiration — the useful life of the real option — may depend on events thatcannot be gauged in advance.

MORE THAN ONE WAY TO GET THERE

In previous chapters we examined capital investment decisions using decision trees, dynamicprogramming, and computer simulation. These offer both competing and complementary meth-ods for dealing with projects that have real options embedded in them. Which method is bestsuited to a particular situation is something that the decision-maker must choose, somethingthat cannot be specified in advance for all cases. Where it is possible, analysis of a project bythe real options approach and one or more other approaches can provide insights that raise theodds for success — for sound decisions — those that foster success of the organization.

Success with the real options approach, as with any analytical technique, rests heavily uponcareful, accurate specification of the problem situation and all the relevant factors. A techniquethat is based on ad hoc, seat-of-the-pants “guesstimates” risks becoming little more than aburlesque that consumes resources and misleads management into costly mistakes. This is nomore a problem with the real options approach than with any of the others. However, whenevera new approach becomes so suddenly popular as the real options way of looking at things,one must resist the temptation to throw data into it haphazardly in the belief or hope that themethod itself will make things right.2

WHERE ARE REAL OPTIONS FOUND?

Real options are found wherever there is potential for management action beyond the presenttime, t = 0. They are found in such diverse decisions as mineral and petroleum exploration;valuation of and investment in a start-up business; research, development and testing of a newmedicine; valuation of raw land; licensing a technology; investment in infrastructure/ancillarysupport operations; and contingency planning for flexibility in responding to competitors’actions. These are often referred to as strategic investments.

Besides finding real options in projects already planned or extant, they may also be deliber-ately engineered into a project from the beginning. In other words, they arise due to the naturalattributes of the project (abandonment, for instance), or they may be created through the designof the project. For example, a firm can make a product that can be retrofitted later in the eventa technology becomes developed sufficiently. This is what happened with the now ubiquitousglobal positioning system (GPS). The cost of engineering the option into the project at theplanning stage essentially makes the initial outlay endogenous.

2The popularity of real options as a tool is attested to by the seminar-workshops on the topic as this chapter is written. One need onlylook to the zeal with which some promoted value at risk (VAR), and before that to value-added accounting and the capital asset pricingmodel (CAPM), to see why some sober caution is in order before leaping onto the bandwagon. That said, the real options approachcan offer genuine value if applied properly and to appropriate problem situations. At the very least it offers a different angle of viewthat can be compared to other measures, such as discounted cash flow.

Real Options 253

Table 23.1 Comparison of Real to Financial Options

Parameter Capital asset project Financial Option

X Initial investment cost Exercise premium (option price)S Value of capital assets Price of underlying securityt Time of viability for option Time to expiration (i.e. days/365)σ 2 Volatility of project cash flows Volatility of security returnsi Risk-free discount rate Risk-free discount rated Annual cash flow return Annual dividend or interest return

In contrast to real options, discounted cash flow (DCF) analysis is like a “buy-and-hold”strategy over the span t = 0 to t = n. DCF analysis implicitly assumes that a decision is takenimmediately, then not revisited before the end of the project’s useful life.

One can find real options embedded in such management decisions as:

1. Selecting a production facility design (plant layout design) and choosing the technologiesthat will be built into it.

2. Building a parking garage or high-rise office or apartment building on a parking lot.3. Investing in new cable TV lines and nodes when a superior technology appears to be

imminent.4. Entering a foreign market by licensing, joint venture, or other means.5. Leasing versus purchasing land, buildings and equipment.6. Investing in a new computer system today or waiting for a likely better one next year.7. Selling a business now or delaying until the economy improves.8. Investing in corporate “capabilities” (training, R&D, distribution channels, customer and

supplier relationships, etc.), allowing the corporation to be better prepared for the businessgame, by creating new real options that can be quickly exercised in favorable scenarios [seee.g. 81].

A project such as a research and development (R&D) program may be required at stage 1 inorder for the organization to be able to undertake a more advanced R&D program later on atstage 2 or 3. In other words, R&D may offer no short cuts; one must ante up early on to be inthe game later. However, each stage of R&D may produce some cash generating opportunities.These might be foreseen or be the result of serendipity, like 3M’s ubiquitous Post-ItTM notes,discovered when an adhesive “failed” to hold tightly.

COMPARISON TO FINANCIAL OPTIONS

It is useful for those who are familiar with financial options to see how real options comparewith them. Table 23.1 contains a comparison of the corresponding terms. Some authoritiesomit the term d, for cash flow, but sometimes this can be important in the analysis of bothfinancial and real options.

FLASHBACK TO PI RATIO — A ROSE BY ANOTHER NAME . . .

Chapter 6 examined the profitability index (PI) as a means for comparing two or more competingcapital assets for which the initial investments were different. The PI is usually defined as the


ratio of the present value of returns to present value of initial project investment cost. In theliterature on real options the PI is used under the label quotient, or NPVq [see e.g. 92]. In optionterminology it is defined as NPVq ≡ S/PV[X ]. The PV of the exercise or initial investmentcost indicates that the exercise may occur some time into the future.

The value of an option depends most heavily on the relationship between X and S, t and σ . Thecumulative variance is given by the quantity σ 2t, and the corresponding standard deviationby σ

√t. Cumulative variance is very important in valuing real options. A low cumulative

variance means that there is a low probability of the option ever achieving a major changefrom its current value over its remaining useful life. A high cumulative variance means thatthere exists a good chance that the option will experience a significant change in value over itsremaining life.

TYPES OF REAL OPTIONS

Real options have been placed into as many as half a dozen categories, based upon the type offlexibility they offer. The categories may conveniently be reduced to the following:

1. The option to delay, to put off a decision until some date in the future. The future datemay be determined by some event or by management calendar. The event might be anaction from a competitor or change in some government regulation or tax, for example. Thedelay is valuable in general because new information may arrive to enhance managerialdecision-making.

2. The option to stage development with a series of investment outlays. The project can beterminated at any stage. Thus the option to abandon is a type of staged development option.So is the option to grow, in which initial R&D and prototyping are required to provide infor-mation for further development, abandonment, or delay until market or technology develop.

3. The option to change scale. This is basically a staged development option, though it is usefulto consider separately. An initial investment decision might be for production equipment thatis less costly, but also incapable of high-volume output. This contrasts with an initial invest-ment decision to adopt higher-cost, high-volume production equipment. Both decisions haveembedded options to change scale, with their own favorable and unfavorable consequences.

4. The option to switch production function and thus either input or output mix, depending onmarket conditions. This is similar to, and can be a special case of, the option to change scale,but is usefully treated separately. For example, a petroleum refinery may be optimized fora particular type of crude oil and desired mix of refined products, or for flexibility in both.

In addition, the real option can be naturally embedded into a project (for example the optionto shut down a plant) or can be created by an investment to embed flexibility into the project(e.g. buying vacant land neighboring the plant, in order to expand operations at low cost in afavorable scenario). In the first case, managers have to identify the options and plan how totake advantage of them. In the latter, managers need to value the real option and compare thatvalue to the costs.

A paper by Jerry Flatto3, presents the results of a questionnaire he sent to the chief infor-mation officers at Life Office Management Association (LOMA) member companies in theUnited States and Canada. He found that very few had ever heard of the term “real option”,

3See http://www.puc-rio.br/marco.ind/LOMA96.html. The article was written for Resource — The Magazine for Life Insurance Man-agement Resource, published by the Life Office Management Association (LOMA), the education arm of the insurance industry.

Real Options 255

but after having it explained to them most said they do include some aspects in their analysisand that in about 60 percent of the cases this made a difference in the approval process.

Remarkably, he found that no companies were using formal models designed to value realoptions. Instead, those who took real options into account said they incorporated real optionconsiderations only qualitatively. It is a tribute to the importance of real options that he foundthat 75 percent of the companies accept projects despite quantitative analysis to the contrary.4

However, the lack of formal models casts a shadow over the procedure under which quantitativemodels are overruled.

REAL OPTION SOLUTION STEPS

There are four steps for formulating and solving a real option problem:

1. Frame the problem.2. Apply the option valuation model.3. Examine the results.4. Reformulate and return to step (1).

The framing step is the most critical for successful application of real options methodologyto a problem situation for which a real options approach is appropriate. If it is done well theprospect for valid analytical results will be greatly improved. But if it is done poorly the processbecomes a burlesque or parody that it would be better to have avoided.

In framing the problem it is necessary to answer a number of questions:

1. What are the critical decision variables?2. How reliable are the estimates, or forecasts of those numbers?3. What are the uncertain elements that could cause the decision(s) to change?4. What is the risk-to-return profile of the investment? How will it affect the firm’s risk

exposure, both market and private?5. Can the option, or a close surrogate, be found at a better price in the market?6. Does the valuation of the option pass a commonsense filter?7. Who are those who have the authority to exercise the option?8. What changes, if any, are required in the organization to adapt real options and use them

profitably?9. What similar projects, if any, has the organization dealt with in the past?

Each source of uncertainty should have its own payoff diagram. And, the decision rule foroption exercise must be specific. Ambiguity does not make for clear management action. Thedecision rule may be modified as time passes and new information is obtained. But at eachstage it must be clearly understood by those who are authorized to act, to exercise the option.It is possible that managerial inertia may lead to systematic overvaluation of real options. Ifso, then specific decision rules could reduce that bias. For example, managers often delayabandonment until a project has gone so far down in value as to be indubitably a lost cause,when they should have recognized the problem earlier and acted then.

4The author recalls a case in which a major US manufacturer leaned toward accepting a project to manufacture a product for com-mercial aircraft that was marginal in terms of the conventional NPV, IRR, and payback criteria. The reason was that by producingand selling the product their sales staff would have access to customers that would facilitate sales of other products, and also that theproduct in question could be further refined as information was obtained from customers based on their experience with it.


The option valuation step may be carried out in a number of ways, depending on the exactproblem specification. Among these are application of the Black–Scholes model for valuingAmerican-style options; use of the binomial option valuation model, and computer simulation.The choice among these is less critical than the careful specification of answers to the framingquestions above.

The difference between the valuation of an American-style and European-style option is notcritical in most instances. If one uses a model for valuing a European-style option in a situationthat really offers an American-style exercise choice, then one should view the valuation as alower bound on the option’s value.

OPTION PHASE DIAGRAMS

Real options may usefully be viewed with a phase diagram, in which cumulative variance isplotted against how far in-the-money or out-of-the-money the option is at the present. ProfessorLuehrman offers a very nice example with his “tomato garden” which has tomatoes at variousstages of ripeness [92, p. 8]. Figure 23.1 contains a similar phase diagram. The density ofthe shading indicates the option value at present; the darker the shading the higher the optionvalue. The line labeled NPVq indicates the project’s value according to the DCF quotientmeasure. This measure and the conventional NPV must converge at expiration to an agreement

HighNPVq

IV. High Cumulative Variance

V. Unlikely Future II. Wait, if possible

III. Maybe, but NPV < 0

VI. Never Exercise the Option

Out of the Money In the Money

I. Exercise the Option NowLow

Cum

ulative Variance

Figure 23.1 Real option value

Real Options 257

on whether or not to undertake the project. But prior to expiration NPVq may be greater thanone, while conventional NPV is negative [92, p. 7].

To see how the phase diagram may be useful consider Alluvial Inc., which owns and operatesa profitable sand and gravel pit in the path of a major city’s expansion. The company can sellnow to the city, which would use the land for a major park and recreation area. The price isconsidered fair in light of current use of the land, but not much better. The company has tofactor in some other considerations.

The company can continue to operate the sand and gravel business until the deposits aredepleted. During this time span the value of the land is likely to increase, but so are taxes,especially since some in the city government want to encourage a sale to the city. Alluvial’smanagement is thinking of converting the gravel pit, now filled with pure fresh water, into acenterpiece lake around which a housing development will be constructed. The pit could alsobe used for a landfill, but this would create problems with the groundwater supply, and thus runafoul of environmental protection laws. A suitable alternative is conversion into a golf courseand private recreation area (boating sports, fishing). This would allow Alluvial to convert lateron to housing or other use with minimal cost.

Under the above description, Alluvial has a project that has a positive NPV and also containsuncertainty. The uncertainty as specified puts the sand and gravel operation into the “wait” cat-egory. It is likely to become worth much more, and in the meantime can continue to be operatedprofitably. This project falls in the middle right-hand portion of the phase diagram, phase II.

Now consider OO Corporation, which for the past five years has leased land it owns in NewJersey to LCN Inc. LCN has broken the terms of the lease, failed to make the last three monthlypayments, and its nominee directors and managers have disappeared. The land is determinedto have a great deal of toxic waste stored on and under it (and possibly some other things). Thesurrounding municipality is concerned about groundwater contamination and noxious fumesemanating from the land, and fears that going to court may delay action that could clean it upunder a federal superfund grant. The municipality has made an offer that its lawyers believeOO Corp. cannot refuse. The offer is to buy the land for the amount owed by LCN Inc. andwithout recourse for the costs of cleanup. The cost of cleanup would be prohibitive to OOCorp. to undertake on its own.

In this case immediate exercise by the company of this put option is indicated. To delay wouldlikely mean costly legal battles with little prospect of winning. And the eventual judgmentagainst OO Corp. could mean its demise. This is a project that will never have a positive NPVand the option to sell should be exercised immediately. It falls in the lower right portion of thephase diagram, phase I.

COMPLEX PROJECTS

A project may have the characteristics of both “assets-in-place” and “growth options.” Suchprojects should be analyzed by separating the parts and analyzing them separately. The assets-in-place may be properly handled with DCF techniques, but the growth components shouldbe handled with an option approach. The process of separation may, however, be difficult toaccomplish. For instance, assets-in-place may well contain embedded options, such as that ofabandonment. Nevertheless, for such projects DCF alone should be avoided because it yieldsincorrect and misleading values.

In framing a real options problem one must be careful to exclude sources of uncertaintythat are not truly relevant and significant. Options with more than two or three sources of


uncertainty are computationally difficult to value, and most numerical solutions cannot handlemore than two, or in some cases three. A new approach to solve several sources of uncertainty,Monte Carlo simulation for American (real) options, offers great practical promise, but is stillunder research development [30].

If the problem frame cannot be explained to experienced, senior management it is probablytoo complex, and should be reformulated. The frame should be capable of explanation toexperienced decision-makers in the industry or area of business affected. And it should bealigned with their experience and understanding. Managers who are too busy to peruse complexdetails should quickly grasp it. Decisions are seldom profitable or unprofitable because of detailbut rather because of the major factors involved.

One of the crucial questions that those framing the problem as one in real options mustask is how the relevant uncertainty affects the payoff function. Under some circumstancesthe problem may lend itself to the Black–Scholes model; in others the binomial model orsimulation may be indicated. Besides the sources of uncertainty, the option model inputs mustbe defined carefully.

In many cases a DCF analysis will have been performed already. If it has, then most ofthe input data for a real options analysis may already be at hand.5 Other than reviewing it foraccuracy and updating to incorporate new information, one would not develop the data overagain from scratch.

The number of options identified in a particular project depends on the number of decisionstages it may be broken into. The greater the number of stages the greater the number ofidentifiable options in most cases.

ESTIMATING THE UNDERLYING VALUE

Estimating the value of the underlying asset for a real option presents a problem not encounteredwith financial options for which the underlying asset is traded actively on the stock or futuresexchange. Sometimes one can estimate the value of the underlying real asset from market data.For example, if the real asset is thought to be worth 15 percent of the value of a competitor’sbusiness that is publicly traded, then one may use that amount as a proxy for the underlyingasset value.

One might argue that the exercise price seldom presents a problem because the cost ofexercise can be determined with relative precision compared to the market value of real assets.However, salvage value of an assembly plant five years into the future is not likely to be knownwith anything close to certainty today. And, in R&D, the exercise prices (cost of developmentfollowed by cost of commercial-scale production) are just wild guesses too.

WHAT TO EXPECT FROM REAL OPTION ANALYSIS

Before committing time and resources to real options analysis one should ask what is reasonableto expect from it. One usually seeks numerical output from the exercise of valuing an investmentproject; that is, a dollar amount that management can gauge for how well it measures up to

5Real options analysis can be considered a sensible generalization on static DCF analysis. After one basic estimation has been donefor one DCF analysis, management can then ask whether any real option is present, and what the parameters of these real optionsare. The value of the real options can then be estimated, and value of the project can be critical when the traditional capital-budgetingcriteria such as NPV and IRR are marginal for a project. Static NPV is a special case of “post modern” NPV where the options havenegligible value.

As parameter estimation is much more problematic with real options, in comparison to financial options, scenario analysis andsensitivity analysis using a range of likely parameters are often helpful.

Real Options 259

benchmarks for such projects. But “ . . . the most important output is not the dollar value ofthe strategy but the three types of decision-making tools shown. These can be used to manageand design investments and to build consensus around the investment strategy” [3, p. 104].The three decision-making tools mentioned are critical values of the asset over time, strategyspace (i.e. abandon, continue, modify), and investment risk profile, the last of which can berelated to probabilities of abandoning, continuing, modifying the investment. In other words,dollar values can be useful, but other factors that management can weave into the fabric of thedecision may be even more important.

Dollar value provides a point estimate, such as the NPV of a project. Knowledge of theinvestment’s risk profile can indicate to management the probabilities associated with rangesof dollar outcomes. And the decisions to abandon, continue, or expand the project are tied tothe probabilities revealed by the risk profile of the project. (For further insight on this, reviewthe discussion of risk profile in Chapter 19, in which computer simulation is covered.)

IDENTIFYING REAL OPTIONS — SOME EXAMPLES

Real options are everywhere, it merely takes practice to develop an eye for seeing them. Forexample:

1. Corel Corporation’s decision to port WordPerfect Suite 2000 to the Linux operating systemhad real options characteristics. Corel in essence decided for early exercise of the option toadapt its sound, but declining-in-popularity office suite to the relatively new, rapidly grow-ing, Unix-like open operating system, Linux. In contrast, Microsoft Corporation decidednot to exercise the same type option. If Corel could have captured a significant share ofthe Linux office suite market before Microsoft moved into it, the early exercise would havepaid off. Microsoft may have difficulty adapting its software to Linux because the Linuxoperating system is open software, basically free to those who wish to use it. Porting itsoffice suite to Linux would assist in making a significant rival to its core operating systembusiness even more popular. However, by delaying what may be the inevitable, Microsoftwas risking widespread adoption of Corel’s rival office suite, which might be difficult torecapture in the Linux market later on.6

2. Rain forest/jungle versus cattle ranches/coffee–tea–cocoa plantations/etc. suggests severalreal options for tropical nations. On the one hand, the wait and see, or wait to exerciseoption for developing jungle or rain forest into use for cattle ranching, or into plantationsfor coffee, bananas, teas, cocoa, or hundreds of other crops, may mean opportunity lossesof export sales, employment growth, and foreign exchange earnings. On the other hand,nonconsumptive development for tourism, biomedical plant research and sustainable har-vesting of timber and medicinal and food plants offers an option that is similar in respect togenerating revenues, yet different in terms of being far more reversible later on if desired.(One cannot turn a cattle ranch or coffee plantation back to the original state of the land, atleast not quickly and at low cost.) And the flexibility to change from one type of reversibledevelopment to another is clearly different than an irreversible conversion of land to a usethat may or may not remain viable in a few years.

3. Farmland in path of city expansion can be — depending upon zoning, which can often bechanged if necessary — converted to a golf course, cemetery, housing or industrial devel-opment. If there are subsurface minerals, or even sand or gravel of good quality that may be

6In 2001 Corel got out of Linux altogether. A visit to its website at http://www.corel.com reveals Corel no longer offers or supportsCorel Linux and is no longer offering an office suite to run under Linux or support for the one it did offer.


extracted, some of the land may be converted to a landfill upon which the golf course can bebuilt later, or as a lake at the center of a luxury housing or high-rise building development.This type of project offers many option possibilities and opportunities for sequencing thatmay improve the overall project value. For instance, conversion to a golf course leaves openthe possibility for later conversion to construction of luxury housing or high-rise buildingseither upon or around the periphery of the golf course. In contrast, an early decision to exer-cise the option of building an industrial park or a cemetery would preclude or at least hinderother opportunities afterward. And, until a decision to do otherwise, the farm might continueto be operated as a farm, earning cash flows until an alternative use decision is exercised.

CONTINGENT CLAIM ANALYSIS

Contingent claim analysis (CCA) is based on two simple, yet powerful ideas of modern financetheory. First, the present value of a risky cash flow stream can be found by discounting thecertainty equivalent of the risky cash flows at the risk-free interest rate. In contrast, conventionalmethods discount the risky cash flow at the required rate of return (risk-free rate plus a riskpremium) of the cash flow stream. Second, the value of a contingent claim on an asset can bederived by finding the value of an equivalent tracking portfolio of similar traded or observedassets. The binomial option pricing model is the best known application of CCA in a binomialtree-based structure.

THE BINOMIAL OPTION PRICING MODEL

The principles behind the binomial option pricing model are elegantly simple. It is based onthe assumption that from one time period to another the asset can take on only one of twovalues. If we represent the value of an asset at time t = 0 as A, then at the end of the first timeperiod the asset will have either value Au for an upward change, or Ad for a downward change.At the end of period 2 the asset will have either value Au2 or Ad2. And the process continuesover all the time periods in the uncertain life of the asset.

This model assumes a risk-neutral or hedged asset. That is, it assumes the asset’s risk ishedged with a tracking portfolio, or tracking instrument. This resolves the question of whatdiscount rate is appropriate to the asset values over time — it is the risk-free rate, r. The reasonit is only this much is that we assume risk has been eliminated by the hedge. For convenience,continuous compounding is used.7 The expected return on the asset is r, the risk-free rate, thevariability is given by the variance, σ 2.

Figure 23.2 shows the process of option evolution. The risk-neutral probability parametersfor upward ( p ) and downward ( 1 − p ) are calculated so that the distribution of final outcomesmatches the conditions of the situation. From the definition of return at the risk-free rate weobtain the equation

p Au + (1 − p)Ad

A= er (23.1)

Next, equating the variance of return to the estimated distribution8 yields

pu2 + (1 − p) d2 − (pu + (1 − p) d)2 = σ 2 (23.2)

7As the number of time periods in a year increases the continuous compounding result is rapidly approached in any case.8It is generally assumed that the normal distribution applies.

Real Options 261

0

12

34

56

7

Binomial Lattice, or Tree

Outcome Distribution Histogram

Figure 23.2 A binomial model of uncertainty

so that

p = Aer − Ad

Au − Ad(23.3)

If we assume symmetry of the upward and downward movements we can obtain one solutionto p from the equations.9 Then

d = 1

eσ= e−σ and u = eσ (23.4)

and

p = er − d

u − d(23.5)

9The point of such an assumption, of course, is to reduce the number of unknowns to the number of unique equations so that a uniquesolution exists.


An advantage of the binomial option pricing model is that it is adaptable to modification to suita particular situation where standard assumptions are inappropriate. It is adaptable to optionpricing situations for which the Black–Scholes model is not.

We will not go into the assumptions of the Black-Scholes model or closed-form, partialdifferential equation (PDE) option pricing here. There are many good references to the former,and the latter is best left to specialists in financial engineering.10 Furthermore, most personswho are using this book already have some acquaintance with the Black–Scholes model.

Application of option pricing models to real options is best shown by some examples.

Example 23.1 Value of Strategic Flexibility in a Ranch/Farm Consider the case of theKaiser Ranch that raises both native and “exotic” game animal species, and whose sharesare publicly traded, and thus its market value is observable. The ranch also operates privatehunting activities that attract a worldwide clientele of wealthy sportsmen.11 Figure 23.3 depictsthe (uncertain) growth path of the value12 of the business (not the cash flows). The value of theranch is driven by uncertainty about the state of the economy, the amount of hunter spendingat the ranch, and the market price of the wild game meat. Further, suppose that the risk of thebusiness requires a k = 17.5 percent required rate of return, while the risk-free rate is r =

S0 � 100

Su1 � 175

Sd1 � 60

Sd2 � 36

Sud2 � 105

Su2 � 306.25

Figure 23.3 The growth path of ranch

10Fisher Black, cocreator of the Black–Scholes model, speaking at an annual meeting in New York of the International Associationof Financial Engineers, where he was honored as 1994 IAFE/Sunguard Financial Engineer of the Year, made some emphatic pointsin his luncheon speech. The gist of it was that while closed-form solutions are elegant, they often do not exist for an option. And,with computers so powerful and available, one should not waste time and effort searching for closed-form solutions when computerapproximations are perfectly adequate.11Those who object to hunting for sport, or to it being done on farms like this, may substitute — without loss of generality — a duderanch cum photo safari operation combined with a ranching operation that raises endangered species for reintroduction in their nativehabitats and for wildlife parks.12The assumption here is that the firm value each period/node has been found by DCF analysis for the given scenario to derive thevalue tree.

Real Options 263

A0 � 300

Au1 � 525

Au2 � 918.75

Aud2 � 315

Ad2 � 108

Ad1 � 180

Figure 23.4 Growth path for new project

5 percent. There is a 50 percent probability that the price will go up by 75 percent each period,and a 50 percent probability that it will go down by 40 percent each period. Kaiser is traded,its asset value is observable. We use Kaiser and the risk-free asset to derive the risk-neutralprobabilities for the contingent claim valuation of a new project with embedded real options.

Suppose there is a similar ranching project available for adoption. Figure 23.4 depictsthe (uncertain) growth path of the asset value of the new project. The new project is exactlythree times the scale of the existing ranch. Using conventional DCF technique, the NPV of theproject is

PV (future asset value, not cash flow13) = (0.5 × 525 + 0.5×180)/(1 + 17.5%) = 300

This is exactly the current value given for the project. If the current owner asks for a price of310, then the NPV of this project is

NPV = 300 − 310 = −10

The CCA approach uses a backward risk-neutral valuation process.14 As derived above, therisk-neutral probability p can be found as15

p = [(1 + r )S0 − Sd ]/(Su − Sd )

We do not here include the complicating factor of dividend yield. However, in the case ofdividend yield (or some other cash flow rate from the underlying asset S, in percent perannum) δ, one may use the preceding equation but include (by summing) the dividend in the

13It is important to recognize that if we were to use cash flows instead of asset values it would be necessary to include each and everycash flow. By using asset values we need only work with the nearest values.14Varian [165] contains a detailed discussion of the theoretical foundation and derivation of CCA in general, and compares risk-neutralprobabilities to decision-tree probabilities.15Note that we use discrete compounding here instead of the er term of continuous compounding introduced earlier.


values of Su and Sd, or alternatively one may use the equation

p = ([(1 + r )/(1 + δ)]S0 − Sd)/(Su − Sd)

Given the risk-neutral probability, the certainty equivalent value is computed by taking inte-gration of the risky asset values over the risk-neutral probabilities, or

AVCE = pAu1 + (1 − p)Ad1

The present value of project cash flows is then found by discounting the certainty equivalentcash flow at the risk-free rate:

A0 = AVCE/(1 + r ) = [pAu1 + (1 − p)Ad1]/(1 + r )

We have:

p = ((1.05) × 100 − 60)/(175 − 60) = 45/115 = 0.3913

(1 − p) = 0.6087

and

A0 = (0.3913 × 525 + 0.6087 × 180)/(1 + 0.05) = 315/1.05 = 300

Again

NPV = 300 − 310 = −10

The CCA result confirms the valuation derived from conventional DCF valuation. In thisbasic scenario, no real option was considered, and the two approaches yield identical results.In situations involving complex options, conventional DCF would generate erroneous value(ignoring the value of the real option), whereas CCA is a straightforward and elegant tool thatcan be used to properly account for all the embedded options in project valuation, as will beillustrated below.

Although our simple two-period tree ends with only two possible outcomes, in practicalapplications it is quite easy to model more subperiods, each of shorter span, and thus gen-erate as many outcomes as required for the analysis. It is important to note that the risk-neutral probability is distinct from the probabilities used in a conventional decision-tree typeof analysis.

For the project, an NPV of −10 would seem to make it unprofitable to undertake. However,considerable operational flexibility is present in this project. This operational flexibility canbe analyzed and valued as real options using the CCA approach. These embedded optionsenhance the value of the project. In using conventional DCF analysis, these valuable optionsare ignored, and consequently misallocation in the form of under-investment results. We define:

Strategic NPV = Passive NPV + Value of operational flexibility (real options)

The correct decision rule for investment then is to accept projects with positive StrategicNPV.

A GAME FARM, RECREATIONAL PROJECT

We now examine a ranch project in more detail. Using the payoff trees given above, assumethat we are hired as consultants to work out for Eric Brand, who is considering conversion of

Real Options 265

his large cattle and alfalfa ranch to a ranch that raises deer and bison for sale of the meat tohealth-conscious consumers. The ranch is surrounded by large tracts of national forest land,bureau of land management lands, and Indian reservation lands. The first payoff tree above(S’s) illustrates the future path of the value of a similar, existing ranch. The Brand project(A’s) requires initial investment of 310 in lease of some adjoining land, purchase of breedingstock, equipment, and infrastructure development costs — the same as the asking price for theKaiser Ranch.16 As illustrated above, the conventional (i.e. passive) NPV is −10, and wouldrender the project unacceptable. There are several interesting, different embedded strategicreal options in this project that Mr Brand is considering.

Option to Switch Operation

It is especially true of farms and ranches, but also true for many other businesses, that anoperation can be switched into a different process/technology, different input combination, ordifferent output mix. In times of economic downturn, managers have the option of running agame ranch as a conventional ranching operation, and selling the meat in more than just thespecialty market. Even in times of high sport hunting spending, the state of the economy couldbe such that meat prices are high enough to justify a switch from the sport hunting operation toconventional game animal farming, or some mix of the two. We discuss two possible scenarios.

Case 1: 100 percent Switch

Assume that the ranch has to be run completely (100 percent) either in sport hunting or inconventional game animal farming. Further, assume the following asset valuations (A′) fromthe conventional game farming operation. Figure 23.5 Illustrates the evolution of asset values.

Under the switched operation, cash flows are lower and less volatile than the sport huntingincomes. The income fluctuation is driven by uncertainty in game animal meat prices and gameanimal farming costs. Current value of the switched operation (A′

o) is lower than that of thesport hunting project (A0). Otherwise management will immediately switch.

The strategic flexibility embedded in this case is that the management has the option ofswitching. The switching decision will be based on year 1 realized state of economy. Themanagement compares the asset values of current use (A) to that of the alternative operation(A′). Year 1 asset value will be the maximum of the two, max (A, A′).

Vu1 = max (Au1, A′u1) = max (525, 412.5) = 525

when sport hunting is chosen, and

Vd1 = max (Ad1, A′d1) = max (180, 233.75) = 233.75,

when operation is switched. Applying CCA, the strategic NPV (value of the project plus theoption to switch operation) is

Strategic NPV = [pVu1 + (1 − p)Vd1]/(1 + r ) − I0

= (0.3913 × 525 + 0.6087 × 233.75)/(1.05) − 310

= (347.72/1.05) − 310 = 21.16

16Results may be scaled to make the analysis easier, and adjusted accordingly afterward.


A'0 � 275

A'u1 � 412.5

A'd1 � 233.75

A'u2 � 618.75

A'd2 � 198.688

A'ud2 �350.625

Figure 23.5 Evolution of asset value, switch operation

The value of the strategic option to switch can be calculated as

Value of option to switch = Strategic NPV − Passive NPV

= 21.16 − (−10) = 31.16

The value of the option to switch accounts for a full 10 percent of the project’s gross value(310). The presence of this strategic option makes the project viable (strategic NPV =21.16 > 0), given a negative conventional NPV.

It is important to note that, although incomes from conventional game animal farming aresubstantially lower than the sport hunting operation in the good states, the lower volatility, andthus higher incomes in the bad states, creates a valuable strategic option for management.

Case 2: Mix Operation

In most manufacturing operations, assets are flexible only on an exclusive basis, capable ofa complete switch in term of inputs or outputs, a situation similar to case 1 analyzed above.In a sport hunting operation, however, a mixed operation is not only possible, but very likelyoptimal in many situations. We next consider a scenario of mixed operation.

Assume that management can reconfigure the operation by separating one part or the ranchfacility, say one or two sections, for conventional game farming and still maintain a certainlevel of sport hunting activity. Specifically, suppose the optimal mix is to run 40 percent ofplanned sport hunting activity and at the same time raise game animals to generate 75 percentof the cash flow from complete (100 percent) conventional game farming. The manager will

Real Options 267

compare a 100 percent sport hunting outcome (A) to the outcomes from the optimal mix(0.4 × A + 0.75 × A′) and determine the strategic choice in year 1. Year 1 project value willbe: max (A, 0.4 × A + 0.75 × A′), or

Vu1 = max (Au1, 0.4 × Au1 + 0.75 × Au′1) = max (525, 0.4 × 525 + 0.75 × 412.5)

= max (525, 519.39) = 525

when 100 percent sport hunting is maintained, and

Vd1 = max (Ad1, 0.4 × Ad1 + 0.75 × A′d1) = max (180, 0.4 × 180 + 0.75 × 233.75)

= max (180, 247.31) = 247.31

when mix operation is chosen. The strategic NPV (value of the project plus the option to mixoperation) is


= (0.3913 × 525 + 0.6087 × 247.31)/(1.05) − 310

= (355.97/1.05) − 310 = 29.02

The value of the strategic option to switch and mix operation can be calculated as

Value of option to switch and mix operation = Strategic NPV − Passive NPV

= 29.02 − (−10) = 39.02

The value of the option to switch to a mixed operation accounts for about 12.6 percent ofthe gross value of the project. To the degree that mixed operation may involve higher cost inreconfiguring the operations, the higher value of the option to mix would justify the strategicchoice if the increase in cost is not more than 7.86 (= 39.02 − 31.16) over that of case 1.

We note that, for the sake of clarity and tractability, we have assumed that the optimal mix ofthe two operations is given. In practice, the optimal mix would be part of the optimal exerciseproblem and is not independent of, nor can be predetermined before, the optimal exerciseproblem. The consideration of various degrees of mixed operation makes the analysis muchmore complex. The complexity, though, comes with a higher value for the strategic flexibility.

Option to Abandon for Salvage Value

Most assets can be sold for salvage value when continued operation is not economical. Assumethat after one year the assets and land can be sold for 70 percent of the total initial costs at 217(= 310 × 70 percent). At year 1, the Brand management would compare the salvage value tothe value from current operations and decide whether to continue the operation or to abandon.Year 1 project value will be: max (A, 217), or

Vu1 = max (Au1, 217) = max (525, 217) = 525

when sport hunting is continued, and

Vd1 = max (Ad1, 217) = max (180, 217) = 217


when the operation is abandoned and assets and land are sold for their salvage value or convertedto “ranchettes”. The strategic NPV (value of the project plus the option to abandon) is

Strategic NPV = [pVu1 + (1 − p)V d1]/(1 + r ) − I0

= (0.3913 × 525 + 0.6087 × 217)/(1.05) − 310

= (337.52/1.05) − 310 = 11.45

The value of the strategic option to abandon can be calculated as

Value of option to abandon = Strategic NPV – Passive NPV

= (11.45) − (−10) = 21.45

The option to abandon a project for its salvage value is a valuable option when a businessprospect turns out unsatisfactory. By not accounting for this option, conventional NPV anal-ysis forces the assumption that once a project is accepted, it will be carried out to the endregardless of the realized state of economy. This is obviously an erroneous assumption, withserious implications. In our example, the abandonment option accounts for over 7 percent ofthe gross value of the project. The abandonment option also makes the project viable (strategicNPV = 21.45 > 0).

The Option to Expand Operation (Growth Option)

Practically all businesses have the flexibility to expand the scale of operation by incurringadditional investment. Growth options are particularly valuable when state of economy turnsout to be good. Assume that management can make a follow-up investment of IG = 300 anddouble the ranching operations. At year 1, management decides whether to expand the scale ofoperation by comparing the asset values from the base-case operation to those of the expandedoperation. Year 1 asset values will be: V1 = max (A1, 2 × A1 − I G

1 ), or,

Vu1 = max (525, 2 × 525 − 300) = max (525, 750) = 750

when the growth option is exercised, and,

Vd1 = max (180, 2 × 180 − 300) = max (180, 60) = 180

when base operation is maintained the strategic NPV (value of the project plus the option togrow) is:

Strategic NPV = [p Vu1 + (1 − p)Vd1]/(1 + r ) − I0

= (0.3913 × 750 + 0.6087 × 180)/(1.05) − 310

= (403.04/1.05) − 310 = 83.85

The value of the strategic option to grow can be calculated as

Value of option to grow = Strategic NPV − Passive NPV

= 73.85 − (−10) = 44.51

The strategic option to grow when business prospects are good is a highly valuable realoption that is embedded in many businesses. In many businesses, growth options account for alarge fraction of firm value. By failing to account for this type of option, conventional NPV can

Real Options 269

lead to serious underinvestment and missed opportunities. In the above example this growthoption accounts for nearly 27 percent of the gross value of the project (= 83.85/310).

Interaction among Strategic Options

It is easy to see that all the real options analyzed above interact with each other. For example,the option to grow interacts with options to abandon and to switch. For the sake of clarity,we have focused on the value of individual real options and on how they enhance the project.When multiple real options are present, which is in most real-world cases, one applies thesame analytical process for determining project cash flows from future operations. Currentvalue (the strategic NPV) of the project can then be derived by using CCA. In this section, weanalyze and illustrate the interaction between the real option to abandon and the real option togrow.

Using data given above, when a manager has both the option to abandon and the option toexpand, year 1 project value is determined by comparing incomes from base operation (A1),expanded operation (2 × A1 − I G

1 ), and salvage value (VS = 217 = 70 percent × 310), or

Vu1 = max (Au1, 2 × Au1 − 300, 217) = max (525, 2 × 525 − 300, 217)

= max (525, 750, 217) = 750

when expansion is chosen, and

AVd1 = max (Ad1, 2 × Ad1 − 300, 217) = max (180, 2 × 180 − 300, 217)

= max (180, 60, 217) = 217

when operation is abandoned. The strategic NPV (value of the project plus the option to switchoperation) is


= (0.3913 × 750 + 0.6087 × 217)/(1.05) − 310

= (425.56/1.05) − 310 = 95.3

The value of the portfolio of the two strategic options can be calculated as

Value of the options = Strategic NPV − Passive NPV

= 95.3 − (−10) = 105.3

Note that the growth option is exercised in the up-state, and the abandonment option is exercisedin the down-state. The recognition of the strategic flexibility has transformed the project to thedegree that it is completely different from the original base-case operation.

The combined value of these two options accounts for nearly 34 percent of the gross valueof the project (= 105.3/310)). Although in this illustration the strategic value of the jointpresence of the two options just equals the sum of the stand-alone value of the two options(105.3 = 21.45 + 83.85), it has been shown [160] that in most cases the value of a portfolioof options can be substantially higher than the sum of the values of the individual stand-aloneoptions.17 The implication is that the presence of multiple real options in a sport hunting

17However, it has been noted that, although in general a portfolio of options has greater value than an option on a portfolio of assets,in the presence of some other real options, the incremental value of an additional option is lower than its isolated value. In essence,one has diminishing marginal value increase as one adds more options.


operation, as analyzed above, makes the miscalculation of conventional NPV, in the form ofundervaluation, even more pronounced.

Our analyses and discussions illustrate the importance of real option in the valuation ofbusiness investment projects. We also provide illustration for the optimal exercise of thesestrategic options. Although the analyses in this example are framed in terms of a game farmand sport hunting project, the analytical tools, strategic conclusions and insights are completelygeneral and should be applied to all business strategic management.

Example 23.2 Paper Moon is a “dot.com” start-up company that has a novel Internet appli-cation. The company has a patent pending on its technological development. The company’spatent attorney estimates it will take a year for the US patent office to issue the patent. There isa probability of 0.5 that the patent will be issued, and 0.5 that the patent office will refuse thepatent because the patent examiner believes it is obvious, or a search reveals the technologyhas already been patented by others. The management of Paper Moon wants to launch theirInternet operation immediately when the patent is issued. To be ready to do that, they mustinvest $11.0 million now for programming, computer and network hardware, and marketingmaterials. If the patent is not issued, or is delayed, they will launch the application anywayafter a year. But the consequences of not having a patent will be that competitors can duplicatetheir application, and Paper Moon’s earnings will be much lower than otherwise.

The payoff diagram (conventional decisions tree, predicated on the base-scale operation)for Paper Moon is shown in Figure 23.6. The risk-free interest rate is rf = 6 percent, and thecost of capital considered appropriate to projects of this risk of k = 20 percent. A venturecapital firm has been approached to fund the $11.0 million of development. The company hastwo strategic options available, abandonment at end of year 1, and growth at end of year 2. Thecompany can abandon the project and sell the firm for $1.5 million salvage value at the endof year 1 in the down-state. At the end of year 2, the company also has the strategic flexibilityto invest an additional $5 million to increase their operation by 60 percent.

DeLong and Tinic, Paper Moon’s consulting firm, has completed the first-stage research forthe contingent claim valuation, and came up with risk-neutral probabilities of

p = 0.625 and (1 − p) = 0.375

They also generated the value tree of the firm with risk-neutral probabilities, as shown inFigure 23.3.

As senior financial staff, you are given the assignment to come up with an in-firm valuationto check against DeLong and Tinic’s number. What is the value of the project? How should thefirm proceed over time? Will the firm be able to acquire the necessary funding from the venturecapitalist? Should they invest or pass on this opportunity? What are the critical uncertaintiesin this situation?

Contingent Claim Valuation

Step 1: Risk-neutral probabilities has been estimated to be

p = 0.625 and (1 − p) = 0.375

Step 2: Valuation by applying the risk-neutral probabilities to the state-dependent projectvalues including optimal exercise of any real options.

Real Options 271

Cost � $11.0MM

Patent Granted;No Revenue Yet

Patent Refused;Competition Now

Value � $-5MM

Value � $-5MM

Value � $-5MM

Value � $-10MM

Value � $5MM

Value = $5MM

Value � $19MM

Value � $24MM

Value � $18MM

Value � $18MM

Value � $4MM

Value � $4MM

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

Figure 23.6 Sample problem: Paper Moon

The Optimal Exercise of the Growth Option and Firm Value

At the end of year 2, the value of the firm would be (1 + 60 percent)∗ S3 − I G if the firm exer-cises the growth option and expands operation. Firm value would be S3 if the firm maintainsthe base-scale of operation. Firm value at t = 3 will be V3 = max (S3, 1.6∗S3 − I G). Firmwill exercise if the former is larger:

Vuuu3 = max (Suuu3, 1.6∗Suuu3 − I G) max (24, 1.6∗24 − 5)

= max (24, 33.4) = 33.4


The growth option is exercised.

Vuud3 = max (Suud3, 1.6∗Suud3 − I G) = max (18, 1.6∗18 − 5)

= max (18, 23.8) = 23.8

The growth Option is exercised.Applying the risk-neutral probabilities, we have

Vuu2 = (0.625∗33.4 + 0.375∗23.8)/(1 + 6%) = 29.8/1.06 = 28.11

Similarly, at Node (t = 2, ud), we have

Vudu3 = max (Sudu3, 1.6∗Sudu3 − I G) = max (18, 1.6∗18 − 5)

= max (18, 23.8) = 23.8

The growth option is exercised.

Vudu3 = max (Sudd3, 1.6∗Sudd3 − I G) = max (4, 1.6∗4 − 5)

= max (4, 1.4) = 4

The growth option is not exercised.Applying the risk-neutral probabilities, we have

Vud2 = (0.625∗23.8 + 0.375∗4)/(1 + 6%) = 16.38/1.06 = 15.45

Applying the risk-neutral probabilities obe more round, we have

Vu1 = (0.625∗28.11 + 0.375∗15.45)/(1 + 6%) = 23.36/1.06 = 22.04

We note that the growth option will never be exercised in any of the down-states, as the valueof the firm in those stated are all worse that the “udd3” state when the growth option was shownabove not to be exercised.

The Abandon Option and Firm Value

The firm will exercise the abandon option — sell the firm for salvage value — when the realizedstate is such that salvage value is higher than the value of the firm as a going concern. This isan option that is available for all businesses, but often ignored in conventional NPV analysis.

Suppose at t = 1, Paper Moon can be sold for $1.5 million salvage value in the down-state.(It can be sold at around $22.4 million as a going concern in the up-state. The abandon optionwill not be exercised in the up-state.)

The manager of Paper Moon will sell the firm for salvage value if firm value as a goingconcern under base-scale of operation is lower.

To find Vu1, we again apply the risk-neutral provabilities to firm value in t = 2. We haveVuu2 = $5 million and Vud2 = −$5 million (Base-scale will be maintained as the growthoption will not be ecercised in the down-states.)

Applying the risk-neutral probabilities, we have

Vd1 = (0.625∗5 + 0.375∗(−5))/(1 + 6%) = 1.25/1.06 = 1.18

Real Options 273

Current Value of the Firm

As discussed above, the firm will exercise the growth option in the up-state, and will exercisethe abandon option in the down-state. Current value of the firm must incorporate these optimalexercises of strategic operational flexibility. In other words, the base-scale value-tree is not areliable basis for valuation when strategic flexibility has not been considered.

Having determined the optimal excise of these real option, we now apply the risk-neutralprobabilities one last time to derive current firm value:

V0 = (0.625∗Vu1 + 0.375∗Vd1)/(1 + 6%)

= (0.625∗22.04 + 0.375∗1.18)/(1 + 6%) = 14.22/1.06 = 13.41

Strategic NPV = 13.41 − 11 = 2.41

The Strategic Dimension of the Real-option Analysis

Our analyses above not only generate the numerical analyses necessary in the process of ac-quiring funding from the venture capitalist, but also provide the required managerial/strategicplanning for the firm to be successfully run under different realized states. As discussed in ear-lier sections, such managerial/strategic insight may be as important as the process of numericalsolution.

The critical uncertainty in this project is whether the firm will acquire the patent at t = 1.This uncertainty in turn drives the optimal exercise of the two real options. We have shown abovethat the growth option will be optimally exercised in three out of four states in the up-branch,but the abandonment option will never be excercised in the up-branch. In the down-branch,the growth option will never be exercised, but the abandonment option will immediately beexercised as soon as the patent is rejected.

With the recognition above, the project is radically transformed and substantially differentfrom the base-scale operation. Accordingly, the process of valuation is also fundamentallydifferent.

The recognition and optimal exercise of such real options create additional valur only if themanagerial/strategic dimensions are properly appreciated.

Given a current value of $13.41 million, and a fully specified optimal exercise of all theavailable strategic options, the firm should have no difficulty convincing the venture capitalistto fund the project. This would not have been possible if only the base-scale operation waspresented.

CONCLUSION

Real options provide a way of looking at capital investments and other decision situations thatoffers insights that traditional DCF methods cannot. Whether or not adopting a real optionsapproach will provide a clear numerical signal to adopt or shun a particular project is not asimportant as the fact that engaging in the exercise of framing the problem in real option terms,and focusing management attention on the framed problem, will facilitate better decision-making.

The numerical solution method used to solve a particular real option problem is not nearlyas important as careful framing of the problem and attention to including only the truly relevantsources of uncertainty and key decision variables. The Black–Scholes model works very well


for those problems for which it is appropriate: problems that have embedded American-styleoptions rather than the European-style option the model was designed to solve. The binomialmodel is more flexible, and can handle the latter type of option as well as option situations thathave more complex characteristics. In truly complex option evaluation computer simulation(Monte Carlo method) may be required. It is beyond the scope of this chapter and book toventure into such details, which can be found in specialized works on option pricing. The focusof this chapter is on what managers need to know to integrate real options thinking into theiroperations, not to take up technical details.

Appendix: Financial Mathematics Tablesand Formulas

Tabl

eA

.1Si

ngle

paym

entc

ompo

und

amou

nt.T

ofin

dF

for

agi

ven

PF

=P

(1+

i)N

N/i

12

34

56

78

910

11.

0100

001.

0200

001.

0300

001.

0400

001.

0500

001.

0600

001.

0700

001.

0800

001.

0900

001.

1000

002

1.02

0100

1.04

0400

1.06

0900

1.08

1600

1.10

2500

1.12

3600

1.14

4900

1.16

6400

1.18

8100

1.21

0000

31.

0303

011.

0612

081.

0927

271.

1248

641.

1576

251.

1910

161.

2250

431.

2597

121.

2950

291.

3310

004

1.04

0604

1.08

2432

1.12

5509

1.16

9859

1.21

5506

1.26

2477

1.31

0796

1.36

0489

1.41

1582

1.46

4100

51.

0510

101.

1040

811.

1592

741.

2166

531.

2762

821.

3382

261.

4025

521.

4693

281.

5386

241.

6105

10

61.

0615

201.

1261

621.

1940

521.

2653

191.

3400

961.

4185

191.

5007

301.

5868

741.

6771

001.

7715

617

1.07

2135

1.14

8686

1.22

9874

1.31

5932

1.40

7100

1.50

3630

1.60

5781

1.71

3824

1.82

8039

1.94

8717

81.

0828

571.

1716

591.

2667

701.

3685

691.

4774

551.

5938

481.

7181

861.

8509

301.

9925

632.

1435

899

1.09

3685

1.19

5093

1.30

4773

1.42

3312

1.55

1328

1.68

9479

1.83

8459

1.99

9005

2.17

1893

2.35

7948

101.

1046

221.

2189

941.

3439

161.

4802

441.

6288

951.

7908

481.

9671

512.

1589

252.

3673

642.

5937

42

111.

1156

681.

2433

741.

3842

341.

5394

541.

7103

391.

8982

992.

1048

522.

3316

392.

5804

262.

8531

1712

1.12

6825

1.26

8242

1.42

5761

1.60

1032

1.79

5856

2.01

2196

2.25

2192

2.51

8170

2.81

2665

3.13

8428

131.

1380

931.

2936

071.

4685

341.

6650

741.

8856

492.

1329

282.

4098

452.

7196

243.

0658

053.

4522

7114

1.14

9474

1.31

9479

1.51

2590

1.73

1676

1.97

9932

2.26

0904

2.57

8534

2.93

7194

3.34

1727

3.79

7498

151.

1609

691.

3458

681.

5579

671.

8009

442.

0789

282.

3965

582.

7590

323.

1721

693.

6424

824.

1772

48

161.

1725

791.

3727

861.

6047

061.

8729

812.

1828

752.

5403

522.

9521

643.

4259

433.

9703

064.

5949

7317

1.18

4304

1.40

0241

1.65

2848

1.94

7900

2.29

2018

2.69

2773

3.15

8815

3.70

0018

4.32

7633

5.05

4470

181.

1961

471.

4282

461.

7024

332.

0258

172.

4066

192.

8543

393.

3799

323.

9960

194.

7171

205.

5599

1719

1.20

8109

1.45

6811

1.75

3506

2.10

6849

2.52

6950

3.02

5600

3.61

6528

4.31

5701

5.14

1661

6.11

5909

201.

2201

901.

4859

471.

8061

112.

1911

232.

6532

983.

2071

353.

8696

844.

6609

575.

6044

116.

7275

00

211.

2323

921.

5156

661.

8602

952.

2787

682.

7859

633.

3995

644.

1405

625.

0338

346.

1088

087.

4002

5022

1.24

4716

1.54

5980

1.91

6103

2.36

9919

2.92

5261

3.60

3537

4.43

0402

5.43

6540

6.65

8600

8.14

0275

231.

2571

631.

5768

991.

9735

872.

4647

163.

0715

243.

8197

504.

7405

305.

8714

647.

2578

748.

9543

0224

1.26

9735

1.60

8437

2.03

2794

2.56

3304

3.22

5100

4.04

8935

5.07

2367

6.34

1181

7.91

1083

9.84

9733

251.

2824

321.

6406

062.

0937

782.

6658

363.

3863

554.

2918

715.

4274

336.

8484

758.

6230

8110

.834

706

261.

2952

561.

6734

182.

1565

912.

7724

703.

5556

734.

5493

835.

8073

537.

3963

539.

3991

5811

.918

177

271.

3082

091.

7068

862.

2212

892.

8833

693.

7334

564.

8223

466.

2138

687.

9880

6110

.245

082

13.1

0999

428

1.32

1291

1.74

1024

2.28

7928

2.99

8703

3.92

0129

5.11

1687

6.64

8838

8.62

7106

11.1

6714

014

.420

994

291.

3345

041.

7758

452.

3565

663.

1186

514.

1161

365.

4183

887.

1142

579.

3172

7512

.172

182

15.8

6309

330

1.34

7849

1.81

1362

2.42

7262

3.24

3398

4.32

1942

5.74

3491

7.61

2255

10.0

6265

713

.267

678

17.4

4940

2

311.

3613

271.

8475

892.

5000

803.

3731

334.

5380

396.

0881

018.

1451

1310

.867

669

14.4

6177

019

.194

342

321.

3749

411.

8845

412.

5750

833.

5080

594.

7649

416.

4533

878.

7152

7111

.737

083

15.7

6332

921

.113

777

331.

3886

901.

9222

312.

6523

353.

6483

815.

0031

896.

8405

909.

3253

4012

.676

050

17.1

8202

823

.225

154

341.

4025

771.

9606

762.

7319

053.

7943

165.

2533

487.

2510

259.

9781

1413

.690

134

18.7

2841

125

.547

670

351.

4166

031.

9998

902.

8138

623.

9460

895.

5160

157.

6860

8710

.676

581

14.7

8534

420

.413

968

28.1

0243

7

361.

4307

692.

0398

872.

8982

784.

1039

335.

7918

168.

1472

5211

.423

942

15.9

6817

222

.251

225

30.9

1268

137

1.44

5076

2.08

0685

2.98

5227

4.26

8090

6.08

1407

8.63

6087

12.2

2361

817

.245

626

24.2

5383

534

.003

949

381.

4595

272.

1222

993.

0747

834.

4388

136.

3854

779.

1542

5213

.079

271

18.6

2527

626

.436

680

37.4

0434

339

1.47

4123

2.16

4745

3.16

7027

4.61

6366

6.70

4751

9.70

3507

13.9

9482

020

.115

298

28.8

1598

241

.144

778

401.

4888

642.

2080

403.

2620

384.

8010

217.

0399

8910

.285

718

14.9

7445

821

.724

521

31.4

0942

045

.259

256

411.

5037

522.

2522

003.

3598

994.

9930

617.

3919

8810

.902

861

16.0

2267

023

.462

483

34.2

3626

849

.785

181

421.

5187

902.

2972

443.

4606

965.

1927

847.

7615

8811

.557

033

17.1

4425

725

.339

482

37.3

1753

254

.763

699

431.

5339

782.

3431

893.

5645

175.

4004

958.

1496

6712

.250

455

18.3

4435

527

.366

640

40.6

7611

060

.240

069

441.

5493

182.

3900

533.

6714

525.

6165

158.

5571

5012

.985

482

19.6

2846

029

.555

972

44.3

3696

066

.264

076

451.

5648

112.

4378

543.

7815

965.

8411

768.

9850

0813

.764

611

21.0

0245

231

.920

449

48.3

2728

672

.890

484

461.

5804

592.

4866

113.

8950

446.

0748

239.

4342

5814

.590

487

22.4

7262

334

.474

085

52.6

7674

280

.179

532

471.

5962

632.

5363

444.

0118

956.

3178

169.

9059

7115

.465

917

24.0

4570

737

.232

012

57.4

1764

988

.197

485

481.

6122

262.

5870

704.

1322

526.

5705

2810

.401

270

16.3

9387

225

.728

907

40.2

1057

362

.585

237

97.0

1723

449

1.62

8348

2.63

8812

4.25

6219

6.83

3349

10.9

2133

317

.377

504

27.5

2993

043

.427

419

68.2

1790

810

6.71

8957

501.

6446

322.

6915

884.

3839

067.

1066

8311

.467

400

18.4

2015

429

.457

025

46.9

0161

374

.357

520

117.

3908

53

511.

6610

782.

7454

204.

5154

237.

3909

5112

.040

770

19.5

2536

431

.519

017

50.6

5374

281

.049

697

129.

1299

3852

1.67

7689

2.80

0328

4.65

0886

7.68

6589

12.6

4280

820

.696

885

33.7

2534

854

.706

041

88.3

4417

014

2.04

2932

531.

6944

662.

8563

354.

7904

127.

9940

5213

.274

949

21.9

3869

836

.086

122

59.0

8252

496

.295

145

156.

2472

2554

1.71

1410

2.91

3461

4.93

4125

8.31

3814

13.9

3869

623

.255

020

38.6

1215

163

.809

126

104.

9617

0817

1.87

1948

551.

7285

252.

9717

315.

0821

498.

6463

6714

.635

631

24.6

5032

241

.315

001

68.9

1385

611

4.40

8262

189.

0591

42

561.

7458

103.

0311

655.

2346

138.

9922

2215

.367

412

26.1

2934

144

.207

052

74.4

2696

512

4.70

5005

207.

9650

5757

1.76

3268

3.09

1789

5.39

1651

9.35

1910

16.1

3578

327

.697

101

47.3

0154

580

.381

122

135.

9284

5622

8.76

1562

581.

7809

013.

1536

245.

5534

019.

7259

8716

.942

572

29.3

5892

750

.612

653

86.8

1161

214

8.16

2017

251.

6377

1959

1.79

8710

3.21

6697

5.72

0003

10.1

1502

617

.789

701

31.1

2046

354

.155

539

93.7

5654

016

1.49

6598

276.

8014

9060

1.81

6697

3.28

1031

5.89

1603

10.5

1962

718

.679

186

32.9

8769

157

.946

427

101.

2570

6417

6.03

1292

304.

4816

40

611.

8348

643.

3466

516.

0683

5110

.940

413

19.6

1314

534

.966

952

62.0

0267

710

9.35

7629

191.

8741

0833

4.92

9803

621.

8532

123.

4135

846.

2504

0211

.378

029

20.5

9380

237

.064

969

66.3

4286

411

8.10

6239

209.

1427

7836

8.42

2784

631.

8717

443.

4818

566.

4379

1411

.833

150

21.6

2349

339

.288

868

70.9

8686

512

7.55

4738

227.

9656

2840

5.26

5062

641.

8904

623.

5514

936.

6310

5112

.306

476

22.7

0466

741

.646

200

75.9

5594

513

7.75

9117

248.

4825

3544

5.79

1568

651.

9093

663.

6225

236.

8299

8312

.798

735

23.8

3990

144

.144

972

81.2

7286

114

8.77

9847

270.

8459

6349

0.37

0725

661.

9284

603.

6949

747.

0348

8213

.310

685

25.0

3189

646

.793

670

86.9

6196

216

0.68

2234

295.

2220

9953

9.40

7798

671.

9477

453.

7688

737.

2459

2913

.843

112

26.2

8349

049

.601

290

93.0

4929

917

3.53

6813

321.

7920

8859

3.34

8578

681.

9672

223.

8442

517.

4633

0714

.396

836

27.5

9766

552

.577

368

99.5

6275

018

7.41

9758

350.

7533

7665

2.68

3435

691.

9868

943.

9211

367.

6872

0614

.972

710

28.9

7754

855

.732

010

106.

5321

4220

2.41

3339

382.

3211

8071

7.95

1779

702.

0067

633.

9995

587.

9178

2215

.571

618

30.4

2642

659

.075

930

113.

9893

9221

8.60

6406

416.

7300

8678

9.74

6957

712.

0268

314.

0795

498.

1553

5716

.194

483

31.9

4774

762

.620

486

121.

9686

5023

6.09

4918

454.

2357

9486

8.72

1652

722.

0470

994.

1611

408.

4000

1716

.842

262

33.5

4513

466

.377

715

130.

5064

5525

4.98

2512

495.

1170

1595

5.59

3818

732.

0675

704.

2443

638.

6520

1817

.515

953

35.2

2239

170

.360

378

139.

6419

0727

5.38

1113

539.

6775

4710

51.1

5320

074

2.08

8246

4.32

9250

8.91

1578

18.2

1659

136

.983

510

74.5

8200

114

9.41

6840

297.

4116

0258

8.24

8526

1156

.268

519

752.

1091

284.

4158

359.

1789

2618

.945

255

38.8

3268

679

.056

921

159.

8760

1932

1.20

4530

641.

1908

9312

71.8

9537

1

cont

inue

sov

erle

af

Tabl

eA

.1(c

onti

nued

)

N/i

12

34

56

78

910

762.

1302

204.

5041

529.

4542

9319

.703

065

40.7

7432

083

.800

336

171.

0673

4134

6.90

0892

698.

8980

7413

99.0

8490

977

2.15

1522

4.59

4235

9.73

7922

20.4

9118

742

.813

036

88.8

2835

618

3.04

2055

374.

6529

6476

1.79

8900

1538

.993

399

782.

1730

374.

6861

2010

.030

060

21.3

1083

544

.953

688

94.1

5805

819

5.85

4998

404.

6252

0183

0.36

0801

1692

.892

739

792.

1947

684.

7798

4210

.330

962

22.1

6326

847

.201

372

99.8

0754

120

9.56

4848

436.

9952

1790

5.09

3274

1862

.182

013

802.

2167

154.

8754

3910

.640

891

23.0

4979

949

.561

441

105.

7959

9322

4.23

4388

471.

9548

3498

6.55

1668

2048

.400

215

812.

2388

824.

9729

4810

.960

117

23.9

7179

152

.039

513

112.

1437

5323

9.93

0795

509.

7112

2110

75.3

4131

822

53.2

4023

682

2.26

1271

5.07

2407

11.2

8892

124

.930

663

54.6

4148

911

8.87

2378

256.

7259

5055

0.48

8119

1172

.122

037

2478

.564

260

832.

2838

845.

1738

5511

.627

588

25.9

2788

957

.373

563

126.

0047

2127

4.69

6767

594.

5271

6812

77.6

1302

027

26.4

2068

684

2.30

6723

5.27

7332

11.9

7641

626

.965

005

60.2

4224

113

3.56

5004

293.

9255

4164

2.08

9342

1392

.598

192

2999

.062

754

852.

3297

905.

3828

7912

.335

709

28.0

4360

563

.254

353

141.

5789

0431

4.50

0328

693.

4564

8915

17.9

3202

932

98.9

6903

0

862.

3530

885.

4905

362.

7057

8029

.165

349

66.4

1707

115

0.07

3639

336.

5153

5174

8.93

3008

1654

.545

912

3628

.865

933

872.

3766

195.

6003

4713

.086

953

30.3

3196

369

.737

925

159.

0780

5736

0.07

1426

808.

8476

4918

03.4

5504

439

91.7

5252

688

2.40

0385

5.71

2354

13.4

7956

231

.545

242

73.2

2482

116

8.62

2741

385.

2764

2687

3.55

5461

1965

.765

998

4390

.927

778

892.

4243

895.

8266

0113

.883

949

32.8

0705

176

.886

062

178.

7401

0541

2.24

5776

943.

4398

9721

42.6

8493

848

30.0

2055

690

2.44

8633

5.94

3133

14.3

0046

734

.119

333

80.7

3036

518

9.46

4511

441.

1029

8010

18.9

1508

923

35.5

2658

253

13.0

2261

2

912.

4731

196.

0619

9614

.729

481

35.4

8410

784

.766

883

200.

8323

8247

1.98

0188

1100

.428

296

2545

.723

975

5844

.324

873

922.

4978

506.

1832

3615

.171

366

36.9

0347

189

.005

227

212.

8823

2550

5.01

8802

1188

.462

560

2774

.839

132

6428

.757

360

932.

5228

296.

3069

0015

.626

507

38.3

7961

093

.455

489

225.

6552

6454

0.37

0118

1283

.539

565

3024

.574

654

7071

.633

096

942.

5480

576.

4330

3816

.095

302

39.9

1479

498

.128

263

239.

1945

8057

8.19

6026

1386

.222

730

3296

.786

373

7778

.796

406

952.

5735

386.

5616

9916

.578

161

41.5

1138

610

3.03

4676

253.

5462

5561

8.66

9748

1497

.120

549

3593

.497

147

8556

.676

047

962.

5992

736.

6929

3317

.075

506

43.1

7184

110

8.18

6410

268.

7590

3066

1.97

6630

1616

.890

192

3916

.911

890

9412

.343

651

972.

6252

666.

8267

9217

.587

771

44.8

9871

511

3.59

5731

284.

8845

7270

8.31

4994

1746

.241

408

4269

.433

960

1035

3.57

8016

982.

6515

186.

9633

2818

.115

404

46.6

9466

411

9.27

5517

301.

9776

4675

7.89

7044

1885

.940

720

4653

.683

016

1138

8.93

5818

992.

6780

337.

1025

9418

.658

866

48.5

6245

012

5.23

9293

320.

0963

0581

0.94

9837

2036

.815

978

5072

.514

488

1252

7.82

9400

100

2.70

4814

7.24

4646

19.2

1863

250

.504

948

131.

5012

5833

9.30

2084

867.

7163

2621

99.7

6125

655

29.0

4079

213

780.

6123

40

N/i

1112

1314

1516

1718

1920

11.

1100

001.

1200

001.

1300

001.

1400

001.

1500

001.

1600

001.

1700

001.

1800

001.

1900

001.

2000

002

1.23

2100

1.25

4400

1.27

6900

1.29

9600

1.32

2500

1.34

5600

1.36

8900

1.39

2400

1.41

6100

1.44

0000

31.

3676

311.

4049

281.

4428

971.

4815

441.

5208

751.

5608

961.

6016

131.

6430

321.

6851

591.

7280

004

1.51

8070

1.57

3519

1.63

0474

1.68

8960

1.74

9006

1.81

0639

1.87

3887

1.93

8778

2.00

5339

2.07

3600

51.

6850

581.

7623

421.

8424

351.

9254

152.

0113

572.

1003

422.

1924

482.

2877

582.

3863

542.

4883

20

61.

8704

151.

9738

232.

0819

522.

1949

732.

3130

612.

4363

962.

5651

642.

6995

542.

8397

612.

9859

847

2.07

6160

2.21

0681

2.35

2605

2.50

2269

2.66

0020

2.82

6220

3.00

1242

3.18

5474

3.37

9315

3.58

3181

82.

3045

382.

4759

632.

6584

442.

8525

863.

0590

233.

2784

153.

5114

533.

7588

594.

0213

854.

2998

179

2.55

8037

2.77

3079

3.00

4042

3.25

1949

3.51

7876

3.80

2961

4.10

8400

4.43

5454

4.78

5449

5.15

9780

102.

8394

213.

1058

483.

3945

673.

7072

214.

0455

584.

4114

354.

8068

285.

2338

365.

6946

846.

1917

36

113.

1517

573.

4785

503.

8358

614.

2262

324.

6523

915.

1172

655.

6239

896.

1759

266.

7766

747.

4300

8412

3.49

8451

3.89

5976

4.33

4523

4.81

7905

5.35

0250

5.93

6027

6.58

0067

7.28

7593

8.06

4242

8.91

6100

133.

8832

804.

3634

934.

8980

115.

4924

116.

1527

886.

8857

917.

6986

798.

5993

599.

5964

4810

.699

321

144.

3104

414.

8871

125.

5347

536.

2613

497.

0757

067.

9875

189.

0074

5410

.147

244

11.4

1977

312

.839

185

154.

7845

895.

4735

666.

2542

707.

1379

388.

1370

629.

2655

2110

.538

721

11.9

7374

813

.589

530

15.4

0702

2

165.

3108

946.

1303

947.

0673

268.

1372

499.

3576

2110

.748

004

12.3

3030

414

.129

023

16.1

7154

018

.488

426

175.

8950

936.

8660

417.

9860

789.

2764

6410

.761

264

12.4

6768

514

.426

456

16.6

7224

719

.244

133

22.1

8611

118

6.54

3553

7.68

9966

9.02

4268

10.5

7516

912

.375

454

14.4

6251

416

.878

953

19.6

7325

122

.900

518

26.6

2333

319

7.26

3344

8.61

2762

10.1

9742

312

.055

693

14.2

3177

216

.776

517

19.7

4837

523

.214

436

27.2

5161

631

.948

000

208.

0623

129.

6462

9311

.523

088

13.7

4349

016

.366

537

19.4

6075

923

.105

599

27.3

9303

532

.429

423

38.3

3760

021

8.94

9166

10.8

0384

813

.021

089

15.6

6757

818

.821

518

22.5

7448

127

.033

551

32.3

2378

138

.591

014

46.0

0512

022

9.93

3574

12.1

0031

014

.713

831

17.8

6103

921

.644

746

26.1

8639

831

.629

255

38.1

4206

145

.923

307

55.2

0614

423

11.0

2626

713

.552

347

16.6

2662

920

.361

585

24.8

9145

830

.376

222

37.0

0622

845

.007

632

54.6

4873

566

.247

373

2412

.239

157

15.1

7862

918

.788

091

23.2

1220

728

.625

176

35.2

3641

743

.297

287

53.1

0900

665

.031

994

79.4

9684

725

13.5

8546

417

.000

064

21.2

3054

226

.461

916

32.9

1895

340

.874

244

50.6

5782

662

.668

627

77.3

8807

395

.396

217

2615

.079

865

19.0

4007

223

.990

513

30.1

6658

437

.856

796

47.4

1412

359

.269

656

73.9

4898

092

.091

807

114.

4754

6027

16.7

3865

021

.324

881

27.1

0927

934

.389

906

43.5

3531

555

.000

382

69.3

4549

787

.259

797

109.

5892

5113

7.37

0552

2818

.579

901

23.8

8386

630

.633

486

39.2

0449

350

.065

612

63.8

0044

481

.134

232

102.

9665

6013

0.41

1208

164.

8446

6229

20.6

2369

126

.749

930

34.6

1583

944

.693

122

57.5

7545

474

.008

515

94.9

2705

112

1.50

0541

155.

1893

3819

7.81

3595

3022

.892

297

29.9

5992

239

.115

898

50.9

5015

966

.211

772

85.8

4987

711

1.06

4650

143.

3706

3818

4.67

5312

237.

3763

14

3125

.410

449

33.5

5511

344

.200

965

58.0

8318

176

.143

538

99.5

8585

712

9.94

5641

169.

1773

5321

9.76

3621

284.

8515

7732

28.2

0559

937

.581

726

49.9

4709

066

.214

826

87.5

6506

811

5.51

9594

152.

0363

9919

9.62

9277

261.

5187

1034

1.82

1892

3331

.308

214

42.0

9153

356

.440

212

75.4

8490

210

0.69

9829

134.

0027

2917

7.88

2587

235.

5625

4731

1.20

7264

410.

1862

7034

34.7

5211

847

.142

517

63.7

7743

986

.052

788

115.

8048

0315

5.44

3166

208.

1226

2727

7.96

3805

370.

3366

4549

2.22

3524

3538

.574

851

52.7

9962

072

.068

506

98.1

0017

813

3.17

5523

180.

3140

7324

3.50

3474

327.

9972

9044

0.70

0607

590.

6682

29

3642

.818

085

59.1

3557

481

.437

412

111.

8342

0315

3.15

1852

209.

1643

2428

4.89

9064

387.

0368

0252

4.43

3722

708.

8018

7537

47.5

2807

466

.231

843

92.0

2427

612

7.49

0992

176.

1246

3024

2.63

0616

333.

3319

0545

6.70

3427

624.

0761

3085

0.56

2250

3852

.756

162

74.1

7966

410

3.98

7432

145.

3397

3120

2.54

3324

281.

4515

1538

9.99

8329

538.

9100

4474

2.65

0594

1020

.674

739

58.5

5934

083

.081

224

117.

5057

9816

5.68

7293

232.

9248

2332

6.48

3757

456.

2980

4563

5.91

3852

883.

7542

0712

24.8

096

4065

.000

867

93.0

5097

013

2.78

1552

188.

8835

1426

7.86

3546

378.

7211

5853

3.86

8713

750.

3783

4510

51.6

675

1469

.771

6

4172

.150

963

104.

2170

8715

0.04

3153

215.

3272

0630

8.04

3078

439.

3165

4462

4.62

6394

885.

4464

4712

51.4

843

1763

.725

942

80.0

8756

911

6.72

3137

169.

5487

6324

5.47

3015

354.

2495

4050

9.60

7191

730.

8128

8110

44.8

268

1489

.266

421

16.4

711

4388

.897

201

130.

7299

1419

1.59

0103

279.

8392

3740

7.38

6971

591.

1443

4185

5.05

1071

1232

.895

617

72.2

270

2539

.765

344

98.6

7589

314

6.41

7503

216.

4968

1631

9.01

6730

468.

4950

1768

5.72

7436

1000

.409

814

54.8

168

2108

.950

130

47.7

183

4510

9.53

0242

163.

9876

0424

4.64

1402

363.

6790

7253

8.76

9269

795.

4438

2611

70.4

794

1716

.683

925

09.6

506

3657

.262

0

4612

1.57

8568

183.

6661

1627

6.44

4784

414.

5941

4261

9.58

4659

922.

7148

3813

69.4

609

2025

.687

029

86.4

842

4388

.714

447

134.

9522

1120

5.70

6050

312.

3826

0647

2.63

7322

712.

5223

5810

70.3

492

1602

.269

323

90.3

106

3553

.916

252

66.4

573

4814

9.79

6954

230.

3907

7635

2.99

2345

538.

8065

4781

9.40

0712

1241

.605

118

74.6

550

2820

.566

542

29.1

603

6319

.748

749

166.

2746

1925

8.03

7669

398.

8813

5061

4.23

9464

942.

3108

1914

40.2

619

2193

.346

433

28.2

685

5032

.700

875

83.6

985

5018

4.56

4827

289.

0021

9045

0.73

5925

700.

2329

8810

83.6

574

1670

.703

825

66.2

153

3927

.356

959

88.9

139

9100

.438

2

cont

inue

sov

erle

af

Tabl

eA

.1(c

onti

nued

)

N/i

1112

1314

1516

1718

1920

5120

4.86

6958

323.

6824

5350

9.33

1595

798.

2656

0712

46.2

061

1938

.016

430

02.4

719

4634

.281

171

26.8

075

1092

0.52

5852

227.

4023

2336

2.52

4347

575.

5447

0391

0.02

2792

1433

.137

022

48.0

990

3512

.892

154

68.4

517

8480

.901

013

104.

6309

5325

2.41

6579

406.

0272

6965

0.36

5514

1037

.426

016

48.1

075

2607

.794

941

10.0

838

6452

.773

010

092.

2722

1572

5.55

7154

280.

1824

0245

4.75

0541

734.

9130

3111

82.6

656

1895

.323

630

25.0

421

4808

.798

076

14.2

721

1200

9.80

3918

870.

6685

5531

1.00

2466

509.

3206

0683

0.45

1725

1348

.238

821

79.6

222

3509

.048

856

26.2

937

8984

.841

114

291.

6666

2264

4.80

23

5634

5.21

2738

570.

4390

7893

8.41

0449

1536

.992

225

06.5

655

4070

.496

665

82.7

636

1060

2.11

2517

007.

0833

2717

3.76

2757

383.

1861

3963

8.89

1768

1060

.403

817

52.1

712

2882

.550

347

21.7

761

7701

.833

412

510.

4928

2023

8.42

9132

608.

5153

5842

5.33

6614

715.

5587

8011

98.2

563

1997

.475

133

14.9

329

5477

.260

290

11.1

451

1476

2.38

1524

083.

7306

3913

0.21

8359

472.

1236

4280

1.42

5833

1354

.029

622

77.1

216

3812

.172

863

53.6

219

1054

3.03

9717

419.

6101

2865

9.63

9446

956.

2620

6052

4.05

7242

897.

5969

3315

30.0

535

2595

.918

743

83.9

987

7370

.201

412

335.

3565

2055

5.14

0034

104.

9709

5634

7.51

44

6158

1.70

3539

1005

.308

617

28.9

604

2959

.347

350

41.5

986

8549

.433

614

432.

3671

2425

5.06

5240

584.

9154

6761

7.01

7262

645.

6909

2811

25.9

456

1953

.725

333

73.6

559

5797

.838

399

17.3

430

1688

5.86

9528

620.

9769

4829

6.04

9381

140.

4207

6371

6.71

6930

1261

.059

122

07.7

096

3845

.967

766

67.5

141

1150

4.11

7819

756.

4673

3377

2.75

2757

472.

2987

9736

8.50

4864

795.

5557

9314

12.3

862

2494

.711

843

84.4

032

7667

.641

213

344.

7767

2311

5.06

6739

851.

8482

6839

2.03

5411

6842

.205

865

883.

0669

3015

81.8

725

2819

.024

349

98.2

196

8817

.787

415

479.

9410

2704

4.62

8147

025.

1809

8138

6.52

2214

0210

.646

9

6698

0.20

4292

1771

.697

231

85.4

975

5697

.970

410

140.

4555

1795

6.73

1531

642.

2149

5548

9.71

3596

849.

9614

1682

52.7

763

6710

88.0

268

1984

.300

935

99.6

122

6495

.686

211

661.

5238

2082

9.80

8537

021.

3914

6547

7.86

1911

5251

.454

120

1903

.331

668

1207

.709

722

22.4

170

4067

.561

874

05.0

823

1341

0.75

2424

162.

5779

4331

5.02

7977

263.

8770

1371

49.2

303

2422

83.9

979

6913

40.5

578

2489

.107

045

96.3

448

8441

.793

815

422.

3653

2802

8.59

0450

678.

5827

9117

1.37

4916

3207

.584

129

0740

.797

470

1488

.019

127

87.7

998

5193

.869

696

23.6

450

1773

5.72

0032

513.

1648

5929

3.94

1710

7582

.222

419

4217

.025

134

8888

.956

9

7116

51.7

012

3122

.335

858

69.0

727

1097

0.95

5320

396.

0780

3771

5.27

1269

373.

9118

1269

47.0

224

2311

18.2

598

4186

66.7

483

7218

33.3

884

3497

.016

166

32.0

521

1250

6.88

9023

455.

4898

4374

9.71

4681

167.

4768

1497

97.4

864

2750

30.7

292

5024

00.0

980

7320

35.0

611

3916

.658

074

94.2

189

1425

7.85

3526

973.

8132

5074

9.66

8994

965.

9479

1767

61.0

340

3272

86.5

677

6028

80.1

176

7422

58.9

178

4386

.657

084

68.4

674

1625

3.95

3031

019.

8852

5886

9.61

6011

1110

.159

020

8578

.020

138

9471

.015

672

3456

.141

175

2507

.398

849

13.0

558

9569

.368

118

529.

5064

3567

2.86

8068

288.

7545

1299

98.8

861

2461

22.0

637

4634

70.5

086

8681

47.3

693

7627

83.2

126

5502

.622

510

813.

3860

2112

3.63

7341

023.

7982

7921

4.95

5315

2098

.696

729

0424

.035

255

1529

.905

210

4177

6.84

3277

3089

.366

061

62.9

372

1221

9.12

6124

080.

9465

4717

7.36

7991

889.

3481

1779

55.4

751

3427

00.3

615

6563

20.5

872

1250

132.

2118

7834

29.1

963

6902

.489

713

807.

6125

2745

2.27

9054

253.

9731

1065

91.6

438

2082

07.9

059

4043

86.4

266

7810

21.4

987

1500

158.

6542

7938

06.4

079

7730

.788

515

602.

6022

3129

5.59

8162

392.

0690

1236

46.3

068

2436

03.2

499

4771

75.9

834

9294

15.5

835

1800

190.

3850

8042

25.1

128

8658

.483

117

630.

9405

3567

6.98

1871

750.

8794

1434

29.7

159

2850

15.8

024

5630

67.6

604

1106

004.

5444

2160

228.

4620

8146

89.8

752

9697

.501

119

922.

9627

4067

1.75

9382

513.

5113

1663

78.4

704

3334

68.4

888

6644

19.8

393

1316

145.

4078

2592

274.

1544

8252

05.7

614

1086

1.20

1222

512.

9479

4636

5.80

5694

890.

5380

1929

99.0

257

3901

58.1

319

7840

15.4

103

1566

213.

0353

3110

728.

9853

8357

78.3

952

1216

4.54

5325

439.

6311

5285

7.01

8310

9124

.118

722

3878

.869

845

6485

.014

392

5138

.184

218

6379

3.51

2037

3287

4.78

2484

6414

.018

613

624.

2908

2874

6.78

3160

257.

0009

1254

92.7

365

2596

99.4

890

5340

87.4

668

1091

663.

0573

2217

914.

2792

4479

449.

7388

8571

19.5

607

1525

9.20

5732

483.

8649

6869

2.98

1014

4316

.647

030

1251

.407

262

4882

.336

112

8816

2.40

7726

3931

7.99

2353

7533

9.68

66

8679

02.7

124

1709

0.31

0436

706.

7674

7830

9.99

8416

5964

.144

034

9451

.632

473

1112

.333

315

2003

1.64

1031

4078

8.41

0864

5040

7.62

3987

8772

.010

719

141.

1476

4147

8.64

7189

273.

3981

1908

58.7

656

4053

63.8

936

8554

01.4

299

1793

637.

3364

3737

538.

2089

7740

489.

1487

8897

36.9

319

2143

8.08

5346

870.

8713

1017

71.6

739

2194

87.5

805

4702

22.1

165

1000

819.

6730

2116

492.

0570

4447

670.

4686

9288

586.

9784

8910

807.

9944

2401

0.65

5652

964.

0845

1160

19.7

082

2524

10.7

176

5454

57.6

552

1170

959.

0175

2497

460.

6272

5292

727.

8576

1114

6304

.374

190

1199

6.87

3826

891.

9342

5984

9.41

5513

2262

.467

429

0272

.325

263

2730

.880

013

7002

2.05

0429

4700

3.54

0162

9834

6.15

0513

3755

65.2

489

9113

316.

5299

3011

8.96

6367

629.

8395

1507

79.2

128

3338

13.1

740

7339

67.8

208

1602

925.

7990

3477

464.

1773

7495

031.

9191

1605

0678

.298

792

1478

1.34

8233

733.

2423

7642

1.71

8717

1888

.302

638

3885

.150

185

1402

.672

118

7542

3.18

4841

0340

7.72

9389

1908

7.98

3819

2608

13.9

585

9316

407.

2965

3778

1.23

1486

356.

5421

1959

52.6

650

4414

67.9

226

9876

27.0

997

2194

245.

1262

4842

021.

1205

1061

3714

.700

723

1129

76.7

502

9418

212.

0991

4231

4.97

9197

582.

8926

2233

86.0

381

5076

88.1

110

1145

647.

4356

2567

266.

7977

5713

584.

9222

1263

0320

.493

827

7355

72.1

002

9520

215.

4301

4739

2.77

6611

0268

.668

625

4660

.083

458

3841

.327

613

2895

1.02

5330

0370

2.15

3367

4203

0.20

8215

0300

81.3

876

3328

2686

.520

2

9622

439.

1274

5307

9.90

9812

4603

.595

529

0312

.495

167

1417

.526

815

4158

3.18

9435

1433

1.51

9479

5559

5.64

5717

8857

96.8

513

3993

9223

.824

397

2490

7.43

1459

449.

4990

1408

02.0

630

3309

56.2

444

7721

30.1

558

1788

236.

4997

4111

767.

8777

9387

602.

8619

2128

4098

.253

047

9270

68.5

891

9827

647.

2488

6658

3.43

8915

9106

.331

137

7290

.118

688

7949

.679

220

7435

4.33

9648

1076

8.41

6911

0773

71.3

771

2532

8076

.921

157

5124

82.3

070

9930

688.

4462

7457

3.45

1517

9790

.154

243

0110

.735

210

2114

2.13

1024

0625

1.03

3956

2859

9.04

7713

0712

98.2

250

3014

0411

.536

169

0149

78.7

683

100

3406

4.17

5383

522.

2657

2031

62.8

742

4903

26.2

381

1174

313.

4507

2791

251.

1994

6585

460.

8858

1542

4131

.905

535

8670

89.7

280

8281

7974

.522

0

N/i

2122

2324

2526

2728

2930

11.

2100

001.

2200

001.

2300

001.

2400

001.

2500

001.

2600

001.

2700

001.

2800

001.

2900

001.

3000

002

1.46

4100

1.48

8400

1.51

2900

1.53

7600

1.56

2500

1.58

7600

1.61

2900

1.63

8400

1.66

4100

1.69

0000

31.

7715

611.

8158

481.

8608

671.

9066

241.

9531

252.

0003

762.

0483

832.

0971

522.

1466

892.

1970

004

2.14

3589

2.21

5335

2.28

8866

2.36

4214

2.44

1406

2.52

0474

2.60

1446

2.68

4355

2.76

9229

2.85

6100

52.

5937

422.

7027

082.

8153

062.

9316

253.

0517

583.

1757

973.

3038

373.

4359

743.

5723

053.

7129

30

63.

1384

283.

2973

043.

4628

263.

6352

153.

8146

974.

0015

044.

1958

734.

3980

474.

6082

744.

8268

097

3.79

7498

4.02

2711

4.25

9276

4.50

7667

4.76

8372

5.04

1895

5.32

8759

5.62

9500

5.94

4673

6.27

4852

84.

5949

734.

9077

075.

2389

095.

5895

075.

9604

646.

3527

886.

7675

237.

2057

597.

6686

288.

1573

079

5.55

9917

5.98

7403

6.44

3859

6.93

0988

7.45

0581

8.00

4513

8.59

4755

9.22

3372

9.89

2530

10.6

0449

910

6.72

7500

7.30

4631

7.92

5946

8.59

4426

9.31

3226

10.0

8568

610

.915

339

11.8

0591

612

.761

364

13.7

8584

9

118.

1402

758.

9116

509.

7489

1410

.657

088

11.6

4153

212

.707

965

13.8

6248

015

.111

573

16.4

6216

017

.921

604

129.

8497

3310

.872

213

11.9

9116

413

.214

789

14.5

5191

516

.012

035

17.6

0535

019

.342

813

21.2

3618

623

.298

085

1311

.918

177

13.2

6410

014

.749

132

16.3

8633

818

.189

894

20.1

7516

522

.358

794

24.7

5880

127

.394

680

30.2

8751

114

14.4

2099

416

.182

202

18.1

4143

220

.319

059

22.7

3736

825

.420

707

28.3

9566

831

.691

265

35.3

3913

739

.373

764

1517

.449

402

19.7

4228

722

.313

961

25.1

9563

328

.421

709

32.0

3009

136

.062

499

40.5

6481

945

.587

487

51.1

8589

3

1621

.113

777

24.0

8559

027

.446

172

31.2

4258

535

.527

137

40.3

5791

545

.799

373

51.9

2296

958

.807

859

66.5

4166

117

25.5

4767

029

.384

420

33.7

5879

238

.740

806

44.4

0892

150

.850

973

58.1

6520

466

.461

400

75.8

6213

786

.504

159

1830

.912

681

35.8

4899

241

.523

314

48.0

3859

955

.511

151

64.0

7222

673

.869

809

85.0

7059

297

.862

157

112.

4554

0719

37.4

0434

343

.735

771

51.0

7367

659

.567

863

69.3

8893

980

.731

005

93.8

1465

810

8.89

0357

126.

2421

8314

6.19

2029

2045

.259

256

53.3

5764

062

.820

622

73.8

6415

086

.736

174

101.

7210

6611

9.14

4615

139.

3796

5716

2.85

2416

190.

0496

38

cont

inue

sov

erle

af

Tabl

eA

.1(c

onti

nued

)

N/i

1112

1314

1516

1718

1920

2154

.763

699

65.0

9632

177

.269

364

91.5

9154

610

8.42

0217

128.

1685

4315

1.31

3661

178.

4059

6221

0.07

9617

247.

0645

2922

66.2

6407

679

.417

512

95.0

4131

811

3.57

3517

135.

5252

7216

1.49

2364

192.

1683

5022

8.35

9631

271.

0027

0532

1.18

3888

2380

.179

532

96.8

8936

411

6.90

0822

140.

8311

6116

9.40

6589

203.

4803

7924

4.05

3804

292.

3003

2734

9.59

3490

417.

5390

5424

97.0

1723

411

8.20

5024

143.

7880

1017

4.63

0639

211.

7582

3725

6.38

5277

309.

9483

3237

4.14

4419

450.

9756

0254

2.80

0770

2511

7.39

0853

144.

2101

3017

6.85

9253

216.

5419

9326

4.69

7796

323.

0454

5039

3.63

4381

478.

9048

5758

1.75

8527

705.

6410

01

2614

2.04

2932

175.

9363

5821

7.53

6881

268.

5120

7133

0.87

2245

407.

0372

6649

9.91

5664

612.

9982

1675

0.46

8500

917.

3333

0227

171.

8719

4821

4.64

2357

267.

5703

6433

2.95

4968

413.

5903

0651

2.86

6956

634.

8928

9378

4.63

7717

968.

1043

6511

92.5

333

2820

7.96

5057

261.

8636

7532

9.11

1547

412.

8641

6051

6.98

7883

646.

2123

6480

6.31

3974

1004

.336

312

48.8

546

1550

.293

329

251.

6377

1931

9.47

3684

404.

8072

0351

1.95

1559

646.

2348

5481

4.22

7579

1024

.018

712

85.5

504

1611

.022

520

15.3

813

3030

4.48

1640

389.

7578

9449

7.91

2860

634.

8199

3380

7.79

3567

1025

.926

713

00.5

038

1645

.504

620

78.2

190

2619

.995

631

368.

4227

8447

5.50

4631

612.

4328

1878

7.17

6717

1009

.742

012

92.6

677

1651

.639

821

06.2

458

2680

.902

534

05.9

943

3244

5.79

1568

580.

1156

5075

3.29

2366

976.

0991

2912

62.1

774

1628

.761

320

97.5

826

2695

.994

734

58.3

642

4427

.792

633

539.

4077

9870

7.74

1093

926.

5496

1012

10.3

629

1577

.721

820

52.2

392

2663

.929

934

50.8

732

4461

.289

857

56.1

304

3465

2.68

3435

863.

4441

3311

39.6

560

1500

.850

019

72.1

523

2585

.821

533

83.1

910

4417

.117

757

55.0

639

7482

.969

635

789.

7469

5710

53.4

018

1401

.776

918

61.0

540

2465

.190

332

58.1

350

4296

.652

556

53.9

106

7424

.032

497

27.8

604

3695

5.59

3818

1285

.150

217

24.1

856

2307

.707

030

81.4

879

4105

.250

154

56.7

487

7237

.005

695

77.0

018

1264

6.21

8637

1156

.268

5215

67.8

833

2120

.748

328

61.5

567

3851

.859

951

72.6

152

6930

.070

992

63.3

671

1235

4.33

2416

440.

0841

3813

99.0

849

1912

.817

626

08.5

204

3548

.330

348

14.8

249

6517

.495

188

01.1

900

1185

7.10

9915

937.

0888

2137

2.10

9439

1692

.892

723

33.6

375

3208

.480

143

99.9

295

6018

.531

182

12.0

438

1117

7.51

1315

177.

1007

2055

8.84

4527

783.

7422

4020

48.4

002

2847

.037

839

46.4

305

5455

.912

675

23.1

638

1034

7.17

5214

195.

4393

1942

6.68

8926

520.

9094

3611

8.86

48

4124

78.5

643

3473

.386

148

54.1

095

6765

.331

794

03.9

548

1303

7.44

0818

028.

2080

2486

6.16

1834

211.

9731

4695

4.52

4342

2999

.062

842

37.5

310

5970

.554

783

89.0

113

1175

4.94

3516

427.

1754

2289

5.82

4131

828.

6871

4413

3.44

5361

040.

8815

4336

28.8

659

5169

.787

873

43.7

823

1040

2.37

4014

693.

6794

2069

8.24

1029

077.

6966

4074

0.71

9556

932.

1445

7935

3.14

6044

4390

.927

863

07.1

411

9032

.852

212

898.

9437

1836

7.09

9226

079.

7837

3692

8.67

4752

148.

1210

7344

2.46

6410

3159

.089

845

5313

.022

676

94.7

122

1111

0.40

8215

994.

6902

2295

8.87

4032

860.

5275

4689

9.41

6966

749.

5949

9474

0.78

1613

4106

.816

7

4664

28.7

574

9387

.548

913

665.

8021

1983

3.41

5828

698.

5925

4140

4.26

4659

562.

2594

8543

9.48

1412

2215

.608

317

4338

.861

747

7778

.796

411

452.

8096

1680

8.93

6524

593.

4356

3587

3.24

0752

169.

3734

7564

4.06

9510

9362

.536

215

7658

.134

722

6640

.520

248

9412

.343

713

972.

4277

2067

4.99

1930

495.

8602

4484

1.55

0965

733.

4105

‘960

67.9

683

1399

84.0

464

2033

78.9

938

2946

32.6

763

4911

388.

9358

1704

6.36

1825

430.

2401

3781

4.86

6656

051.

9386

8282

4.09

7212

2006

.319

717

9179

.579

426

2358

.902

038

3022

.479

250

1378

0.61

2320

796.

5615

3127

9.19

5346

890.

4346

7006

4.92

3210

4358

.362

515

4948

.026

022

9349

.861

633

8442

.983

649

7929

.223

0

Tabl

eA

.2Si

ngle

paym

entp

rese

ntw

orth

fact

or.T

ofin

dP

for

agi

ven

FP

=F

(1+

i)−N

N/i

12

34

56

78

910

10.

9900

990

0.98

0392

20.

9708

738

0.96

1538

50.

9523

810

0.94

3396

20.

9345

794

0.92

5925

90.

9174

312

0.90

9090

92

0.98

0296

00.

9611

688

0.94

2595

90.

9245

562

0.90

7029

50.

8899

964

0.87

3438

70.

8573

388

0.84

1680

00.

8264

463

30.

9705

901

0.94

2322

30.

9151

417

0.88

8996

40.

8638

376

0.83

9619

30.

8162

979

0.79

3832

20.

7721

835

0.75

1314

84

0.96

0980

30.

9238

454

0.88

8487

00.

8548

042

0.82

2702

50.

7920

937

0.76

2895

20.

7350

299

0.70

8425

20.

6830

135

50.

9514

657

0.90

5730

80.

8626

088

0.82

1927

10.

7835

262

0.74

7258

20.

7129

862

0.68

0583

20.

6499

314

0.62

0921

3

60.

9420

452

0.88

7971

40.

8374

843

0.79

0314

50.

7462

154

0.70

4960

50.

6663

422

0.63

0169

60.

5962

673

0.56

4473

97

0.93

2718

10.

8705

602

0.81

3091

50.

7599

178

0.71

0681

30.

6650

571

0.62

2749

70.

5834

904

0.54

7034

20.

5131

581

80.

9234

832

0.85

3490

40.

7894

092

0.73

0690

20.

6768

394

0.62

7412

40.

5820

091

0.54

0268

90.

5018

663

0.46

6507

49

0.91

4339

80.

8367

553

0.76

6416

70.

7025

867

0.64

4608

90.

5918

985

0.54

3933

70.

5002

490

0.46

0427

80.

4240

976

100.

9052

870

0.82

0348

30.

7440

939

0.67

5564

20.

6139

133

0.55

8394

80.

5083

493

0.46

3193

50.

4224

108

0.38

5543

3

110.

8963

237

0.80

4263

00.

7224

213

0.64

9580

90.

5846

793

0.52

6787

50.

4750

928

0.42

8882

90.

3875

329

0.35

0493

912

0.88

7449

20.

7884

932

0.70

1379

90.

6245

970

0.55

6837

40.

4969

694

0.44

4012

00.

3971

138

0.35

5534

70.

3186

308

130.

8786

626

0.77

3032

50.

6809

513

0.60

0574

10.

5303

214

0.46

8839

00.

4149

644

0.36

7697

90.

3261

786

0.28

9664

414

0.86

9963

00.

7578

750

0.66

1117

80.

5774

751

0.50

5068

00.

4423

010

0.38

7817

20.

3404

610

0.29

9246

50.

2633

313

150.

8613

495

0.74

3014

70.

6418

619

0.55

5264

50.

4810

171

0.41

7265

10.

3624

460

0.31

5241

70.

2745

380

0.23

9392

0

160.

8528

213

0.72

8445

80.

6231

669

0.53

3908

20.

4581

115

0.39

3646

30.

3387

346

0.29

1890

50.

2518

698

0.21

7629

117

0.84

4377

50.

7141

626

0.60

5016

40.

5133

732

0.43

6296

70.

3713

644

0.31

6574

40.

2702

690

0.23

1073

20.

1978

447

180.

8360

173

0.70

0159

40.

5873

946

0.49

3628

10.

4155

207

0.35

0343

80.

2958

639

0.25

0249

00.

2119

937

0.17

9858

819

0.82

7739

90.

6864

308

0.57

0286

00.

4746

424

0.39

5734

00.

3305

130

0.27

6508

30.

2317

121

0.19

4489

70.

1635

080

200.

8195

445

0.67

2971

30.

5536

758

0.45

6386

90.

3768

895

0.31

1804

70.

2584

190

0.21

4548

20.

1784

309

0.14

8643

6

210.

8114

302

0.65

9775

80.

5375

493

0.43

8833

60.

3589

424

0.29

4155

40.

2415

131

0.19

8655

70.

1636

981

0.13

5130

622

0.80

3396

20.

6468

390

0.52

1892

50.

4219

554

0.34

1849

90.

2775

051

0.22

5713

20.

1839

405

0.15

0181

70.

1228

460

230.

7954

418

0.63

4155

90.

5066

917

0.40

5726

30.

3255

713

0.26

1797

30.

2109

469

0.17

0315

30.

1377

814

0.11

1678

224

0.78

7566

10.

6217

215

0.49

1933

70.

3901

215

0.31

0067

90.

2469

785

0.19

7146

60.

1576

993

0.12

6404

90.

1015

256

250.

7797

684

0.60

9530

90.

4776

056

0.37

5116

80.

2953

028

0.23

2998

60.

1842

492

0.14

6017

90.

1159

678

0.09

2296

0

260.

7720

480

0.59

7579

30.

4636

947

0.36

0689

20.

2812

407

0.21

9810

00.

1721

955

0.13

5201

80.

1063

925

0.08

3905

527

0.76

4403

90.

5858

620

0.45

0189

10.

3468

166

0.26

7848

30.

2073

680

0.16

0930

40.

1251

868

0.09

7607

80.

0762

777

280.

7568

356

0.57

4374

60.

4370

768

0.33

3477

50.

2550

936

0.19

5630

10.

1504

022

0.11

5913

70.

0895

484

0.06

9343

329

0.74

9342

10.

5631

123

0.42

4346

40.

3206

514

0.24

2946

30.

1845

567

0.14

0562

80.

1073

275

0.08

2154

50.

0630

394

300.

7419

229

0.55

2070

90.

4119

868

0.30

8318

70.

2313

774

0.17

4110

10.

1313

671

0.09

9377

30.

0753

711

0.05

7308

6

cont

inue

sov

erle

af

Tabl

eA

.2(c

onti

nued

)

N/i

12

34

56

78

910

310.

7345

771

0.54

1246

00.

3999

871

0.29

6460

30.

2203

595

0.16

4254

80.

1227

730

0.09

2016

00.

0691

478

0.05

2098

732

0.72

7304

10.

5306

333

0.38

8337

00.

2850

579

0.20

9866

20.

1549

574

0.11

4741

10.

0852

000

0.06

3438

40.

0473

624

330.

7201

031

0.52

0228

70.

3770

262

0.27

4094

20.

1998

725

0.14

6186

20.

1072

347

0.07

8888

90.

0582

003

0.04

3056

834

0.71

2973

30.

5100

282

0.36

6044

90.

2635

521

0.19

0354

80.

1379

115

0.10

0219

30.

0730

453

0.05

3394

80.

0391

425

350.

7059

142

0.50

0027

60.

3553

834

0.25

3415

50.

1812

903

0.13

0105

20.

0936

629

0.06

7634

50.

0489

861

0.03

5584

1

360.

6989

249

0.49

0223

20.

3450

324

0.24

3668

70.

1726

574

0.12

2740

80.

0875

355

0.06

2624

60.

0449

413

0.03

2349

237

0.69

2004

90.

4806

109

0.33

4982

90.

2342

968

0.16

4435

60.

1157

932

0.08

1808

80.

0579

857

0.04

1230

60.

0294

083

380.

6851

534

0.47

1187

20.

3252

262

0.22

5285

40.

1566

054

0.10

9238

90.

0764

569

0.05

3690

50.

0378

262

0.02

6734

939

0.67

8369

70.

4619

482

0.31

5753

50.

2166

206

0.14

9148

00.

1030

555

0.07

1455

00.

0497

134

0.03

4703

00.

0243

044

400.

6716

531

0.45

2890

40.

3065

568

0.20

8289

00.

1420

457

0.09

7222

20.

0667

804

0.04

6030

90.

0318

376

0.02

2094

9

410.

6650

031

0.44

4010

20.

2976

280

0.20

0277

90.

1352

816

0.09

1719

00.

0624

116

0.04

2621

20.

0292

088

0.02

0086

342

0.65

8418

90.

4353

041

0.28

8959

20.

1925

749

0.12

8839

60.

0865

274

0.05

8328

60.

0394

641

0.02

6797

10.

0182

603

430.

6518

999

0.42

6768

80.

2805

429

0.18

5168

20.

1227

044

0.08

1629

60.

0545

127

0.03

6540

80.

0245

845

0.01

6600

244

0.64

5445

50.

4184

007

0.27

2371

80.

1780

463

0.11

6861

30.

0770

091

0.05

0946

40.

0338

341

0.02

2554

50.

0150

911

450.

6390

549

0.41

0196

80.

2644

386

0.17

1198

40.

1112

965

0.07

2650

10.

0476

135

0.03

1327

90.

0206

922

0.01

3719

2

460.

6327

276

0.40

2153

70.

2567

365

0.16

4613

90.

1059

967

0.06

8537

80.

0444

986

0.02

9007

30.

0189

837

0.01

2472

047

0.62

6463

00.

3942

684

0.24

9258

80.

1582

826

0.10

0949

20.

0646

583

0.04

1587

50.

0268

586

0.01

7416

20.

0113

382

480.

6202

604

0.38

6537

60.

2419

988

0.15

2194

80.

0961

421

0.06

0998

40.

0388

668

0.02

4869

10.

0159

782

0.01

0307

449

0.61

4119

20.

3789

584

0.23

4950

30.

1463

411

0.09

1563

90.

0575

457

0.03

6324

10.

0230

269

0.01

4658

90.

0093

704

500.

6080

388

0.37

1527

90.

2281

071

0.14

0712

60.

0872

037

0.05

4288

40.

0339

478

0.02

1321

20.

0134

485

0.00

8518

6

510.

6020

186

0.36

4243

00.

2214

632

0.13

5300

60.

0830

512

0.05

1215

40.

0317

269

0.01

9741

90.

0123

381

0.00

7744

152

0.59

6058

10.

3571

010

0.21

5012

80.

1300

967

0.07

9096

40.

0483

164

0.02

9651

30.

0182

795

0.01

1319

40.

0070

401

530.

5901

565

0.35

0099

00.

2087

503

0.12

5093

00.

0753

299

0.04

5581

60.

0277

115

0.01

6925

50.

0103

847

0.00

6400

154

0.58

4313

40.

3432

343

0.20

2670

20.

1202

817

0.07

1742

70.

0430

015

0.02

5898

60.

0156

717

0.00

9527

30.

0058

183

550.

5785

281

0.33

6504

20.

1967

672

0.11

5655

50.

0683

264

0.04

0567

40.

0242

043

0.01

4510

90.

0087

406

0.00

5289

4

560.

5728

001

0.32

9906

10.

1910

361

0.11

1207

20.

0650

728

0.03

8271

20.

0226

208

0.01

3436

00.

0080

189

0.00

4808

557

0.56

7128

80.

3234

374

0.18

5471

90.

1069

300

0.06

1974

10.

0361

049

0.02

1141

00.

0124

407

0.00

7356

80.

0043

714

580.

5615

137

0.31

7095

50.

1800

698

0.10

2817

30.

0590

229

0.03

4061

20.

0197

579

0.01

1519

20.

0067

494

0.00

3974

059

0.55

5954

10.

3108

779

0.17

4825

10.

0988

628

0.05

6213

30.

0321

332

0.01

8465

30.

0106

659

0.00

6192

10.

0036

127

600.

5504

496

0.30

4782

30.

1697

331

0.09

5060

40.

0535

355

0.03

0314

30.

0172

573

0.00

9875

90.

0056

808

0.00

3284

3

610.

5449

996

0.29

8806

10.

1647

894

0.09

1404

20.

0509

862

0.02

8598

40.

0161

283

0.00

9144

30.

0052

118

0.00

2985

762

0.53

9603

60.

2929

472

0.15

9989

70.

0878

887

0.04

5583

0.02

6979

70.

0150

732

0.00

8467

00.

0047

814

0.00

2714

363

0.53

4261

00.

2872

031

0.15

5329

80.

0845

084

0.04

6246

00.

0254

525

0.01

4087

10.

0078

398

0.00

4386

60.

0024

675

640.

5289

713

0.28

1571

70.

1508

057

0.08

1258

00.

0440

438

0.02

4011

80.

0131

655

0.00

7259

00.

0040

244

0.00

2243

265

0.52

3733

90.

2760

507

0.14

6413

30.

0781

327

0.04

1946

50.

0226

526

0.01

2304

20.

0067

213

0.00

3692

10.

0020

393

660.

5185

484

0.27

0637

90.

1421

488

0.07

5127

60.

0399

490

0.02

1370

40.

0114

993

0.00

6223

50.

0033

873

0.00

1853

967

0.51

3414

30.

2653

313

0.13

8008

50.

0722

381

0.03

8046

70.

0201

608

0.01

0747

00.

0057

625

0.00

3107

60.

0016

853

680.

5083

310

0.26

0128

70.

1339

889

0.06

9459

70.

0362

349

0.01

9019

60.

0100

439

0.00

5335

60.

0028

510

0.00

1532

169

0.50

3298

00.

2550

282

0.13

0086

30.

0667

882

0.03

4509

50.

0179

430

0.00

9386

80.

0049

404

0.00

2615

60.

0013

929

700.

4983

149

0.25

0027

60.

1262

974

0.06

4219

40.

0328

662

0.01

6927

40.

0087

727

0.00

4574

40.

0023

996

0.00

1266

2

710.

4933

810

0.24

5125

10.

1226

188

0.06

1749

40.

0313

011

0.01

5969

20.

0081

988

0.00

4235

60.

0022

015

0.00

1151

172

0.48

8496

10.

2403

187

0.11

9047

40.

0593

744

0.02

9810

60.

0150

653

0.00

7662

50.

0039

218

0.00

2019

70.

0010

465

730.

4836

595

0.23

5606

60.

1155

800

0.05

7090

80.

0283

910

0.01

4212

50.

0071

612

0.00

3631

30.

0018

530

0.00

0951

374

0.47

8870

80.

2309

869

0.01

1221

360.

0548

950

0.02

7039

10.

0134

081

0.00

6692

70.

0033

623

0.00

1700

00.

0008

649

750.

4741

295

0.22

6457

70.

1089

452

0.05

2783

70.

0257

515

0.01

2649

10.

0062

548

0.00

3113

30.

0015

596

0.00

0786

2

760.

4694

351

0.22

2017

40.

1057

721

0.05

0753

50.

0245

252

0.01

1933

10.

0058

457

0.00

2882

70.

0014

308

0.00

0714

877

0.46

4787

30.

2176

641

0.10

2691

30.

0488

015

0.02

3357

40.

0112

577

0.00

5463

20.

0026

691

0.00

1312

70.

0006

498

780.

4601

854

0.21

3396

20.

0997

003

0.04

6924

50.

0222

451

0.01

0620

40.

0051

058

0.00

2471

40.

0012

043

0.00

0590

779

0.45

5629

10.

2092

119

0.09

6796

40.

0451

197

0.02

1185

80.

0100

193

0.00

4771

80.

0022

884

0.00

1104

90.

0005

370

800.

4511

179

0.20

5109

70.

0939

771

0.04

3384

30.

0201

770

0.00

9452

20.

0044

596

0.00

2118

80.

0010

136

0.00

0488

2

810.

4466

514

0.20

1088

00.

0912

399

0.04

1715

70.

0192

162

0.00

8917

10.

0041

679

0.00

1961

90.

0009

299

0.00

0443

882

0.44

2229

10.

1971

451

0.08

8582

40.

0401

112

0.01

8301

10.

0084

124

0.00

3895

20.

0018

166

0.00

0853

20.

0004

035

830.

4378

506

0.19

3279

50.

0860

024

0.03

8568

50.

0174

296

0.00

7936

20.

0036

404

0.00

1682

00.

0007

827

0.00

0366

884

0.43

3515

50.

1894

897

0.08

3497

40.

0370

851

0.01

6599

60.

0074

870

0.00

3402

20.

0015

574

0.00

0718

10.

0003

334

850.

4292

232

0.18

5774

20.

0810

655

0.03

5658

80.

0158

092

0.00

7063

20.

0031

796

0.00

1442

10.

0006

588

0.00

0303

1

860.

4249

735

0.18

2131

60.

0787

043

0.03

4287

30.

0150

564

0.00

6663

40.

0029

716

0.00

1335

20.

0006

044

0.00

0275

687

0.42

0765

80.

1785

604

0.07

6412

00.

0329

685

0.01

4339

40.

0062

862

0.00

2777

20.

0012

363

0.00

0554

50.

0002

505

880.

4165

998

0.17

5059

20.

0741

864

0.03

1700

50.

0136

566

0.00

5930

40.

0025

955

0.00

1144

70.

0005

087

0.00

0227

789

0.41

2475

10.

1716

266

0.07

2025

60.

0304

813

0.01

3006

30.

0055

947

0.00

2425

70.

0010

600

0.00

0466

70.

0002

070

900.

4083

912

0.16

8261

40.

0699

278

0.02

9308

90.

0123

869

0.00

5278

00.

0022

670

0.00

0981

40.

0004

282

0.00

0188

2

910.

4043

477

0.16

4962

20.

0678

911

0.02

8181

60.

0117

971

0.00

4979

30.

0021

187

0.00

0908

70.

0003

928

0.00

0171

192

0.40

0344

30.

1617

276

0.06

5913

60.

0270

977

0.01

1235

30.

0046

974

0.00

1980

10.

0008

414

0.00

0360

40.

0001

556

930.

3963

805

0.15

8556

50.

0639

938

0.02

6055

50.

0107

003

0.00

4431

50.

0018

506

0.00

0779

10.

0003

306

0.00

0141

494

0.39

2455

90.

1554

475

0.06

2129

90.

0250

534

0.01

0190

70.

0041

807

0.00

1729

50.

0007

214

0.00

0303

30.

0001

286

950.

3885

702

0.15

2399

50.

0603

203

0.02

4089

80.

0097

055

0.00

3944

10.

0016

164

0.00

0667

90.

0002

783

0.00

0116

9

cont

inue

sov

erle

af

Tabl

eA

.2(c

onti

nued

)

N/i

12

34

56

78

910

960.

3847

230

0.14

9411

30.

0585

634

0.02

3163

20.

0092

433

0.00

3720

80.

0015

106

0.00

0618

50.

0002

553

0.00

0106

297

0.38

0913

80.

1464

817

0.05

6857

70.

0222

724

0.00

8803

10.

0035

102

0.00

1411

80.

0005

727

0.00

0234

20.

0000

966

980.

3771

424

0.14

3609

50.

0552

016

0.02

1415

70.

0083

840

0.00

3311

50.

0013

194

0.00

0530

20.

0002

149

0.00

0087

899

0.37

3408

30.

1479

360.

0535

938

0.02

0592

00.

0079

847

0.00

3124

10.

0012

331

0.00

0491

00.

0001

971

0.00

0079

810

00.

3697

112

0.13

8033

00.

0520

328

0.01

9800

00.

0076

045

0.00

2947

20.

0011

525

0.00

0454

60.

0001

809

0.00

0072

6

N/i

1112

1314

1516

1718

1920

10.

9009

009

0.89

2857

10.

8849

558

0.87

7193

00.

8695

652

0.86

2069

00.

8547

009

0.84

7457

60.

8403

361

0.83

3333

32

0.81

1622

40.

7971

939

0.78

3146

70.

7694

675

0.75

6143

70.

7431

629

0.73

0513

60.

7181

844

0.70

6164

80.

6944

444

30.

7311

914

0.71

1780

20.

6930

502

0.67

4971

50.

6575

162

0.64

0657

70.

6243

706

0.60

8630

90.

5934

158

0.57

8703

74

0.65

8731

00.

6355

181

0.61

3318

70.

5920

803

0.57

1753

20.

5522

911

0.53

3650

00.

5157

889

0.49

8668

80.

4822

531

50.

5934

513

0.56

7426

90.

5427

599

0.51

9368

70.

4971

767

0.47

6113

00.

4561

112

0.43

7109

20.

4190

494

0.40

1877

6

60.

5346

408

0.50

6631

10.

4803

185

0.45

5586

50.

4323

276

0.41

0442

30.

3898

386

0.37

0431

50.

3521

423

0.33

4898

07

0.48

1658

40.

4523

492

0.42

5060

60.

3996

373

0.37

5937

00.

3538

295

0.33

3195

40.

3139

250

0.29

5917

90.

2790

816

80.

4339

265

0.40

3883

20.

3761

599

0.35

0559

10.

3269

018

0.30

5025

50.

2847

824

0.26

6038

20.

2486

705

0.23

2568

09

0.39

0924

80.

3606

100

0.33

2884

80.

3075

079

0.28

4262

40.

2629

530

0.24

3403

70.

2254

561

0.20

8966

80.

1938

067

100.

3521

845

0.32

1973

20.

2945

883

0.26

9743

80.

2471

847

0.22

6683

60.

2080

374

0.19

1064

50.

1756

024

0.16

1505

6

110.

3172

833

0.28

7476

10.

2606

977

0.23

6617

40.

2149

432

0.19

5416

90.

1778

097

0.16

1919

00.

1475

650

0.13

4588

012

0.28

5840

80.

2566

751

0.23

0705

90.

2075

591

0.18

6907

20.

1684

628

0.15

1974

10.

1372

195

0.12

4004

20.

1121

567

130.

2575

143

0.22

9174

20.

2041

645

0.18

2069

40.

1625

280

0.14

5226

60.

1298

924

0.11

6287

70.

1042

052

0.09

3463

914

0.23

1994

80.

2046

198

0.18

0676

60.

1597

100

0.14

1328

70.

1251

953

0.11

1019

20.

0985

489

0.08

7567

40.

0778

866

150.

2090

043

0.18

2696

30.

1598

908

0.14

0096

50.

1228

945

0.10

7927

00.

0948

882

0.08

3516

00.

0735

861

0.06

4905

5

160.

1882

922

0.16

3121

70.

1414

962

0.12

2891

70.

1068

648

0.09

3040

50.

0811

010

0.07

0776

30.

0618

370

0.05

4087

917

0.16

9632

60.

1456

443

0.12

5217

90.

1077

997

0.09

2925

90.

0802

074

0.06

9317

10.

0599

799

0.05

1963

90.

0450

732

180.

1528

222

0.13

0039

60.

1108

123

0.09

4511

0.08

0805

10.

0691

443

0.05

9245

40.

0508

304

0.04

3667

10.

0375

610

190.

1376

776

0.11

6106

80.

0980

640

0.08

2948

40.

0702

653

0.05

9607

10.

0506

371

0.04

3076

60.

0366

951

0.03

1300

920

0.12

4033

90.

1036

668

0.08

6782

30.

0727

617

0.06

1100

30.

0513

855

0.04

3279

60.

0365

056

0.03

0836

20.

0260

841

210.

1117

423

0.09

2559

60.

0767

985

0.06

3826

10.

0531

307

0.04

4297

80.

0369

911

0.03

0937

00.

0259

128

0.02

1736

722

0.10

0668

70.

0826

425

0.06

7963

30.

0559

878

0.04

6200

60.

0381

878

0.03

1616

30.

0262

178

0.02

1775

40.

0181

139

230.

0906

925

0.07

3788

00.

0601

445

0.04

9112

10.

0401

744

0.03

2920

50.

0270

225

0.02

2218

50.

0182

987

0.01

5094

924

0.08

1705

00.

0658

821

0.05

3225

20.

0430

808

0.03

4934

30.

0283

797

0.02

3096

10.

0188

292

0.01

5377

00.

0125

791

250.

0736

081

0.05

8823

30.

0471

020

0.03

7790

20.

0303

776

0.02

4465

30.

0197

403

0.01

5956

90.

0129

219

0.01

0482

6

260.

0663

136

0.05

2520

80.

0416

831

0.03

3149

30.

0264

153

0.02

1090

80.

0168

720

0.01

3522

80.

0108

587

0.00

8735

527

0.05

9742

00.

0468

936

0.03

6887

70.

0290

783

0.02

2969

90.

0181

817

0.01

4420

50.

0114

600

0.00

9125

00.

0072

796

280.

0538

216

0.04

1869

30.

0326

440

0.02

5507

30.

0199

738

0.01

5673

90.

0123

253

0.00

9711

90.

0076

681

0.00

6066

329

0.04

8487

90.

0373

833

0.02

8888

50.

0223

748

0.01

7368

50.

0135

120

0.01

0534

40.

0082

304

0.00

6443

70.

0050

553

300.

0436

828

0.03

3377

90.

0255

651

0.01

9627

00.

0151

031

0.01

1648

20.

0090

038

0.00

6974

90.

0054

149

0.00

4212

7

310.

0393

539

0.02

9801

70.

0226

239

0.01

7216

70.

0131

331

0.01

0041

60.

0076

955

0.00

5911

00.

0045

503

0.00

3510

632

0.03

5454

00.

0266

087

0.02

0021

20.

0151

024

0.01

1420

10.

0086

565

0.00

6577

40.

0050

093

0.00

3823

80.

0029

255

330.

0319

405

0.02

3757

70.

0177

179

0.01

3247

70.

0099

305

0.00

7462

50.

0056

217

0.00

4245

20.

0032

133

0.00

2437

934

0.02

8775

20.

0212

123

0.01

5679

50.

0116

208

0.00

8635

20.

0064

332

0.00

4804

90.

0035

976

0.00

2700

20.

0020

316

350.

0259

236

0.01

8939

50.

0138

757

0.01

0193

70.

0075

089

0.00

5545

90.

0041

067

0.00

3048

80.

0022

691

0.00

1693

0

360.

0233

546

0.01

6910

30.

0122

794

0.00

8941

80.

0065

295

0.00

4780

90.

0035

100

0.00

2583

70.

0019

068

0.00

1410

837

0.02

1040

20.

0150

985

0.01

0866

70.

0078

437

0.00

5677

80.

0041

215

0.00

3000

00.

0021

896

0.00

1602

40.

0011

757

380.

0189

551

0.01

3480

80.

0096

165

0.00

6880

40.

0049

372

0.00

3553

00.

0025

641

0.00

1855

60.

0013

465

0.00

0979

739

0.01

7076

70.

0120

364

0.00

8510

20.

0060

355

0.00

4293

20.

0030

629

0.00

2191

60.

0015

725

0.00

1131

50.

0008

165

400.

0153

844

0.01

0746

80.

0075

312

0.00

5294

30.

0037

332

0.00

2640

50.

0018

731

0.00

1332

70.

0009

509

0.00

0680

4

410.

0138

598

0.00

9595

40.

0066

647

0.00

4644

10.

0032

463

0.00

2276

30.

0016

010

0.00

1129

40.

0007

991

0.00

0567

042

0.01

2486

30.

0085

673

0.00

5898

00.

0040

738

0.00

2822

90.

0019

623

0.00

1368

30.

0009

571

0.00

0671

50.

0004

725

430.

0112

489

0.00

7649

40.

0052

195

0.00

3573

50.

0024

547

0.00

1691

60.

0011

695

0.00

0811

10.

0005

643

0.00

0393

744

0.01

0134

20.

0068

298

0.00

4619

00.

0031

346

0.00

2134

50.

0014

583

0.00

0999

60.

0006

874

0.00

0474

20.

0003

281

450.

0091

299

0.00

6098

00.

0040

876

0.00

2749

70.

0018

561

0.00

1257

20.

0008

544

0.00

0582

50.

0003

985

0.00

0273

4

460.

0082

251

0.00

5444

70.

0036

174

0.00

2412

00.

0016

140

0.00

1083

80.

0007

302

0.00

0493

70.

0003

348

0.00

0227

947

0.00

7410

00.

0048

613

0.00

3201

20.

0021

158

0.00

1403

50.

0009

343

0.00

0624

10.

0004

184

0.00

0281

40.

0001

899

480.

0066

757

0.00

4340

50.

0028

329

0.00

1856

00.

0012

204

0.00

0805

40.

0005

334

0.00

0354

50.

0002

365

0.00

0158

249

0.00

6014

10.

0038

754

0.00

2507

00.

0016

280

0.00

1061

20.

0006

943

0.00

0455

90.

0003

005

0.00

0198

70.

0001

319

500.

0054

182

0.00

3460

20.

0022

186

0.00

1428

10.

0009

228

0.00

0598

60.

0003

897

0.00

0254

60.

0001

670

0.00

0109

9

510.

0048

812

0.00

3089

40.

0019

634

0.00

1252

70.

0008

024

0.00

0516

00.

0003

331

0.00

0215

80.

0001

403

0.00

0091

652

0.00

4397

50.

0027

584

0.00

1737

50.

0010

989

0.00

0697

80.

0004

448

0.00

0284

70.

0001

829

0.00

0117

90.

0000

763

530.

0039

617

0.00

2462

90.

0015

376

0.00

0963

90.

0006

068

0.00

0383

50.

0002

433

0.00

0155

00.

0000

991

0.00

0063

654

0.00

3569

10.

0021

990

0.00

1360

70.

0008

455

0.00

0527

60.

0003

306

0.00

0208

00.

0001

313

0.00

0083

30.

0000

530

550.

0032

154

0.00

1963

40.

0012

042

0.00

0741

70.

0004

588

0.00

0285

00.

0001

777

0.00

0111

30.

0000

700

0.00

0044

2

560.

0028

968

0.00

1753

00.

0010

656

0.00

0650

60.

0003

990

0.00

0245

70.

0001

519

0.00

0094

30.

0000

588

0.00

0036

857

0.00

2609

70.

0015

652

0.00

0943

00.

0005

707

0.00

0346

90.

0002

118

0.00

0129

80.

0000

799

0.00

0049

40.

0000

307

580.

0023

511

0.00

1397

50.

0008

345

0.00

0500

60.

0003

017

0.00

0182

60.

0001

110

0.00

0067

70.

0000

415

0.00

0025

659

0.00

2118

10.

0012

478

0.00

0738

50.

0004

392

0.00

0262

30.

0001

574

0.00

0094

80.

0000

574

0.00

0034

90.

0000

213

600.

0019

082

0.00

1114

10.

0006

536

0.00

0385

20.

0002

281

0.00

0135

70.

0000

811

0.00

0048

60.

0000

293

0.00

0017

7

cont

inue

sov

erle

af

Tabl

eA

.2(c

onti

nued

)

N/i

1112

1314

1516

1718

1920

610.

0017

190.

0009

950.

0005

780.

0003

380.

0001

980.

0001

170.

0000

690.

0000

410.

0000

250.

0000

1562

0.00

1549

0.00

0888

0.00

0512

0.00

0296

0.00

0172

0.00

0101

0.00

0059

0.00

0035

0.00

0021

0.00

0012

630.

0013

950.

0007

930.

0004

530.

0002

600.

0001

500.

0000

870.

0000

510.

0000

300.

0000

170.

0000

1064

0.00

1257

0.00

0708

0.00

0401

0.00

0228

0.00

0130

0.00

0075

0.00

0043

0.00

0025

0.00

0015

0.00

0009

650.

0011

320.

0006

320.

0003

550.

0002

000.

0001

130.

0000

650.

0000

370.

0000

210.

0000

120.

0000

07

660.

0010

200.

0005

640.

0003

140.

0001

760.

0000

990.

0000

560.

0000

320.

0000

180.

0000

100.

0000

0667

0.00

0919

0.00

0504

0.00

0278

0.00

0154

0.00

0086

0.00

0048

0.00

0027

0.00

0015

0.00

0009

0.00

0005

680.

0008

280.

0004

500.

0002

460.

0001

350.

0000

750.

0000

410.

0000

230.

0000

130.

0000

070.

0000

0469

0.00

0746

0.00

0402

0.00

0218

0.00

0118

0.00

0065

0.00

0036

0.00

0020

0.00

0011

0.00

0006

0.00

0003

700.

0006

720.

0003

590.

0001

930.

0001

040.

0000

560.

0000

310.

0000

170.

0000

090.

0000

050.

0000

03

710.

0006

050.

0003

200.

0001

700.

0000

910.

0000

490.

0000

270.

0000

140.

0000

080.

0000

040.

0000

0272

0.00

0545

0.00

0286

0.00

0151

0.00

0080

0.00

0043

0.00

0023

0.00

0012

0.00

0007

0.00

0004

0.00

0002

730.

0004

910.

0002

550.

0001

330.

0000

700.

0000

370.

0000

200.

0000

110.

0000

060.

0000

030.

0000

0274

0.00

0443

0.00

0228

0.00

0118

0.00

0062

0.00

0032

0.00

0017

0.00

0009

0.00

0005

0.00

0003

0.00

0001

750.

0003

990.

0002

040.

0001

050.

0000

540.

0000

280.

0000

150.

0000

080.

0000

040.

0000

020.

0000

01

760.

0003

590.

0001

820.

0000

920.

0000

470.

0000

240.

0000

130.

0000

070.

0000

030.

0000

020.

0000

0177

0.00

0324

0.00

0162

0.00

0082

0.00

0042

0.00

0021

0.00

0011

0.00

0006

0.00

0003

0.00

0002

0.00

0001

780.

0002

920.

0001

450.

0000

720.

0000

360.

0000

180.

0000

090.

0000

050.

0000

020.

0000

010.

0000

0179

0.00

0263

0.00

0129

0.00

0064

0.00

0032

0.00

0016

0.00

0008

0.00

0004

0.00

0002

0.00

0001

0.00

0001

800.

0002

370.

0001

150.

0000

570.

0000

280.

0000

140.

0000

070.

0000

040.

0000

020.

0000

010.

0000

00

810.

0002

132

0.00

0103

10.

0000

502

0.00

0024

60.

0000

121

0.00

0006

00.

0000

030

0.00

0001

50.

0000

008

0.00

0000

482

0.00

0192

10.

0000

921

0.00

0044

40.

0000

216

0.00

0010

50.

0000

052

0.00

0002

60.

0000

013

0.00

0000

60.

0000

003

830.

0001

731

0.00

0082

20.

0000

393

0.00

0018

90.

0000

092

0.00

0004

50.

0000

022

0.00

0001

10.

0000

005

0.00

0000

384

0.00

0155

90.

0000

734

0.00

0034

80.

0000

166

0.00

0008

00.

0000

039

0.00

0001

90.

0000

009

0.00

0000

50.

0000

002

850.

0001

405

0.00

0065

50.

0000

308

0.00

0014

60.

0000

069

0.00

0003

30.

0000

016

0.00

0000

80.

0000

004

0.00

0000

2

860.

0001

265

0.00

0058

50.

0000

272

0.00

0012

80.

0000

060

0.00

0002

90.

0000

014

0.00

0000

70.

0000

003

0.00

0000

287

0.00

0114

00.

0000

522

0.00

0024

10.

0000

112

0.00

0005

20.

0000

025

0.00

0001

20.

0000

006

0.00

0000

30.

0000

001

880.

0001

027

0.00

0046

60.

0000

213

0.00

0009

80.

0000

046

0.00

0002

10.

0000

010

0.00

0000

50.

0000

002

0.00

0000

189

0.00

0092

50.

0000

416

0.00

0018

90.

0000

086

0.00

0004

00.

0000

018

0.00

0000

90.

0000

004

0.00

0000

20.

0000

001

900.

0000

834

0.00

0037

20.

0000

167

0.00

0007

60.

0000

034

0.00

0001

60.

0000

007

0.00

0000

30.

0000

002

0.00

0000

1

910.

0000

751

0.00

0033

20.

0000

148

0.00

0006

60.

0000

030

0.00

0001

40.

0000

006

0.00

0000

30.

0000

001

0.00

0000

192

0.00

0067

70.

0000

296

0.00

0013

10.

0000

058

0.00

0002

60.

0000

012

0.00

0000

50.

0000

002

0.00

0000

10.

0000

001

930.

0000

609

0.00

0026

50.

0000

116

0.00

0005

10.

0000

023

0.00

0001

00.

0000

005

0.00

0000

20.

0000

001

4.32

66E

-08

940.

0000

549

0.00

0023

60.

0000

102

0.00

0004

50.

0000

020

0.00

0000

90.

0000

004

0.00

0000

20.

0000

001

3.60

55E

-08

950.

0000

495

0.00

0021

10.

0000

091

0.00

0003

90.

0000

017

0.00

0000

80.

0000

003

0.00

0000

10.

0000

001

3.00

46E

-08

960.

0000

446

0.00

0018

80.

0000

080

0.00

0003

40.

0000

015

0.00

0000

60.

0000

003

0.00

0000

10.

0000

001

2.50

38E

-08

970.

0000

401

0.00

0016

80.

0000

071

0.00

0003

00.

0000

013

0.00

0000

60.

0000

002

0.00

0000

14.

6983

E-0

82.

0865

E-0

898

0.00

0036

20.

0000

150

0.00

0006

30.

0000

027

0.00

0001

10.

0000

005

0.00

0000

20.

0000

001

3.94

82E

-08

1.73

88E

-08

990.

0000

326

0.00

0013

40.

0000

056

0.00

0002

30.

0000

010

0.00

0000

40.

0000

002

0.00

0000

13.

3178

E-0

81.

4490

E-0

810

00.

0000

294

0.00

0012

00.

0000

049

0.00

0002

00.

0000

009

0.00

0000

40.

0000

002

0.00

0000

12.

7881

E-0

81.

2075

E-0

8

N/i

2122

2324

2526

2728

2930

10.

8264

463

0.81

9672

10.

8130

081

0.80

6451

60.

8000

000

0.79

3650

80.

7874

016

0.78

1250

00.

7751

938

0.76

9230

82

0.68

3013

50.

6718

624

0.66

0982

20.

6503

642

0.64

0000

00.

6298

816

0.62

0001

20.

6103

516

0.60

0925

40.

5917

160

30.

5644

739

0.55

0706

90.

5373

839

0.52

4487

30.

5120

000

0.49

9906

00.

4881

900

0.47

6837

20.

4658

337

0.45

5166

14

0.46

6507

40.

4513

991

0.43

6897

50.

4229

736

0.40

9600

00.

3967

508

0.38

4401

50.

3725

290

0.36

1111

40.

3501

278

50.

3855

433

0.36

9999

30.

3552

012

0.34

1107

70.

3276

800

0.31

4881

60.

3026

784

0.29

1038

30.

2799

313

0.26

9329

1

60.

3186

308

0.30

3278

10.

2887

815

0.27

5086

90.

2621

440

0.24

9906

00.

2383

294

0.22

7373

70.

2170

010

0.20

7176

27

0.26

3331

30.

2485

886

0.23

4781

70.

2218

443

0.20

9715

20.

1983

381

0.18

7661

00.

1776

357

0.16

8217

80.

1593

663

80.

2176

291

0.20

3761

10.

1908

794

0.17

8906

70.

1677

722

0.15

7411

20.

1477

645

0.13

8777

90.

1304

014

0.12

2589

59

0.17

9858

80.

1670

173

0.15

5186

50.

1442

796

0.13

4217

70.

1249

295

0.11

6350

00.

1084

202

0.10

1086

40.

0942

996

100.

1486

436

0.13

6899

40.

1261

679

0.11

6354

50.

1073

742

0.09

9150

40.

0916

142

0.08

4703

30.

0783

615

0.07

2538

2

110.

1228

460

0.11

2212

70.

1025

755

0.09

3834

30.

0858

993

0.07

8690

80.

0721

372

0.06

6174

40.

0607

454

0.05

5798

612

0.10

1525

60.

0919

776

0.08

3394

70.

0756

728

0.06

8719

50.

0624

530

0.05

6800

90.

0516

988

0.04

7089

40.

0429

220

130.

0839

055

0.07

5391

50.

0678

006

0.06

1026

40.

0549

756

0.04

9565

90.

0447

251

0.04

0389

70.

0365

034

0.03

3016

914

0.06

9343

30.

0617

963

0.05

5122

40.

0492

149

0.04

3980

50.

0393

380

0.03

5216

60.

0315

544

0.02

8297

20.

0253

976

150.

0573

086

0.05

0652

70.

0448

150

0.03

9689

40.

0351

844

0.03

1220

60.

0277

296

0.02

4651

90.

0219

358

0.01

9536

6

160.

0473

624

0.04

1518

60.

0364

350

0.03

2007

60.

0281

475

0.02

4778

30.

0218

344

0.01

9259

30.

0170

045

0.01

5028

217

0.03

9142

50.

0340

316

0.02

9621

90.

0258

126

0.02

2518

00.

0196

653

0.01

7192

40.

0150

463

0.01

3181

80.

0115

601

180.

0323

492

0.02

7894

80.

0240

829

0.02

0816

60.

0180

144

0.01

5607

40.

0135

373

0.01

1754

90.

0102

185

0.00

8892

419

0.02

6734

90.

0228

646

0.01

9579

60.

0167

876

0.01

4411

50.

0123

868

0.01

0659

30.

0091

835

0.00

7921

30.

0068

403

200.

0220

949

0.01

8741

50.

0159

183

0.01

3538

40.

0115

292

0.00

9830

80.

0083

932

0.00

7174

60.

0061

405

0.00

5261

8

cont

inue

sov

erle

af

Tabl

eA

.2(c

onti

nued

)

N/i

2122

2324

2526

2728

2930

210.

0182

603

0.01

5361

90.

0129

417

0.01

0918

00.

0092

234

0.00

7802

20.

0066

088

0.00

5605

20.

0047

601

0.00

4047

522

0.01

5091

10.

0125

917

0.01

0521

70.

0088

049

0.00

7378

70.

0061

922

0.00

5203

80.

0043

791

0.00

3690

00.

0031

135

230.

0124

720

0.01

0321

10.

0085

543

0.00

7100

70.

0059

030

0.00

4914

50.

0040

975

0.00

3421

10.

0028

605

0.00

2395

024

0.01

0307

40.

0084

599

0.00

6954

70.

0057

264

0.00

4722

40.

0039

004

0.00

3226

30.

0026

728

0.00

2217

40.

0018

423

250.

0085

186

0.00

6934

30.

0056

542

0.00

4618

00.

0037

779

0.00

3095

50.

0025

404

0.00

2088

10.

0017

189

0.00

1417

2

260.

0070

401

0.00

5683

90.

0045

969

0.00

3724

20.

0030

223

0.00

2456

80.

0020

003

0.00

1631

30.

0013

325

0.00

1090

127

0.00

5818

30.

0046

589

0.00

3737

30.

0030

034

0.00

2417

90.

0019

498

0.00

1575

10.

0012

745

0.00

1032

90.

0008

386

280.

0048

085

0.00

3818

80.

0030

385

0.00

2422

10.

0019

343

0.00

1547

50.

0012

402

0.00

0995

70.

0008

007

0.00

0645

029

0.00

3974

00.

0031

301

0.00

2470

30.

0019

533

0.00

1547

40.

0012

282

0.00

0976

50.

0007

779

0.00

0620

70.

0004

962

300.

0032

843

0.00

2565

70.

0020

084

0.00

1575

20.

0012

379

0.00

0974

70.

0007

689

0.00

0607

70.

0004

812

0.00

0381

7

310.

0027

143

0.00

2103

00.

0016

328

0.00

1270

40.

0009

904

0.00

0773

60.

0006

055

0.00

0474

80.

0003

730

0.00

0293

632

0.00

2243

20.

0017

238

0.00

1327

50.

0010

245

0.00

0792

30.

0006

140

0.00

0476

70.

0003

709

0.00

0289

20.

0002

258

330.

0018

539

0.00

1412

90.

0010

793

0.00

0826

20.

0006

338

0.00

0487

30.

0003

754

0.00

0289

80.

0002

242

0.00

0173

734

0.00

1532

10.

0011

582

0.00

0877

50.

0006

663

0.00

0507

10.

0003

867

0.00

0295

60.

0002

264

0.00

0173

80.

0001

336

350.

0012

662

0.00

0949

30.

0007

134

0.00

0537

30.

0004

056

0.00

0306

90.

0002

327

0.00

0176

90.

0001

347

0.00

0102

8

360.

0010

465

0.00

0778

10.

0005

800

0.00

0433

30.

0003

245

0.00

0243

60.

0001

833

0.00

0138

20.

0001

044

0.00

0079

137

0.00

0864

90.

0006

378

0.00

0471

50.

0003

495

0.00

0259

60.

0001

933

0.00

0144

30.

0001

080

0.00

0080

90.

0000

608

380.

0007

148

0.00

0522

80.

0003

834

0.00

0281

80.

0002

077

0.00

0153

40.

0001

136

0.00

0084

30.

0000

627

0.00

0046

839

0.00

0590

70.

0004

285

0.00

0311

70.

0002

273

0.00

0166

20.

0001

218

0.00

0089

50.

0000

659

0.00

0048

60.

0000

360

400.

0004

882

0.00

0351

20.

0002

534

0.00

0183

30.

0001

329

0.00

0096

60.

0000

704

0.00

0051

50.

0000

377

0.00

0027

7

410.

0004

035

0.00

0287

90.

0002

060

0.00

0147

80.

0001

063

0.00

0076

70.

0000

555

0.00

0040

20.

0000

292

0.00

0021

342

0.00

0333

40.

0002

360

0.00

0167

50.

0001

192

0.00

0085

10.

0000

609

0.00

0043

70.

0000

314

0.00

0022

70.

0000

164

430.

0002

756

0.00

0193

40.

0001

362

0.00

0096

10.

0000

681

0.00

0048

30.

0000

344

0.00

0024

50.

0000

176

0.00

0012

644

0.00

0227

70.

0001

586

0.00

0110

70.

0000

775

0.00

0054

40.

0000

383

0.00

0027

10.

0000

192

0.00

0013

60.

0000

097

450.

0001

882

0.00

0130

00.

0000

900

0.00

0062

50.

0000

436

0.00

0030

40.

0000

213

0.00

0015

00.

0000

106

0.00

0007

5

460.

0001

556

0.00

0106

50.

0000

732

0.00

0050

40.

0000

348

0.00

0024

20.

0000

168

0.00

0011

70.

0000

082

0.00

0005

747

0.00

0128

60.

0000

873

0.00

0059

50.

0000

407

0.00

0027

90.

0000

192

0.00

0013

20.

0000

091

0.00

0006

30.

0000

044

480.

0001

062

0.00

0071

60.

0000

484

0.00

0032

80.

0000

223

0.00

0015

20.

0000

104

0.00

0007

10.

0000

049

0.00

0003

449

0.00

0087

80.

0000

587

0.00

0039

30.

0000

264

0.00

0017

80.

0000

121

0.00

0008

20.

0000

056

0.00

0003

80.

0000

026

500.

0000

726

0.00

0048

10.

0000

320

0.00

0021

30.

0000

143

0.00

0009

60.

0000

065

0.00

0004

40.

0000

030

0.00

0002

0

Tabl

eA

.3O

rdin

ary

annu

ityco

mpo

und

amou

ntfa

ctor

.To

find

Ffo

ra

give

nR

rece

ived

atth

een

dof

each

peri

odF

=R

(1+i

)N

−1i

N/i

12

34

56

78

910

11.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

02

2.01

0000

02.

0200

000

2.03

0000

02.

0400

000

2.05

0000

02.

0600

000

2.07

0000

02.

0800

000

2.09

0000

02.

1000

000

33.

0301

000

3.06

0400

03.

0909

000

3.12

1600

03.

1525

000

3.18

3600

03.

2149

000

3.24

6400

03.

2781

000

3.31

0000

04

4.06

0401

04.

1216

080

4.18

3627

04.

2464

640

4.31

0125

04.

3746

160

4.43

9943

04.

5061

120

4.57

3129

04.

6410

000

55.

1010

050

5.20

4040

25.

3091

358

5.41

6322

65.

5256

313

5.63

7093

05.

7507

390

5.86

6601

05.

9847

106

6.10

5100

0

66.

1520

151

6.30

8121

06.

4684

099

6.63

2975

56.

8019

128

6.97

5318

57.

1532

907

7.33

5929

07.

5233

346

7.71

5610

07

7.21

3535

27.

4342

834

7.66

2462

27.

8982

945

8.14

2008

58.

3938

376

8.65

4021

18.

9228

034

9.20

0434

79.

4871

710

88.

2856

706

8.58

2969

18.

8923

360

9.21

4226

39.

5491

089

9.89

7467

910

.259

8026

10.6

3662

7611

.028

4738

11.4

3588

819

9.36

8527

39.

7546

284

10.1

5910

6110

.582

7953

11.0

2656

4311

.491

3160

11.9

7798

8712

.487

5578

13.0

2103

6413

.579

4769

1010

.462

2125

10.9

4972

1011

.463

8793

12.0

0610

7112

.577

8925

13.1

8079

4913

.816

4480

14.4

8656

2515

.192

9297

15.9

3742

46

1111

.566

8347

12.1

6871

5412

.807

7957

13.4

8635

1414

.206

7872

14.9

7164

2615

.783

5993

16.6

4548

7517

.560

2934

18.5

3116

7112

12.6

8250

3013

.412

0897

14.1

9202

9615

.025

8055

15.9

1712

6516

.869

9412

17.8

8845

1318

.977

1265

20.1

4071

9821

.384

2838

1313

.809

3280

14.6

8033

1515

.617

7904

16.6

2683

7717

.712

9828

18.8

8213

7720

.140

6429

21.4

9529

6622

.953

3846

24.5

2271

2114

14.9

4742

1315

.973

9382

17.0

8632

4218

.291

9112

19.5

9863

2021

.015

0659

22.5

5048

7924

.214

9203

26.0

1918

9227

.974

9834

1516

.096

8955

17.2

9341

6918

.598

9139

20.0

2358

7621

.578

5636

23.2

7596

9925

.129

0220

27.1

5211

3929

.360

9162

31.7

7248

17

1617

.257

8645

18.6

3928

5320

.156

8813

21.8

2453

1123

.657

4918

25.6

7252

8127

.888

0536

30.3

2428

3033

.003

3987

35.9

4972

9917

18.4

3044

3120

.012

0710

21.7

6158

7723

.697

5124

25.8

4036

6428

.212

8798

30.8

4021

7333

.750

2257

36.9

7370

4640

.544

7028

1819

.614

7476

21.4

1231

2423

.414

4354

25.6

4541

2928

.132

3847

30.9

0565

2533

.999

0325

37.4

5024

3741

.301

3380

45.5

9917

3119

20.8

1089

5022

.840

5586

25.1

1686

8427

.671

2294

30.5

3900

3933

.759

9917

37.3

7896

4841

.446

2632

46.0

1845

8451

.159

0904

2022

.019

0040

24.2

9736

9826

.870

3745

29.7

7807

8633

.065

9541

36.7

8559

1240

.995

4923

45.7

6196

4351

.160

1196

57.2

7499

95

2123

.239

1940

25.7

8331

7228

.676

4857

31.9

6920

1735

.719

2518

39.9

9272

6744

.865

1768

50.4

2292

1456

.764

5304

64.0

0249

9422

24.4

7158

6027

.298

9835

30.5

3678

0334

.247

9698

38.5

0521

4443

.392

2903

49.0

0573

9255

.456

7552

62.8

7333

8171

.402

7494

2325

.716

3018

28.8

4496

3232

.452

8837

36.6

1788

8641

.430

4751

46.9

9582

7753

.436

1409

60.8

9329

5669

.531

9386

79.5

4302

4324

26.9

7346

4930

.421

8625

34.4

2647

0239

.082

6041

44.5

0199

8950

.815

5774

58.1

7667

0866

.764

7592

76.7

8981

3188

.497

3268

2528

.243

1995

32.0

3029

9736

.459

2643

41.6

4590

8347

.727

0988

54.8

6451

2063

.249

0377

73.1

0594

0084

.700

8962

98.3

4705

94

2629

.525

6315

33.6

7090

5738

.553

0423

44.3

1174

4651

.113

4538

59.1

5638

2768

.676

4704

79.9

5441

5193

.323

9769

109.

1817

654

2730

.820

8878

35.3

4432

3840

.709

6335

47.0

8421

4454

.669

1264

63.7

0576

5774

.483

8233

87.3

5076

8410

2.72

3134

812

1.09

9941

928

32.1

2909

6737

.051

2103

42.9

3092

2549

.967

5830

58.4

0258

2868

.528

1116

80.6

9769

0995

.338

8298

112.

9682

169

134.

2099

361

2933

.450

3877

38.7

9223

4545

.218

8502

52.9

6628

6362

.322

7119

73.6

3979

8387

.346

5293

103.

9659

362

124.

1353

565

148.

6309

297

3034

.784

8915

40.5

6807

9247

.575

4157

56.0

8493

7866

.438

8475

79.0

5818

6294

.460

7863

113.

2832

111

136.

3075

385

164.

4940

227

cont

inue

sov

erle

af

Tabl

eA

.3(c

onti

nued

)

N/i

12

34

56

78

910

3136

.132

7404

42.3

7944

0850

.002

6782

59.3

2833

5370

.760

7899

84.8

0167

7410

2.07

3041

412

3.34

5868

014

9.57

5217

018

1.94

3425

032

37.4

9406

7944

.227

0296

52.5

0275

8562

.701

4687

75.2

9882

9490

.889

7780

110.

2181

543

134.

2135

374

164.

0369

865

201.

1377

675

3338

.869

0085

46.1

1157

0255

.077

8413

66.2

0952

7480

.063

7708

97.3

4316

4711

8.93

3425

114

5.95

0620

417

9.80

0315

322

2.25

1544

234

40.2

5769

8648

.033

8016

57.7

3017

6569

.857

9085

85.0

6695

9410

4.18

3754

612

8.25

8764

815

8.62

6670

119

6.98

2343

724

5.47

6698

635

41.6

6027

5649

.994

4776

60.4

6208

1873

.652

2249

90.3

2030

7411

1.43

4779

913

8.23

6878

417

2.31

6803

721

5.71

0754

727

1.02

4368

5

3643

.076

8784

51.9

9436

7263

.275

9443

77.5

9831

3895

.836

3227

119.

1208

667

148.

9134

598

187.

1021

480

236.

1247

226

299.

1268

053

3744

.507

6471

54.0

3425

4566

.174

2226

81.7

0224

6410

1.62

8138

912

7.26

8118

716

0.33

7402

020

3.07

0319

825

8.37

5947

633

0.03

9485

938

45.9

5272

3656

.114

9396

69.1

5944

9385

.970

3363

107.

7095

458

135.

9042

058

172.

5610

202

220.

3159

454

282.

6297

829

364.

0434

344

3947

.412

2509

58.2

3723

8472

.234

2328

90.4

0914

9711

4.09

5023

114

5.05

8458

118

5.64

0291

623

8.94

1221

030

9.06

6463

340

1.44

7777

940

48.8

8637

3460

.401

9832

75.4

0125

9795

.025

5157

120.

7997

742

154.

7619

656

199.

6351

120

259.

0565

187

337.

8824

450

442.

5925

557

4150

.375

2371

62.6

1002

2878

.663

2975

99.8

2653

6312

7.83

9763

016

5.04

7683

621

4.60

9569

828

0.78

1040

236

9.29

1865

148

7.85

1811

242

51.8

7898

9564

.862

2233

82.0

2319

6510

4.81

9597

813

5.23

1751

117

5.95

0544

623

0.63

2239

730

4.24

3523

440

3.52

8133

053

7.63

6992

443

53.3

9777

9467

.159

4678

85.4

8389

2311

0.01

2381

714

2.99

3338

718

7.50

7577

224

7.77

6496

532

9.58

3005

344

0.84

5664

959

2.40

0691

644

54.9

3175

7269

.502

6571

89.0

4840

9111

5.41

2877

015

1.14

3005

619

9.75

8031

926

6.12

0851

335

6.94

9645

748

1.52

1774

865

2.64

0760

845

56.4

8107

4771

.892

7103

92.7

1986

1412

1.02

9392

015

9.70

0155

921

2.74

3513

828

5.74

9310

838

6.50

5617

452

5.85

8734

571

8.90

4836

9

4658

.045

8855

74.3

3056

4596

.501

4572

126.

8705

677

168.

6851

637

226.

5081

246

306.

7517

626

418.

4260

668

574.

1860

206

791.

7953

205

4759

.626

3443

76.8

1717

5810

0.39

6500

913

2.94

5390

417

8.11

9421

824

1.09

8612

132

9.22

4386

045

2.90

0152

162

6.86

2762

587

1.97

4852

648

61.2

2260

7879

.353

5193

104.

4083

960

139.

2632

060

188.

0253

929

256.

5645

288

353.

2700

930

490.

1321

643

684.

2804

111

960.

1723

378

4962

.834

8338

81.9

4058

9710

8.54

0647

914

5.83

3734

319

8.42

6662

627

2.95

8400

637

8.99

8999

553

0.34

2737

474

6.86

5648

110

57.1

8957

1650

64.4

6318

2284

.579

4015

112.

7968

673

152.

6670

837

209.

3479

957

290.

3359

046

406.

5289

295

573.

7701

564

815.

0835

564

1163

.908

5288

5166

.107

8140

87.2

7098

9511

7.18

0773

315

9.77

3767

022

0.81

5395

530

8.75

6058

943

5.98

5954

562

0.67

1768

988

9.44

1076

512

81.2

9938

1752

67.7

6889

2190

.016

4093

121.

6961

965

167.

1647

177

232.

8561

653

328.

2814

224

467.

5049

714

671.

3255

104

970.

4907

734

1410

.429

3198

5369

.446

5811

92.8

1673

7512

6.34

7082

417

4.85

1306

424

5.49

8973

534

8.97

8307

750

1.23

0319

372

6.03

1551

310

58.8

3494

3015

52.4

7225

1854

71.1

4104

6995

.673

0722

131.

1374

949

182.

8453

586

258.

7739

222

370.

9170

062

537.

3164

417

785.

1140

754

1155

.130

0878

1708

.719

4770

5572

.852

4573

98.5

8653

3713

6.07

1619

719

1.15

9173

027

2.71

2618

339

4.17

2026

657

5.92

8592

684

8.92

3201

412

60.0

9179

5718

80.5

9142

47

5674

.580

9819

101.

5582

643

141.

1537

683

199.

8055

399

287.

3482

492

418.

8223

482

617.

2435

941

917.

8370

575

1374

.500

0573

2069

.650

5672

5776

.326

7917

104.

5894

296

146.

3883

814

208.

7977

615

302.

7156

617

444.

9516

891

661.

4506

457

992.

2640

221

1499

.205

0625

2277

.615

6239

5878

.090

0597

107.

6812

182

151.

7800

328

218.

1496

720

318.

8514

448

472.

6487

904

708.

7521

909

1072

.645

1439

1635

.133

5181

2506

.377

1863

5979

.870

9603

110.

8348

426

157.

3334

338

227.

8756

588

335.

7940

170

502.

0077

178

759.

3648

443

1159

.456

7554

1783

.295

5348

2758

.014

9049

6081

.669

6699

114.

0515

394

163.

0534

368

237.

9906

852

353.

5837

179

533.

1281

809

813.

5203

834

1253

.213

2958

1944

.792

1329

3034

.816

3954

6183

.486

3666

117.

3325

702

168.

9450

399

248.

5103

126

372.

2629

038

566.

1158

717

871.

4668

102

1354

.470

3595

2120

.823

4249

3339

.298

0350

6285

.321

2302

120.

6792

216

175.

0133

911

259.

4507

251

391.

8760

490

601.

0828

240

933.

4694

869

1463

.827

9883

2312

.697

5331

3674

.227

8385

6387

.174

4425

124.

0928

060

181.

2637

928

270.

8287

541

412.

4698

514

638.

1477

935

999.

8123

510

1581

.934

2273

2521

.840

3111

4042

.650

6223

6489

.046

1869

127.

5746

622

187.

7017

066

282.

6619

043

434.

0933

440

677.

4366

611

1070

.799

2155

1709

.488

9655

2749

.805

9391

4447

.915

6845

6590

.936

6488

131.

1261

554

194.

3327

578

294.

9683

805

456.

7980

112

719.

0828

608

1146

.755

1606

1847

.248

0828

2998

.288

4736

4893

.707

2530

6692

.846

0153

134.

7486

785

201.

1627

406

307.

7671

157

480.

6379

117

763.

2278

324

1228

.028

0219

1996

.027

9294

3269

.134

4362

5384

.077

9783

6794

.774

4755

138.

4436

521

208.

1976

228

321.

0778

003

505.

6698

073

810.

0215

024

1314

.989

9834

2156

.710

1637

3564

.356

5355

5923

.485

7761

6896

.722

2202

142.

2125

251

215.

4435

515

334.

9209

123

531.

9532

977

859.

6227

925

1408

.039

2823

2330

.246

9768

3886

.148

6236

6516

.834

3537

6998

.689

4424

146.

0567

756

222.

9068

580

349.

3177

488

559.

5509

626

912.

2001

600

1507

.602

0320

2517

.666

7350

4236

.901

9998

7169

.517

7891

7010

0.67

6336

814

9.97

7911

123

0.59

4063

736

4.29

0458

858

8.52

8510

796

7.93

2169

616

14.1

3417

4327

20.0

8007

3846

19.2

2317

9878

87.4

6956

80

7110

2.68

3100

215

3.97

7469

423

8.51

1885

637

9.86

2077

161

8.95

4936

210

27.0

0809

9817

28.1

2356

6429

38.6

8647

9750

35.9

5326

5986

77.2

1652

4872

104.

7099

312

158.

0570

188

246.

6672

422

396.

0565

602

650.

9026

831

1089

.628

5858

1850

.092

2161

3174

.781

3980

5490

.189

0599

9545

.938

1773

7310

6.75

7030

516

2.21

8159

125

5.06

7259

541

2.89

8822

668

4.44

7817

211

56.0

0630

1019

80.5

9867

1234

29.7

6390

9959

85.3

0607

5310

501.

5319

950

7410

8.82

4600

816

6.46

2522

326

3.71

9277

343

0.41

4775

571

9.67

0208

112

26.3

6667

9021

20.2

4057

8237

05.1

4502

2765

24.9

8362

2011

552.

6851

945

7511

0.91

2846

817

0.79

1772

827

2.63

0855

644

8.63

1366

575

6.65

3718

513

00.9

4867

9822

69.6

5741

8740

02.5

5662

4571

13.2

3214

8012

708.

9537

140

7611

3.02

1975

317

5.20

7608

228

1.80

9781

346

7.57

6621

279

5.48

6404

413

80.0

0560

0624

29.5

3343

8043

23.7

6115

4577

54.4

2304

1313

980.

8490

853

7711

5.15

2195

117

9.71

1760

429

1.26

4074

748

7.27

9686

083

6.26

0724

614

63.8

0593

6626

00.6

0077

8746

70.6

6204

6884

53.3

2111

5115

379.

9339

939

7811

7.30

3717

018

4.30

5995

630

1.00

1996

950

7.77

0873

587

9.07

3760

815

52.6

3429

2827

83.6

4283

3250

45.3

1501

0692

15.1

2001

5416

918.

9273

933

7911

9.47

6754

218

8.99

2115

531

1.03

2056

852

9.08

1708

492

4.02

7448

916

46.7

9235

0329

79.4

9783

1554

49.9

4021

1410

045.

4808

168

1861

1.82

0132

680

121.

6715

217

193.

7719

578

321.

3630

185

551.

2449

767

971.

2288

213

1746

.599

8914

3189

.062

6797

5886

.935

4283

1095

0.57

4090

320

474.

0021

459

8112

3.88

8236

919

8.64

7397

033

2.00

3909

157

4.29

4775

810

20.7

9026

2418

52.3

9588

4934

13.2

9706

7363

58.8

9026

2611

937.

1257

584

2252

2.40

2360

482

126.

1271

193

203.

6203

449

342.

9640

264

598.

2665

668

1072

.829

7755

1964

.539

6379

3653

.227

8620

6868

.601

4836

1301

2.46

7076

724

775.

6425

965

8312

8.38

8390

520

8.69

2751

835

4.25

2947

262

3.19

7229

511

27.4

7126

4320

83.4

1201

6239

09.9

5381

2374

19.0

8960

2314

184.

5891

136

2725

4.20

6856

184

130.

6722

744

213.

8666

068

365.

8805

356

649.

1251

187

1184

.844

8275

2209

.416

7372

4184

.650

5792

8013

.616

7705

1546

2.20

2133

829

980.

6275

417

8513

2.97

8997

121

9.14

3939

037

7.85

6951

767

6.09

0123

512

45.0

8706

8923

42.9

8174

1444

78.5

7611

9786

55.7

0611

2116

854.

8003

259

3297

9.69

0295

9

8613

5.30

8787

122

4.52

6817

839

0.19

2660

270

4.13

3728

413

08.3

4142

2324

84.5

6064

5947

93.0

7644

8193

49.1

6260

1118

372.

7323

552

3627

8.65

9325

587

137.

6618

750

230.

0173

541

402.

8984

400

733.

2990

775

1374

.758

4935

2634

.634

2847

5129

.591

7995

1009

8.09

5609

120

027.

2782

672

3990

7.52

5281

8814

0.03

8493

723

5.61

7701

241

5.98

5393

276

3.63

1040

614

44.4

9641

8127

93.7

1234

1754

89.6

6322

5410

906.

9432

579

2183

0.73

3311

243

899.

2777

839

8914

2.43

8878

724

1.33

0055

242

9.46

4955

079

5.17

6282

315

17.7

2123

9029

62.3

3508

2258

74.9

3965

1211

780.

4987

185

2379

6.49

9309

248

290.

2055

623

9014

4.86

3267

524

7.15

6656

344

3.34

8903

782

7.98

3333

515

94.6

0730

1031

41.0

7518

7262

87.1

8542

6812

723.

9386

160

2593

9.18

4247

053

120.

2261

185

9114

7.31

1900

125

3.09

9789

445

7.64

9370

886

2.10

2666

916

75.3

3766

6033

30.5

3969

8467

28.2

8840

6713

742.

8537

053

2827

4.71

0829

358

433.

2487

303

9214

9.78

5019

125

9.16

1785

247

2.37

8851

989

7.58

6773

617

60.1

0454

9335

31.3

7208

0372

00.2

6859

5114

843.

2820

017

3082

0.43

4803

964

277.

5736

034

9315

2.28

2869

326

5.34

5020

948

7.55

0217

493

4.49

0244

518

49.1

0977

6837

44.2

5440

5177

05.2

8739

6816

031.

7445

618

3359

5.27

3936

370

706.

3309

637

9415

4.80

5698

027

1.65

1921

450

3.17

6724

097

2.86

9854

319

42.5

6526

5639

69.9

0966

9482

45.6

5751

4617

315.

2841

268

3661

9.84

8590

577

777.

9640

601

9515

7.35

3755

027

8.08

4959

851

9.27

2025

710

12.7

8464

820

40.6

9352

8942

09.1

0424

9688

23.8

5354

0618

701.

5068

569

3991

6.63

4963

785

556.

7604

661

cont

inue

sov

erle

af

Tabl

eA

.3(c

onti

nued

)

N/i

12

34

56

78

910

9615

9.92

7292

628

4.64

6659

053

5.85

0186

510

54.2

9603

421

43.7

2820

5444

62.6

5050

4694

42.5

2328

8420

198.

6274

054

4351

0.13

2110

494

113.

4365

197

162.

5265

655

291.

3395

922

552.

9256

920

1097

.467

876

2251

.914

6156

4731

.409

5349

1010

4.49

992

2181

5.51

7597

947

427.

0440

004

1035

25.7

802

9816

5.15

1831

129

8.16

6384

057

0.51

3462

811

42.3

6659

123

65.5

1034

6450

16.2

9410

7010

812.

8149

123

561.

7590

057

5169

6.47

7960

411

3879

.358

299

167.

8033

494

305.

1297

117

588.

6288

667

1189

.061

254

2484

.785

8637

5318

.271

7534

1157

0.71

196

2544

7.69

9726

256

350.

1609

768

1252

68.2

9410

017

0.48

1382

931

2.23

2305

960

7.28

7732

712

37.6

2370

526

10.0

2515

6956

38.3

6805

8612

381.

6617

927

484.

5157

043

6142

2.67

5464

713

7796

.123

4

N/i

1112

1314

1516

1718

1920

11.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

02

2.11

0000

02.

1200

000

2.13

0000

02.

1400

000

2.15

0000

02.

1600

000

2.17

0000

02.

1800

000

2.19

0000

02.

2000

000

33.

3421

000

3.37

4400

03.

4069

000

3.43

9600

03.

4725

000

3.50

5600

03.

5389

000

3.57

2400

03.

6061

000

3.64

0000

04

4.70

9731

04.

7793

280

4.84

9797

04.

9211

440

4.99

3375

05.

0664

960

5.14

0513

05.

2154

320

5.29

1259

05.

3680

000

56.

2278

014

6.35

2847

46.

4802

706

6.61

0104

26.

7423

813

6.87

7135

47.

0144

002

7.15

4209

87.

2965

982

7.44

1600

0

67.

9128

596

8.11

5189

08.

3227

058

8.53

5518

78.

7537

384

8.97

7477

09.

2068

482

9.44

1967

59.

6829

519

9.92

9920

07

9.78

3274

110

.089

0117

10.4

0465

7510

.730

4914

11.0

6679

9211

.413

8733

11.7

7201

2412

.141

5217

12.5

2271

2712

.915

9040

811

.859

4343

12.2

9969

3112

.757

2630

13.2

3276

0213

.726

8191

14.2

4009

3114

.773

2546

15.3

2699

5615

.902

0281

16.4

9908

489

14.1

6397

2014

.775

6563

15.4

1570

7216

.085

3466

16.7

8584

1917

.518

5080

18.2

8470

7819

.085

8548

19.9

2341

3520

.798

9018

1016

.722

0090

17.5

4873

5118

.419

7492

19.3

3729

5120

.303

7182

21.3

2146

9222

.393

1082

23.5

2130

8624

.708

8621

25.9

5868

21

1119

.561

4300

20.6

5458

3321

.814

3165

23.0

4451

6424

.349

2760

25.7

3290

4327

.199

9366

28.7

5514

4230

.403

5458

32.1

5041

8512

22.7

1318

7224

.133

1333

25.6

5017

7727

.270

7487

29.0

0166

7430

.850

1690

32.8

2392

5834

.931

0701

37.1

8021

9639

.580

5022

1326

.211

6378

28.0

2910

9329

.984

7008

32.0

8865

3534

.351

9175

36.7

8619

6139

.403

9932

42.2

1866

2845

.244

4613

48.4

9660

2714

30.0

9491

8032

.392

6024

34.8

8271

1937

.581

0650

40.5

0470

5143

.671

9874

47.1

0267

2050

.818

0221

54.8

4090

8959

.195

9232

1534

.405

3590

37.2

7971

4740

.417

4644

43.8

4241

4147

.580

4109

51.6

5950

5456

.110

1262

60.9

6526

6066

.260

6816

72.0

3510

79

1639

.189

9485

42.7

5328

0446

.671

7348

50.9

8035

2155

.717

4725

60.9

2502

6366

.648

8477

72.9

3901

3979

.850

2111

87.4

4212

9417

44.5

0084

2848

.883

6741

53.7

3906

0359

.117

6014

65.0

7509

3471

.673

0305

78.9

7915

1887

.068

0364

96.0

2175

1210

5.93

0555

318

50.3

9593

5555

.749

7150

61.7

2513

8268

.394

0656

75.8

3635

7484

.140

7154

93.4

0560

7610

3.74

0283

011

5.26

5883

912

8.11

6666

419

59.9

3948

8463

.439

6808

70.7

4940

6278

.969

2348

88.2

1181

1098

.603

2298

110.

2845

609

123.

4135

339

138.

1664

019

154.

7399

997

2064

.202

8321

72.0

5244

2480

.946

8290

91.0

2492

7710

2.44

3582

611

5.37

9746

613

0.03

2936

314

6.62

7970

016

5.41

8018

318

6.68

7999

6

2172

.265

1437

81.6

9873

5592

.469

9167

104.

7684

175

118.

8101

200

134.

8405

060

153.

1385

354

174.

0210

046

197.

8474

417

225.

0255

995

2281

.214

3095

92.5

0258

3810

5.49

1005

912

0.43

5996

013

7.63

1638

015

7.41

4987

018

0.17

2086

420

6.34

4785

523

6.43

8455

727

1.03

0719

523

91.1

4788

3510

4.60

2893

912

0.20

4836

713

8.29

7035

415

9.27

6383

718

3.60

1384

921

1.80

1341

124

4.48

6846

828

2.36

1762

232

6.23

6863

324

102.

1741

507

118.

1552

411

136.

8314

654

158.

6586

204

184.

1678

413

213.

9776

065

248.

8075

691

289.

4944

793

337.

0104

971

392.

4842

360

2511

4.41

3307

313

3.33

3870

115

5.61

9555

918

1.87

0827

221

2.79

3017

524

9.21

4023

529

2.10

4855

934

2.60

3485

540

2.04

2491

547

1.98

1083

2

2612

7.99

8771

115

0.33

3934

517

6.85

0098

220

8.33

2743

024

5.71

1970

129

0.08

8267

334

2.76

2681

440

5.27

2112

947

9.43

0564

956

7.37

7299

927

143.

0786

359

169.

3740

066

200.

8406

110

238.

4993

271

283.

5687

656

337.

5023

901

402.

0323

372

479.

2210

933

571.

5223

722

681.

8527

598

2815

9.81

7285

919

0.69

8887

422

7.94

9890

427

2.88

9232

932

7.10

4080

439

2.50

2772

547

1.37

7834

556

6.48

0890

168

1.11

1622

981

9.22

3311

829

178.

3971

873

214.

5827

539

258.

5833

762

312.

0937

255

377.

1696

925

456.

3032

161

552.

5120

664

669.

4474

503

811.

5228

313

984.

0679

742

3019

9.02

0877

924

1.33

2684

329

3.19

9215

135

6.78

6847

043

4.74

5146

453

0.31

1730

764

7.43

9117

779

0.94

7991

396

6.71

2169

211

81.8

8156

90

3122

1.91

3174

527

1.29

2606

533

2.31

5113

040

7.73

7005

650

0.95

6918

361

6.16

1607

675

8.50

3767

793

4.31

8629

811

51.3

8748

1414

19.2

5788

2832

247.

3236

237

304.

8477

192

376.

5160

777

465.

8201

864

577.

1004

561

715.

7474

648

888.

4494

082

1103

.495

9831

1371

.151

1029

1704

.109

4594

3327

5.52

9222

334

2.42

9445

542

6.46

3167

853

2.03

5012

566

4.66

5524

583

1.26

7059

210

40.4

8580

7613

03.1

2526

0116

32.6

6981

2420

45.9

3135

1234

306.

8374

368

384.

5209

790

482.

9033

796

607.

5199

142

765.

3653

532

965.

2697

886

1218

.368

3949

1538

.687

8069

1943

.877

0767

2456

.117

6215

3534

1.58

9554

843

1.66

3496

554

6.68

0819

069

3.57

2702

288

1.17

0156

111

20.7

1295

4814

26.4

9102

2118

16.6

5161

2123

14.2

1372

1329

48.3

4114

58

3638

0.16

4405

848

4.46

3116

161

8.74

9325

479

1.67

2880

510

14.3

4567

9613

01.0

2702

7616

69.9

9449

5821

44.6

4890

2327

54.9

1432

8435

39.0

0937

4937

422.

9824

905

543.

5986

900

700.

1867

377

903.

5070

838

1167

.497

5315

1510

.191

3520

1954

.893

5601

2531

.685

7047

3279

.348

0508

4247

.811

2499

3847

0.51

0564

460

9.83

0532

879

2.21

1013

710

30.9

9807

613

43.6

2216

1217

52.8

2196

8322

88.2

2546

5329

88.3

8913

1639

03.4

2418

0450

98.3

7349

9939

523.

2667

265

684.

0101

967

896.

1984

454

1176

.337

806

1546

.165

4854

2034

.273

4833

2678

.223

7944

3527

.299

1753

4646

.074

7747

6119

.048

1999

4058

1.82

6066

476

7.09

1420

310

13.7

0424

313

42.0

2509

917

79.0

9030

8223

60.7

5724

0631

34.5

2183

9541

63.2

1302

6855

29.8

2898

1973

43.8

5783

98

4164

6.82

6933

786

0.14

2390

811

46.4

8579

515

30.9

0861

320

46.9

5385

427

39.4

7839

936

68.3

9055

249

13.5

9137

265

81.4

9648

888

13.6

2940

842

718.

9778

964

964.

3594

777

1296

.528

948

1746

.235

819

2354

.996

933

3178

.794

943

4293

.016

946

5799

.037

819

7832

.980

821

1057

7.35

529

4379

9.06

5465

1081

.082

615

1466

.077

712

1991

.708

833

2709

.246

473

3688

.402

134

5023

.829

827

6843

.864

626

9322

.247

177

1269

3.82

635

4488

7.96

2666

212

11.8

1252

916

57.6

6781

422

71.5

4807

3116

.633

443

4279

.546

475

5878

.880

897

8076

.760

259

1109

4.47

414

1523

3.59

162

4598

6.63

8559

513

58.2

3003

218

74.1

6463

2590

.564

835

85.1

2846

4965

.273

911

6879

.290

6595

31.5

7710

513

203.

4242

318

281.

3099

4

4610

96.1

6880

115

22.2

1763

621

18.8

0603

229

54.2

4387

241

23.8

9772

957

60.7

1773

780

49.7

7006

111

248.

2609

815

713.

0748

321

938.

5719

347

1217

.747

369

1705

.883

752

2395

.250

816

3368

.838

014

4743

.482

388

6683

.432

575

9419

.230

971

1327

3.94

796

1869

9.55

905

2632

7.28

631

4813

52.6

9958

1911

.589

803

2707

.633

422

3841

.475

336

5456

.004

746

7753

.781

787

1102

1.50

024

1566

4.25

859

2225

3.47

527

3159

3.74

358

4915

02.4

9653

321

41.9

8057

930

60.6

2576

743

80.2

8188

362

75.4

0545

889

95.3

8687

312

896.

1552

818

484.

8251

426

482.

6355

737

913.

4922

950

1668

.771

152

2400

.018

249

3459

.507

117

4994

.521

346

7217

.716

277

1043

5.68

477

1508

9.50

167

2181

3.09

367

3151

5.33

633

4549

7.19

075

5118

53.3

3597

926

89.0

2043

839

10.2

4304

256

94.7

5433

583

01.3

7371

912

106.

3525

817

655.

7169

625

740.

4505

337

504.

2502

354

597.

6289

5220

58.2

0293

730

12.7

0289

144

19.5

7463

764

93.0

1994

195

47.5

7977

714

044.

3689

920

658.

1888

430

374.

7316

244

631.

0577

765

518.

1546

853

2285

.605

2633

75.2

2723

849

95.1

1934

7403

.042

733

1098

0.71

674

1629

2.46

803

2417

1.08

094

3584

3.18

331

5311

1.95

875

7862

2.78

562

5425

38.0

2183

837

81.2

5450

656

45.4

8485

484

40.4

6871

612

628.

8242

518

900.

2629

128

281.

1647

4229

5.95

631

6320

4.23

091

9434

8.34

274

5528

18.2

0424

4236

.005

047

6380

.397

885

9623

.134

336

1452

4.14

789

2192

5.30

498

3308

9.96

2749

910.

2284

475

214.

0347

911

3219

.011

3

5631

29.2

0670

747

45.3

2565

372

10.8

4961

1097

1.37

314

1670

3.77

008

2543

4.35

377

3871

6.25

636

5889

5.06

957

8950

5.70

1413

5863

.813

557

3474

.419

445

5315

.764

731

8149

.260

0612

508.

3653

819

210.

3355

929

504.

8503

845

299.

0199

469

497.

1820

910

6512

.784

716

3037

.576

358

3857

.605

583

5954

.656

499

9209

.663

867

1426

0.53

654

2209

2.88

593

3422

6.62

644

5300

0.85

333

8200

7.67

486

1267

51.2

137

1956

46.0

915

5942

82.9

4219

866

70.2

1527

910

407.

9201

716

258.

0116

525

407.

8188

239

703.

8866

762

011.

9984

9677

0.05

634

1508

34.9

444

2347

76.3

098

6047

55.0

6583

974

71.6

4111

211

761.

9497

918

535.

1332

829

219.

9916

446

057.

5085

372

555.

0381

311

4189

.666

517

9494

.583

828

1732

.571

8

cont

inue

sov

erle

af

Tabl

eA

.3(c

onti

nued

)

N/i

1112

1314

1516

1718

1920

6152

79.1

2308

283

69.2

3804

613

292.

0032

721

131.

0519

433

603.

9903

853

427.

7099

8489

0.39

461

1347

44.8

064

2135

99.5

547

3380

80.0

861

6258

60.8

2662

193

74.5

4661

115

020.

9636

924

090.

3992

238

645.

5889

461

977.

1434

899

322.

7616

915

8999

.871

625

4184

.470

140

5697

.103

363

6506

.517

549

1050

0.49

2216

974.

6889

727

464.

0551

144

443.

4272

871

894.

4864

411

6208

.631

218

7620

.848

530

2480

.519

448

6837

.524

6472

23.2

3447

911

761.

5512

719

182.

3985

431

310.

0228

251

110.

9413

783

398.

6042

713

5965

.098

522

1393

.601

235

9952

.818

158

4206

.028

865

8018

.790

272

1317

3.93

742

2167

7.11

035

3569

4.42

601

5877

8.58

258

9674

3.38

095

1590

80.1

652

2612

45.4

494

4283

44.8

535

7010

48.2

346

6689

01.8

5720

214

755.

8099

124

496.

1346

940

692.

6456

667

596.

3699

711

2223

.321

918

6124

.793

330

8270

.630

350

9731

.375

784

1258

.881

567

9882

.061

494

1652

7.50

7127

681.

6322

4639

0.61

605

7773

6.82

546

1301

80.0

534

2177

67.0

082

3637

60.3

438

6065

81.3

371

1009

511.

658

6810

970.

0882

618

511.

8079

531

281.

2443

952

886.

3023

8939

8.34

928

1510

09.8

6225

4788

.399

642

9238

.205

772

1832

.791

212

1141

4.98

969

1217

7.79

797

2073

4.22

491

3534

8.80

616

6029

1.38

462

1028

09.1

017

1751

72.4

399

2981

03.4

275

5065

02.0

827

8589

82.0

215

1453

698.

987

7013

518.

3557

423

223.

3319

3994

5.15

096

6873

3.17

846

1182

31.4

669

2032

01.0

302

3487

82.0

102

5976

73.4

576

1022

189.

606

1744

439.

785

7115

006.

3748

826

011.

1317

345

139.

0205

878

356.

8234

513

5967

.187

2357

14.1

951

4080

75.9

519

7052

55.6

812

1640

6.63

120

9332

8.74

272

1665

8.07

611

2913

3.46

753

5100

8.09

326

8932

7.77

873

1563

63.2

6527

3429

.466

347

7449

.863

783

2202

.702

414

4752

4.89

2511

995.

4973

1849

1.46

448

3263

0.48

364

5764

0.14

538

1018

34.6

678

1798

18.7

548

3171

79.1

809

5586

17.3

406

9820

00.1

888

1722

555.

6230

1439

5.58

874

2052

6.52

558

3654

7.14

167

6513

4.36

428

1160

92.5

212

2067

92.5

6836

7928

.849

965

3583

.288

411

5876

1.22

320

4984

2.18

736

1727

5.70

575

2278

5.44

339

4093

3.79

867

7360

2.83

163

1323

46.4

742

2378

12.4

532

4267

98.4

658

7646

93.4

475

1367

339.

243

2439

313.

203

4340

731.

847

7625

292.

8421

645

846.

8545

183

172.

1997

515

0875

.980

627

3485

.321

149

5087

.220

489

4692

.333

616

1346

1.30

729

0278

3.71

152

0887

9.21

677

2807

6.05

4851

349.

4770

693

985.

5857

117

1999

.617

931

4509

.119

357

4302

.175

610

4679

1.03

1903

885.

342

3454

313.

617

6250

656.

059

7831

165.

4208

357

512.

4143

1062

04.7

119

1960

80.5

644

3616

86.4

872

6661

91.5

237

1224

746.

505

2246

585.

703

4110

634.

204

7500

788.

271

7934

594.

6171

264

414.

9040

212

0012

.324

422

3532

.843

441

5940

.460

377

2783

.167

514

3295

4.41

126

5097

2.13

4891

655.

703

9000

946.

925

8038

401.

025

7214

5.69

2513

5614

.926

625

4828

.441

547

8332

.529

389

6429

.474

316

7655

7.66

131

2814

8.11

358

2107

1.28

610

8011

37.3

1

8142

626.

1377

580

804.

1756

1532

45.8

6729

0505

.423

355

0083

.408

710

3985

9.19

1961

573.

464

3691

215.

774

6927

075.

8312

9613

65.7

782

4731

6.01

291

9050

1.67

667

1731

68.8

297

3311

77.1

825

6325

96.9

201

1206

237.

661

2295

041.

952

4355

635.

613

8243

221.

238

1555

3639

.93

8352

521.

7743

310

1362

.877

919

5681

.777

637

7542

.988

172

7487

.458

113

9923

6.68

626

8520

0.08

451

3965

1.02

398

0943

4.27

318

6643

68.9

184

5830

0.16

9511

3527

.423

222

1121

.408

743

0400

.006

483

6611

.576

816

2311

5.55

631

4168

5.09

960

6478

9.20

711

6732

27.7

922

3972

43.6

985

6471

4.18

815

1271

51.7

1424

9868

.191

849

0657

.007

396

2104

.313

318

8281

5.04

536

7577

2.56

671

5645

2.26

513

8911

42.0

626

8766

93.4

3

8671

833.

7488

514

2410

.919

728

2352

.056

855

9349

.988

411

0642

0.96

2184

066.

452

4300

654.

902

8444

614.

672

1653

0460

.06

3225

2033

.12

8779

736.

4612

215

9501

.23

3190

58.8

241

6376

59.9

867

1272

385.

104

2533

518.

085

5031

767.

235

9964

646.

313

1967

1248

.47

3870

2440

.74

8888

508.

4719

517

8642

.377

736

0537

.471

372

6933

.384

914

6324

3.87

2938

881.

978

5887

168.

665

1175

8283

.65

2340

8786

.68

4644

2929

.89

8998

245.

4038

720

0080

.463

4074

08.3

425

8287

05.0

588

1682

731.

4534

0910

4.09

568

8798

8.33

813

8747

75.7

127

8564

57.1

555

7315

16.8

790

1090

53.3

983

2240

91.1

185

4603

72.4

271

9447

24.7

6719

3514

2.16

839

5456

1.75

8058

947.

355

1637

2236

.33

3314

9185

6687

7821

.24

9112

1050

.272

125

0983

.052

852

0221

.842

610

7698

7.23

422

2541

4.49

345

8729

2.63

9428

969.

406

1931

9239

.87

3944

7531

.15

8025

3386

.49

9213

4366

.802

2811

02.0

191

5878

51.6

821

1227

766.

447

2559

227.

667

5321

260.

451

1103

1895

.222

7967

04.0

546

9425

63.0

796

3040

64.7

993

1491

48.1

503

3148

35.2

614

6642

73.4

008

1399

654.

7529

4311

2.81

761

7266

3.12

312

9073

18.3

926

9001

11.7

855

8616

51.0

611

5564

878.

894

1655

55.4

468

3526

16.4

927

7506

29.9

429

1595

607.

415

3384

580.

7471

6029

0.22

315

1015

63.5

231

7421

32.9

6647

5365

.76

1386

7785

5.5

9518

3767

.545

939

4931

.471

984

8212

.835

518

1899

3.45

338

9226

8.85

183

0593

7.65

817

6688

30.3

137

4557

17.8

279

1056

86.2

516

6413

427.

6

9620

3982

.976

4423

24.2

485

9584

81.5

041

2073

653.

536

4476

110.

179

9634

888.

684

2067

2532

.47

4419

7748

.03

9413

5767

.64

1996

9611

4.1

9722

6422

.103

349

5404

.158

310

8308

5.1

2363

966.

031

5147

527.

705

1117

6471

.87

2418

6863

.99

5215

3343

.68

1120

2156

4.5

2396

3533

7.9

9825

1329

.534

755

4853

.657

312

2388

7.16

326

9492

2.27

659

1965

7.86

112

9647

08.3

728

2986

31.8

661

5409

46.5

413

3305

662.

728

7562

406.

599

2789

76.7

835

6214

37.0

962

1382

993.

494

3072

212.

394

6807

607.

5415

0390

62.7

133

1094

00.2

872

6183

17.9

215

8633

739.

734

5074

888.

810

030

9665

.229

769

6010

.547

715

6278

3.64

835

0232

3.12

978

2874

9.67

117

4453

13.7

538

7379

99.3

385

6896

16.1

418

8774

151.

241

4089

867.

6

N/i

2122

2324

2526

2728

2930

11.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

01.

0000

000

1.00

0000

02

2.21

0000

02.

2200

000

2.23

0000

02.

2400

000

2.25

0000

02.

2600

000

2.27

0000

02.

2800

000

2.29

0000

02.

3000

000

33.

6741

000

3.70

8400

03.

7429

000

3.77

7600

03.

8125

000

3.84

7600

03.

8829

000

3.91

8400

03.

9541

000

3.99

0000

04

5.44

5661

05.

5242

480

5.60

3767

05.

6842

240

5.76

5625

05.

8479

760

5.93

1283

06.

0155

520

6.10

0789

06.

1870

000

57.

5892

498

7.73

9582

67.

8926

334

8.04

8437

88.

2070

313

8.36

8449

88.

5327

294

8.69

9906

68.

8700

178

9.04

3100

0

610

.182

9923

10.4

4229

0710

.707

9391

10.9

8006

2811

.258

7891

11.5

4424

6711

.836

5664

12.1

3588

0412

.442

3230

12.7

5603

007

13.3

2142

0613

.739

5947

14.1

7076

5114

.615

2779

15.0

7348

6315

.545

7508

16.0

3243

9316

.533

9269

17.0

5059

6617

.582

8390

817

.118

9190

17.7

6230

5518

.430

0411

19.1

2294

4619

.841

8579

20.5

8764

6121

.361

1979

22.1

6342

6422

.995

2697

23.8

5769

079

21.7

1389

2022

.670

0127

23.6

6895

0524

.712

4513

25.8

0232

2426

.940

4340

28.1

2872

1329

.369

1858

30.6

6389

7932

.014

9979

1027

.273

8093

28.6

5741

5530

.112

8091

31.6

4343

9633

.252

9030

34.9

4494

6936

.723

4760

38.5

9255

7940

.556

4282

42.6

1949

73

1134

.001

3092

35.9

6204

6938

.038

7552

40.2

3786

5142

.566

1287

45.0

3063

3147

.638

8146

50.3

9847

4153

.317

7924

56.4

0534

6512

42.1

4158

4244

.873

6973

47.7

8766

8950

.894

9527

54.2

0766

0957

.738

5977

61.5

0129

4565

.510

0468

69.7

7995

2274

.326

9504

1351

.991

3168

55.7

4591

0759

.778

8328

64.1

0974

1468

.759

5761

73.7

5063

3179

.106

6440

84.8

5285

9991

.016

1384

97.6

2503

5514

63.9

0949

3469

.010

0110

74.5

2796

4380

.496

0793

86.9

4947

0293

.925

7977

101.

4654

379

109.

6116

607

118.

4108

185

127.

9125

462

1578

.330

4870

85.1

9221

3492

.669

3961

100.

8151

384

109.

6868

377

119.

3465

050

129.

8611

061

141.

3029

257

153.

7499

559

167.

2863

100

1695

.779

8893

104.

9345

004

114.

9833

572

126.

0107

716

138.

1085

472

151.

3765

964

165.

9236

048

181.

8677

449

199.

3374

431

218.

4722

031

1711

6.89

3666

012

9.02

0090

514

2.42

9529

315

7.25

3356

817

3.63

5683

919

1.73

4511

421

1.72

2978

123

3.79

0713

525

8.14

5301

628

5.01

3864

018

142.

4413

359

158.

4045

104

176.

1883

211

195.

9941

624

218.

0446

049

242.

5854

844

269.

8881

822

300.

2521

133

334.

0074

391

371.

5180

232

1917

3.35

4016

419

4.25

3502

721

7.71

1634

924

4.03

2761

427

3.55

5756

230

6.65

7710

334

3.75

7991

438

5.32

2705

143

1.86

9596

448

3.97

3430

120

210.

7583

598

237.

9892

733

268.

7853

109

303.

6006

241

342.

9446

952

387.

3887

150

437.

5726

490

494.

2130

625

558.

1117

794

630.

1654

592

cont

inue

sov

erle

af

Tabl

eA

.3(c

onti

nued

)

N/i

2122

2324

2526

2728

2930

2125

6.01

7615

429

1.34

6913

433

1.60

5932

537

7.46

4773

942

9.68

0869

489.

1097

809

556.

7172

643

633.

5927

272

0.96

4195

482

0.21

5096

922

310.

7813

147

356.

4432

343

408.

8752

969

469.

0563

196

538.

1010

862

617.

2783

239

708.

0309

256

811.

9986

815

931.

0438

121

1067

.279

626

2337

7.04

5390

743

5.86

0745

950

3.91

6615

258

2.62

9836

367

3.62

6357

877

8.77

0688

290

0.19

9275

610

40.3

5831

212

02.0

4651

813

88.4

6351

424

457.

2249

228

532.

7501

099

620.

8174

367

723.

4609

971

843.

0329

473

982.

2510

671

1144

.253

0813

32.6

5864

1551

.640

008

1806

.002

568

2555

4.24

2156

665

0.95

5134

176

4.60

5447

289

8.09

1636

410

54.7

9118

412

38.6

3634

514

54.2

0141

217

06.8

0305

920

02.6

1561

2348

.803

338

2667

1.63

3009

479

5.16

5263

694

1.46

4700

011

14.6

3362

913

19.4

8898

1561

.681

794

1847

.835

793

2185

.707

916

2584

.374

137

3054

.444

3427

813.

6759

414

971.

1016

216

1159

.001

581

1383

.145

716

50.3

6122

519

68.7

1906

123

47.7

5145

727

98.7

0613

233

34.8

4263

639

71.7

7764

228

985.

5478

891

1185

.743

978

1426

.571

945

1716

.100

668

2063

.951

531

2481

.586

016

2982

.644

3535

83.3

4384

943

02.9

4700

151

64.3

1093

429

1193

.512

946

1447

.607

654

1755

.683

492

2128

.964

828

2580

.939

414

3127

.798

381

3788

.958

324

4587

.680

126

5551

.801

631

6714

.604

214

3014

45.1

5066

417

67.0

8133

721

60.4

9069

526

40.9

1638

732

27.1

7426

839

42.0

2595

948

12.9

7707

258

73.2

3056

271

62.8

2410

487

29.9

8547

9

3117

49.6

3230

421

56.8

3923

226

58.4

0355

532

75.7

3632

4034

.967

835

4967

.952

709

6113

.480

882

7518

.735

119

9241

.043

095

1134

9.98

112

3221

18.0

5508

826

32.3

4386

332

70.8

3637

340

62.9

1303

750

44.7

0979

362

60.6

2041

377

65.1

2072

9624

.980

953

1192

1.94

559

1475

5.97

546

3325

63.8

4665

632

12.4

5951

240

24.1

2873

850

39.0

1216

663

06.8

8724

278

89.3

8172

198

62.7

0331

412

320.

9756

215

380.

3098

119

183.

7681

3431

03.2

5445

439

20.2

0060

549

50.6

7834

862

49.3

7508

678

84.6

0905

299

41.6

2096

812

526.

6332

115

771.

8487

919

841.

5996

624

939.

8985

335

3755

.937

8947

83.6

4473

860

90.3

3436

877

50.2

2510

698

56.7

6131

512

527.

4424

215

909.

8241

720

188.

9664

525

596.

6635

632

422.

8680

8

3645

45.6

8484

658

37.0

4658

174

92.1

1127

396

11.2

7913

212

321.

9516

415

785.

5774

520

206.

4767

2584

2.87

706

3302

0.69

599

4215

0.72

851

3755

01.2

7866

471

22.1

9682

992

16.2

9686

611

918.

9861

215

403.

4395

619

890.

8275

925

663.

2254

133

079.

8826

442

597.

6978

354

796.

9470

638

6657

.547

183

8690

.080

131

1133

7.04

514

1478

0.54

279

1925

5.29

944

2506

3.44

276

3259

3.29

627

4234

3.24

978

5495

2.03

0271

237.

0311

839

8056

.632

092

1060

2.89

776

1394

5.56

553

1832

8.87

306

2407

0.12

4331

580.

9378

841

394.

4862

754

200.

3597

270

889.

1189

692

609.

1405

340

9749

.524

831

1293

6.53

527

1715

4.04

5622

728.

8026

3008

8.65

538

3979

2.98

172

5257

1.99

756

6937

7.46

044

9144

7.96

346

1203

92.8

827

4111

797.

9250

515

783.

5730

321

100.

4760

928

184.

7152

237

611.

8192

350

140.

1569

766

767.

4369

8880

4.14

936

1179

68.8

729

1565

11.7

475

4214

276.

4893

119

256.

9590

925

954.

5855

934

950.

0468

847

015.

7740

363

177.

5977

884

795.

6448

611

3670

.311

215

2180

.846

2034

66.2

718

4317

275.

5520

623

494.

4900

931

925.

1402

743

339.

0581

358

770.

7175

479

604.

7732

110

7691

.469

1454

98.9

983

1963

14.2

913

2645

07.1

533

4420

904.

4179

928

664.

2779

139

268.

9225

353

741.

4320

873

464.

3969

310

0303

.014

213

6769

.165

618

6239

.717

825

3246

.435

834

3860

.299

345

2529

5.34

577

3497

1.41

905

4830

1.77

472

6664

0.37

577

9183

1.49

616

1263

82.7

979

1736

97.8

403

2383

87.8

388

3266

88.9

022

4470

19.3

89

4630

608.

3683

842

666.

1312

459

412.

1829

8263

5.06

596

1147

90.3

702

1592

43.3

254

2205

97.2

572

3051

37.4

337

4214

29.6

838

5811

26.2

058

4737

037.

1257

452

053.

6801

273

077.

9849

710

2468

.481

814

3488

.962

720

0647

.59

2801

59.5

166

3905

76.9

151

5436

45.2

922

7554

65.0

675

4844

815.

9221

563

506.

4897

489

886.

9215

112

7061

.917

417

9362

.203

425

2816

.963

435

5803

.586

149

9939

.451

470

1303

.426

998

2105

.587

749

5422

8.26

5877

478.

9174

811

0561

.913

515

7557

.777

622

4203

.754

331

8550

.373

945

1871

.554

463

9923

.497

890

4682

.420

712

7673

8.26

450

6561

7.20

162

9452

5.27

933

1359

92.1

536

1953

72.6

442

2802

55.6

929

4013

74.4

711

5738

77.8

741

8191

03.0

771

1167

041.

323

1659

760.

743

Tabl

eA

.4O

rdin

ary

annu

itypr

esen

twor

thfa

ctor

.To

find

Pfo

ra

give

nR

rece

ived

atth

een

dof

each

peri

odP

=R

1−(

1+i

)−N

i

N/i

12

34

56

78

910

10.

9900

990

0.98

0392

20.

9708

738

0.96

1538

50.

9523

810

0.94

3396

20.

9345

794

0.92

5925

90.

9174

312

0.90

9090

92

1.97

0395

11.

9415

609

1.91

3469

71.

8860

947

1.85

9410

41.

8333

927

1.80

8018

21.

7832

647

1.75

9111

21.

7355

372

32.

9409

852

2.88

3883

32.

8286

114

2.77

5091

02.

7232

480

2.67

3011

92.

6243

160

2.57

7097

02.

5312

947

2.48

6852

04

3.90

1965

63.

8077

287

3.71

7098

43.

6298

952

3.54

5950

53.

4651

056

3.38

7211

33.

3121

268

3.23

9719

93.

1698

654

54.

8534

312

4.71

3459

54.

5797

072

4.45

1822

34.

3294

767

4.21

2363

84.

1001

974

3.99

2710

03.

8896

513

3.79

0786

8

65.

7954

765

5.60

1430

95.

4171

914

5.24

2136

95.

0756

921

4.91

7324

34.

7665

397

4.62

2879

74.

4859

186

4.35

5260

77

6.72

8194

56.

4719

911

6.23

0283

06.

0020

547

5.78

6373

45.

5823

814

5.38

9289

45.

2063

701

5.03

2952

84.

8684

188

87.

6516

778

7.32

5481

47.

0196

922

6.73

2744

96.

4632

128

6.20

9793

85.

9712

985

5.74

6638

95.

5348

191

5.33

4926

29

8.56

6017

68.

1622

367

7.78

6108

97.

4353

316

7.10

7821

76.

8016

923

6.51

5232

26.

2468

879

5.99

5246

95.

7590

238

109.

4713

045

8.98

2585

08.

5302

028

8.11

0895

87.

7217

349

7.36

0087

17.

0235

815

6.71

0081

46.

4176

577

6.14

4567

1

1110

.367

6282

9.78

6848

09.

2526

241

8.76

0476

78.

3064

142

7.88

6874

67.

4986

743

7.13

8964

36.

8051

906

6.49

5061

012

11.2

5507

7510

.575

3412

9.95

4004

09.

3850

738

8.86

3251

68.

3838

439

7.94

2686

37.

5360

780

7.16

0725

36.

8136

918

1312

.133

7401

11.3

4837

3710

.634

9553

9.98

5647

89.

3935

730

8.85

2683

08.

3576

507

7.90

3775

97.

4869

039

7.10

3356

214

13.0

0370

3012

.106

2488

11.2

9607

3110

.563

1229

9.89

8640

99.

2949

839

8.74

5468

08.

2442

370

7.78

6150

47.

3666

875

1513

.865

0525

12.8

4926

3511

.937

9351

11.1

1838

7410

.379

6580

9.71

2249

09.

1079

140

8.55

9478

78.

0606

884

7.60

6079

5

1614

.717

8738

13.5

7770

9312

.561

1020

11.6

5229

5610

.837

7696

10.1

0589

539.

4466

486

8.85

1369

28.

3125

582

7.82

3708

617

15.5

6225

1314

.291

8719

13.1

6611

8512

.165

6689

11.2

7406

6210

.477

2597

9.76

3223

09.

1216

381

8.54

3631

48.

0215

533

1816

.398

2686

14.9

9203

1313

.753

5131

12.6

5929

7011

.689

5869

10.8

2760

3510

.059

0869

9.37

1887

18.

7556

251

8.20

1412

119

17.2

2600

8515

.678

4620

14.3

2379

9113

.133

9394

12.0

8532

0911

.158

1165

10.3

3559

529.

6035

992

8.95

0114

88.

3649

201

2018

.045

5530

16.3

5143

3314

.877

4749

13.5

9032

6312

.462

2103

11.4

6992

1210

.594

0142

9.81

8147

49.

1285

457

8.51

3563

7

2118

.856

9831

17.0

1120

9215

.415

0241

14.0

2915

9912

.821

1527

11.7

6407

6610

.835

5273

10.0

1680

329.

2922

437

8.64

8694

322

19.6

6037

9317

.658

0482

15.9

3691

6614

.451

1153

13.1

6300

2612

.041

5817

11.0

6124

0510

.200

7437

9.44

2425

48.

7715

403

2320

.455

8211

18.2

9220

4116

.443

6084

14.8

5684

1713

.488

5739

12.3

0337

9011

.272

1874

10.3

7105

899.

5802

068

8.88

3218

424

21.2

4338

7318

.913

9256

16.9

3554

2115

.246

9631

13.7

9864

1812

.550

3575

11.4

6933

4010

.528

7583

9.70

6611

88.

9847

440

2522

.023

1557

19.5

2345

6517

.413

1477

15.6

2207

9914

.093

9446

12.7

8335

6211

.653

5832

10.6

7477

629.

8225

796

9.07

7040

0

2622

.795

2037

20.1

2103

5817

.876

8424

15.9

8276

9214

.375

1853

13.0

0316

6211

.825

7787

10.8

0997

809.

9289

721

9.16

0945

527

23.5

5960

7620

.706

8978

18.3

2703

1516

.329

5857

14.6

4303

3613

.210

5341

11.9

8670

9010

.935

1648

10.0

2657

999.

2372

232

2824

.316

4432

21.2

8127

2418

.764

1082

16.6

6306

3214

.898

1273

13.4

0616

4312

.137

1113

11.0

5107

8510

.116

1284

9.30

6566

529

25.0

6578

5321

.844

3847

19.1

8845

4616

.983

7146

15.1

4107

3613

.590

7210

12.2

7767

4111

.158

4060

10.1

9828

299.

3696

059

3025

.807

7082

22.3

9645

5619

.600

4413

17.2

9203

3315

.372

4510

13.7

6483

1212

.409

0412

11.2

5778

3310

.273

6540

9.42

6914

5

cont

inue

sov

erle

af

Tabl

eA

.4(c

onti

nued

)

N/i

12

34

56

78

910

3126

.542

2854

22.9

3770

1520

.000

4285

17.5

8849

3615

.592

8105

13.9

2908

6012

.531

8142

11.3

4979

9410

.342

8019

9.47

9013

232

27.2

6958

9523

.468

3348

20.3

8876

5517

.873

5515

15.8

0267

6714

.084

0434

12.6

4655

5311

.434

9994

10.4

0624

039.

5263

756

3327

.989

6925

23.9

8856

3620

.765

7918

18.1

4764

5716

.002

5492

14.2

3022

9612

.753

7900

11.5

1388

8410

.464

4406

9.56

9432

434

28.7

0266

5924

.498

5917

21.1

3183

6718

.411

1978

16.1

9290

4014

.368

1411

12.8

5400

9411

.586

9337

10.5

1783

549.

6085

749

3529

.408

5801

24.9

9861

9321

.487

2201

18.6

6461

3216

.374

1943

14.4

9824

6412

.947

6723

11.6

5456

8210

.566

8215

9.64

4159

0

3630

.107

5050

25.4

8884

2521

.832

2525

18.9

0828

2016

.546

8517

14.6

2098

7113

.035

2078

11.7

1719

2810

.611

7628

9.67

6508

237

30.7

9950

9925

.969

4534

22.1

6723

5419

.142

5788

16.7

1128

7314

.736

7803

13.1

1701

6611

.775

1785

10.6

5299

349.

7059

165

3831

.484

6633

26.4

4064

0622

.492

4616

19.3

6786

4216

.867

8927

14.8

4601

9213

.193

4735

11.8

2886

9010

.690

8196

9.73

2651

439

32.1

6303

3026

.902

5888

22.8

0821

5119

.584

4848

17.0

1704

0714

.949

0747

13.2

6492

8511

.878

5824

10.7

2552

269.

7569

558

4032

.834

6861

27.3

5547

9223

.114

7720

19.7

9277

3917

.159

0864

15.0

4629

6913

.331

7088

11.9

2461

3310

.757

3602

9.77

9050

7

4133

.499

6892

27.7

9948

9523

.412

4000

19.9

9305

1817

.294

3680

15.1

3801

5913

.394

1204

11.9

6723

4610

.786

5690

9.79

9137

042

34.1

5810

8128

.234

7936

23.7

0135

9220

.185

6267

17.4

2320

7615

.224

5433

13.4

5244

9012

.006

6987

10.8

1336

609.

8173

973

4334

.810

0081

28.6

6156

2323

.981

9021

20.3

7079

4917

.545

9120

15.3

0617

2913

.506

9617

12.0

4323

9510

.837

9505

9.83

3997

544

35.4

5545

3529

.079

9631

24.2

5427

3920

.548

8413

17.6

6277

3315

.383

1820

13.5

5790

8112

.077

0736

10.8

6050

509.

8490

887

4536

.094

5084

29.4

9015

9924

.518

7125

20.7

2003

9717

.774

0698

15.4

5583

2113

.605

5216

12.1

0840

1510

.881

1973

9.86

2807

9

4636

.727

2361

29.8

9231

3624

.775

4491

20.8

8465

3617

.880

0665

15.5

2436

9913

.650

0202

12.1

3740

8810

.900

1810

9.87

5279

947

37.3

5369

9130

.286

5820

25.0

2470

7821

.042

9361

17.9

8101

5715

.589

0282

13.6

9160

7612

.164

2674

10.9

1759

729.

8866

181

4837

.973

9595

30.6

7311

9625

.266

7066

21.1

9513

0918

.077

1578

15.6

5002

6613

.730

4744

12.1

8913

6510

.933

5755

9.89

6925

549

38.5

8807

8731

.052

0780

25.5

0165

6921

.341

4720

18.1

6872

1715

.707

5723

13.7

6679

8512

.212

1634

10.9

4823

449.

9062

959

5039

.196

1175

31.4

2360

5925

.729

7640

21.4

8218

4618

.255

9255

15.7

6186

0613

.800

7463

12.2

3348

4610

.961

6829

9.91

4814

5

5139

.798

1362

31.7

8784

8925

.951

2272

21.6

1748

5218

.338

9766

15.8

1307

6113

.832

4732

12.2

5322

6510

.974

0210

9.92

2558

652

40.3

9419

4232

.144

9499

26.1

6624

0021

.747

5819

18.4

1807

3015

.861

3925

13.8

6212

4512

.271

5060

10.9

8534

049.

9295

987

5340

.984

3507

32.4

9504

8926

.374

9903

21.8

7267

4918

.493

4028

15.9

0697

4113

.889

8359

12.2

8843

1510

.995

7251

9.93

5998

954

41.5

6866

4132

.838

2833

26.5

7766

0521

.992

9567

18.5

6514

5615

.949

9755

13.9

1573

4512

.304

1033

11.0

0525

249.

9418

171

5542

.147

1922

33.1

7478

7526

.774

4276

22.1

0861

2218

.633

4720

15.9

9054

3013

.939

9388

12.3

1861

4111

.013

9930

9.94

7106

5

5642

.719

9922

33.5

0469

3626

.965

4637

22.2

1981

9418

.698

5447

16.0

2881

4113

.962

5596

12.3

3205

0111

.022

0120

9.95

1915

057

43.2

8712

1033

.828

1310

27.1

5093

5722

.326

7494

18.7

6051

8816

.064

9190

13.9

8370

0612

.344

4908

11.0

2936

889.

9562

864

5843

.848

6347

34.1

4522

6527

.331

0055

22.4

2956

6818

.819

5417

16.0

9898

0214

.003

4585

12.3

5601

0011

.036

1181

9.96

0260

359

44.4

0458

8834

.456

1044

27.5

0583

0622

.528

4296

18.8

7575

4016

.131

1134

14.0

2192

3812

.366

6760

11.0

4231

029.

9638

730

6044

.955

0384

34.7

6088

6727

.675

5637

22.6

2349

0018

.929

2895

16.1

6142

7714

.039

1812

12.3

7655

1811

.047

9910

9.96

7157

3

6145

.500

0380

35.0

5969

2827

.840

3531

22.7

1489

4218

.980

2757

16.1

9002

6114

.055

3095

12.3

8569

6111

.053

2028

9.97

0143

062

46.0

3964

1635

.352

6400

28.0

0034

2822

.802

7829

19.0

2883

4016

.217

0058

14.0

7038

2712

.394

1631

11.0

5798

429.

9728

573

6346

.573

9026

35.6

3984

3228

.155

6726

22.8

8729

1219

.075

0800

16.2

4245

8314

.084

4698

12.4

0200

2911

.062

3708

9.97

5324

864

47.1

0287

3835

.921

4149

28.3

0647

8322

.968

5493

19.1

1912

3816

.266

4701

14.0

9763

5312

.409

2619

11.0

6639

529.

9775

680

6547

.626

6078

36.1

9746

5528

.452

8915

23.0

4668

2019

.161

0703

16.2

8912

2714

.109

9396

12.4

1598

3211

.070

0874

9.97

9607

3

6648

.145

1562

36.4

6810

3528

.595

0403

23.1

2180

9619

.201

0194

16.3

1049

3114

.121

4388

12.4

2220

6711

.073

4747

9.98

1461

267

48.6

5857

0536

.733

4348

28.7

3304

8823

.194

0477

19.2

3906

6116

.330

6539

14.1

3218

5812

.427

9692

11.0

7658

239.

9831

465

6849

.166

9015

36.9

9356

3528

.867

0377

23.2

6350

7419

.275

3010

16.3

4967

3514

.142

2298

12.4

3330

4811

.079

4333

9.98

4678

669

49.6

7019

9537

.248

5917

28.9

9712

4023

.330

2956

19.3

0981

0516

.367

6165

14.1

5161

6612

.438

2452

11.0

8204

899.

9860

715

7050

.168

5143

37.4

9861

9329

.123

4214

23.3

9451

5019

.342

6766

16.3

8454

3914

.160

3893

12.4

4281

9611

.084

4485

9.98

7337

7

7150

.661

8954

37.7

4374

4429

.246

0401

23.4

5626

4419

.373

9778

16.4

0051

3114

.168

5882

12.4

4705

5211

.086

6500

9.98

8488

872

51.1

5039

1537

.984

0631

29.3

6508

7523

.515

6388

19.4

0378

8316

.415

5784

14.1

7625

0612

.450

9770

11.0

8866

979.

9895

353

7351

.634

0510

38.2

1966

9729

.480

6675

23.5

7272

9719

.432

1794

16.4

2979

0914

.183

4118

12.4

5460

8411

.090

5227

9.99

0486

674

52.1

1292

1838

.450

6566

29.5

9288

1123

.627

6247

19.4

5921

8516

.443

1990

14.1

9010

4512

.457

9707

11.0

9222

269.

9913

515

7552

.587

0512

38.6

7711

4329

.701

8263

23.6

8040

8319

.484

9700

16.4

5584

8114

.196

3593

12.4

6108

4011

.093

7822

9.99

2137

7

7653

.056

4864

38.8

9913

1729

.807

5983

23.7

3116

1919

.509

4952

16.4

6778

1214

.202

2050

12.4

6396

6711

.095

2131

9.99

2852

577

53.5

2127

3639

.116

7958

29.9

1028

9623

.779

9633

19.5

3285

2616

.479

0389

14.2

0766

8212

.466

6358

11.0

9652

589.

9935

022

7853

.981

4590

39.3

3019

1930

.009

9899

23.8

2688

7819

.555

0977

16.4

8965

9314

.212

7740

12.4

6910

7211

.097

7300

9.99

4093

079

54.4

3708

8239

.539

4039

30.1

0678

6323

.872

0075

19.5

7628

3516

.499

6786

14.2

1754

5812

.471

3956

11.0

9883

499.

9946

300

8054

.888

2061

39.7

4451

3630

.200

7634

23.9

1539

1819

.596

4605

16.5

0913

0814

.222

0054

12.4

7351

4411

.099

8485

9.99

5118

1

8155

.334

8575

39.9

4560

1630

.292

0033

23.9

5710

7519

.615

6767

16.5

1804

7914

.226

1733

12.4

7547

6311

.100

7785

9.99

5561

982

55.7

7708

6740

.142

7466

30.3

8058

5823

.997

2188

19.6

3397

7816

.526

4603

14.2

3006

8512

.477

2929

11.1

0163

169.

9959

654

8356

.214

9373

40.3

3602

6130

.466

5881

24.0

3578

7319

.651

4074

16.5

3439

6514

.233

7089

12.4

7897

4911

.102

4143

9.99

6332

284

56.6

4845

2840

.525

5158

30.5

5008

5624

.072

8724

19.6

6800

7016

.541

8835

14.2

3711

1112

.480

5323

11.1

0313

249.

9966

656

8557

.077

6760

40.7

1129

0030

.631

1510

24.1

0853

1219

.683

8162

16.5

4894

6714

.240

2908

12.4

8197

4411

.103

7912

9.99

6968

7

8657

.502

6495

40.8

9342

1630

.709

8554

24.1

4281

8419

.698

8726

16.5

5561

0114

.243

2624

12.4

8330

9611

.104

3956

9.99

7244

387

57.9

2341

5441

.071

9819

30.7

8626

7324

.175

7869

19.7

1321

2016

.561

8963

14.2

4603

9612

.484

5459

11.1

0495

019.

9974

948

8858

.340

0152

41.2

4704

1130

.860

4537

24.2

0748

7419

.726

8686

16.5

6782

6714

.248

6352

12.4

8569

0711

.105

4588

9.99

7722

689

58.7

5249

0341

.418

6677

30.9

3247

9424

.237

9687

19.7

3987

4816

.573

4214

14.2

5106

0912

.486

7506

11.1

0592

559.

9979

296

9059

.160

8815

41.5

8692

9231

.002

4071

24.2

6727

7619

.752

2617

16.5

7869

9414

.253

3279

12.4

8773

2011

.106

3537

9.99

8117

8

9159

.565

2292

41.7

5189

1331

.070

2982

24.2

9545

9219

.764

0588

16.5

8367

8714

.255

4467

12.4

8864

0811

.106

7465

9.99

8288

992

59.9

6557

3541

.913

6190

31.1

3621

1824

.322

5569

19.7

7529

4116

.588

3762

14.2

5742

6812

.489

4822

11.1

0710

699.

9984

445

9360

.361

9539

42.0

7217

5431

.200

2057

24.3

4861

2419

.785

9944

16.5

9280

7714

.259

2774

12.4

9026

1311

.107

4375

9.99

8585

994

60.7

5440

9842

.227

6230

31.2

6233

5624

.373

6658

19.7

9618

5116

.596

9884

14.2

6100

6912

.490

9827

11.1

0774

089.

9987

145

9561

.142

9800

42.3

8002

2531

.322

6559

24.3

9775

5619

.805

8906

16.6

0093

2414

.262

6233

12.4

9165

0611

.108

0191

9.99

8831

3

cont

inue

sov

erle

af

Tabl

eA

.4(c

onti

nued

)

N/i

12

34

56

78

910

9661

.527

7030

42.5

2943

3931

.381

2193

24.4

2091

8819

.815

1339

16.6

0465

3214

.264

1339

12.4

9226

9111

.108

2744

9.99

8937

697

61.9

0861

6842

.675

9155

31.4

3807

7024

.443

1912

19.8

2393

7016

.608

1634

14.2

6554

5712

.492

8418

11.1

0850

869.

9990

342

9862

.285

7592

42.8

1952

5031

.493

2787

24.4

6460

6919

.832

3210

16.6

1147

4914

.266

8651

12.4

9337

2011

.108

7235

9.99

9122

099

62.6

5916

7642

.960

3187

31.5

4687

2524

.485

1990

19.8

4030

5716

.614

5990

14.2

6809

8312

.493

8630

11.1

0892

079.

9992

018

100

63.0

2887

8843

.098

3516

31.5

9890

5324

.504

9990

19.8

4791

0216

.617

5462

14.2

6925

0712

.494

3176

11.1

0910

159.

9992

743

N/i

1112

1314

1516

1718

1920

10.

9009

009

0.89

2857

10.

8849

558

0.87

7193

00.

8695

652

0.86

2069

00.

8547

009

0.84

7457

60.

8403

361

0.83

3333

32

1.71

2523

31.

6900

510

1.66

8102

41.

6466

605

1.62

5708

91.

6052

319

1.58

5214

41.

5656

421

1.54

6501

01.

5277

778

32.

4437

147

2.40

1831

32.

3611

526

2.32

1632

02.

2832

251

2.24

5889

52.

2095

850

2.17

4272

92.

1399

168

2.10

6481

54

3.10

2445

73.

0373

493

2.97

4471

32.

9137

123

2.85

4978

42.

7981

806

2.74

3235

02.

6900

618

2.63

8585

52.

5887

346

53.

6958

970

3.60

4776

23.

5172

313

3.43

3081

03.

3521

551

3.27

4293

73.

1993

462

3.12

7171

03.

0576

349

2.99

0612

1

64.

2305

379

4.11

1407

33.

9975

498

3.88

8667

53.

7844

827

3.68

4735

93.

5891

848

3.49

7602

63.

4097

772

3.32

5510

17

4.71

2196

34.

5637

565

4.42

2610

44.

2883

048

4.16

0419

74.

0385

654

3.92

2380

13.

8115

276

3.70

5695

13.

6045

918

85.

1461

228

4.96

7639

84.

7987

703

4.63

8863

94.

4873

215

4.34

3590

94.

2071

625

4.07

7565

83.

9543

657

3.83

7159

89

5.53

7047

55.

3282

498

5.13

1655

14.

9463

718

4.77

1583

94.

6065

439

4.45

0566

24.

3030

218

4.16

3332

54.

0309

665

105.

8892

320

5.65

0223

05.

4262

435

5.21

6115

65.

0187

686

4.83

3227

54.

6586

036

4.49

4086

34.

3389

349

4.19

2472

1

116.

2065

153

5.93

7699

15.

6869

411

5.45

2733

05.

2337

118

5.02

8644

44.

8364

134

4.65

6005

34.

4864

999

4.32

7060

112

6.49

2356

16.

1943

742

5.91

7647

05.

6602

921

5.42

0619

05.

1971

072

4.98

8387

54.

7932

249

4.61

0504

14.

4392

167

136.

7498

704

6.42

3548

46.

1218

115

5.84

2361

55.

5831

470

5.34

2333

85.

1182

799

4.90

9512

64.

7147

093

4.53

2680

614

6.98

1865

26.

6281

682

6.30

2488

16.

0020

715

5.72

4475

65.

4675

291

5.22

9299

15.

0080

615

4.80

2276

84.

6105

672

157.

1908

696

6.81

0864

56.

4623

788

6.14

2168

05.

8473

701

5.57

5456

25.

3241

872

5.09

1577

64.

8758

628

4.67

5472

6

167.

3791

618

6.97

3986

26.

6038

751

6.26

5059

65.

9542

349

5.66

8496

75.

4052

882

5.16

2353

94.

9376

998

4.72

9560

517

7.54

8794

47.

1196

305

6.72

9093

06.

3728

593

6.04

7160

85.

7487

040

5.47

4605

35.

2223

338

4.98

9663

74.

7746

338

187.

7016

166

7.24

9670

16.

8399

053

6.46

7420

56.

1279

659

5.81

7848

35.

5338

507

5.27

3164

25.

0333

309

4.81

2194

819

7.83

9294

27.

3657

769

6.93

7969

36.

5503

688

6.19

8231

25.

8774

554

5.58

4487

85.

3162

409

5.07

0025

94.

8434

957

207.

9633

281

7.46

9443

67.

0247

516

6.62

3130

66.

2593

315

5.92

8840

95.

6277

673

5.35

2746

55.

1008

621

4.86

9579

7

218.

0750

704

7.56

2003

27.

1015

501

6.68

6956

66.

3124

622

5.97

3138

75.

6647

584

5.38

3683

55.

1267

749

4.89

1316

422

8.17

5739

17.

6446

457

7.16

9513

36.

7429

444

6.35

8662

76.

0113

265

5.69

6374

75.

4099

012

5.14

8550

34.

9094

304

238.

2664

316

7.71

8433

77.

2296

578

6.79

2056

56.

3988

372

6.04

4247

05.

7233

972

5.43

2119

75.

1668

490

4.92

4525

324

8.34

8136

67.

7843

158

7.28

2883

06.

8351

373

6.43

3771

46.

0726

267

5.74

6493

35.

4509

489

5.18

2226

14.

9371

044

258.

4217

447

7.84

3139

17.

3299

850

6.87

2927

46.

4641

491

6.09

7092

05.

7662

336

5.46

6905

85.

1951

480

4.94

7587

0

268.

4880

583

7.89

5659

97.

3716

681

6.90

6076

76.

4905

644

6.11

8182

75.

7831

056

5.48

0428

75.

2060

067

4.95

6322

527

8.54

7800

27.

9425

535

7.40

8555

96.

9351

550

6.51

3534

36.

1363

644

5.79

7526

25.

4918

887

5.21

5131

74.

9636

021

288.

6016

218

7.98

4422

87.

4411

999

6.96

0662

36.

5335

081

6.15

2038

35.

8098

514

5.50

1600

65.

2227

997

4.96

9668

429

8.65

0109

88.

0218

060

7.47

0088

46.

9830

371

6.55

0876

66.

1655

503

5.82

0385

95.

5098

310

5.22

9243

54.

9747

237

308.

6937

926

8.05

5184

07.

4956

534

7.00

2664

16.

5659

796

6.17

7198

55.

8293

896

5.51

6806

05.

2346

584

4.97

8936

4

318.

7331

465

8.08

4985

77.

5182

774

7.01

9880

86.

5791

127

6.18

7240

15.

8370

851

5.52

2716

95.

2392

087

4.98

2447

032

8.76

8600

48.

1115

944

7.53

8298

67.

0349

832

6.59

0532

86.

1958

966

5.84

3662

55.

5277

262

5.24

3032

54.

9853

725

338.

8005

409

8.13

5352

17.

5560

164

7.04

8230

86.

6004

633

6.20

3359

85.

8492

842

5.53

1971

35.

2462

458

4.98

7810

434

8.82

9316

18.

1565

644

7.57

1696

07.

0598

516

6.60

9098

56.

2097

924

5.85

4089

15.

5355

689

5.24

8946

14.

9898

420

358.

8552

398

8.17

5503

97.

5855

716

7.07

0045

36.

6166

074

6.21

5338

35.

8581

958

5.53

8617

75.

2512

152

4.99

1535

0

368.

8785

944

8.19

2414

27.

5978

510

7.07

8987

16.

6231

369

6.22

0119

25.

8617

058

5.54

1201

55.

2531

220

4.99

2945

837

8.89

9634

68.

2075

127

7.60

8717

77.

0868

308

6.62

8814

76.

2242

407

5.86

4705

85.

5433

911

5.25

4724

44.

9941

215

388.

9185

897

8.22

0993

57.

6183

343

7.09

3711

26.

6337

519

6.22

7793

75.

8672

699

5.54

5246

75.

2560

709

4.99

5101

339

8.93

5666

48.

2330

299

7.62

6844

57.

0997

467

6.63

8045

16.

2308

566

5.86

9461

55.

5468

192

5.25

7202

44.

9959

177

408.

9510

508

8.24

3776

77.

6343

756

7.10

5040

96.

6417

784

6.23

3497

15.

8713

346

5.54

8151

95.

2581

533

4.99

6598

1

418.

9649

106

8.25

3372

07.

6410

404

7.10

9685

06.

6450

247

6.23

5773

45.

8729

355

5.54

9281

35.

2589

524

4.99

7165

142

8.97

7397

08.

2619

393

7.64

6938

47.

1137

588

6.64

7847

56.

2377

357

5.87

4303

95.

5502

384

5.25

9623

84.

9976

376

438.

9886

459

8.26

9588

77.

6521

579

7.11

7332

36.

6503

022

6.23

9427

35.

8754

734

5.55

1049

55.

2601

881

4.99

8031

344

8.99

8780

18.

2764

185

7.65

6776

97.

1204

669

6.65

2436

76.

2408

565.

8764

730

5.55

1736

85.

2606

623

4.99

8359

445

9.00

7910

08.

2825

165

7.66

0864

57.

1232

166

6.65

4292

86.

2421

428

5.87

7327

35.

5523

193

5.26

1060

74.

9986

329

469.

0161

351

8.28

7961

17.

6644

819

7.12

5628

66.

6559

068

6.24

3226

55.

8780

576

5.55

2813

05.

2613

956

4.99

8860

747

9.02

3545

28.

2928

225

7.66

7683

17.

1277

444

6.65

7310

26.

2441

608

5.87

8681

75.

5532

314

5.26

1676

94.

9990

506

489.

0302

209

8.29

7162

97.

6705

160

7.12

9600

36.

6585

306

6.24

4966

25.

8792

151

5.55

3585

95.

2619

134

4.99

9208

849

9.03

6235

08.

3010

383

7.67

3023

07.

1312

284

6.65

9591

96.

2456

605

5.87

9671

05.

5538

864

5.26

2112

14.

9993

407

509.

0416

532

8.30

4498

57.

6752

416

7.13

2656

56.

6605

147

6.24

6259

15.

8800

607

5.55

4141

05.

2622

791

4.99

9450

6

519.

0465

344

8.30

7587

97.

6772

049

7.13

3909

26.

6613

171

6.24

6775

15.

8803

938

5.55

4356

85.

2624

194

4.99

9542

152

9.05

0931

98.

3103

464

7.67

8942

47.

1350

080

6.66

2014

96.

2472

199

5.88

0678

45.

5545

396

5.26

2537

34.

9996

185

539.

0548

936

8.31

2809

37.

6804

800

7.13

5972

06.

6626

216

6.24

7603

35.

8809

217

5.55

4694

65.

2626

364

4.99

9682

054

9.05

8462

78.

3150

083

7.68

1840

77.

1368

175

6.66

3149

26.

2479

339

5.88

1129

75.

5548

259

5.26

2719

74.

9997

350

559.

0616

781

8.31

6971

77.

6830

449

7.13

7559

26.

6636

080

6.24

8218

95.

8813

074

5.55

4937

25.

2627

896

4.99

9779

2

569.

0645

749

8.31

8724

77.

6841

105

7.13

8209

86.

6640

070

6.24

8464

65.

8814

593

5.55

5031

65.

2628

484

4.99

9816

057

9.06

7184

68.

3202

899

7.68

5053

67.

1387

806

6.66

4353

96.

2486

763

5.88

1589

25.

5551

115

5.26

2897

84.

9998

467

589.

0695

356

8.32

1687

47.

6858

881

7.13

9281

26.

6646

556

6.24

8858

95.

8817

002

5.55

5179

25.

2629

394

4.99

9872

259

9.07

1653

78.

3229

352

7.68

6626

67.

1397

204

6.66

4917

96.

2490

163

5.88

1795

05.

5552

366

5.26

2974

34.

9998

935

609.

0735

619

8.32

4049

37.

6872

802

7.14

0105

66.

6651

460

6.24

9152

05.

8818

761

5.55

5285

35.

2630

036

4.99

9911

3

cont

inue

sov

erle

af

Tabl

eA

.4(c

onti

nued

)

N/i

1112

1314

1516

1718

1920

619.

0752

810

8.32

5044

07.

6878

586

7.14

0443

56.

6653

443

6.24

9269

05.

8819

454

5.55

5326

55.

2630

282

4.99

9926

162

9.07

6829

78.

3259

321

7.68

8370

47.

1407

399

6.66

5516

86.

2493

698

5.88

2004

65.

5553

614

5.26

3048

94.

9999

384

639.

0782

250

8.32

6725

17.

6888

234

7.14

0999

96.

6656

668

6.24

9456

75.

8820

552

5.55

5391

15.

2630

663

4.99

9948

664

9.07

9482

08.

3274

332

7.68

9224

27.

1412

280

6.66

5797

26.

2495

317

5.88

2098

55.

5554

162

5.26

3080

94.

9999

572

659.

0806

144

8.32

8065

37.

6895

790

7.14

1428

16.

6659

106

6.24

9596

35.

8821

354

5.55

5437

45.

2630

932

4.99

9964

3

669.

0816

346

8.32

8629

77.

6898

929

7.14

1603

66.

6660

092

6.24

9651

95.

8821

670

5.55

5455

45.

2631

036

4.99

9970

367

9.08

2553

78.

3291

337

7.69

0170

77.

1417

575

6.66

6095

06.

2496

999

5.88

2194

15.

5554

707

5.26

3112

24.

9999

752

689.

0833

817

8.32

9583

77.

6904

166

7.14

1892

66.

6661

696

6.24

9741

35.

8822

171

5.55

5483

75.

2631

195

4.99

9979

469

9.08

4127

78.

3299

854

7.69

0634

17.

1420

110

6.66

6234

46.

2497

770

5.88

2236

95.

5554

946

5.26

3125

64.

9999

828

709.

0847

997

8.33

0344

17.

6908

267

7.14

2114

96.

6662

908

6.24

9807

85.

8822

537

5.55

5503

95.

2631

308

4.99

9985

7

719.

0854

051

8.33

0664

47.

6909

970

7.14

2206

16.

6663

398

6.24

9834

35.

8822

681

5.55

5511

85.

2631

351

4.99

9988

172

9.08

5950

68.

3309

503

7.69

1147

87.

1422

860

6.66

6382

46.

2498

571

5.88

2280

55.

5555

185

5.26

3138

84.

9999

900

739.

0864

419

8.33

1205

77.

6912

813

7.14

2356

26.

6664

195

6.24

9876

85.

8822

910

5.55

5524

15.

2631

418

4.99

9991

774

9.08

6884

68.

3314

336

7.69

1399

37.

1424

177

6.66

6451

86.

2498

938

5.88

2300

05.

5555

289

5.26

3144

44.

9999

931

759.

0872

835

8.33

1637

27.

6915

038

7.14

2471

76.

6664

798

6.24

9908

55.

8823

077

5.55

5533

05.

2631

465

4.99

9994

2

769.

0876

428

8.33

1818

97.

6915

963

7.14

2519

06.

6665

042

6.24

9921

15.

8823

143

5.55

5536

45.

2631

484

4.99

9995

277

9.08

7966

48.

3319

812

7.69

1678

27.

1425

605

6.66

6525

46.

2499

320

5.88

2319

95.

5555

393

5.26

3149

94.

9999

960

789.

0882

581

8.33

2126

07.

6917

506

7.14

2597

06.

6665

438

6.24

9941

45.

8823

247

5.55

5541

85.

2631

512

4.99

9996

779

9.08

8520

88.

3322

554

7.69

1814

77.

1426

289

6.66

6559

86.

2499

495

5.88

2328

85.

5555

439

5.26

3152

24.

9999

972

809.

0887

575

8.33

2370

97.

6918

714

7.14

2656

96.

6665

738

6.24

9956

45.

8823

323

5.55

5545

75.

2631

531

4.99

9997

7

819.

0889

707

8.33

2474

07.

6919

216

7.14

2681

56.

6665

859

6.24

9962

45.

8823

353

5.55

5547

25.

2631

539

4.99

9998

182

9.08

9162

88.

3325

661

7.69

1966

07.

1427

031

6.66

6596

46.

2499

676

5.88

2337

95.

5555

485

5.26

3154

54.

9999

984

839.

0893

358

8.33

2648

37.

6920

053

7.14

2722

06.

6666

056

6.24

9972

15.

8823

401

5.55

5549

65.

2631

551

4.99

9998

784

9.08

9491

78.

3327

217

7.69

2040

17.

1427

386

6.66

6613

56.

2499

759

5.88

2341

95.

5555

505

5.26

3155

54.

9999

989

859.

0896

322

8.33

2787

27.

6920

709

7.14

2753

26.

6666

205

6.24

9979

35.

8823

435

5.55

5551

25.

2631

559

4.99

9999

1

869.

0897

587

8.33

2845

77.

6920

981

7.14

2765

96.

6666

265

6.24

9982

15.

8823

449

5.55

5551

95.

2631

562

4.99

9999

287

9.08

9872

78.

3328

980

7.69

2122

27.

1427

771

6.66

6631

76.

2499

846

5.88

2346

15.

5555

525

5.26

3156

54.

9999

994

889.

0899

754

8.33

2944

67.

6921

436

7.14

2787

06.

6666

363

6.24

9986

75.

8823

471

5.55

5552

95.

2631

567

4.99

9999

189

9.09

0068

08.

3329

863

7.69

2162

57.

1427

956

6.66

6640

36.

2499

885

5.88

2347

95.

5555

533

5.26

3156

94.

9999

996

909.

0901

513

8.33

3023

57.

6921

792

7.14

2803

16.

6666

437

6.24

9990

15.

8823

486

5.55

5553

75.

2631

571

4.99

9999

6

919.

0902

264

8.33

3056

77.

6921

940

7.14

2809

86.

6666

467

6.24

9991

55.

8823

493

5.55

5554

05.

2631

572

4.99

9999

792

9.09

0294

18.

3330

863

7.69

2207

07.

1428

156

6.66

6649

36.

2499

927

5.88

2349

85.

5555

542

5.26

3157

34.

9999

997

939.

0903

550

8.33

3112

87.

6922

186

7.14

2820

76.

6666

516

6.24

9993

75.

8823

503

5.55

5554

45.

2631

574

4.99

9999

894

9.09

0409

98.

3331

364

7.69

2228

97.

1428

252

6.66

6653

56.

2499

945

5.88

2350

65.

5555

546

5.26

3157

54.

9999

998

959.

0904

594

8.33

3157

57.

6922

379

7.14

2829

16.

6666

552

6.24

9995

35.

8823

510

5.55

5554

75.

2631

575

4.99

9999

8

969.

0905

040

8.33

3176

37.

6922

460

7.14

2832

56.

6666

567

6.24

9995

95.

8823

513

5.55

5554

95.

2631

576

4.99

9999

997

9.09

0544

18.

3331

932

7.69

2253

17.

1428

356

6.66

6658

06.

2499

965

5.88

2351

55.

5555

550

5.26

3157

64.

9999

999

989.

0905

803

8.33

3208

27.

6922

593

7.14

2838

26.

6666

592

6.24

9997

05.

8823

517

5.55

5555

15.

2631

577

4.99

9999

999

9.09

0612

98.

3332

216

7.69

2264

97.

1428

405

6.66

6660

16.

2499

974

5.88

2351

95.

5555

551

5.26

3157

74.

9999

999

100

9.09

0642

28.

3332

336

7.69

2269

87.

1428

426

6.66

6661

06.

2499

978

5.88

2352

05.

5555

552

5.26

3157

74.

9999

999

N/i

2122

2324

2526

2728

2930

10.

8264

463

0.81

9672

10.

8130

081

0.80

6451

60.

8000

000

0.79

3650

80.

7874

016

0.78

1250

00.

7751

938

0.76

9230

82

1.50

9459

71.

4915

345

1.47

3990

31.

4568

158

1.44

0000

01.

4235

324

1.40

7402

81.

3916

016

1.37

6119

21.

3609

467

32.

0739

337

2.04

2241

42.

0113

743

1.98

1303

11.

9520

000

1.92

3438

41.

8955

928

1.86

8438

71.

8419

529

1.81

6112

94

2.54

0441

02.

4936

405

2.44

8271

82.

4042

767

2.36

1600

02.

3201

892

2.27

9994

32.

2409

678

2.20

3064

32.

1662

407

52.

9259

843

2.86

3639

82.

8034

730

2.74

5384

42.

6892

800

2.63

5070

82.

5826

727

2.53

2006

12.

4829

955

2.43

5569

8

63.

2446

152

3.16

6917

83.

0922

545

3.02

0471

32.

9514

240

2.88

4976

82.

8210

021

2.75

9379

72.

6999

965

2.64

2746

07

3.50

7946

43.

4155

064

3.32

7036

13.

2423

156

3.16

1139

23.

0833

149

3.00

8663

12.

9370

154

2.86

8214

42.

8021

123

83.

7255

755

3.61

9267

63.

5179

156

3.42

1222

23.

3289

114

3.24

0726

13.

1564

276

3.07

5793

32.

9986

158

2.92

4701

89

3.90

5434

33.

7862

849

3.67

3102

13.

5655

018

3.46

3129

13.

3656

557

3.27

2777

73.

1842

135

3.09

9702

23.

0190

013

104.

0540

780

3.92

3184

33.

7992

700

3.68

1856

33.

5705

033

3.46

4806

13.

3643

919

3.26

8916

83.

1780

637

3.09

1539

5

114.

1769

239

4.03

5397

03.

9018

455

3.77

5690

63.

6564

026

3.54

3496

93.

4365

290

3.33

5091

33.

2388

091

3.14

7338

112

4.27

8449

54.

1273

746

3.98

5240

33.

8513

634

3.72

5122

13.

6059

499

3.49

3329

93.

3867

900

3.28

5898

53.

1902

601

134.

3623

550

4.20

2766

14.

0530

409

3.91

2389

83.

7800

977

3.65

5515

83.

5380

551

3.42

7179

73.

3224

019

3.22

3277

014

4.43

1698

34.

2645

623

4.10

8163

33.

9616

047

3.82

4078

13.

6948

538

3.57

3271

73.

4587

342

3.35

0699

23.

2486

746

154.

4890

069

4.31

5215

04.

1529

783

4.00

1294

13.

8592

625

3.72

6074

53.

6010

013

3.48

3386

13.

3726

350

3.26

8211

2

164.

5363

693

4.35

6733

64.

1894

132

4.03

3301

73.

8874

100

3.75

0852

73.

6228

357

3.50

2645

43.

3896

396

3.28

3239

417

4.57

5511

84.

3907

653

4.21

9035

24.

0591

143

3.90

9928

03.

7705

180

3.64

0028

13.

5176

917

3.40

2821

43.

2947

995

184.

6078

610

4.41

8660

14.

2431

180

4.07

9930

93.

9279

424

3.78

6125

43.

6535

654

3.52

9446

63.

4130

398

3.30

3692

019

4.63

4595

94.

4415

246

4.26

2697

64.

0967

184

3.94

2353

93.

7985

123

3.66

4224

83.

5386

302

3.42

0961

13.

3105

323

204.

6566

908

4.46

0266

14.

2786

159

4.11

0256

83.

9538

831

3.80

8343

13.

6726

179

3.54

5804

83.

4271

016

3.31

5794

1

cont

inue

sov

erle

af

Tabl

eA

.4(c

onti

nued

)

N/i

2122

2324

2526

2728

2930

214.

6749

511

4.47

5627

94.

2915

577

4.12

1174

83.

9631

065

3.81

6145

33.

6792

267

3.55

1410

03.

4318

617

3.31

9841

622

4.69

0042

24.

4882

196

4.30

2079

44.

1299

797

3.97

0485

23.

8223

375

3.68

4430

53.

5557

891

3.43

5551

73.

3229

551

234.

7025

142

4.49

8540

74.

3106

337

4.13

7080

43.

9763

882

3.82

7252

03.

6885

279

3.55

9210

23.

4384

122

3.32

5350

024

4.71

2821

74.

5070

006

4.31

7588

34.

1428

068

3.98

1110

53.

8311

524

3.69

1754

33.

5618

830

3.44

0629

63.

3271

923

254.

7213

402

4.51

3934

94.

3232

425

4.14

7424

83.

9848

884

3.83

4247

93.

6942

947

3.56

3971

13.

4423

485

3.32

8609

5

264.

7283

804

4.51

9618

84.

3278

395

4.15

1149

13.

9879

107

3.83

6704

73.

6962

950

3.56

5602

43.

4436

810

3.32

9699

627

4.73

4198

64.

5242

777

4.33

1576

84.

1541

525

3.99

0328

63.

8386

545

3.69

7870

13.

5668

769

3.44

4714

03.

3305

382

284.

7390

071

4.52

8096

54.

3346

153

4.15

6574

63.

9922

629

3.84

0202

03.

6991

103

3.56

7872

63.

4455

147

3.33

1183

229

4.74

2981

14.

5312

266

4.33

7085

64.

1585

279

3.99

3810

33.

8414

302

3.70

0086

93.

5686

504

3.44

6135

43.

3316

794

304.

7462

654

4.53

3792

34.

3390

940

4.16

0103

13.

9950

482

3.84

2404

93.

7008

558

3.56

9258

23.

4466

166

3.33

2061

1

314.

7489

797

4.53

5895

34.

3407

268

4.16

1373

53.

9960

386

3.84

3178

53.

7014

613

3.56

9732

93.

4469

896

3.33

2354

732

4.75

1222

94.

5376

191

4.34

2054

34.

1623

980

3.99

6830

93.

8437

924

3.70

1938

03.

5701

039

3.44

7278

83.

3325

805

334.

7530

767

4.53

9032

14.

3431

336

4.16

3224

23.

9974

647

3.84

4279

73.

7023

134

3.57

0393

63.

4475

029

3.33

2754

234

4.75

4608

94.

5401

902

4.34

4011

14.

1638

905

3.99

7971

83.

8446

664

3.70

2609

03.

5706

200

3.44

7676

73.

3328

879

354.

7558

751

4.54

1139

54.

3447

244

4.16

4427

83.

9983

774

3.84

4973

43.

7028

417

3.57

0796

93.

4478

114

3.33

2990

7

364.

7569

216

4.54

1917

64.

3453

044

4.16

4861

13.

9987

019

3.84

5217

03.

7030

250

3.57

0935

13.

4479

158

3.33

3069

737

4.75

7786

44.

5425

554

4.34

5775

94.

1652

106

3.99

8961

53.

8454

103

3.70

3169

33.

5710

430

3.44

7996

73.

3331

306

384.

7585

012

4.54

3078

24.

3461

593

4.16

5492

43.

9991

692

3.84

5563

73.

7032

829

3.57

1127

43.

4480

595

3.33

3177

439

4.75

9091

94.

5435

067

4.34

6471

04.

1657

197

3.99

9335

43.

8456

855

3.70

3372

43.

5711

933

3.44

8108

13.

3332

134

404.

7595

801

4.54

3858

04.

3467

244

4.16

5903

03.

9994

683

3.84

5782

13.

7034

428

3.57

1244

73.

4481

458

3.33

3241

0

414.

7599

835

4.54

4145

94.

3469

304

4.16

6050

83.

9995

746

3.84

5858

83.

7034

983

3.57

1284

93.

4481

751

3.33

3262

342

4.76

0317

04.

5443

819

4.34

7097

94.

1661

700

3.99

9659

73.

8459

197

3.70

3541

93.

5713

164

3.44

8197

73.

3332

787

434.

7605

925

4.54

4575

34.

3472

340

4.16

6266

13.

9997

278

3.84

5968

03.

7035

763

3.57

1340

93.

4482

153

3.33

3291

344

4.76

0820

34.

5447

339

4.34

7344

84.

1663

436

3.99

9782

23.

8460

064

3.70

3603

43.

5713

601

3.44

8228

93.

3333

010

454.

7610

085

4.54

4863

84.

3474

348

4.16

6406

23.

9998

258

3.84

6036

83.

7036

247

3.57

1375

13.

4482

395

3.33

3308

5

464.

7611

640

4.54

4970

34.

3475

079

4.16

6456

63.

9998

606

3.84

6061

03.

7036

415

3.57

1386

83.

4482

476

3.33

3314

247

4.76

1292

64.

5450

577

4.34

7567

44.

1664

972

3.99

9888

53.

8460

801

3.70

3654

73.

5713

959

3.44

8254

03.

3333

186

484.

7613

988

4.54

5129

24.

3476

158

4.16

6530

03.

9999

108

3.84

6095

33.

7036

652

3.57

1403

13.

4482

589

3.33

3322

049

4.76

1486

64.

5451

879

4.34

7655

14.

1665

565

3.99

9928

63.

8461

074

3.70

3673

33.

5714

086

3.44

8262

73.

3333

246

504.

7615

592

4.54

5236

04.

3476

871

4.16

6577

83.

9999

429

3.84

1170

3.70

3679

83.

5714

130

3.44

8265

73.

3333

266

Bibliography

The references cited are those considered the best for a particular insight, irrespective of their date ofpublication. Thus there are many classical references which more recent publications have not improvedupon. This is mentioned for the sake of those who might question vintage references in what purports tobe a modern book.

[1] Alchian, A. A., “The Rate of Interest, Fisher’s Rate of Return over Cost, and Keynes’ — InternalRate of Return,” The American Economic Review (December 1955), pp. 938–948.

[2] Allen, R. G. D., Mathematical Economics, 2nd edn (London: Macmillan Company, 1965).[3] Amram, Martha and Nalin Kulatilaka, Real Options: Managing Strategic Investment in an Uncer-

tain World (Harvard Business School Press, 1999). Their website, http://www.real-options.com, isa rich source of information on real options.

[4] Bacon, Peter, Robert Haessler, and Richard Williams, comment on Trippi’s “Conventional andUnconventional Methods for Evaluating Investments,” Financial Management, Vol. 5 (Spring1975), p. 8.

[5] Balas, Egon, “An Additive Algorithm for Solving the Linear Programs with Zero–One Variables,”Operations Research, Vol. 13 (July–August 1965), pp. 517–546.

[6] Balinski, M. L., “Integer Programming: Methods, Uses, Computation,” Management Science,Vol. 12 (November 1965), pp. 253–309.

[7] Baum, Sanford, Robert C. Carlson, and James V. Jucker, “Some Problems in Applying the Contin-uous Portfolio Selection Model to the Discrete Capital Budgeting Problem,” Journal of Financialand Quantitative Analysis, Vol. XIII, No. 2 (June 1978), pp. 333–344.

[8] Bellman, R. E. and S. E. Dreyfus, Applied Dynamic Programming (Princeton, NJ: PrincetonUniversity Press, 1962).

[9] Berenson, Mark L., David M. Levine, and Matthew Goldstein, Intermediate Statistical Methodsand Applications (Englewood Cliffs, NJ: Prentice-Hall, 1983).

[10] Bernado, John J. and Howard P. Lanser, “A Capital Budgeting Model with Subjective Criteria,”Journal of Financial and Quantitative Analysis, Vol. XII, No. 2 (June 1977), pp. 261–275.

[11] Bierman, Harold, Jr., “Leveraged Leasing: An Alternative Analysis,” The Bankers’ Magazine,Vol. 157 (Summer 1974).

[12] Bierman, Harold, Jr., Charles P. Bonnini, and Warren H. Hausman, Quantitative Analysis forBusiness Decisions, 4th edn (Homewood, Ill.: Irwin, 1973).

[13] Black, Fisher, “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business,Vol. 45 (July 1972), pp. 444–455.

[14] Blatt, John M., “Investment Evaluation under Uncertainty,” Financial Management, Vol. 8, No. 2(Summer 1979), pp. 66–81.

[15] Boulding, K. E., “Time and Investment,” Economica, Vol. 3 (1936), pp. 196–220.[16] Bower, Dorothy H., Richard S. Bower and Dennis E. Logue, “A Primer on Arbitrage Pricing The-

ory,” Midland Corporate Finance Journal (a publication of Stern Stewart Management Services,Inc.), Vol. 2, No. 3 (Fall 1984), pp. 31–40. Reprinted in Financial Management Collection, Vol. 2,No. 3 (Autumn 1987), pp. 1, 6–9.

308 Bibliography

[17] Bower, R., “Issues in Lease Financing,” Financial Management (Winter 1973), pp. 25–34.[18] Brennan, M. J., “An Approach to the Valuation of Uncertain Income Streams,” The Journal of

Finance, Vol. XXVIII, No. 3 (June 1973), pp. 661–674.[19] Brick, John R. and Howard E. Thompson, “The Economic Life of an Investment and the Appropriate

Discount Rate,” Journal of Financial and Quantitative Analysis, Vol. XIII, No. 5 (December 1978),pp. 831–846.

[20] Campanella, Joseph A., “Leveraged Leasing, Financing and the Commercial Bank,” Master’sthesis, The Stonier Graduate School of Banking, Rutgers University, June 1974.

[21] Charnes, A., W. W. Cooper, and Y. Ijiri, “Breakeven Budgeting and Programming to Goals,”Journal of Accounting Research, Vol. 1 (Spring 1963), pp. 16–43.

[22] Childs, C. Rogers, Jr. and William G. Gridley, Jr., “Leveraged Leasing and the Reinvestment RateFallacy,” The Bankers’ Magazine, Vol. 156 (Winter 1973), pp. 53–61.

[23] Converse, A.O., Optimization (New York: Holt, Rinehart and Winston, 1970).[24] Cooper, D. O., L. B. Davidson, and W. K. Denison, “A Tool for More Effective Financial Analysis,”

Interfaces, Vol. 5, No. 2, Pt. 2 (February 1975), pp. 91–103.[25] Copeland, Thomas E., and Fred J. Weston, Financial Theory and Corporate Policy (Reading Mass.:

Addison-Wesley, 1979).[26] Cyert, Richard M. and James G. March, A Behavioral Theory of the Firm (Englewood Cliffs, NJ:

Prentice-Hall, 1963).[27] Daellenbach, Hans C. and Earl J. Bell, User’s Guide to Linear Programming (Englewood Cliffs,

NJ: Prentice-Hall, 1970).[28] Dean, Joel and Winfield Smith, “Has MAP! a Place in a Comprehensive System of Capital

Controls?,” Journal of Business (October 1955).[29] Derman, C., G. J. Lieberman, and S. M. Ross, “A Renewal Decision Problem,” Management

Science, Vol. 24, No. 5 (January 1978), pp. 554–561.[30] Dias, M. A. G., “A Note on Bibliographical Evolution of Real Options,” in L. Trigeorgis (ed.),

Real Options and Business Strategy—Applications to Decision Making (Risk Books, 1999), pp.357–362 or his Monte Carlo page at http://www.puc-rio-br/marco.ind /monte-carlo.html.

[31] Dias, Marco, Real Options in Petroleum http://www.puc-rio.br/marco.ind / (The title of this websiteis somewhat of a misnomer. He has put together what is one of the richest sources of informationabout real options, including presentation graphics and links to other sources.

[32] Durand, David, “The Cost of Capital, Corporation Finance and the Theory of Investment,” AmericanEconomic Review (June 1958), pp. 361–397.

[33] Durand, David, “The Cost of Debt and Equity Funds for Business,” reprinted in E. Solomon,Management of Corporate Capital (Chicago: University of Chicago Press, 1959).

[34] Durand, David, “The Cost of Capital in an Imperfect Market: A Reply to Modigliani and Miller,”American Economic Review (June 1959).

[35] Durand, David, “Time as a Dimension of Investment,” Proceedings of the Eastern Finance Asso-ciation (April 1973), pp. 200–219.

[36] Durand, David, “Problems with Concept of the Cost of Capital,” Journal of Financial and Quan-titative Analysis (December 1978), pp. 847–870.

[37] Dyer, James S., “Interactive Goal Programming,” Management Science (September 1972),pp. 62–70.

[38] Dyl, Edward A. and Hugh W. Long, “Abandonment Value and Capital Budgeting: Comment,” TheJournal of Finance, Vol. 24 (March 1969), pp. 88–95.

[39] Erdogmus, Hakan, A Software Engineering Resource List http://wwwsel.iit.nrc.ca/˜erdogmus/Software Engineering Group of Institute for Information Technology, National Research Council ofCanada. This site has excellent tutorials on the binomial option pricing model, including valua-tion of multiple real options: http://wwwsel.iit.nrc.ca/˜erdogmus/Favorites/SEEFavs.html

[40] Everett, James E. and Bernhard Schwab, “On the Proper Adjustment for Risk through DiscountRates in a Mean–Variance Framework,” Financial Management, Vol. 8, No. 2 (Summer 1979),pp. 7–23.

[41] ExcelTM workbook for Wiar method and MISM, from Actuarial Studies at Macquarie University:http://www.acst.mq.edu.au/unit info/ACST827/levlease.xls

[42] Fama, Eugene F., “Risk, Return, and Equilibrium: Some Clarifying Comments,” The Journal ofFinance, Vol. XXIII, No. 1 (March 1968), pp. 29–40.

Bibliography 309

[43] Fama, Eugene F., Foundations of Finance (New York: Basic Books, 1976).[44] Fama, Eugene F., “Risk Adjusted Discount Rates and Capital Budgeting under Uncertainty,” Jour-

nal of Financial Economics (August 1977), pp. 1–24.[45] Fama, Eugene F. and James D. MacBeth, “Risk, Return and Equilibrium: Empirical Test,” Journal

of Political Economy, Vol. 71 (May–June 1973), pp. 607–636.[46] Farrar, Donald E., The Investment Decision under Uncertainty (Englewood Cliffs, NJ: Prentice-

Hall, 1962).[47] Federal Reserve Bank of St Louis, International Economic Trends, August 2001. Website at

http://www.stls.frb.org[48] Feller, William, An Introduction to Probability Theory and Its Applications (New York: Wiley,

1957).[49] Flatto, Jerry (http://www.puc-rio.br/marco.ind/LOMA96.html). An article written for Resource—

The Magazine for Life Insurance Management Resource. Published by the Life Office ManagementAssociation (LOMA), the education arm of the insurance industry.

[50] Forrester, Jay W., Industrial Dynamics (Cambridge, Mass.: MIT Press, 1961).[51] Galbraith, John K., The New Industrial State (Boston: Houghton Mifflin, 1967).[52] Gaumnitz, Jack E. and Douglas R. Emery, “Asset Growth Abandonment Value and the Replacement

Decision of Like-for-Like Capital Assets,” Journal of Financial and Quantitative Analysis, Vol.XV, No. 2 (June 1980), pp. 407–419.

[53] Geoffrion, A. M., “An Improved Implicit Enumeration Approach for Integer Programming,” Op-erations Research, Vol. 17 (May–June 1969), pp. 437–454.

[54] Glassman, Gerald F., “Discounted Cash Flow versus the Capital Asset Pricing Model,” PublicUtilities Fortnightly, Vol. 102 (September 14, 1978), pp. 30–34.

[55] Gomory, R. E., “All-Integer Programming Algorithm,” in Muth and Thompson (eds)., IndustrialScheduling (Englewood Cliffs, NJ: Prentice-Hall, 1963), Ch. 13.

[56] Gordon, Myron, The Investment, Financing and Valuation of the Corporation (Homewood, Ill.:Irwin, 1962).

[57] Haley, Charles W. and Lawrence D. Schall, The Theory of Financial Decisions, 2d edn (New York:McGraw-Hill, 1979).

[58] Hawkins, Clark A. and Richard A. Adams, “A Goal Programming Model for Capital Budgeting,”Financial Management (Spring 1974), pp. 52–57.

[59] Henderson, Glenn V., “In Defense of the Weighted Average Cost of Capital,” Financial Manage-ment, Vol. 8, No. 3 (Autumn 1979), pp. 57–61.

[60] Herbst, Anthony F., “Truth in Lending, Regulation Z: Comments on Closed-End Contract InterestRate Disclosure Requirements,” Journal of Bank Research, Vol. 3, No. 2 (December 1972).

[61] Herbst, Anthony F., “An Algorithm for Systematic Economic Analysis of Abandonment Value inCapital Budgeting,” The Engineering Economist, Vol. 22 (Fall 1976), pp. 63–71.

[62] Herbst, Anthony F., “The Unique, Real Internal Rate of Return: Caveat Emptor!”. Reprintedwith permission of the Journal of Financial and Quantitative Analysis, copyright c© 1978. Withcorrections.

[63] Herbst, Anthony F., Excel Program for Black–Scholes Option Pricing http://www.utep.edu/futures/Programs-Templates-General/

[64] Herbst, Anthony F. and Sayeed Hasan, “A Simulation Analysis of Primary Recovery OilfieldReinvestment Management,” Proceedings of the Eighth Annual Meeting of the American Institutefor Decision Sciences, November 10–12, 1976, pp. 94–96.

[65] Hertz, David B., “Risk Analysis in Capital Investment,” Harvard Business Review, Vol. 42, No. 1(January–February 1964), pp. 95–106.

[66] Hertz, David B., “Investment Policies That Pay Off,” Harvard Business Review, Vol. 46, No. 1(January–February 1968), pp. 96–108.

[67] Hess, S. W. and Quigley, H. A., Analysis of Risk in Investments Using Monte Carlo Techniques,Chemical Engineering Symposium Series 42: Statistics and Numerical Methods in Chemical En-gineering (New York: American Institute of Chemical Engineers, 1963).

[68] Hillier, Frederick S., “The Derivation of Probabilistic Information for the Evaluation of RiskyInvestments,” Management Science, Vol. 9 (April 1963), pp. 443–457.

[69] Hirschleifer, J., “On Multiple Rates of Return: Comment,” The Journal of Finance, Vol. XXXIV,No. 1 (March 1969).

310 Bibliography

[70] Holthausen, Duncan M. and John S. Hughes, “Commodity Returns and Capital Asset Pricing,”Financial Management, Vol. 7, No. 2 (Summer 1978), pp. 37–44.

[71] Horvath, P. A., “A High Quality Heuristic for the Resolution of Zero–One Quadratic Programswith Interdependence Constraints,” a working paper, June 1978.

[72] Huffman, Stephen, “Capital Budgeting Survey Preliminary Results,” (1997), http://www.uwosh.edu/faculty staff/huffman/CAPBUD1.HTM

[73] Hughes, John S. and Wilbur C. Lewellen, “Programming Solutions to Capital Rationing Problems,”Journal of Business Finance and Accounting, Vol. 1 (Spring 1974), pp. 55–74.

[74] Ignizio, J. P., Goal Programming and Extensions (Lexington, Mass.: Lexington Books, 1976).[75] Ijiri, Y., Management Goals and Accounting for Control (Chicago: Rand-McNally, 1965).[76] Jean, W. H., “On Multiple Rates of Return,” The Journal of Finance, Vol. XXIII, No. 1 (March

1968).[77] Jean, W. H., “Reply,” The Journal of Finance, Vol. XXIV, No. 1 (March 1969).[78] Kamstra, Mark, “Fundamental Valuation of Zero-Dividend Firms”. Working paper, Department of

Economics, Simon Fraser University, June, 2000.[79] Kemna, A., “Case Studies on Real Options,” Financial Management, Vol. 22 (1993), pp. 259–270.

http://www.fma.org/finmgmt/FMFall1993.pdf[80] Knight, Frank H., Risk, Uncertainty and Profit (Boston: Houghton Mifflin, 1921).[81] Kogut, B. and N. Kulatilaka, “Options Thinking and Platform Investments: Investing in Opportu-

nity,” California Management Review (Winter 1994), pp. 52–71.[82] Kryzanowski, L., P. Lusztig, and B. Schwab, “Monte Carlo Simulation and Capital Expenditure

Decisions—a Case Study,” Engineering Economist, Vol. 18 (February 1972).[83] Lawler, E. L. and M. D. Bell, “A Method for Solving Discrete Optimization Problems,” Operations

Research, Vol. 14 (November–December 1966), pp. 1098–1112.[84] Lee, James, The Gold and the Garbage in Management Theories and Prescriptions (Athens, Ohio:

Ohio University Press, 1980).[85] Lee, Sang, Goal Programming for Decision Analysis (Philadelphia: Averback, 1972).[86] Lee, Sang and A. J. Lerro, “Capital Budgeting for Multiple Objectives,” Financial Management

(Spring 1974), pp. 58–66.[87] Lemke, C. E. and K. Spielberg, “Direct Search Algorithms for Zero–One and Mixed Integer

Programming,” Operations Research, Vol. 15 (September–October 1967), pp. 892–914.[88] Levy, Robert E., “Fair Return on Equity for Public Utilities,” Business Economics, Vol. XIII, No.

4 (September 1978), pp. 46–57.[89] Lewellen, Wilbur C. and Michael S. Long, “Simulation versus Single-value Estimates in Capital

Expenditure Analysis,” Decision Sciences, Vol. 3, No. 4 (October 1972), pp. 19–33.[90] Lintner, John, “Security Prices, Risk, and Maximal Gains from Diversification,” The Journal of

Finance, Vol. XX, No. 5 (December 1965), pp. 587–615.[91] Lintner, John, “The Aggregation of Investors’ Diverse Judgments and Preferences in Purely

Competitive Security Markets,” Journal of Financial and Quantitative Analysis, Vol. IV, No. 6(December 1969), pp. 347–400.

[92] Luehrman, Timothy A., “Capital Projects as Real Options: An Introduction,” Harvard BusinessSchool case 9-295-074, Rev. March 22, 1995.

[93] Luken, Jack A. and David P. Stuhr, “Decision Trees and Risky Projects,” The EngineeringEconomist, Vol. 24, No. 2 (Winter 1979), pp. 75–86.

[94] Macaulay, Frederick R., Some Theoretical Problems Suggested by the Movement of Interest Rates,Bond Yields and Stock Prices in the United States since 1856 (New York: National Bureau ofEconomic Research, 1938).

[95] Magee, John F., “Decision Trees for Decision Making,” Harvard Business Review, Vol. 42 (July–August 1964), pp. 126–138.

[96] Makridakis, Spyros, Steven C. Wheelwright, and Rob J. Hyndman, Forecasting: Methods andApplications, 3rd edn. (New York: John Wiley & Sons, 1997).

[97] Mao, J. C.T., “Financing Urban Change in the Great Society: Quantitative Analysis of UrbanRenewal Investment Decisions,” The Journal of Finance, Vol. XXII, No. 2 (May, 1967).

[98] Mao, James C. T., Quantitative Analysis of Financial Decisions (New York: Macmillan, 1969).[99] Mao, James C. T., “Survey of Capital Budgeting: Theory and Practice,” The Journal of Finance,

Vol. XXV, No. 2 (May 1970), pp. 349–360.

Bibliography 311

[100] Mao, James C. T., Corporate Financial Decisions (Palo Alto, Calif: Pavan, 1976).[101] Markowitz, Harry M., Portfolio Selection: Efficient Diversification of Investments (New York:

Wiley, 1962).[102] Masse, Pierre, Optimal Investment Decisions (Englewood Cliffs, NJ: Prentice-Hall, 1962).[103] Meisner, James F. and John W. Labuszewski, “Modifying the Black–Scholes Option Pricing Model

for Alternative Underlying Instruments,” Financial Analysts Journal (November–December 1984)pp. 23–30.

[104] Metzger, Robert W., Elementary Mathematical Programming (New York: Wiley, 1958).[105] Miller, David W. and Martin K. Starr, Executive Decisions and Operations Research, 2nd edn

(Englewood Cliffs, NJ: Prentice-Hall, 1969).[106] Miller, M. H. and F. Modigliani, “Dividend Policy, Growth and the Valuation of Shares,” Journal

of Business, Vol. XXIV (October 1961), pp. 411–433.[107] Miller, Merton and Myron Scholes, “Rates of Return in Relation to Risk: A Re-Examination of

Some Recent Findings,” in M. C. Jensen (ed.), Studies in the Theory of Capital Markets (NewYork: Praeger, 1972), pp. 47–78.

[108] Modigliani, F. and M. H. Miller, “The Cost of Capital, Corporation Finance, and the Theory ofInvestment,” American Economic Review (June 1958), pp. 361–397.

[109] Modigliani, F. and M. H. Miller, “Corporate Income Taxes and the Cost of Capital: A Correction,”American Economic Review, Vol. LIII (June 1963).

[110] Modigliani, Franco and Gerald Pogue, “An Introduction to Risk and Return,” Financial AnalystsJournal (March/April 1974), pp. 68–80 and (May/June 1974), pp. 69–85.

[111] Mossin, Jan, “Equilibrium in a Capital Asset Market,” Econometrica, Vol. 34 (October 1966),pp. 768–783.

[112] Myers, Stewart C., “Interactions of Corporate Financing and Investment Decisions—Implicationsfor Capital Budgeting,” The Journal of Finance, Vol. XXIX, No. 1 (March 1974), pp. 1–25.

[113] Myers, Stewart C., “Postscript: Using Simulation for Risk Analysis,” Modern Developments inFinancial Management (Hinsdale, Ill.: Dryden, 1976), pp. 457–463.

[114] Myers, Stewart C., David A. Dill, and Alberto J. Bautista, “Valuation of Financial Lease Contracts,”Journal of Finance, Vol. XXXI (June 1976), pp. 799–820.

[115] Myers, Stewart C. and Stuart W. Turnbull, “Capital Budgeting and the Capital Asset PricingModel: Good News and Bad News,” The Journal of Finance, Vol. XXXII, No. 2 (May 1977),pp. 321–332.

[116] National Association of Accountants, Financial Analysis Guide to Capital Expenditure Decisions,Research Report 43, 1967, Washington, DC.

[117] Naylor, Thomas, Computer Simulation Experiments with Models of Economic Systems (New York:Wiley, 1971).

[118] Naylor, Thomas, J. L. Balintfy, D. S. Burdick, and K. Chu, Computer Simulation Techniques (NewYork: Wiley, 1966).

[119] Nichols, N., “Scientific Management at Merck: An Interview with Judy Lewent,” Harvard BusinessReview (January/February 1994), pp. 89–99. http://www.cbi.cgey.com/pub/docs/Journal Issue2.pdf

[120] Pappas, James L., “The Role of Abandonment Value in Capital Asset Management,” The Engi-neering Economist, Vol. 22 (Fall 1976), pp. 53–61.

[121] Park, Chan S. and Gerald J. Thuesen, “Combining the Concepts of Uncertainty Resolution andProject Balance for Capital Allocation Decisions,” The Engineering Economist, Vol. 24, No. 2(Winter 1979), pp. 109–127.

[122] Parkinson, C. Northcote, East and West (New York: New American Library, 1963).[123] Parrino, Robert, Allen M. Poteshman, and Michael S. Weisbach (2002), “Measuring Investment

Distortions when Risk-averse Managers Decide Whether to Undertake Risky Projects,” workingpaper: http://papers.ssrn.com/paper.taf?abstract id=299186

[124] Perg, Wayne F. and Anthony F. Herbst, “Lease vs. Conventional Financing: Integrating the Lessor’sGains and Losses,” Proceedings of the 1980 Meeting, Eastern Finance Association.

[125] Pellens, Bernhard, Real Options in Germany http://www.iur.ruhr-uni-bochum.de/forschung/realoptions.html has a downloadable Excel application using the binomial option valuation model. Itis useful for many real options.

312 Bibliography

[126] Perrakis, Stylianas, “Capital Budgeting and Timing Uncertainty within the CAPM,” FinancialManagement, Vol. 8, No. 3 (Autumn 1979), pp. 32–40.

[127] Pettway, Richard H., “Integer Programming in Capital Budgeting: a Note on ComputationalExperience,” Journal of Financial and Quantitative Analysis (September 1973), pp. 665–672.

[128] Redington, F. M., “Review of the Principles of Line-Office Valuations,” Journal of the Institute ofActuaries, Vol. 78 (1952).

[129] Rendleman, Richard J., Jr., “Ranking Errors in CAPM Capital Budgeting Applications,” FinancialManagement, Vol. 7, No. 4 (Winter 1978), pp. 40–44.

[130] Robichek, Alexander A. and James C. Van Home, “Abandonment Value and Capital Budgeting,”The Journal of Finance, Vol. 22 (December 1967), pp. 577–589.

[131] Robichek, Alexander A. and James C. Van Home, “Abandonment Value and Capital Budgeting:Reply,” The Journal of Finance, Vol. 24 (March 1969), pp. 96–97.

[132] Robinson, E. A. G., The Structure of Competitive Industry (Chicago: University of Chicago Press,1958).

[133] Richard Roll, “A Critique of the Asset Pricing Theory’s Tests; Part I: On Past and Poten-tial Testability of the Theory,” Journal of Financial Economics, Vol. 4, No. 2 (March 1977),pp. 129–176.

[134] Ross, Stephen A., “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory,Vol. 13 (1976), pp. 341–360.

[135] Roy, A. D., “Safety First and the Holding of Assets,” Econometrica, Vol. 20 (July 1952),pp. 431–449.

[136] Rychel, Dwight F., “Capital Budgeting with Mixed Integer Linear Programming: An Application,”Financial Management, Vol. 6, No. 4 (Winter 1977), pp. 11–19.

[137] Salazar, Rodolfo and Subrata K. Sen, “A Simulation Model of Capital Budgeting under Uncer-tainty,” Management Science, Vol. 15, No. 4 (December 1968), pp. B161–B179.

[138] Salkin, Harvey M., Integer Programming (Reading, Mass.: Addison-Wesley, 1975).[139] Schall, Lawrence D. and Gary L. Sundem, “Capital Budgeting Methods and Risk,” Financial

Management, Vol. 9, No. 1 (Spring 1980), pp. 7–11.[140] Shamblin, James E. and G. T. Stevens, Jr., Operations Research (New York: McGraw-Hill, 1974).[141] Sharpe, William F., “A Simplified Model for Portfolio Analysis,” Management Science, Vol. IX,

No. 1 (January 1963), pp. 277–293.[142] Sharpe, William F., “Risk Aversion in the Stock Market: Some Empirical Evidence,” The Journal

of Finance, Vol. 20 (September 1963), pp. 416–422.[143] Sharpe, William F., “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of

Risk,” The Journal of Finance, Vol. XIX, No. 3 (September 1964), pp. 425–442.[144] Shillinglaw, Gordon, “Profit Analysis for Abandonment Decisions,” Journal of Business (January

1957). Reprinted in Ezra Solomon (ed.), The Management of Corporate Capital (Chicago: GraduateSchool of Business, University of Chicago, 1959), pp. 269–281.

[145] Simon, Herbert A., Administrative Behavior, 2nd edn (New York: Macmillan, 1957), p. xxiv.[146] Solomon, Ezra, The Theory of Financial Management (New York: Columbia University Press,

1963).[147] Stone, Bernell K., “A General Class of Three-parameter Risk Measures,” The Journal of Finance,

Vol. XXVIII, No. 3 (June 1973), pp. 675–685.[148] Teichroew, Daniel, An Introduction to Management Science: Deterministic Models (New York:

Wiley, 1964).[149] Teichroew, Daniel, Alexander Robichek, and Michael Montalbano, “Mathematical Analysis of

Rates of Return under Certainty,” Management Science, Vol. 11 (January 1965).[150] Teichroew, Daniel, Alexander Robichek, and Michael Montalbano, “An Analysis of Criteria for

Investment and Financing Decisions under Certainty,” Management Science, Vol. 12 (November1965).

[151] Terborgh, George, Dynamic Equipment Policy (Washington: Machinery and Allied ProductsInstitute, 1949).

[152] Terborgh, George, MAPI Replacement Manual (Washington: Machinery and Allied ProductsInstitute, 1950).

[153] Terborgh, George, “Some Comments on the Dean-Smith Article on the MAP! Formula,” Journalof Business (April 1956).

Bibliography 313

[154] Terborgh, George, Business Investment Analysis (Washington: Machinery and Allied ProductsInstitute, 1958).

[155] Terborgh, George, Studies in the Analysis of Business Investment Projects (Washington: Machineryand Allied Products Institute), consisting of a set of six volumes dated 1960–61.

[156] Terborgh, George, Studies in Business Investment Strategy (Washington: Machinery and AlliedProducts Institute), consisting of a set beginning in 1963.

[157] Terborgh, George, Business Investment Management (Washington: Machinery and Allied ProductsInstitute, 1967).

[158] Teweles, Richard J., Charles V. Harlow, and Herbert L. Stone, The Commodity Futures Game (NewYork: McGraw-Hill, 1977).

[159] Tobin, James, “Liquidity Preference as Behavior Towards Risk,” The Review of Economic Studies,Vol. 67 (February 1958), pp. 65–86.

[160] Trigeorgis, Lenos, “The Nature of Option Interactions and the Valuation of Investments withMultiple Real Options,” The Journal of Financial and Quantitative Analysis, Vol. 28, No. 1 (March1993), 1–20.

[161] Trigeorgis, Lenos, Real Options (The MIT Press, 1996).[162] Trippi, Robert R., “Conventional and Unconventional Methods for Evaluating Investments,”

Financial Management, Vol. 3 (Autumn 1974), pp. 31–35.[163] Trippi, Robert R., “Reply,” Financial Management (Spring 1975), pp. 8–9.[164] Van Home, James C., “An Application of the CAPM to Divisional Required Returns,” Financial

Management, Vol. 9, No. 1 (Spring 1980), pp. 14–19.[165] Varian, H. “The Arbitrage Principle in Financial Economics,” Journal of Economic Perspectives

(Winter 1987).[166] Vinso, Joseph D., “A Determination of the Risk of Ruin,” Journal of Financial and Quantitative

Analysis, Vol. XIV, No. 1 (March 1979), pp. 77–100.[167] Von Neumann, John and Oskar Morgenstern, The Theory of Games and Economic Behavior

(Princeton, NJ: Princeton University Press, 1947).[168] Weil, Roman L., “Macaulay’s Duration: An Appreciation,” Journal of Business, Vol. 46 (October

1973), p. 590.[169] Weingartner, H. Martin, “Capital Budgeting of Interrelated Projects: Survey and Synthesis,”

Management Science, Vol. 12 (March 1966), pp. 485–516.[170] Weingartner, H. Martin, “Some New Views on the Payback Period and Capital Budgeting

Decisions,” Management Science, Vol. 15 (August 1969).[171] Wiar, Robert, “Economic Implications of Multiple Rates of Return in the Leveraged Lease Context,”

The Journal of Finance, Vol. 28, No. 5 (1973), 1275–1286.

Index

abandon, real option to, 272abandonment analysis, 149–158accounting method of investment return

measurement, 43adverse minimum, defined, 80algorithm, defined, 41American Council for Capital Formation, 2annuity factor, 48, 56 footnotesAPT and CAPM, 236arbitrage pricing theory, APT, 234–237average discounted rate of return, 142average return on average investment, 43

Bacon, P., 143Balas, E., 162basic valuation model, 11Baum, 221, 222, 248Bell, E.J., 163 footnoteBell, M.D., 162beta, 228binomial option pricing model, 260, 261Black, F., 262 footnoteBlack-Scholes model, 258, 262“borrowing rate”, two-stage method, 117Bower, D.H. and R.S., 237

Campanella, J.A., 131Canada, 1capacity disparities, 82capital budgeting, defined, 5capital cost, defined, 77capital market line, 227capital rationing, defined, 11capital recovery, 118capital recovery factor, 53 footnote, 78capital structure, optimal, 36capital, defined, 1CAPM, capital asset pricing model, 33, 38, 202,

222, 225–237, 252 footnote

CAPM, compared with portfolio approaches, 232CAPM, some criticisms, 233–234Carlson, 221, 222cash flows, 6, 12, 19cash flows, quarterly, 64cash outflows, types of, 22certainty equivalents, 192–194challenger, defined, 77change scale, option to, 254Childs and Gridley leveraged lease, 112coefficient of variation, 186, 242component projects, defined, 20computer simulation, 195–214conflicting rankings, NPV vs. IRR, 68constraint specification, 164constraints, post-optimization, 175contingent claim analysis, CCA, 260contingent projects, 161contingent relationships, 215Corel Corporation, 259correlation coefficient, 239cost minimization, 13cost of capital, 6, 31

and CAPM, 231common stock, 33debt, 31marginal, 47marginal, 35preferred stock, 32

cost, initial, 15cost, opportunity, 66cost, sunk, 15cost-benefit analysis, defined, 5cumulative variance, 254Cyert, R.M., 9

Daellenbach, H.G., 163 footnotedecision trees, 203defender, defined, 77

316 Index

delay, real option to, 254depreciation, 23

ACRS, 24, 27, 28ADR, 23, 26Canadian, 28double-declining balance, 24general guidelines (GG), 23modified accelerated cost recovery (MACRS),

24straight line, 24sum-of-year’s-digits, 24

Descartes’ rule, 55, 59 footnote, 85deterioration, physical, 18deviational variables, 168Dias, M.A.G., 251, footnotedividend capitalization model, 33dominated point, 221, 222dual problem formulation, 13duality, defined, 75Durand, D., 146duration, Macaulay’s, 146Dyl, E.A., 149dynamic programming, 156

efficient frontier, 247efficient set, 220, 225evolution, of real option, 260Excel, answer report, 174Excel, solution model, 173excess present value, defined, 70existing firm assets, need to include, 221

factor loadings, 236factor rotation, 236factors, factor analysis, 235Federal Reserve Bank of St. Louis, 1first standard assumption, MAPI, 80Fisher’s intersection, rate, 68, 70, 71Flatto, J., 254forecasting, 19framing, real option, 255, 257France, 1future value, project, 87

Galbraith, J.K., 9geometric mean rate of return, 141Germany, 1goal programming, 165Gomory, R.E., 162Gordon model, 33Gross Domestic Product (GDP), 1

Haessler, R., 143Harlow, C.V., 184Hasan, S., 207 footnoteHirschleifer, J., 94

Holthausen, D.M., 233Horvath, P.A., 162housing, 3 footnoteHughes, J.S., 233

IIM, 99, 100immunization, 146independence, project, 217, 242indivisibility, project, 217, 243inflation, 30initial cost, components, 16initial investment method rate of return, 99, 100integer linear programming, 162interaction of financing and investment, 38internal rate of return

adjusted or modified IRR, 73deficiencies, 85IRR, 49, 145, 147IRR, calculating, 60IRR, caution and rule for, 58IRR, defined, 55IRR, uniqueness, 94

interval bisection method for IRR, 60investment curve, 184investment tax credit, 29investment, unrecovered, 43Italy, 1

Japan, 1Jean, W.H., 94Jucker, J.V., 221, 222

Koopmans, T.C., 146

lattice or tree, binomial option model, 261Lawler, E.L., 162lease

alternative analysis, 122derivation of generalized valuation, 128financial, 119leveraged, analysis, 132–139leveraged, defined, 131operating, 119traditional analysis, 120

leasing, 119–129alleged advantages, 119practical perspective, 127

Lee, J.A., 9 footnotelevel annuities, 53leveraged lease, see lease, leveragedLevy, R.E., 234Lewellen, W.G., 200–202linear programming formulation, 76linear programming, general approach, 159Linux, 259“loan rate”, two-stage method, 112

Index 317

Logue, D.E., 237Long, H.W., 149Long, M.S., 200–202Luehrman, T.A., 256

Magee, J.F., 203magnitude of capital investment, 1major projects, defined, 20Mao, J.C.T., 33, 72, 73, 87, 94, 95, 187, 188, 190,

220–222MAPI, 75, 151, 158

basic assumptions, 79, 80capacity disparities, 82

March, J.G., 9market portfolio, 227Markov chain, 203Markowitz, H.M., 207, 218–219, 221–223,

232–233, 246Masse, P., 203Microsoft, 33MISFM, 99, 107mixed cash flows, defined, 87mixed project, defined, 88Modigliani and Miller (MM), 36Monte Carlo simulation, 258, 274multiple investment sinking fund method,

MISFM, 99, 107multiple project selection, 218–219mutual exclusivity, 161, 192Myers, Dill, and Bautista lease model, 122, 123,

127Myers, S., 251Myers, S.C., 234

net present value, see NPVNewton-Raphson method for IRR, 61normative model, for capital budgeting, 10NPV, 47

as a function of cost of capital, 105relationship to two-stage method, 115

NPVq, 254

obsolescence, technological, 19operating inferiority, 17, 77, 78optimizer, spreadsheet, 171option exercise, decision rule, 255option, real, see real optionown risk, 202owner trustee, 131

Pappas, J.L., 157payback, 39, 191payback, two-stage method, 112Perg, W.F., 123 footnote, 127perpetuity, 33 footnotePettway, R.H., 162

phase diagram, option, 256physical deterioration, 79polynomial equation, 86portfolio return, overall, 247portfolio risk, 247portfolio selection approaches to selection,

215–224“price of risk”, 228principal components, 236profitability index, 51profitability index, 253project balance equations, defined, 87project

characteristics, 14financing rate, 88investment rate, 88life, 17types, 13types, defined, 20

public sector, cash flows, 12pure investments, defined, 88

qualified investment, under tax code, 29qualitative considerations, 14quotient, 254

rate of return, naıve, 41rates: nominal vs. effective, 62real option, defined, 251real options, types of, 254reinvestment rate, assumptions, 65Rendelman, R.J., 233return on invested capital, RIC, 88risk, 7

attitudes toward, 178, 187–188aversion, 177measures of, 185other considerations, 207of ruin, 188–189

risk-adjusted discount rate, 194–195, 249risk-free asset, 226riskiness, relative, 216risk-neutral asset, 260risk-neutral probability, 260, 263, 264rmin defined, 88Robichek, R.R., 149Robinson, E.A.G., 5Roll, R., 234Ross, S.A., 235Roy, A.D., 220

“satisficing”, 9saving & investment, 1second standard assumption, MAPI, 80securities, 247security market line, 228

318 Index

semi-variance, 186Sharpe, W.F., 222Simon, H., 9simple investments, defined, 88simulation, oilfield example, 207–214sinking fund earnings rate, 99sinking fund method rate of return, 99, 100sinking fund methods, 99solver, spreadsheet, 170spreadsheet optimization, 170standard error of the estimate, 186stochastic independence, 215Stone, H.L., 184strategic investments, 252sunk cost, 15switch production function, mix, etc., 254switch production, option to, 254systematic risk, 229

taxes, 23techological obsolescence, 79, 19Teichroew, D., 157Teichroew, Robicheck & Montalbano, TRM, 87,

97, 111Terborgh, G., 14, 19, 42, 75, 79, 80, 158Teweles, R.J., 184time spread, Boulding’s, 145time-related measures, 145–148tracking, asset or instrument, 260trade-off functions, 167, 168Trippi, R.R., 142, 144, 148TRM algorithm, defined, 89TSFM, 99, 100Turnbull, S.W., 234

two-stage method of analysis, 111, 112two-stage method, relationship to NPV, 115

Uncertainty, 7, 8uncertainty, a brief digression, 118undepreciated balance, 26unequal project lives, 51unequal project size, 51United Kingdom, 1unrecovered investment, 147unrecovered investment, 43unsystematic risk, 229useful life, 17utility, 177

enterprise, 184investor, 223personal, calculating, 180

valuation model, basic, 11valuation, and CAPM, 230value at risk, VAR, 252 footnotevalue of the enterprise, 11Van Horne, J.C., 149Varian, H., 263 footnotevariance, 185

wealth, maximization of as goal, 11Weil, R.L., 146Wiar method, 97Wiar, R., 97–99Williams, R., 143working capital requirements, 16

zero-one integer programming, 162

Documents

Wiley Capital Asset Investment Strategy, Tactics