Evaluating Multiperiod Performance QA-0518

UVA-QA-0518

This technical note was prepared by Sherwood C. Frey Jr., Ethyl Corporation Professor of Business Administration.

Copyright 1996 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to [email protected]. No part of this publication may be

reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any meanselectronic, mechanical, photocopying, recording, or otherwisewithout the permission of the Darden School Foundation.

EVALUATING MULTIPERIOD PERFORMANCE

A tract of land is developed for the resale value of the lots into which it will be

subdivided. A promising new product is nationally introduced on the basis of its future sales and

subsequent profits. A piece of equipment is ordered for the improved operating efficiency and

increased capacity that it will provide relative to the piece it replaces. A corporate bond is

purchased for its coupons and the ultimate repayment of its par value.

These decisions are similar in that each requires the investment of money in anticipation

of benefits whose realization will be spread over time. The value of such an investment depends

on many factors including: the magnitude of the benefits, the timing of those benefits, and the

degree of uncertainty in actually receiving the anticipated benefits. Although the prediction of

future benefits is perhaps the most significant challenge in appraising investments, the careful

and consistent consideration of the effects of time is necessary, even when those benefits are

known. Suppose the monetary return of an investment far exceeds the initial investment, but the

return is delayed into the distant future. Does the magnitude of the return justify the wait?

Suppose one investment yields greater monetary returns than another does, but the returns of the

first extend over a longer period than the second. Which is better or is either desirable?

This note offers a systematic approach to answering these questions. The focus will be

on the measurement of the monetary returns of an investment (cash flow) and on the evaluation

of the effects of timing on the value of those returns (discounted cash flows). The discussion

assumes that the returns are known with certainty. The concepts and techniques for explicitly

addressing uncertainty are discussed in other notes.

Cash Flow

Why is real estate developed, a new product introduced, equipment replaced, or a bond

purchased? In each decision, there are a host of reasons, ranging from the strategic goals of the

firm to the personal desires of the manager. Common to almost all investment decisions,

however, is the objective of earning financial returns from the invested money. This section

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concentrates on the identification and measurement of these financial returns and ignores the

other more subjective considerations that significantly influence investment decisions.

Simply stated, the returns from an investment can either be reinvested in the firm or

distributed to the shareholders. As a result, the returns from an investment should be viewed as

usable funds generated by the investment, and the outlays for an investment should be regarded as the money withdrawn from the pool of usable funds. Usable funds are cash, and thus

investments should be evaluated in terms of cash flowthe inflow and outflow of usable fundsand not in terms of the profits which are reported by the firms accounting system. Profits and cash flow are not the same. As an illustration of this difference, suppose a manager

were offered the choice between prepaying the entire premium of $15,000 for a three-year

insurance program or paying the premium in three annual installments of $5,000 each. Almost

certainly, the manager would select the installment option. But why? For each payment option,

the firms income statement would report the same annual insurance cost of $5,000 (accrual accounting methods require that the prepayment be spread evenly over the three-year life of the

policy), and hence the same profits would be reported for both plans. The difference between the

two options lies in their differing schedules of demand for cash, that is, their differing cash

flows. The installment plan is preferred because it actually spreads the payments over three

years. Focusing on the cash flow reveals the advantage while considering profits obscures it.

The common practice in calculating cash flows for an investment is to calculate the

incremental cash flows of the investmentthe difference between the cash flows from the investment and the cash flows of a do-nothing alternative. If there are several alternative

investments, several cash flow calculations are required, one for each investment relative to the

same do-nothing alternative. In addition to calculating the cash flows on an incremental basis,

one must take care to distinguish between items that are really cash flows and those noncash

items masquerading as cash flows which result from accounting conventions. A simple rule

states that if you write a check for it, its a cash outflow; if you can deposit it in the bank, its a cash inflow. In the above insurance example, the prepayment option has a cash outflow of $15,000 now and no cash outflows in the subsequent years, regardless of how the payment may

be expensed over the life of the policy.

Making an exhaustive list of the sources of cash flows is impossible, but the cash flows

that are most commonly encountered in practice can be grouped into several comprehensive

categories. When calculating the initial outlay of an investment, look first for the obvious initial

purchase or construction cost and then note any changes in working capital (the holding of cash,

inventories, the net of accounts receivable and accounts payable) required to support the project,

the salvage value of any equipment that is being replaced or discarded, and any investment

incentives offered by the tax authorities. For cash flows subsequent to the initial outlay, look for

revenues (sales, dividends, or interest payments if it is a purely financial investment) resulting

from the investment, for cost of goods sold (materials, manufacturing costs), for changes in

selling and administrative expenses, for any subsequent investment costs, and for taxes. Avoid

expenses that do not change if the investment is not done, but are allocated to the investment as

if they were incremental. Do not credit the investment with sales that are cannibalized from

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existing sales. At the end of the life of the investment, be sure to include the recovery of

working capital, any shutdown costs, and the salvage value of any equipment.

Note that neither depreciation nor financing expense is included as a cash flow.

Depreciation is simply an accounting provision whose effects are reflected in the calculation of

taxes, but is not itself a cash flow. The financing costs of an investment are excluded because of

the widely accepted practice of separating the evaluation of the investment itself from the

financing of the investment. There are several reasons for keeping these two considerations

separate. A firms portfolio of investments is generally funded from capital that is raised through a combination of debt, equity, and retained earnings. Without the perspective of the entire

demands for capital within the firm, it would be inappropriate to assume the cost on any one of

these sources (or any mix) in evaluating a specific project. Even if an investment were clearly to

be financed out of either debt or equity but not both, the cost of either means would not reflect

the real cost of financing because the funding of the investment would affect the firms ability to acquire future capital by either means.

1 Thus, the financing decision is a corporation-wide

decision and should not be implicitly made (or assumed) at the level of an individual investment

decision.

An example

As an illustration of these concepts, consider a proposed investment of $128,000 that will

expand production operations for three years and allow the firm to satisfy demand that is

currently being lost. The anticipated revenue from the incremental sales will be $108,000 with a

cost of goods sold of $48,000. Although the company allocates its selling and general

administrative expenses as 12 percent of revenue, there would be no increase in the actual selling

and administrative expenses. To support the increased sales volume, $32,000 must be set aside

at the time of the investment to cover increases in inventories and accounts receivable, all of

which will be recoverable at the end of the third year. At the end of the third year, there will also

be usable equipment with a salvage value of $8,000. The initial investment will be straight-line

depreciated so that the book value is $8,000 at the end of the third year. The marginal tax rate

(federal as well as state and local) is assumed to be 38 percent.

The cash flows for this investment are shown in Table 1. Note that the flows during each

year have been aggregated to give an annual total even though most of them will actually occur

continuously during the year. In addition, revenues are just from the incremental sales, and the

increased allocation of selling and administrative expenses is not included because the actual

selling and administrative expenses do not increase because of the investment. The format of

Table 1 highlights the actual cash flows by never including directly in the calculations any non-

cash items such as depreciation. As a result, taxes are computed as a side calculation. An

1An exception to this statement would occur when the financing of an investment is specific to the investment

itself and the firm is insulated from any risks associated with the investment. Some real estate projects may be

financed in this fashion. When these conditions apply, the investment is de facto a separate entity and the terms of

the financing are integral to the project. As a result, neither the investment nor the financing of the investment can

be isolated from the other and the two must be treated together.

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alternative format that many find useful follows the layout of an income statement. This format

will yield the same results as long as the noncash items that have been included in the calculation

of profits before tax are added back to convert profits after tax into cash flows. Table 2 presents the calculation of the cash flows for the example using the income statement format.

Table 1

Cash Flows

Now Year 1 Year 2 Year 3

Cost of project $(128,000)

Sales $108,000 $108,000 $108,000

Cost of goods sold (48,000) (48,000) (48,000)

Taxes (Note 1) (7,600) (7,600) (7,600)

Changes in working capital (32,000) 32,000

Salvage of equipment 8,000

Taxes on salvage (Note 2) 0

Total cash flow $(160,000) $52,400 $52,400 $92,400

Note 1: Tax computation

Sales $108,000

Cost of goods sold (48,000)

Depreciation (40,000)

Profits before taxes 20,000

Taxes (38% of profit) $7,600

Note 2: Because the equipment is sold at book value, there is no capital gain or loss on the sale

and taxes are zero.

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Table 2

Cash Flows: Income Statement Format

Now Year 1 Year 2 Year 3

Cash flow of initial investment $(128,000)

Sales $108,000 $108,000 $108,000

Cost of goods sold 48,000 48,000 48,000

Depreciation 40,000 40,000 40,000

Total cost $88,000 $88,000 $88,000

Before tax profits from sales 20,000 20,000 20,000

Taxes (38% of profits) 7,600 7,600 7,600

Aftertax profits from sales 12,400 12,400 12,400

PLUS: noncash charges to sales (Note 1) 40,000 40,000 40,000

Cash flow from sales $52,400 $52,400 $52,400

Salvage of equipment $8,000

Book value of equipment 8,000

Profits from equipment salvage 0

Capital gains tax 0

Aftertax profits from equipment salvage 0

PLUS: noncash charges to equipment salvage 8,000

Cash flow from equipment salvage $8,000

Cash flow from working capital (32,000) 32,000

Total cash flow $(160,000) $52,400 $52,400 $92,400

Note 1: Because noncash items (depreciation and book value of equipment) are subtracted in

calculating aftertax profits, these items must be added back to convert profits into cash flow.

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Time Value of Money

Once the cash flows for a proposed investment have been calculated, it still must be

determined whether the proposal is a sound investment. In addition, if a choice must be made

between mutually exclusive alternatives or if there are several attractive alternatives but limited

available capital, managers must evaluate the alternatives to determine which ones make the

most effective use of the invested funds. This section presents a systematic way to make these

evaluations.

Suppose a manager is faced with a choice between two investment opportunities, A and

B, each of which requires an initial investment of $50,000. Investment A produces cash flows of

$22,000 at the end of each of the next three years. The cash flows of investment B are $12,000

at the end of the next two years and $46,000 at the end of the third year. Which investment, if

either, should the manager select?

Accumulated value

Over its lifetime each alternative will return more than the initial investment of $50,000:

$66,000 for investment A, $70,000 for investment B. At first blush, it would appear that both

investments are attractive and that investment B is better because it returns more for the same

initial investment.

Such an initial reaction should be tempered, however, by the realization that in the early

years the cash flows from investment A exceed those from investment B. Because the manager

would certainly not leave the cash flows idle from either of the investments, the larger earnings

from the reinvestment of the larger flows from investment A in years 1 and 2 could offset

investment Bs larger cash flow in year 3. The extent to which the larger reinvestment earnings will benefit investment A depends upon the attractiveness of the reinvestment opportunities.

Assume that the manager will aggressively manage any cash returns so that they would earn 15

percent after taxes. With this reinvestment environment, what would be the total value

(including reinvestment) of each of the alternatives at the end of its lifetime?

A manager can answer this question with a calculation that is identical to the calculation

of the balances in a savings account. For a savings account, the interest rate is applied to the

average balance in the account, and the ending balance for a time period is the total of the

opening balance, the interest earned, and any deposits or withdrawals made during the time

period. In the investment example, the returns from the investments are like the deposits to the savings account, and the earnings from reinvesting the returns are like the interest payments. More specifically, for investment A, there are zero dollars on deposit during year 1, so no

interest is earned, but a deposit of $22,000 is made at the end of the year. For year 2, the opening

balance of $22,000 is invested throughout the year. That balance will earn $3,300 in interest at a

15 percent rate. At the end of year 2, the balance will be $47,300the sum of the opening balance of $22,000, the interest of $3,300, and year-end deposit of $22,000. Similarly, the

balance at the end of year 3 will be $76,395the sum of the opening balance of $47,300, the

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interest earned during the year of $7,095 ($47,300.15), and the year-end deposit of $22,000. Table 3 presents these calculations for investments A and B.

When the two investments are evaluated in light of the 15 percent reinvestment

opportunities for their returns, investment A is preferred because it accumulates $76,395 by the

end of year 3, as compared to the $75,670 accumulated by investment B. Even though

investment A offers a smaller total return, those returns come earlier and permit greater earnings

from reinvestment. The figures in Table 3 show that the earlier cash flows of investment A earn

$10,395 in reinvestment income compared to the $5,670 for investment B. The added

reinvestment earnings of investment A are sufficient to offset the $4,000 difference in total

return.

If the managers reinvestment opportunities were less attractive, investment A might not be the more attractive choice. Suppose the reinvestment opportunities were 10 percent instead of

15 percent, which investment would be the better? Table 4 presents the same calculation as

Table 3 but with a reinvestment rate of 10 percent. Now, investment B has the greater

accumulated value at the end of year 3 and is the more attractive investment. At this lower

reinvestment rate, the accelerated cash flows of investment A do not earn sufficient reinvestment

income to offset the $4,000 difference in total return.

The above examples show that the evaluation of investments whose payoffs extend into

the future depends not only on the magnitude of the cash flows but also on the timing of the

flows and the subsequent use to which those flows can be put. To appropriately evaluate

alternative cash flow streams, one must consider all three aspectsmagnitude, timing, and reinvestment rate.

Thus far, the analysis of the two investments has compared only the two alternatives, but

has not determined if either of them is an attractive use of the $50,000 initial investment. Would

it be better to put the $50,000 into the 15 percent investment opportunities rather than either of

the two alternatives? One way to answer this question is to calculate the accumulated value of

$50,000 after three years and compare it with the accumulated values of the two investment

alternatives. At a 15 percent rate with the earnings from one year reinvested for the next, the

$50,000 will compound to $57,500 at the end of the first year, $66,125 at the end of the second

year, and $76,044 at the end of third year.2 Comparing this final figure to the accumulated value

of investment A ($76,395) shows that investment A is a slightly better use of the $50,000 than

simply investing in the available 15 percent opportunities. On the other hand, investment B

($75,670) is not a sound investment when 15 percent opportunities exist. What do you expect to

happen if the investment opportunity rate were 10 percent? Check your intuition with a

numerical calculation similar to the one just performed.

2The exact number (at the end of the third year) is $76,043.75, but to simplify the presentation, figures will be

rounded to whole dollars.

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Table 3

Comparing Investment A and Investment B

(reinvestment at 15 percent)

Investment A Year 1 Year 2 Year 3

Cash balance at beginning of year $0 $22,000 $47,300

Earnings from reinvestment of 0 3,300 7,095

balance at 15 percent

Cash inflow at end of year from 22,000 22,000 22,000

Investment A

Total cash available at end of year $22,000 $47,300 $76,395

for reinvestment next year

Investment B Year 1 Year 2 Year 3





Investment B



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Table 4

Comparing Investment A and Investment B

(reinvestment at 10 percent)

Investment A

Year 1 Year 2 Year 3





Investment A



Investment B Year 1 Year 2 Year 3





Investment B



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Present value and net present value

In the above discussion, the evaluation of a stream of cash flows was based on the

accumulated value (including reinvestment income) of the cash flows to the end of the streams lifetime. This is a very natural way to approach the evaluation because it so closely parallels the

compounding calculations of a savings account. There are, however, several drawbacks to this

future value approach. First, the value of an investment is associated with a future point in time. For short-lived investments, as in the examples, this is not a problem. For investments

with long lifetimes, say 20 or 40 years, it is very difficult to internalize the significance of their

accumulated values. The numbers will be extraordinarily large and very distant in time. Even if

the investments under consideration have moderate lifetimes, they may be of different lengths.

As a result, their evaluations will be associated with different points in time, making comparison

difficult. Finally, the financial attractiveness of each investment would require the calculation of

two accumulated valuesone for initial investment and one for the future cash flows.

If the perspective of the evaluation were changed from future dollars to today dollars, these difficulties would be eliminated. The frame of reference would be today, not some distant

time; all investments would be evaluated at a common point in time, today, not potentially

different points; the attractiveness of an investment could be based on the simple comparison of

the initial investment and the today value of the future cash flows.

From the future dollars perspective, $50,000 today is worth $76,044 in three years in an

environment of 15 percent reinvestment opportunities. (This figure was calculated in the

previous section.) If we shift our perspective to today, it could be said that $50,000 is the present

value of $76,044 received three years from now when 15 percent opportunities exist. The

$50,000 is the present value in the sense that if it were invested today at 15 percent it would

grow to $76,044 three years in the future. As a result, $50,000 today and $76,044 in three years

are financially equivalent when 15 percent investments are available. This leads to the definition

that the present value of a future payment is the amount today for which the investor is

indifferent between receiving the present value or waiting for the future payment.

This concept can be applied to a stream of future cash flows, for example, the three

annual payments of $22,000 that make up the returns from investment A. The present value of

these flows at a reinvestment rate of 15 percent would be the amount of money required today to

generate the future stream of cash flows. Each of the individual cash flows in the stream has a

present value (the amount required today to generate it). The sum of these individual present

values would be the amount required to generate the entire stream. Because the accumulation

over time of a reinvested dollar is the underlying concept of present value, the following table of

the accumulated value of a reinvested dollar at 15 percent will help in calculating the present

values of each of the cash flows resulting from investment A.

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Year 1 Year 2 Year 3

Beginning amount 1.0000 1.1500 1.3225

Interest at 15 percent .1500 .1725 .1984

Ending amount 1.1500 1.3225 1.5209

The first component of investment A stream is $22,000 one year from now. A dollar that is

invested at 15 percent will accrue to $1.15 one year from now. Thus, $22,000 equals 115

percent of the amount that would be necessary to invest today to have $22,000 at years end. The present value of $22,000 is therefore $19,130 ($22,000/1.15). Similarly, the cash flows of

years 2 and 3 of investment A would require $16,635 (or $22,000/1.3225) and $14,465 (or

$22,000/1.5209), respectively. The entire cash flow stream would require an investment at 15

percent of the sum of these individual investments, $50,230 ($19,130 + $16,635 + $14,465).

This sum is the present value of the stream of future cash flows of investment A.

Because it would take $50,230 to generate the stream of future cash flows of investment

A with 15 percent investments and investment A requires an initial investment of only $50,000,

investment A is an attractive investment when 15 percent alternatives are available. Not

surprisingly, because present value and accumulated value are so tightly related, this is the same

conclusion that was reached when comparing the accumulated value of the cash flow stream of

investment A to the accumulated value of $50,000 after three years. If investment A were made,

it would add today $230 ($50,230 $50,000) in value above of the use of the $50,000 in the managers usual 15 percent investments. The difference between the present value of the future cash flow stream and the initial investment is called the net present value. The net present

value (NPV) is a measure of the attractiveness of an investment. If the NPV is positive, an

investment is attractive, because it would require more money to generate the investments future cash flows through the managers reinvestment opportunities than is required by the investment itself. Value is being added by positive NPV investments, and the more positive the NPV, the

more attractive the investment. If the NPV is negative, the investment is unattractive, and value

is depleted. If the NPV is zero, the investment is equivalent to earning the reinvestment rate, and

the investment neither adds nor depletes value. As such, a zero NPV investment is an indifferent

opportunity.

Note that in the above calculations, the present value of a future cash flow is less than the

cash flow itself. The future flows have been discounted to account for the time value of money. At a 15 percent reinvestment rate, $22,000 received one year from now has a present

value of $19,130 or .8696 of its future value. To account for the time value of money, the flow

one year from now must be multiplied by the factor, .8696, to bring the flow to its present value.

This factor is the one year discount factor at 15 percent. Similarly, the discount factors at 15

percent are .7561 for two years and .6575 for three years. Each of these can be easily calculated

as the reciprocal of the accumulated values of a dollar: .8696=1/1.15, .7561=1/1.3225, and

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.6575=1/1.5209. Discount factors can be interpreted as the present values of future one dollar

payments and can be used to calculate the present value of future streams of cash flows. For

investment A, the calculation is:

($22,000 . .8696) + ($22,000 .7561) + ($22,000 .6575) = $50,230

Formulas for accumulated and present value calculations

Underlying the calculations of accumulated value and present value are several straight-

forward equations. The accumulated value, A, at reinvestment rate, r, of a single payment, P,

after one year is P (1+r), the payment P plus the interest rP. After two years, the accumulated

value is [P (1+r)] (1+r), the amount after one year multiplied by 1+r. The accumulated values

in subsequent years would be calculated by the successive multiplications of the ending value by

1+r. This would yield the general formula for the accumulated value after n years, An, of a

payment, P, at a reinvestment rate, r:

An = P (1 + r)n.

This formula can then be used to express the present value, P, of an amount An that is

available n years in the future with a reinvestment rate, r, as:

P = An / (1 + r)n.

Although these formulas were developed on the basis of the cash flows occurring at the

end of their respective years, they apply just as well to flows that occur within years. For

example, the calculation of the present value of a cash flow that occurs after 2 years and 3

months would use the above formula with n=2.25.

Streams in perpetuity

There is one cash flow stream worthy of special considerationa stream of equal year-end cash flows continuing forever. Such a stream is called a perpetuity. At a reinvestment rate

of 15 percent, $20,000 will produce a cash flow stream of $3,000 per year forever. The $20,000

will not accumulate because the annual interest of $3,000 is taken out as a cash flow. By

definition, the present value of this perpetuity of $3,000 per year is the amount required today to

generate that never-ending stream. With a 15 percent rate, the stream can be generated with an

initial amount of $20,000, so the streams present value is $20,000. Note that the present value is the annual payment divided by the rate ($3,000/.15). In general, a stream of equal annual cash

flows has a present value equal to the annual payment, C, divided by the reinvestment rate; this

is C/r. If investment A were a perpetuity with an annual payment of $22,000, one can see it

would have a present value at a reinvestment rate of 15 percent of $146,667 ($22,000/.15). One

can see the exponential effects of discounting from this figure because the present value of the

first three years of the perpetuity ($50,340) accounts for more than one-third of the present value

of the entire perpetuity.

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Pretax versus aftertax analyses

On some occasions, calculations can be simplified by working with pretax monetary

flows and avoiding tax calculations. Pretax analyses are appropriate when the pretax monetary

flows are proportional to their cash flows, which are always aftertax. This proportionality will

not be the case when the investments under consideration involve differences in depreciation,

investment tax incentives, or working capital. When these latter conditions exist, a doubling of

the pretax monetary flow will not result in a doubling of the cash flow.

When a pretax analysis is appropriate, at what reinvestment rate should the monetary

flows be discounted? It is tempting to think that a pretax reinvestment rate would be consistent

with the pretax nature of the flows, but this is incorrect. Because a full cash flow analysis will

always be appropriate, it is necessary to have any pretax (short-cut) analysis be consistent with it.

Tax considerations will affect the numerator of the present value calculation in a proportional

fashion (remember this was the condition necessary for a pretax analysis to be appropriate), but

an adjustment to the reinvestment rate will not affect the denominator in the same fashion. The

denominator of the present value calculation involves one plus the reinvestment rate and, as a

result, increasing the reinvestment rate proportional to the tax rate will not increase the

denominator proportionally. The non-proportional changes to the denominator will lead to an

inconsistency between the always correct cash flow analysis and the pretax analysis with a pretax

reinvestment rate.

As an illustration of the above discussion, suppose the tax rate is 50 percent, the

reinvestment rate is 10 percent, and a pretax monetary flow in year 1 is $100. The present value

of the cash flow associated with the monetary flow would be $45.45 ($100.50/1.10). The

correct pretax analysis would result in a present value of $90.90 ($100/1.10), which has the

pretax and the cash flow analyses in proportion to the tax rate. An incorrect pretax analysis

would be $83.33 ($100/1.20), which does not preserve the proportionality of the results. The

only meaningful reinvestment rate is the aftertax reinvestment rate, and it should be used in all

circumstances.

The Reinvestment Rate

Present value calculations require a reinvestment rate. Throughout this chapter, the rate

has been assumed to be known, but the methodology suggests the fundamental principle upon

which such a rate should be based. Cash that becomes available does not lie idle but is

recommitted to other activities throughout the firm. Strictly speaking, the appropriate rate is the

amount that must be earned on a dollar so the investor/manager is indifferent between receiving

the dollar now and receiving the dollar plus its earnings a year from now. Because the

estimation of this rate is difficult, the objective of this section is to provide a flavor, but not a

complete exposition, of approaches to the issue.

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Hurdle rate

One approach to the estimation of the reinvestment rate is to estimate the opportunity

rate, that is, the marginal rate of return of the pool of investment opportunities that the firm

might undertake with its available cash spinoffs. It was in this context that the 15 percent rate

was established for the assessment of investment A and investment B. In practice, where the

profile of potential investment opportunities is complex, the opportunity rate is a difficult

number to estimate.

An alternative approach is to seek the rate from the perspective of the companys cost of capital. The prices that investors are willing to pay for a firms securities and the yields that they demand from those securities determine a market cost of capital raised through debt and

equity. The cost of debt is simply the interest that must be paid, but because interest expenses

are tax deductible, the effective cost of debt is the aftertax interest rate. There is a corresponding

cost for shareholders equity because investors want to earn a satisfactory rate of return as compensation for the use of their money and for the risks that they take by investing in the firm.

This cost applies both to the new investments made in the firm through the purchases of stock

and to the earnings that are retained in the firm rather than paid out in dividends. These costs,

combined with the capital structure of the firm, result in a weighted average cost of capital

(WACC). Because the WACC is the average rate demanded by the capital markets for

investment funds, the firm should consider only those investments whose cash flows will yield at

least that rate. Thus the value of the investment, from the perspective of the capital markets, is

the present value of the cash flows using the WACC.

In a perfect environment, where both the firm and the investors have complete

information, the WACC and the opportunity rate will be identical because the firm will invest in

all projects with a positive net present value at the weighted average cost of capital. As a result,

the marginal project will have a net present value equal to zero so its rate of return will be the

WACC. Even though the capital markets are not perfect, the WACC is commonly used as the

reinvestment rate. It is often referred to as the hurdle rate because investments with a positive

net present value at this rate are judged to be financially attractive; that is, to have passed the

hurdle. In contrast, investments with negative net present values fail the hurdle.

Internal Rate of Return

Frequently the accept/reject decision is not particularly sensitive to the exact value of the hurdle rate and there may be a comfortable leeway for error in its specification. To find out

how much leeway there may be, the hurdle rate could be compared to the reinvestment rate, for

which the NPV of the investment is zero. At this break-even reinvestment rate the decision will

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change from accept to reject or from reject to accept. This break-even reinvestment rate is called

the internal rate of return (IRR).3

There is no formula for computing the internal rate of return; it must be found by trial and

error. For example, consider investment A. At a 15 percent reinvestment rate, the net present

value is $230. If the reinvestment rate were 10 percent, the investment would be even more

attractive and would have a net present value of $4,711. On the other hand, if the reinvestment

rate were increased to 17 percent, the net present value would be $1,389. As the reinvestment rate changes from 10 percent to 17 percent, the net present value changes from being very

positive to very negative. Figure 1 is a graph of the relationship and shows that the net present

value is zero somewhere between 15 and 16 percent. Thus the internal rate of return is between

15 and 16 percent. By continuing a trial and error process, values in this range can be tested

until the reinvestment rate that results in a zero net present value is found. For investment A, the

internal rate of return is approximately 15.3 percent. If the reinvestment rate is less than 15.3

percent, the net present value will be positive and the investment will be judged to be attractive.

If the reinvestment rate is greater than 15.3 percent, the net present value will be negative. There

is little leeway between the reinvestment rate of 15 percent and the point where the investment

changes from attractive to unattractive, so a careful consideration of the appropriate reinvestment

rate is necessary.

In the accept/reject decision for an investment, the internal rate of return and the net present value are equivalent. If an investments net present value at the reinvestment rate is positive, the investments internal rate of return must be greater than the reinvestment rate. Refer to Figure 1 for a visual confirmation of this statement. Similarly, if the internal rate of return is

greater than the reinvestment rate, the investments net present value at the reinvestment rate must be positive. Regardless of the perspective, the investment is attractive, so the two figures

can be used interchangeably in this case.

Although net present value and internal rate of return are equivalent in the accept/reject

decision, internal rate of return should not be used to rank alternative, mutually exclusive

investments. The internal rate of return for an investment is calculated on the basis of the

investments stream of cash flows and is divorced from the actual reinvestment opportunities in which the cash flows of the investment could be put. As its name states, the internal rate of

return is internal to the investment and does not reflect the reality of the reinvestment environment facing the investor. As a result, internal rate of return does not apply a common

standard of comparison to each of the investments under consideration. Consequently, the

Selection of a project with a larger internal rate of return does not guarantee that it will have the

larger net present value at the appropriate reinvestment rate (rather than its own internal rate).

3In certain circumstances there may be more than one reinvestment rate that results in a zero net present value

for an investment. Such cases may arise when the cash flow stream has more than one change in sign between

successive cash flows (that is, more than one time when successive cash flows change from negative to positive or

positive to negative). For these cases, the internal rate of return is not a useful concept.

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Figure 1

NPV of Investment A for Different Reinvestment Rates

To illustrate this point, consider the two investments whose net present values are graphed in

Figure 2. If a choice between these two alternatives is made on the basis of the larger internal

rate of return, Investment 1 would be selected. If the reinvestment rate is 15 percent, however,

Investment 2 is the better choice because it has the larger net present value at the actual

reinvestment rates available to the investor. Because of its inward-looking nature, the internal

rate of return can be a misleading criterion for selecting among mutually exclusive investments.

Figure 2

NPV and IRR

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Nominal versus effective rates of return

Suppose an annual reinvestment rate is quoted as 12 percent. Does this mean each dollar

invested will earn one cent at the end of the first month? It all depends on whether the annual

rate is being quoted as an effective annual rate or a nominal annual rate and on how frequently

earnings are compounded. There is considerable room for confusion unless terms and

assumptions are carefully specified.

Let us suppose that a 12 percent annual rate results in a 1 percent payment each month.

Because of the compounding of the monthly payments (interest being earned on interest), the year-end value of an investment will be more than 112 percent of the initial investment. A

$1,000 investment would have a year-end value of

$1,127 = $1,000 1.01 1.01 1.01 . . . 1.01 = $1,000 (1.01)12

.

This is equivalent to a 12.7 percent annual rate, even though a 12 percent annual rate was quoted.

A vocabulary that would clarify the situation is to state that the nominal annual rate is 12 percent

compounded monthly and the effective annual rate is 12.7 percent. Reinvestment rates are

quoted as effective annual rates.

Not only is it important to distinguish between nominal and effective annual rates, but

also care must be taken when stating equivalent rates for periods of less than a year. For a 12

percent nominal annual rate that is compounded monthly, the equivalent monthly rate is 1

percent (the annual rate divided by 12). For an effective annual rate of 12 percent, the monthly

rate must be calculated taking into account the compounding during the year.

If the effective annual rate is 12 percent, the value of a dollar at the end of the year is $1.12. If

returns are made on a monthly basis, the monthly rate must satisfy the following equation:

$1.12 = $1.00 (1 + monthly rate)12

.

The solution of this equation is a monthly rate of .95 percent (.0095). Note that because of the

compounding that is inherent in effective rates, the equivalent monthly rate for an effective

annual rate is less than the equivalent monthly rate for a nominal annual rate. In general, the

equivalent periodic rate of an effective annual rate is

(1 + effective annual rate)1/number of periods

1.

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Evaluating Multiperiod Performance QA-0518