Continuous-Time Finance

Robert C. Merton

Revised Edition

Foreword

A great economist of an earlier generation said that, useful though economic theoryis for understanding the world, no one would go to an economic theorist for advice onhow to run a brewery or produce a mousetrap. Today that sage would have to changehis tune: economic principles really do apply and woe to the accountant or marketerwho runs counter to economic Law. Paradoxically, one of our most elegant andcomplex sectors of economic analysisthe modern theory of financeis confirmeddaily by millions of statistical observations. When todays associate professor ofsecurity analysis is asked, Young man, if youre so smart, why aint you rich?,he replies by laughing all the way to the bank or to his appointment as a high-paidconsultant to Wall Street.

Among connoisseurs, Robert C. Merton is known as an expert among experts,a giant who stands on the shoulders of such giants as Louis Bachelier, John BurrWilliams, George Terborgh, Keynes, James Tobin and Harry Markowitz, KennethArrow and Gerard Debreu, John Lintner and William Sharpe, Eugene Fama, BenoitMandelbrot, and the ubiquitous Black-Scholes. (Benjamin Graham occupies anotherpart of the forest.) The pole that propelled Merton to Byronic eminence was themathematical tool of continuous probability a` la Norbert Wiener and Kiyoshi Ito.Suddenly what had been complex approximation became beautifully simple truth.

The present book patiently explicates for readers with imperfect mathematicalbackground the essentials of efficient-market asset pricing. Many of its chaptersreproduce articles that have become classics in the literature. Several chapters arenew to this book and break novel ground.

I am proud to have figured in the Mertonian march to fame. When a youngster,with an electrical engineering bachelor degree and beginning Cal Tech graduate workin applied mathematics, decided to be an economist, he applied to several graduateschools for admission in economics. All but one, he says, turned him down: MIT,mirabile dictu, offered him a fellowship! He worked with me and was a joy to workwith. One of the great pleasures in academic life is to see a younger savant develop,evolving into a colleague and co-authorand then, best of all, is the rare sight of thecompanion at arms who forges ahead of you as you were able to do at the inflectionpoint of your own career. Robert K. Merton, anthropological observer of the zooof scientists (and fond mentor of Robert C.), will want to add this saga to his casestudies of how science actually evolves.

To the reader I repeat:

ii Foreword

Bon Appetit!

Paul A. SamuelsonMIT

Preface

This book develops the mathematics and economic theory of finance from the per-spective of a model in which agents can revise their decisions continuously in time.Time and uncertainty are the central elements that influence financial economic be-havior. It is the complexity of their interaction that provides intellectual challengeand excitement to the study of finance. To analyze the effects of this interactionproperly often requires sophisticated analytical tools. Indeed, advanced mathemat-ical training has become a prerequisite for researchers in the field. Yet, for all itsmathematical complexity, finance theory has had a direct and significant influenceon finance practice. A casual comparison of current practices with those of 20 yearsago is enough to note the impact of efficient market theory, portfolio selection, riskanalysis, and contingent-claim pricing theory on money management, financial inter-mediation, investment banking, and corporate financing and capital budgeting proce-dures. The effects of this theory have even been observed in legal proceedings suchas appraisal cases, hearings on rates of return for regulated industries, and revisionsof the prudent person laws governing behavior for fiduciaries. The role of financetheory in the current wave of financial innovations in capital markets has been oftendocumented in the financial press. See especially the carefully researched book byBernstein (1992). Evidence that this influence on practice will continue can be foundin the curricula of the best-known schools of management, where the fundamentalfinancial research papers (with their mathematics included) are routinely assigned toMBA students. Although not unique, this conjoining of intrinsic intellectual interestwith extrinsic application is a distinctive and prevailing theme of research in finance.

It was not always thus. Finance was first treated as a separate field of study early inthis century, and for the next 40 years it was almost entirely a descriptive disciplinewith a focus on institutional and legal matters. As recently as a generation ago.finance theory was still little more than a collection of anecdotes, rules of thumb,and manipulations of accounting data. The most sophisticated tool of analysis wasdiscounted value and the central intellectual controversy centered on whether to usepresent value or internal rate of return to rank corporate investments. The subsequentevolution from this conceptual potpourri to a rigorous economic theory subjected toscientific empirical examination was, of course, the work of many, but most observerswould agree that Arrow, Debreu, Lintner, Markowitz, Miller, Modigliani, Samuelson,Sharpe, and Tobin were the early pioneers in this transformation.

Since the continuous-time model is the mode of analysis used throughout the

iv Preface

book, a few background remarks on the model as both synthesis and watershed offinance theory are perhaps in order. It was in 1900 at the Sorbonne that Louis Bache-lier wrote his magnificent dissertation on the theory of speculation. This work marksthe simultaneous births of both the continuous-time mathematics of stochastic pro-cesses and the continuous- time economics of option and derivative-security pricing.Although Bachehers research was unknown in the economics and finance literaturefor more than a half-century and although from todays perspective his economicsand mathematics are flawed, the lineage from Bachelier to modern continuous-timefinance is direct and indisputable.

Over the past two decades, the continuous-time model has proved to be a ver-satile and productive tool in the development of finance. Although mathematicallymore complex, the continuous-time formulation often provides just enough addi-tional specificity to produce both more precise theoretical solutions and more refinedempirical hypotheses than can otherwise be derived from its discrete-time counter-part. As a case in point, we need only consider an early version of the continuous-time model of portfolio selection first published in 1969 and reprinted here as Chapter4. In the late 1960s, the primary capital-market models in finance were the one-period mean-variance model of Markowitz and Tobin and its equilibrium version,the Sharpe-Lintner-Mossin Capital Asset Pricing Model. Although quite feasiblefor practical use and remarkably elegant in their simplicity, these models neverthe-less received relatively limited application in the broader community of economicresearch. The main reason was a widespread belief that the mean-variance criterionis not consistent with the generally accepted von Neumann-Morgenstern axioms ofchoice unless either asset prices have Gaussian probability distributions or investorpreferences are quadratic. In addition to being rather specialized conditions, nor-mal distributions for prices violate the fundamental provision of limited liability forowners of financial assets. Furthermore, quadratic utility appears to be grossly incon-sistent with observed behavior. However, by replacing the generic one period witha definite specification of the time interval between successive portfolio revisions, thecontinuous-time model with log-normally distributed asset prices (which do satisfylimited liability) produces optimal portfolio rules that are identical in form with thoseprescribed by the mean-variance model and the Capital Asset Pricing Model. More-over, this result obtains for general von Neumann-Morgenstern preferences. Thus,somewhat paradoxically, introducing the greater realism of a dynamic intertemporalmodel served to make more plausible the optimal rules derived from these classicstatic models. In this sense, the continuous-time model is a watershed between thestatic and dynamic models of finance. As we shall see in the chapters to follow,continuous-time analysis shows that those other classic pillars of finance theorytheArrow-Debreu complete-markets model and the Modigliani-Miller theoremsarealso far more robust than had been believed.

While reaffirming old insights, the continuous-time model also provides new ones.Perhaps no better example is the seminal contribution of Black and Scholes that,

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virtually on the day it was published, brought the field to closure on the subjects ofoption and corporate-liability pricing. As the Black-Scholes work was closing gateson fundamental research in these areas, it was simultaneously opening new gates:in applied and empirical study and in setting the foundation for a new branch offinance called contingent-claims analysis. As shown in Part IV, the applications ofcontingent-claims analysis range from the pricing of complex financial securities tothe evaluation of corporate capital budgeting and strategic decisions. As we shallalso see, it has an important place in the theory of financial intermediation.

As surely exemplified by the more than a thousand equations that appear in thisvolume, finance is a highly analytical subject, and nowhere more so than in continuous-time analysis. Indeed, the mathematics of the continuous-time finance model con-tains some of the most beautiful applications of probability and optimization theory.But, of course, not all that is beautiful in science need also be practical. And surely,not all that is practical in science is beautiful. Here we have both. With all its seem-ingly abstruse mathematics, the continuous-time model has nevertheless found itsway into the mainstream of finance practice. Perhaps its most visible influence onpractice has been in the pricing and hedging of financial instruments, an area thathas experienced an explosion of real-world innovations over the last decade. In fact,much of the applied research on using the continuous-time model in this area nowtakes place within practicing financial institutions.

Any virtue can readily become a vice merely by being carried to excess, and justso with application of the continuous-time paradigm. Its powerful analytics are atemptation to excessive focus on mathematical rigor with the unhappy consequenceof leaving the accompanying substantive economics inaccessible to all but a few.Attention to formal technique without equal attention to the underlying economicsassumptions leads to misplaced concreteness by confusing rigor in the mathematicalsense with rigor in the economic sense. In Chapter 3, the mathematics and basic eco-nomics assumptions of the continuous-time model are developed with this in mind.The focus is on those mathematical concepts such as the Ito calculus that are essentialfor applying continuous-time analysis in finance. By taking elementary probabilitytheory and ordinary calculus as its only prerequisites, the chapter sacrifices somemathematical rigor and generality in return for greater accessibility and, I hope, clar-ity. In the same spirit, the derivation of stochastic dynamic programming in Chapter4 also requires no more than these prerequisites. Ample mathematical references are,of course, provided for those who would prefer a broader and more formal treatmentof the subject.

As in mathematics, emphasis on compactness of models and their presentation is avirtue highly prized in mathematical economics. But, when taken to an extreme, suchemphasis has the surely unintended consequence of reducing the substantive richnessof the analysis by deleting institutional settings and interpretations of the models thatare seemingly redundant from the mathematical perspective but that are not at all re-dundant in the economics domain. It will be granted that the model development and

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analysis presented here are not immune to excessive austerity of this particular kind.Yet, especially but not exclusively in the chapters on financial intermediation andgeneral equilibrium theory, I try my hand at correcting such omissions, by includingconsiderably more institutional interpretation of the models than is the usual practicein neoclassical economics treatments of finance theory. Perhaps I will be forgiven if,in places, my attempts to avoid excessive laconism lead to sheer verbosity.

The core of the book is a collection of 15 previously published papers and a widelycirculated working paper, written over the period spanning from 1969 to the present.These chapters are organized into six parts according to subject matter rather thandate of original publication. In reprinting these articles, I have made minor revi-sions in language and have corrected misprints and technical errors without indicat-ing changes from the original. An asterisk identifies added footnotes that describeeither more extensive revisions or citations to subsequent research on the topic. Pub-lished references are substituted for original citations of unpublished manuscriptsand working papers that have subsequently appeared in print. The original notationof each paper is preserved and notation is therefore not entirely uniform across thechapters. And since each chapter is largely self-contained, some repetition of ana-lyzes does occur. Perhaps some readers will find these repetitions a useful form ofemphasis to underscore important concepts in the theory.

I need hardly say that when these papers were first written they were not intendedas chapters of a single volume on the continuous-time theory of finance. It wouldbe foolish therefore to suggest that this format of synthesis is an unconstrained op-timal design. Temporal differences in the original publication of the papers wouldalone dictate a path of development for the subject that follows its historical evolu-tion more closely than might otherwise be optimal. I nevertheless harbor the hopethat a logical coherence will be found in the organization of the parts and the selec-tion of papers. To facilitate the continuity and expand the scope of coverage, I haveadded five essays to the core of reprinted papers. Written expressly for the book,these essays include a short introductory chapter and four other chapters in the areasof optimal consumption and portfolio selection, option pricing, financial intermedi-ation, and general equilibrium theory. To provide unity with the core, each of thesechapters follows the same self-contained style of a separate journal article. Other newmaterials on intertemporal capital asset pricing and contingent-claims analysis havebeen added. These appear not as separate essays but, to preserve the continuity ofsubject matter further, have been placed in new sections added to the original paperson those topics reprinted as Chapters 13 and 15.

Many economists and mathematicians have contributed to the development of thecontinuous-time theory of finance. Whenever known, they are referenced in the textand in the numerous notes to the individual chapters. There are, undoubtedly, otherswhose contributions are not cited but should be, and to them, I offer both blanketapology and acknowledgment. But, even if the list of citations were complete, itwould still be inadequate to trace all the sources of influence that over the years led

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me to the conceptions set forth in this book. Among those sources, there are a few towhom I owe special and long-standing debts that I want to acknowledge here.

The earliest of these debts, incurred before my ever studying economics, is to theColumbia School of Engineering and the California Institute of Technology. Withits small and flexible program and fine faculty, Columbia was a great place for anundergraduate to explore mathematics and its applications. It was there that I firstbecame intrigued with stochastic processes and optimal control theory. I rememberwith particular fondness John Chus course on heat transfer that turned on the light forme regarding the power of advanced mathematics for solving real-world problems.A brief year of graduate study at Cal Tech added much to my stock of mathematics.Most valuable was the Cal Tech creed of immediately involving students in playingwith the subject in a quasi-research mode rather than just passively absorbing infor-mation. I am grateful to Gerald Whitham (then department head) who, although hethought it was crazy, supported my decision to leave Cal Tech (and mathematics)to study economics. Among other things, he allowed me to take economics courseswith Alan Sweezy and Horace Gilbert (surely an interesting combination for onesintroduction to the subject).

To members of the Economics Department at Massachusetts Institute of Technol-ogy in 1967, I am deeply indebted for their taking a flyer and admitting one with nobackground in economics, when departments elsewhere would not. I am especiallythankful to Harold Freeman who (as 1 discovered only later) was instrumental in theDepartments decision and who, on my arrival in Cambridge, advised me to eschewthe standard first-term curriculum and to take Paul Samuelsons course in mathemat-ical economics.

To that teacher, mentor, colleague, co-researcher, and long-time friend, Paul A.Samuelson, I have come to owe an incalculable debt. Again, as before (cf. Merton,1983a), I cannot find the words to pay adequate tribute to him. Reiteration wouldonly dull the insufficient thanks expressed on those occasions. Instead, I say onlythis: because of Paul, no one could have had a better introduction to the study ofeconomics than mine. After taking that course of Pauls (which erased any lingeringdoubts that Whitham was wrong), I had the great good fortune to spend the rest ofmy graduate student years living in his office as both research assistant and tutee.

As readers of the following pages will soon recognize, I owe a great debt to a circleof brilliant colleagues in the Finance Group at MIT Sloan School of Management:Fischer Black, John Cox, Chi-fu Huang, Franco Modigliani, Stewart Myers, My-ron Scholes, andas always, gladly included as ex officio members of the GroupStanley Fischer and Paul Samuelson. The tenure and tenor of this small collegiumvaried considerably during my 18 years at Sloan. Indeed, the entire finance facultyrarely numbered more than a half dozen. But, throughout those years, there werereliable constants among my colleagues: quality of mind, diversity of thought, and agenuine affection for one another. Sloan was not only a stimulating place in which todo research; it was also a happy place for research.

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Added to this array of long-term debts, there are other, current debts. I am espe-cially indebted to Peter Dougherty for proposing the book and for his unwaveringconfidence that a selected combination of my published papers, together with somenew material, would have a value exceeding the sum of its parts. I trust only thatthe final result approximates his expectations. As acknowledged in later notes, I amgrateful to various publishers for permission to reprint papers appearing in their jour-nals and to Paul Samuelson for agreeing to have our joint paper reprinted as Chapter7. Aid from the National Science Foundation, which supported the work reported inseveral of the chapters, is also gratefully acknowledged. I am thankful to the MITSloan School of Management for a reflective sabbatical year during which much ofthe book was written and to the Harvard Graduate School of Business Administrationfor providing research facilities and support that year and since.

I have been blessed by having a remarkable and still young applied mathematicianserve as a research assistant on the book. With great skill and verve, Arnout Eike-boom checked each of the dozens of theorems and hundreds of equations. It was pureserendipity when he also showed an equal talent for uncovering and correcting poorlyphrased text, confusing notation, and even mere errors of spelling. My special thanksto Deborah Hannon for typing a difficult manuscript and for general administrativeassistance in preparing the volume. To one of the six to whom this book is dedicated,I am deeply grateful for editorial suggestions, and so much more.

R. C. MertonHarvard University

Contents

I Introduction to Finance and the Mathematics of Continuous-Time Models xv

1 Modern Finance 1

2 Introduction to Portfolio Selection and Capital Market Theory: StaticAnalysis 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 One-Period Portfolio Selection . . . . . . . . . . . . . . . . . . . . . . 132.3 Risk Measures for Securities and Portfolios in the One-Period Model 202.4 Spanning, Separation, and Mutual-Fund Theorems . . . . . . . . . . . 26

3 On the Mathematics and Economics Assumptions of Continuous-TimeModels 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Continuous-Sample-Path Processes with No Rare Events . . . . . . 523.3 Continuous-Sample-Path Processes with Rare Events . . . . . . . . 643.4 Discontinuous-Sample-Path Processes with Rare Events . . . . . . 68

II Optimum Consumption and Portfolio Selection in Continuous-Time Models 75

4 Lifetime Portfolio Selection Under Uncertainty: The Continuous-TimeCase 764.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2 Dynamics of the Model: The Budget Equation . . . . . . . . . . . . . 764.3 The Two-Asset Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4 Constant Relative Risk Aversion . . . . . . . . . . . . . . . . . . . . . 814.5 Dynamic Behavior and the Bequest Valuation Function . . . . . . . . 834.6 Infinite Time Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.7 Economic Interpretation of the Optimal Decision Rules for Portfolio

Selection and Consumption . . . . . . . . . . . . . . . . . . . . . . . . 864.8 Extension to Many Assets . . . . . . . . . . . . . . . . . . . . . . . . 904.9 Constant Absolute Risk Aversion . . . . . . . . . . . . . . . . . . . . 914.10 Other Extensions of the Model . . . . . . . . . . . . . . . . . . . . . . 92

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5 Optimum Consumption and Portfolio Rules in a Continuous-time Model 945.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2 A Digression on Ito Processes . . . . . . . . . . . . . . . . . . . . . . 955.3 Asset-Price Dynamics and the Budget Equation . . . . . . . . . . . . 975.4 Optimal Portfolio and Consumption Rules: The Equations of Opti-

mality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.5 Log-Normality of Prices and the Continuous-Time Analog to Tobin-

Markowitz Mean-Variance Analysis . . . . . . . . . . . . . . . . . . . 1035.6 Explicit Solutions for a Particular Class of Utility Functions . . . . . 1075.7 Noncapital Gains Income: Wages . . . . . . . . . . . . . . . . . . . . 1115.8 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.9 Alternative Price Expectations to the Geometric Brownian Motion . . 1175.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6 Further Developments in the Theory of Optimal Consumption and Port-folio Selection 1286.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.2 The Cox-Huang Alternative to Stochastic Dynamic Programming . . 130

6.2.1 The Growth-Optimum Portfolio Strategy . . . . . . . . . . . . 1306.2.2 The Cox-Huang Solution of the Intertemporal Consumption-

Investment Problem . . . . . . . . . . . . . . . . . . . . . . . 1336.2.3 The Relation Between the Cox-Huang and Dynamic Pro-

gramming Solutions . . . . . . . . . . . . . . . . . . . . . . . 1396.3 Optimal Portfolio Rules when the Nonnegativity Constraint on Con-

sumption is Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.4 Generalized Preferences and Their Impact on Optimal Portfolio De-

mands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

III Warrant and Option Pricing Theory 165

7 A Complete Model of Warrant Pricing that Maximizes Utility 1667.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.2 Cash-Stock Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . 1667.3 Recapitulation of the 1965 Model . . . . . . . . . . . . . . . . . . . . 1707.4 Determining Average Stock Yield . . . . . . . . . . . . . . . . . . . . 1727.5 Determining Warrant Holdings and Prices . . . . . . . . . . . . . . . . 1737.6 Digression: General Equilibrium Pricing . . . . . . . . . . . . . . . . 1757.7 Utility-Maximizing Warrant Pricing: The Important Incipient Case 1767.8 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.9 Warrants Never to be Converted . . . . . . . . . . . . . . . . . . . . . 1807.10 Exact Solution to the Perpetual Warrant Case . . . . . . . . . . . . . . 1827.11 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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7.12 Proof of the Superiority of Yield of Warrants Over Yield of CommonStock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8 Theory of rational option pricing 1968.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.2 Restrictions on Rational Option Pricing . . . . . . . . . . . . . . . . . 1978.3 Effects of Dividends and Changing Exercise Price . . . . . . . . . . . 2078.4 Restrictions on Rational Put Option Pricing . . . . . . . . . . . . . . . 2138.5 Rational Option Pricing Along Black-Scholes Lines . . . . . . . . . . 2168.6 An Alternative Derivation of the Black-Scholes Model . . . . . . . . 2188.7 Extension of the Model to Include Dividend Payments And Exercise

Price Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2268.8 Valuing an American Put Option . . . . . . . . . . . . . . . . . . . . . 2308.9 Valuing the Down-and-Out Call Option . . . . . . . . . . . . . . . . 2328.10 Valuing a Callable Warrant . . . . . . . . . . . . . . . . . . . . . . . . 2348.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

9 Option Pricing When Underlying Stock Returns are Discontinuous 2399.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2399.2 The Stock Price and Option Price Dynamics . . . . . . . . . . . . . . 2419.3 An Option Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . . 2469.4 A Possible Answer to an Empirical Puzzle . . . . . . . . . . . . . . . 251

10 Further Developments in Option Pricing Theory 25610.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25610.2 Cox-Ross Risk-Neutral Pricing and the Binomial Option Pricing

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25910.3 Pricing Options on Futures Contracts . . . . . . . . . . . . . . . . . . 270

IV Contingent-Claims Analysis in the Theory of Corporate Finance and Finan-cial Intermediation 277

11 A Dynamic General Equilibrium Model of the Asset Market and Its Ap-plication to the Pricing of the Capital Structure of the Firm 27811.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27811.2 A Partial-Equilibrium One-Period Model . . . . . . . . . . . . . . . . 27811.3 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28111.4 A General Intertemporal Equilibrium Model of the Asset Market . . . 28611.5 Model I: A Constant Interest Rate Assumption . . . . . . . . . . . . . 29111.6 Model II: The No Riskless Asset Case . . . . . . . . . . . . . . . . 29611.7 Model III: The General Model . . . . . . . . . . . . . . . . . . . . . . 297

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11.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

12 On the Pricing of Corporate Debt: The Risk Structure of Interest Rates 30312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30312.2 On The Pricing of Corporate Liabilities . . . . . . . . . . . . . . . . . 30412.3 On Pricing Risky Discount Bonds . . . . . . . . . . . . . . . . . . . 30712.4 A Comparative Statics Analysis of the Risk Structure . . . . . . . . . 30912.5 On the Modigliani-Miller Theorem with Bankruptcy . . . . . . . . . . 31712.6 On the Pricing of Risky Coupon Bonds . . . . . . . . . . . . . . . . . 32012.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

13 On the Pricing of Contingent Claims and the Modigliani-Miller Theorem32313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32313.2 A general derivation of a contingent claim price . . . . . . . . . . . . 32413.3 On the Modigliani-Miller Theorem with Bankruptcy . . . . . . . . . . 32813.4 Applications of Contingent-Claims Analysis in Corporate Finance . . 331

14 Financial Intermediation in the Continuous-Time Model 33714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33714.2 Derivative-Security Pricing with Transactions Costs . . . . . . . . . . 34114.3 Production Theory for Zero-Transaction-Cost Financial Intermediaries 34714.4 Risk Management for Financial Intermediaries . . . . . . . . . . . . . 35414.5 On the Role of Efficient Financial Intermediation in the Continuous-

Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36014.6 Afterword: Policy and Strategy in Financial Intermediation . . . . . . 368

V An Intertemporal Equilibrium Theory of Finance 373

15 An Intertemporal Capital Asset Pricing Model 37415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37415.2 Capital Market Structure . . . . . . . . . . . . . . . . . . . . . . . . . 37615.3 Asset Value and Rate of Return Dynamics . . . . . . . . . . . . . . . 37715.4 Preference Structure and Budget Equation Dynamics . . . . . . . . . 38115.5 The Equations of Optimality: The Demand Functions for Assets . . . 38215.6 Constant Investment Opportunity Set . . . . . . . . . . . . . . . . . . 38515.7 Generalized Separation: A Three-Fund Theorem . . . . . . . . . . . . 38615.8 The Equilibrium Yield Relationship among Assets . . . . . . . . . . . 38815.9 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39115.10An (m+ 2)-Fund Theorem and the Security Market Hyperplane . . . 39315.11The Consumption-Based Capital Asset Pricing Model . . . . . . . . . 40315.12Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

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16 A Complete-Markets General Equilibrium Theory of Finance in Contin-uous Time 41316.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41316.2 Financial Intermediation with Dynamically-Complete Markets . . . . 41616.3 Optimal Consumption and Portfolio Rules with Dynamically-Complete

Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42416.4 General Equilibrium: The Case of Pure Exchange . . . . . . . . . . . 43216.5 General Equilibrium: The Case of Production . . . . . . . . . . . . . 43716.6 A General Equilibrium Model in which the Capital Asset Pricing

Model Obtains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44016.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

VI Applications of the Continuous-Time Model to Selected Issues in Public Fi-nance: Long-Run Economic Growth, Public Pension Plans, Deposit Insurance,Loan Guarantees, and Endowment Management for Universities 455

17 An Asymptotic Theory of Growth Under Uncertainty 45617.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45617.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45717.3 The Steady-State Distribution for k . . . . . . . . . . . . . . . . . . . 46117.4 The Cobb-Douglas/Constant Savings Function Economy . . . . . . . 46217.5 The Stochastic Ramsey Problem . . . . . . . . . . . . . . . . . . . . . 466

18 On Consumption-Indexed Public Pension Plans 47718.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47718.2 A Simple Intertemporal Equilibrium Model . . . . . . . . . . . . . . . 48018.3 On the Merits and Feasibility of a Consumption-Indexed Public Plan . 486

19 An Analytic Derivation of the Cost of Loan Guarantees and Deposit In-surance: An Application of Modern Option Pricing Theory 49319.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49319.2 A model for pricing deposit insurance . . . . . . . . . . . . . . . . . . 495

20 On the Cost of Deposit Insurance When There are Surveillance Costs 50120.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50120.2 Assumptions of the Model . . . . . . . . . . . . . . . . . . . . . . . . 50220.3 The Evaluation of FDIC Liabilities . . . . . . . . . . . . . . . . . . . 50320.4 The Evaluation of Bank Equity . . . . . . . . . . . . . . . . . . . . . . 50820.5 On the Equilibrium Deposit Rate . . . . . . . . . . . . . . . . . . . . . 50920.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

xiv Preface

21 Optimal Investment Strategies for University Endowment Funds 51321.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51321.2 Overview of Basic Insights and Prescriptions for Policy . . . . . . . . 51421.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52021.4 Optimal Endowment Management with Other Sources of Income . . 526

Part I

Introduction to Finance and theMathematics of Continuous-Time

Models

Chapter 1

Modern Finance

It is generally agreed that financial management of firms and households, interme-diation, capital market and microinvestment theory, and much of the economics ofuncertainty fall within the sphere of modern finance. As is evident from its influenceon other branches of economics including public finance, industrial organization, andmonetary theory, the boundaries of this sphere, like other specialties, are both per-meable and flexible.1 The theoretical and empirical literatures covering this largeand imperfectly defined discipline are truly vast. Furthermore, the theme of exten-sive interplay between research and practical application is quite distinctive of the fi-nance corpus. Thus, even with the continuous-time qualifier in its title, this bookspromise of a general synthesis does not suffer from undue modesty of dimension.

We know that synthesis involves abstraction from the complex whole. Here, wemust be severely selective in our abstractions, since the wide-ranging scope andseemingly unbounded volume of finance researches allows only a few aspects ofthe work to be developed. Moreover, the law of comparative disadvantage rules outmy examining either empirical or applied matters in fitting detail. I therefore useannotated references and notes to only touch upon those important elements of thesubject. But, even with this understandable focus on theory, further stringent abstrac-tion remains necessary. It is surely better to abstract by concentrating in detail on asmall subset of the theory than by attempting to summarize the whole of it: hence thenotion of an anthology on finance theory from the perspective of the continuous-timemodel. And so by way of a general introduction, I survey the limited set of topics tobe covered in this book.

The core of the theory is the study of how best to allocate and deploy resourcesacross time in an uncertain environment and of the role of economic organizations infacilitating these allocations. The key organizations in finance are households, busi-ness firms, financial intermediaries, and capital markets. The tradition in neoclassicaleconomics is to take the existence of households, their tastes, and their endowmentsas exogenous to the theory. However, this tradition does not extend to other economicorganizations and institutions. They are regarded as existing primarily because of thefunctions they serve and are therefore endogenous to the theory. Thus, optimal fi-

Portions of this chapter draw heavily on Merton (1983a, 1989, 1990a).1Cf. Fischer and Merton (1984) for a discussion of topics in finance that intersect with macroeconomics.

2 Modern Finance

nancial behavior of households is derived from individual and exogenously specifiedpreference functions that rank-order alternative programs of lifetime consumptionand bequests for each household. In contrast, optimal management decisions forbusiness firms and financial intermediaries are derived from criteria determined bythe functions of those organizations within the financial economic system.

In the models of this book, business firms serve the principal functions of owningthe physical (real) assets and operating the production technologies of the econ-omy. The real assets held by firms include both tangible assets, such as machinery,factories, and land, and intangible assets, such as an ongoing organizational structure,trademarks, and patents. Purchases of these assets are financed by issuing financialsecurities, such as stocks and bonds, either to households or to financial intermedi-aries. The managers who make the capital budgeting and financing decisions for thefirms are treated as agents of the owners who are the current stockholders.2 It followsthat the primary objective of management is to operate the firm in the best interestsof the stockholders.3

In both theory and practice, financial intermediaries often act as agents of theirhousehold and business-firm customers to provide transactional services includingmoney transfer. However, their main function in the analyzes here is to act as prin-cipals and create financial instruments that, because of scale and specificity of terms,cannot be efficiently supported by direct trading in organized financial markets. De-mand deposits, commercial loans, private placement of corporate securities, mort-gages, mutual funds, annuities, and a wide range of insurance contracts are among thefinancial products offered by real-world intermediaries such as commercial banks,investment banks, thrift institutions, and insurance companies. Although some ofthese specific products and institutions are used for descriptive purposes in the text,the focus is on the economic function of financial intermediaries and their productsrather than on their particular institutional forms.

Like business firms, financial intermediaries raise capital for operations by issuingstock and debt to investors. The theory therefore assumes that managers of intermedi-

2A stereotypical corporate structure in which ownership and active management of firms are generally sep-arated is assumed throughout the book. Division of labor, comparative advantage, and economies of scaleincluding reductions of redundant information processing provide the well-known arguments for gains in eco-nomic efficiency from such separations in large and diverse economies. However, this separation inevitablyleads to conflicts between owners and managers interests that can also cause losses in efficiency. Jensen andMeckling (1976) discuss this basic principal-agent problem and present a theory of agency analyze the richset of alternative organizational forms, management incentive systems, and approaches to financial contractingthat we observe in the real world. Williamson (1988) pursues a different line of inquiry to explain these sameobservations.

3It is my understanding that, under US laws, managers are viewed as fiduciaries, which implies an evenstronger legal obligation to shareholders than if they were simply the owners agents. This stronger legalbonding does not, of course, obligate managers to benefit owners by violating other laws, nor does it prohibitmanagers from giving due consideration to other stakeholders of the firm. Common practice in finance theoryis to translate this obligation to shareholders into the simple operating imperative: maximize the current valueof the equity of the firm. Although in efficiency terms this rule works for entirely equity-financed firms in anenvironment with frictionless, competitive, and complete financial markets it does not apply in general. SeeMerton and Subrahmanyam (1974), Merton (1982a, Section 6) and Duffie (1988, Section 13) for referencesand a discussion of conditions under which value maximization produces efficient economic allocations.

3aries and business firms share the same primary objective of maximizing the interestsof current stockholders. But, unlike business firms, intermediaries hold only financialassets and, more importantly, they create explicit new liabilities whenever they selltheir products.4 Indeed, we know that in the real world the vast bulk of insurancecompany and bank liabilities are held by their customers, not their investors. Arisingfrom the specific economic function of intermediaries, this distinctive characteristicleads to a theory of managerial and regulatory behavior for intermediaries that differsfrom that derived for business firms.

The capital market is the collection of organized financial markets for trading stan-dardized securities such as stocks, bonds, futures contracts, and options.5 It providesthe central external environment connecting the financial activities of households,business firms, and intermediaries. By transacting in the capital market, firms raisethe funds necessary for investment and households deploy their savings to be used forfuture consumption. In the basic cash-flow cycle, firms sell securities to householdsand use the cash to purchase real assets for their production operations. Later, thefirms make dividend and interest payments and undertake repurchases of their secu-rities to return the cash generated by these operations to the household sector. A por-tion of these cash flows is used for current consumption and the balance is recycledthrough the capital market back to the business firms to support further investment.As with the products offered by financial intermediaries, the function of the often-elaborate menu of securities traded in the capital market is to provide householdswith risk-pooling and risk-sharing opportunities that facilitate the efficient allocationof resources.

In addition to these manifest functions, the capital market serves an important, per-haps more nearly latent, function as a key source of information that helps coordinatedecentralized decision-making in the various sectors of the economy. Security pricesand interest rates are used by households in making their consumption-saving deci-sions and in choosing the allocation of their wealth among the available assets. Thesesame prices provide critical signals to managers of firms in their selection of invest-ment projects and financings. Thus, even managers of firms with no anticipated needto transact in the capital market will nevertheless use that market to acquire infor-mation for decisions.6 Efficient separation of financial functions among specializedorganizations can obtain only if there is a corresponding separation of the informa-

4For a typical financial product, a customer pays cash to the intermediary now in return for a contractpromising a stream of state-contingent cash payments in the future. Issuance of this contract thus creates aliability for the intermediary. Of course, business firms almost always incur some implicit liabilities wheneverthey sell their products, and these become explicit in the case of product warranties. However, the magni-tude of these liabilities as a fraction of total product sales is considerably smaller for business firms than forintermediaries.

5In places, we expand this definition to include financial intermediaries and their financial products.6The conscious motivation for creating a capital market is to provide the means for financial transactions.

However, an objective consequence of this action is to produce a flow of information that is essential for allagents decision-making, including that of those agents who only rarely transact in the market. The manifestand latent functions of social behavior and organizations as a general analytical idea is developed by R. K.Merton (1957, Ch. 1).

4 Modern Finance

tion sets needed to perform each function. A common theme threaded throughout thechapters of this book, therefore, is the influence of information requirements on thedesign of financial instruments and organizations.

The exogenous-endogenous asymmetry of the treatment of households and othereconomic organizations makes examination of the optimal financial behavior of house-holds a natural entry point for study of the financial economic system. Householdsare both consumers and investors, and their financial decisions reflect those dualroles. As consumer, the household chooses how much of its income and wealth toallocate to current consumption, and thereby how much to save for future consump-tion including bequests. As investor, the household solves the portfolio-selectionproblem to determine the fractional allocations of its savings among the availableinvestment opportunities. In general, the optimal consumption-saving and portfolio-selection decisions cannot be made independently of each other. For that reason,the continuous-time model is applied first in Part II to solve the combined lifetimeconsumption and portfolio-selection problem for the individual household.

Two reference chapters precede the analyzes in Part II. Designed to provide back-ground on basic terminology, concepts, and theorems in capital-market theory, Chap-ter 2 formulates and solves the portfolio-selection problem in the classical staticframework, taking as given the households consumption decision. To help locateand connect the findings of the continuous-time model with those of the one-periodmodels, emphasis is placed on derivations of spanning and mutual-fund theorems, theRoss Arbitrage Pricing Theory (AFT) Model, and the Sharpe-Lintner-Mossin CapitalAsset Pricing Model (CAPM). Chapter 3 contains an introduction to the mathemat-ics of continuous-time processes including Itos calculus and stochastic differentialequations for mixtures of diffusion and Poisson-driven random variables.

In Chapter 4, the basic two-asset version of the lifetime consumption and portfolio-selection problem is solved by using stochastic dynamic programming. Since model-ing ideas often originate in a discrete-time setting, the analysis begins with a discrete-time formulation of the intertemporal model and then derives the continuous-timeformulation as a limiting case. This approach also permits a development of the opti-mality equations that requires no more mathematical background than Taylors theo-rem and elementary probability theory. By assuming a risky asset with log-normallydistributed returns and a riskless asset with a constant interest rate, we derive ex-plicit optimal consumption and portfolio rules for households with preferences thatexhibit either constant relative risk aversion or constant absolute risk aversion. Theintertemporal age-dependent behavior of optimal consumption is shown to be con-sistent with the Modigliani-Brumberg Life-Cycle Hypothesis. The derived optimalportfolio rules have the same structure as those prescribed in the Markowitz-Tobinmean-variance model.

In Chapter 5, the model is expanded to include wage income, uncertain lifetimes,and several assets with more general probability distributions of returns. Itos lemmais introduced as a tool for analyzing the dynamics of asset prices, wealth, and con-

5sumption. In the prototypal case of joint log-normally distributed asset returns, thederived structure of each households optimal demands for assets is such that all op-timal portfolios can be generated by simple combinations of just two portfolios. Thismutual-fund theorem is identical in form with the well-known separation theoremof the static mean-variance model. Closed-form solutions for optimal consumptionfunctions are found for members of the family of utility functions with hyperbolicabsolute risk aversion. It is further shown that these are the only time-additive andindependent preference orderings that lead to optimal consumption functions whichare linear in wealth.

In the determination of these optimal policies, the analyzes of Chapters 4 and 5 donot explicitly impose the feasibility constraints that neither consumption nor wealthcan be negative. Moreover, the models posit a single consumption good and furtherassume that preferences are time additive with no intertemporal complementarity ofconsumption. Chapter 6 focuses on exploring the robustness of the derived resultswith respect to relaxation of these assumptions. It also provides the occasion to in-troduce the important Cox-Huang application of martingale theory to the solutionof the lifetime consumption and portfolio problem. As an alternative to stochasticdynamic programming, the Cox-Huang technique is especially well suited for in-corporating these particular nonnegativity constraints. Unlike the analyzes in theunconstrained case, this analysis shows that with mild regularity conditions on pref-erences the optimal consumption and portfolio strategies of households never riskpersonal bankruptcy. Although the details of the optimal strategies are affected bythese constraints, the fundamental spanning and mutual-fund theorems of the pre-ceding unconstrained analyzes are left unchanged. This same preservation of the es-sential structure of optimal portfolio demands is also shown to obtain for householdswith preferences that depend on other variables in addition to current consumptionand age. Specific cases examined include multiple consumption goods, money-in-the-utility function, and preferences with nonzero intertemporal complementarity ofconsumption.

Having developed the optimal investment behavior of households, the analysisturns next to the pricing of financial instruments traded in the capital markets. War-rant and option pricing theory marks the earliest example of the application of continuous-time analysis.7 During the last 15 years, these highly specialized securities have be-come increasingly more important in the real world with the creation and successfuldevelopment of organized option markets. However, neither history nor commercialsuccess is the reason for the extensive treatment of option pricing in Part III. Instead,the prominent role for options analysis evolves from the fact that option-like con-tracts are found in just about every sector of the economy. As a prototypal structure,options serve as the simplest examples of securities with nonlinear sharing rules.

Just as Chapter 2 provides a bridge between static and dynamic models of portfo-lio selection, so Chapter 7 serves to connect the one-period preference-based mod-

7Cf. Bachelier (1900) and Samuelson (1965a).

6 Modern Finance

els of warrant and option pricing to the dynamic arbitrage-based pricing models.Development of modern option pricing theory begins in Chapter 8 with the deriva-tion of price restrictions that are necessary to rule out arbitrage opportunities. Notsurprisingly, these restrictions are insufficient to determine a unique set of optionprices. However, conditional on the twin assumptions of continuous trading and acontinuous-sample-path stochastic process for the underlying stock price dynamics,the seminal Black-Scholes model applies and a unique set of option prices can bederived from arbitrage considerations alone. The chapter concludes with applicationof this conditional-arbitrage model to the pricing of several types of options andwarrants.

In Chapter 9, option pricing is examined for the case in which discontinuouschanges or gaps in the underlying stock price are possible. With such jumps in pricesa possibility, the conditional-arbitrage argument used to derive the Black-Scholesmodel is no longer valid. However, with the further assumption that the discontinu-ous components of stock-price changes are diversifiable risks, an equilibrium modelof option prices is derived.

Since the publication of the Black-Scholes model in 1973, there has been an ex-plosion of theoretical, applied, and empirical research on option pricing. The impactof that explosion has even been felt beyond the borders of economics. In appliedstatistics, it created renewed interest in the estimation of variance rates for stochasticprocesses. In numerical methods, it stimulated extensive new research on numericalintegration of the partial differential equations that must be solved to determine theBlack-Scholes option prices. Hence, in Chapter 10 on further developments in optionpricing analysis, we can do little more than give references to the many directions ofexpansions and extensions of the subject. There are three topics, however, with-out which even the most rudimentary presentation of modern option pricing theorywould be conspicuously incomplete. Thus, Chapter 10 contains a more detailed de-velopment of the Cox-Ross risk-neutral pricing methodology, the binomial optionpricing model, and the basic theory for pricing options on futures contracts.

Contingent-claims analysis (CCA) combines the dynamic portfolio theory of PartII with the Black-Scholes option pricing model of Part III to create one of the mostpowerful tools of analysis in modern finance. In Part IV, CCA is used to study awide range of topics in the theory of corporate finance and financial intermediation.Continuing the study of financial instruments initiated in the preceding part, Chapters11-13 develop a unified theory for the pricing of corporate liabilities. Like Chapter 7,Chapter 11 provides a transition from the one-period utility-based theory of pricingto an intertemporal theory based on conditional arbitrage. In the process, it also pro-vides an introduction to the pricing of general capital assets and to the determinationof the term structure of interest rates, topics which are developed more fully in PartV.

The most common division of a firms capital structure is between debt and equity.Chapter 12 investigates the pricing of corporate debt and levered equity, beginning

7with the simplest, nontrivial capital structure of a single homogeneous zero-couponbond issue and equity. The analysis shows that corporate debt can be representedfunctionally as a combination of default-free debt and unlevered equity. Just as theterm structure uses maturity to distinguish among bond yields, so the risk structureof interest rates uses default risk to distinguish among promised yields. The derivedpricing model for corporate bonds is used to define the risk structure and compara-tive statics are applied to demonstrate the effects of parameter changes on that struc-ture. The analysis is extended to include corporate coupon bonds. The importantModigliani-Miller theorem that the value of the firm is invariant to its choice of debt-equity mix is shown to obtain even in the presence of bankruptcy possibilities.

In Chapter 13, a model for pricing general contingent claims or derivative securi-ties8 is developed using continuous-time dynamic portfolio theory. With frictionlessmarkets and no taxes, the Modigliani-Miller theorem is proved for firms with generalcorporate-liability structures. It is well known that the Modigliani-Miller theoremgenerally fails in an environment of corporate taxes and deductibility of interest pay-ments. Therefore, the chapter includes a discussion of modifications to the pricingmodel that are sufficient to accommodate the effects of taxes on financing choice.The analysis concludes with a survey of the applications of CCA to corporate fi-nance issues that range from investment and financing decisions to the evaluation ofcorporate strategy.

The subject of Chapter 14 is the theory of financial intermediation with focus onthe risk-pooling and risk-sharing services provided by intermediaries. The discussionis organized around three categories of contributions of continuous-time analysis tothe theory and practice of financial intermediation: product identification, productimplementation and pricing, and risk management and control for an intermediarysentire portfolio.

The theory holds that financial intermediaries and derivative-security markets areessentially redundant organizations in an idealized environment in which all investorspay neither transactions costs nor taxes, have the same information, and can tradecontinuously. To provide an important economic function for these activities in thetheory, therefore, the posited environment must include some type of transaction orinformation cost structure in which financial intermediaries and market makers havea comparative advantage with respect to some investors and corporate issuers of se-curities. Thus, the formal analysis of Chapter 14 begins with a simple binomialmodel of derivative-security pricing in the presence of transactions costs. This anal-ysis serves to demonstrate that bid-ask spreads in these prices can be substantial andthat significant economic benefits can accrue from efficient intermediation. With thisestablished in a simple model, we then explore the theory of intermediation usinga continuous-time model in which many agents cannot trade without cost, but thelowest-cost transactors, financial intermediaries, can. This model both provides a

8Derivative securities are securities with contractual payoff structures that are contingent on the prices ofone or more traded securities.

8 Modern Finance

raison detre for derivative-security products and allows the use of standard CCA todetermine the costs for intermediaries to produce them.

The theory of optimal consumption and portfolio choice is used to identify cus-tomer demands for a list of intermediary products ranging from generic multipurposemutual funds to custom-designed financial contracts tailored to fit each investorsspecific needs. The analysis also contributes to the theory of product implementationby specifying in detail the production technologies and costs for intermediaries tomanufacture these products. The same CCA and dynamic portfolio-selection toolsare used to examine the problem of overall risk management for an intermediary. Inpreparation for the equilibrium analysis in Part V, we discuss the role that an efficientfinancial intermediation system plays in justifying models that assume dichotomybetween the real and financial sectors of the economy. The chapter concludes witha few observations on policy and strategy issues drawn from the continuous-timetheory of intermediation.

The separate investigations of the main organizations comprising the financialeconomy are brought together in Part V to provide an equilibrium analysis of theentire system. Optimal financial behavior of individual agents and organizations isdescribed by a set of contingent decisions or plans that depend on current prices andprobability beliefs about the future evolution of the economy. The conditions sat-isfied by an equilibrium set of prices and beliefs are that all optimal plans can beimplemented at each point in time (market clearing) and that the resulting ex posttime path of the economy is consistent with ex ante probability assessments (ratio-nal expectations). In the analyzes, it is assumed that all agents and organizations areprice-takers in the capital markets and that financial intermediation is a competitiveindustry. A standard Wairasian setting is posited as the mechanism to clear marketsat each point in time.

The centerpiece of the classic CAPM is the Security Market Line, a linear equationthat relates the equilibrium expected return on each asset (or portfolio of assets) toa single identifiable risk measure. Reduced-form equations of this genus are amongthe most widely and frequently used analytical tools in both applied and empiricalfinance. The focus of Chapter 15 is on establishing similar types of necessary con-ditions to be satisfied by intertemporal equilibrium prices of capital assets. We showthat in the intertemporal Capital Asset Pricing Model, equilibrium expected returnsare linearly related to a vector of risk measures. This reduced-form equation of thecontinuous-time model is called the Security Market Hyperplane. Under somewhatmore restrictive conditions, the Breeden Consumption-Based Capital Asset PricingModel (CCAPM) is shown to obtain. In this important version of the intertemporalmodel, equilibrium expected returns can again be expressed in terms of a single riskmeasure. That risk measure, however, is different from the one in the static CAPM.

Further results on investor hedging behavior and mutual-fund theorems derivedin Chapter 15 add important detail to the product identification part of the financialintermediation theory in Chapter 14. However, the model itself does not explicitly

9include either derivative-security markets or much of the intermediation sector, a sim-plification justified by the quasi-dichotomy findings of that chapter. Moreover, as inthe development of the original CAPM, the analysis emphasizes the demand side ofthe capital markets and thus treats as largely exogenous the dynamics of the supplycurves for securities. Therefore the model does not provide all the structural equa-tions of endogenous behavior required for a full equilibrium analysis of the system.Such is the subject matter of Chapter 16.

For this general equilibrium study of the financial economy, the continuous-timemodel is reformulated to fit the framework of the Arrow-Debreu model of completemarkets. As we know, an Arrow-Debreu pure security provides a positive payoff atonly one point in time and in only one state of the economy. In an Arrow-Debreueconomy, pure securities are traded for each possible state of the economy at everyfuture date. It follows that the continuous-time version of an Arrow-Debreu economyrequires a continuum of such securities. However, as we shall see, financial interme-diaries and other zero-cost transactors can synthetically create a complete set of puresecurities by trading continuously in a finite number of securities. The continuous-time model thus provides a concrete demonstration of the Arrow and Radner observa-tion that dynamic trading in securities can be a substitute for a full set of pure-securitymarkets. Building on the theory of intermediation in Chapter 14, we derive explicitformulas for the trading technologies and production costs required to manufacturepure securities. The specificity of these theoretical findings suggests the possibility offeasible approximations in real-world financial markets to the seemingly unrealisticrequirements of complete markets in an Arrow-Debreu world.

Having established the mechanism by which a complete set of pure securities iscreated, we then solve the lifetime consumption allocation problem for households,using static-optimization techniques as in the original Arrow-Debreu analysis. Thesestate-contingent optimal demand functions are used to derive equilibrium allocations,prices, and rational expectations for the case of a pure-exchange economy. Produc-tion and optimal investment behavior of business firms are added in a second compet-itive equilibrium model. A third, more specific version shows that the prescriptionsof the original CAPM can obtain in an intertemporal general equilibrium model withproduction.

The formal analyzes of these 16 chapters can be used in the study of internationalfinance and comparative financial systems. The posited structure of organizationsand their functions, however, is modeled after the US system. Moreover, the insti-tutional discussions surrounding those analyzes abstract from the special issues thatarise with intersystem transactions which cross sovereign borders. Happily, there isan extensive and accessible literature on applications of the continuous-time financemodel that addresses the particular characteristics of an international setting.9

9Adler and Dumas (1983) and Branson and Henderson (1984) provide excellent surveys on the applica-tion of the continuous-time model in international finance, an application pioneered by Solnik (1973, 1974).See also Kouri (1976, 1977), Kouri and de Macedo (1978), Stulz (1981, 1984), de Macedo, Goldstein andMeerschwam (1984), Eun (1985), Dumas (1988), Dixit (1989a), and Penati and Pennacchi (1989).

10 Modern Finance

The general equilibrium analysis of Part V follows the traditional line of separa-tion between macroeconomics and finance and excludes the public-sector componentfrom its model of the financial system. That analysis, therefore, does not explicitlycapture the effects of central bank and other government activities on the financialeconomy. However, the concluding part of the book contains five essays that applycontinuous-time analysis to selected topics on the border between private and publicfinance.

In Chapter 17, a Ramsey-Solow macro model is developed to analyze the dy-namics of long-run economic growth in an economy with uncertainty about eitherdemographics or technological progress. The focus of the analysis is on the biasesintroduced in long-run forecasting and planning by neglecting uncertainty. Closed-form solutions for the steady-state probability distributions of the relevant economicvariables are derived for the case of a Cobb-Douglas production function and a con-stant proportional savings function. The analysis concludes with an examination ofthe stochastic Ramsey problem of central planning.

The subject of Chapter 18 is the design of public pension plans. Both academicsand practitioners have often suggested that pension annuities be indexed to protectretirees against inflation. However, such annuities do not provide protection againstreal gains in the standard of living. A new type of plan that indexes benefits to aggre-gate per capita consumption is investigated as a possible solution for protecting pen-sioners against both of these risks. The analysis posits a simple model for mortalitywith the standard assumption that individual mortality risk is diversifiable across thepopulation. The equilibrium prices of consumption-indexed annuities are derived byapplying the competitive arbitrage techniques of Chapter 9. Under the condition thathousehold preferences satisfy the Life-Cycle Hypothesis, the optimal consumption-saving model of Chapter 5 is used to determine the required contribution rate to thepension plan that ensures an optimal level of retirement benefits. The formal analysisis followed by a brief discussion of the feasibility of implementing such a plan.

An important topic in both private and public finance is the evaluation of loan guar-antees. The next two chapters use CCA to develop generic models for pricing theseguarantees and apply them to the analysis of deposit insurance. Third-party guar-antees of financial performance on loans and other debt-related contracts are widelyused throughout the USA and other well-developed economies. Parent corporationsoften guarantee the debt obligations of their subsidiaries. In return for fees, com-mercial banks offer guarantees on a broad spectrum of financial instruments rangingfrom letters of credit to interest rate and currency swaps. More specialized firms sellguarantees of interest and principal payments on tax-exempt municipal bonds. How-ever, the largest provider of financial guarantees is the federal government, eitherdirectly or through its agencies. In the corporate sector, it has guaranteed loans tosmall businesses and on occasion, as with Lockheed Aircraft and the Chrysler Cor-poration, it has done so for very large businesses. Established in 1980, the UnitedStates Synthetic Fuels Corporation was empowered to grant loan guarantees to assist

11

the financing of commercial projects that involve the development of alternative fueltechnologies. Through the Pension Benefit Guaranty Corporation, the governmentprovides limited insurance of corporate pension-plan benefits. Residential mortgagesand farm and student loans are examples of noncorporate obligations that the govern-ment has guaranteed. But perhaps the most important of its liability guarantees, botheconomically and politically, is deposit insurance.

The Federal Deposit Insurance Corporation (FDIC) and the Federal Savings andLoan Insurance Corporation (FSLIC) insure the deposits of commercial banks andthrift institutions up to a maximum of US $100,000 per account.10 With such in-surance, the economic responsibility for monitoring a banks activities shifts fromits depositors to the insurer. In a frictionless world with only liquid assets and nosurveillance costs, the insurer would continuously monitor the value of those assetsin relation to deposits. It could thereby avoid any losses by simply forcing liquidationof the assets before the insolvency point is reached. But, of course, most bank assetsare not liquid and there are surveillance costs, and so our models for determining thecost of deposit insurance take account of both the potential losses from insolvencyand the cost of monitoring. The evaluation potential losses from insolvency andthe cost of monitoring. The evaluation of deposit insurance is further complicatedin the real world because, unlike some private-sector guarantees, deposit insuranceobligations are not traded in markets. The government, therefore, must estimate theactuarial cost of providing this insurance, without the benefit of market prices.11 Aswe shall see, implementation of CCA generally does not require historical price dataon the security to be priced. This evaluation technique of Chapters 19 and 20 is thusa well-suited appraisal tool for estimating the costs of deposit guarantees.

The concluding chapter of the book analyzes optimal portfolio and expenditurerules for university endowment funds. A common approach to the management ofendowment is to treat it in isolation as if it were the only asset of the university.This approach leads to rather uniform prescriptions for optimal investment and ex-penditure policies. In contrast, the model in Chapter 21 follows a more integrativeapproach to endowment management that takes account of the universitys overallobjectives and total resources. Inclusion of other university assets (in addition toendowment) in the analysis leads to significantly different optimal portfolios andexpenditure patterns among universities with similar objectives and similar sized en-dowments, but different non-endowment sources of cash flow. The model also takesaccount of the uncertainty surrounding the costs of the various activities such as edu-cation, research, and knowledge storage that define the purpose of the university. Asa result, the analysis reveals a perhaps somewhat latent role for endowment: namely,

10After passage of the Financial Institutions, Reform, Recovery and Enforcement Act in August 1989,FSLIC was replaced by the Savings Association Insurance Fund and the Resolution Trust Corporation.

11Even if the government were not to charge for deposit insurance, it would still require such cost estimatesto determine the amount of the subsidy given to banks their depositors. The federal and provincial governmentsof Canada use loan guarantees to subsidize local corporations (cf. Baldwin, Lessard, and Mason, 1983). Itis my understanding that they have built models using the CCA methodology to evaluate the costs of thesesubsidies.

12 Modern Finance

hedging against unanticipated changes in those costs.This completes the itinerary for our journey through the continuous-time theory

of finance. I hope that, by journeys end, some readers find it a compact and cod-ified method for organizing their understanding of financial activities and the inter-actions among them. But, before embarkation, I must add a word of caution againstabsolutizing the theory. As with much of neoclassical economics, the foundationof modern finance theory rests on the perfect-market paradigm of rational behaviorand frictionless, competitive, and informationally efficient capital markets. With itsfurther assumption of continuous trading, the base of our theory should perhaps belabeled the super perfect-market paradigm. The conditions of this paradigm are not,of course, literally satisfied in the real world. Furthermore, its accuracy as a usefulapproximation to that world varies considerably across time and place.12 The prac-titioner should therefore apply the continuous-time theory only tentatively, assessingits limitations in each application. Just so, the researcher should treat it as a point ofdeparture for both problem finding and problem solving.

12A general discussion of the limits of the perfect-market paradigm can be found Merton (1987a, b). Caveatson the specific assumptions of the continuous-time model are dispersed throughout the book. However, seethe concluding sections of Chapters 14 and 16, especially. Information efficiency in the sense of the Efficient-Market Hypothesis of Samuelson (1965b, 1973) and Fama (1965a, 1970a, 1991) requires that the dynamicsof speculative prices satisfy rational expectations. In the models here, we impose the rational expectationsassumption only in the equilibrium analyzes of Part V and Chapter 20.

Chapter 2

Introduction to Portfolio Selection andCapital Market Theory: Static Analysis

2.1 Introduction

It is convenient to view the investment decision by households as having two parts:(a) the consumption-saving choice where the individual decides how much incomeand wealth to allocate to current consumption and how much to save for future con-sumption; and (b) the portfolio-selection choice where the investor decides how toallocate savings among the available investment opportunities. In general, the twodecisions cannot be made independently. However, many of the important findingsin portfolio theory can be more easily derived in a one-period environment where theconsumption-savings allocation has little substantive impact on the results. Thus, webegin in Section 2.2 with the formulation and solution of the basic portfolio-selectionproblem in a static framework, taking as given the individuals consumption deci-sion.

Using the analysis of Section 2.2, we derive necessary conditions for static finan-cial equilibrium that are used to determine restrictions on equilibrium security pricesand returns in Sections 2.3 and 2.4. In Section 2.4, these restrictions are used to de-rive spanning and mutual-fund theorems that provide a basis for an elementary theoryof financial intermediation.1

2.2 One-Period Portfolio Selection

The basic investment-choice problem for an individual is to determine the optimalallocation of his or her wealth among the available investment opportunities. The so-lution to the general problem of choosing the best investment mix is called portfolio-selection theory. The study of portfolio- selection theory begins with its classic one-period or static formulation.

Reproduced from Handbook of Monetary Economics, H. Friedman and F. Hahn, eds, 1990, Amsterdam:North-Holland. This chapter includes Sections 2-4 of Merton (1990a), which is a revised and expanded versionof Merton (1982a).

1Spanning and mutual-fund theorems for the intertemporal continuous-time model are developed in Chap-ters 5, 15, and 16. The role of financial intermediation in this model is discussed in Chapter 14.

14 Introduction to Portfolio Selection and Capital Market Theory: Static Analysis

There are n different investment opportunities called securities and the randomvariable one-period return per dollar on security j is denoted Zj (j = 1, . . . , n) where adollar is the unit of account. Any linear combination of these securities which hasa positive market value is called a portfolio. It is assumed that the investor choosesat the beginning of a period that feasible portfolio allocation which maximizes theexpected value of a von Neumann-Morgenstern utility function2 for end-of-periodwealth. Denote this utility function by U(W ), where W is the end-of-period value ofthe investors wealth measured in dollars. It is further assumed that U is an increasingstrictly concave function on the range of feasible values for W and that U is twicecontinuously differentiable.3 Because the criterion function for choice depends onlyon the distribution of end-of-period wealth, the only information about the securitiesthat is relevant to the investors decision is his subjective joint probability distributionfor (Z1, . . . , Zn).

In addition, the following assumptions are made.

Assumption 1 (Frictionless Markets).There are no transactions costs or taxes, and all securities are perfectly divisible.

Assumption 2 (Price-Taker).The investor believes that his actions cannot affect the probability distribution of re-turns on the available securities. Hence, if wj is the fraction of the investors initialwealth W0 allocated to security j, then {w1, . . . , wn} uniquely determines the proba-bility distribution of his terminal wealth.

A riskless security is defined to be a security or feasible portfolio of securitieswhose return per dollar over the period is known with certainty.

Assumption 3 (No-Arbitrage Opportunities).All riskless securities must have the same return per dollar. This common return willbe denoted by R.

Assumption 4 (No-Institutional Restrictions).Short-sales of all securities, with full use of proceeds, are allowed without restriction.If there exists a riskless security, then the borrowing rate equals the lending rate.4

2von Neumann and Morgenstern (1947). For an axiomatic description, see Herstein and Milnor (1953)and Machina (1982). Although the original axioms require that U be bounded, the continuity axiom can beextended to allow for unbounded functions. See Samuelson (1977) for a discussion of this and the St Petersburgparadox.

3The strict concavity assumption implies that investors are everywhere risk averse. Although strictly con-vex or linear utility functions on the entire range imply behavior that is grossly at variance with observedbehavior, the strict concavity assumption also rules out Friedman-Savage type utility functions whose behav-ioral implications are reasonable. The strict concavity also implies U (W ) > 0, which rules out individualsatiation.

4Borrowings and short-sales are demand loans collateralized by the investors total portfolio. The borrow-ing rate is the rate on riskless-in-terms-of-default loans. Although virtually every individual loan involvessome chance of default, the empirical spread in the rate on actual margin loans to investors suggests thatthis assumption is not a bad approximation for portfolio-selection analysis. However, explicit analyzes ofrisky-loan evaluation are provided in Chapters 11-14 and Chapters 19 and 20. See also Merton (1990b, pp.272-85).

2.2 One-Period Portfolio Selection 15

Hence, the only restriction on the choice for the {wj} is the budget constraint thatn1 wj = 1.

Given these assumptions, the portfolio-selection problem can be formally statedas

max{w1,...,wn}

E

U n

j=1

wjZjW0

(2.1)subject to

n1 wj = 1, where E is the expectation operator for the subjective joint

probability distribution. If (w1, . . . , wn) is a solution to (2.1), then it will satisfy thefirst-order conditions

E{U (ZW0)Zj

}=

W0j = 1, 2, . . . , n (2.2)

where the prime denotes derivative, Z n1 wjZj is the random variable returnper dollar on the optimal portfolio, and is the Lagrange multiplier for the budgetconstraint. Together with the concavity assumptions on U , if the n n variance-covariance matrix of the returns (Z1, . . . , Zn) is non-singular and an interior solutionexists, then the solution is unique.5 This nonsingularity condition on the returnsdistribution eliminates redundant securities (i.e. securities whose returns can beexpressed as exact linear combinations of the returns on other available securities).6

It also rules out that any one of the securities is a riskless security.If a riskless security is added to the menu of available securities (call it the (n+1)th

security), then it is the convention to express (2.1) as the following unconstrainedmaximization problem:

max{w1,...,wn}

E

U n

j=1

wj(Zj R) +RW0

(2.3)where the portfolio allocations to the risky securities are unconstrained because thefraction allocated to the riskless security can always be chosen to satisfy the budgetconstraint (i.e. wn+1 = 1

n1 w

j ). The first-order conditions can be written as

E{U (ZW0)(Zj R)} = 0 j = 1, 2, . . . , n (2.4)

where Z can be rewritten asn

1 wj (Zj R) + R. Again, if it is assumed that the

variance-covariance matrix of the returns on the risky securities is nonsingular andan interior solution exists, then the solution is unique.

As formulated, neither (2.1) nor (2.3) reflects the physical constraint that end-of-period wealth cannot be negative. That is, no explicit consideration is given to thetreatment of bankruptcy. To rule out bankruptcy, the additional constraint that, with

5The existence of an interior solution is assumed throughout the analyzes. For a discussion of necessaryand sufficient conditions for the existence of an interior solution, see Leland (1972) and Berlsekas (1974).

6For a trivial example, shares of IBM with odd serial numbers are distinguishable from ones with evenserial numbers and are therefore technically different securities. However, because their returns are identical,they are perfect substitutes from the point of view of investors. In portfolio theory, securities are operationallydefined by their return distributions, and therefore two securities with identical returns are indistinguishable.

16 Introduction to Portfolio Selection and Capital Market Theory: Static Analysis

probability one, Z 0 could be imposed on the choices for (w1, . . . , wn).7 If, how-ever, the purpose of this constraint is to reflect institutional restrictions designed toavoid individual bankruptcy, then it is too weak, because the probability assessmentson the {Zj} are subjective. An alternative treatment is to forbid borrowing and short-selling in conjunction with limited-liability securities where, by law, Zj 0. Theserules can be formalized as restrictions on the allowable set of {wj}, such that wj 0,j = 1, 2, . . . , n + 1, and (2.1) or (2.3) can be solved using the methods of Kuhn andTucker (1951) for inequality constraints. In Chapters 6, 13, 14, and 16, we formallyanalyze portfolio behavior and the pricing of securities when both investors and secu-rity lenders recognize the prospect of default. Thus, until those chapters, it is simplyassumed that there exists a bankruptcy law which allows for U(W ) to be defined forW < 0, and that this law is consistent with the continuity and concavity assumptionson U .

The optimal demand functions for risky securities, {wjW0}, and the resultingprobability distribution for the optimal portfolio will, of course, depend on the riskpreferences of the investor, his initial wealth, and the joint distribution for the secu-rities returns. It is well known that the von Neumann-Morgenstern utility functioncan only be determined up to a positive affine transformation. Hence the prefer-ence orderings of all choices available to the investor are completely specified by thePratt-Arrow8 absolute risk-aversion function, which can be written as

A(W ) U(W )

U (W )(2.5)

and the change in absolute risk aversion with respect to a change in wealth is thereforegiven by

dA

dW= A(W ) = A(W )

[A(W ) +

U (W )U (W )

](2.6)

By the assumption that U(W ) is increasing and strictly concave, A(W ) is positive,and such investors are called risk averse. An alternative, but related, measure of riskaversion is the relative risk-aversion function defined by

R(W ) U(W )WU (W )

= A(W )W (2.7)

and its change with respect to a change in wealth is given by

R(W ) = A(W )W + A(W ) (2.8)

The certainty-equivalent end-of-period wealth Wc, associated with a given portfo-lio for end-of-period wealth whose random variable value is denoted by W , is definedto be such that

U(Wc) = E{U(W )} (2.9)7If U is such that U (0) = and, by extension, U (W ) =, W < 0, then from (2.2) or (2.4) it is easy to

show that the probability of Z 0 is a set of measure zero. Mason (1981) has studied the effects of variousbankruptcy rules on portfolio behavior. See also Chapter 6.

8The behavior associated with the utility functions V (W ) aU(W ) + b, a > 0, is identical with thatassociated with U(W ). Note that A(W ) is invariant to any positive affine transformation of U(W ). See Pratt(1964).

2.2 One-Period Portfolio Selection 17

i.e. Wc is the amount of money such that the investor is indifferent between havingthis amount of money for certain or the portfolio with random variable outcome W .The term risk averse as applied to investors with strictly concave utility functions isdescriptive in the sense that the certainty-equivalent end-of-period wealth is alwaysless than the expected value E{W} of the associated portfolio for all such investors.The proof follows directly by Jensens inequality: if U is strictly concave, then

U(Wc) = E{U(W )} < U(E{W})

whenever W has positive dispersion, and because U is an increasing function of W ,Wc < E{W}.

The certainty equivalent can be used to compare the risk aversions of two in-vestors. An investor is said to be more risk averse than a second investor if, for everyportfolio, the certainty-equivalent end-of-period wealth for the first investor is lessthan or equal to the certainty-equivalent end-of-period wealth associated with thesame portfolio for the second investor with strict inequality holding for at least oneportfolio.

While the certainty equivalent provides a natural definition for comparing riskaversions across investors, Rothschild and Stiglitz9 have in a corresponding fashionattempted to define the meaning of increasing risk for a security so that the risk-iness of two securities or portfolios can be compared. In comparing two portfolioswith the same expected values, the first portfolio with random variable outcome de-noted by W1 is said to be less risky than the second portfolio with random variableoutcome denoted by W2 if

E{U(W1)} E{U(W2)} (2.10)

for all concave U with strict inequality holding for some concave U . They bolstertheir argument for this definition by showing its equivalence to the two followingdefinitions:

There exists a random variable Z such that W2 has the same distribution as

W1 + Z where the conditional expectation of Z given the outcome of W1 is

zero (i.e. W2 is equal in distribution to W1 plus some noise).(2.11)

If the points of F and G, the distribution functions of W1 and W2, are

confined to the closed interval [a, b], and T (y) ya[G(x) F (x)]dx, then

T (y) 0 and T (b) = 0 (i.e. W2 has more weight in its tails than W1).(2.12)

A feasible portfolio with return per dollar Z will be called an efficient portfolio ifthere exists an increasing strictly concave function V such that E{V (Z)(ZjR)} = 0,j = 1, 2, . . . , n. Using the Rothschild-Stiglitz definition of less risky, a feasibleportfolio will be an efficient portfolio only if there does not exist another feasible

9Rothschild and Stiglitz (1970, 1971). There is an extensive literature, not discussed here, that uses thistype of risk measure to determine when one portfolio stochasticafly dominates another. Cf. Hadar andRussell (1969, 1971), Hanoch and Levy (1969), and Bawa (1975).

18 Introduction to Portfolio Selection and Capital Market Theory: Static Analysis

portfolio which is less risky than it is. All portfolios that are not efficient are calledinefficient portfolios.

From the definition of an efficient portfolio, it follows that no two portfolios inthe efficient set can be ordered with