TheTheoryofFinance

Eugene F. FamaMerton H. MillerGraduate School of Business

The University of Chicago

Holt , Rinehart and WinstonNew York Chicago San Francisco AtlantaDallas Montreal Toronto London Sydney

CONTENTS

Preface vii

I CERTAINTY MODELS 1

Chapter 1 A Model of the Accumulation andAllocation of Wealth by Individuals 3

I . THE ECONOMIC THEORY OF CHOICE 3I .A. Opportunities and Preferences 3I .B. Representation of Preferences : The Utility Function and

Indifference Curves 4I .B.l . The axioms of choice and the principle of maximum utility 5I .B.2. Indifference curves and the geometrical representation of

preferences 7I .B.3. The axioms of nonsatiation and convexity 9I .C. The Opportunity Set 12I .D. Choice Subject to Constraints 13

*I.E . The Solution in Mathematical Form 15II . THE APPLICATION OF THE THEORY OF CHOICE

TO THE ALLOCATION OF FINANCIAL RESOURCESOVER TIME 16

II .A. The Two-Period Case 16II .A.l . The objects of choice: standards of living at different points

in time 17*11.A.2. The properties of U(o, c2, . . ., Ct,. . .) 19II .A.3. Opportunities : resources and capital markets 20II .A.4. The opportunity set under perfect capital markets 22II .A.5. Interest rates and present values 24II .A.6. The preferred allocation 27II .B. Extension to Three or More Time Periods 28II .B.l . The structure of prices for claims 29II .B.2. An equivalent representation in terms of interest rates 31II .B.3. Multiperiod rates of interest and the concept of the term

structure 34II .B.4. The equal rate of return principle 36II .B.5. The re-period opportunity set 38

*11.B.6. The optimal allocation in the multiperiod case; mathematicaltreatment 39

II .C. Conclusion 40

Appendix : An Illustrative Application of the Wealth AllocationModel : The Life Cycle of Savings 42

I . THE BASIC FRAMEWORK 42I .A . Smoothing Irregularities in the Income Stream 42

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I .B. Some Specifications and Simplifications 43I .C. Consumption , Saving , and Wealth over the Life Cycle 44I .D . Consumption and Saving with a Growing Income 47I .E . The Effects of Changes in Income and Wealth 51

II . CONSUMPTION , SAVING , AND THE RATE OF INTEREST 51II .A. Some Additional Assumptions : Homogeneity and Its Implications 51II .B. Interest Rates and Consumption Decisions over the Life Cycle 53

Chapter 2 Extension of the Model to Durable Commodities,Production , and Corporations 58

I . DURABLE COMMODITIES AND INVESTMENT 59I .A. Representation of the Carry-over Opportunities Provided

by Commodity Storage and Production 59I .B. The Opportunity Set When Both Commodity Carry-overs

and Capital Markets Are Available 61I .C. The Case of Production and Investment 62I .C.l . The double-tangency solution .62I .C.2. Some remaining issues 64

*I.D. The Solution in the «-Period Case 64II . EXTENSION TO THE CASE OF CORPORATIONS 67II .A. Management Objectives and Stockholder Preferences 67II .B. The Market Value Criterion 69II .B.l . The case of many owners 70

*11.C. The Market Value Criterion in the General «-Period Case 72II .D. The Market Value Criterion: Some Problems and

Limitations 73II .D.l . The market value rule and profit maximization 74II .D.2. The market value rule and management motivation 75II .D.3. The criterion problem under imperfect capital markets 76

III . MARKET VALUE, DIVIDENDS , ANDSTOCKHOLDER RETURNS 78

III .A. The Effect of Dividend Policy 78III .A.l . Assumptions, notation, and timing conventions 78III .A.2. The equal rate of return principle once again 79III .B. The Effects of Dividend Policy on the Market Value of the

Shares 80III .C. A Graphical Illustration 81III .D, The Independence Proposition Further Considered 84III .D.l . The bird-in-the-hand argument 84III .D.2. Dividend policy and the internal rate of return 85III .E. Some Equivalent Alternative Valuation Formulas 86III .E.l . The stream of dividends approach 86III .E.2. Cash flow and earnings approaches 87III .E.3. Investment opportunities, growth, and valuation 89III .F. Growth Potential and Stockholder Returns 92III .F.l . The constant growth model 94III .F.2. The growth of total earnings and the growth of dividends

and price per share 94III .F.3. Corporate earnings and investor returns 96IV. SUMMARY 97

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Appendix : Valuation Formulas : Some Numerical Illustrations 100

I . FINANCIAL POLICY , RETURNS , AND VALUATION 100II . AN EXAMPLE USING THE CONSTANT GROWTH

MODEL 104

Chapter 3 Criteria for Optimal Investment Decisions 108

I . THE REPRESENTATION OF OPTIMALINVESTMENT DECISIONS 109

I .A. The Case of a Single Capital Good and Two Time Periods 109I .B. Graphical Representation of the Complete Solution 111I . C. An Alternative Representation Highlighting the Investment

Decision 115I . C.l . Alternative forms for the optimizing conditions 117I . C.2. Optimal capital stock and optimal investment 118I .D. Extension to the Case of Many Different Machines 119I .E . The Investment Decision and the Transformation Curve 121I .F . Extension to More than Two Time Periods 122I . F .l . The case of perfect markets for capital goods 122I . F .2. The case of fixed capital 123

II . INVESTMENT DECISIONS AND CAPITALBUDGETING 126

II .A. Problems in the Application of the Present Value Criterion 126II .A.l . Marginal versus average present values 127II .A.2. Comparing investments with different lives 128II .B. Replacement Policies and the Optimal Economic Life of

Equipment 130II .C. Maximizing Present Value Subject to Constraints 134II .D . The Rate of Return Criterion : Uses and Abuses 137II .D .l . The discounted cash flow rate of return 137II .D .2. Mutually exclusive investments and multiple rates of return 139II .D .3. A modified rate of return rule for the mutually exclusive case 140II .E . The Rate of Return , Rankings of Projects , and Financial

Constraints 142II . F . Conclusion 143

Chapter 4 Financing Decisions , Investment Decisions,and the Cost of Capital 147

I . INTRODUCTION 147II . MARKET SETTING 148

III . CAPITAL STRUCTURE AND MARKET VALUES 150III .A. Two-Period Market Equilibrium Model 150III .B . Multiperiod Model 156III .C. Two Partial Equilibrium Treatments 157III .C.l . Two-period states of the world model 157III . C.2. The market value of a firm in the two-period risk class model 160III .C.3. The market value of a firm in a multiperiod risk class model 164III .C.4. The effects of financing decisions on the firm’s bondholders

and shareholders 157III .C.5. Summary 170III .D . Market Imperfections : The Effects of Tax Laws 170

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III .D.2. The tax deductibility of personal interest payments 174III .D.3. Some closing comments on market imperfections 175IV. THK FIRM ’S OBJECTIVE FUNCTION : THE

MARKET VALUE RULE 176IV.A. The Market Value Rule: Derivation 176IV.B. The Market Value Rule: Implementation 178V. THE COST OF CAPITAL AND THE RETURN ON

A FIRM ’S SHARES 181V.A. Discount Rates and the Cost of Capital 181V.B. The Expected Return on Common Stock 184

VI. SUMMARY AND CONCLUSIONS 187

Chapter 5 The Expected Utility Approach to the Problem ofChoice Under Uncertainty 189

I. INTRODUCTION 189II . THE EXPECTED UTILITY MODEL: GENERAL

AXIOMATIC TREATMENT 190II .A. The Axiom System 192II .B. Derivation of the Expected Utility Rule 194II .C. Some Properties of the Utility Functions Implied by the

Expected Utility Model 196III . THE TIMELESS EXPECTED UTILITY OF

WEALTH MODEL 198III .A. Obtaining the Utility of Wealth Function from Axiom4 199III .B. Usual Types of Utility of Wealth Functions: Risk Aversion,

Risk Preference, and Risk Neutrality 200IV. EXPECTED UTILITY AND THE THEORY OF

FINANCE 203IV.A. The Multiperiod Expected Utility of Consumption Model 203IV.B. Utilities for Consumption Dollars from Utilities for

Consumption Goods 206V. CONCLUSION 207

Appendix : Statistical Review 209

I . INTRODUCTION 209II . EXPECTED VALUES OF WEIGHTED SUMS OF

RANDOM VARIABLES 209III . THE VARIANCE OF A WEIGHTED SUM OF

RANDOM VARIABLES 211

Chapter 6 The Two-Period Consumption-Investment Model 215I . INTRODUCTION 215I .A. The Mean- Standard Deviation Model: An Overview 216I .B. A Familiar Picture 220

II . THE CONSUMER’S TASTES^ _ 222*11.A. Portfolio Decisions Based on E (R) and a (R) 223*11.B. Marginal Expected Utilities and the Efficient Set Theorem 224*11.C. Properties of Indifference Curves in the Two-Parameter Model 226III . THE INVESTMENT OPPORTUNITY SET:

TWO-ASSET CASE 228

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IV . THE EFFICIENT SET AND CONSUMEREQUILIBRIUM WITH N ASSETS 233

IV .A. Definitions and Elementary Expressions for the V-Asset Model 234IV .B. Geometric Representation of Consumer Equilibrium 236

*IV.C. Algebraic Representation of Consumer Equilibrium 243IV .D . Some Odds and Ends of the Mean - Standard Deviation Model 250IV .D .l . The investment assets to be included in portfolio models 250IV .D .2. The effects of diversification : algebraic treatment 253IV .D .3. Quadratic utility functions 256IV .D .4. Why normality ? 259V. THE TWO -PARAMETER MODEL WITH

SYMMETRIC -STABLE DISTRIBUTIONS OFPORTFOLIO RETURNS 261

V.A. Properties of Stable Distributions : A Brief Review 261V.B . Representation of the Consumer ’s Tastes : The Efficient

Set Theorem 265V.C. The Opportunity Set with Stable Return Distributions 267V.C.l . The market model 267V.C.2. Diversification and the distribution of portfolio return 269V.C.3. The efficient set 271V.C.4. Consumer equilibrium 273

VI . CONCLUSIONS 274

Chapter 7 Risk, Return, and Market Equilibrium 276I . INTRODUCTION 276

II . THE MARKET SETTING 277III . RISK AND EXPECTED RETURN FROM THE

VIEWPOINT OF A CONSUMER 279III .A. The Risks of Assets and Portfolios 279III .B . The Relationship between Risk and Expected Return 281IV . RISK AND EXPECTED RETURN FOR THE MARKET 286IV.A. Homogeneous Expectations and Portfolio Opportunities 287IV.B. The Role of a Riskless Asset 288IV.C. Interpretation 290IV.C.l . Risk and expected return in the market portfolio 290IV.C.2. Homogeneous expectations and the riskless asset : a closer

look 292V. EQUILIBRIUM EXPECTED RETURN AND THE

MARKET VALUE OF A FIRM 295VI . OPTIMAL PRODUCTION DECISIONS BY FIRMS 299VI.A. The Firm ’s Objective Function 299VI. B. Optimal Production Decisions : Single-Activity Firms 301VI. C. Optimal Production Decisions : Multiple -Activity Firms 304VI .D . Optimal Production Decisions : Some Comments 307

*VH. ALGEBRAIC TREATMENT OF CAPITAL MARKETEQUILIBRIUM 308

VII .A. Consumer Equilibrium 309VII .B. Equilibrium for the Firm 311VII .C. Market Equilibrium 312

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Vili . MARKET EQUILIBRIUM WITH SYMMETRIC-STABLE RETURN DISTRIBUTIONS 313

VIH.A. The Market Model Return -Generating Process 313Vili .B. Consumer Equilibrium and the Measurement of Risk 314Vili .C. Risk and Return for the Market 317IX . CONCLUSIONS 318

Chapter 8 Multiperiod Models 321

I . INTRODUCTION 321II . MULTIPERIOD CONSUMPTION-INVESTMENT

DECISIONS 322II .A. The Problem 322II .B. The Wealth Allocation Model 323II .C. Implications : Bridging the Gap between Two-Period and

Multiperiod Models 326II .C.l . The utility of money function 327II .C.2. Two-period and multiperiod models: general treatment 327II .C.3. A multiperiod setting for two-period two-parameter portfolio

models 329II .C.4 Theorems and proofs 332

III . EFFICIENT CAPITAL MARKETS 335III .A. Expected Return or Fair Game Models 336III .B. The Submartingale Model 338III .C. The Random Walk Model 339

Index 343