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Interview study book Principles of Financial Economics of Stephen F. LeRoy, Jan Werner - ISBN: 9781316060872 (Study Notes)
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Principles of Financial EconomicsStephen F. LeRoy
University of California, Santa Barbara
and
Jan Werner
University of Minnesota
@ March 10, 2000, Stephen F. LeRoy and Jan Werner
,Contents
I Equilibrium and Arbitrage 1
1 Equilibrium in Security Markets 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Security Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Consumption and Portfolio Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 First-Order Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Left and Right Inverses of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Existence and Uniqueness of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 8
1.9 Representative Agent Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Linear Pricing 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Law of One Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 The Payoff Pricing Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Linear Equilibrium Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 State Prices in Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Recasting the Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Arbitrage and Positive Pricing 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Arbitrage and Strong Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 A Diagrammatic Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Positivity of the Payoff Pricing Functional . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Positive State Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6 Arbitrage and Optimal Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.7 Positive Equilibrium Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Portfolio Restrictions 29
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Short Sales Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Portfolio Choice under Short Sales Restrictions . . . . . . . . . . . . . . . . . . . . . 30
4.4 The Law of One Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5 Limited and Unlimited Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 Diagrammatic Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.7 Bid-Ask Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.8 Bid-Ask Spreads in Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
i
,ii CONTENTS
II Valuation 39
5 Valuation 41
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 The Fundamental Theorem of Finance . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Bounds on the Values of Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 The Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.5 Uniqueness of the Valuation Functional . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 State Prices and Risk-Neutral Probabilities 51
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 State Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3 Farkas-Stiemke Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4 Diagrammatic Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.5 State Prices and Value Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.6 Risk-Free Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.7 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 Valuation under Portfolio Restrictions 61
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.2 Payoff Pricing under Short Sales Restrictions . . . . . . . . . . . . . . . . . . . . . . 61
7.3 State Prices under Short Sales Restrictions . . . . . . . . . . . . . . . . . . . . . . . 62
7.4 Diagrammatic Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.5 Bid-Ask Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
III Risk 71
8 Expected Utility 73
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.2 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.3 Von Neumann-Morgenstern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.4 Savage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.5 Axiomatization of State-Dependent Expected Utility . . . . . . . . . . . . . . . . . . 74
8.6 Axiomatization of Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.7 Non-Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.8 Expected Utility with Two-Date Consumption . . . . . . . . . . . . . . . . . . . . . 77
9 Risk Aversion 83
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.2 Risk Aversion and Risk Neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.3 Risk Aversion and Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
9.4 Arrow-Pratt Measures of Absolute Risk Aversion . . . . . . . . . . . . . . . . . . . . 85
9.5 Risk Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.6 The Pratt Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.7 Decreasing, Constant and Increasing Risk Aversion . . . . . . . . . . . . . . . . . . . 88
9.8 Relative Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
9.9 Utility Functions with Linear Risk Tolerance . . . . . . . . . . . . . . . . . . . . . . 89
9.10 Risk Aversion with Two-Date Consumption . . . . . . . . . . . . . . . . . . . . . . . 90
, CONTENTS iii
10 Risk 93
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.2 Greater Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.3 Uncorrelatedness, Mean-Independence and Independence . . . . . . . . . . . . . . . . 94
10.4 A Property of Mean-Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
10.5 Risk and Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
10.6 Greater Risk and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
10.7 A Characterization of Greater Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
IV Optimal Portfolios 103
11 Optimal Portfolios with One Risky Security 105
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.2 Portfolio Choice and Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.3 Optimal Portfolios with One Risky Security . . . . . . . . . . . . . . . . . . . . . . . 106
11.4 Risk Premium and Optimal Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.5 Optimal Portfolios When the Risk Premium Is Small . . . . . . . . . . . . . . . . . . 108
12 Comparative Statics of Optimal Portfolios 113
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
12.2 Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
12.3 Expected Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
12.4 Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12.5 Optimal Portfolios with Two-Date Consumption . . . . . . . . . . . . . . . . . . . . 117
13 Optimal Portfolios with Several Risky Securities 123
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
13.2 Optimal Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
13.3 Risk-Return Tradeoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
13.4 Optimal Portfolios under Fair Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 124
13.5 Risk Premia and Optimal Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
13.6 Optimal Portfolios under Linear Risk Tolerance . . . . . . . . . . . . . . . . . . . . . 127
13.7 Optimal Portfolios with Two-Date Consumption . . . . . . . . . . . . . . . . . . . . 129
V Equilibrium Prices and Allocations 133
14 Consumption-Based Security Pricing 135
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
14.2 Risk-Free Return in Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
14.3 Expected Returns in Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
14.4 Volatility of Marginal Rates of Substitution . . . . . . . . . . . . . . . . . . . . . . . 137
14.5 A First Pass at the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
15 Complete Markets and Pareto-Optimal Allocations of Risk 143
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.2 Pareto-Optimal Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.3 Pareto-Optimal Equilibria in Complete Markets . . . . . . . . . . . . . . . . . . . . . 144
15.4 Complete Markets and Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
15.5 Pareto-Optimal Allocations under Expected Utility . . . . . . . . . . . . . . . . . . . 146
15.6 Pareto-Optimal Allocations under Linear Risk Tolerance . . . . . . . . . . . . . . . . 148
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