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Principles of Financial Economics
Stephen 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
iv CONTENTS
16 Optimality in Incomplete Security Markets 153
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
16.2 Constrained Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
16.3 Effectively Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
16.4 Equilibria in Effectively Complete Markets . . . . . . . . . . . . . . . . . . . . . . . 155
16.5 Effectively Complete Markets with No Aggregate Risk . . . . . . . . . . . . . . . . . 157
16.6 Effectively Complete Markets with Options . . . . . . . . . . . . . . . . . . . . . . . 157
16.7 Effectively Complete Markets with Linear Risk Tolerance . . . . . . . . . . . . . . . 158
16.8 Multi-Fund Spanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
16.9 A Second Pass at the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
VI Mean-Variance Analysis 165
17 The Expectations and Pricing Kernels 167
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
17.2 Hilbert Spaces and Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
17.3 The Expectations Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
17.4 Orthogonal Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
17.5 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
17.6 Diagrammatic Methods in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 170
17.7 Riesz Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
17.8 Construction of the Riesz Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
17.9 The Expectations Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
17.10The Pricing Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
18 The Mean-Variance Frontier Payoffs 179
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
18.2 Mean-Variance Frontier Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
18.3 Frontier Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
18.4 Zero-Covariance Frontier Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
18.5 Beta Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
18.6 Mean-Variance Efficient Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
18.7 Volatility of Marginal Rates of Substitution . . . . . . . . . . . . . . . . . . . . . . . 183
19 CAPM 187
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
19.2 Security Market Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
19.3 Mean-Variance Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
19.4 Equilibrium Portfolios under Mean-Variance Preferences . . . . . . . . . . . . . . . . 190
19.5 Quadratic Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
19.6 Normally Distributed Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
20 Factor Pricing 197
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
20.2 Exact Factor Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
20.3 Exact Factor Pricing, Beta Pricing and the CAPM . . . . . . . . . . . . . . . . . . . 199
20.4 Factor Pricing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
20.5 Factor Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
20.6 Mean-Independent Factor Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
20.7 Options as Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
CONTENTS v
VII Multidate Security Markets 209
21 Equilibrium in Multidate Security Markets 211
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
21.2 Uncertainty and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
21.3 Multidate Security Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
21.4 The Asset Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
21.5 Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
21.6 Portfolio Choice and the First-Order Conditions . . . . . . . . . . . . . . . . . . . . 214
21.7 General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
22 Multidate Arbitrage and Positivity 219
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
22.2 Law of One Price and Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
22.3 Arbitrage and Positive Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
22.4 One-Period Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
22.5 Positive Equilibrium Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
23 Dynamically Complete Markets 225
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
23.2 Dynamically Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
23.3 Binomial Security Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
23.4 Event Prices in Dynamically Complete Markets . . . . . . . . . . . . . . . . . . . . . 227
23.5 Event Prices in Binomial Security Markets . . . . . . . . . . . . . . . . . . . . . . . . 227
23.6 Equilibrium in Dynamically Complete Markets . . . . . . . . . . . . . . . . . . . . . 228
23.7 Pareto-Optimal Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
24 Valuation 233
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
24.2 The Fundamental Theorem of Finance . . . . . . . . . . . . . . . . . . . . . . . . . . 233
24.3 Uniqueness of the Valuation Functional . . . . . . . . . . . . . . . . . . . . . . . . . 235
VIII Martingale Property of Security Prices 239
25 Event Prices, Risk-Neutral Probabilities and the Pricing Kernel 241
25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
25.2 Event Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
25.3 Risk-Free Return and Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 243
25.4 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
25.5 Expected Returns under Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . 245
25.6 Risk-Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
25.7 Value Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
25.8 The Pricing Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
26 Security Gains As Martingales 251
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
26.2 Gain and Discounted Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
26.3 Discounted Gains as Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
26.4 Gains as Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
vi CONTENTS
27 Conditional Consumption-Based Security Pricing 257
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
27.2 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
27.3 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
27.4 Conditional Covariance and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
27.5 Conditional Consumption-Based Security Pricing . . . . . . . . . . . . . . . . . . . . 259
27.6 Security Pricing under Time Separability . . . . . . . . . . . . . . . . . . . . . . . . 260
27.7 Volatility of Intertemporal Marginal Rates of Substitution . . . . . . . . . . . . . . . 261
28 Conditional Beta Pricing and the CAPM 265
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
28.2 Two-Date Security Markets at a Date-t Event . . . . . . . . . . . . . . . . . . . . . . 265
28.3 Conditional Beta Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
28.4 Conditional CAPM with Quadratic Utilities . . . . . . . . . . . . . . . . . . . . . . . 267
28.5 Multidate Market Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
28.6 Conditional CAPM with Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . 269
Introduction
Financial economics plays a far more prominent role in the training of economists than it did even
a few years ago.
This change is generally attributed to the parallel transformation in capital markets that has
occurred in recent years. It is true that trillions of dollars of assets are traded daily in financial
markets—for derivative securities like options and futures, for example—that hardly existed a
decade ago. However, it is less obvious how important these changes are. Insofar as derivative
securities can be valued by arbitrage, such securities only duplicate primary securities. For example,
to the extent that the assumptions underlying the Black-Scholes model of option pricing (or any of
its more recent extensions) are accurate, the entire options market is redundant, since by assumption
the payoff of an option can be duplicated using stocks and bonds. The same argument applies to
other derivative securities markets. Thus it is arguable that the variables that matter most—
consumption allocations—are not greatly affected by the change in capital markets. Along these
lines one would no more infer the importance of financial markets from their volume of trade than
one would make a similar argument for supermarket clerks or bank tellers based on the fact that
they handle large quantities of cash.
In questioning the appropriateness of correlating the expanding role of finance theory to the
explosion in derivatives trading we are in the same position as the physicist who demurs when
journalists express the opinion that Einstein’s theories are important because they led to the devel-
opment of television. Similarly, in his appraisal of John Nash’s contributions to economic theory,
Myerson [13] protested the tendency of journalists to point to the FCC bandwidth auctions as
indicating the importance of Nash’s work. At least to those with some curiosity about the phys-
ical and social sciences, Einstein’s and Nash’s work has a deeper importance than television and
the FCC auctions! The same is true of finance theory: its increasing prominence has little to
do with the expansion of derivatives markets, which in any case owes more to developments in
telecommunications and computing than in finance theory.
A more plausible explanation for the expanded role of financial economics points to the rapid
development of the field itself. A generation ago finance theory was little more than institutional
description combined with practitioner-generated rules of thumb that had little analytical basis
and, for that matter, little validity. Financial economists agreed that in principle security prices
ought to be amenable to analysis using serious economic theory, but in practice most did not devote
much effort to specializing economics in this direction.
Today, in contrast, financial economics is increasingly occupying center stage in the economic
analysis of problems that involve time and uncertainty. Many of the problems formerly analyzed
using methods having little finance content now are seen as finance topics. The term structure of
interest rates is a good example: formerly this was a topic in monetary economics; now it is a topic
in finance. There can be little doubt that the quality of the analysis has improved immensely as a
result of this change.
Increasingly finance methods are used to analyze problems beyond those involving securities
prices or portfolio selection, particularly when these involve both time and uncertainty. An example
is the “real options” literature, in which finance tools initially developed for the analysis of option
vii
viii CONTENTS
markets are applied to areas like environmental economics. Such areas do not deal with options
per se, but do involve problems to which the idea of an option is very much relevant.
Financial economics lies at the intersection of finance and economics. The two disciplines are
different culturally, more so than one would expect given their substantive similarity. Partly this
reflects the fact that finance departments are in business schools and are oriented towards finance
practitioners, whereas economics departments typically are in liberal arts divisions of colleges and
universities, and are not usually oriented toward any single nonacademic community.
From the perspective of economists starting out in finance, the most important difference is that
finance scholars typically use continuous-time models, whereas economists use discrete time models.
Students do not fail to notice that continuous-time finance is much more difficult mathematically
than discrete-time finance, leading them to ask why finance scholars prefer it. The question is
seldom discussed. Certainly product differentiation is part of the explanation, and the possibility
that entry deterrence plays a role cannot be dismissed. However, for the most part the preference
of finance scholars for continuous-time methods is based on the fact that the problems that are
most distinctively those of finance rather than economics—valuation of derivative securities, for
example—are best handled using continuous-time methods. The reason is technical: it has to
do with the effect of risk aversion on equilibrium security prices in models of financial markets.
In many settings risk aversion is most conveniently handled by imposing a certain distortion on
the probability measure used to value payoffs. It happens that (under very weak restrictions)
in continuous time the distortion affects the drifts of the stochastic processes characterizing the
evolution of security prices, but not their volatilities (Girsanov’s Theorem). This is evident in the
derivation of the Black-Scholes option pricing formula.
In contrast, it is easy to show using examples that in discrete-time models distorting the un-
derlying measure affects volatilities as well as drifts. As one would expect given that the effect
disappears in continuous time, the effect in discrete time is second-order in the time interval. The
presence of these higher-order terms often makes the discrete-time versions of valuation problems
intractable. It is far easier to perform the underlying analysis in continuous time, even when one
must ultimately discretize the resulting partial differential equations in order to obtain numerical
solutions. For serious students of finance, the conclusion from this is that there is no escape from
learning continuous-time methods, however difficult they may be.
Despite this, it is true that the appropriate place to begin is with discrete-time and discrete-
state models—the maintained framework in this book—where the economic ideas can be discussed
in a setting that requires mathematical methods that are standard in economic theory. For most
of this book (Parts I - VI) we assume that there is one time interval (two dates) and a single
consumption good. This setting is most suitable for the study of the relation between risk and
return on securities and the role of securities in allocation of risk. In the rest (Parts VII - VIII),
we assume that there are multiple dates (a finite number). The multidate model allows for gradual
resolution of uncertainty and retrading of securities as new information becomes available.
A little more than ten years ago the beginning student in Ph.D.-level financial economics had
no alternative but to read journal articles. The obvious disadvantage of this is that the ideas
are not set out systematically, so that authors typically presuppose, often unrealistically, that the
reader already understands prior material. Alternatively, familiar material may be reviewed, often
in painful detail. Typically notation varies from one article to the next. The inefficiency of this
process is evident.
Now the situation is the reverse: there are about a dozen excellent books that can serve as
texts in introductory courses in financial economics. Books that have an orientation similar to
ours include Krouse [9], Milne [12], Ingersoll [8], Huang and Litzenberger [5], Pliska [16] and
Ohlson [15]. Books that are oriented more toward finance specialists, and therefore include more
material on valuation by arbitrage and less material on equilibrium considerations, include Hull [7],
Dothan [3], Baxter and Rennie [1], Wilmott, Howison and DeWynne [18], Nielsen [14] and Shiryaev
CONTENTS ix
[17]. Of these, Hull emphasizes the practical use of continuous-finance tools rather than their
mathematical justification. Wilmott, Howison and DeWynne approach continuous-time finance
via partial differential equations rather than through risk-neutral probabilities, which has some
advantages and some disadvantages. Baxter and Rennie give an excellent intuitive presentation of
the mathematical ideas of continuous-time finance, but do not discuss the economic ideas at length.
Campbell, Lo and MacKinlay [2] stress empirical and econometric issues. The authoritative text
is Duffie [4]. However, because Duffie presumes a very thorough mathematical preparation, that
book may not be the place to begin.
There exist several worthwhile books on subjects closely related to financial economics. Excel-
lent introductions to the economics of uncertainty are Laffont [10] and Hirshleifer and Riley [6].
Magill and Quinzii [11] is a fine exposition of the economics of incomplete markets in a more general
setting than that adopted here.
Our opinion is that none of the finance books cited above adequately emphasizes the connection
between financial economics and general equilibrium theory, or sets out the major ideas in the
simplest and most direct way possible. We attempt to do so. We understand that some readers
have a different orientation. For example, finance practitioners often have little interest in making
the connection between security pricing and general equilibrium, and therefore want to proceed to
continuous-time finance by the most direct route possible. Such readers might do better beginning
with books other than ours.
This book is based on material used in the introductory finance field sequence in the economics
departments of the University of California, Santa Barbara and the University of Minnesota, and in
the Carlson School of Management of the latter. At the University of Minnesota it is now the basis
for a two-semester sequence, while at the University of California, Santa Barbara it is the basis for
a one-quarter course. In a one-quarter course it is unrealistic to expect that students will master
the material; rather, the intention is to introduce the major ideas at an intuitive level. Students
writing dissertations in finance typically sit in on the course again in years following the year they
take it for credit, at which time they digest the material more thoroughly. It is not obvious which
method of instruction is more efficient.
Our students have had good preparation in Ph.D.-level microeconomics, but have not had
enough experience with economics to have developed strong intuitions about how economic models
work. Typically they had no previous exposure to finance or the economics of uncertainty. When
that was the case we encouraged them to read undergraduate-level finance texts and the introduc-
tions to the economics of uncertainty cited above. Rather than emphasizing technique, we have
tried to discuss results so as to enable students to develop intuition.
After some hesitation we decided to adopt a theorem-proof expository style. A less formal
writing style might make the book more readable, but it would also make it more difficult for us
to achieve the level of analytical precision that we believe is appropriate in a book such as this.
We have provided examples wherever appropriate. However, readers will find that they will
assimilate the material best if they make up their own examples. The simple models we consider
lend themselves well to numerical solution using Mathematica or Mathcad; although not strictly
necessary, it is a good idea for readers to develop facility with methods for numerical solution of
these models.
We are painfully aware that the placid financial markets modeled in these pages bear little
resemblance to the turbulent markets one reads about in the Wall Street Journal. Further, attempts
to test empirically the models described in these pages have not had favorable outcomes. There is
no doubt that much is missing from these models; the question is how to improve them. About
this there is little consensus, which is why we restrict our attention to relatively elementary and
noncontroversial material. We believe that when improved models come along, the themes discussed
here—allocation and pricing of risk—will still play a central role. Our hope is that readers of this
book will be in a good position to develop these improved models.
x CONTENTS
We wish to acknowledge conversations about these ideas with many of our colleagues at the
University of California, Santa Barbara and University of Minnesota. The second author has
also taught material from this book at Pompeu Fabra University and University of Bonn. Jack
Kareken read successive drafts of parts of this book and made many valuable comments. The book
has benefited enormously from his attention, although we do not entertain any illusions that he
believes that our writing is as clear and simple as it could and should be. Our greatest debt is to
several generations of Ph.D. students at the University of California, Santa Barbara and University
of Minnesota. Comments from Alexandre Baptista have been particularly helpful. They assure us
that they enjoy the material and think they benefit from it. Remarkably, the assurances continue
even after grades have been recorded and dissertations signed. Our students have repeatedly and
with evident pleasure accepted our invitations to point out errors in earlier versions of the text.
We are grateful for these corrections. Several ex-students, we are pleased to report, have gone
on to make independent contributions to the body of material introduced here. Our hope and
expectation is that this book will enable others who we have not taught to do the same.
Bibliography
[1] Martin Baxter and Andrew Rennie. Financial Calculus. Cambridge University Press, Cam-
bridge, 1996.
[2] John Y. Campbell, Andrew W. Lo, and A. Craig MacKinlay. The Econometrics of Financial
Markets. Princeton University Press, Princeton, NJ, 1996.
[3] Michael U. Dothan. Prices in Financial Markets. Oxford U. P., New York, 1990.
[4] Darrell Duffie. Dynamic Asset Pricing Theory, Second Edition. Princeton University Press,
Princeton, N. J., 1996.
[5] Chi fu Huang and Robert Litzenberger. Foundations for Financial Economics. North-Holland,
New York, 1988.
[6] Jack Hirshleifer and John G. Riley. The Analytics of Uncertainty and Information. Cambridge
University Press, Cambridge, 1992.
[7] John C. Hull. Options, Futures and Other Derivative Securities. Prentice-Hall, 1993.
[8] Jonathan E. Ingersoll. Theory of Financial Decision Making. Rowman and Littlefield, Totowa,
N. J., 1987.
[9] Clement G. Krouse. Capital Markets and Prices: Valuing Uncertain Income Stream. North-
Holland, New York, 1986.
[10] Jean-Jacques Laffont. The Economics of Uncertainty and Information. MIT Press, Cambridge,
MA., 1993.
[11] Michael Magill and Martine Quinzii. Theory of Incomplete Markets. MIT Press, 1996.
[12] Frank Milne. Finance Theory and Asset Pricing. Clarendon Press, Oxford, UK, 1995.
[13] Roger Myerson. Nash equilibrium and the history of economic theory. Journal of Economic
Literature, XXXVII:1067–1082, 1999.
[14] Lars T. Nielsen. Pricing and Hedging of Derivative Securities. Oxford University Press, Oxford,
U. K., 1999.
[15] Jomes A. Ohlson. The Theory of Financial Markets and Information. North-Holland, New
York, 1987.
[16] Stanley R. Pliska. Introduction to Mathematical Finance: Discrete Time Models. Oxford
University Press, Oxford, 1997.
[17] Albert N. Shiryaev. Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific
Publishing Co., River Edge, NJ, 1999.
xi
xii BIBLIOGRAPHY
[18] P. Wilmott, S. Howison, and H. DeWynne. The Mathematics of Financial Derivatives. Cam-
bridge University Press, Cambridge, UK, 1995.
Part I
Equilibrium and Arbitrage
1
Chapter 1
Equilibrium in Security Markets
1.1 Introduction
The analytical framework in the classical finance models discussed in this book is largely the same
as in general equilibrium theory: agents, acting as price-takers, exchange claims on consumption
to maximize their respective utilities. Since the focus in financial economics is somewhat different
from that in mainstream economics, we will ask for greater generality in some directions, while
sacrificing generality in favor of simplification in other directions.
As an example of the former, it will be assumed that markets are incomplete: the Arrow-Debreu
assumption of complete markets is an important special case, but in general it will not be assumed
that agents can purchase any imaginable payoff pattern on security markets. Another example is
that uncertainty will always be explicitly incorporated in the analysis. It is not asserted that there
is any special merit in doing so; the point is simply that the area of economics that deals with the
same concerns as finance, but concentrates on production rather than uncertainty, has a different
name (capital theory).
As an example of the latter, it will generally be assumed in this book that only one good is
consumed, and that there is no production. Again, the specialization to a single-good exchange
economy is adopted only in order to focus attention on the concerns that are distinctive to finance
rather than microeconomics, in which it is assumed that there are many goods (some produced),
or capital theory, in which production economies are analyzed in an intertemporal setting.
In addition to those simplifications motivated by the distinctive concerns of finance, classical
finance shares many of the same restrictions as Walrasian equilibrium analysis: agents treat the
market structure as given, implying that no one tries to create new trading opportunities, and
the abstract Walrasian auctioneer must be introduced to establish prices. Markets are competitive
and free of transactions costs (except possibly costs of certain trading restrictions, as analyzed in
Chapter 4), and they clear instantaneously. Finally, it is assumed that all agents have the same
information. This last assumption largely defines the term “classical”; much of the best work now
being done in finance assumes asymmetric information, and therefore lies outside the framework of
this book.
However, even students whose primary interest is in the economics of asymmetric information
are well advised to devote some effort to understanding how financial markets work under symmetric
information before passing to the much more difficult general case.
1.2 Security Markets
Securities are traded at date 0 and their payoffs are realized at date 1. Date 0, the present, is
certain, while any of S states can occur at date 1, representing the uncertain future.
3
4 CHAPTER 1. EQUILIBRIUM IN SECURITY MARKETS
Security j is identified by its payoff x
j
, an element of R
S
, where x
js
denotes the payoff the holder
of one share of security j receives in state s at date 1. Payoffs are in terms of the consumption
good. They may be positive, zero or negative. There exists a finite number J of securities with
payoffs x
1
, . . . , x
J
, x
j
∈ R
S
, taken as given.
The J × S matrix X of payoffs of all securities
X =
x
1
x
2
.
.
.
x
J
(1.1)
is the payoff matrix . Here, vectors x
j
are understood to be row vectors. In general, vectors are
understood to be either row vectors or column vectors as the context requires.
A portfolio is composed of holdings of the J securities. These holdings may be positive, zero or
negative. A positive holding of a security means a long position in that security, while a negative
holding means a short position (short sale). Thus short sales are allowed (except in Chapters 4 and
7).
A portfolio is denoted by a J-dimensional vector h, where h
j
denotes the holding of security j.
The portfolio payoff is
P
j
h
j
x
j
, and can be represented as hX.
The set of payoffs available via trades in security markets is the asset span, and is denoted by
M:
M = {z ∈ R
S
: z = hX for some h ∈ R
J
}. (1.2)
Thus M is the subspace of R
S
spanned by the security payoffs, that is, the row span of the payoff
matrix X. If M = R
S
, then markets are complete. If M is a proper subspace of R
S
, then markets
are incomplete. When markets are complete, any date-1 consumption plan—that is, any element
of R
S
—can be obtained as a portfolio payoff, perhaps not uniquely.
1.2.1 Theorem
Markets are complete iff the payoff matrix X has rank S.
1
Proof: Asset span M equals the whole space R
S
iff the equation z = hX, with J unknowns
h
j
, has a solution for every z ∈ R
S
. A necessary and sufficient condition for that is that X has
rank S.
2
A security is redundant if its payoff can be generated as the payoff of a portfolio of other
securities. There are no redundant securities iff the payoff matrix X has rank J.
The prices of securities at date 0 are denoted by a J-dimensional vector p = (p
1
, . . . , p
J
). The
price of portfolio h at security prices p is ph =
P
j
p
j
h
j
.
The return r
j
on security j is its payoff x
j
divided by its price p
j
(assumed to be nonzero; the
return on a payoff with zero price is undefined):
r
j
=
x
j
p
j
. (1.3)
Thus “return” means gross return (“net return” equals gross return minus one). Throughout we
will be working with gross returns.
Frequently the practice in the finance literature is to specify the asset span using the returns
on the securities rather than their payoffs, so that the asset span is the subspace of R
S
spanned by
the returns of the securities.
The following example illustrates the concepts introduced above:
1
Here and throughout this book, “A iff B”, an abbreviation for “A if and only if B”, has the same meaning as “A
is equivalent to B” and as “for A to be true, B is a necessary and sufficient condition”. Therefore proving necessity
in “A iff B” means proving “A implies B”, while proving sufficiency means proving “B implies A”.
1.3. AGENTS 5
1.2.2 Example
Let there be three states and two securities. Security 1 is risk free and has payoff x
1
= (1, 1, 1).
Security 2 is risky with x
2
= (1, 2, 2). The payoff matrix is
"
1 1 1
1 2 2
#
.
The asset span is M = {(z
1
, z
2
, z
3
) : z
1
= h
1
+h
2
, z
2
= h
1
+2h
2
, z
3
= h
1
+2h
2
, for some (h
1
, h
2
)}—
a two-dimensional subspace of R
3
. By inspection, M = {(z
1
, z
2
, z
3
) : z
2
= z
3
}. At prices p
1
= 0.8
and p
2
= 1.25, security returns are r
1
= (1.25, 1.25, 1.25) and r
2
= (0.8, 1.6, 1.6).
2
1.3 Agents
In the most general case (pending discussion of the multidate model), agents consume at both
dates 0 and 1. Consumption at date 0 is represented by the scalar c
0
, while consumption at date
1 is represented by the S-dimensional vector c
1
= (c
11
, . . . , c
1S
), where c
1s
represents consumption
conditional on state s. Consumption c
1s
will be denoted by c
s
when no confusion can result.
At times we will restrict the set of admissible consumption plans. The most common restriction
will be that c
0
and c
1
be positive.
2
However, when using particular utility functions it is generally
necessary to impose restrictions other than, or in addition to, positivity. For example, the loga-
rithmic utility function presumes that consumption is strictly positive, while the quadratic utility
function u(c) = −
P
S
s=1
(c
s
− α)
2
has acceptable properties only when c
s
≤ α. However, under the
quadratic utility function, unlike the logarithmic function, zero or negative consumption poses no
difficulties.
There is a finite number I of agents. Agent i’s preferences are indicated by a continuous utility
function u
i
: R
S+1
+
→ R, in the case in which admissible consumption plans are restricted to be
positive, with u
i
(c
0
, c
1
) being the utility of consumption plan (c
0
, c
1
). Agent i’s endowment is w
i
0
at date 0 and w
i
1
at date 1.
A securities market economy is an economy in which all agents’ date-1 endowments lie in the
asset span. In that case one can think of agents as endowed with initial portfolios of securities (see
Section 1.7)
Utility function u is increasing at date 0 if u(c
0
0
, c
1
) ≥ u(c
0
, c
1
) whenever c
0
0
≥ c
0
for every c
1
,
and increasing at date 1 if u(c
0
, c
0
1
) ≥ u(c
0
, c
1
) whenever c
0
1
≥ c
1
for every c
0
. It is strictly increasing
at date 0 if u(c
0
0
, c
1
) > u(c
0
, c
1
) whenever c
0
0
> c
0
for every c
1
, and strictly increasing at date 1 if
u(c
0
, c
0
1
) > u(c
0
, c
1
) whenever c
0
1
> c
1
for every c
0
. If u is (strictly) increasing at date 0 and at date
1, then u is (strictly) increasing .
Utility functions and endowments typically differ across agents; nevertheless, the superscript i
will frequently be deleted when no confusion can result.
2
Our convention on inequalities is as follows: for two vectors x, y ∈ R
n
,
x ≥ y means that x
i
≥ y
i
∀I; x is greater than y
x > y means that x ≥ y and x 6= y; x is greater than but not equal to y
x À y means that x
i
> y
i
∀i; x is strictly greater than y.
For a vector x, positive means x ≥ 0, positive and nonzero means x > 0, and strictly positive means x À 0. These
definitions apply to scalars as well. For scalars, “positive and nonzero” is equivalent to “strictly positive”.
6 CHAPTER 1. EQUILIBRIUM IN SECURITY MARKETS
1.4 Consumption and Portfolio Choice
At date 0 agents consume their date-0 endowments less the value of their security purchases. At
date 1 they consume their date-1 endowments plus their security payoffs. The agent’s consumption
and portfolio choice problem is
max
c
0
,c
1
,h
u(c
0
, c
1
) (1.4)
subject to
c
0
≤ w
0
− ph (1.5)
c
1
≤ w
1
+ hX, (1.6)
and a restriction that consumption be positive, c
0
≥ 0, c
1
≥ 0, if that restriction is imposed.
When, as in Chapters 11 and 13, we want to analyze an agent’s optimal portfolio abstracting
from the effects of intertemporal consumption choice, we will consider a simplified model in which
date-0 consumption does not enter the utility function. The agent’s choice problem is then
max
c
1
,h
u(c
1
) (1.7)
subject to
ph ≤ w
0
(1.8)
and
c
1
≤ w
1
+ hX. (1.9)
1.5 First-Order Conditions
If utility function u is differentiable, the first-order conditions for a solution to the consumption
and portfolio choice problem 1.4 – 1.6 (assuming that the constraint c
0
≥ 0, c
1
≥ 0 is imposed) are
∂
0
u(c
0
, c
1
) − λ ≤ 0, (∂
0
u(c
0
, c
1
) − λ)c
0
= 0 (1.10)
∂
s
u(c
0
, c
1
) − µ
s
≤ 0, (∂
s
u(c
0
, c
1
) − µ
s
)c
s
= 0 , ∀s (1.11)
λp = Xµ, (1.12)
where λ and µ = (µ
1
, . . . , µ
S
) are positive Lagrange multipliers .
3
If u is quasi-concave, then these conditions are sufficient as well as necessary. Assuming that
the solution is interior and that ∂
0
u > 0, inequalities 1.10 and 1.11 are satisfied with equality. Then
1.12 becomes
p = X
∂
1
u
∂
0
u
(1.13)
with typical equation
p
j
=
X
s
x
js
µ
∂
s
u
∂
0
u
¶
, (1.14)
3
If f is a function of a single variable, its first derivative is indicated f
0
(x) or, when no confusion can result, f
0
.
Similarly, the second derivative is indicated f
00
(x) or f
00
. The partial derivative of a function f of two variables x
and y with respect to the first variable is indicated ∂
x
f(x, y) or ∂
x
f.
Frequently the function in question is a utility function u, and the argument is (c
0
, c
1
) where, as noted above, c
0
is a scalar and c
1
is an S-vector. In that case the partial derivative of the function u with respect to c
0
is denoted
∂
0
u(c
0
, c
1
) or ∂
0
u and the partial derivative with respect to c
s
is denoted ∂
s
u(c
0
, c
1
) or ∂
s
u. The vector of S partial
derivatives with respect to c
s
for all s is denoted ∂
1
u(c
0
, c
1
) or ∂
1
u.
Note that there exists the possibility of confusion: the subscript “1” can indicate either the vector of date-1 partial
derivatives or the (scalar) partial derivative with respect to consumption in state 1. The context will always make
the intended meaning clear.
1.6. LEFT AND RIGHT INVERSES OF X 7
where we now—and henceforth—delete the argument of u in the first-order conditions. Eq. 1.14
says that the price of security j (which is the cost in units of date-0 consumption of a unit increase in
the holding of the j-th security) is equal to the sum over states of its payoff in each state multiplied
by the marginal rate of substitution between consumption in that state and consumption at date
0.
The first-order conditions for the problem 1.7 with no consumption at date 0 are:
∂
s
u − µ
s
≤ 0, (∂
s
u − µ
s
)c
s
= 0 , ∀s (1.15)
λp = Xµ. (1.16)
At an interior solution 1.16 becomes
λp = X∂
1
u (1.17)
with typical element
λp
j
=
X
s
x
js
∂
s
u. (1.18)
Since security prices are denominated in units of an abstract numeraire, all we can say about
security prices is that they are proportional to the sum of marginal-utility-weighted payoffs.
1.6 Left and Right Inverses of X
The payoff matrix X has an inverse iff it is a square matrix (J = S) and of full rank. Neither
of these properties is assumed to be true in general. However, even if X is not square, it may
have a left inverse , defined as a matrix L that satisfies LX = I
S
, where I
S
is the S × S identity
matrix. The left inverse exists iff X is of rank S, which occurs if J ≥ S and the columns of X are
linearly independent. Iff the left inverse of X exists, the asset span M coincides with the date-1
consumption space R
S
, so that markets are complete.
If markets are complete, the vectors of marginal rates of substitution of all agents (whose optimal
consumption is interior) are the same, and can be inferred uniquely from security prices. To see
this, premultiply 1.13 by the left inverse L to obtain
Lp =
∂
1
u
∂
0
u
. (1.19)
If markets are incomplete, the vectors of marginal rates of substitution may differ across agents.
Similarly, X may have a right inverse, defined as a matrix R that satisfies XR = I
J
. The right
inverse exists if X is of rank J, which occurs if J ≤ S and the rows of X are linearly independent.
Then no security is redundant. Any date-1 consumption plan c
1
such that c
1
− w
1
belongs to the
asset span is associated with a unique portfolio
h = (c
1
− w
1
)R, (1.20)
which is derived by postmultiplying 1.6 by R.
The left and right inverses, if they exist, are given by
L = (X
0
X)
−1
X
0
(1.21)
R = X
0
(XX
0
)
−1
, (1.22)
where
0
indicates transposition. As these expressions make clear, L exists iff X
0
X is invertible,
while R exists iff XX
0
is invertible.
The payoff matrix X is invertible iff both the left and right inverses exist. Under the assumptions
so far none of the four possibilities: (1) both left and right inverses exist, (2) the left inverse exists
but the right inverse does not exist, (3) the right inverse exists but the left inverse does not exist,
or (4) neither directional inverse exists, is ruled out.
8 CHAPTER 1. EQUILIBRIUM IN SECURITY MARKETS
1.7 General Equilibrium
An equilibrium in security markets consists of a vector of security prices p, a portfolio allocation
{h
i
}, and a consumption allocation {(c
i
0
, c
i
1
)} such that (1) portfolio h
i
and consumption plan
(c
i
0
, c
i
1
) are a solution to agent i’s choice problem 1.4 at prices p, and (2) markets clear, that is
X
i
h
i
= 0, (1.23)
and
X
i
c
i
0
≤ ¯w
0
≡
X
i
w
i
0
,
X
i
c
i
1
≤ ¯w
1
≡
X
i
w
i
1
. (1.24)
The portfolio market-clearing condition 1.23 implies, by summing over agents’ budget con-
straints, the consumption market-clearing condition 1.24. If agents’ utility functions are strictly
increasing so that all budget constraints hold with equality, and if there are no redundant securities
(X has a right inverse), then the converse is also true. If, on the other hand, there are redundant se-
curities, then there exist many portfolio allocations associated with a market-clearing consumption
allocation. At least one of these portfolio allocations is market-clearing.
In the simplified model in which date-0 consumption does not enter utility functions, each
agent’s equilibrium portfolio and date-1 consumption plan is a solution to the choice problem 1.7.
Agents’ endowments at date 0 are equal to zero so that there is zero demand and zero supply of
date-0 consumption.
As the portfolio market-clearing condition 1.23 indicates, securities are in zero supply. This is
consistent with the assumption that agents’ endowments are in the form of consumption endow-
ments. However, our modeling format allows consideration of the case when agents have initial
portfolios of securities and there exists positive supply of securities. In that case, equilibrium port-
folio allocation {h
i
} should be interpreted as an allocation of net trades in securities markets. To
be more specific, suppose (in a securities market economy) that each agent’s endowment at date
1 equals the payoff of an initial portfolio
ˆ
h
i
so that w
i
1
=
ˆ
h
i
X. Using total portfolio holdings, an
equilibrium can be written as a vector of security prices p, an allocation of total portfolios {
¯
h
i
}, and
a consumption allocation {(c
i
0
, c
i
1
)} such that the net portfolio holding h
i
=
¯
h
i
−
ˆ
h
i
and consumption
plan (c
i
0
, c
i
1
) are a solution to 1.4 for each agent i, and
X
i
¯
h
i
=
X
i
ˆ
h
i
, (1.25)
and
X
i
c
i
0
≤
X
i
w
i
0
,
X
i
c
i
1
≤
X
i
ˆ
h
i
X. (1.26)
1.8 Existence and Uniqueness of Equilibrium
The existence of a general equilibrium in security markets is guaranteed under the standard as-
sumptions of positivity of consumption and quasi-concavity of utility functions.
1.8.1 Theorem
If each agent’s admissible consumption plans are restricted to be positive, his utility function is
strictly increasing and quasi-concave, his initial endowment is strictly positive, and there exists a
portfolio with positive and nonzero payoff, then there exists an equilibrium in security markets.
The proof is not given here, but can be found in the sources cited in the notes at the end of
this chapter.
1.9. REPRESENTATIVE AGENT MODELS 9
Without further restrictions on agents’ utility functions, initial endowments or security payoffs,
there may be multiple equilibrium prices and allocations in security markets. If all agents’ utility
functions are such that they imply gross substitutability between consumption at different states
and dates, and if security markets are complete, then the equilibrium consumption allocation and
prices are unique. This is so because, as we will show in Chapter 15, equilibrium allocations in
complete security markets are the same as Walrasian equilibrium allocations. The corresponding
equilibrium portfolio allocation is unique as long as there are no redundant securities. Otherwise,
if there are redundant securities, then there are infinitely many portfolio allocations that generate
the equilibrium consumption allocation.
1.9 Representative Agent Models
Many of the points to be made in this book are most simply illustrated using representative agent
models: models in which all agents have identical utility functions and endowments. With all agents
alike, security prices at which no agent wants to trade are equilibrium prices, since then markets
clear. Equilibrium consumption plans equal endowments.
In representative agent models specification of securities is unimportant: in equilibrium agents
consume their endowments regardless of what markets exist. It is often most convenient to assume
complete markets, so as to allow discussion of equilibrium prices of all possible securities.
Notes
As noted in the introduction, it is a good idea for the reader to make up and analyze as many
examples as possible in studying financial economics. There arises the question of how to represent
preferences. It happens that a few utility functions are used in the large majority of cases, this
because of their convenient properties. Presentation of these utility functions is deferred to Chapter
9 since a fair amount of preliminary work is needed before these properties can be presented in a
way that makes sense. However, it is worthwhile looking ahead now to find out what these utility
functions are.
The purpose of specifying security payoffs is to determine the asset span M. It was observed
that the asset span can be specified using the returns on the securities rather than their payoffs.
This requires the assumption that M does not consist of payoffs with zero price alone, since in
that case returns are undefined. As long as M has a set of basis vectors of which at least one has
nonzero price, then another basis of M can always be found of which all the vectors have nonzero
price. Therefore these can be rescaled to have unit price. It is important to bear in mind that
returns are not simply an arbitrary rescaling of payoffs. Payoffs are given exogenously; returns,
being payoffs divided by equilibrium prices, are endogenous.
The model presented in this chapter is based on the theory of general equilibrium as formulated
by Arrow [1] and Debreu [3]. In some respects, the present treatment is more general than that of
Arrow-Debreu: most significantly, we assume that agents trade securities in markets that may be
incomplete,
whereas Arrow and Debreu assumed complete markets. On the other hand, our specification
involves a single good whereas the Arrow-Debreu model allows for multiple goods. Accordingly, our
framework can be seen as the general equilibrium model with incomplete markets (GEI ) simplified
to the case of a single good; see Geanakoplos [4] for a survey of the literature on GEI models; see
also Magill and Quinzii [8] and Magill and Shafer [9].
The proof of Theorem 1.8.1 can be found in Milne [11], see also Geanakoplos and Polemarchakis
[5]. Our maintained assumptions of symmetric information (agents anticipate the same state-
contingent security payoffs) and a single good are essential for the existence of an equilibrium
when short sales are allowed. There exists an extensive literature on the existence of a security
10 CHAPTER 1. EQUILIBRIUM IN SECURITY MARKETS
markets equilibrium when agents have different expectations about security payoffs. See Hart [7],
Hammond [6], Neilsen [13], Page [14], and Werner [15]. On the other hand, the assumption of
strictly positive endowments can be significantly weakened. Consumption sets other than the set of
positive consumption plans can also be included, see Neilsen [13], Page [14], and Werner [15]. For
discussions of the existence of an equilibrium in a model with multiple goods (GEI), see Geanakoplos
[4] and Magill and Shafer [9].
A sufficient condition for satisfaction of the gross substitutes condition mentioned in Section
1.8 is that agents have strictly concave expected utility functions with common probabilities and
with the Arrow-Pratt measure of relative risk aversion (see Chapter 4) that is everywhere less
than one. There exist a few further results on uniqueness. It follows from a results of Mitiushin
and Polterovich [12] (in Russian) that if agents have strictly concave expected utility functions
with common probabilities and relative risk aversion that is everywhere less than four, if their
endowments are collinear (that is, each agent’s endowment is a fixed proportion (the same in all
states) of the aggregate endowment) and security markets are complete, then equilibrium is unique.
See Mas-Colell [10] for a discussion of the Mitiushin-Polterovich result and of uniqueness generally.
See also Dana [2] on uniqueness in financial models.
As noted in the introduction, throughout this book only exchange economies are considered.
The reason is that production theory—or, in intertemporal economies, capital theory—does not lie
within the scope of finance as usually defined, and not much is gained by combining exposition of
the theory of asset pricing with that of resource allocation. The theory of the equilibrium allocation
of resources is modeled by including production functions (or production sets), and assuming that
agents have endowments of productive resources instead of, or in addition to, endowments of
consumption goods. Because these production functions share most of the properties of utility
functions, the theory of allocation of productive resources is similar to that of consumption goods.
In the finance literature there has been much discussion of the problem of determining firm
behavior under incomplete markets when firms are owned by stockholders with different utility
functions. There is, of course, no difficulty when markets are complete: even if stockholders
have different preferences, they will agree that that firm should maximize profit. However, when
markets are incomplete and firm output is not in the asset span, firm output cannot be valued
unambiguously. If this output is distributed to stockholders in proportion to their ownership
shares, stockholders will generally disagree about the ordering of different possible outputs.
This is not a genuine problem, at least in the kinds of economies modeled in these notes.
The reason is that in the framework considered here, in which all problems of scale economies,
externalities, coordination, agency issues, incentives and the like are ruled out, there is no reason
for nontrivial firms to exist in the first place. As is well known, in such neoclassical production
economies the zero-profit condition guarantees that there is no difference between an agent renting
out his own resource endowment and employing other agents’ resources, assuming that all agents
have access to the same technology. Therefore there is no reason not to consider each owner of
productive resources as operating his or her own firm. Of course, this is saying nothing more than
that if firms play only a trivial role in the economy, then there can exist no nontrivial problem
about what the firm should do. In a setting in which firms do play a nontrivial role, these issues of
corporate governance become significant.
Bibliography
[1] Kenneth J. Arrow. The role of securities in the optimal allocation of risk bearing. Review of
Economic Studies, pages 91–96, 1964.
[2] Rose-Anne Dana. Existence, uniqueness and determinacy of Arrow-Debreu equilibria in finance
models. Journal of Mathematical Economics, 22:563–579, 1993.
[3] Gerard Debreu. Theory of Value. Wiley, New York, 1959.
[4] John Geanakoplos. An introduction to general equilibrium with incomplete asset markets.
Journal of Mathematical Economics, 19:1–38, 1990.
[5] John Geanakoplos and Heraklis Polemarchakis. Existence, regularity, and constrained sub-
optimality of competitive allocations when the asset markets is incomplete. In Walter Heller
and David Starrett, editors, Essays in Honor of Kenneth J. Arrow, Volume III. Cambridge
University Press, 1986.
[6] Peter Hammond. Overlapping expectations and Hart’s condition for equilibrium in a securities
model. Journal of Economic Theory, 31:170–175, 1983.
[7] Oliver D. Hart. On the existence of equilibrium in a securities model. Journal of Economic
Theory, 9:293–311, 1974.
[8] Michael Magill and Martine Quinzii. Theory of Incomplete Markets. MIT Press, 1996.
[9] Michael Magill and Wayne Shafer. Incomplete markets. In Werner Hildenbrand and Hugo
Sonnenschein, editors, Handbook of Mathematical Economics, Vol. 4. North Holland, 1991.
[10] Andreu Mas-Colell. On the uniqueness of equilibrium once again. In William A. Barnett,
Bernard Cornet, Claude d’Aspremont, Jean Gabszewicz, and Andreu Mas-Colell, editors,
Equilibrium Theory and Applications: Proceedings of the Sixth International Symposium in
Economic Theory and Econometrics. Cambridge University Press, 1991.
[11] Frank Milne. Default risk in a general equilibrium asset economy with incomplete markets.
International Economic Review, 17:613–625, 1976.
[12] L. G. Mitiushin and V. W. Polterovich. Criteria for monotonicity of demand functions, vol.
14. In Ekonomika i Matematicheskie Metody. 1978.
[13] Lars T. Nielsen. Asset market equilibrium with short-selling. Review of Economic Studies,
56:467–474, 1989.
[14] Frank Page. On equilibrium in Hart’s securities exchange model. Journal of Economic Theory,
41:392–404, 1987.
[15] Jan Werner. Arbitrage and the existence of competitive equilibrium. Econometrica, 55:1403–
1418, 1987.
11
12 BIBLIOGRAPHY
Chapter 2
Linear Pricing
2.1 Introduction
In analyzing security prices, two concepts are central: linearity and positivity. Linearity of pricing,
treated in this chapter, is a consequence of the law of one price. The law of one price says that
portfolios that have the same payoff must have the same price. It holds in a securities market
equilibrium under weak restrictions on agents’ preferences. Positivity of pricing is treated in the
next chapter.
2.2 The Law of One Price
The law of one price says that all portfolios with the same payoff have the same price. That is,
if hX = h
0
X , then ph = ph
0
, (2.1)
for any two portfolios h and h
0
. If there exist no redundant securities, only one portfolio generates
any given payoff, so the law of one price is trivially satisfied.
A necessary and sufficient condition for the law of one price to hold is that every portfolio with
zero payoff has zero price. If the law of one price does not hold, then every payoff in the asset span
can be purchased at any price. To see this note first that the zero payoff can be purchased at any
price, since any multiple of a portfolio with zero payoff is also a portfolio with zero payoff. If the
zero payoff can be purchased at any price, then any payoff can be purchased at any price.
2.3 The Payoff Pricing Functional
For any security prices p we define a mapping q : M → R that assigns to each payoff the price(s)
of the portfolio(s) that generate(s) that payoff. Formally,
q(z) ≡ {w : w = ph for some h such that z = hX}. (2.2)
In general the mapping q is a correspondence rather than a single-valued function. If the law of
one price holds, then q is single-valued.
Further, it is a linear functional:
2.3.1 Theorem
The law of one price holds iff q is a linear functional on the asset span M.
Proof: If the law of one price holds, then, as just noted, q is single-valued. To prove linearity,
consider payoffs z, z
0
∈ M such that z = hX and z
0
= h
0
X for some portfolios h and h
0
. For
13
14 CHAPTER 2. LINEAR PRICING
arbitrary λ, µ ∈ R, the payoff λz + µz
0
can be generated by the portfolio λh + µh
0
with price
λph + µph
0
. Since q is single-valued, definition 2.2 implies that
q(λz + µz
0
) = λph + µph
0
. (2.3)
The right-hand side of 2.3 equals λq(z) + µq(z
0
), so q is linear.
Conversely, if q is a functional, then the law of one price holds by definition.
2
Whenever the law of one price holds, we call q the payoff pricing functional .
The payoff pricing functional q is one of three operators that are related in a triangular fashion.
Each portfolio is a J-dimensional vector of holdings of all securities. The set of all portfolios, R
J
, is
termed the portfolio space . A vector of security prices p can be interpreted as the linear functional
(portfolio pricing functional) from the portfolio space R
J
to the reals,
p : R
J
→ R (2.4)
assigning price ph to each portfolio h. Note that we are using p to denote either the functional or
the price vector as the context requires. Similarly, payoff matrix X can be interpreted as a linear
operator (payoff operator) from the portfolio space R
J
to the asset span M,
X : R
J
→ M (2.5)
assigning payoff hX to each portfolio h. Assuming that q is a functional, we have
p = q ◦ X, (2.6)
or, more explicitly,
ph = q(hX), (2.7)
for every portfolio h.
If there exist no redundant securities, then the right inverse R of the payoff matrix X is well
defined. Then we can write
q(z) = zRp (2.8)
for every payoff z ∈ M.
2.4 Linear Equilibrium Pricing
The payoff pricing functional associated with equilibrium security prices is the equilibrium payoff
pricing functional . If the law of one price holds in equilibrium then, by Theorem 2.3.1, the
equilibrium payoff pricing functional is a linear functional on the asset span M. We have
2.4.1 Theorem
If agents’ utility functions are strictly increasing at date 0, then the law of one price holds in an
equilibrium, and the equilibrium payoff pricing functional is linear.
Proof: If the law of one price does not hold at equilibrium prices p, then there is a portfolio
h
0
with zero payoff, h
0
X = 0, and nonzero price. We can assume that ph
0
< 0. For every
budget-feasible portfolio h and consumption plan (c
0
, c
1
), portfolio h + h
0
and consumption plan
(c
0
− ph
0
, c
1
) are budget feasible and strictly preferred. Therefore there cannot exist an optimal
consumption and portfolio choice for any agent.
2
2.5. STATE PRICES IN COMPLETE MARKETS 15
Note that Theorem 2.4.1 holds whether or not consumption is restricted to be positive. We
will see in Chapter 4 that the law of one price may fail in the presence of restrictions on portfolio
holdings.
If date-0 consumption does not enter agents’ utility functions, the strict monotonicity condition
in Theorem 2.4.1 fails. In that case the law of one price is satisfied under the conditions established
in the following:
2.4.2 Theorem
If agents’ utility functions are strictly increasing at date 1 and there exists a portfolio with positive
and nonzero payoff, then the law of one price holds in an equilibrium, and the equilibrium payoff
pricing functional is linear.
Proof: If the law of one price does not hold, then, as in the proof of Theorem 2.4.1, we
consider portfolio h
0
with zero payoff and nonzero price, and an arbitrary budget-feasible date-1
consumption plan c
1
and portfolio h. Let
ˆ
h be a portfolio with positive and nonzero payoff. There
exists a number α such that αph
0
= p
ˆ
h. But then portfolio h+
ˆ
h−αh
0
and date-1 consumption plan
c
1
+
ˆ
hX are budget feasible and strictly preferred. Thus there cannot exist an optimal consumption
and portfolio choice for any agent.
2
The following examples illustrate the possibility of failure of the law of one price in equilibrium
if the conditions of Theorems 2.4.1 and 2.4.2 are not satisfied.
2.4.3 Example
Suppose that there are two states and three securities with payoffs x
1
= (1, 0), x
2
= (0, 1) and
x
3
= (1, 1). The utility function of the representative agent is given by
u(c
0
, c
1
, c
2
) = −(c
0
− 1)
2
− (c
1
− 1)
2
− (c
2
− 2)
2
. (2.9)
His endowment is 1 at date 0 and (1, 2) at date 1. Since the endowment is a satiation point, any
prices p
1
, p
2
and p
3
of the securities are equilibrium prices. When p
1
+ p
2
6= p
3
, the law of one
price does not hold. Here the condition of strictly increasing utility functions is not satisfied.
2
2.4.4 Example
Suppose that there are two states and two securities with payoffs x
1
= (1, −1) and x
2
= (2, −2).
The utility function of the representative agent depends only on date-1 consumption and is given
by
u(c
1
, c
2
) = ln(c
1
) + ln(c
2
), (2.10)
for (c
1
, c
2
) À 0. His endowment is 0 at date 0 and (1, 1) at date 1.
Let the security prices be p
1
= p
2
= 1. The agent’s optimal portfolio at these prices is the zero
portfolio. Therefore these prices are equilibrium prices even though the law of one price does not
hold. Here the condition of strictly increasing utility functions at date 1 is satisfied but there exists
no portfolio with positive and nonzero payoff.
2
2.5 State Prices in Complete Markets
Let e
s
denote the s-th basis vector in the space R
S
of contingent claims, with 1 in the s-th place
and zeros elsewhere. Vector e
s
is the state claim or the Arrow security of state s. It is the claim
16 CHAPTER 2. LINEAR PRICING
to one unit of consumption contingent on the occurrence of state s. If markets are complete and
the law of one price holds, then the payoff pricing functional assigns a unique price to each state
claim. Let
q
s
≡ q(e
s
) (2.11)
denote the price of the state claim of state s. We call q
s
the state price of state s.
Since any linear functional on R
S
can be identified by its values on the basis vectors of R
S
, the
payoff pricing functional q can be represented as
q(z) = qz (2.12)
for every z ∈ R
S
, where q on the right-hand side of 2.12 is an S-dimensional vector of state prices.
Observe that we use the same notation for the functional and the vector that represents it.
Since the price of each security equals the value of its payoff under the payoff pricing functional,
we have
p
j
= qx
j
, (2.13)
or, in matrix notation,
p = Xq. (2.14)
Eq. 2.14 is a system of linear equations that associates state prices with given security prices.
Using the left inverse of the payoff matrix, it follows that
q = Lp. (2.15)
The results of this section depend on the assumption of market completeness, since otherwise
state claim e
s
may not be in the asset span M, and so q(e
s
) may not be defined. In Chapter 5 we
will introduce state prices in incomplete markets.
2.6 Recasting the Optimization Problem
When the law of one price is satisfied, the payoff pricing functional provides a convenient way
of representing the agent’s consumption and portfolio choice problem. Substituting z = hX and
q(z) = ph, the problem 1.4 – 1.6 can be written as
max
c
0
,c
1
,z
u(c
0
, c
1
) (2.16)
subject to
c
0
≤ w
0
− q(z) (2.17)
c
1
≤ w
1
+ z (2.18)
z ∈ M. (2.19)
This formulation makes clear that the agent’s consumption choice in security markets depends only
on the asset span and the payoff pricing functional. Any two sets of security payoffs and prices that
generate the same asset span and the same payoff pricing functional induce the same consumption
choice.
If markets are complete, restriction 2.19 is vacuous. Further, we can use state prices in place
of the payoff pricing functional. The problem 2.16 – 2.19 then simplifies to
max
c
0
,c
1
,z
u(c
0
, c
1
) (2.20)
subject to
c
0
≤ w
0
− qz (2.21)
2.6. RECASTING THE OPTIMIZATION PROBLEM 17
c
1
≤ w
1
+ z. (2.22)
This problem can be interpreted as the consumption and portfolio choice problem with Arrow
securities.
The first-order conditions for the problem 2.20 (at an interior solution) imply that
q =
∂
1
u
∂
0
u
. (2.23)
Thus state prices are equal to marginal rates of substitution . Security prices can be obtained from
state prices using 2.14. Eq. 2.23 can also be obtained by premultiplying 1.13 by L and using 2.15.
The following example illustrates the use of state prices for determining equilibrium security
prices in complete markets.
2.6.1 Example
Suppose that there are two states and two securities with payoffs x
1
= (1, 1) and x
2
= (2, 0). The
representative agent’s utility function is given by
u(c
0
, c
1
, c
2
) = ln(c
0
) +
1
2
ln(c
1
) +
1
2
ln(c
2
), (2.24)
for (c
0
, c
1
, c
2
) À 0. His endowment is 1 at date 0 and (1, 2) at date 1. Equilibrium security
prices are such that the agent’s optimal portfolio is the zero portfolio. Using simple substitution of
variables, the agent’s problem 1.4 – 1.6 can be written
max
h
1
,h
2
ln(1 − p
1
h
1
− p
2
h
2
) +
1
2
ln(1 + h
1
+ 2h
2
) +
1
2
ln(2 + h
1
). (2.25)
The first-order condition for problem 2.25 evaluated at h
1
= h
2
= 0 yields equilibrium security
prices p
1
= 3/4 and p
2
= 1.
The same prices can be calculated by using the payoff pricing functional. Since markets are
complete, the payoff pricing functional is given by the state prices which, by 2.23, are equal to the
marginal rates of substitution at the equilibrium consumption plan. The equilibrium consumption
plan is (1, 1, 2), and the marginal utilities are 1 for date-0 consumption, 1/2 for state-1 consumption,
and 1/4 for state-2 consumption. Marginal rates of substitution are (1/2, 1/4), hence
q = (
1
2
,
1
4
). (2.26)
Equilibrium security prices are p
1
= qx
1
= 3/4 and p
2
= qx
2
= 1.
2
Notes
As an inspection of the proof of Theorem 2.4.1 reveals, linear equilibrium pricing obtains under
nonsatiation of agents’ utility functions at equilibrium consumption plans. Nonsatiation is a weaker
restriction than strict monotonicity.
The linearity of payoff pricing is a very important result. It is much discussed in elementary
finance texts under the name “value additivity.” One implication of value additivity is the Miller-
Modigliani theorem (Miller and Modigliani [3]) which says that two firms that generate the same
future profits have the same market value regardless of their debt-equity structure. Another im-
plication is that corporate managers have no motive to diversify into unrelated activities: if a firm
pays market value for an acquisition, then the value of the two cash flows together is the sum of
18 CHAPTER 2. LINEAR PRICING
their values separately, and no more. Thus acquisitions do not create value by making the firm
more attractive to stockholders via, say, reduced cash flow volatility. It remains true, though, that
if the summed cash flows increase due to reduced costs or “synergies” of management, then value
is created.
Other important implications of the law of one price are parity relations such as interest rate
parity, put-call parity, and others.
For papers emphasizing the role of state prices in the analysis of security pricing, see Hirshleifer
[1], [2].
Bibliography
[1] Jack Hirshleifer. Investment decision under uncertainty: Choice theoretic approaches. Quarterly
Journal of Economics, 79:509–536, 1965.
[2] Jack Hirshleifer. Investment decision under uncertainty: Application of the state preference
approach. Quarterly Journal of Economics, 80:252–277, 1966.
[3] Merton Miller and Franco Modigliani. The cost of capital, corporation finance and the theory
of investment. American Economic Review, 48:261–297, 1958.
19
20 BIBLIOGRAPHY
Chapter 3
Arbitrage and Positive Pricing
3.1 Introduction
The principle that there cannot exist arbitrage opportunities in security markets is one of the
most basic ideas of financial economics. Whether there exists an arbitrage opportunity or not
depends on security prices. We show in this chapter that, if security prices exclude arbitrage, then
the payoff pricing functional is strictly positive. Further, exclusion of arbitrage is necessary (and
sufficient, when consumption is restricted to be positive) for the existence of optimal portfolios for
agents with strictly increasing utility functions. In particular, equilibrium prices exclude arbitrage
opportunities when agents have strictly increasing utility functions.
Conditions on security prices under which there exists no arbitrage are derived in this chapter
in special cases (complete markets, or two securities). The complete characterization will be given
in Chapter 5.
3.2 Arbitrage and Strong Arbitrage
A strong arbitrage is a portfolio that has a positive payoff and a strictly negative price. An arbitrage
is a portfolio that is either a strong arbitrage or has a positive and nonzero payoff and zero price.
Formally, a strong arbitrage is a portfolio h that satisfies hX ≥ 0 and ph < 0, and an arbitrage is
a portfolio h that satisfies hX ≥ 0 and ph ≤ 0 with at least one strict inequality.
There may exist a portfolio that is an arbitrage but not a strong arbitrage:
3.2.1 Example
Let there be two securities with payoffs x
1
= (1, 1) and x
2
= (1, 2), and prices p
1
= p
2
= 1. Then
portfolio h = (−1, 1) is an arbitrage, but not a strong arbitrage. In fact, there exists no strong
arbitrage.
2
If there exists no portfolio with positive and nonzero payoff, then any arbitrage is a strong
arbitrage. Further, a strong arbitrage exists iff the law of one price does not hold, and it is a
portfolio with zero payoff and strictly negative price.
3.2.2 Example
Suppose that the securities have payoffs x
1
= (−1, 2, 0) and x
2
= (2, 2, −1). A portfolio h = (h
1
, h
2
)
has a positive payoff if
−h
1
+ 2h
2
≥ 0, (3.1)
h
1
+ h
2
≥ 0, (3.2)
21
22 CHAPTER 3. ARBITRAGE AND POSITIVE PRICING
and
−h
2
≥ 0. (3.3)
These inequalities are satisfied by the zero portfolio alone. Therefore there exists no portfolio with
positive and nonzero payoff. Since there are no redundant securities, the law of one price holds for
any security prices. Consequently, there exists no arbitrage for any security prices.
2
3.3 A Diagrammatic Representation
It is helpful to have a diagrammatic representation of the set of security prices that exclude arbi-
trage. Suppose that there are two securities with payoffs x
1
and x
2
, and consider the payoff pairs
x
·s
= (x
1s
, x
2s
) in each state s = 1, 2, 3. Figure 3.1 is drawn assuming x
js
> 0 for each j and s, but
the analysis does not depend on this restriction.
Now interpret the coordinate axes as portfolio weights h
1
and h
2
, so that any point in the
diagram is associated with a portfolio h = (h
1
, h
2
). For each x
·s
, construct a line perpendicular
to x
·s
through the origin. The set of portfolios h with positive payoff in state s is the set of
points northeast of this line. If this construction is performed in each state, the intersection of the
indicated portfolio sets gives the set of portfolios with positive payoffs in all states. The indicated
portfolios are those for which the ray through the point h intersects the arc.
Suppose that security prices are given by p = (p
1
, p
2
), as shown in Figure 3.2. Then the set of
zero-price portfolios consists of the line through the origin perpendicular to p. Figure 3.3, which
combines Figures 3.1 and 3.2, shows that the set of positive-payoff portfolios intersects the set of
negative-price portfolios only at the origin, so there is no arbitrage.
This conclusion is a consequence of the fact that p lies in the interior of the cone defined by
the x
·s
. If p lies on the boundary of the cone, then there exists arbitrage but not strong arbitrage
(Figure 3.4), while if p lies outside the cone, then there exists strong arbitrage (Figure 3.5).
The above construction, being two-dimensional, is necessarily restricted to the case in which
agents take nonzero positions in at most two securities. It is worth noticing that, if there are
more than two securities, then nonexistence of an arbitrage if portfolios are restricted to contain
at most two securities is consistent with existence of arbitrage if portfolios are unrestricted. This
is illustrated by the following example.
3.3.1 Example
Consider three securities with payoffs x
1
= (1, 1, 0), x
2
= (0, 1, 1), x
3
= (1, 0, 1), and with prices
p
1
= 1, and p
2
= p
3
= 1/2. There exists no arbitrage with nonzero positions in any two of these
securities but portfolio h = (−1, 1, 1) is an arbitrage.
2
3.4 Positivity of the Payoff Pricing Functional
A functional is positive if it assigns positive value to every positive element of its domain. It
is strictly positive if it assigns strictly positive value to every positive and nonzero element of
its domain. Note that if there is no positive (positive and nonzero) element in the domain of a
functional, then the functional is trivially positive (strictly positive). Our terminology of positive
and strictly positive functionals is consistent with the terminology of positive and strictly positive
vectors in the following sense: A linear functional F : R
l
→ R has a representation in the form of
a scalar product F (x) = f x for some vector f ∈ R
l
. Functional F is strictly positive (positive) iff
the corresponding vector f is strictly positive (positive).
3.5. POSITIVE STATE PRICES 23
Absence of arbitrage or strong arbitrage at given security prices corresponds to the payoff pricing
functional being strictly positive or positive.
3.4.1 Theorem
The payoff pricing functional is linear and strictly positive iff there is no arbitrage.
Proof: The necessity of the condition is obvious. To prove sufficiency, note that exclusion
of arbitrage implies satisfaction of the law of one price, which in turn implies that q is a linear
functional (Theorem 2.3.1). If z ∈ M, then q(z) = ph for h such that hX = z. Exclusion of
arbitrage implies that q(z) > 0 if z > 0, so that q is strictly positive.
2
We also have
3.4.2 Theorem
The payoff pricing functional is linear and positive iff there is no strong arbitrage.
The proof is similar to that of Theorem 3.4.1.
3.5 Positive State Prices
In Chapter 2 we showed that if markets are complete, so that the asset span coincides with the
date-1 contingent claims space, then the law of one price implies the existence of a state price vector
q such that
p = Xq. (3.4)
Since the payoff matrix X is left-invertible under complete markets, the vector q that solves 3.4 is
unique. In view of
q(z) = qz, (3.5)
the absence of arbitrage is equivalent to state prices being strictly positive (q À 0), and the absence
of strong arbitrage is equivalent to those prices being positive (q ≥ 0).
We have demonstrated the role of state prices in characterizing security prices that exclude
arbitrage in complete markets. It turns out that this characterization generalizes to the case of
incomplete markets, but that requires separate treatment.
3.6 Arbitrage and Optimal Portfolios
If an agent’s utility function is strictly increasing, absence of arbitrage is necessary for the existence
of an optimal portfolio.
We have
3.6.1 Theorem
If at given security prices an agent’s optimal portfolio exists, and if the agent’s utility function is
strictly increasing, then there is no arbitrage.
Proof: Suppose that there exists a portfolio
ˆ
h that is an arbitrage at given prices p. For every
budget feasible portfolio h and consumption plan (c
0
, c
1
), portfolio h +
ˆ
h is budget feasible. The
resulting consumption plan (c
0
−p
ˆ
h, c
1
+
ˆ
hX) is strictly preferred to (c
0
, c
1
) since the agent’s utility
function is strictly increasing. Therefore there cannot exist an optimal portfolio.
2
If the agent’s utility function is increasing but not strictly increasing, the conclusion of Theorem
3.6.1 may fail to hold.
24 CHAPTER 3. ARBITRAGE AND POSITIVE PRICING
3.6.2 Example
Consider two securities with payoffs in two states given by x
1
= (1, 0) and x
2
= (0, 1). An agent’s
utility function is given by
u(c
0
, c
1
, c
2
) = c
0
+ min{c
1
, c
2
}. (3.6)
His endowment is 1 at date 0, and (1, 2) at date 1. At prices p
1
= 1 and p
2
= 0, the zero portfolio
is an optimal portfolio. Security 2 is an arbitrage. Utility function 3.6 is increasing but not strictly
increasing.
2
The absence of strong arbitrage is necessary for the existence of an optimal portfolio under a
weaker monotonicity assumption.
3.6.3 Theorem
If at given security prices an agent’s optimal portfolio exists, and if the agent’s utility function is
strictly increasing at date 0 and increasing at date 1, then there is no strong arbitrage.
The proof is the same as in Theorem 3.6.1.
The need for strict monotonicity in date-0 consumption is indicated by the following example.
3.6.4 Example
As in Example 2.4.4 there are two securities with payoffs x
1
= (1, −1) and x
2
= (2, −2). The utility
function of the representative agent depends only on date-1 consumption and is given by
u(c
1
, c
2
) = ln(c
1
) + ln(c
2
), (3.7)
for (c
1
, c
2
) À 0. His endowment is 0 at date 0 and (1, 1) at date 1. At prices p
1
= p
2
= 1, portfolio
h = (−2, 1) is a strong arbitrage. However, there exists an optimal portfolio: the zero portfolio.
Utility function 3.7 is not strictly increasing at date 0 since date 0 consumption does not enter the
utility function.
2
Both Theorems 3.6.1 and 3.6.3 require strictly increasing utility function at date 0, and therefore
do not apply to settings with no date-0 consumption, see Example 3.6.4. As in Theorem 2.4.2,
the assumption that the utility function is strictly increasing at date 0 can be replaced by the
assumptions that there exists a portfolio with positive and nonzero payoff and that the utility
function is strictly increasing at date 1.
If consumption is restricted to be positive, then the absence of arbitrage is also a sufficient
condition for the existence of an optimal portfolio.
3.6.5 Theorem
If at given security prices there is no arbitrage, and if the agent’s consumption is restricted to be
positive, then there exists an optimal portfolio.
Proof: Absence of arbitrage implies that the law of one price holds. If there exist redundant
securities, then their prices must equal the prices of the portfolios of other securities that have equal
payoffs. A solution to the consumption and portfolio problem with a smaller subset of nonredundant
securities is also a solution with the full set of securities. Therefore we can assume without loss of
generality that there are no redundant securities.
Since the agent’s utility function is continuous, the Weierstrass theorem (which states that every
continuous function on a compact set has a maximum) implies that it is sufficient to prove that
the agent’s budget set given by 1.5 and 1.6 is compact (that is, closed and bounded). It is clearly
closed, so we only have to demonstrate that it is bounded. Suppose, by contradiction, that it is
3.7. POSITIVE EQUILIBRIUM PRICING 25
not bounded. Then there exists an unbounded sequence of budget feasible consumption plans and
portfolios {c
n
, h
n
}. The inequalities 0 ≤ c
n
0
≤ w
0
− ph
n
and 0 ≤ c
n
1
≤ w
1
+ h
n
X imply that the
sequence of portfolios {h
n
} must be unbounded, for otherwise the sequences of prices {ph
n
} and
payoffs {h
n
X} would be bounded, and consequently the sequence of consumption plans would be
bounded as well.
Let k h
n
k denote the Euclidean norm of h
n
. We have that lim k h
n
k = +∞. Each portfolio
h
n
/ k h
n
k has unit norm, and therefore the sequence {h
n
/ k h
n
k} is bounded and, by switching
to a subsequence if necessary, can be assumed convergent to a nonzero portfolio
ˆ
h.
Using the positivity of consumption plan c
n
, it follows from budget constraints 1.5 and 1.6 that
ph
n
≤ w
0
, (3.8)
and
h
n
X + w
1
≥ 0. (3.9)
Dividing both sides of 3.8 and 3.9 by k h
n
k and taking limits as n goes to infinity, we obtain
p
ˆ
h ≤ 0, (3.10)
and
ˆ
hX ≥ 0. (3.11)
Since portfolio
ˆ
h is nonzero and there are no redundant securities, its payoff is nonzero and 3.10
and 3.11 imply that
ˆ
h is an arbitrage.
2
If consumption is unrestricted, exclusion of arbitrage does not guarantee existence of an optimal
portfolio. This is illustrated by the following example.
3.6.6 Example
Suppose that there are two states and a single security with payoff (1, 1). The agent’s utility
function is given by
u(c
0
, c
1
, c
2
) = c
0
+ c
1
+ c
2
. (3.12)
If consumption is unrestricted, then there exists no optimal portfolio unless the price of the security
equals 2 (in which case all portfolios are optimal). However, there is no arbitrage at any strictly
positive price of the security. If consumption is restricted to be positive, an optimal portfolio exists
for every strictly positive price.
2
3.7 Positive Equilibrium Pricing
Each agent’s equilibrium portfolio is by definition an optimal portfolio. We can apply Theorem
3.6.1 to equilibrium security prices. Combining this result with Theorem 3.4.1, we obtain
3.7.1 Theorem
If agents’ utility functions are strictly increasing, then there is no arbitrage at equilibrium security
prices. Further, the equilibrium payoff pricing functional is linear and strictly positive.
Again, Example 3.6.2 demonstrates the need for strict monotonicity. The assumption of strictly
increasing utility functions at date 0 in Theorem 3.7.1 can be replaced by assuming that utility
functions are strictly increasing at date 1 and there exists a portfolio with positive and nonzero
payoff.
Similarly, Theorems 3.4.2 and 3.6.3 imply
26 CHAPTER 3. ARBITRAGE AND POSITIVE PRICING
3.7.2 Theorem
If agents’ utility functions are strictly increasing at date 0 and increasing at date 1, then there is
no strong arbitrage at equilibrium security prices and the equilibrium payoff pricing functional is
linear and positive.
Notes
The assumption of no arbitrage plays a central role in finance. For example, in analyzing the
valuation of derivative securities the financial analyst takes security returns as primitives and derives
prices of derivative securities in such a way that there is no arbitrage. Imposing the requirement of
no arbitrage makes the analysis consistent with agents’ having strictly increasing utility functions
without explicitly specifying these functions. Thus, even though an equilibrium model of security
markets is not explicitly employed, the requirement of no arbitrage makes the analysis consistent
with an equilibrium.
The assumption of no arbitrage plays a much lesser role in economics than in finance. The
reason is that in economics the focus is on equilibrium analysis. Accordingly, the economist takes
preferences, endowments, and so on to be the primitives. There is no need to make a separate
assumption that there is no arbitrage since the assumption of strictly increasing utility functions,
which is generally made explicitly, guarantees that there will be no arbitrage in equilibrium.
Thus the assumption of no arbitrage is the finance counterpart of the economic assumption of
strictly increasing utility functions; one assumption is appropriate in the context of a valuation
analysis, the other in the context of an equilibrium analysis.
Arbitrage sometimes means “risk-free arbitrage” : a portfolio with state-independent positive
and nonzero payoff and a negative price, or a zero payoff and strictly negative price. This notion of
arbitrage is clearly much stronger than that defined in the text, so exclusion of risk-free arbitrage is
a very weak restriction. In fact, if the risk-free payoff is not in the asset span, then there cannot exist
a risk-free arbitrage with nonzero payoff. In that case exclusion of risk-free arbitrage is equivalent
to assuming satisfaction of the law of one price. Absence of arbitrage or strong arbitrage at given
security prices corresponds to the payoff pricing functional being strictly positive or positive. If
the risk-free payoff is in the asset span, then risk-free arbitrage is excluded as long as the sum of
the state prices is strictly positive; this condition may be satisfied even if some state prices are
negative, so that there exists arbitrage as we have defined it. The most interesting consequences of
absence of arbitrage do not obtain if only risk-free arbitrage is excluded.
Financial analysts recognized the central role of the assumption of absence of arbitrage only
gradually. Major papers developing the arbitrage theme were Black and Scholes [2] and Ross [5],
[6]. A clear and intuitive discussion of arbitrage can be found in Varian [7] where attention is
restricted to what we call strong arbitrage. Werner [8] studied the relation between the absence of
arbitrage and the existence of an equilibrium in a general class of markets.
The diagrammatic analysis of Section 3.3 is apparently due to Garman [3]. Theorem 3.6.5 is
closely related to the results of Bertsekas [1] and Leland [4].
Bibliography
[1] Dimitri P. Bertsekas. Necessary and sufficient conditions for existence of an optimal portfolio.
Journal of Economic Theory, 8:235–247, 1974.
[2] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of
Political Economy, 81:637–654, 1973.
[3] Mark B. Garman. A synthesis of the pure theory of arbitrage. reproduced, University of
California, Berkeley, 1978.
[4] Hayne E. Leland. On the existence of optimal policies under uncertainty. Journal of Economic
Theory, 4:35–44, 1972.
[5] Stephen A. Ross. Risk, return and arbitrage. In Irwin Friend and James Bicksler, editors, Risk
and Return in Finance. Ballinger, Cambridge, Massachusetts, 1976.
[6] Stephen A. Ross. A simple approach to the valuation of risky streams. Journal of Business,
51:453–475, 1978.
[7] Hal R. Varian. The arbitrage principle in financial economics. Journal of Economic Perspectives,
1:55–72, 1987.
[8] Jan Werner. Arbitrage and the existence of competitive equilibrium. Econometrica, 55:1403–
1418, 1987.
27
28 BIBLIOGRAPHY
Chapter 4
Portfolio Restrictions
4.1 Introduction
So far we have assumed that agents can trade without explicit portfolio restrictions, meaning
that they can choose any portfolio provided that the resulting consumption satisfies the agent’s
restriction on admissible consumptions (for example, positivity). In particular, the only limits on
short selling were those implied by restrictions on consumption, if any were imposed.
Short sales restrictions and transaction costs are important features of real-world security mar-
kets. In this chapter we introduce explicit portfolio restrictions and discuss the validity of the
results of Chapters 2 and 3 under such restrictions. The simplest example of an explicit portfolio
restriction arises when short sales of securities are limited. The general treatment of portfolio re-
strictions in this chapter allows us to determine the consequences of short sales restrictions, and
also to model more complex portfolio restrictions, such as bid-ask spreads.
4.2 Short Sales Restrictions
The most typical short sales restriction takes the form of a lower bound on holdings of a security.
That is,
h
j
≥ −b
j
, (4.1)
where b
j
is a positive number and may be different for different agents. The short sales restrictions
may apply to one or a few securities, not necessarily to all securities. The set of securities subject
to short sales restrictions will be denoted by J
0
. The set of portfolios that satisfy restriction 4.1
for every j ∈ J
0
is the agent’s feasible portfolio set.
Our use of the term “short sales restriction” in regard to 4.1 requires clarification. Strictly,
the distinction between sales of a security and short sales is appropriate only when agents have
nonzero endowments of the security. Suppose that agents’ consumption endowments are interpreted
as payoffs on initial portfolios. As in Section 1.7, we set w
1
=
ˆ
hX, where
ˆ
h is an agent’s initial
portfolio. Assume that
ˆ
h ≥ 0. Any negative holding h
j
of security j such that h
j
< −
ˆ
h
j
means
that the agent sells more of the security than he initially owns. Consequently the restriction 4.1
with the bound b
j
set equal to
ˆ
h
j
states that the agent is prohibited from selling more of security
j than he is endowed with. With a bound b
j
smaller than
ˆ
h
j
, the agent is permitted to sell only a
fraction of his endowment of the security. With a bound that exceeds
ˆ
h
j
, the agent can sell more
than his endowment of a security, but the size of the sale is limited. We will employ the term “short
sales restriction” to denote any lower bound 4.1 on portfolios, so that all these cases are covered.
Another case of a short sales restriction of the form 4.1 is as follows: suppose that commitments
to security holdings involving strictly negative payoffs in some states—these would result from
short positions in securities with strictly positive payoffs in those states—are unenforceable in the
29
30 CHAPTER 4. PORTFOLIO RESTRICTIONS
absence of collateral. However, agents can precommit to fulfill the obligations implied by their
security holdings by pledging their endowments as collateral. In such a setting an agent would
divide his date-1 consumption endowment into collateral against each security position—that is,
would choose w
1j
for each j to satisfy w
1j
≥ 0 and
P
j
w
1j
= w
1
—and would choose security holding
h
j
subject to
h
j
x
j
+ w
1j
≥ 0 (4.2)
for all j. It can be easily seen that if x
j
is positive and nonzero such a restriction reduces to 4.1
for some bound b
j
.
Portfolio restrictions 4.2 are more stringent than the requirement that consumption be positive.
Positivity of consumption can also be cast as a collateral requirement: an agent’s date-1 endowment
is a collateral against the payoff of his portfolio so that the portfolio payoff must equal or exceed
the negative of the agent’s endowment. Clearly, restriction 4.2 implies that the payoff of portfolio
h is equal to or exceeds −w
1
, but the converse implication is not true.
4.2.1 Example
Suppose that there are two securities with payoffs x
1
= (1, 0) and x
2
= (1, 1). The restriction
that consumption be positive imposes no limit on the long (positive) position the agent can take in
portfolio h = (−1, 1), since this portfolio’s payoff is positive. In contrast, restriction 4.2 for security
1 requires that the holding of this security is limited by the agent’s collateral in state 1 so that the
agent cannot take an arbitrary position in portfolio h.
2
4.3 Portfolio Choice under Short Sales Restrictions
The agent’s consumption-portfolio choice problem in the presence of short sales restrictions is
max
c
0
,c
1
,h
u(c
0
, c
1
) (4.3)
subject to
c
0
≤ w
0
− ph (4.4)
c
1
≤ w
1
+ hX, (4.5)
and
h
j
≥ −b
j
, ∀j ∈ J
0
. (4.6)
As usual, the agent’s choice problem may involve an additional constraint on admissible consump-
tion.
The presence of short sales restrictions in the consumption and portfolio choice 4.3 leads to
first-order conditions that are slightly different from those of Section 1.5. In particular, if the
optimal consumption plan is interior, we have
p
j
≥
S
X
s=1
x
js
µ
∂
s
u
∂
0
u
¶
, ∀j ∈ J
0
, (4.7)
and
p
j
=
S
X
s=1
x
js
µ
∂
s
u
∂
0
u
¶
, ∀j /∈ J
0
. (4.8)
Inequality 4.7 can be strict only if the short sale restriction is binding on the holding of security j
at the optimal portfolio. If inequality 4.7 is strict, then the price of security j is greater than the
4.4. THE LAW OF ONE PRICE 31
sum over states of its payoff in each state multiplied by the marginal rate of substitution between
consumption in that state and consumption at date 0.
When date-0 consumption does not enter the agent’s utility function, the first-order conditions
corresponding to 4.7 and 4.8 are
λp
j
≥
S
X
s=1
x
sj
∂
s
u, ∀j ∈ J
0
, (4.9)
and
λp
j
=
S
X
s=1
x
sj
∂
s
u, ∀j /∈ J
0
, (4.10)
where λ is the Lagrange multiplier.
4.4 The Law of One Price
If there are no redundant securities—that is, if payoff matrix X has rank J—then the law of one
price holds trivially with or without portfolio restrictions. We also saw in Theorems 2.4.1 and 2.4.2
that the law of one price holds in equilibrium in the absence of portfolio restrictions under weak
monotonicity assumptions even if there exist redundant securities. This latter result fails in the
presence of portfolio restrictions: there may exist two portfolios with the same payoff and different
prices in an equilibrium.
4.4.1 Example
There are two states at date 1 and two agents who consume only at date 1 and have the same
utility function
u(c
i
1
, c
i
2
) =
1
2
ln(c
i
1
) +
1
2
ln(c
i
2
), (4.11)
for i = 1, 2. Their endowments are zero at date 0 and (3, 0) and (0, 3), respectively, at date 1. The
are three securities with payoffs
x
1
= (1, 1), x
2
= (1, 0) and x
3
= (0, 1). (4.12)
Note that the payoff of security 1 can be generated by a portfolio of one share each of securities
2 and 3. In the absence of short sales restrictions, we can calculate an equilibrium by finding
first an equilibrium with security 1 deleted from the model. That equilibrium is an equilibrium
for the model with three securities with agents’ holdings of security 1 set at zero and the price of
security 1 given by p
1
= p
2
+ p
3
, so that the law of one price holds. It involves portfolio allocation
(0, −3/2, 3/2) for agent 1 and (0, 3/2, −3/2) for agent 2, and security prices p
1
= 2/3, p
2
= 1/3, and
p
3
= 1/3. This portfolio allocation gives both agents the same risk-free consumption (3/2, 3/2). It
is easily checked that these portfolios and prices satisfy the first-order condition 1.18 that security
prices are proportional to their payoffs multiplied by agents’ marginal utilities, summed over states.
Suppose now that agents can short sell at most one share of each security, so that restriction
4.1 in the form h
j
≥ −1, for each j, is imposed. Portfolios (0, −3/2, 3/2) and (0, 3/2, −3/2) are
no longer feasible. We conjecture that portfolio allocation (0, −1, 1) for agent 1 and (0, 1, −1) for
agent 2, and security prices p
1
= 3/4, and p
2
= p
3
= 1/2 are an equilibrium under the assumed
short sales restrictions.
We can check that the first-order conditions 4.9 hold for both agents in the conjectured equilib-
rium with the Lagrange multiplier λ equal to 1. The portfolio (0, −1, 1) results in the consumption
plan (2, 1) for agent 1, so the vector of marginal utilities equals (1/4, 1/2). The holdings of securities
1 and 3 in that portfolio are strictly greater than the bound −1, so 4.9 holds with equality for these
32 CHAPTER 4. PORTFOLIO RESTRICTIONS
securities. Multiplying the payoffs of security 1 by agent 1’s marginal utilities and summing over
states, 1/4 · 1 + 1/2 · 1, we obtain p
1
= 3/4. Similarly, 1/2 ·1 equals p
3
.
Agent 1’s holding of security 2 equals the bound −1. For that security, the payoffs multiplied
by agent 1’s marginal utilities and summed over states give 1/4, which is strictly less than p
2
, so
4.9 is satisfied. Thus portfolio (0, −1, 1) is optimal for agent 1 at prices p
1
= 3/4, p
2
= 1/2 and
p
3
= 1/2 in the presence of short sales restrictions.
Checking that portfolio (0, 1, −1) is optimal for agent 2 involves the same calculations as for
agent 1 due to the symmetry of payoffs, endowments and utility functions across the states. Since
we have
p
1
6= p
2
+ p
3
, (4.13)
the law of one price fails here.
2
4.5 Limited and Unlimited Arbitrage
The fundamental result of Theorems 3.6.1 and 3.6.3—that there exists no arbitrage in equilibrium
under suitable monotonicity assumptions—does not extend to the case of portfolio restrictions. In
Example 4.4.1 the portfolio (1, −1, −1) has zero payoff and negative price, and is therefore a strong
arbitrage under the equilibrium prices.
In the presence of short sales restrictions it is necessary to distinguish unlimited arbitrages from
limited arbitrages. An unlimited arbitrage is an arbitrage that involves a long (or zero) position
in each of the securities that is subject to a short sales restriction; that is, a portfolio h such that
hX ≥ 0, ph ≤ 0, with at least one strict inequality, and h
j
≥ 0 for every j ∈ J
0
. Similarly, an
unlimited strong arbitrage is a strong arbitrage that involves a long (or zero) position in each security
that is subject to a short sales restriction; that is, a portfolio h such that hX ≥ 0, ph < 0, and
h
j
≥ 0 for every j ∈ J
0
. A limited arbitrage (limited strong arbitrage) is an arbitrage that is not
an unlimited arbitrage (strong arbitrage). In the absence of short sales restrictions, all arbitrages
are unlimited arbitrages.
The reason for the distinction is that unlimited arbitrages (unlimited strong arbitrages) can
be operated on any scale desired, whereas limited arbitrages (limited strong arbitrages) cannot.
Portfolio (1, −1, −1) is feasible under the short sales restrictions of Example 4.4.1 but scale multiples
(with the scale parameter greater than one) of it are not. It is a limited strong arbitrage under
equilibrium prices.
In the presence of short sales restrictions, under strict monotonicity the proof of Theorem 3.7.1
implies the nonexistence of unlimited arbitrages, but does not rule out limited arbitrages. Similarly,
under monotonicity the proof of Theorem 3.7.2 rules out unlimited strong arbitrages, but does not
rule out limited strong arbitrages.
4.6 Diagrammatic Representation
In Chapter 3 we presented a diagrammatic method of determining the set of security prices that
exclude arbitrage when there are no short sales restrictions. In the presence of short sales restrictions
we are interested in determining the set of security prices that exclude unlimited arbitrage.
The diagrammatic treatment is readily extended to this case. Suppose that there are two
securities, and that short selling of security 2 is restricted. If a vector of security prices p = (p
1
, p
2
)
lies in the convex cone generated by x
·1
and x
·2
, as in Figure 3.3, then there is no limited or
unlimited arbitrage portfolio. However, if p is as shown in Figure 4.1, then there is no unlimited
arbitrage portfolio, but the portfolios in the shaded region are limited arbitrages. These portfolios
involve a long position in security 1 and a short position in security 2. As the figure suggests,
4.7. BID-ASK SPREADS 33
the set of security prices excluding unlimited arbitrage is larger than the set of prices excluding
arbitrage.
If short sales of both securities 1 and 2 are restricted, then any positive p excludes unlimited
arbitrage.
4.7 Bid-Ask Spreads
In most real-world financial markets each traded security has two prices, a bid price and an ask
price. These two prices are quoted by a specialist who matches buying and selling orders on each
security. Agents buy securities from the specialist at ask prices (the prices the specialist is asking)
and sell securities to the specialist at bid prices (the prices the specialist is bidding). The difference
between the two prices is the bid-ask spread.
We can use the foregoing analysis of short sales restrictions to analyze bid-ask spreads. We shall
not attempt to formulate a full analysis of bid-ask spreads, which would include an explanation of
why they exist, but rather discuss (here, and in Chapter 7) some implications of the absence of
(unlimited) arbitrage opportunities.
Let p
bj
denote the bid price and p
aj
the ask price of security j. It is convenient to describe an
agent’s portfolio choice by two portfolios: portfolio h
a
∈ R
J
, h
a
≥ 0 purchased by the agent from
the specialist at ask prices and portfolio h
b
∈ R
J
, h
b
≥ 0 sold by the agent to the specialist at bid
prices. The agent’s consumption-portfolio choice problem is
max
c
0
,c
1
,h
a
,h
b
u(c
0
, c
1
) (4.14)
subject to
c
0
≤ w
0
− p
a
h
a
+ p
b
h
b
, (4.15)
c
1
≤ w
1
+ (h
a
− h
b
)X, (4.16)
h
b
≥ 0, h
a
≥ 0. (4.17)
Security markets with bid-ask spreads can be viewed as markets with short sales restrictions.
One only needs to consider each security j as two securities, each with a distinct price: one with
payoff x
j
and price p
aj
, the other with payoff −x
j
and price −p
bj
. Agents’ holdings of such securities
are limited by zero short sales restrictions h
aj
≥ 0 and h
bj
≥ 0.
A strong unlimited arbitrage under bid and ask price vectors p
b
and p
a
is a portfolio (h
b
, h
a
)
satisfying h
b
≥ 0, h
a
≥ 0 and such that p
a
h
a
−p
b
h
b
< 0 and (h
a
−h
b
)X ≥ 0. An unlimited arbitrage
under bid and ask price vectors p
b
and p
a
is a portfolio (h
b
, h
a
) satisfying h
b
≥ 0, h
a
≥ 0 that is
either a strong arbitrage or is such that p
a
h
a
− p
b
h
b
= 0 and (h
a
− h
b
)X > 0. The exclusion of
strong unlimited arbitrage implies that
p
aj
≥ p
bj
, (4.18)
that is, the bid-ask spread is positive for every security. To see this, note that if p
bj
> p
aj
, then a
simultaneous purchase and sale of security j would constitute a strong unlimited arbitrage.
4.8 Bid-Ask Spreads in Equilibrium
Suppose that there are I agents whose portfolio-consumption decisions are as described in 4.14.
The specialist who matches buying and selling orders for each security and imposes bid and ask
prices earns a profit equal to the sum over all securities of the quantity of traded shares multiplied
by the bid-ask spread. Suppose that the specialist consumes his profit at date 0.
An equilibrium for given bid-ask spreads {t
j
} consists of bid and ask security prices (p
b
, p
a
) that
satisfy p
aj
− p
bj
= t
j
for each j, a portfolio allocation {h
i
b
, h
i
a
} and a consumption allocation {c
i
}
34 CHAPTER 4. PORTFOLIO RESTRICTIONS
such that portfolio (h
i
a
, h
i
b
) and consumption plan c
i
are a solution to agent i’s choice problem 4.14
at prices (p
b
, p
a
), and markets clear. The market-clearing conditions are
X
i
h
i
b
=
X
i
h
i
a
, (4.19)
X
i
c
i
0
≤
X
i
w
i
0
−
X
j
[t
j
(
X
i
h
i
bj
)], (4.20)
and
X
i
c
i
1
≤
X
i
w
i
1
, (4.21)
Condition 4.20 reflects the assumption that the specialist consumes his profit at date 0. Note that
the market-clearing conditions 4.20 and 4.21 follow from 4.19 and the budget constraints 4.15–4.17.
The bid-ask spreads are exogenously given, but one could specify an objective function for the
specialist and derive his optimal choice of bid-ask spreads.
4.8.1 Example
There are two states at date 1 and two agents who have the same utility function
u(c
i
0
, c
i
1
, c
i
2
) = ln(c
i
0
) +
1
2
ln(c
i
1
) +
1
2
ln(c
i
2
), (4.22)
for i = 1, 2. Agent 1’s endowment is (1, 2, 0) and agent 2’s endowment is (1, 0, 2). The securities
traded are x
1
= (1, 0) and x
2
= (0, 1). The bid-ask spread is set exogenously at t for both securities:
that is, p
1
a
− p
1
b
= p
2
a
− p
2
b
= t.
If t = 0, so there is no bid-ask spread, agents will exchange one unit of each security so as to
reach the risk-free consumption (1, 1, 1) for each agent. When t is strictly positive, agents will not
eliminate individual risk completely due to the transactions cost.
To determine the equilibrium prices and portfolios, write agent 1’s portfolio choice problem as
max
h
1
1b
,h
1
2a
ln(1 + p
1b
h
1
1b
− p
2a
h
1
2a
) +
1
2
ln(2 − h
1
1b
) +
1
2
ln(h
1
2a
). (4.23)
Here the notation anticipates that agent 1 will set h
1
1a
= h
1
2b
= 0; that is, he will not sell security
2 or buy security 1. The first-order conditions are
1 + p
1b
h
1
1b
− p
2a
h
1
2a
= 2(2 − h
1
1b
)p
1b
(4.24)
and
1 + p
1b
h
1
1b
− p
2a
h
1
2a
= 2h
1
2a
p
2a
. (4.25)
By a similar calculation, the first-order conditions of agent 2 are
1 − p
1a
h
2
1a
+ p
2b
h
2
2b
= 2h
2
1a
p
1a
(4.26)
and
1 − p
1a
h
2
1a
+ p
2b
h
2
2b
= 2(2 − h
2
2b
)p
2b
(4.27)
for agent 2.
The symmetry of payoffs, endowments and utilities across the states implies that equilibrium
prices satisfy
p
1b
= p
2b
≡ p
b
(4.28)
p
1a
= p
2a
≡ p
a
, (4.29)
4.8. BID-ASK SPREADS IN EQUILIBRIUM 35
and equilibrium portfolios satisfy
h
1
1b
= h
2
2b
h
1
2a
= h
2
1a
. (4.30)
Market-clearing 4.19 implies that h
1
1b
= h
2
1a
and h
1
2a
= h
2
2b
. Summing up, we have
h
1
1b
= h
1
2a
= h
2
1a
= h
2
2b
≡ h. (4.31)
Further, we have
p
a
= p
b
+ t. (4.32)
Substituting 4.28 – 4.32 into 4.24 and 4.25, there results
1 − th = 2(2 − h)p
b
, (4.33)
and
1 − th = 2h(p
b
+ t). (4.34)
Eq. 4.33 implies that
p
b
=
1 − th
2(2 − h)
. (4.35)
Substituting 4.35 into 4.34, there result the quadratic equation
2th
2
− (3t + 1)h + 1 = 0, (4.36)
which has real roots. The smaller of these gives equilibrium security holding h.
1
Solution values for (h, p
b
) are (1, 0.5) when t = 0, (0.990, 0.490) when t = 0.01, (0.892, 0.411)
when t = 0.1, (0.5, 0.25) when t = 0.5, and (0.293, 0.207) when t = 1. Thus the higher t, the lower
the quantity of shares traded, as one would expect.
2
Our analysis of the effects of bid-ask spreads on security prices and volume of trade in the
preceding example should be regarded as provisional at best. As already noted, the model does not
explain why bid-ask spreads exist. It is seldom possible to obtain a reliable analysis of the effects
of any economic institution from a model that does not give an account of why that institution
exists.
Notes
In Section 4.4 we used the term “redundant security”, carrying over its meaning from Chapter 1.
Strictly, the term is a misnomer in the presence of portfolio restrictions: the fact that the payoff of
a security can be duplicated by a portfolio of other securities does not mean that it is redundant,
since the duplicating portfolio may be infeasible due to portfolio restrictions. That being the case,
the presence of portfolio restrictions implies that deleting a “redundant” security from the model
may change the equilibrium.
A model of an equilibrium with transaction costs and trading constraints has been developed
by Hahn [4]. Glosten and Milgrom [3] showed that bid-ask spreads can arise due to differences in
information about security payoffs between specialists and agents. Foley [1], Garman and Ohlson
[2], Prisman [7], Luttmer [6], and He and Modest [5] explored implications of transaction costs and
trading constraints on security prices.
1
The larger root implies negative values of p
b
, from 4.35, and negative values of date-0 consumption. It decreases
from infinity at t = 0 to 1.5 at t = ∞.
36 CHAPTER 4. PORTFOLIO RESTRICTIONS
Bibliography
[1] Duncan K. Foley. Economic equilibrium with costly marketing. In Ross M. Starr, editor,
General Equilibrium Models of Monetary Economies. Academic Press, 1989.
[2] Mark Garman and James Ohlson. Valuation of risky assets in arbitrage-free economies with
transactions costs. Journal of Financial Economics, 9:271–280, 1981.
[3] Lawrence R. Glosten and Paul R. Milgrom. Bid, ask and transaction prices in a specialist model
with heterogeneously informed traders. Journal of Financial Economics, 14:71–100, 1985.
[4] Frank Hahn. Equilibrium with transaction costs. Econometrica, 39:417–439, 1971.
[5] Hua He and David M. Modest. Market frictions and consumption-based asset pricing. Journal
of Political Economy, 103:94–117, 1995.
[6] Erzo Luttmer. Asset pricing in economies with frictions. Econometrica, 64:1439–1467, 1996.
[7] Eliezer Prisman. Valuation of risky assets in arbitrage free economies with frictions. Journal of
Finance, 41:545–560, 1986.
37
38 BIBLIOGRAPHY
Part II
Valuation
39
Chapter 5
Valuation
5.1 Introduction
In this and the next chapter we assume again that agents can trade without any portfolio re-
strictions. As established in Chapter 2, security prices can be characterized by a payoff pricing
functional mapping the asset span into the reals. The payoff pricing functional is linear and strictly
positive (positive) iff security prices exclude arbitrage (strong arbitrage). A valuation functional
is an extension of the payoff pricing functional to the entire contingent claim space R
S
. Thus the
valuation functional is a linear functional
Q : R
S
→ R (5.1)
that coincides with the payoff pricing functional on the asset span M; that is
Q(z) = q(z) for every z ∈ M. (5.2)
The valuation functional assigns values to all contingent claims, not just to payoffs. Of special
interest is a valuation functional that is strictly positive (positive) since, as shown in Chapter 3
in the case of complete markets, this property is equivalent to the absence of arbitrage (strong
arbitrage). A strictly positive (positive) valuation functional will be used in Chapter 6 to derive
important representations of security prices.
The following simple example illustrates a positive valuation functional:
5.1.1 Example
Suppose that there are two states and a single security with payoff x
1
= (1, 2) and price p
1
= 1.
The asset span is M = span{(1, 2)} = {(α, 2α) : α ∈ R}, and the payoff pricing functional is given
by q(α, 2α) = α. Each functional Q : R
2
→ R defined by Q(z) = q
1
z
1
+ q
2
z
2
, where q
1
, q
2
≥ 0 and
q
1
+ 2q
2
= 1 is a positive valuation functional.
2
5.2 The Fundamental Theorem of Finance
In equilibrium the vector ∂
1
u/∂
0
u of marginal rates of substitution of an agent whose consumption
is interior defines a linear functional that maps each contingent claim z ∈ R
S
to (∂
1
u/∂
0
u)z. This
functional coincides with the equilibrium payoff pricing functional on the asset span (in particular,
p
j
= (∂
1
u/∂
0
u)x
j
; see 1.14). The functional given by the marginal rates of substitution is strictly
positive (positive) if utility functions are strictly increasing (increasing). Of course, unless markets
are complete, different agents may have different marginal rates of substitution; these give rise to
different valuation functionals.
41
42 CHAPTER 5. VALUATION
If we consider an arbitrary vector of security prices, can we be assured that a strictly positive
(positive) valuation functional exists? It cannot exist if security prices permit arbitrage (strong
arbitrage) since then either the payoff pricing functional does not exist or it is not strictly positive
(positive).
We come now to a critical question: If security prices are such as to exclude arbitrage, does a
strictly positive valuation functional exist? The answer is provided in the following theorem.
5.2.1 Fundamental Theorem of Finance
Security prices exclude arbitrage iff there exists a strictly positive valuation functional.
Suppose now only that security prices exclude strong arbitrage. This weakening of the condition
implies a weakening of the conclusion:
5.2.2 Fundamental Theorem of Finance, Weak Form
Security prices exclude strong arbitrage iff there exists a positive valuation functional.
For both theorems, sufficiency follows from Theorems 3.4.2 and 3.4.1, since existence of a strictly
positive (positive) valuation functional implies existence of a strictly positive (positive) payoff
pricing functional, the payoff pricing functional being a restriction of the valuation functional. The
proof of necessity will occupy us in the remainder of this chapter.
The extension of the payoff pricing functional q from the asset span to the entire commodity
space is achieved by extending q one dimension at a time. In the first step we choose a contingent
claim ˆz not in the asset span M and extend q to the subspace spanned by M and ˆz. This extended
subspace has dimension equal to the dimension of M plus one. The extension of the payoff pricing
functional is achieved by specifying a value π for the contingent claim ˆz. For the extension to
remain strictly positive (positive), the chosen value π must be such that all payoffs in M strictly
greater (greater) than ˆz have prices that are strictly greater (greater) than π, and all payoffs in
M strictly less (less) than ˆz have prices that are strictly less (less) than π. These restrictions
define an interval in which π must lie. The extension is the payoff pricing functional for security
markets consisting of J securities with payoffs {x
1
, . . . , x
J
} and prices {p
1
, . . . , p
J
} and a security
with payoff ˆz and price π.
In the second step, we choose a contingent claim not in the span of the J + 1 securities of step
1 and extend the payoff pricing functional to the subspace spanned by the J + 1 securities of step 1
and the new contingent claim. After S − J steps we achieve an extension to the entire commodity
space. Since all of the steps in this construction are the same, we present only the first.
5.3 Bounds on the Values of Contingent Claims
We now define the upper and lower bounds on the value of a contingent claim z ∈ R
S
that can be
inferred from the prices of the payoffs in M. The upper bound
q
u
(z) ≡ min
h
{ph : hX ≥ z} (5.3)
is the lowest price of a portfolio the payoff of which dominates the contingent claim. The lower
bound
q
`
(z) ≡ max
h
{ph : hX ≤ z} (5.4)
is the highest price of a portfolio the payoff of which is dominated by the contingent claim.
1
For a payoff in the asset span, the lower and the upper bounds coincide with the value under
the payoff pricing functional as long as there exists no strong arbitrage:
1
If {h : hX ≥ z} is empty, we set q
u
(z) = ∞. This occurs if, for example, M = span{(1, 0)} and z = (1, 1).
Similarly, if {h : hX ≤ z} is empty, we set q
`
(z) = −∞.
5.3. BOUNDS ON THE VALUES OF CONTINGENT CLAIMS 43
5.3.1 Proposition
If security prices exclude strong arbitrage, then q
u
(z) = q
`
(z) = q(z) for every z ∈ M.
Proof: By the definitions of the bounds we have q
u
(z) ≤ q(z) and q
`
(z) ≥ q(z) for z ∈ M.
Suppose that q
u
(z) < q(z) for some z ∈ M. Then there exists a portfolio h
0
such that
h
0
X ≥ z (5.5)
and
ph
0
< q(z). (5.6)
Let h be a portfolio such that hX = z and ph = q(z). Then portfolio h
0
− h is a strong arbitrage.
This contradicts the assumption. The proof that q
`
(z) = q(z) is similar.
2
The following two examples illustrate the bounds on the values of contingent claims that are
not in the asset span.
5.3.2 Example
In Example 5.1.1, the contingent claim z = (1, 1) is not in the asset span. We have
q
u
(z) = min{h : (h, 2h) ≥ (1, 1)} = 1 (5.7)
q
`
(z) = max{h : (h, 2h) ≤ (1, 1)} =
1
2
. (5.8)
Thus the bounds on the value of z are 1/2 and 1.
2
5.3.3 Example
Let there be two securities: security 1, a bond with risk-free payoff x
1
= (1, 1, 1); and security 2, a
stock with payoff x
2
= (1, 2, 4). The prices of the bond and stock are, respectively, p
1
= 1/2 and
p
2
= 1. A nontraded call option on the stock with strike price of 3 has the payoff z = (0, 0, 1).
That payoff is not in the span of the payoffs on the stock and the bond and hence cannot be priced
using the payoff pricing functional.
A lower bound on the value of the call is determined by solving
max
h
1
,h
2
(p
1
h
1
+ p
2
h
2
) (5.9)
subject to
h
1
x
1
+ h
2
x
2
≤ z. (5.10)
The constraint implies that h
1
and h
2
satisfy
h
1
+ h
2
≤ 0, (5.11)
h
1
+ 2h
2
≤ 0, (5.12)
h
1
+ 4h
2
≤ 1. (5.13)
The linear program 5.9 can easily be solved graphically.
One can also argue as follows: since there are two choice variables, it is permissible to assume
that at the solution at least two of the constraints are satisfied with equality. Constraints 5.11 and
5.12 are satisfied at equality by h
1
= h
2
= 0, at which point 5.13 is satisfied. Constraints 5.11 and
5.13 are satisfied at equality by h
1
= −1/3, h
2
= 1/3, at which point 5.12 is violated. Constraints
5.12 and 5.13 are satisfied at equality by h
1
= −1, h
2
= 1/2, at which point 5.11 is satisfied.
44 CHAPTER 5. VALUATION
The two points at which two of the constraints are satisfied as equalities and the third constraint
is satisfied both give portfolios with zero price, so zero is the lower bound for the value of the call.
The upper bound on the value of the call is determined by solving
min
h
1
,h
2
(p
1
h
1
+ p
2
h
2
) (5.14)
subject to
h
1
+ h
2
≥ 0, (5.15)
h
1
+ 2h
2
≥ 0, (5.16)
h
1
+ 4h
2
≥ 1. (5.17)
As above, the minimum is attained at a point where at least two of the constraints are satisfied
with equality. Since constraints 5.15 – 5.17 are the reverse inequalities to 5.11 – 5.13, the only
point that satisfies two of the constraints with equality is h
1
= −1/3, h
2
= 1/3. The price of this
portfolio is 1/6. Thus the bounds on the value of the call option are zero and 1/6.
2
Important properties of the bounds q
`
and q
u
are given in the following propositions.
5.3.4 Proposition
If security prices exclude strong arbitrage, then q
u
(z) ≥ q
`
(z) for every contingent claim z ∈ R
S
.
Proof: Suppose that q
u
(z) < q
`
(z) for some z ∈ R
S
. By the definitions of the bounds q
u
and
q
`
, there exist portfolios h
0
and h
00
such that
h
0
X ≤ z ≤ h
00
X (5.18)
and
ph
0
> ph
00
. (5.19)
But then the portfolio h
00
− h
0
satisfies (h
00
− h
0
)X ≥ 0 and p(h
00
− h
0
) < 0, so that it is a strong
arbitrage. This contradicts the assumption.
2
Also
5.3.5 Proposition
If security prices exclude arbitrage, then q
u
(z) > q
`
(z) for every contingent claim z not in the asset
span.
Proof: In view of Proposition 5.3.4, we only have to prove that q
u
(z) 6= q
`
(z) for every z /∈ M.
Suppose that q
u
(z) = q
`
(z) for some z /∈ M. Then there exist portfolios h
0
and h
00
such that
h
0
X ≤ z ≤ h
00
X (5.20)
and
ph
0
= ph
00
= q
u
(z). (5.21)
Neither of the weak inequalities in 5.20 can be an equality since z is not in the asset span; that is,
it cannot be generated by a portfolio. Consequently, (h
00
− h
0
)X > 0, and p(h
00
− h
0
) = 0, so that
the portfolio h
00
− h
0
is an arbitrage. This is a contradiction.
2
5.4. THE EXTENSION 45
5.4 The Extension
Having derived upper and lower bounds on the value of any contingent claim, we turn now to how
these bounds are used to extend the payoff pricing functional.
Fix a contingent claim ˆz /∈ M. Define N by
N = {z + λˆz : z ∈ M and λ ∈ R}. (5.22)
Thus N is the subspace of R
S
that has dimension equal to the dimension of M plus one and
contains M and ˆz. It is the asset span of J + 1 securities with payoffs {x
1
, . . . , x
J
} and ˆz.
If there is no strong arbitrage—equivalently, if the payoff pricing functional q is positive—then
Proposition 5.3.4 implies that a finite value π can be chosen to satisfy
2
q
`
(ˆz) ≤ π ≤ q
u
(ˆz). (5.23)
We extend q to a linear functional on N in that we define Q : N → R by
Q(z + λˆz) ≡ q(z) + λπ. (5.24)
We now prove that Q, as just defined, is the desired positive extension of q.
5.4.1 Proposition
If q : M → R is positive, so is Q : N → R.
Proof: Let y ∈ N. Then
y = z + λˆz (5.25)
for some z ∈ M and some λ ∈ R. Of the three possibilities for λ, suppose first that λ > 0. Then
y ≥ 0 implies
ˆz ≥ −
z
λ
. (5.26)
Applying q
`
to both sides of 5.26 and using the implication of 5.4 that q
`
is an increasing function,
there results
q
`
(ˆz) ≥ q
`
(−
z
λ
). (5.27)
By Proposition 5.3.1, the functions q and q
`
coincide on M. Since −z/λ ∈ M, we have q
`
(−z/λ) =
q(−z/λ). Therefore 5.27 becomes
q
`
(ˆz) ≥ q(−
z
λ
). (5.28)
Since π ≥ q
`
(ˆz), 5.28 implies that
π ≥ q(−
z
λ
), (5.29)
or alternatively that
q(z) + λπ ≥ 0. (5.30)
Since the left-hand side of 5.30 equals Q(y), we obtain that Q(y) ≥ 0.
If λ < 0, a similar argument, but using q
u
and the fact that π ≤ q
u
(ˆz), also gives Q(y) ≥ 0.
Finally, if λ = 0, then y = z and Q(y) = q(z). The positivity of q implies that if y ≥ 0, then
Q(y) ≥ 0.
2
If there is no arbitrage—equivalently, if q is strictly positive—then Proposition 5.3.5 implies
that π can be chosen to satisfy
q
`
(ˆz) < π < q
u
(ˆz). (5.31)
Then
2
One can show that the assumption of no strong arbitrage implies that the lower and upper bounds cannot be
both equal to +∞ or both equal to −∞.
46 CHAPTER 5. VALUATION
5.4.2 Proposition
If q : M → R is strictly positive, so is Q : N → R.
The proof is essentially the same as the proof of Proposition 5.4.1.
For the prices {p
1
, . . . , p
J
} and π, functional Q, as defined in 5.24, is the payoff pricing functional
on N. Therefore Q is strictly positive (positive) on N iff the indicated prices exclude arbitrage
(strong arbitrage) in J + 1 securities markets with payoffs {x
1
, . . . , x
J
} and ˆz.
5.4.3 Example
In example 5.3.2, define
N = {z + λˆz : z ∈ M, λ ∈ R}, (5.32)
where M = span{(1, 2)}, and ˆz = (1, 1). Thus N = R
2
. We have the following bounds on the
value π of ˆz (see 5.7 and 5.8):
1
2
≤ π ≤ 1. (5.33)
We choose π = 3/4 and define Q : N → R by
Q(z + λˆz) = q(z) +
3
4
λ (5.34)
for z ∈ M and λ ∈ R. Recall that q(z) = α for z = (α, 2α). One can easily check that
Q(1, 0) =
1
2
and Q(0, 1) =
1
4
(5.35)
and hence that
Q(y
1
, y
2
) =
1
2
y
1
+
1
4
y
2
. (5.36)
Thus Q is strictly positive.
2
5.5 Uniqueness of the Valuation Functional
The construction of Section 5.4 indicates that extending the payoff pricing functional does not
result in a unique valuation functional. Indeed, as was proved in Proposition 5.3.5, there exists a
continuum of values of π that define extensions with the desired properties. An exception is the
case of complete markets. Then the asset span M equals the contingent claim space R
S
and the
payoff pricing functional is the valuation functional. It turns out that this is the only case of unique
valuation functional.
5.5.1 Theorem
Suppose that security prices exclude arbitrage (strong arbitrage). Then security markets are com-
plete iff there exists a unique strictly positive (positive) valuation functional.
Proof: Necessity is obvious. Sufficiency follows from Proposition 5.3.5 (Proposition 5.3.4). If
markets are not complete, so that there exists a contingent claim not in the asset span, then there
exists a nondegenerate interval of values of that contingent claim that give rise to different strictly
positive (positive) valuation functionals.
2
We pointed out in Section 5.1 that, if security prices are equilibrium prices, then the marginal
rates of substitution of an agent define a valuation functional. If markets are incomplete, the
marginal rates may be different for different agents and the associated valuation functionals are
different. Otherwise, if markets are complete, there is a unique valuation functional. Hence the
marginal rates of substitution of all agents have to be the same.
5.5. UNIQUENESS OF THE VALUATION FUNCTIONAL 47
Notes
The term “Fundamental Theorem of Finance” is due to Dybvig and Ross [3]. The first statement
and proof of the Fundamental Theorem of Finance appears in [4] and [5]. See also Beja [1].
The derivation of the valuation functional by extending the payoff pricing functional is due
to Clark [2]. Note, though, that Clark does not restrict himself, as we do, to finite-dimensional
contingent claim spaces.
Theorem 5.5.1 demonstrates that markets are complete iff the valuation functional is unique.
Clark [2] shows that this result does not carry over to the infinite-dimensional case. The valuation
functional may be unique even if markets are incomplete.
48 CHAPTER 5. VALUATION
Bibliography
[1] Avraham Beja. The structure of the cost of capital under uncertainty. Review of Economic
Studies, 38:359–369, 1971.
[2] Stephen A. Clark. The valuation problem in arbitrage price theory. Journal of Mathematical
Economics, 22:463–478, 1993.
[3] Philip Dybvig and Stephen A. Ross. Arbitrage. In M. Milgate J. Eatwell and P. Newman,
editors, The New Palgrave: A Dictionary of Economics. McMillan, 1987.
[4] Stephen A. Ross. Risk, return and arbitrage. In Irwin Friend and James Bicksler, editors, Risk
and Return in Finance. Ballinger, Cambridge, Massachusetts, 1976.
[5] Stephen A. Ross. A simple approach to the valuation of risky streams. Journal of Business,
51:453–475, 1978.
49
50 BIBLIOGRAPHY
Chapter 6
State Prices and Risk-Neutral
Probabilities
6.1 Introduction
By the Fundamental Theorem of Finance, the payoff pricing functional can be extended to a strictly
positive (positive) valuation functional iff security prices exclude arbitrage (strong arbitrage). We
show in this chapter that each strictly positive (positive) valuation functional can be represented
by a vector of strictly positive (positive) state prices. State prices can be easily calculated as a
strictly positive (positive) solution to a system of linear equations relating security prices and their
payoffs. An implication of the existence of strictly positive (positive) state prices is the absence of
arbitrage (strong arbitrage). An implication of the uniqueness of state prices is that markets are
complete.
The valuation functional can also be represented by strictly positive (positive) probabilities of
the states. These probabilities, commonly known as risk-neutral probabilities, are simple transforms
of the state prices and therefore just as useful as those prices. Under the risk-neutral probabilities
representation, the price of each security equals its expected payoff discounted by the risk-free
return.
6.2 State Prices
In Chapter 3 we derived the state prices associated with given security prices under the assumption
of complete markets. If markets are complete, the payoff pricing functional q is defined on the entire
contingent claim space R
S
, and the state price vector q = (q
1
, . . . , q
S
) provides a representation
of the functional q as q(z) = qz for every payoff z ∈ R
S
. The derivation of Chapter 3 can now
be extended to incomplete markets using the valuation functional rather than the payoff pricing
functional.
A valuation functional, being a linear functional on R
S
, can be identified by its values on the
basis vectors of that space. Let
q
s
≡ Q(e
s
), (6.1)
for every s, where e
s
is the state claim for state s. The value q
s
is the state price of state s. If Q
is strictly positive (positive), then each state price q
s
is strictly positive (positive).
Since every contingent claim z ∈ R
S
can be written as z =
P
s
z
s
e
s
, we have
Q(z) =
X
s
z
s
Q(e
s
) =
X
s
z
s
q
s
, (6.2)
or
Q(z) = qz. (6.3)
51
52 CHAPTER 6. STATE PRICES AND RISK-NEUTRAL PROBABILITIES
Eq. 6.3 is the state-price representation of the valuation functional Q. It defines a one-to-one
relation between valuation functionals and state price vectors. Since the valuation functional in
incomplete markets is not unique (Theorem 5.5.1), state prices are not unique either.
Eq. 6.3 provides a simple method for pricing payoffs without determining a portfolio that
generates the payoff under consideration. Once state prices are known, the price of every payoff
can be obtained. Eq. 6.3 can also be applied to contingent claims not in the asset span, although
for any such claim the derived value will depend on the state price vector used. It follows from the
proof of the Fundamental Theorem of Finance, provided in Section 5.4, that the derived value is
independent of the state price vector iff the contingent claim lies in the asset span.
State prices can be characterized as solutions to a system of linear equations, just as under
complete markets (recall 2.14). To see this we apply 6.3 to the payoff x
j
of security j. Since
Q(x
j
) = p
j
, we obtain
p
j
= qx
j
, (6.4)
or in vector-matrix notation
p = Xq. (6.5)
State prices are a solution to the system of J equations 6.4 with S unknowns q
s
. Strictly positive
state prices are a strictly positive solution; positive state prices are a positive solution. If markets
are incomplete, then the payoff matrix X has rank less than S and the independent equations of
6.4 are fewer in number than the number of unknowns. If markets are complete, then state prices
are unique. Of course, if markets are incomplete there are also nonpositive solutions to 6.4, but
they do not qualify as state prices.
Eq. 6.5 provides a complete characterization of state-price vectors and hence valuation func-
tionals as well.
6.2.1 Theorem
There exists a strictly positive valuation functional iff there exists a strictly positive solution to
equations 6.5. Each strictly positive solution q defines a strictly positive valuation functional Q
satisfying Q(z) = qz for every z ∈ R
S
.
Proof: It was proved in 6.1 – 6.5 that state prices associated with a strictly positive valuation
functional are a solution to 6.5. Existence of a valuation functional follows from the fact that, if
q is a strictly positive solution to 6.5, then the functional Q defined by Q(z) = qz is linear and
strictly positive. Whenever z ∈ M, then z = hX for some portfolio h, and Q(z) = qz = hXq = ph,
that is, Q coincides with the payoff pricing functional on M. Thus Q is a strictly positive valuation
functional.
2
Similarly,
6.2.2 Theorem
There exists a positive valuation functional iff there exists a positive solution to equations 6.5. Each
positive solution q defines a positive valuation functional Q satisfying Q(z) = qz for every z ∈ R
S
.
Theorems 6.2.1 and 6.2.2 say that state price vectors can be defined either as the values of the
state claims under valuation functionals, as in 6.1, or as a strictly positive (positive) solution to 6.5.
The Fundamental Theorem of Finance can be restated to say that security prices exclude arbitrage
(strong arbitrage) iff there exists a strictly positive (positive) state-price vector.
6.2.3 Example
In Example 5.3.3, there were two securities: a risk-free bond with payoff x
1
= (1, 1, 1) and price
p
1
= 1/2, and a risky stock with payoff x
2
= (1, 2, 4) and price p
2
= 1. Positive state prices q
1
, q
2
, q
3
6.3. FARKAS-STIEMKE LEMMA 53
are a positive solution to the system of two equations
q
1
+ q
2
+ q
3
=
1
2
(6.6)
and
q
1
+ 2q
2
+ 4q
3
= 1. (6.7)
Using q
3
as a parameter (we have two equations and three unknowns), the solution is
q
1
= 2q
3
, q
2
=
1
2
− 3q
3
. (6.8)
For state prices to be positive, we must have 0 ≤ q
3
≤ 1/6. If 0 < q
3
< 1/6, then state prices are
strictly positive. The existence of a strictly positive solution verifies that security prices p
1
= 1/2
and p
2
= 1 exclude arbitrage.
It is worth noticing that the value of a call option on the stock with exercise price 3 is q
3
under the valuation functional given by q
1
, q
2
and q
3
. The condition 0 ≤ q
3
≤ 1/6 is precisely the
condition that the value of the option has to lie between the lower and upper bounds derived in
Example 5.3.3.
2
6.3 Farkas-Stiemke Lemma
The equivalence of the absence of strong arbitrage and the existence of positive state prices can be
derived directly from a well-known mathematical result, Farkas’ Lemma. This result is essential in
deriving state prices under portfolio restrictions. A derivation will be provided in Chapter 7.
Let y and a be m-dimensional vectors, b an n-dimensional vector, and Y an m × n matrix for
arbitrary m, n.
6.3.1 Theorem (Farkas’ Lemma)
There does not exist a ∈ R
m
such that
aY ≥ 0 and ay < 0 (6.9)
iff there exists b ∈ R
n
such that
y = Y b and b ≥ 0. (6.10)
With Y = X, y = p, a = h and b = q, Farkas’ Lemma says that no strong arbitrage and the
existence of positive state prices are equivalent. That result was proved in Theorems 5.2.2 and
6.2.1.
The equivalence of the absence of arbitrage and the existence of strictly positive state prices can
be derived directly from Stiemke’s Lemma, a version of Farkas’ Lemma under which b is strictly
positive.
6.3.2 Theorem (Stiemke’s Lemma )
There does not exist a ∈ R
m
such that
aY ≥ 0 and ay ≤ 0, with at least one strict inequality (6.11)
iff there exists b ∈ R
n
such that
y = Y b and b À 0. (6.12)
With Y = X, y = p, a = h and b = q, Stiemke’s Lemma says that the no arbitrage is equivalent
to the existence of strictly positive state prices. That result was proved in Theorems 5.2.1 and
6.2.2.
54 CHAPTER 6. STATE PRICES AND RISK-NEUTRAL PROBABILITIES
6.4 Diagrammatic Representation
In Chapter 3 we presented a diagrammatic analysis of security prices for two securities. It was
shown that security prices exclude strong arbitrage whenever the price vector lies in the convex
cone generated by the vectors of payoffs of the two securities in each state. Security prices exclude
arbitrage whenever the vector of security prices lies in the interior of that cone. That is precisely
the diagrammatic interpretation of the existence of strictly positive (positive) state prices. Eq. 6.5
with positive state prices q
s
means that the vector of security prices p lies in the cone generated by
vectors x
.s
= (x
1s
, . . . , x
Js
) in R
J
. If the state prices are strictly positive, then vector p lies in the
interior of that cone.
6.5 State Prices and Value Bounds
In the proof of the Fundamental Theorem of Finance in Section 5.4 we showed that for any value
lying between the lower bound q
`
(z) and the upper bound q
u
(z) of a contingent claim z, it is possible
to define a positive valuation functional that maps z onto this assumed value. It follows that the set
of values of z under all positive valuation functionals is the interval with q
`
(z) as the lower limit and
q
u
(z) as the upper limit. Since each valuation functional has a state-price representation 6.3, the
same set of values of z obtains when applying all positive state prices associated with given security
prices to z. Using the characterization 6.5 of state prices, we obtain the following expressions for
the upper and the lower bounds:
q
u
(z) = max
q≥0
{qz : p = Xq}, (6.13)
and
q
`
(z) = min
q≥0
{qz : p = Xq}. (6.14)
The use of these expressions for calculating bounds is illustrated by the following example.
6.5.1 Example
Value bounds for the contingent claim (1, 1) of Example 5.3.2 can be calculated using 6.13 and
6.14. We have
q
u
(1, 1) = max
(q
1
,q
2
)≥0
{q
1
+ q
2
: q
1
+ 2q
2
= 1}, (6.15)
and
q
`
(1, 1) = min
(q
1
,q
2
)≥0
{q
1
+ q
2
: q
1
+ 2q
2
= 1}. (6.16)
The maximum equals 1 and is attained at q = (1, 0). The minimum equals 1/2 and is attained at
q = (0, 1/2).
2
6.5.2 Example
The value bounds in Example 5.3.3 can be derived using 6.13 and 6.14 as
q
u
(0, 0, 1) = max
(q
1
,q
2
,q
3
)≥0
{q
3
: q
1
+ q
2
+ q
3
=
1
2
; q
1
+ 2q
2
+ 4q
3
= 1}, (6.17)
and
q
`
(0, 0, 1) = min
(q
1
,q
2
,q
3
)≥0
{q
3
: q
1
+ q
2
+ q
3
=
1
2
; q
1
+ 2q
2
+ 4q
3
= 1}. (6.18)
The maximum equals 1/6 and is attained at q = (1/3, 0, 1/6). The minimum equals 0 and is
attained at q = (0, 1/2, 0).
2
6.6. RISK-FREE PAYOFFS 55
6.6 Risk-Free Payoffs
A contingent claim that does not depend on the state is risk free. If markets are complete, risk-free
claims are necessarily in the asset span. If markets are incomplete it may or may not be possible
to construct a portfolio with a risk-free payoff.
Given the presence of Treasury debt, which is free of default risk, it might seem that there is
no reason to consider the possibility that risk-free claims are not in the asset span. However, the
payoff on nominal debt is subject to inflation risk, and therefore is random in real terms. Since
we are not modeling monetary economies we will not attempt to explain inflation risk, but we do
not want to restrict the analysis to the case in which investors are guaranteed to have access to
investments that are completely risk free.
If a nonzero risk-free payoff lies in the asset span, then all risk-free payoffs lie in the asset span
and, as long as the law of one price holds, they all have the same return. We denote that risk-free
return by ¯r. It follows from 6.2 that ¯r satisfies
¯r =
1
P
s
q
s
. (6.19)
6.7 Risk-Neutral Probabilities
Suppose that security prices exclude arbitrage (strong arbitrage) and that a risk-free claim with
strictly positive return ¯r lies in the asset span. Let q be a strictly positive (positive) state price
vector. Define
π
∗
s
≡ ¯rq
s
=
q
s
P
s
q
s
, (6.20)
for every s. So defined, the π
∗
s
’s are strictly positive (positive) and sum to one. It is natural to
interpret them as probabilities. We call them risk-neutral probabilities. The motivation for this
term will be presented in Chapter 14.
When equipped with risk-neutral probabilities, the set of states S can be regarded as a prob-
ability space. Date-1 consumption plans, security payoffs, contingent claims and others, which we
have thus far regarded as vectors with S components, can now be regarded as random variables on
the probability space S. Here and throughout this book we make no distinction between a random
variable and the vector of values the random variables takes on.
Let E
∗
denote the expectation with respect to the probabilities π
∗
. Then E
∗
(z) =
P
s
π
∗
s
z
s
for
a contingent claim z. We have
qz =
X
s
q
s
z
s
=
1
¯r
X
s
π
∗
s
z
s
=
1
¯r
E
∗
(z). (6.21)
Substituting 6.21 in 6.4 we obtain
p
j
=
1
¯r
E
∗
(x
j
) (6.22)
for every security j.
Eq. 6.22 says that the price of each security equals the expectation of its payoff with respect to
probabilities π
∗
discounted by the risk-free return. We emphasize that the expectation is taken with
respect to probabilities π
∗
derived from state prices rather than agents’ subjective probabilities.
Eq. 6.22 can also be written in terms of returns as
¯r = E
∗
(r
j
). (6.23)
Substituting 6.21 in 6.4 we obtain
Q(z) =
1
¯r
E
∗
(z) (6.24)
56 CHAPTER 6. STATE PRICES AND RISK-NEUTRAL PROBABILITIES
for every z ∈ R
S
. Eq. 6.24 is the representation of the valuation functional Q by risk-neutral
probabilities. The value of each contingent claim equals the discounted expectation of the claim
with respect to risk-neutral probabilities.
Since risk-neutral probabilities are rescaled state prices, they have all the properties of those
prices. They are characterized as strictly positive (positive) solutions to equations 6.22. Their exis-
tence and strict positivity (positivity) are equivalent to the absence of arbitrage (strong arbitrage);
their uniqueness is equivalent to market completeness.
Using risk-neutral probabilities instead of state prices, the upper and lower bounds on values
of a contingent claim 6.13 and 6.14 can be written as
q
u
(z) =
1
¯r
max
π
∗
E
∗
(z) (6.25)
and
q
`
(z) =
1
¯r
min
π
∗
E
∗
(z), (6.26)
where the maximum and minimum are taken over all risk-neutral probabilities.
Risk-neutral probabilities play an important role in multidate security markets. A natural
extension of the pricing relationship 6.22 is the martingale property of security prices; see Chapter
26.
6.7.1 Example
The risk-neutral probabilities of Example 6.2.3 can be derived by multiplying state prices by the
risk-free return ¯r. Since ¯r = 2, we have
π
∗
1
= 2π
∗
3
, π
∗
2
= 1 − 3π
∗
3
, and 0 ≤ π
∗
3
≤
1
3
. (6.27)
Since state prices are not unique, neither are risk-neutral probabilities.
Risk-neutral probabilities can also by derived directly from the system of equations 6.22, that
is,
1 = π
∗
1
+ π
∗
2
+ π
∗
3
, (6.28)
and
2 = π
∗
1
+ 2π
∗
2
+ 4π
∗
3
. (6.29)
2
Notes
State prices and risk-neutral probabilities were first introduced by Ross [4] and [5]. Further discus-
sion of state prices and risk-neutral probabilities can be found in Dybvig and Ross [2] and Varian [6].
Green and Srivastava [3] studied the relation between state prices and agents’ optimal consumption
plans.
We presented two ways of deriving state prices under the assumption that security prices exclude
arbitrage or strong arbitrage. One uses the extension of the payoff pricing functional (Section 5.4);
the other applies the Farkas-Stiemke Lemma (Section 6.3). There are two other ways of deriving
state prices: the first, by making use of the duality theorem of linear programming; the second, by
making use of the separating hyperplane theorem (see Duffie [1]).
The duality theorem of linear programming says that linear programs come in pairs: with every
constrained maximization problem that has a solution there is associated a constrained minimiza-
tion problem that also has a solution, and the optimized values of the objective functions in the
two problems are the same. Absence of strong arbitrage implies that a certain primal problem
6.7. RISK-NEUTRAL PROBABILITIES 57
has a solution, and the duality theorem therefore implies the existence of positive state prices as a
solution to a dual problem. The result of Section 6.5 that the upper (lower) bound on the value of
a contingent claim can be derived either by minimizing (maximizing) over payoffs or maximizing
(minimizing) over state prices associated with given security prices is also an implication of duality
of linear programming.
A risk-free payoff that equals the expectation of a risky payoff with respect to the risk-neutral
probabilities is called the certainty-equivalent payoff . By construction, the certainty-equivalent
payoff associated with a given risky payoff is a risk-free payoff with the same price as the risky
payoff.
The derivation of risk-neutral probabilities in Section 6.7 relies on the assumption that the
risk-free payoff is in the asset span. If it is not, then the return on any security or portfolio, if
strictly positive, can be substituted for the risk-free return. Using the return on security k as the
deflator, the price of security j can be written
p
j
=
X
s
q
s
r
ks
x
js
r
ks
=
X
s
ν
s
x
js
r
ks
, (6.30)
where
ν
s
≡ q
s
r
ks
. (6.31)
Since
P
s
ν
s
= 1, the ν
s
’s can be interpreted as probabilities, and 6.30 can therefore be rewritten as
p
j
= E
ν
µ
x
j
r
k
¶
. (6.32)
The probabilities ν depend on the choice of deflator security. If one security is substituted for
another, then, unless the returns are the same, ν will change.
58 CHAPTER 6. STATE PRICES AND RISK-NEUTRAL PROBABILITIES
Bibliography
[1] Darrell Duffie. Dynamic Asset Pricing Theory, Second Edition. Princeton University Press,
Princeton, N. J., 1996.
[2] Philip Dybvig and Stephen A. Ross. Arbitrage. In M. Milgate J. Eatwell and P. Newman,
editors, The New Palgrave: A Dictionary of Economics. McMillan, 1987.
[3] Richard C. Green and Sanjay S. Srivastava. Risk aversion and arbitrage. Journal of Finance,
40:257–268, 1985.
[4] Stephen A. Ross. Risk, return and arbitrage. In Irwin Friend and James Bicksler, editors, Risk
and Return in Finance. Ballinger, Cambridge, Massachusetts, 1976.
[5] Stephen A. Ross. A simple approach to the valuation of risky streams. Journal of Business,
51:453–475, 1978.
[6] Hal R. Varian. The arbitrage principle in financial economics. Journal of Economic Perspectives,
1:55–72, 1987.
59
60 BIBLIOGRAPHY
Chapter 7
Valuation under Portfolio Restrictions
7.1 Introduction
The valuation theory of Chapters 5 and 6 relied on linearity of pricing in security markets or, in
other words, on the law of one price. We observed in Chapter 4 that the law of one price may fail in
an equilibrium in the presence of portfolio restrictions. We show in this chapter that, nevertheless,
many of the results of valuation theory in the absence of portfolio restrictions can be extended,
although generally in altered form, to security markets with such portfolio restrictions as short
sales restrictions or bid-ask spreads. In particular, there exist strictly positive (positive) state
prices iff security prices exclude unlimited arbitrage (unlimited strong arbitrage). The existence
of strictly positive (positive) state prices therefore provides a simple test of whether or not there
exists unlimited arbitrage (unlimited strong arbitrage).
7.2 Payoff Pricing under Short Sales Restrictions
As in Chapter 4, we consider short sales restrictions of the form
h
j
≥ −b
j
(7.1)
for every security j ∈ J
0
, with positive b
j
.
The payoff pricing functional as introduced in Chapter 2 is a single-valued functional if security
prices satisfy the law of one price. As noted above, in the presence of short sales restrictions the
law of one price may fail in an equilibrium, as long as the implied strong arbitrage is a limited
arbitrage (recall Example 4.4.1). It follows that in the presence of short sales restrictions the payoff
pricing functional should be defined in a way that does not presume satisfaction of the law of one
price. The appropriate definition of the price of a payoff is as the minimal price of a portfolio that
generates that payoff. An agent whose utility function is increasing at date 0 will always select a
portfolio that generates its payoff at minimum cost.
Let
˜
M be the set of payoffs that can be generated by portfolios satisfying short sales restriction
7.1:
˜
M ≡ {z ∈ R
S
: z = hX for some h such that h
j
≥ −b
j
, ∀j ∈ J
0
}. (7.2)
The payoff pricing functional ˜q :
˜
M → R is defined by
˜q(z) ≡ min
h
{ph : hX = z, h
j
≥ −b
j
, ∀j ∈ J
0
} (7.3)
for z ∈
˜
M, whenever the minimum exists.
The set
˜
M is convex but in general it is not a linear subspace. The payoff pricing functional ˜q
is a convex function but it may be nonlinear.
61
62 CHAPTER 7. VALUATION UNDER PORTFOLIO RESTRICTIONS
The price of any security is greater than or equal to the value of its payoff under the payoff
pricing functional. Inequality can be strict, so that there exists a portfolio that generates the same
payoff as a particular security, but at strictly lower cost.
7.2.1 Example
In Example 4.4.1 there were three securities with payoffs x
1
= (1, 1), x
2
= (1, 0), and x
3
= (0, 1).
When holdings of securities were restricted by h
j
≥ −1 for each j, equilibrium prices were p
1
= 3/4,
p
2
= p
3
= 1/2. The payoff pricing functional associated with these prices is defined by the
minimization problem
˜q(z
1
, z
2
) = min
h
µ
3
4
h
1
+
1
2
h
2
+
1
2
h
3
¶
(7.4)
subject to
h
1
+ h
2
= z
1
, h
1
+ h
3
= z
2
, (7.5)
h
1
≥ −1, h
2
≥ −1, h
3
≥ −1, (7.6)
for any (z
1
, z
2
) ∈
˜
M, where
˜
M consists of all payoffs (z
1
, z
2
) for which there exists a portfolio
h = (h
1
, h
2
, h
3
) that satisfies constraints 7.5 and 7.6. Using 7.5 to eliminate h
2
and h
3
in 7.4, the
latter becomes
˜q(z
1
, z
2
) = min
h
µ
1
2
z
1
+
1
2
z
2
−
1
4
h
1
¶
(7.7)
subject to 7.5 and 7.6. If z
1
≥ z
2
, then the minimum in 7.7 is attained at h
1
= z
2
+1, h
2
= z
1
−z
2
−1
and h
3
= −1. If z
1
< z
2
, it is attained at h
1
= z
1
+ 1, h
2
= −1 and h
3
= z
2
−z
1
−1. Summing up,
we have
˜q(z
1
, z
2
) =
1
2
z
1
+
1
2
z
2
−
1
4
min{z
1
, z
2
} −
1
4
. (7.8)
The functional ˜q is nonlinear.
Note that the price (measured by ˜q) of the payoff of each security is strictly less than the security
price; for instance, ˜q(x
1
) = 1/2 < 3/4 = p
1
.
2
If the law of one price holds, then the payoff pricing functional ˜q coincides on
˜
M with the
functional q defined in Chapter 2 and is linear. In particular, if there are no redundant securities
(that is, if each payoff is generated by a unique portfolio), then ˜q is linear.
Using the payoff pricing functional, an agent’s consumption choice problem 4.3 – 4.6 can be
written
max
c
0
,c
1
,z
u(c
0
, c
1
) (7.9)
subject to
c
0
≤ w
0
− ˜q(z) (7.10)
c
1
≤ w
1
+ z (7.11)
z ∈
˜
M, (7.12)
whenever u is increasing in c
0
, so that when making their portfolio and consumption decisions agents
evaluate payoffs using the payoff pricing functional. This representation of agents’ consumption-
portfolio choice problem coincides with that of Section 2.6 in the absence of portfolio restrictions.
7.3 State Prices under Short Sales Restrictions
Even though the payoff pricing functional may fail to be linear or positive in the presence of short
sales restrictions, there exist positive state prices that satisfy a weaker form of 6.5 whenever security
prices exclude unlimited arbitrage opportunities. The existence of positive state prices therefore
provides a useful characterization of security prices that exclude unlimited arbitrage.
7.3. STATE PRICES UNDER SHORT SALES RESTRICTIONS 63
7.3.1 Theorem
Security prices p exclude unlimited strong arbitrage under short sales restrictions iff there exists a
positive vector q ∈ R
S
such that
p
j
≥ x
j
q ∀j ∈ J
0
, (7.13)
and
p
j
= x
j
q ∀j /∈ J
0
. (7.14)
Proof: Let J
0
be the number of securities in the set J
0
. Let Y be a J × (S + J
0
) matrix
consisting of the J ×S payoff matrix X augmented by J
0
column vectors corresponding to securities
in the set J
0
. For each j ∈ J
0
, the (S + j)-th column of Y is a J-dimensional vector with the j-th
coordinate equal to one and all other coordinates equal to zero. Denoting the matrix of such J
0
column vectors by K
0
, we can write
Y = [X K
0
]. (7.15)
The inequality hY ≥ 0 is equivalent to
hX ≥ 0, (7.16)
and
h
j
≥ 0 for every j ∈ J
0
. (7.17)
Thus hY ≥ 0 and ph < 0 is equivalent to h being an unlimited strong arbitrage portfolio. Farkas’
Lemma 6.3.1 says that nonexistence of h with hY ≥ 0 and ph < 0 is equivalent to existence of a
vector b ∈ R
S+J
0
such that
p = Y b and b ≥ 0. (7.18)
Let us partition vector b as b = (q, ²) with q ∈ R
S
and ² ∈ R
J
0
. Using 7.15 we can write 7.18 as
p
j
= x
j
q (7.19)
for j /∈ J
0
, and
p
j
= x
j
q + ²
j
(7.20)
for j ∈ J
0
. Since q ≥ 0 and ²
j
≥ 0, 7.19 and 7.20 are equivalent to 7.13 and 7.14.
2
The strict version of Theorem 7.3.1 is the following
7.3.2 Theorem
Security prices p exclude unlimited arbitrage under short sales restrictions iff there exists a strictly
positive vector q ∈ R
S
such that
p
j
≥ x
j
q ∀j ∈ J
0
, (7.21)
and
p
j
= x
j
q ∀j /∈ J
0
. (7.22)
See the chapter notes for discussion of the proof.
Any positive or strictly positive vector q satisfying 7.21 and 7.22 will be referred to as a vector
of state prices under short sales restrictions. According to 7.22 the price of a security that is not
subject to a short sales restriction equals the value of its payoff under state prices. For a security
that is subject to a short sales restriction, the price exceeds the value of the payoff under state
prices.
It follows from the first-order conditions 4.7 under short sales restrictions that the vector of
marginal rates of substitution of an agent with strictly increasing utility function and interior
optimal consumption is one of the vectors of strictly positive state prices.
64 CHAPTER 7. VALUATION UNDER PORTFOLIO RESTRICTIONS
If there exists a risk-free security and that security is not subject to a short sales restriction,
then the risk-free return satisfies ¯r = 1/
P
s
q
s
and risk-neutral probabilities π
∗
can be defined by
π
∗
s
= ¯rq
s
, as in Section 6.7 in the absence of portfolio restrictions. Using risk-neutral probabilities,
we can rewrite 7.13 and 7.14 as
p
j
≥
1
¯r
E
∗
(x
j
) ∀j ∈ J
0
, (7.23)
and
p
j
=
1
¯r
E
∗
(x
j
) ∀j /∈ J
0
. (7.24)
Thus the price of a security that is subject to a short sales constraint exceeds its expected payoff
discounted by the risk-free return while the price of a security that is not subject to a short sales
constraint equals its expected payoff discounted by the risk-free return when the expectations are
taken with respect to the risk-neutral probabilities.
It is important to note that in the presence of short sales restrictions state prices do not in general
have the strong association with the prices of Arrow securities that they have in the absence of
portfolio restrictions: Theorem 7.5.5 implies that state prices merely provide lower bounds on the
prices of Arrow securities. Further, the positive linear functional that can be defined by a vector of
positive state prices via z 7→ qz on the space R
S
of contingent claims does not in general coincide
with the payoff pricing functional ˜q on the set
˜
M and hence it is not a valuation functional in the
sense of Chapter 5.
7.3.3 Example
In Example 7.2.1 security prices p
1
= 3/4, p
2
= p
3
= 1/2 are equilibrium prices under short sales
restrictions. Consequently, these prices exclude unlimited arbitrage. Strictly positive state prices
are all pairs (q
1
, q
2
) of numbers satisfying
3
4
≥ q
1
+ q
2
,
1
2
≥ q
1
> 0, and
1
2
≥ q
2
> 0. (7.25)
Note that the Arrow security for state 1 is traded at the price of 1/2. The range of state prices of
state 1 is 1/2 ≥ q
1
> 0.
2
7.4 Diagrammatic Representation
In Chapter 4 we presented a diagrammatic analysis of prices of two securities that are subject to
short sales restrictions. With a short sales restriction only on security 2, the set of prices that
exclude unlimited arbitrage was seen to be the area within and to the north of the convex cone
generated by vectors of payoffs of the securities in each state. This is precisely the diagrammatic
interpretation of the existence of positive state prices in this case. Equation 7.14, for the unrestricted
security 1, and inequality 7.13, for the restricted security 2, mean that a vector of security prices
dominates in its second coordinate some vector in the convex cone generated by payoffs.
If short sales of both securities 1 and 2 are restricted, then any positive vector of security prices
excludes unlimited arbitrage. This is also the diagrammatic interpretation of inequalities 7.13 for
both securities.
7.5 Bid-Ask Spreads
The foregoing analysis of valuation under short sales restrictions can be applied to security markets
with bid and ask spreads. As explained in Section 4.7, if one considers each security j with bid
7.5. BID-ASK SPREADS 65
price p
bj
and ask price p
aj
as two securities each with a single price—one with payoff x
j
and price
p
aj
, the other with payoff −x
j
and price −p
bj
, and both with a zero short sales restriction—bid-ask
spreads can be viewed as a special case of short sales restrictions. The fact that the implied short
sales restrictions involve zero bounds leads to a specialization of the results in the general case
analyzed earlier.
The set of payoffs that can be generated by arbitrary portfolios under bid-ask spreads coincides
with the asset span M and is a linear subspace. The payoff pricing functional ˜q is given by
˜q(z) = min
h
a
,h
b
{p
a
h
a
− p
b
h
b
: (h
a
− h
b
)X = z, h
a
≥ 0, h
b
≥ 0}, (7.26)
for z ∈ M. It follows that ˜q satisfies
˜q(z + z
0
) ≤ ˜q(z) + ˜q(z
0
) (7.27)
for every z, z
0
∈ M, and
˜q(λz) = λ˜q(z) (7.28)
every z ∈ M and λ ≥ 0. Properties 7.27 and 7.28 establish that the payoff pricing functional ˜q is
sublinear on M.
7.5.1 Example
In Example 4.8.1 there were two securities with payoffs x
1
= (1, 0) and x
2
= (0, 1). Ask prices
p
a1
= p
a2
= 0.75 and bid prices p
b1
= p
b2
= 0.25 were shown to be equilibrium prices for bid-ask
spreads of 0.5.
Since the asset span M equals R
2
, the payoff pricing functional associated with equilibrium
security prices is defined for every z = (z
1
, z
2
) ∈ R
2
as the value of the minimization problem
min
h
a
,h
b
(0.75h
a1
− 0.25h
b1
+ 0.75h
a2
− 0.25h
b2
) (7.29)
subject to
h
a1
− h
b1
= z
1
, h
a2
− h
b2
= z
2
, (7.30)
h
a1
≥ 0, h
b1
≥ 0, h
a2
≥ 0, h
b2
≥ 0. (7.31)
for (z
1
, z
2
) ∈ R
2
. It follows that
˜q(z
1
, z
2
) = 0.75 max{z
1
, 0} − 0.25 min{z
1
, 0} + 0.75 max{z
2
, 0} − 0.25 min{z
2
, 0}. (7.32)
Since each term 0.75 max{z
s
, 0}−0.25 min{z
s
, 0} is sublinear (but not linear) in z
s
for s = 1, 2, the
functional ˜q is sublinear.
2
Since the short sales restrictions implied by bid-ask spreads involve zero bounds, bid and ask
security prices (p
b
, p
a
) exclude strong unlimited arbitrage iff the payoff pricing functional ˜q is
positive, that is, ˜q(z) ≥ 0 for every z ≥ 0. Further, bid and ask prices (p
b
, p
a
) exclude unlimited
arbitrage iff the payoff pricing functional ˜q is strictly positive. Note that the payoff pricing functional
in Example 7.5.1 is strictly positive.
Bid and ask prices that exclude strong unlimited arbitrage can be characterized by the existence
of positive state prices.
66 CHAPTER 7. VALUATION UNDER PORTFOLIO RESTRICTIONS
7.5.2 Theorem
Bid and ask security prices (p
b
, p
a
) exclude strong unlimited arbitrage iff there exists a positive
vector q ∈ R
S
such that
p
aj
≥ x
j
q ≥ p
bj
(7.33)
for each security j.
Proof: As indicated above, bid-ask spreads can be viewed as a special case of short sales
restrictions by considering each security as two securities with single price and zero short sales
restriction. Applying Theorem 7.3.1, we obtain that the exclusion of strong unlimited arbitrage is
equivalent to the existence of a vector q ∈ R
S
, q ≥ 0 such that
p
aj
≥ x
j
q, (7.34)
and
−p
bj
≥ −x
j
q, (7.35)
for each security j. Inequalities 7.34 and 7.35 are equivalent to 7.33.
2
The strict version of Theorem 7.5.2 is the following
7.5.3 Theorem
Bid and ask security prices (p
b
, p
a
) exclude unlimited arbitrage iff there exists a strictly positive
vector q ∈ R
S
such that
p
aj
≥ x
j
q ≥ p
bj
(7.36)
for every security j.
Any positive or strictly positive vector q satisfying 7.33 will be referred to as a vector of state
prices under bid-ask spreads. If there exists a risk-free security and that security has the same bid
and ask price, then the risk-free return satisfies ¯r = 1/
P
s
q
s
and risk-neutral probabilities π
∗
can
be defined by π
∗
s
= ¯rq
s
. Using risk-neutral probabilities, we can rewrite 7.33 as
p
aj
≥
1
¯r
E
∗
(x
j
) ≥ p
bj
, (7.37)
for every security j. Thus the expected payoff of a security discounted by the risk-free return lies
between the bid and the ask prices of the security when the expectation is taken with respect to
the risk-neutral probabilities.
7.5.4 Example
In Example 7.5.1 ask prices p
a1
= p
a2
= 0.75 and bid prices p
b1
= p
b2
= 0.25 exclude unlimited
arbitrage. Strictly positive state prices are pairs (q
1
, q
2
) of strictly positive numbers satisfying 7.33,
that is
0.75 ≥ q
1
≥ 0.25, and 0.75 ≥ q
2
≥ 0.25. (7.38)
2
Any vector of positive (strictly positive) state prices q can be used to define a positive (strictly
positive) linear functional on the contingent claim space R
S
by z 7→ qz. Again, this functional is
not a valuation functional in the sense of Chapter 5. However, it provides a lower bound on the
payoff pricing functional ˜q on the asset span M.
7.5. BID-ASK SPREADS 67
7.5.5 Theorem
For any vector of positive state prices q under bid-ask spreads, we have
˜q(z) ≥ qz, (7.39)
for every payoff z ∈ M.
Proof: Let (h
a
, h
b
) be any portfolio such that (h
a
−h
b
)X = z with h
a
≥ 0 and h
b
≥ 0. Using
7.33 we obtain
(p
a
h
a
− p
b
h
b
) ≥ h
a
Xq − h
b
Xq = qz. (7.40)
Taking the minimum over (h
a
, h
b
) on the left hand side of 7.40, there results ˜q(z) ≥ qz.
2
If there exists a risk-free security with the same bid and ask price so that the risk-neutral
probabilities π
∗
can be defined by π
∗
s
= ¯rq
s
, then inequality 7.39 can be written as
˜q(z) ≥
1
¯r
E
∗
(z), (7.41)
for every z ∈ M.
Notes
The proof of Theorem 7.3.2 is similar to that of Theorem 7.3.1. Instead of applying Farkas’ Lemma
one has to apply a strict version of it. However, the required strict version is not Stiemke’s Lemma
6.3.2, but a slightly different variant of Farkas’ Lemma. To see this, observe that an application of
Stiemke’s Lemma in place of Farkas’ Lemma in the proof of Theorem 7.3.1 would give the following
equivalence: there exists q À 0 such that p
j
= x
j
q for every j /∈ J
0
and p
j
> x
j
q for every j ∈ J
0
,
iff there does not exist h satisfying (i) hX ≥ 0, (ii) ph ≤ 0, and (iii) h
j
≥ 0 for every j ∈ J
0
,
with at least one strict inequality in (i), (ii), or (iii). This is a different equivalence than that of
Theorem 7.3.2. Observe that the condition that security prices exclude unlimited arbitrage says
that there is no portfolio h satisfying (i), (ii) and (iii) with at least one strict inequality required
to hold in (i) or (ii). A version of Farkas’ Lemma that can be used to prove Theorem 7.3.2 can be
found in Luenberger [4].
The existence of positive state prices in security markets with bid-ask spreads was demonstrated
by Garman and Ohlson [1]. The payoff pricing functional as defined in Section 7.2 was introduced
by Prisman [6]. Ross [7] studied implications of the exclusion of arbitrage in securities markets
with taxation. General results on valuation and the existence of state prices under so-called cone
constraints (that is, when the set of agent’s feasible portfolios forms a convex cone, as it is the case
under zero short sales restrictions or bid-ask spreads) can be found in Luttmer [5] and Jouini and
Kallal [3].
Luttmer [5] and He and Modest [2] examined empirical implications of portfolio restrictions in
security markets.
68 CHAPTER 7. VALUATION UNDER PORTFOLIO RESTRICTIONS
Bibliography
[1] Mark Garman and James Ohlson. Valuation of risky assets in arbitrage-free economies with
transactions costs. Journal of Financial Economics, 9:271–280, 1981.
[2] Hua He and David M. Modest. Market frictions and consumption-based asset pricing. Journal
of Political Economy, 103:94–117, 1995.
[3] Elyes Jouini and Hedi Kallal. Martingales and arbitrage in securities markets with transaction
costs. Journal of Economic Theory, 66:178–197, 1995.
[4] David G. Luenberger. Optimization by Vector Space Methods. Wiley, New York, 1969.
[5] Erzo Luttmer. Asset pricing in economies with frictions. Econometrica, 64:1439–1467, 1996.
[6] Eliezer Prisman. Valuation of risky assets in arbitrage free economies with frictions. Journal of
Finance, 41:545–560, 1986.
[7] Stephen A. Ross. Arbitrage and martingales with taxation. Journal of Political Economy,
195:371–393, 1987.
69
70 BIBLIOGRAPHY
Part III
Risk
71
Chapter 8
Expected Utility
8.1 Introduction
Up to now preferences over uncertain consumption plans have been handled in the most general
fashion: we have merely assumed the existence of a utility function on the set of admissible con-
sumption plans. The canonical model of preferences under uncertainty is the expected utility model.
Expected utility is based on axiomatic foundations and provides a framework for the analysis of
agents’ attitudes toward risk. Consequently, expected utility plays a central role in the analysis of
portfolio choice.
It is assumed (except in Section 8.8) that date-0 consumption does not enter agents’ utility
functions. There are no restrictions on admissible state-contingent consumption plans, so that
utility functions are defined on the entire date-1 consumption space. However, the results to be
presented remain valid if agents’ admissible consumption plans are restricted to being positive.
8.2 Expected Utility
An agent’s utility function u : R
S
→ R on state-contingent consumption plans has a state-dependent
expected utility representation if there exist functions v
s
: R → R (one for each state s) and a
probability measure π on S such that
u(c
1
, . . . , c
S
) ≥ u(c
0
1
, . . . , c
0
S
) iff
S
X
s=1
π
s
v
s
(c
s
) ≥
S
X
s=1
π
s
v
s
(c
0
s
). (8.1)
Utility function u has a state-independent expected utility representation if the functions v
s
can be
taken to be the same in all states; that is, if
u(c
1
, . . . , c
S
) ≥ u(c
0
1
, . . . , c
0
S
) iff
S
X
s=1
π
s
v(c
s
) ≥
S
X
s=1
π
s
v(c
0
s
) (8.2)
for some probability measure π and some function v : R → R. Hereafter “expected utility” will
mean “state-independent expected utility.” The utility function v in 8.2 will be referred to as the
von Neumann-Morgenstern utility function.
The probability measure in the state-dependent expected utility of 8.1 is indeterminate; one
can rescale functions v
s
to associate u with any probability measure. Condition 8.1 therefore says
nothing more than that u has an additively separable representation. In contrast, the probability
measure in the state-independent expected utility of 8.2 is unique. The von Neumann-Morgenstern
utility function v is unique up to a strictly increasing affine transformation. That is, v can be
replaced by a + bv for any constants a and b > 0 without changing the preference ordering of u.
73
74 CHAPTER 8. EXPECTED UTILITY
When equipped with the probability measure π of expected utility representation 8.2, the set
of states S can be regarded as a probability space. State-contingent consumption plans can then
be regarded as random variables. The expected value of a random variable with respect to the
probability measure π is indicated by E
π
, or simply by E when there is no ambiguity about the
probability measure. Expected utility in 8.2 is written as E[v(c)].
Under either 8.1 or 8.2 the marginal rate of substitution between consumption in any two states
is independent of consumption in other states. In the context of choice among many goods under
certainty, independence of the marginal rate of substitution between two goods from the level of
consumption of other goods would be a restrictive assumption, but in the present context it is
reasonable since one state can occur only if other states do not occur.
8.3 Von Neumann-Morgenstern
The first derivation of an expected utility representation of preferences under uncertainty was
provided by von Neumann and Morgenstern. They assumed that agents choose among lotteries.
A lottery is by definition a random variable with specified payoffs and specified probabilities. The
critical assumption of the von Neumann-Morgenstern approach is that agents know the relevant
probabilities. Thus the approach is relevant to situations like games of chance where the existence
of objective probabilities can be assumed. In settings characterized by what has become known as
“Knightian uncertainty”, meaning settings in which agents cannot specify probability distributions,
the von Neumann-Morgenstern approach does not apply since agents are not assumed to be able
to characterize the available choices as lotteries.
8.4 Savage
Savage’s subjective expected utility theory takes as the object of choice state-contingent outcomes
rather than lotteries. The difference between Savage and von Neumann-Morgenstern is that under
Savage’s approach probabilities are derived rather than taken as given. Specifically, Savage proved
that if agents’ preferences on state-contingent outcomes obey certain axioms, then they have an
expected utility representation, where the probabilities as well as the utility function are derived
from the assumed ordering on outcomes. Thus Savage’s approach, unlike that of von Neumann-
Morgenstern, is immune to the objection that agents may not know the relevant probabilities;
if agents are able to choose consistently (and in conformity with the Savage axioms), then they
act as if they know the probabilities, which is all that is relevant for economic problems. These
probabilities, being subjective, may, of course, differ across agents.
From our point of view, Savage’s derivation of expected utility has one shortcoming. It requires
that there be an infinite number of states. This conflicts with the assumption here that the number
of states is finite. We present an alternative axiomatization that applies to the case of finitely many
states.
8.5 Axiomatization of State-Dependent Expected Utility
The principal axiom that implies that an agent’s utility function u : R
S
→ R has a state-dependent
expected utility representation is the independence axiom. The independence axiom requires that
u(c
−s
y) ≥ u(d
−s
y) iff u(c
−s
w) ≥ u(d
−s
w) (8.3)
for all c, d ∈ R
S
and y, w ∈ R. Here c
−s
y refers to the consumption plan c with consumption c
s
in
state s replaced by y.
8.6. AXIOMATIZATION OF EXPECTED UTILITY 75
The independence axiom states that the preference between c
−s
y and d
−s
y, will be unaffected
if y is replaced by w. This must be true for any c, d, y and w. That is, the independence axiom
implies that the level of consumption in state s does not interact with consumption in other states
in such a way as to reverse the preference.
Assume that u is strictly increasing and continuous. We have
8.5.1 Theorem
Assume that there are at least three states, S ≥ 3. Utility function u has a state-dependent expected
utility representation iff it obeys the independence axiom.
Proof: It can be easily verified that a state-dependent expected utility satisfies the indepen-
dence axiom. The proof that the independence axiom implies the representation is not presented.
2
An example of a utility function that does not satisfy the independence axiom, and hence does
not have a state-dependent expected utility representation, is the following.
8.5.2 Example
Consider the utility function u : R
3
+
→ R given by u(c
1
, c
2
, c
3
) = c
1
+
√
c
2
c
3
. Since u(2, 1, 1) >
u(0, 1, 4), we would have that u(2, w, 1) > u(0, w, 4) for every w ≥ 0, if the independence axiom
held. However, for w = 25 we have u(2, 25, 1) < u(0, 25, 4). Thus u does not have a state-dependent
expected utility representation.
2
The sufficiency part of Theorem 8.5.1 does not hold in the case of two states. In that case
every strictly increasing utility function u on R
2
satisfies the independence axiom. To see this,
note that u(c
1
, y) ≥ u(d
1
, y) iff c
1
≥ d
1
, regardless of y. However, not every utility function of
state-contingent consumption in two states has a state-dependent expected utility representation.
An axiomatization of state-dependent expected utility with two states can be found in sources cited
in the notes.
8.6 Axiomatization of Expected Utility
A strengthening of the independence axiom implies that preferences have a state-independent ex-
pected utility representation. The strengthened version is called the cardinal coordinate indepen-
dence axiom. To understand this axiom, suppose that c and d are consumption plans such that
u(c
−s
y) ≤ u(d
−s
w), (8.4)
so that the plan including w is preferred to that including y. Now assume that if y is replaced by
y
0
and w by w
0
, the preference is reversed:
u(c
−s
y
0
) ≥ u(d
−s
w
0
). (8.5)
Further, consider any other pair of consumption plans c
0
and d
0
that provide the consumptions y
and w, respectively, in state t, and are such that c
0
−t
y is preferred to d
0
−t
w:
u(c
0
−t
y) ≥ u(d
0
−t
w). (8.6)
Then the axiom of cardinal coordinate independence states that if y
0
and w
0
are substituted for y
and w, the preference is preserved:
u(c
0
−t
y
0
) ≥ u(d
0
−t
w
0
). (8.7)
This must be true for any s, t, c, d, c
0
, d
0
, w, w
0
, y, and y
0
.
Cardinal coordinate independence is a stronger assumption than independence. This is worth
proving explicitly.
76 CHAPTER 8. EXPECTED UTILITY
8.6.1 Proposition
Cardinal coordinate independence implies independence.
Proof: Set c = d, and replace both by c in 8.4 and 8.5. Set y = w, and replace both by w in
8.4 and 8.6. Set y
0
= w
0
and replace both by w
0
in 8.5 and 8.7. Then 8.4 and 8.5 become trivial.
Further 8.6 and 8.7 become
u(c
0
−t
w) ≥ u(d
0
−t
w) (8.8)
and
u(c
0
−t
w
0
) ≥ u(d
0
−t
w
0
). (8.9)
If cardinal coordinate independence holds, then 8.8 implies 8.9. Since w, w
0
and t are arbitrary,
we actually have an equivalence of 8.8 and 8.9. This equivalence coincides with the independence
axiom 8.3.
2
Again, assume that u is strictly increasing and continuous. We have
8.6.2 Theorem
Utility function u has a state-independent expected utility representation iff it obeys the cardinal
coordinate independence axiom.
Proof: As with Theorem 8.5.1, it can be easily verified that an expected utility satisfies the
cardinal coordinate independence axiom. The proof of the reverse implication is not given here.
2
In contrast to Theorem 8.5.1, the assumption of at least three states is not needed in Theorem
8.6.2. Cardinal coordinate independence is not vacuous in the case of two states.
8.6.3 Example
Consider the utility function u : R
2
+
→ R given by u(c
1
, c
2
) = c
1
+
√
c
2
. The following three
pairs of consumption plans are indifferent under this utility function: (2, 1) and (1, 4), (2, 4) and
(1, 9), (1, 16) and (4, 1). If the cardinal coordinate independence axiom held, we would have that
(4, 16) and (9, 1) are indifferent. However, 4 +
√
16 < 9 +
√
1. Consequently, cardinal coordinate
independence fails, implying that u does not have an expected utility representation.
2
8.7 Non-Expected Utility
Despite its simplicity and intuitive appeal, expected utility theory has proven to be a poor descrip-
tion of preferences over uncertain consumption plans. There exists ample evidence of behavior that
violates the axioms of expected utility. Much effort has been devoted to developing alternatives.
Rather than surveying this work, we present a class of non-expected utility functions that have
strong intuitive appeal.
Agents whose preferences have an expected utility representation know, or act as if they knew,
probabilities of all states. One might argue that agents do not know exact probabilities of each
state but instead have a vague assessment of the probabilities. This leads us to consider agents’
beliefs not as a single probability measure π on S but rather as a set P of probability measures on
S. The set P is assumed to be closed and convex. An agent’s utility function is then defined as
u(c) = min
π∈P
E
π
[v(c)], (8.10)
8.8. EXPECTED UTILITY WITH TWO-DATE CONSUMPTION 77
for some function v : R → R. The preferences represented by u of 8.10 exhibit uncertainty aversion
in the following sense: a smaller set of probabilities increases the agent’s utility. Thus, more precise
information about probabilities is utility-increasing.
The case of an agent who is completely uninformed about probabilities of the states can be
described as using the set ∆ = {π :
P
S
1
π
s
= 1} of all probability measures on S. In this case
min
π∈∆
E
π
[v(c)] = min
s
v(c
s
), (8.11)
and represents “maxmin” behavior with extreme uncertainty aversion. Another simple example is
the following.
8.7.1 Example
Suppose that the set P of probability measures is given by P = {π : π
s
≥ η
s
,
P
S
1
π
s
= 1}, where
η
s
≥ 0 is a lower bound on the probability of state s, and
P
S
1
η
s
≤ 1. One can easily show that
min
π∈P
E
π
[v(c)] = (1 −η) min
s
v(c
s
) + ηE
π
∗
[v(c)], (8.12)
where η =
P
S
1
η
s
and the probability measure π
∗
is given by π
∗
s
= η
s
/η.
2
Such non-expected utility functions are not everywhere differentiable. For instance, u is non-
differentiable when consumption is state-independent.
8.8 Expected Utility with Two-Date Consumption
In the case of consumption at both dates 0 and 1 the (state-independent) expected utility function
takes the form
S
X
s=1
π
s
v(c
0
, c
s
), (8.13)
for some function v : R
2
→ R. Specification 8.13, which will be written as E[v(c
0
, c
1
)], displays
separability across states but not over time. A general form of expected utility that is additively
separable over time is
v
0
(c
0
) +
S
X
s=1
π
s
v
1
(c
s
), (8.14)
for some functions v
0
: R → R and v
1
: R → R. A frequently-used form of time-separable expected
utility is
v(c
0
) + δ
S
X
s=1
π
s
v(c
s
), (8.15)
with time-invariant period utility function v : R → R and δ > 0.
Variations of the cardinal coordinate independence axiom allow derivation of expected utility
representations when consumption occurs at more than one date. If the cardinal coordinate in-
dependence axiom holds for a strictly increasing and continuous utility function u : R
S+1
→ R
with date-0 consumption treated like any other coordinate of a consumption plan, then u has a
time-separable expected utility representation 8.15.
The more general representation 8.14 involves additive separability over time and an expected
utility representation for date-1. Axiomatization of additive separability with two dates is similar
to that of state-dependent expected utility with two states (neither of which is presented here).
Once a time separable representation is achieved, an expected utility representation of the utility
of date-1 consumption results when the cardinal coordinate independence axiom is satisfied.
78 CHAPTER 8. EXPECTED UTILITY
To obtain the representation 8.13 one assumes that agents’ preferences are described by a
utility function u : R
2S
→ R on state-contingent consumption plans. Here both date-0 and
date-1 consumption are state-dependent. The cardinal coordinate independence axiom with two-
date consumption in each of S states implies a representation of u in the form
P
S
s=1
π
s
v(c
0s
, c
s
).
In this setting, a consumption plan with deterministic date-0 consumption is identified by state-
independent date-0 consumption, c
0s
= c
0
for every s. Restricting attention to such consumption
plans, we obtain the expected utility representation 8.13.
Notes
For a general discussion of expected utility theory, see Fishburn [4] and Karni and Schmeidler [9].
The major sources for Sections 8.3 and 8.4 are von Neumann-Morgenstern [17] and Savage
[15]. The results of Sections 8.5 and 8.6 and their proofs can be found in Debreu [2] and Wakker
[18], [19]. An axiomatization of state-dependent expected utility with two states can also be found
in Debreu [2] and Wakker [18], [19]. Leontief [11] proved that a differentiable utility function
has a state-dependent expected utility representation iff the marginal rate of substitution between
consumption in any two states is independent of consumption in other states. For an alternative
axiomatization of expected utility with finitely many states (different from the one given in Section
8.6), see Gul [6].
Questionnaires readily elicit responses that are inconsistent with expected utility theory from
the large majority of those surveyed. The best-known of these responses are the “Allais paradox”
(Allais [1]) and the “Ellsberg paradox” (Ellsberg [3]). For a collections of papers attempting to
account for these paradoxes, mostly from a psychological point of view, see Kahneman, Slovic and
Tversky [8].
For a generalization of expected utility theory, and also a general discussion of expected utility
theory, see Machina [13]. Axiomatic foundations of the non-expected utility theory of Section 8.7
can be found in Schmeidler [16] and Gilboa and Schmeidler [5].
The axioms of expected utility do not imply that probability measure π and function v are the
same across agents. Nevertheless, we will almost always assume below that π is common across
agents, since the characterizations of security prices and portfolios are much weaker when agents
are assumed to disagreed about state probabilities. Hereafter, π is assumed to be common across
agents, except as noted.
On a more methodological level, there is something unsatisfying about simply taking as ex-
ogenous state probabilities that differ across agents. Suppose that one agent wants to hold long
a security which another wants to sell short, where the difference in the desired holdings reflects
differing state probabilities. Expected utility theory with agent-specific probabilities implies that
the transaction will increase both agents’ expected utilities. Agents who are not completely naive
will, however, be aware that they are able to complete a desirable trade only because they dis-
agrees about state probabilities. They will be led to reassess the reliability of the evidence on
which their probabilities are based, and perhaps revise these probabilities based on the fact that
differently-informed agents are arriving at different probabilities.
This line is pursued by assuming that agents start out with common prior probabilities, but have
differing “naive” posterior distributions—derived by applying Bayesian updating to the priors—
because they have differing information. These posteriors are naive because rational agents will
condition their posterior probabilities not only on their own information, but also on the knowledge
about the information of others as revealed by security prices. In many settings this sophisticated
processing of information results in common state probabilities. This suggests that simply assuming
differing state probabilities and an absence of sophisticated learning from prices imputes an element
of irrationality to agents. The analysis just summarized was originated by Harsanyi [7], and has
been developed considerably in recent years.
8.8. EXPECTED UTILITY WITH TWO-DATE CONSUMPTION 79
The association of the term “Knightian uncertainty” with settings in which agents do not act
as if they attach subjective probabilities to outcomes—equivalently, under the axioms of choice, in
which agents are unable to choose among nondeterministic outcomes—is all but universal in the
economics literature. In fact Knight [10] went to some pains to point out that, in his opinion,
nothing was to be learned by modeling agents as unable to form subjective probabilities.
LeRoy and Singell [12] documented that Knight, by distinguishing between risk and uncertainty,
wished to focus attention on whether markets fail due to moral hazard and adverse selection, not
on whether agents can form subjective probabilities.
In fact, in later work Knight substituted the term “non-insurable risk” for “uncertainty” (Netter
[14]).
80 CHAPTER 8. EXPECTED UTILITY
Bibliography
[1] Maurice Allais. Le comportement de l’homme rationnnel devant le risque: Critique des pos-
tulats et axiomes de l’ecole Americaine. Econometrica, 21:503–546, 1953.
[2] Gerard Debreu. Topological methods in cardinal utility theory. In Kenneth J. Arrow, Samuel
Karlin, and Patrick Suppes, editors, Mathematical Methods in Social Sciences. Stanford Uni-
versity Press, 1959.
[3] Daniel Ellsberg. Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics,
75:643–669, 1961.
[4] Peter C. Fishburn. Utility Theory for Decision Making. Wiley, Inc., 1970.
[5] Itzhak Gilboa and David Schmeidler. Maximum expected utility with nonunique prior. Journal
of Mathematical Economics, 18:141–153, 1989.
[6] Faruk Gul. Savage’s theorem with a finite number of states. Journal of Economic Theory,
57:99–110, 1992.
[7] John C. Harsanyi. Games with incomplete information played by ‘Bayesian’ players. Manage-
ment Science, 14:159–182, 1967.
[8] Daniel Kahneman, Paul Slovic, and Amos Tversky. Judgment under Uncertainty: Heuristics
and Biases. Cambridge University Press, Cambridge, 1982.
[9] Edi Karni and David Schmeidler. Utility theory with uncertainty. In Werner Hildenbrand and
Hugo Sonnenschein, editors, Handbook of Mathematical Economics, Vol. 4. North-Holland,
1991.
[10] Frank H. Knight. Risk, Uncertainty and Profit. Houghton Mifflin, Boston, 1921.
[11] Wassily Leontief. A note on interrelation of subsets of independent variables of a continuous
function with continuous first derivatives. Bulletin of the American Mathematical Society,
53:343–350, 1947.
[12] Stephen F. LeRoy and Larry D. Singell. Knight on risk and uncertainty. Journal of Political
Economy, 95:394–406, 1987.
[13] Mark Machina. Expected utility without the independence axiom. Econometrica, 50:277–323,
1982.
[14] Maurice Netter. Radical uncertainty and its economic scope according to Knight and according
to Keynes. In Christian Schmidt, editor, Uncertainty in Economic Thought. Edward Elgar,
1996.
[15] Leonard J. Savage. The Foundations of Statistics. Wiley, New York, 1954.
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82 BIBLIOGRAPHY
[16] David Schmeidler. Subjective probability and expected utility without additivity. Economet-
rica, 57:571–587, 1989.
[17] John von Neumann and Oskar Morgenstern. Theory of Games and Economic Behavior. Prince-
ton University Press, Princeton, 1947.
[18] Peter P. Wakker. Cardinal coordinate independence for expected utility. Journal of Mathe-
matical Psychology, pages 110–117, 1984.
[19] Peter P. Wakker. Additive Representations of Preferences. Kluwer, 1989.
Chapter 9
Risk Aversion
9.1 Introduction
Expected utility provides a framework for the analysis of agents’ attitudes toward risk. In this
chapter we present a formal definition of risk aversion and introduce measures of the intensity of
risk aversion such as the Arrow-Pratt measures and risk compensation. The main result of this
chapter, the Pratt Theorem, establishes the equivalence of these different measures of risk aversion.
Agents’ preferences over risky consumption plans are assumed to have an expected utility rep-
resentation with continuous von Neumann-Morgenstern utility functions. The consumption plans
in the domain of an expected utility function may be defined either narrowly or broadly. The
axioms of expected utility imply that any consumption plan can be viewed as a random variable
on the set S of states equipped with an agent’s subjective probability measure. Thus if the objects
of choice are specified as the consumption plans that emerge from the axioms of expected utility,
these are appropriately defined narrowly as random variables that can take S values with given
probabilities. However, the analysis of this chapter applies equally well if consumption plans are
broadly interpreted as arbitrary random variables (that is, as random variables with an arbitrary
number of realizations and arbitrary probabilities). The choice between these interpretations is a
matter of taste.
Except in Section 9.10, it is assumed that date-0 consumption does not enter the utility func-
tions, and throughout it is assumed that there are at least two states at date 1, S ≥ 2.
9.2 Risk Aversion and Risk Neutrality
An agent’s attitude toward risk is characterized by his preference between a risky consumption plan
and the deterministic consumption plan equal to the expectation of the risky plan.
An agent with von Neumann-Morgenstern utility function v : R → R is risk averse if he prefers
the expectation of any consumption plan to the consumption plan itself; that is,
E[v(c)] ≤ v(E(c)) (9.1)
for every consumption plan c.
An agent is risk neutral if
E[v(c)] = v(E(c)) (9.2)
for every consumption plan c.
An agent is strictly risk averse if
E[v(c)] < v[E(c)] (9.3)
for every nondeterministic consumption plan c.
83
84 CHAPTER 9. RISK AVERSION
Our term “risk aversion” means “weak risk aversion” as only weak preference is required in 9.1.
Note that in this usage risk neutrality is a special case of risk
aversion.
An agent may be neither risk averse nor risk neutral nor strictly risk averse: he may prefer
some nondeterministic consumption plans to their expectations. Also, an agent may be risk averse,
but neither risk neutral nor strictly risk averse; he may strictly prefer the expectation of some
nondeterministic consumption plans to the plans themselves, but be indifferent for others.
9.3 Risk Aversion and Concavity
Risk aversion, risk neutrality and strict risk aversion can be characterized by, respectively: concavity
, linearity, and strict concavity of the von Neumann-Morgenstern utility function:
9.3.1 Theorem
(i)An agent is risk averse iff his von Neumann-Morgenstern utility function v is concave.
(ii) An agent is risk neutral iff his von Neumann-Morgenstern utility function v is linear.
(iii)An agent is strictly risk averse iff his von Neumann-Morgenstern utility function v is strictly
concave.
Proof: (i) If v is concave, then 9.1 holds—it is Jensen’s inequality—and the agent is risk
averse. To show the converse, suppose that the agent is risk averse but v is not concave. Then
there exist y
1
, y
2
and λ
∗
satisfying 0 < λ
∗
< 1 such that
v(λ
∗
y
1
+ (1 −λ
∗
)y
2
) < λ
∗
v(y
1
) + (1 − λ
∗
)v(y
2
), (9.4)
(Figure 9.1). Consider the set of λ satisfying 0 ≤ λ ≤ λ
∗
and
v(λy
1
+ (1 −λ)y
2
) = λv(y
1
) + (1 − λ)v(y
2
). (9.5)
This set is nonempty (λ = 0 is an element) and closed since v is continuous. Therefore, there exists
a supremum denoted by λ
. Similarly, there exists an infimum of the set of λ satisfying λ
∗
≤ λ ≤ 1
and 9.5. Let
¯
λ denote that infimum. We have λ < λ
∗
<
¯
λ and
v(λy
1
+ (1 −λ)y
2
) < λv(y
1
) + (1 − λ)v(y
2
) (9.6)
for every λ < λ <
¯
λ.
Let y = λy
1
+ (1 −λ)y
2
and ¯y =
¯
λy
1
+ (1 −
¯
λ)y
2
. It follows from 9.6 that
v(γy + (1 − γ)¯y) < γv(y) + (1 − γ)v(¯y) (9.7)
for every 0 < γ < 1.
Consider consumption plan c that takes value y in some (but not all) states and value ¯y in
the remaining states. Note that the deterministic consumption plan E(c) lies in the interval (y, ¯y).
Using 9.7 we obtain
v[E(c)] < E[v(c)], (9.8)
which contradicts the assumption of risk aversion.
(ii) If v is linear of the form v(y) = ay + b, then 9.2 holds, and the agent is risk neutral.
The proof of the converse is very similar to the proof in part (i). The only difference is that the
assumption of v being nonlinear implies that either 9.4 holds or the opposite strict inequality holds.
Both cases lead to a contradiction of risk neutrality.
9.4. ARROW-PRATT MEASURES OF ABSOLUTE RISK AVERSION 85
(iii) If v is strictly concave, then 9.3 holds (it is Jensen’s strict inequality), and the agent is
strictly risk averse. To show the converse, suppose that the agent is strictly risk averse but v is not
strictly concave. If v is linear on some interval [y
1
, y
2
] with y
1
< y
2
, then it follows from part (ii)
that v[E(c)] = E[v(c)] for any consumption plan that takes values in that interval. This contradicts
strict risk aversion. Otherwise, if v is not linear on any nondegenerate interval in its domain, then
the strict inequality 9.4 must hold. The proof in part (i) leads to a contradiction with strict risk
aversion in this case.
2
9.4 Arrow-Pratt Measures of Absolute Risk Aversion
Risk aversion affects agents’ portfolio choices and equilibrium security prices. It is useful to have
a measure of the intensity of risk aversion. In light of Theorem 9.3.1, the candidate that comes to
mind is the second derivative v
00
of the von Neumann-Morgenstern utility function. However, the
second derivative is not invariant to affine transformations of v. As noted in Chapter 8, a strictly
increasing affine transformation of the von Neumann-Morgenstern utility function does not change
preferences. Therefore such a transformation should not change the measure of risk aversion. The
Arrow-Pratt measure of absolute risk aversion is defined by
A(y) ≡ −
v
00
(y)
v
0
(y)
(9.9)
for a scalar variable y such that v
0
(y) 6= 0. It is invariant to strictly increasing affine transformations
of the utility function v.
If nonzero, the reciprocal of the Arrow-Pratt measure of absolute risk aversion
T (y) ≡
1
A(y)
(9.10)
can be used as a measure of risk tolerance.
9.5 Risk Compensation
There exists another measure of risk aversion that is closely related to the Arrow-Pratt measure of
absolute risk aversion: risk compensation. We define risk compensation as the amount of determin-
istic consumption one would have to charge an agent in exchange for relieving him of a risk (Figure
9.2, where the risk has payoff plus or minus 1 with equal probability). In non-finance applications
of the theory of choice under uncertainty, this variable is almost always referred to as the “risk
premium.” Here and in other finance applications, however, the term “risk premium” refers to the
expected return on a security less the risk-free return.
The risk compensation for the additional consumption plan (“risk”) z at deterministic initial
consumption y is the value ρ(y, z) that satisfies
E[v(y + z)] = v(y − ρ(y, z)), (9.11)
so that the deterministic consumption y − ρ(y, z) is the certainty equivalent of risky consumption
y + z.
Note that an agent is risk averse iff risk compensation ρ(y, z) is positive (that is, strictly positive
or zero) for every y and every risk z with E(z) = 0. An agent is risk neutral iff risk compensation
is zero for all risks z with E(z) = 0.
For small risk z, risk compensation ρ(y, z) equals approximately half the product of the variance
σ
2
z
of z and the Arrow-Pratt measure of absolute risk aversion at y.
86 CHAPTER 9. RISK AVERSION
9.5.1 Theorem
For small z with E(z) = 0
ρ(y, z)
∼
=
A(y)σ
2
z
2
. (9.12)
Proof: The quadratic approximation of v(y + z) is
v(y + z)
∼
=
v(y) + v
0
(y)z + v
00
(y)
z
2
2
. (9.13)
Taking expectations, we obtain
E[v(y + z)]
∼
=
v(y) + v
00
(y)
σ
2
z
2
. (9.14)
Similarly, a linear expansion of the right-hand side of 9.11 yields
v(y − ρ(y, z))
∼
=
v(y) − v
0
(y)ρ(y, z). (9.15)
If the right-hand sides of 9.14 and 9.15 are set equal and use is made of the definition of the measure
of absolute risk aversion A, 9.12 results.
The forms of approximation used in 9.13 and 9.15 reveal the meaning of “small” in the statement
of Theorem 9.5.1. For random variable z, small means that the variance is of first-order significance.
Approximations 9.13 and 9.15 take into account only the first-order significant terms.
2
9.6 The Pratt Theorem
The two measures of risk aversion—the Arrow-Pratt measure and risk compensation—can be used
to compare the risk aversion of two agents. An important theorem says that comparisons using
the Arrow-Pratt measure and risk compensation always give the same result. Further, one agent
is more risk-averse than another if the von Neumann-Morgenstern utility function of the first is a
concave transformation of that of the second.
Let v
1
and v
2
be two von Neumann-Morgenstern utility functions on R, and let ρ
i
and A
i
denote
the risk compensation and the Arrow-Pratt measure of absolute risk aversion of v
i
for i = 1, 2.
We have
9.6.1 Theorem
Suppose that utility functions v
1
and v
2
are twice-differentiable with continuous second derivatives,
and strictly increasing. Then the following conditions are equivalent:
(i) A
1
(y) ≥ A
2
(y) for every y.
(ii) ρ
1
(y, z) ≥ ρ
2
(y, z) for every y and every random variable z.
(iii) v
1
is a concave transformation of v
2
, that is, v
1
= f ◦ v
2
for f concave and strictly
increasing.
Proof: We first show that (i) implies (iii). Since v
2
is strictly increasing, the inverse function
v
−1
2
exists and the function f of (iii) is defined by f(t) = v
1
(v
−1
2
(t)).
We have to show now that f is strictly increasing and concave. The derivative of f is
f
0
(t) =
v
0
1
(v
−1
2
(t))
v
0
2
(v
−1
2
(t))
(9.16)
9.6. THE PRATT THEOREM 87
and is strictly positive since v
0
i
> 0 for i = 1, 2. Calculation of the second derivative of f yields
f
00
(t) =
v
00
1
(y) − (v
00
2
(y)v
0
1
(y))/v
0
2
(y)
[v
0
2
(y)]
2
, (9.17)
where y = v
−1
2
(t). This can be rewritten as
f
00
(t) = (A
2
(y) − A
1
(y))
v
0
1
(y)
[v
0
2
(y)]
2
. (9.18)
Thus f
00
≤ 0, and hence f is concave.
Next we show that (iii) implies (ii). By the definition of risk compensation we have
E[v
1
(y + z)] = v
1
(y − ρ
1
(y, z)). (9.19)
Since v
1
= f ◦ v
2
and f is concave, application of Jensen’s inequality yields
E[v
1
(y + z)] = E[f(v
2
(y + z))] ≤ f(E[v
2
(y + z)]). (9.20)
The right-hand side of 9.20 equals f(v
2
(y − ρ
2
(y, z))). Combining 9.20 with 9.19 yields
v
1
(y − ρ
1
(y, z)) ≤ v
1
(y − ρ
2
(y, z)). (9.21)
Since v
1
is strictly increasing, 9.21 implies ρ
1
(y, z) ≥ ρ
2
(y, z).
Finally, we show that (ii) implies (i). Suppose that
A
1
(y
∗
) < A
2
(y
∗
) (9.22)
for some y
∗
. Since A
1
and A
2
S: NOTE CHANGE are continuous, there is an interval around y
∗
such that A
1
(y) < A
2
(y) S: NOTE CHANGE for every y in this interval. Using the arguments
of the proofs above with interchanged roles of v
1
and v
2
, it can be shown that ρ
1
(y, z) < ρ
2
(y, z),
whenever y + z takes values in that interval. This contradicts (ii).
2
We emphasize again that the set of random variables z in Theorem 9.6.1 (condition (ii)) can
be either the set all random variables on the set of states S with given probabilities, or the set of
all arbitrary random variables. Note also that no restriction on consumption has been imposed in
Theorem 9.6.1. Therefore the theorem is valid as stated only for utility functions defined on the
entire real line. However, the same equivalence holds for utility functions defined only for positive
(strictly positive) consumption when risk z in (ii) is such that y + z is positive (strictly positive).
There is also a strict version of Theorem 9.6.1. The equivalence of conditions (i), (ii), and (iii)
remains valid if the inequalities in (i) and (ii) are strict, and the transformation f in (iii) is strictly
concave as well as strictly increasing.
Further, there is an equality version of Theorem 9.6.1: conditions (i), (ii), and (iii) remain
equivalent with equalities in (i) and (ii), and strictly increasing affine transformation f in (iii).
This version is a simple corollary to 9.6.1. It implies that if two utility functions have equal
Arrow-Pratt measures of risk aversion, then each is a strictly increasing affine transformation of
the other. For instance, the only constant absolute risk aversion utility function is (up to a strictly
increasing affine transformation) the negative exponential function. Since a strictly increasing affine
transformation of a utility function describes the same expected utility preferences, the Arrow-Pratt
measure completely characterizes preferences.
88 CHAPTER 9. RISK AVERSION
9.7 Decreasing, Constant and Increasing Risk Aversion
If absolute risk aversion A(y) of an agent is decreasing in y, then he has decreasing absolute risk
aversion. If A(y) is constant (increasing) in y, the agent has constant (increasing) absolute risk
aversion.
Pratt’s Theorem implies that an equivalent expression of decreasing (constant, increasing) ab-
solute risk aversion is that risk compensation ρ(y, z) is decreasing (constant, increasing) in y for
every z.
9.7.1 Corollary
For a strictly increasing and twice-differentiable (with continuous second derivative) utility function
v,
(i) ρ(y, z) is increasing in y for every z iff A(y) is increasing in y.
(ii) ρ(y, z) is constant in y for every z iff A(y) is constant in y.
(iii) ρ(y, z) is decreasing in y for every z iff A(y) is decreasing in y.
Proof: Let us define utility function v
1
by v
1
(y) ≡ v(y + ∆y) for some ∆y ≥ 0. The Arrow-
Pratt measure of absolute risk-aversion and the risk compensation of v
1
are A
1
(y) = A(y + ∆y) and
ρ
1
(y, z) = ρ(y + ∆y, z). Applying Pratt’s Theorem 9.6.1 to v
1
and v yields that A(y + ∆y) ≥ A(y)
iff ρ(y + ∆y, z) ≥ ρ(y, z). Since ∆y is arbitrary, (i) follows.
The proofs of (ii) and (iii) are similar.
2
9.8 Relative Risk Aversion
Sometimes it is of interest to measure risk relative to the initial consumption. There are two
measures of relative risk aversion: the Arrow-Pratt measure of relative risk aversion, and relative
risk compensation.
The Arrow-Pratt measure of relative risk aversion is defined by
R(y) ≡ −
v
00
(y)
v
0
(y)
y, (9.23)
so that R(y) = yA(y).
The relative risk compensation for the relative risk z at deterministic initial consumption y is
the value ρ
r
(y, z) that satisfies
E[v(y + yz)] = v(y − yρ
r
(y, z)). (9.24)
Relative risk compensation ρ
r
is related to (absolute) risk compensation ρ via
ρ
r
(y, z) =
ρ(y, yz)
y
. (9.25)
For small relative risk z with E(z) = 0, it follows from Theorem 9.5.1 that
ρ
r
(y, z)
∼
=
R(y)σ
2
z
2
. (9.26)
The parallel forms of 9.12 and 9.26 provide a motivation for definition 9.23 of the measure R of
relative risk aversion.
A version of Pratt’s Theorem holds for relative risk aversion: comparisons of relative risk
aversion of two agents using the Arrow-Pratt measure and the relative risk compensation always
give the same result. A reference is given in the notes.
9.9. UTILITY FUNCTIONS WITH LINEAR RISK TOLERANCE 89
9.9 Utility Functions with Linear Risk Tolerance
The functions most often used as von Neumann-Morgenstern utility functions in applied work and
as examples are linear utility and the following utility functions:
• Negative Exponential Utility. The utility function
v(y) = −e
−αy
, (9.27)
for α > 0, has absolute risk aversion that is constant and equal to α.
• Logarithmic utility. The utility function
v(y) = ln(y + α), −α < y, (9.28)
has absolute risk aversion that is decreasing and equal to 1/(y + α). If α equals zero, relative
risk aversion equals 1.
• Power utility. The utility function
v(y) =
1
γ − 1
(α + γy)
1−
1
γ
, −α < γy, (9.29)
for γ 6= 0 and γ 6= 1, has absolute risk aversion equal to 1/(α + γy). If γ > 0, absolute risk
aversion is decreasing. Otherwise, if γ < 0, it is increasing. If α equals zero, relative risk
aversion equals γ.
A special case of power utility is quadratic utility. For γ = −1
v(y) = −
1
2
(α − y)
2
, y < α, (9.30)
with absolute risk aversion that is increasing and equal to 1/(α −y).
Logarithmic and negative exponential utility can be viewed as limiting cases of power utility
when γ approaches 1 or 0. If the power utility function is written as
v(y) =
1
γ − 1
((α + γy)
1−
1
γ
− 1), (9.31)
which is an affine transformation of 9.29, then using l’Hopital’s rule it can be shown that v(y)
converges to ln(y +α) as γ approaches one. If a different affine transformation of 9.29 is considered,
v(y) =
1
γ − 1
(1 +
γy
α
)
1−
1
γ
, (9.32)
where α > 0, then v(y) converges to −e
−y/α
as γ approaches zero.
All these utility functions are strictly increasing, strictly concave, and have risk tolerance that
depends linearly on consumption (strictly, the dependence is affine, not linear). For the negative
exponential utility function 9.27, risk tolerance is constant, T (y) = 1/α; for the logarithmic utility
function 9.28, risk tolerance is T (y) = y + α; for the power utility function 9.29, risk tolerance
is T (y) = α + γy. These utility functions are called linear risk tolerance (LRT) utility functions
(alternatively, HARA utility functions, where HARA stands for hyperbolic absolute risk aversion,
since A(y) defines a hyperbola).
The domain of a LRT utility function can be conveniently written as {y : T (y) > 0}. Note
that the parameter γ (with γ = 0 for the negative exponential utility function, and γ = 1 for the
logarithmic utility function) is the slope of the risk tolerance function.
LRT utility functions have many attractive properties, as will be seen in Chapters 13 and 16.
90 CHAPTER 9. RISK AVERSION
9.10 Risk Aversion with Two-Date Consumption
The definitions of risk aversion and risk neutrality can easily be adapted to the case when date-0
consumption enters agents’ utility functions.
An agent with von Neumann-Morgenstern utility function v : R
2
→ R is risk averse if
E[v(c
0
, c
1
)] ≤ v(c
0
, E(c
1
)), (9.33)
for every c
0
and every c
1
, and is risk neutral if
E[v(c
0
, c
1
)] = v(c
0
, E(c
1
)), (9.34)
for every c
0
and every c
1
.
By Theorem 9.3.1 an agent is risk averse iff the von Neumann-Morgenstern utility function
v(y
0
, y
1
) is concave in y
1
for every y
0
and risk neutral iff v(y
0
, y
1
) is linear in y
1
for every y
0
. For
instance, utility functions v(y
0
, y
1
) = y
0
+ δy
1
and v(y
0
, y
1
) = y
0
y
1
imply risk neutrality.
When v is not additively separable over time, the measures of date-1 risk aversion of Section 9.4
depend on date-0 consumption. Consequently, an agent can be risk neutral in date-1 consumption
for some values of c
0
and strictly risk averse for others, for example. In the case of time-separable
expected utility 8.14, an agent’s attitude toward date-1 risk depends only on the form of the date-1
utility function v
1
; the level of date-0 consumption is irrelevant.
For a time-separable power utility function (with α = 0)
v(y
0
, y
1
) =
1
γ − 1
((γy
0
)
1−
1
γ
+ (γy
1
)
1−
1
γ
), (9.35)
where γ 6= 0, 1, the measure of absolute (date-1) risk aversion is 1/γy
1
and depends only on y
1
; the
measure of relative (date-1) risk aversion is γ. Note that the marginal rate of substitution between
date-0 consumption and date-1 consumption under this power utility function is (y
1
/y
0
)
−1/γ
and
the elasticity of substitution is (y
1
/y
0
)
−1/γ−1
. Thus the elasticity of substitution depends on the
coefficient of relative risk aversion. In general, the intertemporal elasticity of substitution and the
coefficient of risk aversion are interdependent under the expected utility representation.
Notes
The equivalences proved in Theorem 9.3.1 between risk aversion (strict risk aversion, risk neutrality)
and concavity (strict concavity, linearity) of utility function are also an implication of the Pratt
Theorem (take v
1
= v and linear v
2
). However, the Pratt Theorem applies only to differentiable
utility functions, while Theorem 9.3.1 applies to all continuous utility functions.
The Arrow-Pratt measures of absolute and relative risk aversion were proposed in Arrow [1], [2]
and Pratt [6]. The Pratt theorem is due to Pratt [6]. A version of the Pratt theorem for relative
risk aversion can also be found in Pratt [6].
An illuminating discussion of measures of risk aversion can be found in Yaari [8].
Measures of risk aversion introduced in this chapter are based on the assumption that a risk-free
payoff is attainable. More general measures that apply when a risk-free payoff is not attainable
have been proposed by Ross [7]; see also Machina and Nielsen [5]. Cohen [3] discussed concepts of
risk aversion without the expected utility representation of preferences.
Kihlstrom and Mirman [4] addressed problems in extending the Arrow-Pratt theory of risk aver-
sion to multivariate risks (for example, state-contingent consumption plans with multiple goods).
Bibliography
[1] Kenneth J. Arrow. Comment. Review of Economics and Statistics, 45, Supplement:24–27, 1963.
[2] Kenneth J. Arrow. Aspects of the Theory of Risk Bearing. Yrjo Jahnssonin Saatio, Helsinki,
1965.
[3] Michele D. Cohen. Risk-aversion concepts in expected- and non-expected utility models. The
Geneva Papers on Risk and Insurance Theory, 20:73–91, 1995.
[4] Richard E. Kihlstrom and Leonard J. Mirman. Risk aversion with many commodities. Journal
of Economic Theory, 8:361–388, 1974.
[5] Mark J. Machina and William S. Neilson. The Ross characterization of risk aversion: Strength-
ening and extension. Econometrica, 55:1139–1149, 1987.
[6] John W. Pratt. Risk aversion in the small and in the large. Econometrica, 32:122–136, 1964.
[7] Stephen A. Ross. Some stronger measures of risk aversion in the small and in the large with
applications. Econometrica, 49:621–638, 1981.
[8] Menahem Yaari. Some remarks on measures of risk aversion and on their uses. Journal of
Economic Theory, 55:95–115, 1969.
91
92 BIBLIOGRAPHY
Chapter 10
Risk
10.1 Introduction
In Chapter 9 we defined an agent as risk averse if he prefers the expectation of a consumption plan
to the consumption plan itself. The consumption plan is obviously riskier than its expectation, and
a risk-averse agent prefers the latter.
A natural extension of this discussion is to consider a risk-averse agent who compares two
consumption plans neither of which is deterministic. In general, without more information about an
agent’s preferences, two risky consumption plans cannot be ranked: some risk-averse agents prefer
one and some the other. However, in the spirit of the discussion of Chapter 9, it is appropriate to
ask whether there is some condition on the distribution of two consumption plans such that if the
two consumption plans have the same expectation, then all risk-averse agents do prefer one to the
other. In Section 10.2 an ordering on consumption plans is defined which, as will be seen in Section
10.5, has the desired property.
In this chapter we assume that agents consume only at date 1.
10.2 Greater Risk
Let y and z be two (date-1) consumption plans. As in Chapter 9 these consumption plans can be
viewed narrowly as random variables on the set of states S with given probabilities, or broadly as
arbitrary random variables (with finite expectations).
Consumption plan y is riskier than consumption plan z if there exists a random variable ² such
that
y − E(y) =
d
z − E(z) + ² and E(²|z) = E(²) = 0. (10.1)
If 10.1 holds, and in addition ² is not the zero random variable, then y is strictly riskier than z.
The symbol =
d
means that the left-hand side equals the right-hand side in distribution—that is,
the left-hand side is a random variable which assumes the same values with the same probabilities
as the random variable defined by the right-hand side. The condition E(²|z) = E(²) states that ²
is mean-independent of z. That is, the expectation of ² conditional on (any realization of) z does
not depend on z.
Equality in distribution is a much weaker condition than equality: two random variables are
equal if they take on the same value in every state, a condition that is sufficient, but not necessary,
for equality in distribution. For example, a payoff consisting of 0 in state 1 and 1 in state 2 is equal
in distribution to a payoff of 1 in state 1 and 0 in state 2 if the two states are equally probable.
These payoffs are not equal since they do not coincide in every state.
93
94 CHAPTER 10. RISK
10.2.1 Example
Let z take on values of plus or minus 1 with equal probabilities and ² take on values of 1 and −3
with probabilities 3/4 and 1/4 when z = 1, and values of 3 and −1 with probabilities 1/4 and 3/4
when z = −1. Then 2z and z + ² have the same distributions. Since ² is mean-independent of z,
10.1 is satisfied, with y equal to 2z. Therefore 2z is strictly riskier than z. Obviously 2z and z + ²
are not equal as random variables, for then z would equal ², which is not the case.
2
Our definition of one consumption plan being riskier than another is a condition on the de-
viations of those plans from the respective expectations. Therefore it is not necessary that the
consumption plans have the same expectation. Note that y is riskier than z iff y − E(y) is riskier
than z − E(z) or, equivalently, iff y is riskier than z − E(z) + E(y). Any consumption plan is
riskier than its expectation, and any nondeterministic consumption plan is strictly riskier than its
expectation.
10.3 Uncorrelatedness, Mean-Independence and Independence
The condition of mean-independence defined in Section 10.1 is a stronger restriction than uncorre-
latedness. However, it is less strong than independence. Independence implies mean-independence,
but the converse is not true. In Example 10.2.1 ² is mean-independent of z, but not independent
of z. This is so because the distribution of ² conditional on z depends on the realization of z, even
though the conditional expectation of ² is zero for both values of z. Similarly, mean-independence
implies uncorrelatedness, but again the converse is not true. For example, suppose that the pair
(z, ²) takes on values (1, 1), (2, 0) and (3, 1) with equal probabilities. Here ² is uncorrelated with z,
but not mean-independent of z.
Uncorrelatedness and independence are symmetric. If z is uncorrelated with (independent of)
², then ² is uncorrelated with (independent of) z. Mean-independence, however, is not symmetric.
The fact that z is mean-independent of ² does not imply that ² is mean-independent of z.
When the joint distribution of z and ² is bivariate normal, then uncorrelatedness, mean-
independence and independence are all equivalent.
10.4 A Property of Mean-Independence
A useful property of mean-independence is the following:
10.4.1 Proposition
If ² is mean-independent of z, then
E[f(z)²] = E[f(z)]E(²). (10.2)
for any function f.
Proof: The expectation of f(z)² over the joint distribution of z and ² can be taken first over
the distribution of ² conditional on z, and then over the marginal distribution of z:
E[f(z)²] = E[E(f (z)²|z)]. (10.3)
Here f(z) can be passed out of the inner expectation, resulting in
E[f(z)²] = E[f(z)E(²|z)]. (10.4)
The right-hand side equals E[f(z)]E(²), by mean-independence.
10.5. RISK AND RISK AVERSION 95
2
If ² is uncorrelated with z, then 10.2 holds for any linear function f. The stronger assumption
of mean-independence is needed to assure that 10.2 is valid even when f is nonlinear. It is worth
pointing out that if ² is mean-independent of z, then it is also mean-independent of f(z).
10.5 Risk and Risk Aversion
The motivation for our definition of risk is that every risk-averse agent prefers a less risky con-
sumption plan to a more risky one if the two have the same expectation:
10.5.1 Theorem
For consumption plans y and z that have the same expectation, y is riskier than z iff every risk-
averse agent prefers z to y.
Proof: If y is riskier than z and they have the same expectation, 10.1 becomes y =
d
z + ²,
where E(²|z) = 0. For utility function v (the domain of which includes the values that y and z take
on) we have
E[v(y)] = E[v(z + ²)] = E[E[v(z + ²)|z]]. (10.5)
If v is concave, so that the agent is risk-averse, Jensen’s inequality implies that
E[v(z + ²|z)] ≤ v(E[z + ²|z]) = v(z). (10.6)
Taking expectations, there results
E[v(y)] ≤ E[v(z)]. (10.7)
The proof of the converse, that if every risk-averse agent prefers z to y, where E(y) = E(z),
then y is riskier than z, is much more difficult. It can be found in the sources cited in the notes at
the end of this chapter.
2
Note that risk-averse agents’ utility functions in Theorem 10.5.1 are not assumed to be increas-
ing. However, the result remains true if one takes only risk-averse agents with increasing utility
functions. For a discussion of this point see the notes.
There is a strict version of Theorem 10.5.1.
10.5.2 Theorem
For consumption plans z and y that have the same expectation, y is strictly riskier than z iff every
strictly risk-averse agent strictly prefers z to y.
Both parts of the equivalence of Theorem 10.5.2 are useful: sometimes one knows that y is
strictly riskier than z and uses the necessity part of Theorem 10.5.2 to infer that all strictly risk-
averse agents strictly prefer z to y, while sometimes one knows that all strictly risk-averse agents
strictly prefer z to y, and uses the sufficiency part of the theorem to infer that y is strictly riskier
than z.
The following two examples illustrate the use of Theorem 10.5.2.
10.5.3 Example
Let y and z be two nondeterministic consumption plans with independent and identical distribu-
tions. We show here that every strictly risk-averse agent strictly prefers the equally weighted average
(y + z)/2 to any other weighted average of y and z (and also, therefore, to y and z themselves).
96 CHAPTER 10. RISK
Let ay + (1 −a)z denote an arbitrary weighted average of y and z (which equals y when a = 1
and z when a = 0). We can write
ay + (1 − a)z =
y + z
2
+ (a −
1
2
)(y − z). (10.8)
We have
E(y − z|y + z) = E(y|y + z) − E(z|y + z), (10.9)
and
E(y|y + z) = E(z|y + z), (10.10)
since y and z are independent and have identical distributions. Therefore (a −
1
2
)(y − z) is mean-
independent of (y + z)/2 and has zero expectation. By 10.1, if a 6= 1/2, then ay + (1−a)z is strictly
riskier than (y + z)/2. By the necessity part of Theorem 10.5.2, every strictly risk-averse agent
strictly prefers the equally weighted average.
2
10.5.4 Example
For any nondeterministic consumption plan z, 2z is strictly riskier than z. To see this, observe first
that
v(z + E(z)) >
1
2
v(2z) +
1
2
v(2E(z)), (10.11)
for every strictly concave v, since z + E(z) is an (equally-weighted) average of 2z and 2E(z). Here
10.11 is to be interpreted as a vector inequality rather than state-by-state (strict inequality holds
only in states s for which z
s
6= E(z)). Taking expectations on both sides of 10.11 results in
E[v(z + E(z))] >
1
2
E[v(2z)] +
1
2
v(2E(z)). (10.12)
Jensen’s inequality implies that
v(2E(z)) > E(v(2z)). (10.13)
Substituting 10.13 in 10.12 results in
E[v(z + E(z))] > E[v(2z)]. (10.14)
The sufficiency part of Theorem 10.5.2 implies that 2z is strictly riskier than z + E(z). Since
expectations do not matter, it follows that 2z is strictly riskier than z.
2
An argument similar to that of Example 10.5.4 can be used to prove a result that will be used
later.
10.5.5 Proposition
For any consumption plan z, if ² 6= 0 is mean-independent of z and E(²) = 0, then z + λ² is strictly
riskier than z + γ² for every λ > γ ≥ 0.
Proof: Let a = γ/λ. Then
z + γ² = a(z + λ²) + (1 −a)z. (10.15)
Since 0 ≤ a < 1, for every strictly concave utility function v we have
v(z + γ²) > av(z + λ²) + (1 − a)v(z) (10.16)
10.6. GREATER RISK AND VARIANCE 97
(again, this inequality is to be interpreted as a vector inequality). Taking expectations on both
sides of 10.16 we obtain
E[v(z + γ²)] > aE[v(z + λ²)] + (1 − a)E[v(z)]. (10.17)
Since z + λ² is strictly riskier than z, we have E[v(z)] > E[v(z + λ²)]. Using this inequality in
10.17, there results
E[v(z + γ²)] > E[v(z + λ²)]. (10.18)
Theorem 10.5.2 implies that z + λ² is strictly riskier than z + γ².
2
Note that, since expectations do not matter in orderings by riskiness, Proposition 10.5.5 remains
true for any ² 6= 0 that is mean-independent of z even if E(²) 6= 0. A corollary to Proposition 10.5.5
provides an extension of Example 10.5.4.
10.5.6 Corollary
For any nondeterministic consumption plan z, λz is strictly riskier than z for every λ > 1.
Proof: Proposition 10.5.5 implies that 0 + λ(z −E(z)) is strictly riskier than 0 +(z −E(z)) for
every λ > 1 and nondeterministic z. Since expectations do not matter, λz is strictly riskier than z.
2
10.6 Greater Risk and Variance
A simple and frequently used measure of risk is variance. It follows from the definition of greater
risk 10.1 that if one consumption plan is riskier than another then it also has higher variance.
The converse is not true: a consumption plan that has higher variance than another consumption
plan need not be riskier. We present an example of two consumption plans that have the same
expectation such that there exists a risk-averse agent who prefers the consumption plan with higher
variance. In view of Theorem 10.5.1, this implies that the consumption plan with higher variance
is not riskier than the one with lower variance.
10.6.1 Example
Let z take on the values 1, 3, 4, 6 with equal probabilities, and let y take value 2 with probability
1/2 and values 3 and 7, each with probability 1/4. We have
E(z) = E(y) = 3.5, and var(y) = 4.25 > var(z) = 3.25. (10.19)
Consider the logarithmic utility function v(c) = ln(c). The expected utilities of z and y are
E[v(z)] =
1
4
(ln(1) + ln(3) + ln(4) + ln(6)) =
1
4
ln(72), (10.20)
and
E[v(y)] =
1
2
ln(2) +
1
4
(ln(3) + ln(7)) =
1
4
ln(84). (10.21)
Thus,
E[v(z)] < E[v(y)], (10.22)
implying that y is not riskier than z.
2
Example 10.6.1 also illustrates that y need not be riskier than z if y = z + ² for some ² that is
uncorrelated with z and has zero expectation. To see this note that ², which takes on value 1 if z
98 CHAPTER 10. RISK
equals 1 or 6 and value −1 if z equals 3 or 4, is uncorrelated with z. Also, y = z + ². We have seen
that there exists a risk-averse agent—the agent with logarithmic utility—who prefers z to y.
According to Theorem 10.5.1, greater risk is an ordering of consumption plans with equal
expectation generated by all concave utility functions. Similarly, one can think of the ranking
according to variance as one generated by all quadratic utility functions. To see this, recall that a
quadratic von Neumann-Morgenstern utility function takes the form
v(c) = −(c −α)
2
, for c ≤ α, (10.23)
for some α. The expected utility of consumption plan z is
E[v(z)] = −[var(z) + (E(z) −α)
2
], (10.24)
and depends only on the expectation and variance of z. For two consumption plans y and z that
have the same expectation, y has higher variance than z iff every agent with quadratic utility
function prefers z to y. Since the class of quadratic utility functions is much smaller than the class
of all concave utility functions, the ranking according to variance is stronger than that according
to risk. In fact, the former is a complete ordering, while the latter is a partial ordering.
The two rankings coincide for normally distributed consumption plans. We have
10.6.2 Proposition
Let y and z be two normally distributed consumption plans with variances σ
2
y
and σ
2
z
, respectively.
Then y is strictly riskier than z iff σ
2
y
> σ
2
z
.
Proof: Define λ = σ
y
/σ
z
, and note that λ > 1. The random variable λ(z − E(z)) is normally
distributed with zero mean and variance equal to λ
2
σ
2
z
= σ
2
y
. Therefore λ(z − E(z)) has the same
distribution as y − E(y). It follows from Corollary 10.5.6 that λ(z − E(z)), and therefore also
y − E(y), is strictly riskier than z − E(z). Since expectations do not matter, y is strictly riskier
than z.
2
10.7 A Characterization of Greater Risk
A useful condition characterizing two consumption plans, one of which is riskier than the other,
involves their cumulative distribution functions. Let F
z
and F
y
be the cumulative distribution
functions of consumption plans z and y (that is, F
z
(w) = prob(z ≤ w), and F
y
(w) = prob(y ≤ w)).
We have
10.7.1 Proposition
For consumption plans y and z that have the same expectations, y is riskier than z iff
Z
w
−∞
F
z
(t)dt ≤
Z
w
−∞
F
y
(t)dt (10.25)
for every w.
Proof: For simplicity we assume that there exist a and b such that F
y
(a) = F
z
(a) = 0 and
F
y
(b) = F
z
(b) = 1. The more general case is treated in sources cited in the notes.
We shall prove that the integral condition 10.25 is equivalent to
Z
b
a
v(t)dF
z
(t) ≥
Z
b
a
v(t)dF
y
(t) (10.26)
10.7. A CHARACTERIZATION OF GREATER RISK 99
for every concave function v on the interval [a, b]. Since
R
b
a
v(t)dF
z
(t) = E[v(z)], the conclusion
follows from Theorem 10.5.1.
We first prove that 10.25 implies 10.26 for every concave v. For a twice differentiable function
v, we can use integration by parts (twice) as follows:
Z
b
a
v(t)dF
y
(t) = v(b) −
Z
b
a
F
y
(w)v
0
(w)dw (10.27)
= v(b) − v
0
(b)
Z
b
a
F
y
(w)dw +
Z
b
a
v
00
(w)
µ
Z
w
a
F
y
(t)dt
¶
dw. (10.28)
Since
R
b
a
F
y
(w)dw = b − E(y) (as can be verified by integrating by parts) and E(y) = E(z), the
first two terms of 10.28 are the same for F
y
and F
z
. Since v
00
≤ 0, 10.25 implies that the last term
in 10.28 is greater for F
z
than for F
y
, and hence that 10.26 holds. This argument can be extended
to nondifferentiable concave utility functions by approximation.
We now assume that 10.26 is true for any concave function v and prove 10.25. In particular,
for the concave function
v
w
(t) =
(
t, t ≤ w
w, w ≤ t
(10.29)
we have
Z
b
a
v
w
(t)dF
z
(t) ≥
Z
b
a
v
w
(t)dF
y
(t). (10.30)
We can use integration by parts again to obtain
Z
b
a
v
w
(t)dF
y
(t) =
Z
w
a
tdF
y
(t) + w(1 −F
y
(w)) = w −
Z
w
a
F
y
(t)dt. (10.31)
Inequality 10.25 follows from 10.30 and 10.31 for every w.
2
The following example illustrates Proposition 10.7.1.
10.7.2 Example
Let z take on values −1 and 1, each with probability π, and value 0 with probability 1 −2π where
0 < π < 1/2 —a symmetric three-point distribution. The cumulative distribution function of z is
given by
F
z
(w) =
0, w < −1
π, −1 ≤ w < 0
1 − π, 0 ≤ w < 1
1, 1 ≤ w
(10.32)
The integral of the cumulative distribution function of z is
Z
w
−∞
F
z
(t)dt =
0, w < −1
πw + π, −1 ≤ w < 0
(1 − π)w + π, 0 ≤ w < 1
w, 1 ≤ w
(10.33)
If y takes values −1 and 1 with equal probability φ and value 0 with probability 1 − 2φ for φ > π,
then the integral of F
y
is everywhere greater than or equal to that of F
z
. Thus y is riskier than z.
The distribution of y puts more (probability) weight at the tails than does the distribution of z.
2
100 CHAPTER 10. RISK
For two consumption plans y and z that may have different expectations, y is riskier than
z iff the deviation of y from its expectation is riskier than the deviation of z from its expecta-
tion. Since the deviations from the expectations have zero expectations, Proposition 10.7.1 can be
applied. Consequently, the characterization of greater risk by the integral condition 10.25 holds
for consumption plans that have different expectations, provided that the cumulative distribution
functions of y and z in 10.25 are replaced by those of the deviations of y and z from their respective
expectations.
Notes
The risk of a security or portfolio can be defined as the the risk of its payoff.
In the proof of Proposition 10.7.1 we demonstrated that the integral condition 10.25 is equivalent
to z being preferred to y by every risk-averse agent. An inspection of this proof shows that this holds
true independently of whether agents’ von Neumann-Morgenstern utility functions are increasing.
Therefore z is preferred to y for every risk-averse agent iff the same holds for every risk-averse agent
with an increasing utility function. Consequently, Theorems 10.5.1 and 10.5.2 remain true if one
takes a risk-averse agent to mean an agent with an increasing and concave (strictly concave) utility
function.
The concept of greater risk is that of Rothschild and Stiglitz [4] generalized to apply to random
variables with unequal expectations. It is closely related to the concept of second-order stochastic
dominance: if z and y have the same expectations, then z second-order stochastically dominates
y iff y is riskier than z. On stochastic dominance (of the first and second order), see Hadar and
Russell [2] and Bawa [1]. The proof of Theorem 10.5.1 can be found in Rothschild and Stiglitz [4].
A proof of Proposition 10.7.1 without the assumption of a bounded set of values of the two random
variables can be found in Tesfatsion [5]. See also Hanoch and Levy [3].
Bibliography
[1] Vijay S. Bawa. Optimal rules for ordering uncertain prospects. Journal of Financial Economics,
2:95–121, 1975.
[2] Joseph Hadar and William R. Russell. Rules of ordering uncertain prospects. American Eco-
nomic Review, 59:25–34, 1969.
[3] Giora Hanoch and Haim Levy. Efficiency analysis of choices involving risk. Review of Economic
Studies, pages 335–346, 1969.
[4] Michael Rothschild and Joseph Stiglitz. Increasing risk I: A definition. Journal of Economic
Theory, 2:225–243, 1970.
[5] Leigh Tesfatsion. Stochastic dominance and the maximization of expected utility. Review of
Economic Studies, XLIII:301–315, 1976.
101
102 BIBLIOGRAPHY
Part IV
Optimal Portfolios
103
Chapter 11
Optimal Portfolios with One Risky
Security
11.1 Introduction
An agent’s willingness to invest in a risky security depends, among other things, on the expected
return of that security. In this chapter we analyze agents’ optimal portfolios in a simple setting of
two securities: a single risky security and a risk-free security.
Agents’ utility functions are assumed to have an expected utility representation with strictly
increasing and twice differentiable von Neumann-Morgenstern utility functions. It is also assumed
that date-0 consumption does not enter agents’ utility functions. Further, their endowments at
date 1 are assumed to lie in the asset span (securities market economy).
11.2 Portfolio Choice and Wealth
The consumption-portfolio choice problem of an agent with strictly increasing expected utility
function that depends only on date-1 consumption can be written as
max
c
1
,h
E[v(c
1
)] (11.1)
subject to
ph = w
0
(11.2)
and
c
1
= w
1
+
J
X
j=1
x
j
h
j
, (11.3)
with an additional restriction on consumption if such is imposed. Date-1 consumption plan c
1
,
date-1 endowment w
1
, and security payoff x
j
in 11.1 – 11.3 are understood as random variables on
the set of states S with probability measure π. If, as assumed, the agent’s date-1 endowment lies
in the asset span so that w
1
=
P
j
x
j
ˆ
h
j
for some portfolio
ˆ
h, then we can substitute the agent’s
total portfolio holding
˜
h for the net trade portfolio h plus the portfolio
ˆ
h and rewrite 11.1 – 11.3 as
max
c
1
,
˜
h
E[v(c
1
)] (11.4)
subject to
p
˜
h = w
0
+ p
ˆ
h (11.5)
105
106 CHAPTER 11. OPTIMAL PORTFOLIOS WITH ONE RISKY SECURITY
c
1
=
J
X
j=1
x
j
˜
h
j
. (11.6)
An agent’s wealth is defined as the sum of his date-0 endowment plus the price of the portfolio
generating his date-1 endowment:
w ≡ w
0
+ p
ˆ
h. (11.7)
Note that the price of portfolio
ˆ
h equals the value of the date-1 endowment w
1
under the payoff
pricing functional, that is, p
ˆ
h = q(w
1
). Unless the agent’s date-1 endowment is zero, his wealth w
depends on security prices.
Using wealth w and substituting 11.6 in the expected utility function, we obtain the portfolio
choice problem
max
h
E[v(
X
j
x
j
h
j
)] (11.8)
subject to
ph = w, (11.9)
where portfolio
˜
h has been relabeled h.
If the agent is strictly risk averse, then the optimal consumption plan c
1
=
P
j
x
j
h
j
is unique.
Two consumption plans cannot both be optimal, since any strictly convex combination of the two
would also be budget-feasible and would yield strictly higher expected utility. If in addition there
are no redundant securities, then his optimal portfolio is also unique.
11.3 Optimal Portfolios with One Risky Security
Let there be two securities: a risky security with return denoted by r and a risk-free security with
return ¯r. The difference r − ¯r, assumed nonzero, is the excess return on the risky security.
It is convenient to describe a budget-feasible portfolio of two securities in terms of wealth
invested in each security, instead of in terms of security holdings. For a portfolio (h
1
, h
2
) such that
p
1
h
1
+p
2
h
2
= w, let a = p
2
h
2
denote the amount invested in the risky security. The amount invested
in the risk-free security is w − a = p
1
h
1
, and the payoff of investment (w − a, a) is w¯r + (r − ¯r)a.
The agent’s optimal investment,
1
denoted by a
∗
, is a solution to the problem
max
a
E[v(w¯r + (r − ¯r)a)] (11.10)
which, as noted, may involve an additional restriction that consumption be positive: w¯r+(r−¯r)a ≥
0. The agent’s wealth w is assumed to be strictly positive.
If security prices exclude arbitrage and if consumption is restricted to be positive, then Theo-
rem 3.6.5 implies that maximization problem 11.10 has a solution. In the present context of two
securities, one of which is risk free, the condition that there be no arbitrage has a simple character-
ization in terms of securities’ returns. The risky return r must be lower than the risk-free return ¯r
in some states and higher in other states. Otherwise—if r is uniformly above ¯r, say—then r − ¯r is
an arbitrage.
If the agent is strictly risk averse, the optimal investment is unique, since in the present setting
neither security is redundant. The optimal investment a
∗
is then a function of the agent’s wealth
w, the risk-free return ¯r, and the (distribution of the) risky return r. Further, since utility function
v is twice differentiable, a
∗
is a differentiable function of its arguments whenever the consumption
plan generated by a
∗
is interior.
1
Up to now we have not found it necessary to adopt a separate notation to distinguish optimum values of variables
from non-optimum values. Here, however, we discuss both optimum and non-optimum portfolios, so the distinction
must be made.
11.4. RISK PREMIUM AND OPTIMAL PORTFOLIOS 107
The interior optimal investment a
∗
satisfies the first-order condition
E[v
0
(w¯r + a
∗
(r − ¯r))(r − ¯r)] = 0. (11.11)
11.3.1 Example
One of the attractive features of quadratic utility is that there exists a closed-form expression for
the optimal investment. For
v(y) = −(α − y)
2
, y < α (11.12)
the first-order condition 11.11 is
E[(α − w¯r − a
∗
(r − ¯r))(r − ¯r)] = 0. (11.13)
Evaluating the expectation and solving for a
∗
results in
a
∗
=
(α − w¯r)(µ − ¯r)
σ
2
+ (µ − ¯r)
2
, (11.14)
where µ = E(r) and σ
2
= var(r). Note that, if µ > ¯r, then the optimal investment a
∗
is a decreasing
function of variance σ
2
and of wealth w.
2
11.4 Risk Premium and Optimal Portfolios
The risk premium on a security is defined as its expected excess return; that is, its expected return
less the risk-free return. If the risk premium is zero, then the security is priced fairly, meaning that
the excess return on the security is a fair game (that is, a random variable with zero expectation).
Of course, there is no suggestion that there is anything unfair about nonzero risk premia.
A risk-neutral agent is indifferent among all investments if the risk premium on the risky
security is zero. If the risk premium is nonzero and there are no restrictions on consumption, then
his optimal investment does not exist. If his consumption is restricted to be positive, then the agent
will hold long the security with high expected return and sell short the security with low expected
return until the positivity restriction becomes binding.
Whether a strictly risk-averse agent chooses a positive or a negative investment in the risky
security depends on the risk premium on the risky security.
11.4.1 Theorem
If an agent is strictly risk averse, then the optimal investment in the risky security is strictly
positive, zero or strictly negative iff the risk premium on the risky security is strictly positive, zero
or strictly negative.
Proof: Since w is strictly positive, zero investment in the risky security results in a strictly
positive risk-free consumption. Therefore a = 0 is an interior point of the interval of the investment
choices whether or not consumption is restricted to be positive. The derivative of expected utility
in 11.10 with respect to a at a = 0 is v
0
(w¯r)(µ − ¯r), where µ ≡ E(r). Since v
0
(w¯r) is strictly
positive, the derivative is strictly positive, zero or strictly negative iff µ − ¯r is strictly positive,
zero or strictly negative. Since expected utility is strictly concave in a, the sign of the derivative at
zero investment determines whether the optimal investment is positive, zero or negative (see Figure
11.1).
2
It is important to keep in mind that the optimal investment a
∗
characterized in Theorem 11.4.1
is the part of total wealth w invested in the risky security. Since w consists of date-0 endowment
108 CHAPTER 11. OPTIMAL PORTFOLIOS WITH ONE RISKY SECURITY
and the price of the portfolio the agent is endowed with (see 11.7), zero investment a
∗
means that
the agent sells all of the shares of the risky security that he is endowed with and invests the proceeds
in the risk-free security.
If the risk premium is zero, then any nonzero investment in the risky security has a strictly
riskier return than the risk-free return and the same expected return. It follows from Theorem
10.5.1 that the optimal investment must be the risk-free investment. Thus this part of Theorem
11.4.1 holds even in the absence of the maintained assumption that the agent’s utility function is
differentiable. This is not the case for other parts of Theorem 11.4.1. For instance, the optimal
investment in the risky security may be zero when the risk premium is strictly positive.
11.4.2 Example
There are two states with equal probabilities. The risk-free return is ¯r = 1 and the return on
the risky security is r = (1.3, 0.8) so that the risk premium is strictly positive. The agent’s von
Neumann-Morgenstern utility function v, given by
v(y) =
(
2y, y ≤ 5
y + 5, y ≥ 5
(11.15)
is strictly increasing and concave. The expected utility
E[v(c)] =
1
2
v(c
1
) +
1
2
v(c
2
) (11.16)
is nondifferentiable whenever c
1
= 5 or c
2
= 5. If the agent’s wealth is w = 5, then his optimal
choice is to invest his entire wealth in the risk-free security (Figure 11.2).
2
Theorem 11.4.1 implies that the return on the optimal portfolio of a strictly risk-averse agent
(with differentiable utility function) is risk free iff the risky security is priced fairly: E(r) = ¯r.
Otherwise, if the risk premium is nonzero, the return on the optimal portfolio is risky. The expected
return on the optimal portfolio equals
¯r +
a
∗
w
(E(r) − ¯r) (11.17)
and is strictly higher than the risk-free return. Thus the risk of the optimal return is compensated
by a relatively high expected return.
11.5 Optimal Portfolios When the Risk Premium Is Small
It follows from Theorem 11.4.1 and continuity of the optimal portfolio as a function of the risk
premium that if the risk premium is small, then the amount invested in the risky security is small.
Much more can be said. If the risk premium is small—that is, if the risky security is priced
approximately fairly—then the optimal investment a
∗
is approximately proportional to the risk
premium E(r) − ¯r, inversely proportional to the Arrow-Pratt measure of absolute risk-aversion,
and inversely proportional to the variance σ
2
of the risky return.
11.5.1 Theorem
If an agent is strictly risk-averse, his date-1 endowment is zero, and the risk premium on the risky
security is small, then the optimal investment in the risky security is
a
∗
∼
=
E(r) − ¯r
σ
2
A(w¯r)
. (11.18)
11.5. OPTIMAL PORTFOLIOS WHEN THE RISK PREMIUM IS SMALL 109
Proof: If the risk premium is zero so that ¯r = µ where µ ≡ E(r), then, by Theorem 11.4.1,
the optimal investment a
∗
equals zero. For small risk premium, the linear approximation of a
∗
is
a
∗
∼
=
(¯r − µ)∂
¯r
a
∗
, (11.19)
where ∂
¯r
a
∗
is the partial derivative of a
∗
with respect to ¯r at ¯r = µ.
To find the partial derivative ∂
¯r
a
∗
at ¯r = µ, we differentiate 11.11 with respect to ¯r. Since the
agent has zero date-1 endowment, the wealth w does not depend on ¯r, and we obtain
E[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)(w + (r − ¯r)∂
¯r
a
∗
− a
∗
) − v
0
(w¯r + a
∗
(r − ¯r))] = 0. (11.20)
Setting ¯r = µ and using the fact that a
∗
is zero when ¯r = µ, we can solve 11.20 for
∂
¯r
a
∗
= −
1
A(w¯r)σ
2
. (11.21)
Substituting the right-hand side of 11.21 in 11.19, we get 11.18.
2
The form of approximation used in 11.19 reveals the meaning of “small” risk premium. “Small”
means that the terms of second and higher order of Taylor’s expansion of a
∗
as a function of ¯r around
¯r = µ are negligible. Further, the positivity constraint on consumption, which is nonbinding when
the risk premium is zero, remains nonbinding at a “small” risk premium.
11.5.2 Example
For the quadratic utility function of Example 11.3.1, the Arrow-Pratt measure of absolute risk
aversion is
A(w¯r) =
1
α − w¯r
. (11.22)
Expression 11.14 for the optimal investment can be written as
a
∗
=
µ − ¯r
(σ
2
+ (µ − ¯r)
2
)A(w¯r)
. (11.23)
The approximate expression 11.18 differs from the exact solution 11.23 in that the former
neglects the second-order term (µ − ¯r)
2
in the denominator.
2
Notes
The portfolio choice problem with single risky security was first analyzed in Tobin [3], Arrow [1]
and Pratt [2].
Extending a result by Pratt [2], Wang and Werner [4] showed that the optimal investment in
a single risky security provides a measure of risk aversion equivalent to the Arrow-Pratt measure.
One risk-averse agent is less risk-averse than another iff the investment in the risky security of the
first is higher than that of the second for all levels of wealth and all risky returns with strictly
positive risk premium.
110 CHAPTER 11. OPTIMAL PORTFOLIOS WITH ONE RISKY SECURITY
Bibliography
[1] Kenneth J. Arrow. Aspects of the Theory of Risk Bearing. Yrjo Jahnssonin Saatio, Helsinki,
1965.
[2] John W. Pratt. Risk aversion in the small and in the large. Econometrica, 32:122–136, 1964.
[3] James Tobin. Liquidity preference as behavior towards risk. Review of Economic Studies,
25:65–86, 1958.
[4] Zhenyu Wang and Jan Werner. Portfolio characterization of risk aversion. Economics Letters,
45:259–265, 1994.
111
112 BIBLIOGRAPHY
Chapter 12
Comparative Statics of Optimal
Portfolios
12.1 Introduction
In this chapter we investigate how optimal portfolios depend on agents’ wealth, on the risk-free
return, and on the expectation and the riskiness of the risky return. As in chapter 11 our analysis
is restricted to the simple setting of two securities: a single risky security and a risk-free security.
We assume that agents’ wealth consists only of date-0 endowment; date-1 endowments are
assumed zero. This implies that the wealth does not depend on security prices or returns and
allows us to abstract from the effects of price or return changes on wealth. For most of this chapter
it is assumed that date-0 consumption does not enter agents’ utility functions. An exception is
Section 12.5 in which we analyze optimal portfolios with intertemporal consumption.
Our analysis of optimal portfolios in this chapter draws on methods and results of comparative
statics in consumer theory.
12.2 Wealth
We recall that the optimal investment a
∗
in the risky security is a solution to the problem
max
a
E[v(w¯r + (r − ¯r)a)], (12.1)
where utility function v is strictly increasing and twice-differentiable. The first-order condition for
an interior optimal investment is
E[v
0
(w¯r + a
∗
(r − ¯r))(r − ¯r)] = 0. (12.2)
Our concern in this section is with the response of optimal investment a
∗
to changes in wealth.
Whether a
∗
increases, decreases, or remains unchanged when wealth increases depends on how the
absolute risk aversion changes as a function of wealth.
12.2.1 Theorem
If an agent is strictly risk averse, if his absolute risk aversion is decreasing, and if the risk premium
on the risky security is positive, then the optimal investment a
∗
in the risky security is increasing
in wealth.
Proof: Differentiating the first-order condition 12.2 with respect to w results in
E[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)(¯r + (r − ¯r)∂
w
a
∗
)] = 0, (12.3)
113
114 CHAPTER 12. COMPARATIVE STATICS OF OPTIMAL PORTFOLIOS
or
∂
w
a
∗
= −
¯rE[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)]
E[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)
2
]
. (12.4)
The denominator in expression 12.4 is strictly negative. We shall prove that the numerator is
positive. Since the measure of absolute risk aversion A is decreasing, we have
A(w¯r + a
∗
(r
s
− ¯r)) ≤ A(w¯r), (12.5)
for all states s such that r
s
> ¯r. Note that a
∗
≥ 0 follows from Theorem 11.5.1. Substituting the
definition of A in the left-hand side of 12.5 and multiplying both sides by r
s
− ¯r, there results
v
00
(w¯r + a
∗
(r
s
− ¯r))(r
s
− ¯r) ≥ −A(w¯r)v
0
(w¯r + a
∗
(r
s
− ¯r))(r
s
− ¯r). (12.6)
In those states in which r
s
≤ ¯r, we have
A(w¯r + a
∗
(r
s
− ¯r)) ≥ A(w¯r). (12.7)
Performing the same calculations as above (and noting that multiplying by r
s
−¯r now reverses the
sign of the inequality), there results 12.6, which is therefore true for all values of r
s
. Taking the
expectation of 12.6 and using 12.2 results in
E[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)] ≥ 0. (12.8)
Thus the numerator on the right-hand side of 12.4 is positive, implying that
∂
w
a
∗
≥ 0. (12.9)
2
Thus under the conditions of the theorem the risky security is a normal good. Results analogous
to Theorem 12.2.1 hold under increasing and constant absolute risk aversion. If an agent is strictly
risk averse and his absolute risk aversion is increasing, then his optimal investment in a risky
security with strictly positive risk premium is decreasing in wealth, so that the risky security is an
inferior good. This is the case for the quadratic utility function, see 11.14. If an agent’s absolute
risk aversion is constant (negative exponential utility), his optimal investment is independent of
wealth.
We also have
12.2.2 Theorem
If an agent is strictly risk averse, if his relative risk aversion is decreasing, and if the risk premium
on the risky security is f positive, then the fraction of wealth a
∗
/w invested in the risky security is
increasing in wealth.
Proof: The first-order condition 12.2 can be written as
E[v
0
(w¯r + w(
a
∗
w
)(r − ¯r))(r − ¯r)] = 0. (12.10)
Evaluation of ∂
w
(a
∗
/w) is precisely analogous to evaluation of ∂
w
a
∗
in the proof of Theorem 12.2.1.
Here the measure of relative risk aversion replaces the measure of absolute risk aversion used in
Theorem 12.2.1.
2
Analogous results hold under increasing and constant relative risk aversion. Thus under constant
relative risk aversion (power and logarithmic utilities with α = 0) the fraction of wealth invested
in the risky security is invariant to wealth.
12.3. EXPECTED RETURN 115
12.3 Expected Return
Our concern in this section is with changes of optimal investment in response to changes in the
risk-free return or the expected return of the risky security. We begin with the risk-free return.
12.3.1 Theorem
If an agent is strictly risk averse, if his absolute risk aversion is increasing, if his optimal investment
in the risk-free security is positive and if the risk premium on the risky security is positive, then
the optimal investment a
∗
in the risky security is strictly decreasing in the risk-free return.
Proof: Differentiating the first-order condition 12.2 with respect to ¯r (see 11.20) results in
∂
¯r
a
∗
=
E[v
0
(w¯r + a
∗
(r − ¯r))] − E[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)](w − a
∗
)
E[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)
2
]
. (12.11)
Using 12.4, we obtain
∂
¯r
a
∗
=
E[v
0
(w¯r + a
∗
(r − ¯r))]
E[v
00
(w¯r + a
∗
(r − ¯r))(r − ¯r)
2
]
+
w − a
∗
¯r
∂
w
a
∗
. (12.12)
The numerator of the first term on the right-hand side of 12.12 is strictly positive, while the
denominator is strictly negative. Therefore the first term is strictly negative. The counterpart
of Theorem 12.2.1 for increasing absolute risk aversion implies that under the assumed conditions
∂
w
a
∗
is negative. Since w − a
∗
is positive by assumption, it follows that ∂
¯r
a
∗
< 0.
2
The effect of a change in the risk-free return on the investment in the risky security can be
decomposed into a substitution effect and an income effect . The first term on the right-hand side
of 12.12 expresses the substitution effect . As shown, the substitution effect is always negative. If
the risk-free return increases, the risk-free security becomes more attractive and the risky security
less attractive, leading to a decrease in the investment in the risky security.
The second term on the right-hand side of 12.12 expresses the income effect. A marginal unit
increase in the risk-free return generates a date-1 consumption increase that equals the investment
in the risk-free security w − a
∗
. This date-1 consumption increase is equivalent to date-0 wealth
increase of (w − a
∗
)/¯r. The effect of this wealth increase on the optimal investment in the risky
security is ((w − a
∗
)/¯r)∂
w
a
∗
, and is the income effect.
In general the income effect may be positive or negative. Under the assumptions of Theorem
12.3.1 it is negative and reinforces the substitution effect. In the following theorem alternative
assumptions are imposed under which the income effect may be positive but it is always dominated
by the negative substitution effect.
12.3.2 Theorem
If an agent is strictly risk averse, if his relative risk aversion is less than or equal to one, and if the
risky return is positive, then the optimal investment a
∗
in the risky security is strictly decreasing
in the risk-free return.
Proof: Let c
∗
1
denote the optimal date-1 consumption
c
∗
1
= w¯r + a
∗
(r − ¯r). (12.13)
The numerator in expression 12.11 for ∂
¯r
a
∗
can be written using the measure of absolute risk
aversion A as
E[v
0
(c
∗
1
)(1 + A(c
∗
1
)(r − ¯r)(w − a
∗
))]. (12.14)
116 CHAPTER 12. COMPARATIVE STATICS OF OPTIMAL PORTFOLIOS
Using 12.13 we can rewrite expression 12.14 as
E[v
0
(c
∗
1
)(1 − A(c
∗
1
)c
∗
1
+ A(c
∗
1
)wr)]. (12.15)
Substituting the measure of relative risk aversion R(c
∗
1
) for A(c
∗
1
)c
∗
1
in 12.15, we obtain
E[v
0
(c
∗
1
)(1 − R(c
∗
1
) + A(c
∗
1
)wr)]. (12.16)
Since the agent is strictly risk averse and the risky return r is positive and nonzero, the term
A(c
∗
1
)wr is positive and nonzero. If, as assumed, R is less than or equal to one, then 12.16 is
strictly positive. Thus the numerator in 12.11 is strictly positive. Since the denominator is strictly
negative, it follows that ∂
¯r
a
∗
< 0.
2
Examples of utility functions with relative risk aversion less than or equal to one include power
utility functions with γ > 1 and α ≥ 0, and logarithmic utility functions with α ≥ 0.
The dependence of the optimal investment on the expected return of the risky security is the
opposite of its dependence on the risk-free return. To determine the effect of changes in the
expected return we write r = µ + ∆r, where µ = E(r), and we consider variations in µ keeping the
distribution of ∆r unchanged. Using the same arguments as in the proof of Theorem 12.3.1 one can
show that if an agent is strictly risk averse, if his absolute risk aversion A is decreasing, and if the
risk premium on the risky security is positive, then the optimal investment a
∗
is strictly increasing
in the expected return of the risky security. If the agent’s absolute risk aversion is increasing (as
for quadratic utilities), then nothing can be said in general as to whether the investment in the
risky security will increase or decrease.
The counterpart to Theorem 12.3.2 when the expected return on the risky security changes is
similar.
12.4 Risk
One might expect that the investment in the risky security would decrease if its return becomes
more risky (in the sense of Chapter 10) but its expected return remains unchanged. This is the case
for a quadratic utility function: increased risk with no change in the expected return implies that
the variance of the return increases, and the investment in the risky security decreases as indicated
by 11.14. However, this need not be the case in general for a strictly risk-averse utility function.
To investigate the effect on the optimal investment in the risky security of an increase in its
riskiness, we consider the first-order condition 12.2 and introduce a function g of two scalar variables
a and y given by
g(a, y) ≡ v
0
(w¯r + a(y − ¯r))(y − ¯r). (12.17)
If the agent is strictly risk averse, then g is a strictly decreasing function of investment a for any
y. Eq. 12.2 can now be written as
E[g(a
∗
, r)] = 0. (12.18)
Suppose that the risky return r is replaced by the more risky return ˜r with the same expectation.
Suppose also (pending discussion below) that g(a
∗
, y) is a concave function of y. Theorem 10.5.1
can be applied to function g(a
∗
, ·) in place of a utility function, and we obtain
E[g(a
∗
, ˜r)] ≤ E[g(a
∗
, r)] = 0. (12.19)
If inequality in 12.19 is strict, so that a
∗
is not the optimal investment with the return ˜r, then the
investment a has to be decreased in order to restore the first-order condition. The opposite holds
if g is a convex function of y.
12.5. OPTIMAL PORTFOLIOS WITH TWO-DATE CONSUMPTION 117
One can show (see the sources cited in the notes) that a sufficient condition for function g of
12.17 to be concave in y is that the relative risk aversion be increasing and less than or equal to one,
and the absolute risk aversion be decreasing. If the risk premium on the risky security is strictly
positive, then this condition implies that the investment in the risky security decreases when the
risky return becomes more risky. Power utility functions with γ > 1 and α ≥ 0, and logarithmic
utility functions with α ≥ 0 satisfy all these conditions on risk aversion.
12.5 Optimal Portfolios with Two-Date Consumption
So far the analysis of optimal portfolios has proceeded under the assumption that date-0 consump-
tion does not enter the agent’s utility function. If it does, then the agent has to choose the division
of wealth between securities and date-0 consumption, in addition to choosing optimal investments
in each security.
The portfolio choice problem with two-date consumption can be written as
max
a
1
,a
2
E[v(w − a
1
− a
2
, ¯ra
1
+ ra
2
)], (12.20)
where a
1
and a
2
are the amounts of wealth invested in the risk-free and the risky security, respec-
tively. The optimal investments are denoted by a
∗
1
and a
∗
2
.
The result of Theorem 11.4.1 that the optimal investment in the risky security is strictly positive,
zero or strictly negative as the risk premium on the risky security is strictly positive, zero or strictly
negative if the agent is strictly risk averse extends to the setting of two-date consumption. To
see this, let c
∗
0
= w − a
∗
1
− a
∗
2
denote the optimal date-0 consumption and let ¯w = w − c
∗
0
and
¯v(c
s
) = v(c
∗
0
, c
s
). Then a
∗
2
is the optimal investment in the risky security for the single-date utility
function ¯v with wealth ¯w. Since ¯v is strictly concave, Theorem 11.4.1 implies the conclusion.
Optimal portfolios can be easily characterized when the agent is risk neutral . For instance, if
the utility function takes the form
v(c
0
, c
s
) = c
0
+ δc
s
(12.21)
for some δ > 0, and if the risk-free return equals δ
−1
and the risk premium on the risky security is
zero, then this risk-neutral agent is indifferent among all portfolios. If one or both securities have
expected return not equal to δ
−1
and there are no restrictions on consumption, then his optimal
portfolio does not exist. If his consumption is restricted to be positive, then there exists an optimal
portfolio. This portfolio is a solution to a linear programming problem. For instance, if the risk-free
return equals δ
−1
and there is a strictly positive risk premium, then the risk-neutral agent will sell
short the risk-free security and invest his entire wealth in the risky security. Since the risk-free
return has to be higher than the risky return in at least one state (otherwise there is an arbitrage
opportunity), the restriction that consumption be positive implies a limit on the short position in
the risk-free security. This limiting short position determines the agent’s optimal portfolio.
We present comparative statics analysis of optimal portfolios with two-date consumption under
an additional restriction that there is only one security. Suppose first that the security has a risk-
free payoff. Then the the agent faces no uncertainty in his portfolio-consumption choice and his
optimal investment a
∗
is a solution to the problem
max
a
v(w − a, ¯ra). (12.22)
The maximization problem 12.22 is the standard saving problem under certainty.
The first-order condition for an interior solution to 12.22 is
∂
0
v(w − a
∗
, ¯ra
∗
) = ¯r∂
1
v(w − a
∗
, ¯ra
∗
). (12.23)
118 CHAPTER 12. COMPARATIVE STATICS OF OPTIMAL PORTFOLIOS
To investigate the effect of an increase in the agent’s wealth on the optimal saving a
∗
we
differentiate the first-order condition 12.23 to find that
∂
w
a
∗
=
∂
00
v − ¯r∂
01
v
D
, (12.24)
where ∂
tτ
v denotes the second-order partial derivative of v at (w − a
∗
, ¯ra
∗
) for t, τ = 0, 1, and
D = (¯r)
2
∂
11
v −2¯r∂
01
v + ∂
00
v. If the agent is strictly risk averse so that v is strictly concave, then,
by the second-order condition, D is strictly negative. However, the sign of the numerator in 12.24,
and hence the sign of the derivative ∂
w
a
∗
, cannot be determined without further assumptions on the
utility function. If the utility function is time-separable, then ∂
01
v = 0 and consequently ∂
w
a
∗
> 0;
that is, the agent’s optimal saving increases when wealth increases.
Differentiating the first-order condition 12.23 with respect to the risk-free return ¯r results in
∂
¯r
a
∗
= −
∂
1
v
D
+
a
∗
(∂
01
v − ¯r∂
11
v)
D
. (12.25)
If the utility function is time-separable so that ∂
01
v = 0 and if a
∗
≥ 0, then ∂
¯r
a
∗
> 0; that is, the
agent’s optimal saving increases when the risk-free return increases. The effect of a change in the
risk-free return on the optimal saving can be decomposed into an income effect and a substitution
effect. Substituting ∂
01
v − ¯r∂
11
v = (1/¯r)(∂
00
v − ¯r∂
01
v − D) in 12.25 and using 12.24, we obtain
∂
¯r
a
∗
= −
∂
1
v
D
−
a
∗
¯r
+
a
∗
¯r
∂
w
a
∗
. (12.26)
The first two terms on the right-hand side of 12.26 add up to the substitution effect and the third
term is the income effect. The sign of the substitution effect is ambiguous (see Figure 12.1).
For a time-separable utility function, the optimal investment in a single security increases with
wealth not only when the payoff of the security is risk-free but also when the payoff is risky.
The optimal investment in a single risky security with return r for an agent with utility function
v(y
0
, y
1
) = v
0
(y
0
) + v
1
(y
1
) is a solution to
max
a
v
0
(w − a) + E[v
1
(ra)]. (12.27)
The first-order condition for an interior solution to 12.27 is
v
0
0
(w − a
∗
) = E[rv
0
1
(ra
∗
)]. (12.28)
Differentiating 12.28 with respect to w results in
∂
w
a
∗
=
v
00
0
v
00
0
+ E(r
2
v
00
1
)
> 0. (12.29)
We investigate now the effect on the optimal investment in the risky security of an increase in
its riskiness. We use the method of Section 12.4. Define function g by
g(a, y) ≡ yv
0
1
(ya) − v
0
0
(w − a). (12.30)
The first-order condition 12.28 can now be written
E[g(a
∗
, r)] = 0 (12.31)
If both period utility functions v
0
and v
1
are strictly concave, then g is a strictly decreasing function
of a. If we assume (pending discussion below) that g(a
∗
, y) is a concave function of y, then we can
conclude that replacing risky return r by a more risky return with the same expectation leads to a
decrease in the optimal investment a
∗
.
One can show that a sufficient condition for function g(a
∗
, y) to be concave in y is that the
third-order derivative v
000
1
be strictly negative and a
∗
> 0. Strictly negative third-order derivative
implies strictly increasing absolute risk aversion.
12.5. OPTIMAL PORTFOLIOS WITH TWO-DATE CONSUMPTION 119
Notes
The literature on comparative statics of the portfolio choice problem with single date consumption
is rich. A few of the relevant references are Tobin [10], Fishburn and Porter [3], Cheng, Magill and
Shafer [1]. A detailed analysis of the dependence of an optimal portfolio on the riskiness of the risky
return can be found in Rothschild and Stiglitz [8]. Gollier [4], [5] derives necessary and sufficient
conditions for a change in the return of the risky security to induce a decrease of the investment in
the risky security for every risk-averse agent.
The literature on saving decisions and portfolio choice with intertemporal consumption is equally
large. Main references include Leland [7], Dreze and Modigliani [2] and Sandmo [9]. Kimball [6]
derived a characterization of the negative third-order derivative of utility function (see Section 12.5)
in terms of prudence.
120 CHAPTER 12. COMPARATIVE STATICS OF OPTIMAL PORTFOLIOS
Bibliography
[1] Harrison Cheng, Michael Magill, and Wayne Shafer. Some results on comparative statics under
uncertainty. International Economic Review, 28:493–509, 1987.
[2] Jacques H. Dreze and Franco Modigliani. Consumption decisions under uncertainty. Journal
of Economic Theory, 5:308–335, 1972.
[3] Peter C. Fishburn and R. Burr Porter. Optimal portfolios with one safe and one risky asset:
Effects of changes in rate of return and risk. Management Science, 22:1064–1072, 1976.
[4] Christian Gollier. The comparative statics of changes in risk revisited. Journal of Economic
Theory, 66:522–535, 1995.
[5] Christian Gollier. A note on portfolio dominance. Review of Economic Studies, 64:147–150,
1997.
[6] Miles Kimball. Precautionary saving in the small and in the large. Econometrica, 58:53–73,
1990.
[7] Hayne E. Leland. Saving and uncertainty: The precautionary demand for saving. Quarterly
Journal of Economics, 82:465–473, 1968.
[8] M. Rothschild and J. Stiglitz. Increasing risk I: A definition. Journal of Economic Theory,
2:225–243, 1970.
[9] Agnar Sandmo. Capital risk, consumption, and portfolio choice. Econometrica, 37:586–599,
1969.
[10] James Tobin. Liquidity preference as behavior towards risk. Review of Economic Studies,
25:65–86, 1958.
121
122 BIBLIOGRAPHY
Chapter 13
Optimal Portfolios with Several Risky
Securities
13.1 Introduction
In this chapter we characterize optimal portfolios in a setting with several risky securities. For
the most part, the comparative statics results of the preceding chapter cannot be extended when
there are several risky securities. We present below the few results that can be extended and derive
some further results under additional restrictions on either securities returns or on agents’ utility
functions.
The assumptions of Chapter 11 are maintained in this chapter: agents’ utility functions have
expected utility representations, are strictly increasing and differentiable and, with the exception
of Section 13.7, depend only on date-1 consumption. Endowments lie in the asset span (securities
market economy). It is also assumed that there are no redundant securities.
13.2 Optimal Portfolios
As in Chapters 11 and 12, it is convenient to describe the portfolio choice problem in terms of
wealth invested in each security. Let a
j
= p
j
h
j
denote the amount of wealth invested in security j
and let a = (a
1
, . . . , a
J
). The portfolio choice problem 11.8 of an agent with a strictly increasing
utility function can be restated as
max
a
E[v(
J
X
j=1
a
j
r
j
)] (13.1)
subject to
J
X
j=1
a
j
= w (13.2)
and possibly the additional constraint of positivity of the resulting consumption.
The agent’s optimal investment will be denoted by a
∗
= (a
∗
1
, . . . , a
∗
J
) and its return by r
∗
. Thus
r
∗
=
P
J
j=1
a
∗
j
r
j
w
. (13.3)
If one of the securities, say security 1, is risk free with return ¯r, then the portfolio choice problem
13.1 can be written as
max
a
2
,...,a
J
E[v(w¯r +
J
X
j=2
a
j
(r
j
− ¯r))]. (13.4)
123
124 CHAPTER 13. OPTIMAL PORTFOLIOS WITH SEVERAL RISKY SECURITIES
The optimal investment a
∗
is given by a solution (a
∗
2
, . . . , a
∗
J
) to 13.4 and the investment in the
risk-free security given by a
∗
1
= w −
P
J
j=2
a
∗
j
.
13.3 Risk-Return Tradeoff
It was shown in Chapter 11 that, with one risky security, an optimal portfolio of a strictly risk-averse
agent is risky iff its expected return is strictly higher than the risk-free return. The portfolio risk
is compensated for by a relatively high expected return. This tradeoff between risk and expected
return holds in the more general setting of many risky securities:
13.3.1 Theorem
If r
∗
is the return on an optimal portfolio of a risk-averse agent and if r
∗
is riskier than the return
r, then E(r
∗
) ≥ E(r).
Proof: Let v be the agent’s von Neumann-Morgenstern utility function. Optimality of the
return r
∗
implies that
E[v(wr
∗
)] ≥ E[v(wr)]. (13.5)
If r
∗
is riskier than r, then so is r
∗
− E(r
∗
) + E(r). Since r
∗
− E(r
∗
) + E(r) and r have the same
expectations and since the agent is risk-averse, we can apply Theorem 10.5.2 to obtain
E[v(wr)] ≥ E[v(wr
∗
− wE(r
∗
) + wE(r))]. (13.6)
Inequalities 13.5 and 13.6 imply that E(r
∗
) ≥ E(r), since v is strictly increasing.
2
Note that Theorem 13.3.1 holds true even in the absence of the maintained assumption of the
differentiability of the utility function.
As usual, there is also a strict version:
13.3.2 Theorem
If r
∗
is the return on an optimal portfolio of a strictly risk-averse agent and if r
∗
is strictly riskier
than a return r, then E(r
∗
) > E(r).
Theorems 13.3.1 and 13.3.2 give an expression of the risk-return tradeoff: the greater the
expected return on an optimal portfolio, the greater the risk of that portfolio. What is interesting
about this result is that the “return” in the “risk-return tradeoff” is identified with the first moment
of the return distribution (the expectation), but “risk” is measured by the ordering introduced in
Chapter 10 and not by the second moment of the return distribution (variance).
13.4 Optimal Portfolios under Fair Pricing
If all securities are priced fairly, then a risk-neutral agent is indifferent among all (budget-feasible)
portfolios, and a strictly risk-averse agent chooses a portfolio with a risk-free payoff (see Theorem
13.3.2) if one is available. Under the assumption of differentiability of the utility function, the
converse is also true: only under fair pricing is the payoff of an optimal portfolio of a strictly
risk-averse agent risk free.
13.4.1 Theorem
Suppose that security 1 is risk free with return ¯r. Then the payoff of an optimal portfolio of a
strictly risk-averse agent is risk free iff all securities are priced fairly; that is, iff
E(r
j
) = ¯r ∀ j. (13.7)
13.5. RISK PREMIA AND OPTIMAL PORTFOLIOS 125
Proof: The first-order condition for optimal investment a
∗
is
E[v
0
(w¯r +
J
X
j=2
a
∗
j
(r
j
− ¯r))(r
k
− ¯r)] = 0 ∀ k ≥ 2 (13.8)
whenever the resulting consumption is interior.
If the payoff of optimal investment a
∗
is risk free, then (since there are no redundant securities)
a
∗
j
= 0 for each j ≥ 2 and a
∗
1
= w. The resulting consumption plan w¯r is strictly positive. The
first-order condition 13.8 with a
∗
j
= 0 for each j ≥ 2 implies fair pricing 13.7.
Conversely, since v is differentiable and 13.7 holds, then a
∗
j
= 0 for each j ≥ 2 satisfies the
first-order conditions 13.8. These conditions are sufficient for optimality, and if v is strictly concave
the optimal portfolio is unique.
2
13.5 Risk Premia and Optimal Portfolios
When there is only one risky security, the optimal holding of the risky security is strictly positive,
zero or strictly negative according to whether the risk premium on that security is strictly positive,
zero or strictly negative (Theorem 11.4.1). One might expect that this relation continues to hold
when there are several risky securities. It does not. For instance, an optimal portfolio can involve a
long position in a security with strictly negative risk premium if the payoff on that security covaries
strongly and negatively with the payoff on another security with a strictly positive risk premium. In
the Capital Asset Pricing Model of Chapter 19, this is exactly the case for a negative-beta security.
As this reasoning suggests, the arguments of the proof of Theorem 11.4.1 do not extend to the
case of several risky securities. As before, the sign of the risk premium E(r
j
) − ¯r determines the
sign of the partial derivative of expected utility with respect to investment in that security at zero.
Without further knowledge of the agent’s utility function and/or security returns, the signs of the
partial derivatives at zero are not enough to determine the location of the optimal investment in
the case of many risky securities.
Of course, if the risk premium is zero on every security then, as seen in Theorem 13.4.1, the
optimal investment of a strictly risk-averse agent in every risky security is zero.
If the return of a security can written as the return on some portfolio of other securities plus a
mean-independent term, then the sign of a strictly risk-averse agent’s optimal investment in that
security is the same as that of the expectation of the mean-independent term.
13.5.1 Theorem
Suppose that the return on security
k satisfies
r
k
=
X
j6=k
η
j
r
j
+ ²
k
, (13.9)
where
P
j6=k
η
j
= 1 and ²
k
is mean-independent of the returns on securities other than security k,
that is,
E(²
k
|r
1
, . . . , r
k−1
, r
k+1
, . . . , r
J
) = E(²
k
). (13.10)
Then the optimal investment in security k for a strictly risk-averse agent is strictly positive, zero
or strictly negative as E(²
k
) is strictly positive, zero or strictly negative.
Proof: Consider the maximization problem
max
λ
E[v(
X
j6=k
a
∗
j
r
j
+ λr
k
+ (a
∗
k
− λ)
X
j6=k
η
j
r
j
)]. (13.11)
126 CHAPTER 13. OPTIMAL PORTFOLIOS WITH SEVERAL RISKY SECURITIES
The value of expected utility in 13.11 cannot exceed E[v(
P
j
a
∗
j
r
j
)] and the latter value is achieved
at λ = a
∗
k
. Thus λ = a
∗
k
is the solution to the maximization problem 13.11. Whether a
∗
k
is strictly
positive, zero or strictly negative depends on the sign of the derivative of the (strictly concave)
expected utility in 13.11 with respect to λ evaluated at λ = 0.
The derivative of the expected utility in 13.11 with respect to λ evaluated at zero is
E[v
0
(
X
j6=k
(a
∗
j
+ a
∗
k
η
j
)r
j
)(r
k
−
X
j6=k
η
j
r
j
)]. (13.12)
Assumptions 13.9 and 13.10, and Proposition 10.4.1 imply that the expression 13.12 is equal to
E[v
0
(
X
j6=k
(a
∗
j
+ a
∗
k
η
j
)r
j
)]E(²
k
). (13.13)
From 13.13 we can see that the sign of the derivative of the expected utility in 13.11 at λ = 0 is
determined by the sign of E(²
k
). Consequently, the sign of the optimal investment a
∗
k
is determined
by the sign of E(²
k
).
2.
A simple but useful corollary to Theorem 13.5.1 relates the risk premium on a security to the
optimal investment if the return on that security is mean independent of the returns on other
securities.
13.5.2 Corollary
Suppose that security 1 is risk free with return ¯r and that the return on security k is mean inde-
pendent of the returns on other securities; that is,
E(r
k
|r
1
, . . . , r
k−1
, r
k+1
, . . . , r
J
) = E(r
k
). (13.14)
Then the optimal investment in security k for a strictly risk-averse agent is strictly positive, zero
or strictly negative as the risk premium E(r
k
) − ¯r is strictly positive, zero or strictly negative.
Proof: We can write the return on security k as
r
k
= ¯r + ²
k
. (13.15)
If 13.14 holds, then ²
k
is mean independent of returns on securities other than security k. Theorem
13.5.1 implies that the optimal investment in security k is strictly positive, zero or strictly negative
as E(²
k
) is strictly positive, zero or strictly negative. Since E(²
k
) equals the risk premium E(r
k
)−¯r,
the conclusion follows.
2
The intuitive explanation for Corollary 13.5.2 is simple. If the return on a security is mean-
independent of other returns and the risk premium is zero, then every portfolio with a nonzero
holding of that security is strictly riskier than a portfolio in which the investment in that security
has been replaced by an investment (of equal value) in the risk-free security. A strictly positive
risk premium is required to induce a strictly risk-averse agent to invest a strictly positive amount
of wealth in that security.
Corollary 13.5.2 can be viewed an extension of Theorem 11.4.1. If there is a single risky security,
then condition 13.14 is trivially satisfied.
The following example illustrates the results of this section.
13.6. OPTIMAL PORTFOLIOS UNDER LINEAR RISK TOLERANCE 127
13.5.3 Example
There are three states with probabilities 1/2, 1/4, and 1/4, and three securities with returns
r
1
= ¯r = (1, 1, 1), r
2
= (0, 3, 3), and r
3
= (1,
3
2
,
1
2
). (13.16)
The risk premium on security 3 is zero. Further, the return on security 3 is mean independent of the
returns on securities 1 and 2. To see this, note that the expected returns on security 3 conditional
on each of the two possible realizations (1, 0) and (1, 3) of the returns on securities 1 and 2 are
the same and equal to the expected return E(r
3
) = 1. Corollary 13.5.2 implies that every strictly
risk-averse agent will invest zero in security 3.
If the return on security 3 were
r
3
= (
5
4
, 2,
1
2
) (13.17)
instead of the return specified in 13.16, then the risk premium on security 3 would be strictly
positive. Mean independence would still hold, and an optimal investment in security 3 would be
strictly positive for a strictly risk-averse agent.
2
13.6 Optimal Portfolios under Linear Risk Tolerance
Optimal portfolios have a particularly simple form for the linear risk tolerance utility functions
introduced in Section 9.9. For the negative exponential utility function, the optimal investment
in a single risky security is independent of wealth (see Theorem 12.2.1). We have already shown
that for the quadratic utility function, the optimal investment in a single risky security is linear in
wealth (see 11.14). For other LRT utility functions and when there are many risky securities, the
optimal investment in each security is linear in wealth.
13.6.1 Theorem
If an agent’s risk tolerance is linear
T (y) = α + γy, (13.18)
then the optimal investment in each risky security is given by
a
∗
j
(w) = (α + γw¯r)b
j
, for j = 2, . . . , J, (13.19)
for some b
j
which is independent of wealth and of parameter α. Hence the optimal investment in
each security is a linear function of wealth.
Proof: Let v be the agent’s von Neumann-Morgenstern utility function with linear risk toler-
ance given by 13.18. Fix wealth ˆw, and let ˆa = a
∗
( ˆw) be the associated optimal investment. We
show that the optimal investment a
∗
(w) for arbitrary wealth w satisfies
a
∗
j
(w) =
α + γw¯r
α + γ ˆw¯r
ˆa
j
(13.20)
for j ≥ 2, so that b
j
in 13.19 is given by
b
j
=
ˆa
j
α + γ ˆw¯r
. (13.21)
The first-order condition for ˆa is
E[v
0
( ˆw¯r +
J
X
j=2
ˆa
j
(r
j
− ¯r))(r
k
− ¯r)] = 0 ∀ k ≥ 2. (13.22)
128 CHAPTER 13. OPTIMAL PORTFOLIOS WITH SEVERAL RISKY SECURITIES
We consider first the case when γ 6= 0. Differentiating 9.29, marginal utility v
0
is given by
v
0
(y) = (α + γy)
−
1
γ
. (13.23)
Substituting 13.23 in 13.22 we obtain
E[(α + γ ˆw¯r + γ
J
X
j=2
ˆa
j
(r
j
− ¯r))
−
1
γ
(r
k
− ¯r)] = 0 ∀ k ≥ 2. (13.24)
Dividing both sides of 13.24 by (α + γ ˆw¯r)
−
1
γ
we obtain
E[(1 + γ
J
X
j=2
ˆa
j
α + γ ˆw¯r
(r
j
− ¯r))
−
1
γ
(r
k
− ¯r)] = 0 ∀ k ≥ 2. (13.25)
Multiplying both sides of 13.25 by (α + γw¯r)
−
1
γ
gives
E[(α + γw¯r + γ
J
X
j=2
ˆa
j
(
α + γw¯r
α + γ ˆw¯r
)(r
j
− ¯r))
−
1
γ
(r
k
− ¯r)] = 0 ∀ k ≥ 2. (13.26)
Thus a
∗
(w), as given by 13.20, satisfies the first order condition when the wealth is w, and hence
it is an optimal portfolio.
In the case when γ = 0, marginal utility is v
0
(y) = αe
−αy
. The first-order condition 13.22
becomes
E[(αe
−α( ˆw¯r+
P
j
ˆa
j
(r
j
−¯r))
)(r
k
− ¯r)] = 0 ∀ k ≥ 2. (13.27)
Multiplying both sides of 13.27 by e
−α¯r(w− ˆw)
we obtain
E[(αe
−α(w¯r+
P
j
ˆa
j
(r
j
−¯r))
)(r
k
− ¯r)] = 0 ∀ k ≥ 2, (13.28)
which indicates that ˆa is also the optimal investment at wealth w, in accordance with 13.20 when
γ = 0.
Clearly, b
j
given by 13.21 does not depend on wealth w. Further, substituting 13.21 in 13.25,
when γ 6= 0, or 13.28, when γ = 0, it can be seen that b
j
does not depend on α.
2
Theorem 13.6.1 implies that the ratio of optimal investments in risky securities is independent
of wealth for an agent with linear risk tolerance. That is,
a
∗
j
(w)
a
∗
k
(w)
=
b
j
b
k
, (13.29)
for each j, k ≥ 2 and every w. Consequently, optimal investments at different levels of wealth differ
only by the amounts of wealth invested in risky securities, and not by the compositions of the
portfolios of risky securities. In other words, the optimal investment a
∗
(w) can be written as
a
∗
(w) = (a
∗
1
(w), (α + γw¯r)b), (13.30)
where b = (b
2
, . . . , b
J
) is the wealth-independent portfolio of risky securities, and
a
∗
1
(w) = w − (α + wγ¯r)
J
X
j=2
b
j
. (13.31)
Theorem 13.6.1 also implies that portfolios b of risky securities in 13.30 are the same for all
agents with linear risk tolerance with common slope γ. This remark will be useful in the analysis
of equilibrium allocations when agents have linear risk tolerance in Chapters 15 and 16.
13.7. OPTIMAL PORTFOLIOS WITH TWO-DATE CONSUMPTION 129
13.7 Optimal Portfolios with Two-Date Consumption
Theorems 13.3.1 and 13.4.1 continue to hold when the agent’s utility function depends on date-0
consumption.
If the agent is risk-neutral with utility function
v(c
0
, c
s
) = c
0
+ δc
s
, δ > 0, (13.32)
the risk-free return equals 1/δ and all securities are priced fairly, then the agent is indifferent among
all portfolios. If the risk premium is non-zero on at least one security, or if the risk-free return is
different from 1/δ and there are no restrictions on consumption, then no optimal portfolio exists for
the risk-neutral agent. But if his consumption is restricted to be positive and there is no arbitrage,
then for that agent an optimal portfolio does exist (Theorem 3.6.5) and can be obtained by solving
a linear programming problem.
Notes
Further results on optimal portfolios with many risky securities can be found in Merton [4], see also
Cass and Stiglitz [2]. Theorem 13.5.1 is closely related to separation theorems of Ross [8]. If the
expectation E(²
k
) is zero in Theorem 13.5.1, then security returns exhibit (J −1)-fund separation.
The results on portfolio demand under linear risk tolerance are originally due to Rubinstein [9],
with a partial anticipation by Pye [7] and Cass and Stiglitz [1]. Milne [5] showed that linear risk
tolerance is a necessary condition for linear portfolio demand for arbitrary security returns. Linear
portfolio demand implies linear consumption demand. Linear consumption demands for the class
of LRT utility functions have been known in consumer theory since Gorman [3] and Pollak [6] as
linear Engel curves.
130 CHAPTER 13. OPTIMAL PORTFOLIOS WITH SEVERAL RISKY SECURITIES
Bibliography
[1] David Cass and Joseph E. Stiglitz. The structure of investor preferences and asset returns and
separability in portfolio allocation: A contribution to the pure theory of mutual funds. Journal
of Financial Economics, 2:122–160, 1970.
[2] David Cass and Joseph E. Stiglitz. Risk aversion and wealth effects on portfolios with many
assets. Review of Economic Studies, 2:331–354, 1973.
[3] W. M. Gorman. Community preference fields. Econometrica, 21:63–80, 1953.
[4] Robert C. Merton. Capital market theory and the pricing of financial securities. In Frank H.
Hahn and Benjamin M. Friedman, editors, Handbook of Monetary Economics. North-Holland,
1990.
[5] Frank Milne. Consumer preferences, linear demand functions and aggregation in competitive
asset markets. Review of Economic Studies, 46:407–417, 1979.
[6] Robert A. Pollak. Additive utility functions and linear engel curves. Review of Economic
Studies, 1971.
[7] Gordon Pye. Portfolio selection and security prices. Review of Economics and Statistics, 49:111–
115, 1967.
[8] Stephen A. Ross. Mutual fund separation in financial theory—the separating distributions.
Journal of Economic Theory, 17:254–286, 1978.
[9] Mark Rubinstein. An aggregation theorem for securities markets. Journal of Financial Eco-
nomics, 1:225–244, 1974.
131
132 BIBLIOGRAPHY
Part V
Equilibrium Prices and Allocations
133
Chapter 14
Consumption-Based Security Pricing
14.1 Introduction
The first-order conditions 1.13 for the consumption-portfolio choice problem relate prices of securi-
ties to their payoffs and to the marginal rates of substitution between the agent’s consumption at
date 0 and in each state at date 1. In equilibrium this relation holds for every agent. Consumption-
based security pricing is derived from this relation when agents’ utility functions are differentiable
and have an expected utility representation.
14.2 Risk-Free Return in Equilibrium
For an agent whose utility function has an expected utility representation E[v(c
0
, c
1
)], the marginal
utility of consumption at date 0 is
P
S
s=1
π
s
∂
0
v(c
0
, c
s
) and the marginal utility of consumption at date
1 in state s is π
s
∂
1
v(c
0
, c
s
), where ∂
0
v(c
0
, c
s
) and ∂
1
v(c
0
, c
s
) denote partial derivatives of the von
Neumann-Morgenstern utility function v. The marginal utility of date-0 consumption will be de-
noted E(∂
0
v). Further, ∂
1
v will be understood to be a random variable with realizations ∂
1
v(c
0
, c
s
).
If the von Neumann-Morgenstern utility function v is time-separable, v(c
0
, c
s
) = v
0
(c
0
) + v
1
(c
s
),
then the marginal utility of date-0 consumption is v
0
0
(c
0
) or v
0
0
for short.
Assuming that optimal consumption is interior, the first-order condition for the consumption-
portfolio choice problem is
p
j
E(∂
0
v) = E(∂
1
v x
j
) (14.1)
for each security j. Eq. 14.1 corresponds to 1.13 specialized to expected utility.
In terms of returns, 14.1 takes the form
E(∂
0
v) = E(∂
1
v r
j
). (14.2)
Assuming that a risk-free security (or portfolio) is traded, 14.2 implies that the return ¯r on this
security satisfies
¯r =
E(∂
0
v)
E(∂
1
v)
. (14.3)
If an agent is risk neutral with von Neumann-Morgenstern utility function v(c
0
, c
s
) = c
0
+ δc
s
, then
(assuming interior consumption) ¯r = δ
−1
, as was shown in Section 12.5.
14.3 Expected Returns in Equilibrium
The expectation of the product of any two random variables y and z can be written as their
covariance plus the product of their expectations:
E(yz) = cov(y, z) + E(y)E(z). (14.4)
135
136 CHAPTER 14. CONSUMPTION-BASED SECURITY PRICING
Using this result, 14.2 becomes
cov(∂
1
v, r
j
) + E(∂
1
v)E(r
j
) = E(∂
0
v). (14.5)
Solving for the expected return E(r
j
) and using 14.3, there results
E(r
j
) = ¯r −
cov(∂
1
v, r
j
)
E(∂
1
v)
= ¯r − ¯r
cov(∂
1
v, r
j
)
E(∂
0
v)
. (14.6)
Eq. 14.6 is the equation of consumption-based security pricing. It says that the risk premium
(that is, the expected excess return) on any security is proportional to the covariance of its return
with the marginal rate of substitution between consumption at date 0 and at date 1 (with a negative
constant of proportionality). Strictly, the expression ∂
1
v/E(∂
0
v) seen in 14.6 is not the marginal
rate of substitution between state-contingent consumption at date 1 and consumption at date 0
because of the absence of probabilities. Similarly, we will refer below to the term ∂
1
v as the marginal
utility of consumption despite the absence of probabilities. There is no reason to take issue with
this imprecision in the terminology, but one should be aware of it.
For a strictly risk-averse agent ∂
1
v(c
0
, c
s
) is a decreasing function of consumption at date 1.
Thus a security that has a high payoff when consumption is high and a low payoff when consumption
is low will have an expected return that is greater than the risk-free return. On the other hand, a
security that has high payoff when consumption is low and low payoff when consumption is high
will have an expected return that is less than the risk-free return. Such a security could be used
to decrease the risk of the agent’s consumption. Its relatively low return reflects a relatively high
price. A security the return on which has zero covariance with the marginal rate of substitution
will have an expected return equal to the risk-free return.
According to 14.6 the risk premium for a security depends solely on the covariance of its return
with the marginal rate of substitution between consumption at dates 0 and 1. This covariance may
be considered as a measure of the risk of a security. This measure of risk differs in several respects
from that of Chapter 10. First, it applies to returns of securities in an equilibrium. In contrast, the
analysis of Chapter 10 applies to contingent claims that are not necessarily in the asset span, and
does not require that there be an equilibrium. Second, the covariance measure gives a complete
ordering of the riskiness of returns, not just a partial ordering.
If the marginal rate of substitution is deterministic, then consumption-based security pricing
14.6 implies fair pricing. There are two cases in which the marginal rate of substitution is deter-
ministic: when the agent’s consumption is deterministic, and when the agent is risk neutral.
The equation of consumption-based security pricing holds for any portfolio return r:
E(r) = ¯r − ¯r
cov(∂
1
v, r)
E(∂
0
v)
. (14.7)
The following example illustrates the dependence of the expected return on a security on the
covariance of its return with the marginal rate of substitution.
14.3.1 Example
Consider a representative-agent economy with two equally probable states at date 1. The agent’s
endowment is 1 at date 0 and (2, 1) at date 1. His expected utility is
E[v(c
0
, c
1
)] = ln(c
0
) +
1
2
ln(c
1
) +
1
2
ln(c
2
). (14.8)
The two Arrow securities, x
1
= (1, 0), x
2
= (0, 1) and the risk-free security x
3
= (1, 1)
are traded. The agent’s marginal utility of date-0 consumption evaluated at the endowment is
14.4. VOLATILITY OF MARGINAL RATES OF SUBSTITUTION 137
E(∂
0
v) = 1. The values of ∂
1
v are 1/2 in state 1 and 1 in state 2. The prices of the securities,
calculated using 14.1, are
p
1
=
1
4
, p
2
=
1
2
, p
3
=
3
4
. (14.9)
Security returns are
r
1
=
x
1
p
1
= (4, 0), r
2
=
x
2
p
2
= (0, 2), r
3
=
x
3
p
3
= (
4
3
,
4
3
) (14.10)
and expected returns are
E(r
1
) = 2, E(r
2
) = 1, E(r
3
) ≡ ¯r =
4
3
. (14.11)
Security 1 has an expected return that is greater than the risk-free return because its payoff occurs
when consumption is least valued. Security 2 has an expected return that is less than the risk-free
return because otherwise its holder would use it to insure against low consumption at date 1.
2
14.4 Volatility of Marginal Rates of Substitution
Consumption-based security pricing provides a link between observable equilibrium security prices
and unobservable marginal rates of substitution between consumption at date 0 and at date 1.
Several inferences about marginal rates of substitution can be drawn from the characteristics of
observed equilibrium prices. An obvious inference is that if risk premia are strictly positive, agents
cannot be risk neutral. More interesting is the inference that a lower bound on the standard devi-
ation of agents’ marginal rates of substitution can be derived from expected returns and standard
deviations of returns on portfolios of securities.
Eqs. 14.2 and 14.3 imply
E[∂
1
v (r
j
− ¯r)] = 0. (14.12)
Let ρ be the correlation between ∂
1
v and r
j
− ¯r, given by
ρ =
E[∂
1
v (r
j
− ¯r)] − E(∂
1
v)E(r
j
− ¯r)
σ(∂
1
v)σ(r
j
)
, (14.13)
where σ denotes the standard deviation. substituting from 14.12 and using |ρ| ≤ 1, there results
σ(∂
1
v) ≥
E(∂
1
v)|E(r
j
) − ¯r|
σ(r
j
)
. (14.14)
Dividing both sides of 14.14 by E(∂
0
v) and using 14.3 for the risk-free return, we obtain
σ
µ
∂
1
v
E(∂
0
v)
¶
≥
|E(r
j
) − ¯r|
¯rσ(r
j
)
. (14.15)
The ratio of the risk premium to the standard deviation of return is called the Sharpe ratio.
Inequality 14.15 says that the volatility of the marginal rate of substitution between consumption
at date 0 and date 1 in equilibrium is greater than the (absolute value of the) Sharpe ratio of
each security divided by the risk-free return. Again, because of missing probabilities the expression
∂
1
v/E(∂
0
v) is not exactly the marginal rate of substitution.
Eq. 14.12—and consequently also inequality 14.15—holds for any portfolio return r, not just
for security returns. Taking the supremum over all returns (other than the risk-free return), we
obtain the following lower bound on the volatility of the marginal rate of substitution:
σ
µ
∂
1
v
E(∂
0
v)
¶
≥ sup
r
|E(r) − ¯r|
¯rσ(r)
. (14.16)
138 CHAPTER 14. CONSUMPTION-BASED SECURITY PRICING
Inequality 14.16 produces surprising results when confronted with aggregate stock market data.
On the one hand, it has been observed that the risk premium on a broad stock market index is
high relative to the volatility of the index returns. Consequently, the Sharpe ratio on that index
is high and the bound on the volatility of the marginal rate of substitution is high. On the other
hand, observed consumption volatility is low. Low volatility of consumption can be reconciled
with high volatility of the marginal rate of substitution only if agents are extremely risk averse.
To see this, recall that risk aversion is identified with curvature of the utility function, so that
high risk aversion means that the marginal utility of consumption undergoes wide variations even
when consumption has little variation. Correspondingly, low risk aversion implies that the marginal
utility of consumption differs very little for different levels of consumption. The conclusion that
agents are highly risk averse is widely regarded as puzzling since it contradicts much empirical
evidence, and also common sense, both of which appear to imply moderate risk aversion. This
anomaly is the “equity premium puzzle”.
14.5 A First Pass at the CAPM
Consumption-based security pricing can be used to derive the Capital Asset Pricing Model. For an
agent whose von Neumann-Morgenstern utility function is quadratic in date-1 consumption,
v(c
0
, c
s
) = v
0
(c
0
) − (c
s
− α)
2
, c
s
< α, (14.17)
where v
0
is some utility function of date-0 consumption, the marginal utility ∂
1
v is
∂
1
v = 2(α −c
1
). (14.18)
Eq. 14.6 becomes
E(r
j
) = ¯r +
cov(c
1
, r
j
)
α − E(c
1
)
. (14.19)
In a securities market economy the aggregate endowment is in the asset span, meaning that it is a
payoff of some portfolio of securities. This portfolio is termed the market portfolio and its return is
denoted by r
m
. Eq. 14.19 holds for returns on portfolios (see 14.7). In particular, it holds for the
market return so that
E(r
m
) = ¯r +
cov(c
1
, r
m
)
α − E(c
1
)
. (14.20)
Moving ¯r to the left-hand side of 14.19 and 14.20 and dividing the former by the latter, it follows
that
E(r
j
) − ¯r
E(r
m
) − ¯r
=
cov(c
1
, r
j
)
cov(c
1
, r
m
)
, (14.21)
where, as we assume, the market risk premium is nonzero.
In a securities market economy an agent’s equilibrium date-1 consumption is in the asset span.
If in addition the agent’s equilibrium consumption is in the span of the market return and the risk-
free return, then the agent’s date-1 consumption and the market return are perfectly correlated.
Accordingly, c
1
can be replaced by r
m
in 14.21, resulting in
E(r
j
) − ¯r
E(r
m
) − ¯r
=
cov(r
m
, r
j
)
var(r
m
)
. (14.22)
Using β
j
to denote cov(r
m
, r
j
)/var(r
m
), we obtain the equation of the security market line of the
CAPM:
E(r
j
) = ¯r + β
j
(E(r
m
) − ¯r). (14.23)
The assumption that equilibrium consumption is in the span of the market payoff and the
risk-free payoff holds trivially in a representative-agent economy, since in that case the equilibrium
14.5. A FIRST PASS AT THE CAPM 139
consumption of each agent equals the payoff of the per capita market portfolio. In the general
discussion of CAPM in Chapter 19 we dispense with the assumption of a representative agent
economy.
Notes
The bound on volatility of the marginal rate of substitution of consumption is due to Hansen and
Jagannathan [1]. The Sharpe ratio was first proposed in Sharpe [4]. For the equity premium puzzle,
see Mehra and Prescott [3] and Kocherlakota [2].
The treatment of risk premia outlined here appears to be very general, yet it conflicts with much
informal discussion of risk premia. For example, it is often recommended that the government do
all of its financing at short maturity, so as to eliminate the risk premium paid on long-maturity
debt relative to short-maturity debt. Under consumption-based security pricing, the risk premium
on long-term debt can exceed that on short-term debt only insofar as the one-period return on long-
term bonds has smaller covariance with the marginal rate of substitution than does the return on
short-term debt. Therefore if debt payments are weighted by marginal utilities, as is appropriate,
shortening the maturity of the debt will not diminish taxpayers’ cost.
140 CHAPTER 14. CONSUMPTION-BASED SECURITY PRICING
Bibliography
[1] Lars P. Hansen and Ravi Jagannathan. Implications of security market data for models of
dynamic economies. Journal of Political Economy, 99:225–262, 1991.
[2] Narayana R. Kocherlakota. The equity premium: It’s still a puzzle. Journal of Economic
Literature, XXXIV:42–71, 1996.
[3] Rajnish Mehra and Edward C. Prescott. The equity premium: A puzzle. Journal of Monetary
Economics, 15:145–161, 1985.
[4] William F. Sharpe. Mutual fund performance. Journal of Business, 39:119–138, 1966.
141
142 BIBLIOGRAPHY
Chapter 15
Complete Markets and
Pareto-Optimal Allocations of Risk
15.1 Introduction
A basic criterion of efficiency of a consumption allocation is Pareto optimality. A consumption
allocation is Pareto optimal if it is impossible to reallocate the aggregate endowment so as to make
any agent better off without making some other agent worse off. In an economy under uncertainty,
the aggregate endowment represents the economy’s aggregate consumption risk. Whether or not a
consumption allocation is optimal depends on how the aggregate consumption risk is shared among
agents.
The classical welfare theorems state that a competitive equilibrium allocation in complete mar-
kets is Pareto optimal and that each Pareto-optimal allocation is an equilibrium allocation under
an appropriate distribution of the aggregate endowment.
In this chapter we provide characterizations of Pareto-optimal allocations of risk and prove the
first welfare theorem.
15.2 Pareto-Optimal Allocations
Consumption allocation {˜c
i
} weakly Pareto dominates another allocation {c
i
} if every agent i
weakly prefers consumption plan ˜c
i
to c
i
, that is,
u
i
(˜c
i
) ≥ u
i
(c
i
). (15.1)
If {˜c
i
} weakly Pareto dominates {c
i
} and in addition at least one agent i strictly prefers ˜c
i
to c
i
(so that 15.1 holds with strict inequality for at least one i), then allocation {c
i
} Pareto dominates
allocation {˜c
i
}. A feasible consumption allocation {c
i
} is Pareto optimal if there does not exist
an alternative feasible allocation {˜c
i
} that Pareto dominates {c
i
}. Feasibility of an allocation {c
i
}
means that
I
X
i=1
c
i
≤ ¯w, (15.2)
where ¯w =
P
I
i=1
w
i
denotes the aggregate endowment.
An important representation of a Pareto-optimal allocation is as the solution to the optimization
problem of a social planner, where the social welfare function being maximized is a weighted sum
of the agents’ utilities. The planner’s problem is
max
{c
i
}
I
X
i=1
µ
i
u
i
(c
i
) (15.3)
143
144 CHAPTER 15. COMPLETE MARKETS
subject to the feasibility constraint
I
X
i=1
c
i
≤ ¯w, (15.4)
for some positive weights {µ
i
}.
Every consumption allocation that solves the planner’s problem for strictly positive weights
is Pareto optimal. Conversely, if agents’ utility functions are concave, then every Pareto-optimal
allocation is a solution to the planner’s problem for some weights µ
i
, all positive with at least
one nonzero. Further, if the Pareto-optimal allocation is interior and utility functions are strictly
increasing, then the weights are all strictly positive.
The planner’s problem has a solution if the set of feasible allocations is compact and under the
assumed continuity of utility functions. A sufficient condition for the compactness of the the set of
feasible allocations is that agents’ consumption sets be closed and bounded below.
If consumption sets are unbounded, then there may not exist a solution to the planner’s problem
for any positive weights; consequently, there may not exist a Pareto-optimal allocation.
15.2.1 Example
Suppose that there is no uncertainty and that two agents have utility functions u
1
(c
0
, c
1
) = c
0
+δ
1
c
1
and u
2
(c
0
, c
1
) = c
0
+ δ
2
c
1
. If δ
1
6= δ
2
and consumption sets are unrestricted, Pareto optimal
allocations do not exist for any specification of endowments.
2
Sufficient conditions for the existence of Pareto-optimal allocations with unbounded consump-
tions sets can be found in sources cited in the notes.
The first-order conditions for an interior solution to the planner’s problem 15.3 are
µ
i
∂
s
u
i
= ν
s
, ∀s, ∀i, (15.5)
where ν
s
is the Lagrange multiplier associated with the feasibility constraint on consumption at
date 1 in state s, or at date 0 when s = 0. Eq. 15.5 states that at a Pareto-optimal allocation
the marginal contribution to social welfare of an increase in agent i’s consumption in state s is the
same for all agents, and equals the Lagrange multiplier associated with consumption in state s.
The first-order conditions 15.5 imply that the marginal rates of substitution
∂
s
u
i
∂
0
u
i
(15.6)
at an interior Pareto-optimal allocation are the same for all agents.
15.3 Pareto-Optimal Equilibria in Complete Markets
The first welfare theorem holds when security markets are complete.
15.3.1 Theorem
If security markets are complete and agents’ utility functions are strictly increasing, then every
equilibrium consumption allocation is Pareto optimal.
Proof: Let p be a vector of equilibrium security prices and {c
i
} an equilibrium consumption
allocation in complete security markets. Using the framework of Section 2.6, the consumption plan
c
i
= (c
i
0
, c
i
1
) maximizes utility u
i
(c
0
, c
1
) subject to the budget constraints
c
0
≤ w
i
0
− qz (15.7)
15.4. COMPLETE MARKETS AND OPTIONS 145
and
c
1
≤ w
i
1
+ z, z ∈ R
S
, (15.8)
where q is the (unique) vector of state prices associated with p. Note that q is strictly positive.
Suppose that the consumption plan c = (c
0
, c
1
) satisfies budget constraints 15.7 and 15.8.
Multiplying 15.8 by q and adding the result to 15.7, we obtain
c
0
+ qc
1
≤ w
i
0
+ qw
i
1
. (15.9)
Conversely, suppose that c satisfies the budget constraint 15.9. Then c also satisfies 15.7 and 15.8
with z = c
1
− w
i
1
. Thus budget constraints 15.7 and 15.8 are equivalent to 15.9. Consequently the
optimal consumption plan c
i
maximizes utility u
i
subject to 15.9.
Suppose that allocation {c
i
} is not Pareto optimal, and let {˜c
i
} be a feasible allocation that
Pareto dominates {c
i
}. Since the utility function u
i
is strictly increasing and c
i
maximizes utility
u
i
subject to 15.9, we have
˜c
i
0
+ q˜c
i
1
≥ w
i
0
+ qw
i
1
(15.10)
for every agent i, with strict inequality for agents who are strictly better off with ˜c
i
than with c
i
.
Summing over all agents, we obtain
I
X
i=1
˜c
i
0
+
I
X
i=1
q˜c
i
1
> ¯w
0
+ q ¯w
1
, (15.11)
which contradicts the assumption that allocation {˜c
i
} is feasible.
2
The second welfare theorem also holds: if every agent’s utility function is concave and strictly
increasing, and if security markets are complete, then every Pareto-optimal allocation is an equi-
librium allocation under an appropriate distribution of the aggregate endowment.
We observed in Section 2.6 that if markets are complete, then the first-order conditions at an
(interior) equilibrium consumption allocation are
q
s
=
∂
s
u
i
∂
0
u
i
(15.12)
for all agents i and all states s. Eq. 15.12 says that marginal rates of substitution are equal to
state prices. Consequently, marginal rates of substitution must be the same for all agents in all
states. This is the requirement for a Pareto-optimal allocation.
15.4 Complete Markets and Options
The only example of securities that generate complete markets we have thus far is the set of state
claims. State claims cannot be regarded as real-world securities, but there is a close connection
between state claims and real-world options. The suggestion is that options can do what state
claims can do.
Suppose that there exists a payoff z in the asset span that takes on different values in different
states; that is, z
s
6= z
s
0
for every pair of states s, s
0
. Payoff z can be the payoff of a security or
a portfolio of securities. Suppose further that call options on payoff z with arbitrary strike prices
can be traded. A call option with strike price k matures out-of-the-money (has zero payoff) in all
states in which the payoff of z is less than or equal to k and matures in-the-money (has strictly
positive payoff) in all other states. As can easily be shown, if the payoff z and S − 1 options with
strike prices z
s
for all values of z
s
(other than the greatest) are traded, then markets are complete.
All securities other than that with payoff z and the S − 1 options are redundant.
146 CHAPTER 15. COMPLETE MARKETS
If payoff z takes on the same value in two states, then all options have equal payoffs in these
states. It follows that markets will not be complete even if options with arbitrary strike prices can
be traded. Options on payoff z do, however, span all payoffs that are state independent in any
subset of states in which payoff z is state independent.
That options can imply completeness of markets is illustrated by the following example.
15.4.1 Example
Let there be three states and let the payoff z be (1, 3, 6). The payoff of a call with strike price 3
is (0, 0, 3) and the payoff of a call with strike price 1 is (0, 2, 5). With trading in z and these two
calls, markets are clearly complete.
Now let there be four states and let the payoff z be (1, 3, 3, 6). The payoffs of z in states 2 and
3 are the same. Options must therefore have the same payoffs in those states. The same is true of
a portfolio made up of z and options on z. Thus markets are incomplete even if all options with
arbitrary strike prices are traded.
2
15.5 Pareto-Optimal Allocations under Expected Utility
We provide now a characterization of Pareto-optimal allocations of risk when agents’ utility func-
tions have expected utility representations with, as assumed throughout, common probabilities.
Suppose that each agent’s von Neumann-Morgenstern utility function v
i
is strictly concave,
strictly increasing and differentiable. Thus agents are strictly risk averse. As noted in Section
15.2, an interior Pareto-optimal allocation {c
i
} is a solution to the optimization problem 15.3 with
strictly positive weights {µ
i
}. The first-order conditions 15.5 imply that
µ
i
∂
1
v
i
(c
i
0
, c
i
s
) = µ
k
∂
1
v
k
(c
k
0
, c
k
s
) (15.13)
for any two agents i and k and any state s.
For any two states s and t such that consumption of agent i is greater in state s than in state t,
c
i
s
> c
i
t
, (15.14)
we have that
∂
1
v
i
(c
i
0
, c
i
s
) < ∂
1
v
i
(c
i
0
, c
i
t
), (15.15)
since the marginal utility ∂
1
v
i
is strictly decreasing in date-1 consumption. It follows from 15.13
and 15.15 that the same relation holds for agent k:
∂
1
v
k
(c
k
0
, c
k
s
) < ∂
1
v
k
(c
k
0
, c
k
t
), (15.16)
and hence that the consumption of agent k is higher in state s than in state t,
c
k
s
> c
k
t
. (15.17)
Thus, if one agent consumes more in state s than state t, all other agents do so as well.
We have demonstrated that agents’ date-1 consumption plans at an interior Pareto-optimal
allocation are strictly co-monotone, that is, c
i
s
> c
i
t
iff c
k
s
> c
k
t
for all agents i and k, and all states
s and t. Since the aggregate consumption equals the aggregate endowment, each agent’s date-1
consumption plan is strictly co-monotone with the aggregate endowment.
The argument above required the assumption that utility functions be differentiable and it
applied only to interior Pareto-optimal allocations. We now prove that a weaker form of co-
monotonicity holds for all Pareto-optimal allocations and without the assumption of differentiability
of utility functions. This proof draws on the concept of greater risk, as defined in Chapter 10.
We say that agents’ date-1 consumption plans {c
i
1
} are co-monotone if c
i
s
≥ c
i
t
iff c
k
s
≥ c
k
t
for all
agents i and k, and all states s and t.
15.5. PARETO-OPTIMAL ALLOCATIONS UNDER EXPECTED UTILITY 147
15.5.1 Theorem
If all agents are strictly risk averse, then at every Pareto-optimal allocation their date-1 consumption
plans are co-monotone.
Proof: To simplify notation, we assume that no agent values date-0 consumption. Suppose by
contradiction that the consumption plans at a Pareto-optimal allocation {c
i
} are not co-monotone.
Then there exist states s and t and agents i and k such that
c
i
s
< c
i
t
and c
k
s
> c
k
t
. (15.18)
Define the consumption plan ˜c
i
by
˜c
i
s
= ˜c
i
t
= E(c
i
|{s, t}), (15.19)
and ˜c
i
s
0
= c
i
s
0
for every s
0
6= s, t. Consumption plan ˜c
i
differs from c
i
in that the consumptions in
states s and t are replaced by their conditional expectation. Define the consumption plan ˜c
k
for
agent k just as for agent i in 15.19. Let
²
i
= c
i
− ˜c
i
and ²
k
= c
k
− ˜c
k
. (15.20)
Since ²
k
and ²
i
are nonzero only in two states and have zero expectation, they must be collinear,
that is
²
k
= −λ²
i
, (15.21)
where, as it follows from 15.18, λ > 0.
Suppose first that λ ≥ 1. We show that transferring ²
i
from agent i to agent k makes both
better off. By construction, ²
i
is mean-independent of ˜c
i
. Similarly, ²
k
, and hence −²
i
, is mean-
independent of ˜c
k
. Taking ²
i
away from agent i leaves him with consumption plan ˜c
i
. Since
c
i
= ˜c
i
+ ²
i
, consumption plan c
i
is more risky than ˜c
i
and agent i is better off after the transfer.
Giving ²
i
to agent k leaves him with consumption plan c
k
+²
i
= ˜c
k
+(λ−1)(−²
i
). Since 0 ≤ λ−1 < λ
and ˜c
k
+ λ(−²
i
) = c
k
, consumption plan c
k
is more risky than c
k
+ ²
i
(see Proposition 10.5.5) and
agent k is also better off after the transfer.
If λ < 1 then instead of transferring ²
i
from agent i to agent k, we transfer ²
k
from agent k
to agent i, thereby making both better off. That these transfers are possible contradicts Pareto-
optimality of the allocation {c
i
}.
2
It follows from Theorem 15.5.1 that if the aggregate date-1 endowment is constant for a subset of
states, then at each Pareto-optimal allocation every agent’s date-1 consumption is state independent
for that subset of states.
15.5.2 Corollary
If agents are strictly risk averse and the aggregate date-1 endowment is state independent for a
subset of states, then each agent’s date-1 consumption at every Pareto-optimal allocation is state
independent for that subset of states.
If the aggregate date-1 endowment is state-independent for all states (risk-free), then we say
that there is no aggregate risk in the economy. Individual endowments, of course, may be risky,
but their risky components are offsetting in the aggregate. It follows from Corollary 15.5.2 that, in
a no-aggregate-risk economy, if agents are strictly risk averse then their date-1 consumption plans
at any Pareto-optimal allocation are risk free.
148 CHAPTER 15. COMPLETE MARKETS
15.6 Pareto-Optimal Allocations under Linear Risk Tolerance
A simple characterization of Pareto-optimal allocations emerges under the assumption that all
agents have linear risk tolerance (LRT utilities) with the same slope. Agents’ date-1 consumption
plans at a Pareto-optimal allocation lie in the span of two payoffs: the risk-free payoff and the
aggregate endowment.
Agents with LRT utilities are assumed to consume only at date 1, although the result also holds
when agents consume at both date 0 and date 1 and have time-separable utility functions. Each
agent’s risk tolerance is
T
i
(y) = α
i
+ γy, (15.22)
where γ is the common slope. The consumption set of agent i is given by {c ∈ R
S
: T
i
(c
s
) >
0 for every s}.
The assumption of the common slope γ implies that all agents either have negative exponential
utility (γ = 0), all have logarithmic utility (γ = 1), or all have power utility with the same exponent
(γ 6= 0, 1). This specification is restrictive, but note that agents can have different degrees of risk
aversion within the restriction, and their endowments can differ.
15.6.1 Theorem
If every agent’s risk tolerance is linear with common slope γ, then date-1 consumption plans at any
Pareto-optimal allocation lie in the span of the risk-free payoff and the aggregate endowment.
Proof: Let {c
i
} be a Pareto-optimal allocation. Then, as follows from 15.5,
µ
i
v
0i
(c
i
s
) = µ
k
v
0k
(c
k
s
) (15.23)
for any two agents i and k. Since every agent’s consumption set is open, the allocation {c
i
} is
interior and therefore the weights µ
i
and µ
k
are strictly positive. Taking logarithms of both sides
of 15.23 results in
ln(µ
i
) + ln(v
0i
(c
i
s
)) = ln(µ
k
) + ln(v
0k
(c
k
s
)). (15.24)
But
ln(v
0i
(c
i
s
)) = ln(v
0i
(¯y
i
)) −
Z
c
i
s
¯y
i
A
i
(y)dy (15.25)
for an arbitrary ¯y
i
in the domain of v
i
, where A
i
(y) = 1/T
i
(y) is the Arrow-Pratt measure of
absolute risk aversion. Thus, if use is made of 15.22 and 15.25, 15.24 can be rewritten as
ln(µ
i
) −
Z
c
i
s
¯y
i
1
(α
i
+ γy)
dy + ln(v
0i
(¯y
i
)) = ln(µ
k
) −
Z
c
k
s
¯y
k
1
(α
k
+ γy)
dy + ln(v
0k
(¯y
k
)). (15.26)
Solving for the integrals in 15.26 when γ 6= 0 and simplifying, there results
ln(µ
i
v
0i
(¯y
i
))−
1
γ
ln(α
i
+γc
i
s
)+
1
γ
ln(α
i
+γ ¯y
i
) = ln(µ
k
v
0k
(¯y
k
))−
1
γ
ln(α
k
+γc
k
s
)+
1
γ
ln(α
k
+γ ¯y
k
). (15.27)
Multiplying 15.27 by −γ, exponentiating both sides and using D
i
≡ (µ
i
v
0i
(¯y
i
))
γ
(α
i
+γ ¯y
i
) where
D
i
6= 0, we obtain
1
D
i
(α
i
+ γc
i
s
) =
1
D
k
(α
k
+ γc
k
s
). (15.28)
Then, multiplying both sides of 15.28 by D
k
, summing over k, and using
P
i
c
i
s
= ¯w
s
, there results
P
k
D
k
D
i
(α
i
+ γc
i
s
) =
X
k
α
k
+ γ ¯w
s
. (15.29)
15.6. PARETO-OPTIMAL ALLOCATIONS UNDER LINEAR RISK TOLERANCE 149
Eq. 15.29 can be solved for
c
i
s
= F
i
¯w
s
+ G
i
(15.30)
where F
i
> 0 and G
i
are constants.
For γ = 0 (negative exponential utility), 15.27 is replaced by
ln(µ
i
v
0i
(¯y
i
)) −
1
α
i
c
i
s
+
1
α
i
¯y
i
= ln(µ
k
v
0k
(¯y
k
)) −
1
α
k
c
k
s
+
1
α
k
¯y
k
. (15.31)
Like 15.27, 15.31 leads to the conclusion 15.30 that the date-1 consumption plan of every agent i
lies in the span of the aggregate endowment and the risk-free payoff.
2
The fact that all Pareto-optimal consumption plans lie in the span of the risk-free payoff and
the aggregate endowment is known as two-fund spanning. The social planner’s problem 15.3 can
be simplified to the planner’s assigning to agents claims on two mutual funds: one consists of the
risk-free payoff and the other is a claim on the aggregate endowment.
Notes
The first welfare theorem 15.3.1 for complete security markets is due to Arrow [1]. The assumption
of strict monotonicity is stronger than necessary; all that is needed is nonsatiation. We used strict
monotonicity because we have not introduced nonsatiation. A modern statement of the welfare
theorems with no uncertainty can be found in Debreu [3].
The characterization of Pareto-optimal allocations as solutions to the optimization problem
15.3 of a social planner can be found in Mas-Colell, Whinston and Green [4]. Sufficient conditions
for the existence of Pareto-optimal allocations with unbounded consumptions sets can be found in
Page and Wooders [5].
The analysis of Section 15.4 is based on Ross [7].
The discussion of Pareto-optimal allocations when agents have LRT utilities follows Pye [6],
Rubinstein [8], Borch [2] and Wilson [9].
150 CHAPTER 15. COMPLETE MARKETS
Bibliography
[1] Kenneth J. Arrow. The role of securities in the optimal allocation of risk bearing. Review of
Economic Studies, pages 91–96, 1964.
[2] Karl Borch. General equilibrium in the economics of uncertainty. In Karl Borch and Jan
Mossin, editors, Proceedings of a Conference Held by the International Economic Association.
MacMillan and St. Martin’s Press, 1968.
[3] Gerard Debreu. Theory of Value. Wiley, New York, 1959.
[4] Andreu Mas-Colell, Michael D. Whinston, and Jerry Green. Microeconomic Theory. Oxford
University Press, New York, 1995.
[5] Frank H. Page and Myrna Holtz Wooders. A necessary and sufficient condition for the com-
pactness of individually rational and feasible outcomes and the existence of an equilibrium.
Economic Letters, 52:153–162, 1996.
[6] Gordon Pye. Portfolio selection and security prices. Review of Economics and Statistics, 49:111–
115, 1967.
[7] Stephen A. Ross. Options and efficiency. Quarterly Journal of Economics, 90:75–89, 1976.
[8] Mark Rubinstein. An aggregation theorem for securities markets. Journal of Financial Eco-
nomics, 1:225–244, 1974.
[9] Robert Wilson. The theory of syndicates. Econometrica, 36:119–131, 1968.
151
152 BIBLIOGRAPHY
Chapter 16
Optimality in Incomplete Security
Markets
16.1 Introduction
If security markets are incomplete, equilibrium consumption allocations are in general not Pareto
optimal. Agents generally cannot implement the trades required to attain a Pareto-optimal alloca-
tion. Equilibrium consumption allocations are, however, optimal in a restricted sense. If realloca-
tions are constrained to those that are attainable through security markets, then it is impossible to
reallocate the aggregate endowment so as to make any agent better off without making some other
agent worse off. We introduce and discuss the concept of constrained optimality in this chapter.
There are particular preferences, endowments and security payoffs for which equilibrium con-
sumption allocations are Pareto optimal despite markets being incomplete. Those preferences,
endowments, and payoffs are also discussed in this chapter.
16.2 Constrained Optimality
A consumption allocation {c
i
} is attainable through security markets if the net trade c
i
1
−w
i
1
lies in
the asset span M for every agent i. A feasible consumption allocation {c
i
} is constrained optimal if
it is attainable through security markets and if there does not exist an alternative feasible allocation
{˜c
i
}, also attainable through security markets, that Pareto dominates the allocation {c
i
}.
16.2.1 Theorem
If agents’ utility functions are strictly increasing, then every security markets equilibrium consump-
tion allocation is constrained optimal.
Proof: The proof is very similar to that of Theorem 15.3.1. Let p be a vector of equilibrium
security prices and {c
i
} be an equilibrium consumption allocation. It follows that consumption
plan c
i
of agent i maximizes utility u
i
subject to the constraints
c
0
≤ w
i
0
− qz, (16.1)
c
1
≤ w
i
1
+ z, z ∈ M, (16.2)
where q is any of the vectors of strictly positive state prices associated with security prices p.
Since u
i
is strictly increasing, the optimal consumption plan c
i
satisfies the budget constraints with
equality. Therefore c
i
1
− w
i
1
∈ M.
Suppose now that {c
i
} is not constrained optimal. Then there exists a feasible allocation {˜c
i
}
that Pareto dominates {c
i
} and is attainable through security markets, that is, ˜c
i
1
− w
i
1
∈ M for
153
154 CHAPTER 16. OPTIMALITY IN INCOMPLETE SECURITY MARKETS
every i. Setting z
i
= ˜c
i
1
− w
i
1
, consumption plan ˜c
i
1
satisfies date-1 budget constraint 16.2. Since
u
i
(˜c
i
) ≥ u
i
(c
i
) and u
i
is strictly increasing, we have
˜c
i
0
≥ w
i
0
− q(˜c
i
1
− w
i
1
) (16.3)
for every agent i, with strict inequality for at least one agent. Summing 16.3 over all i, we obtain
a contradiction to the assumption that {˜c
i
} is a feasible allocation.
2
16.3 Effectively Complete Markets
If security markets are complete, then every allocation is attainable through security markets.
Clearly then, constrained optimal allocations are Pareto optimal. In particular, security markets
equilibrium allocations are Pareto optimal. We show in this section that a weaker sufficient condi-
tion for constrained optimal allocations to be Pareto optimal is that Pareto-optimal allocations be
attainable through security markets.
Security markets are effectively complete if every Pareto-optimal allocation is attainable through
security markets.
16.3.1 Theorem
If security markets are effectively complete and if for every feasible allocation there exists a Pareto-
optimal allocation that weakly Pareto dominates that allocation, then every constrained optimal
allocation is Pareto optimal.
Proof: Let {c
i
} be a constrained optimal allocation. By assumption, there exists a Pareto-
optimal allocation {˜c
i
} that weakly Pareto dominates allocation {c
i
}. Because markets are effec-
tively complete, the allocation {˜c
i
} can be obtained through security markets. If {c
i
} is not Pareto
optimal, then {˜c
i
} (strictly) Pareto dominates {c
i
}. This contradicts the constrained Pareto opti-
mality of {c
i
}.
2
A sufficient condition for the existence of a Pareto-optimal allocation that weakly dominates an
arbitrary feasible allocation is that consumption sets be bounded below and closed (an alternative
sufficient condition will be given in Section 16.7).
16.3.2 Proposition
If agents’ consumption sets are bounded below and closed, then for every feasible allocation there
exists a Pareto-optimal allocation that weakly Pareto dominates that allocation.
Proof: Let {c
i
} be a feasible allocation and suppose that it is not Pareto optimal. Since
utility functions are continuous, the set of feasible allocations that weakly Pareto dominate {c
i
} is
a closed subset of the set of all feasible allocations. The latter set is compact since consumption
sets are bounded below and closed (Section 15.2). Therefore the set of feasible allocations that
weakly Pareto dominate {c
i
} is compact. Maximizing the social welfare function 15.3 with strictly
positive weights over this set generates the required Pareto-optimal allocation.
2
The most important instances of effectively complete markets are to be found in security markets
economies (that is, when agents’ endowments lie in the asset span). Markets are effectively complete
in a security markets economy iff agents’ date-1 consumption plans at any Pareto-optimal allocation
lie in the asset span. Thus if markets are effectively complete for one allocation of endowments
that lie in the asset span, then they are effectively complete for all endowment allocations in the
asset span.
16.4. EQUILIBRIA IN EFFECTIVELY COMPLETE MARKETS 155
16.4 Equilibria in Effectively Complete Markets
Combining Theorems 16.2.1 and 16.3.1 we obtain the first welfare theorem for effectively complete
security markets.
16.4.1 Theorem
If agents’ utility functions are strictly increasing and if the assumption of Theorem 16.3.1 is satis-
fied, then every equilibrium consumption allocation in effectively complete security markets is Pareto
optimal.
It is natural to inquire whether equilibrium consumption allocations in effectively complete
markets are the same as the equilibrium allocations that would result if security markets were
complete.
Even though there are many distinct sets of security payoffs that generate complete markets,
equilibria under complete security markets can be identified by a consumption allocation and a
vector of state prices without any reference to a particular set of securities. Equilibrium prices
of a particular set of securities can be obtained using the usual relation between state prices and
security prices. The existence of the corresponding equilibrium portfolio allocation follows from
the feasibility of the equilibrium consumption allocation, as noted in Section 1.7. When comparing
equilibrium allocations in effectively complete security markets and complete security markets we
will not specify a particular set of securities generating complete markets but rather specify an
equilibrium in complete markets by state prices and a consumption allocation. An equilibrium
in effectively complete security markets will be specified by security prices and a consumption
allocation.
16.4.2 Theorem
Suppose that security markets are effectively complete. If a vector of state prices q and a consump-
tion allocation {c
i
} are a complete markets equilibrium, then security prices given by
p
j
= qx
j
, ∀j, (16.4)
and allocation {c
i
} are a security markets equilibrium.
Proof: The vector q is a vector of state prices associated with security prices defined by 16.4.
It follows from the representation 16.1 – 16.2 of the budget constraints in security markets that the
set of budget feasible consumption plans in security markets at prices p is a subset of the budget
set in complete markets at state prices q.
By the first welfare theorem 15.3.1, consumption allocation {c
i
} is Pareto optimal. Since security
markets are effectively complete, allocation {c
i
} is attainable through security markets, that is, the
net trade c
i
1
− w
i
1
lies in the asset span of security markets for every agent i. Therefore, the
consumption plan c
i
lies in the set of budget feasible consumption plans in security markets and
hence it remains optimal. Consequently, allocation {c
i
} is an equilibrium allocation in security
markets.
2
A partial converse to Theorem 16.4.2 holds if agents’ utility functions are differentiable.
16.4.3 Theorem
Suppose that security markets are effectively complete, agents’ utility functions are strictly increas-
ing and quasi-concave, and the assumption of Theorem 16.3.1 is satisfied. If a vector of security
156 CHAPTER 16. OPTIMALITY IN INCOMPLETE SECURITY MARKETS
prices p and a consumption allocation {c
i
} are a security markets equilibrium such that {c
i
} is
interior, then state prices given by
q
s
=
∂
s
u
i
∂
0
u
i
, ∀s, (16.5)
and the allocation {c
i
} are a complete markets equilibrium.
Proof: It follows from Theorems 16.2.1 and 16.3.1 that the security markets equilibrium
allocation {c
i
} is Pareto optimal. Since it is interior, the marginal rates of substitution in 16.5 are
the same for all agents. Setting the state prices equal to the marginal rates of substitution implies
that the first-order conditions for the consumption choice in complete markets are satisfied for each
agent at the allocation {c
i
}. Since utility functions are quasi-concave, the first-order conditions are
sufficient and the allocation {c
i
} and state price vector q are a complete markets equilibrium.
2
This result provides the rationale for the term “effectively complete markets”: if the condi-
tions of the theorem are satisfied, addition of the missing markets will not substantively change
equilibrium plans or security prices.
The need for interiority of the equilibrium allocation in Theorem 16.4.3 is illustrated by the
following example.
16.4.4 Example
Suppose that there are two states and a single security with payoff x = (1, −1). There are two
agents with utility functions
u
1
(c
0
, c
1
, c
2
) = c
0
+ 2c
1
, and u
2
(c
0
, c
1
, c
2
) = c
0
+ c
2
, (16.6)
and endowments w
1
= (2, 0, 1) and w
2
= (2, 1, 0). Consumption at each state and date is restricted
to being positive.
Pareto-optimal allocations are of the form c
1
= (a, 1, 0) and c
2
= (4 − a, 0, 1) where 0 ≤ a ≤ 4.
Clearly, markets are effectively complete.
To find all security markets equilibria, we derive the two agents’ optimal holdings of the security
as functions of its price p. Agent 1’s optimal holding is 1 for any price p < 2 and 0 for any p > 2.
At p = 2 any holding greater than or equal to 0 and less than or equal to 1 is optimal for agent
1. Agent 2’s optimal holding is 0 for any p < −1, it is −1 (short-sale) for any p > −1, and any
value greater than or equal to −1 and less than or equal to 0 at p = −1. The security market
clears at any price p such that −1 ≤ p ≤ 2. The associated equilibrium consumption allocations
are (2 − p, 1, 0) for agent 1 and (2 + p, 0, 1) for agent 2. There is a continuum of equilibria and all
equilibrium allocations are Pareto optimal.
Now consider complete markets. At state prices q
1
= 2 and q
2
= 1 consumption plan (1, 1, 0)
for agent 1 and consumption plan (3, 0, 1) for agent 2 maximize their respective utilities subject to
the budget constraints. Note that agent 1’s marginal rate of substitution between consumption at
date 0 and in state 1 equals q
1
, since his consumption at date 0 and in state 1 is interior. Agent
2’s marginal rate of substitution between consumption at date 0 and in state 2 equals q
2
. Since
markets clear, we have an equilibrium. It is easy to verify that there are no other complete markets
equilibria.
The set of equilibrium allocations under complete markets is thus a proper subset of the set of
equilibrium allocations in security markets.
2
16.5. EFFECTIVELY COMPLETE MARKETS WITH NO AGGREGATE RISK 157
16.5 Effectively Complete Markets with No Aggregate Risk
In the rest of this chapter we study examples of effectively complete markets. In all these examples
agents’ preferences are assumed to have expected utility representations with strictly increasing
von Neumann-Morgenstern utility functions.
The first example arises when there is no aggregate risk, agents are strictly risk averse and their
date-1 endowments lie in the asset span. We refer to such economy as a security markets economy
with no aggregate risk.
In a security markets economy with no aggregate risk agents’ date-1 consumption plans at any
Pareto-optimal allocation are risk free (Corollary 15.5.2). Since the risk-free payoff lies in the asset
span, these consumption plans lie in the asset span and markets are effectively complete. If agents’
consumptions are restricted to being positive (so that consumption sets are closed and bounded
below), then equilibrium allocations are Pareto optimal (Theorem 16.4.1 and Proposition 16.3.2)
and hence risk free. Further, interior equilibrium allocations are the same as with complete markets
(Theorems 16.4.2 and 16.4.3). In an interior equilibrium (assuming that agents’ utility functions
are differentiable) securities are priced fairly:
E(r
j
) = ¯r ∀j, (16.7)
see Theorem 13.4.1. If date-0 consumption does not enter agents’ utility functions, then equilibrium
consumption plans equal the expectations of endowments E(w
i
).
16.5.1 Example
There are three states and two securities with payoffs
x
1
= (1, 1, 1) and x
2
= (1, 0, 0). (16.8)
There are two agents whose preferences depend only on date-1 consumption and have an expected
utility representation with strictly increasing and differentiable von Neumann-Morgenstern utility
functions and common probabilities (1/4, 1/2, 1/4). Both agents are strictly risk averse. Their
respective endowments are w
1
= (0, 1, 1) and w
2
= (1, 0, 0).
Since each agent’s endowment lies in the asset span and there is no aggregate risk, markets are
effectively complete. In equilibrium securities must be priced fairly. Setting p
1
= 1, which yields
¯r = 1, we obtain p
2
= E(x
2
)/¯r = 1/4. The equilibrium consumption plans of both agents are
risk free and equal to the expectations of their endowments. They are c
1
= (3/4, 3/4, 3/4) and
c
2
= (1/4, 1/4, 1/4).
Note that no use was made of any particular functional form of the utility functions in computing
the equilibrium.
2
16.6 Effectively Complete Markets with Options
The second example arises when all options on the aggregate endowment lie in the asset span,
agents are strictly risk averse and their date-1 endowments lie in the asset span. We refer to such
economy as a security markets economy with options on the market payoff since the aggregate
endowment is the market payoff.
In a security markets economy with options on the market payoff agents’ date-1 consumption
plans at any Pareto-optimal allocation are state independent in every subset of states in which the
aggregate endowment is state independent (Corollary 15.5.2). Such consumption plans lie in the
span of options on the market payoff and hence markets are effectively complete. If consumption
is restricted to being positive, then all equilibrium allocations are Pareto optimal (Theorem 16.4.1
158 CHAPTER 16. OPTIMALITY IN INCOMPLETE SECURITY MARKETS
and Proposition 16.3.2). Every complete markets equilibrium allocation is an equilibrium allocation
in security markets with options (Theorem 16.4.2), and interior equilibrium allocations in security
markets with options are the same as with complete markets (Theorem 16.4.3).
Note that if the market payoff is different in every state, then as observed in Section 15.4,
markets are complete in a security markets economy with options on the market payoff. Otherwise,
if the market payoff takes the same value in two or more states, markets are effectively complete
but not complete.
16.7 Effectively Complete Markets with Linear Risk Tolerance
The third example arises when agents have linear risk tolerance (LRT utilities) with common slope
and the risk-free claim and agents’ endowments lie in the asset span. We refer to such economy
as a security markets economy with LRT utilities. We assume that date-0 consumption does not
enter agents’ utility functions.
In a security markets economy with LRT utilities agents’ consumption plans at any Pareto-
optimal allocation lie in the span of the risk-free payoff and the aggregate endowment (Theorem
15.6.1). Therefore they lie in the asset span and markets are effectively complete. Theorem 16.4.2
implies that every complete markets equilibrium allocation is a security markets equilibrium allo-
cation. To apply Theorem 16.4.3 implying the converse, we need to show that for every feasible
allocation in security markets economy with LRT utilities there exists a Pareto-optimal allocation
that weakly Pareto dominates that allocation. Proposition 16.3.2 cannot be applied because con-
sumption sets of agents with LRT utilities (as specified in Section 15.6) are either not closed or
unbounded below. We recall that the consumption set of an agent with linear risk tolerance of the
form T (y) = α + γy is {c ∈ R
S
: α + γc
s
> 0, for every s} (see Section 9.9).
As an inspection of the proof of Theorem 16.4.3 reveals, it suffices to show that for every
individually rational allocation (that is, every feasible allocation that weakly Pareto dominates the
initial endowment allocation) there exists a Pareto-optimal allocation that weakly Pareto dominates
that allocation. In the following proposition we show that a security markets economy with LRT
utilities has this property. For LRT utilities with strictly negative slope of risk tolerance we impose
an additional condition that assures that individually rational allocations are bounded away from
the boundaries of consumption sets. When the slope γ of risk tolerance is strictly negative, the
consumption sets are bounded above and unbounded below.
16.7.1 Proposition
Suppose that each agent’s risk tolerance is linear with common slope γ. For γ < 0 assume that
there exists ² > 0 such that α
i
+ γc
i
s
≥ ² for every individually rational allocation {c
i
}, every i
and s. Then for every individually rational allocation there exists a Pareto-optimal allocation that
weakly Pareto dominates that allocation.
Proof: Let {c
i
} be an individually rational allocation and let A denote the set of allocations
that weakly Pareto dominate allocation {c
i
}. Thus
A = {(˜c
1
, . . . , ˜c
I
) ∈ R
SI
:
X
i
˜c
i
≤ ¯w, ˜c
i
∈ C
i
, E[v
i
(˜c
i
)] ≥ E[v
i
(c
i
)]}, (16.9)
where C
i
= {c ∈ R
S
: α
i
+ γc
s
> 0, for every s}.
With exception of γ = 1 (logarithmic utility), all LRT utility functions are well defined on the
boundary of the set C
i
. Assuming first (pending a separate discussion below) that γ 6= 1, we define
the set
¯
A in the same way as A in 16.9 replacing C
i
by its closure
¯
C
i
= {c ∈ R
S
: α
i
+ γc
s
≥
0, for every s}. Clearly,
¯
A is the closure of A and hence is a closed set. It is also nonempty and
convex.
16.7. EFFECTIVELY COMPLETE MARKETS WITH LINEAR RISK TOLERANCE 159
Consider the problem of maximizing the social welfare function 15.3 (with strictly positive
weights) over all allocations in
¯
A. If
¯
A is compact, then that problem has a solution. We show that
¯
A is compact.
A basic criterion for compactness of a closed and convex set is that its only direction of recession
(or asymptotic direction) is the zero vector. A vector z is a direction of recession of a convex set
Y ∈ R
n
if y
0
+ λz ∈ Y for every y
0
∈ Y and λ ≥ 0. It is to be noted that convexity of Y implies
that if y
0
+ λz ∈ Y for some y
0
∈ Y and every λ ≥ 0, then the same is true for all y
0
∈ Y . If the
set Y is bounded below, then z ≥ 0 for every direction of recession z of Y .
To show that the only direction of recession of
¯
A is zero, we consider two cases: when γ is
strictly positive and when it is negative. If γ > 0, then the set
¯
C
i
is bounded below for each i.
Consequently, if z = (z
1
, . . . , z
I
) ∈ R
SI
is a direction of recession of
¯
A, then z
i
≥ 0 for each i. The
feasibility constraint implies that
X
i
z
i
≤ 0, (16.10)
for every direction of recession z of
¯
A. It follows from 16.10 and z
i
≥ 0, that z = 0.
If γ ≤ 0, then the set
¯
C
i
is unbounded below, but we prove that the preferred set {˜c
i
∈
¯
C
i
:
E[v
i
(˜c
i
)] ≥ E[v
i
(c
i
)]} is bounded below. The same argument as for γ > 0 implies that the only
direction of recession of
¯
A is the zero vector.
That the preferred set is bounded below follows from the fact that the LRT utility function
with γ ≤ 0 is bounded above and unbounded below (see Section 9.9). A more precise argument is
as follows: Let ¯v
i
be the upper bound on the values that the utility function v
i
can take. Denote
E[v
i
(c
i
)] by ¯u
i
. Then
E[v
i
(˜c
i
)] ≥ ¯u
i
(16.11)
implies
π
s
v
i
(˜c
i
s
) ≥ ¯u
i
−
X
s
0
6=s
π
s
0
v
i
(˜c
i
s
0
) ≥ ¯u
i
− ¯v
i
. (16.12)
Consequently,
v
i
(˜c
i
s
) ≥ ¯u
i
− ¯v
i
. (16.13)
or
˜c
i
s
≥ (v
i
)
−1
(¯u
i
− ¯v
i
). (16.14)
The right-hand side of 16.14 (which is well defined since function v
i
is strictly increasing and
unbounded below) constitutes a lower bound on the preferred set.
Let {ˆc
i
} be a solution to the problem of maximizing the social welfare function 15.3 over the
set
¯
A. We have to show that {ˆc
i
} is a feasible allocation, that is, that {ˆc
i
} ∈ A. Consider first the
case of γ < 0. Since allocation {c
i
} is individually rational, all allocations in A are individually
rational and, by the assumption of Proposition 16.7.1, bounded away from the boundaries of sets
C
i
by ². Therefore, one can replace the set C
i
in the definition 16.9 of A by {c ∈ R
S
: α
i
+ γc
s
≥
², for every s}. It follows that A is closed and hence A =
¯
A. For γ = 0, we also have A =
¯
A
since C
i
=
¯
C
i
= R
S
. Finally, for γ > 0 the marginal utility of consumption at the boundary
of
¯
C
i
is infinity (Inada condition) implying that the allocation {ˆc
i
} that solves the social welfare
maximization problem cannot lie on the boundary of the set
¯
A, and hence it lies in A.
It remains to consider the case of logarithmic utilities, that is, γ = 1. The set C
i
is not closed
but the utility function diverges to negative infinity at the boundary of C
i
. This implies that the
preferred set {˜c
i
∈ C
i
: E[v
i
(˜c
i
)] ≥ E[v
i
(c
i
)]} is closed for each i and hence that A is closed. The
same argument as for other strictly positive values of γ implies that A is compact. The welfare
maximizing allocation is the desired Pareto-optimal allocation.
2
160 CHAPTER 16. OPTIMALITY IN INCOMPLETE SECURITY MARKETS
Since all equilibrium allocations in an economy with LRT utilities are interior, Proposition
16.7.1 and Theorems 16.4.2 and 16.4.3 imply that equilibrium allocations in security markets are
the same as complete markets equilibrium allocations.
16.8 Multi-Fund Spanning
A common feature of the above three examples of effectively complete markets is that agents’ date-1
consumption plans at each Pareto-optimal allocation lie in a low-dimensional subspace of the asset
span. These cases are usually referred to as multi-fund spanning since equilibrium consumption
plans are in the span of payoffs of relatively few portfolios (mutual funds). In an economy with
no aggregate risk each agent’s equilibrium consumption plan is risk free and we have one-fund
spanning. In the case of LRT utilities, each agent’s equilibrium consumption plan lies in the span
of the market payoff and the risk-free payoff, and we have two-fund spanning. In the case of options
on the market payoff, each agent’s equilibrium consumption plan lies in the span of options, and we
have multi-fund spanning with as many funds as the number of distinct values the market payoff
can take.
16.9 A Second Pass at the CAPM
We demonstrated in Section 14.5 that, if there exists at least one agent with quadratic utility
function and whose equilibrium consumption is in the span of the market payoff and the risk-
free payoff, then the equation of the security market line of the CAPM holds in equilibrium. In
particular, the CAPM holds in a representative-agent economy in which the representative agent
has a quadratic utility.
Consider a security markets economy with the risk-free payoff in the asset span. If all agents
have quadratic utility functions, then their risk tolerance is linear with common slope −1 and the
results of Section 16.7 imply that equilibrium consumption plans lie in the span of the market
payoff and the risk-free payoff. Consequently, the CAPM holds.
We have thus extended the CAPM to a security markets economy with a risk-free security and
with many agents with different quadratic utility functions (agents’ quadratic utility functions can
have different parameter α.) A further extension of the CAPM that dispenses with the assumptions
of the security markets economy and the presence of a risk-free security will be presented in Chapter
19.
Notes
The notion of constrained Pareto optimality was introduced by Diamond [3]. A general discussion
of the optimality of equilibrium allocations in incomplete markets (with many goods) can be found
in Geanakoplos and Polemarchakis [5]. When there are more than one good, or in the multidate
model of security markets considered in Part VII, the notion of constrained Pareto optimality is of
limited usefulness because of the endogeneity of the asset span (due to the dependence of security
payoffs on future prices). Hart [6] provided an example of an economy with incomplete markets
and two goods in which there exist two equilibrium allocations, one of which Pareto dominates the
other. Each allocation is constrained optimal with respect to its asset span. Evidently this cannot
happen when there is a single good.
Constrained optimality of a consumption allocation can be viewed as Pareto optimality of the
corresponding portfolio allocation when agents’ rank portfolios according to the utility of consump-
tion they generate. More precisely, if the utility function u
i
is strictly increasing, then one can define
the indirect utility of portfolio h and date-0 consumption c
0
by setting v
i
(c
0
, h) ≡ u
i
(c
0
, w
i
1
+ hX).
16.9. A SECOND PASS AT THE CAPM 161
A feasible allocation of portfolios and date-0 consumptions {(c
i
0
, h
i
)} is Pareto optimal if there is
no alternative feasible allocation {(c
0
i
0
, h
0
i
)} such that v
i
(c
i
0
, h
i
) ≥ v
i
(c
0
i
0
, h
0
i
) for every agent i with
strict inequality for at least one agent. An allocation {(c
i
0
, h
i
)} is Pareto optimal iff the consumption
allocation {(c
i
0
, c
i
1
)} is constrained optimal where c
i
1
= w
i
1
+ h
i
X.
The definition of effectively complete markets presented in Section 16.3 is not standard. An
alternative definition is that markets are effectively complete if every equilibrium allocation is
Pareto optimal, see Elul [4]. Theorem 16.4.1 says that every equilibrium allocation in security
markets that are effectively complete in the sense of Section 16.3 is Pareto optimal if agents’ utility
functions are strictly increasing and their consumption sets are bounded below and closed. Thus
under these assumptions on agents’ utility functions and consumption sets the alternative definition
of effectively complete markets is weaker than the definition of Section 16.3.
The analysis of efficient allocation of risk in the case of LRT utilities is due to Rubinstein [8].
The case of options on the market payoff is due to Breeden and Litzenberger [2].
A excellent exposition of the concept of direction of recession of a set can be found in Rockafellar
[7]. The result that a closed and convex set is compact if its only direction of recession is the zero
vector can also be found in Rockafellar [7]. For a characterization of directions of recession of a
preferred set of expected utility, see Bertsekas [1].
162 CHAPTER 16. OPTIMALITY IN INCOMPLETE SECURITY MARKETS
Bibliography
[1] Dimitri P. Bertsekas. Necessary and sufficient conditions for existence of an optimal portfolio.
Journal of Economic Theory, 8:235–247, 1974.
[2] Douglas T. Breeden and Robert Litzenberger. Prices of state-contingent claims implicit in
option prices. Journal of Business, 51:621–651, 1978.
[3] Peter Diamond. The role of a stock market in a general equilibrium model with technological
uncertainty. American Economic Review, 48:759–776, 1967.
[4] Ronel Elul. Effectively complete equilibria—a note. Journal of Mathematical Economics,
32:113–119, 1999.
[5] John Geanakoplos and Heraklis Polemarchakis. Existence, regularity, and constrained subop-
timality of competitive allocations when the asset markets is incomplete. In Walter Heller
and David Starrett, editors, Essays in Honor of Kenneth J. Arrow, Volume III. Cambridge
University Press, 1986.
[6] Oliver Hart. On the optimality of equilibrium when the market structure is incomplete. 1975,
11:418–443, Journal of Economic Theory.
[7] R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, Princeton, NJ, 1970.
[8] Mark Rubinstein. An aggregation theorem for securities markets. Journal of Financial Eco-
nomics, 1:225–244, 1974.
163
164 BIBLIOGRAPHY
Part VI
Mean-Variance Analysis
165
Chapter 17
The Expectations and Pricing Kernels
17.1 Introduction
In Chapter 6 we showed that the payoff pricing functional—and also its extension, the valuation
functional—can be represented either by state prices or by risk-neutral probabilities. In this chapter
we derive another representation of the payoff pricing functional, the pricing kernel. The existence
of the pricing kernel is a consequence of the Riesz Representation Theorem, which says that any
linear functional on a vector space can be represented by a vector in that space.
We begin by introducing the concepts of inner product, orthogonality and orthogonal projection.
These concepts are associated with an important class of vector spaces, the Hilbert spaces, to
which the Riesz Representation Theorem applies. In the finance context, the Riesz Representation
Theorem implies that any linear functional on the asset span can be represented by a payoff. Two
linear functionals are of particular interest: the payoff pricing functional, and the expectations
functional which maps every payoff into its expectation. Their representations are the pricing
kernel and the expectations kernel, respectively.
Hilbert space methods are important for the study of the Capital Asset Pricing Model and factor
pricing in the following chapters. Our treatment of these methods here is mathematically superficial,
for our interest is in arriving quickly at results that are applicable in finance. In particular, the
finite-dimensional contingent claims space R
S
is for us the primary example of a Hilbert space.
The most important applications of Hilbert space methods come when the payoff space is infinite-
dimensional. Readers who plan to study the infinite-dimensional case are encouraged to read the
sources cited at the end of this chapter.
17.2 Hilbert Spaces and Inner Products
An inner product on a vector space H is a function from H × H to R usually indicated by a dot,
that obeys the the following properties for all x, y ∈ H and all a, b ∈ R:
• symmetry: x ·y = y · x,
• linearity: x · (ay + bz) = a (x · y) + b(x ·z),
• strict positivity: x · x > 0 when x 6= 0.
The inner product is also referred to as a scalar product or as a dot product.
The inner product defines a norm of a vector in the vector space H as
k x k ≡
q
(x · x) . (17.1)
The norm satisfies the following important properties for all x, y ∈ H:
167
168 CHAPTER 17. THE EXPECTATIONS AND PRICING KERNELS
• triangle inequality: k x + y k ≤ k x k + k y k,
• Cauchy-Schwarz inequality: |x · y| ≤ k x k k y k.
Further, the norm defines the convergence of a sequence of vectors in H, and therefore the continuity
of functionals on H.
A Hilbert space is a vector space H which is equipped with an inner product and is complete
with respect to the norm induced by its inner product. In this context, completeness means that
any Cauchy sequence of elements of the vector space H converges to an element of that space.
17.3 The Expectations Inner Product
The space R
S
of state-contingent date-1 consumption plans is a Hilbert space. The most familiar
inner product in that space is the Euclidean inner product:
x · y =
X
s
x
s
y
s
. (17.2)
Another inner product, important in the derivation of the Capital Asset Pricing Model, is the
expectations inner product:
x · y = E(xy) (17.3)
where, as usual, E(xy) =
P
s
π
s
x
s
y
s
for a probability measure π on S. The norm induced by the
expectations inner product is
k x k =
q
E(x
2
) =
q
var(x) + (E(x))
2
. (17.4)
17.4 Orthogonal Vectors
Two vectors x, y ∈ H are orthogonal, denoted by x ⊥ y, iff their inner product is zero:
x ⊥ y iff x · y = 0. (17.5)
A collection of vectors {z
1
, . . . , z
n
} in a Hilbert space H is an orthogonal system if z
i
⊥ z
j
for
all i 6= j. If in addition k z
i
k = 1 for every i, then the collection {z
1
, . . . , z
n
} is an orthonormal
system. An orthonormal system is an orthonormal basis for its linear span.
17.4.1 Pythagorean Theorem
If {z
1
, . . . , z
n
} is an orthogonal system in a Hilbert space H, then
k
n
X
i=1
z
i
k
2
=
n
X
i=1
k z
i
k
2
. (17.6)
Proof: Write the left-hand side using the inner product and apply the definition of orthogo-
nality.
2
A useful implication of the Pythagorean Theorem is the following:
17.5. ORTHOGONAL PROJECTIONS 169
17.4.2 Corollary
Any orthogonal system of nonzero vectors is linearly independent.
Proof: Let {z
1
, . . . , z
n
} be an orthogonal system with z
i
6= 0 for each i. Suppose that
n
X
i=1
λ
i
z
i
= 0 (17.7)
for some λ
i
∈ R. Since {λ
1
z
1
, . . . , λ
n
z
n
} is also an orthogonal system, it follows from 17.6 and 17.7
that
n
X
i=1
λ
2
i
k z
i
k = k
n
X
i=1
λ
i
z
i
k = 0. (17.8)
This implies that λ
i
= 0 for every i and thus that the vectors z
1
, . . . , z
n
are linearly independent.
2
17.5 Orthogonal Projections
A vector x ∈ H is orthogonal to a linear subspace Z ⊂ H iff it is orthogonal to every vector in
z ∈ Z:
x ⊥ Z iff x · z = 0 ∀z ∈ Z. (17.9)
If the subspace Z is the linear span of vectors z
1
, . . . , z
n
, then a vector x is orthogonal to Z iff it
is orthogonal to every z
i
for i = 1, . . . , n. The set of all vectors orthogonal to a subspace Z is the
orthogonal complement of Z and is denoted Z
⊥
. It is a linear subspace of H.
17.5.1 Projection Theorem
For any finite-dimensional subspace Z of a Hilbert space H and any vector x ∈ H, there exist
unique vectors x
Z
∈ Z and y ∈ Z
⊥
such that x = x
Z
+ y.
1
Proof: Let {z
1
, . . . , z
n
} be an orthogonal system that spans Z, and define
x
Z
=
n
X
i=1
x · z
i
z
i
· z
i
z
i
, (17.10)
and
y = x − x
Z
. (17.11)
The vector x
Z
so defined is in Z. We have
y · z
j
= (x −
n
X
i=1
x · z
i
z
i
· z
i
z
i
) · z
j
(17.12)
= (x −
x · z
j
z
j
· z
j
z
j
) · z
j
= 0. (17.13)
Therefore y ⊥ z
j
for every j = 1, . . . , n. Hence y ∈ Z
⊥
.
To see that x
Z
is unique, suppose that x = x
Z
1
+ y
1
= x
Z
2
+ y
2
for some x
Z
1
, x
Z
2
∈ Z and
y
1
, y
2
∈ Z
⊥
. The Pythagorean Theorem implies
k y
2
k
2
= k x
Z
1
− x
Z
2
k
2
+ k y
1
k
2
, (17.14)
1
The projection theorem holds for every closed (and possibly infinite-dimensional) subspace of H. Our proof applies
only in the finite-dimensional case. In the finance applications to be discussed below only the finite-dimensional version
of the theorem is needed.
170 CHAPTER 17. THE EXPECTATIONS AND PRICING KERNELS
and
k y
1
k
2
= k x
Z
1
− x
Z
2
k
2
+ k y
2
k
2
. (17.15)
Eqs. 17.14 and 17.15 imply that
k x
Z
1
− x
Z
2
k
2
= 0 (17.16)
so, by the strict positivity of inner products, x
Z
1
= x
Z
2
.
2
If Z is a (finite-dimensional) subspace of a Hilbert space H, then Theorem 17.5.1 implies that
H can be decomposed as H = Z + Z
⊥
, with Z ∩ Z
⊥
= {0}.
Vector x
Z
of the unique decomposition of Theorem 17.5.1 is the orthogonal projection of x on
Z. If the projection is taken with respect to the expectations inner product, then the coefficients
of the representation 17.10 of the orthogonal projection are
x · z
i
z
i
· z
i
=
E(xz
i
)
E(z
2
i
)
, (17.17)
and we have
x
Z
=
n
X
i=1
E(xz
i
)
E(z
2
i
)
z
i
. (17.18)
Thus the projection with respect to the expectations inner product is the same as the linear regres-
sion. Eq. 17.18 is the equation for the predicted value of the dependent variable for given values
of the independent variables.
17.5.2 Example
In the Hilbert space R
2
with the expectations inner product given by probabilities (1/4, 3/4), let
Z = span {(1, 1)} and x = (1, 2). The orthogonal projection x
Z
is
x
Z
=
(1, 2) · (1, 1)
(1, 1) · (1, 1)
(1, 1) =
7
4
(1, 1) = (7/4, 7/4). (17.19)
2
17.6 Diagrammatic Methods in Hilbert Spaces
One of the most appealing features of Hilbert spaces is that they lend themselves well to diagram-
matic representations. To see this, consider a two-dimensional Hilbert space in which coordinates
are expressed in terms of an orthonormal basis ²
1
, ²
2
. The inner product of two vectors x and y is
given by
x · y = (x
1
²
1
+ x
2
²
2
) · (y
1
²
1
+ y
2
²
2
). (17.20)
Since ²
1
and ²
2
are orthonormal, we have
x · y = x
1
y
1
+ x
2
y
2
, (17.21)
so we can represent the Hilbert space by the Euclidean plane of ordered pairs of real numbers with
the “natural basis” (1, 0), (0, 1) and in which the inner product is the Euclidean inner product.
Therefore x and y are orthogonal if they are perpendicular, that is, if x
1
y
1
+ x
2
y
2
= 0.
In finance applications the basis vectors are state claims {e
s
}. Although these are orthogonal
under the expectations inner product, they do not constitute an orthonormal basis because they
do not have unit norm:
e
s
· e
s
= E(e
2
s
) = π
s
6= 1. (17.22)
17.7. RIESZ REPRESENTATION THEOREM 171
If we use state claims as the basis in a diagrammatic representation, then orthogonal payoffs need
not be perpendicular (unless probabilities of all states are the same). Orthogonal projections
are skewed. For instance, the orthogonal projection x
Z
= (7/4, 7/4) of vector x = (1, 2) on
Z = span {(1, 1)} in Example 17.5.2 differs from the perpendicular projection (3/2, 3/2). Of
course, it is easy to eliminate this skewness by rescaling the basis vectors.
17.7 Riesz Representation Theorem
A linear and (norm) continuous functional on a Hilbert space has a simple form; it is the inner
product with a vector in that space.
17.7.1 Theorem (Riesz-Frechet)
If F : H → R is a continuous linear functional on a Hilbert space H, then there exists a unique
vector k
f
in H such that
F (x) = k
f
· x ∀x ∈ H. (17.23)
Proof: If F is the zero functional, then we take k
f
= 0. Suppose that F is a nonzero functional.
Let N = {x ∈ H : F (x) = 0} be the null space of F and N
⊥
the orthogonal complement of N.
We have H = N + N
⊥
, and N
⊥
6= {0}.
Choose a nonzero vector z in N
⊥
. By multiplying z by a scalar we can have F (z) = 1. Any
vector x ∈ H can be written as
x = (x − F (x)z) + F (x)z. (17.24)
Note that (x − F (x)z) ∈ N. Since z ∈ N
⊥
, it follows that
z · x = z · (F (x)z). (17.25)
Now set
k
f
=
z
(z · z)
. (17.26)
Then 17.25 implies
k
f
· x =
F (x)(z · z)
z · z
= F (x), (17.27)
so that k
f
satisfies 17.23.
It remains to show that k
f
is unique. If there are k
f
and k
0
f
satisfying 17.23, then
(k
f
− k
0
f
) · x = 0 (17.28)
holds for every x ∈ H, hence (k
f
− k
0
f
) = 0.
2
The vector k
f
in the representation 17.23 is called the Riesz kernel corresponding to F .
17.8 Construction of the Riesz Kernel
Finding the Riesz kernel for a linear functional on the Hilbert space R
S
with the Euclidean inner
product is easy. The kernel is obtained from k
fs
= F (e
s
), which implies by linearity that F(x) =
P
s
k
fs
x
s
. Obtaining the kernel with for the expectations inner product is equally easy. The
functional F can first be written F (x) =
P
s
k
s
x
s
. Then k
fs
= k
s
/π
s
gives the desired representation
F (x) =
P
s
π
s
k
fs
x
s
= E(k
f
x).
Any complete subspace of a Hilbert space is a Hilbert space in its own right under the same
inner product. The Riesz Representation Theorem can therefore be applied to linear functionals
172 CHAPTER 17. THE EXPECTATIONS AND PRICING KERNELS
on complete subspaces of a Hilbert space. Thus if Z is a complete subspace of a Hilbert space H
and F is a continuous linear functional on Z, then there exists a unique kernel k
f
in Z such that
F (z) = k
f
· z holds for every z ∈ Z.
If the subspace Z is a linear span of a finite collection of vectors {z
1
, . . . , z
n
}, then kernel k
f
of
a linear functional F : Z → R can be constructed as follows.
Let
w
i
= F (z
i
), (17.29)
for i = 1, . . . , n be the values of F on the basis vectors of Z. The kernel k
f
has to satisfy n equations
w
i
= k
f
· z
i
i = 1, . . . , n. (17.30)
Since k
f
∈ Z, we have k
f
=
P
n
j=1
a
j
z
j
. Substituting in 17.30, we obtain n equations
w
i
=
n
X
j=1
a
j
z
j
· z
i
i = 1, . . . , n (17.31)
with n unknowns a
j
which can be solved using standard methods.
The following example illustrates the above construction:
17.8.1 Example
Let Z = span {(1, 1)} ⊂ R
2
, and let the inner product be the expectations inner product given by
probabilities (1/4, 3/4). Let F : Z → R be given by
F (z) = 2z
1
, (17.32)
for z = (z
1
, z
2
) ∈ Z.
Vector (1, 1) constitutes a basis of Z. The kernel k
f
has to satisfy k
f
= a(1, 1) for some scalar
a. Since F (1, 1) = 2 we can solve for a from the single equation
2 = a(1, 1) ·(1, 1) = a(1/4 + 3/4). (17.33)
Thus a = 2 and
k
f
= (2, 2). (17.34)
2
17.9 The Expectations Kernel
The asset span is a subspace of the Hilbert space R
S
with the expectations inner product, and
hence is a Hilbert space in its own right. Consequently the Riesz Representation Theorem applies
to linear functionals defined on the asset span. Two linear functionals on the asset span M are
of particular interest: the expectations functional, discussed in this section, and the payoff pricing
functional, discussed in Section 17.10. The probability measure π defining the expectations inner
product is taken to be agents’ subjective probability measure. If agents’ preferences have expected
utility representations, then π is the probability measure (assumed common to all agents) of the
expected utility.
The expectations functional E maps every payoff z ∈ M into its expectation E(z). The Riesz
kernel k
e
associated with the expectations functional is the unique payoff that satisfies
E(z) = E(k
e
z), ∀z ∈ M. (17.35)
17.10. THE PRICING KERNEL 173
We emphasize that 17.35 is valid only when z is in the asset span and need not be valid for
contingent claims outside the asset span. The expectations kernel can be constructed using the
method of Section 17.8 with security payoffs x
1
, . . . , x
n
as the basis of M.
If the risk-free payoff is in the asset span M, then the expectations kernel k
e
is risk-free and
equal to one in every state. If the risk-free payoff is not in the asset span, then the kernel k
e
is the
orthogonal projection of the risk-free payoff on M. To see this, observe that
E[(e − k
e
)z] = 0 (17.36)
for every z in M, where e denotes the payoff of one in every state. Therefore e − k
e
is orthogonal
to M. Since e = (e − k
e
) + k
e
, it follows that k
e
is the projection of e onto M.
17.9.1 Example
Assume that there are three states and two securities with payoffs x
1
= (1, 1, 0) and x
2
= (0, 1, 1).
The probability of each state is 1/3.
To find the expectations kernel we consider the following two equations for expected payoffs:
2
3
= E(k
e
x
1
) (17.37)
and
2
3
= E(k
e
x
2
). (17.38)
Since the expectations kernel k
e
lies in the asset span, we have
k
e
= h
1
x
1
+ h
2
x
2
= (h
1
, h
1
+ h
2
, h
2
) (17.39)
for some portfolio (h
1
, h
2
). Substituting 17.39 in 17.37 and 17.38 we obtain
2
3
=
1
3
h
1
+
1
3
(h
1
+ h
2
), (17.40)
and
2
3
=
1
3
(h
1
+ h
2
) +
1
3
h
2
. (17.41)
The solution is h
1
= h
2
= 2/3 which gives
k
e
=
µ
2
3
,
4
3
,
2
3
¶
. (17.42)
Note that k
e
is not the risk-free payoff since the the risk-free payoff is not in the asset span.
2
17.10 The Pricing Kernel
The Riesz kernel associated with the payoff pricing functional q on the asset span M is the pricing
kernel k
q
. It is the unique payoff in M that satisfies
q(z) = E(k
q
z), ∀z ∈ M. (17.43)
The pricing kernel can be constructed using the method of Section 17.8 with security payoffs
x
1
, . . . , x
n
as the basis of M.
The expectation E(k
q
z) is well-defined for contingent claims z not in the asset span, but it
does not in general define a positive valuation functional on R
S
. This is so because the pricing
174 CHAPTER 17. THE EXPECTATIONS AND PRICING KERNELS
kernel need not be positive (or strictly positive) even if there is no strong arbitrage (arbitrage).
For example, if there is no portfolio with strictly positive payoff, then the pricing kernel cannot be
strictly positive.
If there is no arbitrage (strong arbitrage), then there exists a strictly positive (positive) state
price vector q = (q
1
, . . . , q
S
) such that
q(z) =
X
s
q
s
z
s
(17.44)
for every z ∈ M. Consider the vector of state prices rescaled by the probabilities of states, denoted
by q/π = (q
1
/π
1
, . . . , q
s
/π
S
). We can rewrite 17.44 as
q(z) = E(
q
π
z). (17.45)
Eqs. 17.43 and 17.45 imply that
E[(
q
π
− k
q
)z] = 0 (17.46)
for every z ∈ M, and hence that q/π − k
q
is orthogonal to M. Since q/π = (q/π − k
q
) + k
q
, it
follows that the pricing kernel k
q
is the projection of q/π on M.
The pricing kernel is unique regardless of whether markets are complete or incomplete. If
markets are incomplete, then there exist multiple state price vectors. When rescaled by probabilities
all these vectors have the same projection on the asset span, and that projection is the pricing kernel
k
q
. If markets are complete, then there exists a unique state price vector q and the pricing kernel
k
q
equals q/π.
If q is an equilibrium payoff pricing functional, then
q(z) = E
µ
∂
1
v
∂
0
v
z
¶
(17.47)
for every z ∈ M (see 14.1), where ∂
1
v/∂
0
v is the vector of marginal rates of substitution of an
agent whose utility function has an expected utility representation E[v(c)] and whose equilibrium
consumption is interior. The projection of the vector ∂
1
v/∂
0
v on the asset span M equals the
pricing kernel k
q
. If markets are complete, the vector of marginal rates of substitution equals k
q
,
and this holds for all agents with interior consumption.
Substituting z = k
e
in 17.46 we obtain
E(
q
π
) = E(k
q
). (17.48)
It follows that if the state price vector q is positive and nonzero, then the expectation of the pricing
kernel is strictly positive. If the risk-free payoff is in the asset span, then
E(k
q
) = E(k
q
k
e
) =
1
ˆr
, (17.49)
which is used in the following chapter.
17.10.1 Example
In Example 17.9.1, assume that security prices are p
1
= 1, p
2
= 4/3. To find the pricing kernel, we
consider the equations for prices of securities
1 = E(k
q
x
1
) (17.50)
and
4/3 = E(k
q
x
2
). (17.51)
17.10. THE PRICING KERNEL 175
The pricing kernel k
q
lies in the asset span, so we have
k
q
= h
1
x
1
+ h
2
x
2
= (h
1
, h
1
+ h
2
, h
2
) (17.52)
for some portfolio (h
1
, h
2
). The solution is h
1
= 2/3, h
2
= 5/3, which gives
k
q
=
µ
2
3
,
7
3
,
5
3
¶
. (17.53)
2
Notes
Comprehensive treatments of the theory of Hilbert spaces can be found in Luenberger [5], Dudley
[3], and Young [6]. Hilbert space methods were introduced in financial economics by Harrison and
Kreps [4], Chamberlain [1] and Chamberlain and Rothschild [2]
In Section 17.2 we noted without discussion that a space on which an inner product has been
defined must be complete to be a Hilbert spaces. This requirement is innocuous in finite-dimensional
spaces with the Euclidean or the expectations inner product, but not in infinite-dimensional spaces.
For example, let Φ be the space of finitely nonzero sequences of real numbers, i.e., sequences with
only a finite number of nonzero terms. The expectations inner product defined by probabilities
1/2, 1/4, 1/8, ... has all the properties of Section 17.2, but the space is not complete and hence is
not a Hilbert space. To see this, consider the sequence {z
n
} of elements of Φ where z
n
is a sequence
of ones in the first n places and zeros thereafter. Sequence {z
n
} converges in the norm to (1, 1, ....)
(and hence is a Cauchy sequence), but the limit is not an element of Φ.
176 CHAPTER 17. THE EXPECTATIONS AND PRICING KERNELS
Bibliography
[1] Gary Chamberlain. Funds, factors and diversification in arbitrage pricing models. Econometrica,
51:1305–1323, 1983.
[2] Gary Chamberlain and Michael Rothschild. Arbitrage, factor structure and mean variance
analysis in large asset markets. Econometrica, 51:1281–1304, 1983.
[3] Richard M. Dudley. Real Analysis and Probability. Wadsworth and Brooks, Pacific Grove, Ca.,
1989.
[4] J. Michael Harrison and David M. Kreps. Martingales and arbitrage in multidate securities
markets. Journal of Economic Theory, 20:381–408, 1979.
[5] David G. Luenberger. Optimization by Vector Space Methods. Wiley, New York, 1969.
[6] Nicholas Young. An Introduction to Hilbert Space. Cambridge University Press, Cambridge,
1988.
177
178 BIBLIOGRAPHY
Chapter 18
The Mean-Variance Frontier Payoffs
18.1 Introduction
Despite the fact that variance does not in general provide an accurate measure of risk (see Chapter
10), the analysis of expected returns and variances of returns plays an important role in the theory
and applications of finance. It leads to identification of returns that have minimal variance for a
given expected return.
The analysis relies on Hilbert space methods developed in Chapter 17; in particular, on the
representations of the payoff pricing functional by the pricing kernel, and the expectations functional
by the expectations kernel. The returns that attain minimum variance for a given expected return
lie on a line passing through the returns on the pricing kernel and the expectations kernel. The
analysis of expected returns and variances of returns has a simple diagrammatic representation.
18.2 Mean-Variance Frontier Payoffs
A payoff is a mean-variance frontier payoff if there is no other payoff with the same price and
the same expectation, but a smaller variance. In other words, the mean-variance frontier payoffs
minimize variance subject to constraints on price and expectation.
Let E be the subspace of M spanned by the expectations kernel k
e
and the pricing kernel k
q
.
The central result of this chapter is the following:
18.2.1 Theorem
A payoff is a mean-variance frontier payoff iff it lies in the span of the expectations kernel and the
pricing kernel.
Proof: Taking the orthogonal projection (with respect to the expectations inner product) of
an arbitrary payoff z ∈ M onto E results in
z = z
E
+ ², (18.1)
with z
E
∈ E and ² ∈ E
⊥
. In particular, ² is orthogonal to both k
e
and k
q
. Therefore ² has zero
expectation and zero price, implying that z and z
E
have the same expectation and the same price.
Further, since ² is orthogonal to z
E
and E(²) = 0, it follows that cov(², z
E
) = E(²)E(z
E
) = 0.
Consequently, var(z) = var(z
E
) + var(²) and thus var(z
E
) ≤ var(z), with strict inequality if ² 6= 0.
This implies that every mean-variance frontier payoff lies in E.
For the converse, we have to show that every payoff in E is a mean-variance frontier payoff.
Suppose, to the contrary, that there exists a payoff z in E that is not a mean-variance frontier
payoff. Then there must exist another payoff z
0
with the same price and the same expectation,
but smaller variance than z. Using the argument of the first part of the proof we can assume that
179
180 CHAPTER 18. THE MEAN-VARIANCE FRONTIER PAYOFFS
z
0
∈ E. Since z and z
0
have the same price and the same expectation, we have E[k
q
(z − z
0
)] = 0
and E[k
e
(z − z
0
)] = 0. This implies that z − z
0
∈ E
⊥
. Since also z − z
0
∈ E, it follows that z = z
0
.
This is a contradiction to the assumption that z
0
has smaller variance than z.
2
If the expectations kernel and the pricing kernel are collinear, that is, k
q
= γk
e
for some γ 6= 0,
then the set of mean-variance frontier payoffs E is a line. The expectations kernel and the pricing
kernel are collinear iff all portfolios have the same expected return (equal to 1/γ). If the risk-free
payoff lies in the asset span, then k
e
and k
q
are collinear iff fair pricing holds. Under fair pricing,
that is when E(r
j
) = ¯r for every security j, the kernels are k
e
= u and k
q
= (1/¯r)u, where u is the
risk-free unit payoff.
Since the case of fair pricing has already been extensively discussed in Section 13.4 and in
Section 16.5, we are more interested in the case when k
e
and k
q
are not collinear. Then the set of
mean-variance frontier payoffs E is a plane, see Figure 18.1.
If there are only two nonredundant securities, then the asset span is a plane. Further, if the
expectations and pricing kernels are not collinear, then the asset span coincides with the set of
mean-variance frontier payoffs. Thus every payoff is a mean-variance frontier payoff if there are
two securities. Note that the number of states is irrelevant.
For brevity, “frontier payoff” is often used in place of “mean-variance frontier payoff.”
18.3 Frontier Returns
The return associated with any payoff having a nonzero price equals that payoff divided by its
price. Frontier returns are the returns on the frontier payoffs or, equivalently, frontier payoffs with
unit price.
It follows from Theorem 18.2.1 that the return r
q
on the pricing kernel and the return r
e
on
the expectations kernel are frontier returns. They are
r
e
=
k
e
E(k
q
)
and r
q
=
k
q
E(k
2
q
)
, (18.2)
where the pricing kernel was used to derive the prices of k
q
and k
e
.
If the expectations kernel and the pricing kernel are collinear, then returns r
q
and r
e
are equal.
The set of frontier returns consists of the single return r
e
. If the risk-free claim lies in the asset
span, that single return equals the risk-free return ¯r.
We assume throughout the rest of this chapter that the expectations kernel and the pricing
kernel are not collinear. If k
e
and k
q
are not collinear, then the set of frontier returns is the line
passing through the return r
q
and the return r
e
, see Figure 18.2. This line can be indexed by a
single parameter λ, so that
r
λ
= r
e
+ λ(r
q
− r
e
), (18.3)
where −∞ < λ < ∞.
18.3.1 Example
Suppose that there are three equally likely states and that three securities are traded. The security
returns are
r
1
= (3, 0, 0) (18.4)
r
2
= (0, 6, 0) (18.5)
r
3
= (
6
7
,
3
7
,
9
7
). (18.6)
We wish to know which, if any, of these returns are on the mean-variance frontier.
18.3. FRONTIER RETURNS 181
To see if any of the security returns are mean-variance frontier returns, we locate the set of
frontier returns. We first find the returns on the expectations and pricing kernels. Since markets
are complete, the expectations kernel is the risk-free payoff (1, 1, 1) and the pricing kernel is the
state price vector q rescaled by the probabilities of states. The state price vector is the unique
solution to the equations
1 = 3q
1
(18.7)
1 = 6q
2
(18.8)
1 =
6
7
q
1
+
3
7
q
2
+
9
7
q
3
. (18.9)
The solution is q
1
= 1/3, q
2
= 1/6, q
3
= 1/2. The pricing kernel equals q/π, that is (1, 1/2, 3/2).
The prices of the expectations and pricing kernels are obtained using the pricing kernel. The
price of the expectations kernel (1, 1, 1) is 1 and the return r
e
is therefore (1, 1, 1). The price of
the pricing kernel (1, 1/2, 3/2) is 7/6 and the return r
q
equals r
3
. Return r
3
is therefore a frontier
return. Returns r
1
and r
2
are not, since they are not on the line generated by r
e
and r
q
.
2
The expectation of the frontier return r
λ
defined by 18.3 is
E(r
λ
) = E(r
e
) + λ[E(r
q
) − E(r
e
)]. (18.10)
The variance of r
λ
is
var(r
λ
) = var(r
e
) + 2λcov(r
e
, r
q
− r
e
) + λ
2
var(r
q
− r
e
) (18.11)
and its standard deviation σ(r
λ
) is the square root of var(r
λ
). The expectations and standard
deviations of frontier returns are shown in Figures 18.3 and 18.4.
If the expectations kernel is risk free, then E(r
e
) equals the risk-free return ¯r; and as follows
from 18.10, the expectation of the frontier return r
λ
is then
E(r
λ
) = ¯r + λ[E(r
q
) − ¯r]. (18.12)
For use later, note that
¯r > E(r
q
). (18.13)
To see this, we first observe that
E(k
2
q
) = [E(k
q
)]
2
+ var(k
q
) > [E(k
q
)]
2
, (18.14)
since the pricing kernel k
q
is not risk free (under the maintained assumption that k
q
and k
e
are not
collinear). Taking expectations in the right-hand equation of 18.2, using 18.14 and the fact that
¯r = 1/E(k
q
) (17.49), we obtain
E(r
q
) =
E(k
q
)
E(k
2
q
)
<
1
E(k
q
)
= ¯r. (18.15)
If the expectations kernel is risk-free, then, as follows from 18.11, the variance of the frontier
return r
λ
is
var(r
λ
) = λ
2
var(r
q
) (18.16)
and the standard deviation is
σ(r
λ
) = |λ|σ(r
q
), (18.17)
see Figure 18.5.
There always exists a frontier return with minimum variance. Of course, if the risk-free claim
lies in the asset span, then the minimum-variance frontier return is the risk-free return. But if
182 CHAPTER 18. THE MEAN-VARIANCE FRONTIER PAYOFFS
the risk-free payoff is not in the asset span, then all returns have strictly positive variances. The
minimum-variance frontier return may be obtained by minimizing 18.11 with respect to λ. Since
the var(r
λ
) of 18.11 is quadratic in λ, the unique solution λ
0
to that minimization problem can be
obtained from the first-order condition. It is given by
λ
0
= −
cov(r
e
, r
q
− r
e
)
var(r
q
− r
e
)
. (18.18)
Given the above results, the set of expected returns and standard deviations of returns are as
indicated in Figures 18.6 and 18.7.
18.4 Zero-Covariance Frontier Returns
Since the set of frontier returns is a line, any two distinct frontier returns can be used in place
of r
e
and r
q
to describe this line. In deriving the beta pricing relation in the next section we use
two frontier returns that are uncorrelated, i.e., have zero covariance. We show here that for every
frontier return r
λ
other than the minimum-variance return there is another frontier return that it
and r
λ
have zero covariance.
Consider a frontier return r
λ
given by 18.3. Another frontier return r
µ
, given by 18.3 with index
µ, has zero covariance with r
λ
and is the zero-covariance frontier return associated with r
λ
if
cov(r
λ
, r
µ
) = var(r
e
) + (λ + µ)cov(r
e
, r
q
− r
e
) + λµ var(r
q
− r
e
) = 0. (18.19)
Solving for µ results in
µ =
var(r
e
) + λcov(r
e
, r
q
− r
e
)
cov(r
e
, r
q
− r
e
) + λvar(r
q
− r
e
)
. (18.20)
So µ is well-defined if the denominator is not equal to zero. The denominator of 18.20 equals
zero when λ = λ
0
(see 18.18), i.e., when r
λ
is the minimum-variance return. There exists no
zero-covariance frontier return associated with the minimum-variance frontier return.
If the risk-free payoff lies in the asset span, then the zero-covariance return associated with
every frontier return (other than the risk-free return) is the risk-free return.
18.5 Beta Pricing
Let r
λ
be a frontier return other than the minimum-variance return and let r
µ
be the associated
zero-covariance frontier return. Taking the orthogonal projection (using the expectations inner
product) of the return r
j
of security j onto the plane of frontier payoffs E results in
r
j
= r
E
j
+ ²
j
, (18.21)
with r
E
j
∈ E and ²
j
∈ E
⊥
. In particular, ²
j
is orthogonal to both k
e
and k
q
and therefore has zero
expectation and zero price.
Since ²
j
has zero price, r
E
j
is a frontier return. Using the returns r
λ
and r
µ
to describe the
frontier line, return r
E
j
can be written r
µ
+ β
j
(r
λ
− r
µ
) for some β
j
. Consequently,
r
j
= r
µ
+ β
j
(r
λ
− r
µ
) + ²
j
. (18.22)
Since ²
j
has zero expectation, taking expectations of both sides of 18.22 we obtain
E(r
j
) = E(r
µ
) + β
j
[E(r
λ
) − E(r
µ
)]. (18.23)
18.6. MEAN-VARIANCE EFFICIENT RETURNS 183
Taking the covariances of both sides of 18.22 with r
λ
, and then solving the resulting equation for
β
j
, using the facts that r
λ
is uncorrelated with r
µ
and with ²
j
, we find
β
j
=
cov(r
j
, r
λ
)
var(r
λ
)
. (18.24)
Thus β
j
is the regression coefficient of r
j
on r
λ
.
If the risk-free payoff lies in the asset span, 18.23 becomes
E(r
j
) = ¯r + β
j
[E(r
λ
) − ¯r]. (18.25)
Relations 18.24 and 18.25 are the beta pricing equations. They say that the risk premium on
any security is proportional to the covariance of its return with a reference frontier return. It was
seen in Chapter 14 that a similar relation, with the market return substituted for the return r
λ
,
is the equation of the security market line of the Capital Asset Pricing Model. In the following
chapter we will demonstrate that the market return is a frontier return in CAPM, implying that
the equation of the security market line is a special case of beta pricing.
For the arbitrary security markets of this chapter, the market return is generally not a frontier
return. There is thus no justification for substituting the market return for r
λ
in 18.25.
Relations 18.24 and 18.25 hold for portfolio returns as well. If the risk-free return lies in the
asset span, the expectation E(r) of an arbitrary return r satisfies
E(r) = ¯r + β[E(r
λ
) − ¯r]. (18.26)
where
β =
cov(r, r
λ
)
var(r
λ
)
. (18.27)
18.6 Mean-Variance Efficient Returns
A return is mean-variance efficient if there is no other return with the same variance, but greater
expectation. In other words, the mean-variance efficient returns maximize expected return subject
to a constraint on variance.
As Figures 18.6 and 18.7 indicate, the mean-variance efficient returns are the frontier returns
that have expected return equal to or greater than that of the minimum-variance return. If the
expectations kernel is risk free, then they are all frontier returns r
λ
with λ ≤ 0. In that case the
return on the pricing kernel is, in view of 18.13, inefficient.
18.7 Volatility of Marginal Rates of Substitution
In Section 14.4 we derived the following bound on the standard deviation of an agent’s marginal
rate of substitution:
σ
µ
∂
1
v
E(∂
0
v)
¶
≥ sup
r
|E(r) − ¯r|
¯rσ(r)
, (18.28)
where the supremum is taken over all returns other than the risk-free return. The bound is the
greatest absolute value of the Sharpe ratio divided by the risk-free return.
We are now in a position to interpret this inequality at a deeper level. We observe first that
the supremum in 18.28 must be attained at a frontier return, since for every return that is not
a frontier return there exists another return with the same expectation but smaller variance, and
hence a greater absolute value of the Sharpe ratio. Second, all frontier returns other than the
184 CHAPTER 18. THE MEAN-VARIANCE FRONTIER PAYOFFS
risk-free return have the same absolute value of the Sharpe ratio. For a frontier return r
λ
where
λ 6= 0, 18.12 and 18.17 imply that
|E(r
λ
) − ¯r|
σ(r
λ
)
=
|λ(E(r
q
) − ¯r)|
|λ|σ(r
q
)
=
|E(r
q
) − ¯r|
σ(r
q
)
. (18.29)
Therefore the supremum in 18.28 is attained at any frontier return other than the risk-free return.
In particular, it is attained at the return r
q
of the pricing kernel.
It turns out that the absolute value of the Sharpe ratio of r
q
divided by the risk-free return
equals the standard deviation of the pricing kernel k
q
. Substituting r
q
= k
q
/E(k
2
q
) and ¯r = 1/E(k
q
)
(see 18.2) in the leftmost term below, we have
|E(r
q
) − ¯r|
¯rσ(r
q
)
=
|[E(k
q
)]
2
− E(k
2
q
)|
σ(k
q
)
=
σ
2
(k
q
)
σ(k
q
)
= σ(k
q
). (18.30)
In sum, then, we have
sup
r
|E(r) − ¯r|
¯rσ(r)
= σ(k
q
) (18.31)
and
σ
µ
∂
1
v
E(∂
0
v)
¶
≥ σ(k
q
) (18.32)
for any agent. Thus the standard deviation of the pricing kernel is a lower bound for the volatility
of agents’ marginal rates of substitution. Eq. 18.32 can, of course, be verified directly, since the
projection of any agent’s marginal rate of substitution onto the asset span is k
q
(Figure 18.8).
18.7.1 Example
In Example 18.3.1, the pricing kernel k
q
equals (1, 1/2, 3/2) and its standard deviation is
σ(k
q
) =
1
√
6
. (18.33)
The risk-free return ¯r equals 1 and the Sharpe ratios of returns r
1
and r
2
are
E(r
1
) − 1
σ(r
1
)
= 0, (18.34)
and
E(r
2
) − 1
σ(r
2
)
=
1
√
8
, (18.35)
respectively. Both numbers are smaller than σ(k
q
) as they must be given 18.31. That also confirms
that the returns r
1
and r
2
are not frontier returns.
2
Notes
The mean-variance analysis of portfolio returns has been extensively used in finance since its de-
velopment by Markowitz [1] and [2]. An analytical characterization of the mean-variance frontier
was first derived by Merton [3].
Bibliography
[1] Harry Markowitz. Portfolio selection: Efficient diversification of investments. Journal of Fi-
nance, 7:77–91, 1952.
[2] Harry Markowitz. Portfolio Selection: Efficient Diversification of Investments. Wiley, New
York, 1959.
[3] Robert C. Merton. An analytic derivation of the efficient portfolio frontier. Journal of Financial
and Quantitative Analysis, 7:1851–1871, 1972.
185
186 BIBLIOGRAPHY
Chapter 19
CAPM
19.1 Introduction
Beta pricing (see Section 18.5) implies that the risk premium on any security or portfolio is propor-
tional to the covariance of its return with a frontier return. However, beta pricing by itself gives no
guidance as to which returns are frontier returns. We will use the term Capital Asset Pricing Model
if the market return is a frontier return. Note that the CAPM is here identified with a property
of equilibrium security returns, not with a class of models of security markets. Therefore it will
be necessary to determine what restrictions on the primitives of security markets, preferences or
payoffs give rise to equilibria that conform to the CAPM definition.
Under the CAPM the market return, being a frontier return, can be taken as the reference
portfolio in the beta pricing equation, resulting in the security market line, which relates the risk
premium on any security to the covariance between the return on that security and the market
return.
In Chapter 14 we derived the equation of the security market line applying consumption-based
security pricing under the assumption that agents have quadratic utilities. The derivation was
generalized in Chapter 16. In this chapter we derive the CAPM in an equilibrium under the
assumption that agents take variance as a measure of consumption risk (mean-variance preferences).
This condition is satisfied when agents’ preferences have an expected utility representation with
quadratic utilities, and also when security payoffs are multivariate normally distributed. We relax
two of the assumptions of the Chapter 14 derivation: that agents’ endowments lie in the asset span
(securities market economy), and that the risk-free payoff is in the asset span.
19.2 Security Market Line
In Chapter 14 we defined the market payoff in a securities market economy as the aggregate date-1
endowment ¯w
1
, and the market portfolio as a portfolio with payoff equal to the market payoff. We
now extend these definitions to the general case when agents’ endowments, and therefore also the
aggregate endowment, need not lie in the asset span.
Each agent’s date-1 endowment w
i
1
can be decomposed into the sum of two orthogonal com-
ponents. Using the expectations inner product we project w
i
1
onto the asset span in order to
distinguish the tradable component of the aggregate endowment from a nontradable component
which is orthogonal to the asset span. We have
w
i
1
= w
i
1M
+ w
i
1N
, (19.1)
where w
i
1M
∈ M is the tradable component of agent i’s endowment, and w
i
1N
∈ N = M
⊥
is the
nontradable component. The Projection Theorem 17.5.1 guarantees that there is no ambiguity
187
188 CHAPTER 19. CAPM
about this decomposition. The corresponding decomposition for the aggregate endowment is
¯w
1
= ¯w
1M
+ ¯w
1N
. (19.2)
The market payoff m is defined as the tradable component of the aggregate endowment, that
is,
m = ¯w
1M
. (19.3)
The market return r
m
is the market payoff m divided by its equilibrium price q(m), assumed
nonzero.
By the definition of the CAPM, the market return r
m
is a frontier return. Assuming that r
m
is
not the minimum-variance return, there exists another frontier return, denoted r
m0
, that has zero
covariance with r
m
. These two frontier returns can be used in the equation 18.23 of beta pricing.
Thus we have
19.2.1 Theorem
If the market return lies on the mean-variance frontier, then
E(r
j
) = E(r
m0
) + β
j
[E(r
m
) − E(r
m0
)], (19.4)
for every security j, where β
j
= cov(r
j
, r
m
)/var(r
m
).
Eq. 19.4 is the equation of the security market line. If the risk-free payoff is in the asset span,
then r
m0
is risk-free and equal to ¯r, and 19.4 becomes
E(r
j
) = ¯r + β
j
[E(r
m
) − ¯r]. (19.5)
Eq. 19.5 says that the risk premium E(r
j
) − ¯r is proportional to the coefficient β
j
, with
the factor of proportionality being the risk premium E(r
m
) − ¯r on the market return (market
risk premium). Thus coefficient β
j
—the regression coefficient of r
j
on the market return—is the
appropriate measure of security risk in the CAPM.
The equation of the security market line holds for portfolio returns as well. Substituting r and
β for r
j
and β
j
in 19.4, we obtain
E(r) = E(r
m0
) + β[E(r
m
) − E(r
m0
)], (19.6)
where β is the regression coefficient of the return r on the market return. For the market return,
β equals one; for the zero-covariance return r
m0
, β equals zero. Return r
m0
is called zero-beta
return. The following example illustrates the use of 19.5 for pricing securities.
19.2.2 Example
There are three equally probable states at date 1. The aggregate date-1 endowment is (2,3,4).
There are three securities: the first is risk-free and has a return ¯r = 1; the second has a return
r
2
= (0, 3/2, 3); the third security has a payoff x
3
= (0, 0, 1). The problem is to find the price p
3
of
the third security assuming the CAPM.
We observe that the aggregate endowment lies in the span of the first and the second securities.
This allows us to find the market return using the prices of those two securities. The price of the
third security can be found using the security market line.
The price of the market payoff is 8/3, and its return is r
m
= (3/4, 9/8, 3/2). The expected
return on the market portfolio is E(r
m
) = 9/8.
The security market line gives the following:
E(x
3
)
p
3
= ¯r +
cov(x
3
, r
m
)
p
3
var(r
m
)
(E(r
m
) − ¯r) (19.7)
19.3. MEAN-VARIANCE PREFERENCES 189
or
p
3
=
1
¯r
[E(x
3
) −
cov(x
3
, r
m
)
var(r
m
)
(E(r
m
) − ¯r)]. (19.8)
Substituting E(r
m
) = 9/8, E(x
3
) = 1/3, ¯r = 1, cov(x
3
, r
m
) = 1/8, var(r
m
) = 3/32 in 19.8, we
obtain p
3
= 1/6.
An alternative way of calculating p
3
is to note that the pricing kernel lies in the frontier plane.
Since the market payoff is in the frontier plane, the pricing kernel lies in the span of the market
payoff and the risk-free payoff or, equivalently, in the span of r
2
and the risk-free return. Writing
the equations 17.30 for pricing the risk-free return and r
2
, the pricing kernel can be calculated as
(3/2, 1, 1/2). Applying the kernel to x
3
results in p
3
= 1/6.
2
The simplest case in which 19.5 (the securities market line) holds is when there are only two
securities. We observed in Section 18.2 that with two securities every return is a mean-variance
frontier return. In particular, the market return lies on the frontier and the CAPM holds.
19.3 Mean-Variance Preferences
The Capital Asset Pricing Model obtains in equilibrium when agents have mean-variance prefer-
ences. An agent has mean-variance preferences if his utility function u(c
0
, c
1
) is strictly increasing
and has the representation
u(c
0
, c
1
) = v
0
(c
0
) + f(E(c
1
), var(c
1
)) (19.9)
for some functions v
0
: R → R and f : R × R
+
→ R. Under 19.9, agents’ preferences are time
separable with preferences over date-1 consumption plans depending only on the expectation and
variance. The agent therefore takes variance as a measure of consumption risk. An agent with
mean-variance preferences is strictly variance averse if f in 19.9 is strictly decreasing in variance.
Two important cases that lead to mean-variance preferences—quadratic utilities and normally
distributed payoffs and date-1 endowments—are discussed in the next two sections.
19.3.1 Theorem
If every agent has mean-variance preferences and is strictly variance averse, then in an equilibrium
the market return lies on the mean-variance frontier.
Proof: Let c
i
1
be an equilibrium date-1 consumption plan of agent i. We decompose c
i
1
into
the tradable component and the nontradable component (see 19.2) so that
c
i
1
= c
i
1M
+ c
i
1N
, (19.10)
where c
i
1M
∈ M and c
i
1N
∈ N.
It is sufficient to show that the tradable component c
i
1M
of each agent’s date-1 consumption lies
on the mean-variance frontier E, since if that is so then the tradable component of the aggregate
consumption is also a frontier payoff. But the tradable component of aggregate consumption equals
the tradable component of the aggregate endowment, which by definition is the market payoff.
Therefore the market return is a frontier return.
To show that c
i
1M
∈ E, we decompose c
i
1M
by projecting it on the frontier plane E so that
c
i
1M
= c
i
1E
+ c
i
1I
, (19.11)
where c
i
1E
∈ E is the frontier component, and c
i
1I
∈ E
⊥
is the component of c
i
1M
orthogonal to the
frontier plane (here I stands for “inefficient” and E
⊥
is the orthogonal complement of E in M).
190 CHAPTER 19. CAPM
Suppose by contradiction that c
i
1M
does not lie on the frontier plane and hence that c
i
1I
6= 0,
for some i.. Consider the alternative date-1 consumption plan given by
˜c
i
1
≡ c
i
1E
+ c
i
1N
. (19.12)
Note that ˜c
i
1
= c
i
1
− c
i
1I
. Since the agent’s utility function is strictly increasing, the optimal
consumption satisfies the budget constraints with equality, implying that c
i
1
− w
i
1
∈ M. Using
˜c
i
1
− w
i
1
= (c
i
1
− w
i
1
) − c
i
1I
, it follows that
˜c
i
1
− w
i
1
∈ M, (19.13)
so that the consumption plan ˜c
i
1
can be attained by a net trade in the asset span.
By Theorem 18.2.1 the equilibrium pricing kernel k
q
lies in the frontier plane E. Therefore
q(c
i
1I
) = E(k
q
c
i
1I
) = 0, (19.14)
and the net trade ˜c
i
1
−w
i
1
has the same price as c
i
1
−w
i
1
, that is, q(˜c
i
1
−w
i
1
) = q(c
i
1
−w
i
1
). This and
19.13 imply that the date-1 consumption plan ˜c
i
1
and the date-0 plan c
i
0
satisfy agent i’s budget
constraint.
Since the expectations kernel also lies in the frontier plane (Theorem 18.2.1) we have
E(c
i
1I
) = E(k
e
c
i
1I
) = 0. (19.15)
Therefore ˜c
i
1
and c
i
1
have the same expectation. Since c
i
1E
, c
i
1I
and c
i
1N
are mutually orthogonal
and E(c
i
1I
) = 0, it follows that cov(c
i
1E
, c
i
1I
) = cov(c
i
1I
, c
i
1N
) = 0. Using 19.12, we obtain that
cov(˜c
i
1
, c
i
1I
) = 0, and consequently that
var(c
i
1
) = var(˜c
i
1
) + var(c
i
1I
) > var(˜c
i
1
), (19.16)
where the last strict inequality follows from the assumption that c
i
1I
6= 0.
Consumption plan ˜c
i
1
has smaller variance than c
i
1
and the two have the same expectation.
Since the agent has mean-variance preferences and is strictly variance averse, consumption plan ˜c
i
1
is strictly preferred to c
i
1
. This contradicts the optimality of c
i
1
. Therefore the tradable component
c
i
1M
of every agent’s equilibrium consumption lies in the mean-variance frontier plane. Since in
equilibrium the market payoff equals the sum over agents of the tradable components of agents’
consumption plans, the market return lies on the mean-variance frontier as well.
2
It follows from Theorems 19.3.1 and Theorem 19.2.1 that if agents measure consumption risk
by variance, then the equation of the security market line holds in equilibrium.
19.4 Equilibrium Portfolios under Mean-Variance Preferences
In the proof of Theorem 19.3.1 we demonstrated that the tradable component of the date-1 equi-
librium consumption plan of an agent with mean-variance preferences lies on the mean-variance
frontier. The nontradable component of the equilibrium consumption plan is equal to the nontrad-
able component of the endowment. To see this, note that since c
i
1
− w
i
1
∈ M, 19.1 and 19.2 imply
that
c
i
1N
= w
i
1N
. (19.17)
If the risk-free payoff lies in the asset span, then c
i
1N
has zero expectation since is orthogonal to
the asset span.
Summing up, the equilibrium date-1 consumption plan satisfies
c
i
1
= c
i
1M
+ w
i
1N
, with c
i
1M
∈ E. (19.18)
19.4. EQUILIBRIUM PORTFOLIOS UNDER MEAN-VARIANCE PREFERENCES 191
Let
w
i
≡ w
i
0
+ q(w
i
1M
), (19.19)
be the agent’s wealth at date 0 consisting of his date-0 endowment and the value of the tradable
component of his date-1 endowment. Since the mean-variance frontier is spanned by the mar-
ket return r
m
and the zero-covariance return r
m0
, the tradable component of date-1 equilibrium
consumption plan can be written as
c
i
1M
= a
i
r
m
+ (w
i
− c
i
0
− a
i
)r
m0
, (19.20)
where a
i
denotes the amount of date 0 wealth invested in the market portfolio. A simple charac-
terization of the equilibrium investment a
i
can be given when the risk-free payoff lies in the asset
span. Then r
m0
= ¯r and the expectation and variance of date-1 equilibrium consumption plan can
be written using 19.18 and 19.20 as
E(c
i
1
) = (w
i
− c
i
0
)¯r + a
i
[E(r
m
) − ¯r)], (19.21)
and
var(c
i
1
) = (a
i
)
2
var(r
m
) + var(w
i
1N
). (19.22)
The equilibrium investment a
i
and consumption plan c
i
(assumed interior and with strictly
positive variance) satisfy the following first-order conditions obtained from substituting 19.21 and
19.22 in 19.9 and maximizing with respect to c
i
0
and a
i
:
v
0
0
= ¯rδ
E
f (19.23)
a
i
= −
(E(r
m
) − ¯r)δ
E
f
2var(r
m
)δ
v
f
. (19.24)
Here δ
E
f and δ
v
f are the partial derivatives of f with respect to its first and second arguments eval-
uated at the equilibrium date-1 consumption; v
0
0
is the derivative of v
0
evaluated at the equilibrium
date-0 consumption.
Eq. 19.23 states that the marginal rate of substitution between date-0 consumption and the
expectation of date-1 consumption equals the risk-free return. Eq. 19.24 relates the equilibrium
investment in the market portfolio to the risk premium and the variance of the market return, and
also to the marginal rate of substitution between expected return and variance of return.
If each agent’s mean-variance utility function is strictly increasing in the expectation of date-1
consumption and strictly decreasing in its variance, then all agents whose optimal consumption is
not risk-free have investments in the market portfolio that are of the same sign as the risk premium
on the market return. It follows that the market risk premium must be strictly positive since
otherwise the total wealth invested in the market portfolio would be negative. Thus
E(r
m
) > ¯r. (19.25)
Consequently, each agent’s investment in the market portfolio is strictly positive or zero implying
that the expected return on equilibrium investment exceeds the risk-free return. Since every mean-
variance frontier return with expectation that exceeds the risk-free return is mean-variance efficient,
returns on agents’ equilibrium investments are mean-variance efficient.
The foregoing discussion provides a characterization of an equilibrium portfolio net of the port-
folio that generates the tradable component of an agent’s date-1 endowment. The agent’s equilib-
rium portfolio is equal to the difference between the portfolio described above and the portfolio
that generates w
i
1M
.
192 CHAPTER 19. CAPM
19.5 Quadratic Utilities
If an agent’s preferences have an expected utility representation with a quadratic von Neumann-
Morgenstern utility function of the form
v
i
(c
0
, c
s
) = v
i
0
(c
0
) + v
i
1
(c
s
) = v
i
0
(c
0
) − (c
s
− α
i
)
2
, for c
s
≤ α
i
, (19.26)
then the expected utility of consumption (c
0
, c
1
) is
E[v
i
(c
0
, c
1
)] = v
i
0
(c
0
) − [var(c
1
) + (E(c
1
) − α
i
)
2
]. (19.27)
As usual, we assume common probability expectations. The agent’s expected utility 19.27 depends
only on c
0
and the expectation and variance of c
1
. Thus he has mean-variance preferences and is
variance averse. Theorem 19.3.1 therefore applies when utility functions are quadratic.
In Chapter 14, with the subsequent generalization in Chapter 16, we derived the equation of
the security market line in an equilibrium with quadratic utility functions 19.26 under additional
assumptions not appearing in Theorem 19.3.1: that agents’ endowments lie in the asset span, and
that the risk-free payoff is in the asset span. Further, we proved in Chapter 16 that under these
assumptions, markets are effectively complete and equilibrium consumption allocations are Pareto
optimal. From the analysis of this chapter we conclude that the equation of security market line
holds in an equilibrium with quadratic utility functions even when either agents’ endowments or
the risk-free payoff (or both) lie outside of the asset span. However, the Pareto optimality of
equilibrium consumption allocations does not in general hold under the less strict assumptions.
19.6 Normally Distributed Payoffs
If security payoffs and an agent’s date-1 endowment are multivariate normally distributed,
1
then
his date-1 consumption plans that can be generated by portfolios are normally distributed. Since
the normal distribution is completely characterized by its expectation and variance, the agent’s
utility function depends only on date-0 consumption c
0
and the expectation and variance of date-1
consumption plan c
1
. If his utility functions is time separable and strictly increasing, the agent has
mean-variance preferences 19.9.
In particular, if an agent’s preferences have an expected utility representation with a time
separable von Neumann-Morgenstern utility function, the mean-variance representation obtains
when security payoffs and his date-1 endowment are multivariate normally distributed. Further,
if the agent is risk averse, then he is also variance averse. To see this, recall from Section 10.3
that if two random variables are normally distributed, then that with strictly greater variance is
strictly riskier. Thus Theorem 19.3.1 applies when security payoffs and agents’ date-1 endowments
are multivariate normally distributed and agents are risk averse.
Normal payoff distributions can be justified by appeal to the central limit theorem. But that is
only if security payoffs are not subject to limited liability. For instance, the payoff of an option is
a truncated version of the payoff on the underlying security.
Notes
A first expression of the risk-return tradeoff was given in Theorem 13.3.1. In a world of risk-averse
investors, the greater is the expected return the greater is the risk. We observed in Chapter 10 that
even if no assumptions about the form of the utility function are made (other than risk aversion), a
1
Strictly, normal distribution of payoffs cannot be incorporated in the model adopted in this book since we assumed
that there exist only a finite number of states. However, no harm results if we temporarily trespass into a richer
setting.
19.6. NORMALLY DISTRIBUTED PAYOFFS 193
specific measure of return was available: expected return. We also remarked that variance could not
be used as a measure of risk, that it had to be associated with the partial ordering defined in Chapter
10. In the CAPM, in contrast, risk is associated with the complete ordering of return distributions
induced by beta, and the security market line implies that the relation between expected return
and risk is linear.
If the risk-free payoff and agents’ endowments lie in the asset span, the CAPM shares with LRT
utilities a property of equilibrium, that date-1 consumption plans lie in the plane spanned by the
aggregate endowment and the risk-free payoff. However, the pricing relationship of the CAPM—the
security market line—does not apply in the general LRT utilities case (with exception, of course, of
quadratic utilities). Nothing about the assumption that agents have LRT utilities with a common
slope of risk tolerance implies that the market payoff is mean-variance efficient. As was shown in
Theorem 19.3.1, mean-variance efficiency of the market payoff is a consequence of the assumption
that agents measure consumption risk by variance.
In proving Theorem 19.3.1 we assumed that agents’ consumption plans were unrestricted. If
there are restrictions on consumption (such as positivity), the theorem is still true provided that
the equilibrium allocation is interior. But the proof requires a minor modification. Instead of using
˜c
i
1
= c
i
1
− c
i
1I
as an alternative consumption plan it is necessary to use ˜c
i
1
= c
i
1
− δc
i
1I
for small
positive δ. Although the first of these consumption plans may not be in the consumption set even
if c
i
1
is interior, the latter will be for small enough δ.
The portfolio theory under mean-variance preferences is due to Markowitz [3]. The CAPM pric-
ing results were derived independently by Sharpe [10], Lintner, [2], Mossin [5], and (in unpublished
notes) Treynor [11].
Derivation of the CAPM without the assumption that the risk-free payoff is traded is due to
Black [1]. Sufficient conditions for the existence of a CAPM equilibrium when agents have mean-
variance preferences, with and without a risk-free security, can be found in Nielsen [7] and [6].
The testable content of the CAPM is the assertion that the market return is mean-variance
efficient, implying the equation of the security market line. In his critique, Roll [8] observed that
if one uses a proxy for the market portfolio that is not mean-variance efficient, testing the relation
between beta and risk premia is pointless. That is because the CAPM generates a prediction about
this relation only when the reference portfolio is mean-variance efficient.
As noted by Ross [9], if the proxy for the market portfolio is mean-variance efficient, the equation
of the security market line will be satisfied regardless of whether the CAPM is true or not. We
showed this in Chapter 18.
Milne and Smith [4] analyzed the CAPM in the presence of transactions costs.
194 CHAPTER 19. CAPM
Bibliography
[1] Fischer Black. Capital market equilibrium with restricted borrowing. Journal of Business,
45:444–455, 1972.
[2] John Lintner. The valuation of risk assets and the selection of risky investments in stock
portfolios and capital budgets. Review of Economics and Statistics, 47:13–37, 1965.
[3] Harry Markowitz. Portfolio Selection: Efficient Diversification of Investments. Wiley, New
York, 1959.
[4] Frank Milne and Clifford W. Smith. Capital asset pricing with proportional transaction cost.
Journal of Financial and Quantitative Analysis, XV:253–266, 1980.
[5] Jan Mossin. Equilibrium in a capital asset market. Econometrica, 35:768–783, 1968.
[6] Lars T. Neilsen. Equilibrium in CAPM without a riskless asset. Review of Economic Studies,
57:315–324, 1990.
[7] Lars T. Neilsen. Existence of equilibrium in CAPM. Journal of Economic Theory, 52:223–231,
1990.
[8] Richard Roll. A critique of the asset pricing theory’s tests: Part I. Journal of Financial
Economics, 4:129–176, 1977.
[9] Stephen A. Ross. Risk, return and arbitrage. In Irwin Friend and James Bicksler, editors,
Risk and Return in Finance. Ballinger, Cambridge, Massachusetts, 1976.
[10] William F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of
risk. Journal of Finance, 19:425–442, 1964.
[11] John L. Treynor. Toward a theory of market value of risky assets. reproduced, 1961.
195
196 BIBLIOGRAPHY
Chapter 20
Factor Pricing
20.1 Introduction
In the CAPM beta is the measure of the sensitivity of a security’s return to the market return.
The equation of the security market line 19.5 shows that the relation between the risk premium
and beta is linear.
The CAPM relies on restrictive assumptions about agents’ preferences or security returns, and
certainly its empirical implications have not been confirmed by data. In this chapter we consider
models of security markets all with a pricing relation similar to that of the CAPM, but with a factor
(or factors) replacing the market return. These factors are typically taken to be proxies for such
macroeconomic variables as GDP, the rate of inflation, and so on. The relation between expected
return and the measure of the sensitivity of a security’s return to factor risk, like the corresponding
relation in the case of the CAPM, is linear.
20.2 Exact Factor Pricing
There are K contingent claims f
1
, . . . , f
K
, called factors. Each factor is normalized so as to have
zero expectation. The number K of factors is small relative to the number of securities, and the
factors may or may not lie in the asset span. The span of the factors and the risk-free claim e is the
factor span, denoted by F ≡ span{e, f
1
, . . . , f
K
}. It is assumed that all K factors and the risk-free
claim are linearly independent.
Projecting the payoff x
j
of each security on the factor span F (using the expectations inner
product) results in the following decomposition:
x
j
= E(x
j
) +
K
X
k=1
b
jk
f
k
+ δ
j
(20.1)
for every j, where δ
j
is uncorrelated with f
k
for all k and has zero expectation. The coefficient b
jk
in 20.1 is the factor loading of payoff x
j
: it measures the exposure (sensitivity) of that payoff to
the factor f
k
.
Eq. 20.1 can be written using security returns rather than payoffs. If all security prices are
nonzero, then
r
j
= E(r
j
) +
K
X
k=1
β
jk
f
k
+ ²
j
, (20.2)
where β
jk
= b
jk
/p
j
and ²
j
= δ
j
/p
j
. Coefficient β
jk
in 20.2 is the factor loading of return r
j
.
197
198 CHAPTER 20. FACTOR PRICING
Exact factor pricing with factors f
1
, . . . , f
K
holds if security prices satisfy
p
j
= E(x
j
)τ
0
+
K
X
k=1
b
jk
τ
k
∀j (20.3)
for some scalars τ
0
, . . . , τ
K
. Eq. 20.3 is a linear relation between security prices and factor loadings.
Exact factor pricing can be expressed using expected returns. Dividing 20.3 by p
j
and rear-
ranging terms yields
E(r
j
) = γ
0
+
K
X
k=1
β
jk
γ
k
, (20.4)
where γ
0
= 1/τ
0
and γ
k
= −τ
k
/τ
0
. In this form exact factor pricing is a linear relation between
expected returns and factor loadings of returns.
If the risk-free claim and the K factors lie in the asset span, so does the residual δ
j
. Then exact
factor pricing obtains if the residual δ
j
in 20.1, or equivalently ²
j
in 20.2, has zero price; that is, if
q(δ
j
) = 0, (20.5)
where q is the payoff pricing functional associated with security prices p. To see this, apply the
functional q to both sides of 20.1 and use 20.5 to obtain 20.3 with coefficients
τ
0
=
1
¯r
and τ
k
= q(f
k
) (20.6)
equal to factor prices. The coefficients of exact factor pricing for returns are
γ
0
= ¯r, and γ
k
= −¯rq(f
k
). (20.7)
If the risk-free claim and the K factors are payoffs, then the asset span can be decomposed
into M = F + span{²
1
, . . . , ²
J
}. The assumption that each residual δ
j
has zero price implies that
k
q
∈ F. It turns out that the condition that the pricing kernel lies in the factor span is sufficient
for exact factor pricing independent of whether the risk-free claim and the factors lie in the asset
span.
20.2.1 Theorem
If the pricing kernel k
q
lies in the factor span, then exact factor pricing
E(r
j
) = γ
0
+
K
X
k=1
β
jk
γ
k
(20.8)
holds with γ
0
= 1/E(k
q
) and γ
k
= −E(k
q
f
k
)/E(k
q
). If in addition the risk-free claim lies in the
asset span, then γ
0
= ¯r.
Proof: Multiplying 20.2 by k
q
and taking expectations, we obtain
1 = E(r
j
)E(k
q
) +
K
X
k=1
β
jk
E(k
q
f
k
) + E(k
q
²
j
). (20.9)
Dividing both sides of 20.9 by E(k
q
) and rearranging, gives us
E(r
j
) =
1
E(k
q
)
+
K
X
k=1
β
jk
"
−
E(k
q
f
k
)
E(k
q
)
#
−
E(k
q
²
j
)
E(k
q
)
. (20.10)
20.3. EXACT FACTOR PRICING, BETA PRICING AND THE CAPM 199
Since k
q
lies in the factor span F, it is orthogonal to ²
j
. Thus E(k
q
²
j
) = 0 and, as follows from
20.10,
E(r
j
) =
1
E(k
q
)
+
K
X
k=1
β
jk
"
−
E(k
q
f
k
)
E(k
q
)
#
. (20.11)
Therefore exact factor pricing 20.8 holds with γ
0
= 1/E(k
q
) and γ
k
= −E(k
q
f
k
)/E(k
q
). Finally, if
the risk-free claim lies in the asset span, then
E(k
q
) =
1
¯r
, (20.12)
and γ
0
= ¯r.
2
If the risk-free claim lies in the asset span, then a necessary and sufficient condition for the
pricing kernel to lie in the factor span is that the plane of mean-variance frontier payoffs is contained
in the factor span. To see this, recall (Theorem 18.2.1) that the mean-variance frontier plane E is
spanned by the risk-free payoff and the pricing kernel. Thus k
q
∈ F iff E ⊂ F.
20.3 Exact Factor Pricing, Beta Pricing and the CAPM
Suppose that there is a single factor which is a mean-variance frontier return r normalized so as to
have zero expectation:
f = r − E(r) (20.13)
for an arbitrary frontier return r other than the risk-free return.
Suppose also that the risk-free claim lies in the asset span. Then the factor f and the risk-free
return span the plane of mean-variance frontier payoffs. Consequently, the pricing kernel lies in the
factor span. Theorem 20.2.1 implies exact factor pricing:
E(r
j
) = ¯r − β
j
¯rq(f). (20.14)
Since β
j
of 20.14 is the coefficient in the projection of return r
j
on the factor span, it is given by
β
j
=
cov(r
j
, f)
var(f)
=
cov(r
j
, r)
var(r)
(20.15)
and hence is the same as the β
j
of the beta pricing relation 18.25. Proceeding further, we multiply
20.13 by k
q
and take expectations to get
q(f) = E(k
q
f) = 1 −
E(r)
¯r
. (20.16)
Using 20.16, we can rewrite 20.14 as
E(r
j
) = ¯r + β
j
[E(r) − ¯r]. (20.17)
This is the beta pricing relation 18.25. Thus beta pricing with respect to a frontier return r is the
same as exact factor pricing with a single factor equal to return r normalized so as to have zero
expectation.
In the CAPM of Chapter 19, the market return r
m
lies on the mean-variance frontier. Exact
factor pricing with a single factor given by
f = r
m
− E(r
m
) (20.18)
is equivalent to the equation of the security market line.
200 CHAPTER 20. FACTOR PRICING
20.4 Factor Pricing Errors
Even if it does not hold exactly, the factor pricing relation 20.4 provides a point of departure for
developing a definition of pricing errors.
The pricing error of security j is
ψ
j
≡ E(r
j
) − γ
0
−
K
X
k=1
β
jk
γ
k
, (20.19)
where γ
0
= 1/E(k
q
) and γ
k
= −E(k
q
f
k
)/E(k
q
). If pricing errors are zero, then exact factor pricing
holds.
Using 20.10 we can write
ψ
j
= −
E(k
q
²
j
)
E(k
q
)
. (20.20)
If the risk-free claim and the K factors lie in the asset span, then ²
j
∈ M. Thus E(k
q
²
j
) = q(²
j
),
and, using 20.12,
ψ
j
= −¯rq(²
j
), (20.21)
Eq. 20.21 says that the pricing error equals the price of the residual ²
j
multiplied by (the negative
of) the risk-free return.
A bound on the pricing error can be obtained as follows: projecting k
q
on the factor span F,
we obtain the following decomposition:
k
q
= k
F
q
+ η, (20.22)
where k
F
q
∈ F and η ⊥ F. Since each ²
j
is orthogonal to the factors, it follows that
E(k
q
²
j
) = E(η²
j
). (20.23)
Applying the Cauchy-Schwarz inequality (Section 17.2), we obtain
|E(k
q
²
j
)| ≤ k η kk ²
j
k . (20.24)
Using 20.20, 20.22, and E(²
j
) = 0, there results the following bound on the pricing error:
|ψ
j
| ≤
1
E(k
q
)
σ(²
j
) k k
q
− k
F
q
k . (20.25)
The norm k k
q
− k
F
q
k measures the distance between the pricing kernel k
q
and the factor span.
Thus inequality 20.25 indicates that if k
q
is close to the factor span, then the pricing error on
security j is small. When the pricing kernel lies in the factor span, then exact factor pricing holds,
as seen in Theorem 20.2.1.
20.5 Factor Structure
Security returns have a factor structure with factors f
1
, . . . , f
K
if the residuals ²
j
in the decompo-
sition
r
j
= E(r
j
) +
K
X
k=1
β
jk
f
k
+ ²
j
(20.26)
are uncorrelated with each other,
E(²
i
²
j
) = 0 for i 6= j, (20.27)
20.5. FACTOR STRUCTURE 201
in addition to being uncorrelated with factors and having zero expectations. The condition 20.27
is a substantive restriction on security returns and factors. In general, residuals of the projection
of security returns on the factor span need not be uncorrelated with each other.
When returns have the factor structure given by 20.26 and 20.27, factors are called systematic
risk since they affect all security returns, while residuals are called idiosyncratic risk since each
residual is specific to the security in the sense that it is unaffected by the factor risk and other
security returns. If returns do not have a factor structure (so that the residuals may be correlated
with each other), then the terms “systematic risk” and “idiosyncratic risk” are inappropriate: there
is no presumption that the residuals are any less pervasive across securities than are the factors.
The term “systematic risk” is sometimes used in the context of the CAPM to mean market
risk. This usage is different from systematic risk as defined here. The CAPM does not require that
security returns have a factor structure in the sense of 20.26 and 20.27 with the market return as
a factor.
A bound on the summed squared pricing errors obtains when security returns have a factor
structure.
20.5.1 Theorem
If security returns have a factor structure, then
J
X
j=1
ψ
2
j
≤
1
[E(k
q
)]
2
max
j
[σ
2
(²
j
)] k k
q
− k
F
q
k
2
. (20.28)
Proof: We can assume that all ²
j
are nonzero. If some were zero, then the proof to follow would
apply to all securities with nonzero ²
j
. Since the pricing error on a security with zero idiosyncratic
risk equals zero (see 20.20), 20.28 holds for all securities.
The pricing kernel k
q
lies in the asset span M, a subspace of F + span{²
1
, . . . , ²
J
}. Since
the residual η of 20.22 is orthogonal to F, it must lie in span{²
1
, . . . , ²
J
}. The assumption of
factor structure (20.26 and 20.27) implies (recall Corollary 17.4.2) that the idiosyncratic risks ²
j
are linearly independent and hence are a basis for span{²
1
, . . . , ²
J
}. Consequently, η can be written
as
η =
J
X
j=1
a
j
²
j
, (20.29)
for some scalars a
1
, . . . , a
J
. It follows from 20.22 and 20.29 that
E(k
q
²
j
) = a
j
E(²
2
j
). (20.30)
Making use of E(²
2
j
) = σ
2
(²
j
), 20.20 and 20.30 imply
ψ
j
= −
1
E(k
q
)
a
j
σ
2
(²
j
). (20.31)
Further, the Pythagorean Theorem 17.6 and 20.29 imply
J
X
j=1
a
2
j
E(²
2
j
) = k η k
2
. (20.32)
Using η = k
q
− k
F
q
and E(²
2
j
) = σ
2
(²
j
), 20.32 can be written as
J
X
j=1
a
2
j
σ
2
(²
j
) = k k
q
− k
F
q
k
2
. (20.33)
202 CHAPTER 20. FACTOR PRICING
Now, if 20.33 is multiplied by (1/[E(k
q
)]
2
) max
j
[σ
2
(²
j
)] and if use is made of σ
2
(²
j
) ≤ max
j
[σ
2
(²
j
)],
then
J
X
j=1
1
[E(k
q
)]
2
a
2
j
σ
4
(²
j
) ≤
1
[E(k
q
)]
2
max
j
h
σ
2
(²
j
)
i
k k
q
− k
F
q
k
2
. (20.34)
The sought-after result 20.28 follows from 20.31 and 20.34.
2
Theorem 20.5.1 has several important implications. It implies—and hence confirms the finding
of Section 20.4—that if the pricing kernel is close to the factor span, then pricing errors are small.
The theorem also implies that if the number of securities is large, then, independent of the location
of the pricing kernel, most pricing errors are small. We can be more precise. Let ρ > 0 be a small
number and let N
ρ
be the smallest integer greater than M/ρ where M denotes the right hand side
of 20.28. If J > N
ρ
, then at least J − N
ρ
securities have squared pricing errors ψ
2
j
smaller than
ρ. If not, there is a contradiction to 20.28, for then there are more that N
ρ
securities with squared
pricing errors greater than ρ.
If the number J of securities is so large that J − N
ρ
is also large, then for a large number of
securities pricing errors must be small. This justifies the term approximate factor pricing.
In the limit, if there are infinitely many securities (this specification takes us beyond the finite
setting of this book; but see the chapter notes) with a factor structure characterized by bounded
variance of idiosyncratic risks, then, as implied by Theorem 20.5.1, all but a finite number of
securities have (squared) pricing errors that are arbitrarily small. This is the fundamental conclusion
of the Arbitrage Pricing Theory (APT).
20.6 Mean-Independent Factor Structure
Exact factor pricing obtains in a security markets equilibrium under a more restrictive definition
of factor structure. This definition is stated in terms of security payoffs.
In general, the residual δ
j
determined by the projection of x
j
on the factor span,
x
j
= E(x
j
) +
K
X
k=1
b
jk
f
k
+ δ
j
, (20.35)
is uncorrelated with the factors. Security payoffs have a mean-independent factor structure if
uncorrelatedness can be strengthened to mean-independence; that is, to
E(δ
j
|f
1
, . . . , f
K
) = 0, (20.36)
for every j.
In the next theorem we consider securities markets with agents whose preferences have an
expected utility representation with common probabilities and with differentiable von Neumann-
Morgenstern utility functions.
20.6.1 Theorem
If security payoffs have a mean-independent factor structure, if the risk-free claim, the factors, and
agents’ date-1 endowments lie in the asset span, if the aggregate date-1 endowment lies in the factor
span, and if agents are strictly risk averse, then exact factor pricing holds in any equilibrium in
which the consumption allocation is interior.
Proof: Let {c
i
} be a security markets equilibrium consumption allocation, which by Theorem
16.2.1 is constrained optimal. We first prove that the date-1 allocation {c
i
1
} lies in the factor span
F.
20.7. OPTIONS AS FACTORS 203
Since the risk-free claim and the factors lie in the asset span M, we have that M = F +
span{δ
1
, . . . , δ
J
}. Further, since all agents’ date-1 endowments lie in the asset span M, their date-1
equilibrium consumption plans c
i
1
lie in M as well. Therefore each c
i
1
can be decomposed into
c
i
1
= ˆc
i
1
+ ∆
i
, (20.37)
where ˆc
i
1
∈ F and ∆
i
∈ span{δ
1
, . . . , δ
J
}. It follows that
E(∆
i
|f
1
, . . . , f
K
) = 0, (20.38)
since the residuals δ
j
are mean-independent of the factors. Using 20.38 and ˆc
i
1
∈ F, we obtain
E(∆
i
|ˆc
i
1
) = 0. (20.39)
Equations 20.37 and 20.39 say that the consumption plan c
i
1
is more risky than ˆc
i
1
(and strictly so
if ∆
i
6= 0).
Since
I
X
i=1
c
i
1
= ¯w
1
∈ F, (20.40)
we have that
I
X
i=1
ˆc
i
1
= ¯w
1
, and
I
X
i=1
∆
i
= 0. (20.41)
Thus unless ∆
i
= 0 holds for every i, allocation {ˆc
i
} Pareto dominates {c
i
}, conflicting with the
constrained optimality of {c
i
}. Therefore ∆
i
= 0, which implies that
c
i
1
∈ F, (20.42)
for every i.
Since the consumption plan c
i
is interior and the von Neumann-Morgenstern utility function
is differentiable, the marginal rate of substitution ∂
1
v
i
/∂
0
v
i
is well-defined and is a function of
date-1 consumption. By Proposition 10.4.1, the marginal rate of substitution is uncorrelated with
residuals δ
j
; that is
E
Ã
∂
1
v
i
∂
0
v
i
δ
j
!
= 0, (20.43)
for every j. We observed in Section 17.10 that the pricing kernel equals the projection of the
marginal rate of substitution ∂
1
v
i
/∂
0
v
i
on the asset span. Taking into account that M = F +
span{δ
1
, . . . , δ
J
} and using 20.43, we obtain
k
q
∈ F. (20.44)
Theorem 20.2.1 implies now that exact factor pricing holds.
2
Note that if payoffs have mean-independent factor structure, then the assumption that the δ
i
are uncorrelated with each other is not needed for the proof of exact factor pricing.
20.7 Options as Factors
An important example of contingent claims that form a mean-independent factor structure is the
set of payoffs of options on the aggregate endowment. Let n be the number of different values
that the aggregate date-1 endowment ¯w
1
can take. Let ¯w
1k
denote the k-th value of the aggregate
204 CHAPTER 20. FACTOR PRICING
date-1 endowment, with ¯w
1k
< ¯w
1,k+1
, 1 ≤ k < n, and S
k
denote the subset of states s such that
¯w
1s
= ¯w
1k
.
Suppose that 1 < n so that the aggregate date-1 endowment is not risk-free. We consider
K ≡ n − 1 nonredundant call options on the aggregate date-1 endowment ¯w
1
. That number of
options, it should be noted, is one less than the maximal number of nonredundant options. For
concreteness, we choose strike prices a
k
= ¯w
1k
for k = 1, . . . , K, and we denote by z
k
the payoff of
the call option with strike price a
k
. We have
z
ks
= max{¯w
1s
− a
k
, 0}, (20.45)
so that z
ks
is nonzero for s ∈ S
`
and all ` > k. Define factor f
k
by
f
k
= z
k
− E(z
k
). (20.46)
The aggregate date-1 endowment lies in the span of factors 20.46 and the risk-free payoff (the
factor span). To see this, note that ¯w
1
= a
1
+ E(z
1
) + f
1
and therefore ¯w
1
lies in the span of factor
f
1
and the risk-free payoff. If the factors and the risk-free payoff lie in the asset span, then the
aggregate date-1 endowment lies in the asset span and is the market payoff. Note further that the
payoffs of all options on ¯w
1
lie in the factor span.
20.7.1 Proposition
Contingent claims 20.46 form a mean-independent factor structure.
Proof: Let δ
j
denote the residual of projection 20.35 of the payoff x
j
on the factor span of
factors 20.46. We have to show that
E(δ
j
|f
1
, . . . , f
K
) = 0 (20.47)
for every j.
The random vector (f
1
, . . . , f
K
) takes the same value in all states within each set S
k
, and
different values across sets S
k
. The latter follows from the observation that f
k
takes different
values in S
k
and S
k+1
. Therefore, 20.47 is equivalent to
E(δ
j
|S
k
) = 0 (20.48)
for every k. Let e
k
denote the contingent claim equal to one in each state of the set S
k
and zero in
all other states. Then 20.48 can be written as
E(δ
j
e
k
) = 0. (20.49)
It should be clear that contingent claim e
k
lies in the factor span F (see Section 15.4). Therefore
20.49 follows from the fact that δ
j
∈ F
⊥
.
2
If the factors and the risk-free claim lie in the asset span and if all agents are strictly risk averse,
then, as follows from Theorem 20.6.1, exact factor pricing holds in equilibrium. Further, it follows
from Section 15.4 that the equilibrium allocation is Pareto optimal.
Notes
Our analysis of Sections 20.2 and 20.5, based on general Hilbert space methods, can be extended
to the case of infinitely many securities with only minor modification. It remains true that exact
factor pricing holds iff the pricing kernel lies in the factor span. The approximate factor pricing
20.7. OPTIONS AS FACTORS 205
result says that all but a finite number of securities have arbitrarily small pricing errors. For more
discussion, see Chamberlain [2], Chamberlain and Rothschild [3], and Gilles and LeRoy [5].
The first systematic study of factor pricing is due to Ross [9] and [10] (see also Huberman [6]).
Ross developed what he referred to as the Arbitrage Pricing Theory (APT). The term “Arbitrage
Pricing Theory” is, however, a misnomer. The absence of arbitrage, or equivalently the strict posi-
tivity of the payoff pricing functional, is nowhere needed in this chapter. For example, approximate
factor pricing holds if security returns have factor structure independent of whether there exists an
arbitrage opportunity.
A factor structure with the market return (normalized so as to have zero expectation) as the
single factor was first analyzed by Sharpe [11], who referred to it as the market model . Exact
factor pricing in the market model is equivalent to the security market line of the CAPM.
The model of Section 20.6 is due to Connor [4], who referred to it as the Equilibrium APT
(see also Milne [8] and Werner [12]). The model with options on the aggregate endowment is
due to Breeden and Litzenberger [1]. The observation that this model is a special case of the
Equilibrium APT with mean-independent factor structure is due to Kim [7]. Kim proved that the
factor structure of options on the market payoff is in a precise sense minimal.
In Section 20.7 the term “options” was used to describe contingent claims that may or may
not lie in the asset span, that is, may or may not be traded. Evidently the term is completely
appropriate only in the former case.
The idea of portfolio diversification has often been brought up in connection with factor pricing
(Ross [9], Chamberlain [2], Chamberlain and Rothschild [3]). One usually thinks of a diversified
portfolio as a portfolio which contains small holdings of each of a large number of securities. When
security returns have a factor structure (Section 20.5), diversification can be used to reduce id-
iosyncratic risk in portfolios (that is, the risk in portfolio payoffs that reflects idiosyncratic risk in
securities’ payoffs). Of course, with a finite number of securities diversification cannot entirely elim-
inate idiosyncratic risk, but with an infinite number complete diversification is possible. Portfolios
can be constructed that have only factor risk.
When there is infinitely many securities and the security returns have a factor structure, the
possibility of constructing portfolios completely free of idiosyncratic risk provides a justification for
the assumption that factors lie in the asset span (see Werner [12]).
Note that, as shown, portfolio diversification plays no role in the derivation of approximate
factor pricing.
206 CHAPTER 20. FACTOR PRICING
Bibliography
[1] Douglas T. Breeden and Robert Litzenberger. Prices of state-contingent claims implicit in
option prices. Journal of Business, 51:621–651, 1978.
[2] Gary Chamberlain. Funds, factors and diversification in arbitrage pricing models. Economet-
rica, 51:1305–1323, 1983.
[3] Gary Chamberlain and Michael Rothschild. Arbitrage, factor structure and mean variance
analysis in large asset markets. Econometrica, 51:1281–1304, 1983.
[4] Gregory Connor. A unified beta pricing theory. Journal of Economic Theory, 34:13–31, 1984.
[5] Christian Gilles and Stephen F. LeRoy. On the arbitrage pricing theory. Economic Theory,
1:213–229, 1991.
[6] Gur Huberman. A simple approach to arbitrage pricing theory. Journal of Economic Theory,
28:183–192, 1982.
[7] Chongmin Kim. Stochastic dominance, Pareto optimality, and equilibrium asset pricing. Re-
view of Economic Studies, 65(2):341–356, 1998.
[8] Frank Milne. Arbitrage and diversification in a general equilibrium asset economy. Economet-
rica, 56:815–840, 1988.
[9] Stephen A. Ross. The arbitrage theory of capital asset pricing. Journal of Economic Theory,
13:341–360, 1976.
[10] Stephen A. Ross. Risk, return and arbitrage. In Irwin Friend and James Bicksler, editors,
Risk and Return in Finance. Ballinger, Cambridge, Massachusetts, 1976.
[11] William F. Sharpe. A simplified model of portfolio analysis. Management Science, 1963.
[12] Jan Werner. Diversification and equilibrium in securities markets. Journal of Economic Theory,
75:89–103, 1997.
207
208 BIBLIOGRAPHY
Part VII
Multidate Security Markets
209
Chapter 21
Equilibrium in Multidate Security
Markets
21.1 Introduction
We have thus far limited ourselves to a model of two-date security markets in which securities are
traded only once before their payoffs are realized. This model is most suitable for the study of the
risk-return relation for securities and the role of securities in the equilibrium allocation of risk.
In the two-date model all uncertainty is resolved at once. It is more realistic to assume that
uncertainty is resolved only gradually. As the uncertainty is resolved, agents trade securities again
and again. The multidate model of this and the following chapters allows for the gradual resolution
of uncertainty and the retrading of securities as new information about security prices and payoffs
becomes available.
21.2 Uncertainty and Information
In the multidate model, just as in the two-date model, uncertainty is specified by a set of states
S. Each of the states is a description of the economic environment for all dates t = 0, 1, . . . , T . At
date 0 agents do not know which state will be realized. But as time passes, they obtain more and
more information about the state. Then at date T the actual state becomes known to them.
Formally, the information of agents at date t is described by a partition F
t
of the set of states
S (a partition F
t
of S is a collection of subsets of S such that each state s belongs to exactly
one element of F
t
). The interpretation is that at date t agents know the element of the date-t
partition to which the actual state belongs. They do not know which state of the known element
of the date-t partition is the actual state, but they do know that states that do not belong to that
element cannot be realized. The partitions are assumed to be common across agents; that is, all
agents have the same information.
At date 0 agents have no information about the state, so that the date-0 partition is the trivial
partition F
0
= {S}. At date T agents have full information, so that the date-T partition is the
total partition F
T
= {{s} : s ∈ S}. At dates 1, . . . , T − 1 agents have intermediate amounts of
information. The partition F
t+1
is finer (but not necessarily strictly finer) than partition F
t
; that
is, the element of date-(t + 1) partition to which a state belongs is a subset of the element of date-t
partition to which it belongs. Equivalently, if two states belong to different elements of the date-t
partition, they cannot belong to the same element of the partition at any date after t. Thus agents
never forget anything they once knew; their information about the state is nondecreasing. The
(T + 1)-tuple of partitions {F
0
, F
1
, . . . , F
T
} is the information filtration F.
Another term for an information filtration (in the finite case studied here) is event tree. Each
211
212 CHAPTER 21. EQUILIBRIUM IN MULTIDATE SECURITY MARKETS
element of partition F
t
is called a date-t event and is a node of the event tree. The event ξ
0
= F
0
is the root node. The successors of the event ξ
t
are the events ξ
τ
⊂ ξ
t
, for τ > t. The immediate
successors of ξ
t
are the events ξ
t+1
⊂ ξ
t
. The predecessors of ξ
t
are the events ξ
τ
⊃ ξ
t
, for τ < t. The
unique immediate predecessor of ξ
t
is the event ξ
t−1
such that ξ
t−1
⊃ ξ
t
. Sometimes the immediate
predecessor of ξ
t
will be denoted ξ
−
t
.
The set of all events at all future dates t = 1, . . . , T is denoted Ξ, and k = #(Ξ) is the number
of events in Ξ. The number of events including ξ
0
is thus k + 1.
21.2.1 Example
Suppose that the only relevant information is the profit reports of two firms. Each of the reports
is either good (G) or bad (B). One firm issues its report at date 1, the other at date 2. The set
of states S consists of the four possible outcomes of the two reports: {GG, GB, BG, BB}. The
information filtration is
F
0
= {{GG, GB, BG, BB}}, (21.1)
F
1
= {{GG, GB}, {BG, BB}}, (21.2)
F
2
= {{GG}, {GB}, {BG}, {BB}}, (21.3)
so that at date 0 agents know nothing, at date 1 they know the profit report of the first firm, and
at date 2 they know the profit reports of both firms.
Since this example will come up again, it is convenient to introduce a compact notation for
events. Thus we let
ξ
g
≡ {GG, GB}, ξ
b
≡ {BG, BB} (21.4)
be the two date-1 events and
ξ
gg
≡ {GG}, ξ
gb
≡ {GB}, ξ
bg
≡ {BG}, ξ
bb
≡ {BB}, (21.5)
be the four date-2 events. The set of all future events is Ξ = {ξ
g
, ξ
b
, ξ
gg
, ξ
gb
, ξ
bg
, ξ
bb
}.
2
Agents’ information about the state has to be properly reflected in all economic variables such
as endowments, security prices and dividends, portfolio holdings, consumption plans, and so forth.
Specifically, it would not make sense to consider consumption plans or security prices at date t
that differ in states that cannot be distinguished based on the information available to agents at
date t. One way to specify these variables is to represent them as functions on the set of states
S and require that they be measurable with respect to the partition F
t
. If consumption at date t
is represented by a function c
t
: S → R that takes value c
t
(s) in state s, then measurability of c
t
with respect to partition F
t
requires that c
t
(s) = c
t
(s
0
) for each s and s
0
that belong to a common
element ξ
t
of F
t
.
The measurability requirement can be embedded in the notation by using events rather than
states to distinguish different values of functions. If c
t
is measurable with respect to F
t
then, by
definition, c
t
(s) = c
t
(s
0
) for all s, s
0
in a given date-t event ξ
t
and we can denote this common value
by c(ξ
t
).
1
At times we will use c
t
to denote the vector (of dimension equal to the number of events at
date t) of values c(ξ
t
) for all ξ
t
∈ F
t
. Thus we use the same notation c
t
for the consumption plan
as an F
t
-measurable function and as a vector. The distinction often does not matter; when it does
the intended meaning will always be clear from the context. Similarly, we use c to denote either
a (T + 1)-tuple of F
t
-measurable functions c
t
or a (k + 1)-dimensional vector of values c(ξ) for all
ξ ∈ Ξ.
1
Note that we write c(ξ
t
) instead of c
t
(ξ
t
) to simplify notation.
21.3. MULTIDATE SECURITY MARKETS 213
The importance of the distinction between functions and vectors will become evident when
probabilities are associated with the states (Chapter 25) . When that it done, measurable func-
tions on S will be identified with random variables. In order to verify conformability for matrix
operations, it is necessary to be clear when a scalar random variable (for example) is intended, as
opposed to the vector of values the random variable takes on.
If every function c
t
in the (T + 1)-tuple c is F
t
-measurable, then c is adapted to the information
filtration F.
21.3 Multidate Security Markets
There exist J securities. Examples of securities include bonds, stocks, options, and futures con-
tracts. Each security is characterized by the dividends it pays at each date. By the dividend we
mean any payment to which a security holder is entitled. For stocks, dividends are firms’ profit
distributions to stockholders; for bonds, dividends are coupon payments and payments at maturity.
The dividend on security j in event ξ
t
is denoted by x
j
(ξ
t
). We use x
jt
to denote the vector of
dividends x
j
(ξ
t
) in all date-t events ξ
t
, and x
t
to denote the vector of dividends on all J securities
in all date-t events. There are no dividends at date 0. It is possible that a security has nonzero
dividend only at a single date. For instance, a zero-coupon bond that matures at date t with face
value 1 has dividends equal to 1 in each date-t event and zero dividends at all other dates.
Securities are traded at all dates except the terminal date T . The price of security j in event ξ
t
is denoted by p
j
(ξ
t
) . For notational convenience we have date-T prices p
j
(ξ
T
) even though trade
does not take place at date T . These prices are all set equal to zero. We use p
jt
to denote the
vector of prices p
j
(ξ
t
) in all date-t events ξ
t
, and p
t
to denote the vector of prices of all J securities
in all date-t events.
The holding of security j in event ξ
t
is denoted by h
j
(ξ
t
), and the portfolio of J securities in
event ξ
t
is denoted by the vector h(ξ
t
). The holding of each security may be positive, zero or (unless
a short sales constraint has been imposed) negative. We have again, for notational convenience,
a date-T portfolio h(ξ
T
), which, though, is set equal to zero. We use h
t
to denote the vector of
portfolios h(ξ
t
) in all date-t events ξ
t
. The (T + 1)-tuple h = (h
0
, . . . , h
T
) is a portfolio strategy.
The payoff of a portfolio strategy h in event ξ
t
, denoted by z(h, p)(ξ
t
), is the cum-dividend
payoff of the portfolio chosen at immediate predecessor event ξ
−
t
minus the price of the portfolio
chosen in ξ
t
. Thus
z(h, p)(ξ
t
) ≡ (p(ξ
t
) + x(ξ
t
))h(ξ
−
t
) − p(ξ
t
)h(ξ
t
). (21.6)
We use z
t
(h, p) to denote the vector of payoffs z(h, p)(ξ
t
) in all date-t events ξ
t
. The price at date
0 of a portfolio strategy h is p(ξ
0
)h(ξ
0
).
We present two examples of portfolio strategies and their payoffs.
21.3.1 Example
Consider the portfolio strategy that involves buying one share of security j in event ξ
t
at date t ≥ 1
and selling it in every immediate successor event of ξ
t
. This portfolio strategy is represented by
the vector h which has 1 in the position associated with the holding of security j in event ξ
t
and
zeros elsewhere. It has payoff −p
j
(ξ
t
) in ξ
t
, p
j
(ξ
t+1
) + x
j
(ξ
t+1
) in each immediate successor event
ξ
t+1
⊂ ξ
t
, and zero elsewhere. The date-0 price of this portfolio strategy is zero.
A buy-and-hold strategy involves holding one share of security j in every event of the event
tree. It is represented by a vector with 1 in the position associated with the holding of security j
in all events except those at the terminal date, and zeros elsewhere. Its payoff equals the dividend
x
j
(ξ
t
) in each event ξ
t
for every t ≥ 1. Its date-0 price equals the date-0 price of security j, p
j
(ξ
0
).
2
214 CHAPTER 21. EQUILIBRIUM IN MULTIDATE SECURITY MARKETS
As discussed in section 21.2, date-t dividend x
jt
, price p
jt
, portfolio h
t
and payoff z
t
(h, p) can
also be understood as F
t
-measurable functions.
21.4 The Asset Span
The set of payoffs available via trades on security markets is the asset span and is defined by
M(p) = {(z
1
, . . . , z
T
) ∈ R
k
: z
t
= z
t
(h, p) for some h, and all t ≥ 1}. (21.7)
The payoffs of the portfolio strategies of Example 21.3.1 belong to the asset span. In particular,
dividends (x
j1
, . . . , x
jT
) of each security j belong to the asset span M(p) for arbitrary security
prices p.
An important distinction between the two-date model and the multidate model is that in the
former the asset span is exogenous, depending only on specified security payoffs. In the latter, on
the other hand, the asset span depends on security prices, which are endogenous.
Security markets are dynamically complete (at prices p) if any consumption plan for future dates
(dates 1 to T ) can be obtained as the payoff of a portfolio strategy, that is if M(p) = R
k
. Markets
are incomplete if M(p) is a proper subspace of R
k
.
21.5 Agents
Measures of consumption c(ξ
t
), c
t
and c were defined in Section 21.2.
Agents are assumed to have utility functions defined on the set of all consumption plans c =
(c
0
, c
1
, . . . , c
T
). As in Chapter 1, we assume most of the time that consumption is positive. In
that case the utility function of agent i is u
i
: R
k+1
+
→ R. Utility functions are assumed to be
continuous and increasing.
2
The endowment of agent i is w
i
= (w
i
0
, . . . , w
i
T
) ∈ R
k+1
+
.
21.6 Portfolio Choice and the First-Order Conditions
The consumption-portfolio choice problem of an agent with the utility function u is
max
c,h
u(c) (21.8)
subject to
c(ξ
0
) = w(ξ
0
) − p(ξ
0
)h(ξ
0
) (21.9)
c(ξ
t
) = w(ξ
t
) + z(h, p)(ξ
t
) ∀ξ
t
t = 1, . . . , T, (21.10)
and the restriction that consumption be positive, c ≥ 0, if this restriction is imposed. Budget con-
straints 21.9 and 21.10 are written as equalities since utility functions are assumed to be increasing.
Budget constraints 21.9 and 21.10 can be written as
c
0
= w
0
− p
0
h
0
(21.11)
and
c
t
= w
t
+ z
t
(h, p), t = 1, . . . , T. (21.12)
2
Utility function u is increasing at date t if u(c
0
, . . . , c
0
t
, . . . , c
T
) ≥ u(c
0
, . . . , c
t
, . . . , c
T
) whenever c
0
t
≥ c
t
for
every (c
0
, . . . , c
T
); u is increasing if it is increasing at every date. Further, u is strictly increasing at date t if
u(c
0
, . . . , c
0
t
, . . . , c
T
) > u(c
0
, . . . , c
t
, . . . , c
T
) whenever c
0
t
> c
t
for every (c
0
, . . . , c
T
); and u is strictly increasing if it is
strictly increasing at every date.
21.7. GENERAL EQUILIBRIUM 215
If the utility function u is differentiable, the necessary first-order conditions for an interior
solution to the consumption-portfolio choice problem 21.8 are
∂
ξ
t
u − λ(ξ
t
) = 0 , ∀ξ
t
t = 0, . . . , T, (21.13)
λ(ξ
t
)p(ξ
t
) =
X
ξ
t+1
⊂ξ
t
(p(ξ
t+1
) + x(ξ
t+1
))λ(ξ
t+1
), ∀ξ
t
t = 0, . . . , T −1, (21.14)
where λ(ξ
t
) is the Lagrange multiplier associated with budget constraint 21.10. Here ∂
ξ
t
u denotes
the partial derivative of u with respect to c(ξ
t
) evaluated at the optimal consumption. If u is
quasi-concave, then these conditions together with budget constraints 21.9 and 21.10 are sufficient
to determine an optimal consumption-portfolio choice.
Assuming that ∂
ξ
t
u > 0, 21.14 becomes
p(ξ
t
) =
X
ξ
t+1
⊂ ξ
t
(p(ξ
t+1
) + x(ξ
t+1
))
∂
ξ
t+1
u
∂
ξ
t
u
(21.15)
with typical element
p
j
(ξ
t
) =
X
ξ
t+1
⊂ ξ
t
(p
j
(ξ
t+1
) + x
j
(ξ
t+1
))
∂
ξ
t+1
u
∂
ξ
t
u
. (21.16)
Eq. 21.16 says that the price of security j in event ξ
t
equals the sum over immediate successor
events ξ
t+1
of cum-dividend payoffs of security j multiplied by the marginal rate of substitution
between consumption in event ξ
t+1
and consumption in event ξ
t
. Thus the relation between the
price of a security at any date and its payoff at the next date is the same in the multidate model
as in the two-date model.
21.7 General Equilibrium
An equilibrium in multidate security markets consists of a vector of security prices p, an allocation
of portfolio strategies {h
i
} and a consumption allocation {c
i
} such that (1) portfolio strategy h
i
and consumption plan c
i
are a solution to agent i’s choice problem 21.8 at prices p, and (2) markets
clear; that is
X
i
h
i
= 0, (21.17)
and
X
i
c
i
=
X
i
w
i
. (21.18)
The portfolio market-clearing condition 21.17 implies, by summing over agents’ budget con-
straints, the consumption market-clearing condition 21.18. If there are no redundant securities
(that is, if z(h, p) = 0 implies h = 0), then the converse is also true. If there are redundant se-
curities, then at least one of the multiple portfolio allocations associated with a market-clearing
consumption allocation is market-clearing.
As in the two-date model, securities are in zero supply, as seen in the market-clearing condition
21.17. However, a reinterpretation of notation can be used to accommodate the case in which
securities are in positive supply. Specifically, suppose that each agent is endowed with an initial
portfolio
¯
h
i
0
but (for simplicity) with no consumption endowments at any future event. The market-
clearing condition for optimal portfolio strategies
ˆ
h
i
under that specification of endowments is
X
i
ˆ
h
i
(ξ
t
) =
X
i
¯
h
i
0
(ξ
t
), ∀ ξ
t
. (21.19)
This agrees with 21.17 if h
i
is interpreted as a net trade: h
i
≡
ˆ
h
i
0
−
¯
h
i
0
.
216 CHAPTER 21. EQUILIBRIUM IN MULTIDATE SECURITY MARKETS
Notes
The event-tree model of gradual resolution of uncertainty is inadequate when time is continuous and
the set of states is infinite. In a continuous-time setting agents’ information at date t is described
by a sigma-algebra (sigma-field) of events instead of a partition.
The notion of general equilibrium in multidate security markets is due to Radner [5]. Radner
referred to the equilibrium of Section 21.7 as an equilibrium of plans, prices and price expectations.
This term emphasizes the fact that future security prices are to be thought of as agents’ price
anticipations, with rational expectations assumed. All agents have the same price anticipations;
these anticipations are correct in the sense that the anticipated prices turn out to be equilibrium
prices when an event is realized.
As in the two-date model, our specification is restricted to the case of a single good. The
multiple-goods generalization of the model analyzed here is the general equilibrium model with
incomplete markets (GEI); see Geanakoplos [3] and Magill and Quinzii [4]. Unlike in the two-
date model, the existence of a general equilibrium in security markets is not guaranteed under
the standard assumptions. The reason is the dependence of the asset span on security prices. As
prices change the asset span may change in dimension, inducing discontinuity of agents’ portfolio
and consumption demands. For an example of nonexistence of an equilibrium in multidate security
markets see Magill and Quinzii [4]. The nonexistence examples are in some sense rare. Results of
Duffie and Shafer [2] (see also Duffie [1]) imply that for a generic set of agents’ endowments and
securities’ dividends an equilibrium exists.
Bibliography
[1] Darrell Duffie. Stochastic equilibria with incomplete financial markets. Journal of Economic
Theory, 41:405–416, 1987.
[2] Darrell Duffie and Wayne Shafer. Equilibrium in incomplete markets ii: Generic existence in
stochastic economies. Journal of Mathematical Economics, 15:199–216, 1986.
[3] John Geanakoplos. An introduction to general equilibrium with incomplete asset markets.
Journal of Mathematical Economics, 19:1–38, 1990.
[4] Michael Magill and Martine Quinzii. Theory of Incomplete Markets. MIT Press, 1996.
[5] Roy Radner. Existence of equilibrium of plans, prices and price expectations in a sequence
economy. Econometrica, 40:289–303, 1972.
217
218 BIBLIOGRAPHY
Chapter 22
Multidate Arbitrage and Positivity
22.1 Introduction
In multidate security markets, just as in two-date markets, there are two properties of the rela-
tion between future payoffs and their current prices that are of special importance: linearity and
positivity. We can be brief here because the central concepts were presented in our discussion in
Chapters 2 and 3 of that relation in the two-date model.
22.2 Law of One Price and Linearity
The law of one price holds in multidate markets if any two portfolio strategies that have the same
payoff have the same date-0 price, that is
if z(h, p) = z(h
0
, p), then p
0
h
0
= p
0
h
0
0
. (22.1)
Condition 22.1 holds iff p
0
h
0
= 0 for every portfolio strategy h with payoff z(h, p) equal to zero.
As in two-date security markets (recall Theorems 2.4.1 and 2.4.2), the law of one price holds
in equilibrium in multidate security markets if agents’ utility functions are strictly increasing at
date-0.
1
Henceforth we assume that the law of one price holds.
The payoff pricing functional is a mapping
q : M(p) → R (22.2)
defined by
q(z) = p
0
h
0
, (22.3)
where h is such that z = z(h, p) for z ∈ M(p). The law of one price guarantees that the date-0
price p
0
h
0
is the same for every portfolio h that generates payoff z.
The payoff pricing functional q assigns to each payoff the date-0 price of a portfolio strategy
that generates it. The law of one price implies that q a linear functional on M(p).
Since the dividends of each security are generated by a buy-and-hold portfolio strategy (recall
Example 21.3.1), we have x
j
∈ M(p) for any p. The date-0 price of the buy-and-hold strategy is
p
j0
, so
q(x
j
) = p
j0
. (22.4)
1
An alternative sufficient condition is that (1) there exists a portfolio strategy with positive and nonzero payoff,
and (2) utility functions are strictly increasing at any date at which that payoff is nonzero.
219
220 CHAPTER 22. MULTIDATE ARBITRAGE AND POSITIVITY
22.3 Arbitrage and Positive Pricing
A strong arbitrage in multidate security markets is a portfolio strategy h that has positive payoff
z(h, p) and strictly negative date-0 price p
0
h
0
. An arbitrage is a portfolio strategy that either is a
strong arbitrage or has a positive and nonzero payoff and zero date-0 price.
As in two-date markets, there can exist a portfolio strategy that is an arbitrage but not a strong
arbitrage:
22.3.1 Example
Going back to Example 21.2.1, suppose that there exists a single security with dividend equal to 1
in events ξ
gg
and ξ
gb
at date 2 and zero otherwise. This security is risky as of date 0, but it becomes
risk-free at date 1. If its prices are p(ξ
0
) = 0, p(ξ
g
) = −1 and p(ξ
b
) = 0, then the portfolio strategy
of buying the security in event ξ
g
and selling it at both subsequent events, with zero holdings at
all other events, is an arbitrage but not a strong arbitrage.
2
We recall that payoff pricing functional q is positive if q(z) ≥ 0 for every z ≥ 0, z ∈ M(p).
It is strictly positive if q(z) > 0 for every z > 0, z ∈ M(p). The equivalence between positivity
(strict positivity) of the payoff pricing functional and the exclusion of strong arbitrage (arbitrage)
also holds in multidate security markets (compare Theorems 3.4.1 and 3.4.2 ).
22.3.2 Theorem
The payoff pricing functional is strictly positive iff there is no arbitrage.
Proof: Exclusion of arbitrage means that p
0
h
0
> 0 whenever z(h, p) > 0. Since q(z(h, p)) =
p
0
h
0
, this is precisely the property of q being strictly positive on M(p).
2
22.3.3 Theorem
The payoff pricing functional is positive iff there is no strong arbitrage.
The following example illustrates the possibility of a payoff pricing functional that is positive
but not strictly positive.
22.3.4 Example
The payoff pricing functional associated with the prices of the single security of Example 22.3.1
assigns zero to every payoff. This is a consequence of the security price at date 0 being equal to
zero. The zero functional is positive but not strictly positive.
2
22.4 One-Period Arbitrage
The definitions of strong arbitrage and arbitrage of the two-date model can be applied to any
nonterminal event of the multidate model. This leads us to the concepts of one-period strong
arbitrage and one-period arbitrage which are closely related to the concepts of Section 22.3.
A one-period strong arbitrage in event ξ
t
at date t < T is a portfolio h(ξ
t
) that has a positive
one-period payoff
(p(ξ
t+1
) + x(ξ
t+1
))h(ξ
t
) ≥ 0 for every ξ
t+1
⊂ ξ
t
, (22.5)
and a strictly negative price
p(ξ
t
)h(ξ
t
) < 0. (22.6)
22.5. POSITIVE EQUILIBRIUM PRICING 221
A one-period arbitrage in event ξ
t
is a portfolio h(ξ
t
) that either is a one-period strong arbitrage or
has a positive and nonzero one-period payoff and a zero price.
The exclusion of one-period arbitrage at every nonterminal event is equivalent to the exclusion of
multidate arbitrage in the sense of Section 22.3. Only one direction of the corresponding equivalence
holds for strong arbitrage. The exclusion of one-period strong arbitrage at every nonterminal event
implies the exclusion of multidate strong arbitrage. However, the converse is not true. In Example
22.3.1 there exists one-period strong arbitrage at ξ
g
but there is no multidate strong arbitrage.
22.5 Positive Equilibrium Pricing
The payoff pricing functional associated with equilibrium security prices is referred to as the equi-
librium payoff pricing functional. Under appropriate monotonicity properties of agents’ utility
functions, there cannot be an arbitrage or a strong arbitrage at equilibrium prices. The equilib-
rium pricing functional is then strictly positive or positive.
22.5.1 Theorem
If agents’ utility functions are strictly increasing, then there is no arbitrage at equilibrium security
prices. Further, the equilibrium payoff pricing functional is strictly positive.
Proof: Suppose that there exists a portfolio strategy h that is an arbitrage. Thus z(h, p) ≥ 0
and p
0
h
0
≤ 0, with at least one strict inequality. Let h
i
and c
i
be agent i’s equilibrium portfolio
strategy and consumption plan. Then h
i
+h and c
i
+(−p
0
h
0
, z(h, p)) satisfy the budget constraints
and, since utility u
i
is strictly increasing, the latter consumption plan is strictly preferred to c
i
. We
obtain a contradiction. Theorem 22.3.2 implies now that the equilibrium payoff pricing functional
is strictly positive.
2
22.5.2 Theorem
If agents’ utility functions are increasing, and are strictly increasing at date 0, then there is no
strong arbitrage at equilibrium security prices. Further, the equilibrium payoff pricing functional is
positive.
The proof is similar to that for Theorem 22.5.1.
It is sometimes convenient to assume that consumption in a multidate model takes place only
at the initial and terminal dates. Theorem 22.5.1 cannot be applied if that is the case since utility
is not strictly increasing at intermediate dates. A variation that does apply is the following:
22.5.3 Theorem
If agents’ utility functions are increasing, and are strictly increasing at date T , and if there exists
a portfolio the payoff of which is positive at every date and strictly positive at date T , then there
is no arbitrage at equilibrium security prices. Further, the equilibrium payoff pricing functional is
strictly positive.
Proof: Let security j be such that x
jt
≥ 0 for every t ≥ 1 and x
jT
> 0. The equilibrium
price p
jt
must be strictly positive at every date t < T in every event, for otherwise an agent could
purchase security j in an event in which the price is negative, hold it through date T and thereby
strictly increase his consumption at date T .
Let h
i
and c
i
be agent i’s equilibrium portfolio strategy and consumption plan. Suppose that
there exists a portfolio strategy h that is an arbitrage. Thus z(h, p) ≥ 0 and p
0
h
0
≤ 0, with
at least one strict inequality. If z
T
(h, p) > 0, then we obtain a contradiction to the optimality
of h
i
and c
i
in exactly the same way as in the proof of Theorem 22.5.1. If z
T
(h, p) = 0 but
222 CHAPTER 22. MULTIDATE ARBITRAGE AND POSITIVITY
p
0
h
0
< 0, then purchasing security j at the cost equal to −p
0
h
0
, holding it (and portfolio h)
through date T strictly increases an agent’s consumption at date T . Specifically, for portfolio
ˆ
h = h + (0, . . . , α, . . . , 0) where α is the jth coordinate and is defined by αp
j0
= −p
0
h
0
, we have
that h
i
+
ˆ
h and c
i
+ (−p
0
ˆ
h
0
, z(
ˆ
h, p)) satisfy the budget constraints and the latter consumption plan
is strictly preferred to c
i
. If z
T
(h, p) = 0 and p
0
h
0
= 0 but z(h, p)(ξ
t
) > 0 for some ξ
t
, then a
similar argument as in the case of p
0
h
0
< 0 applies. Purchasing security j in event ξ
t
and holding
it (and portfolio h) through date T increases the agent’s utility. Thus we have a contradiction.
2
Thus Theorems 3.6.3 and 3.6.1 extend from the two-date to the multidate model. Note that the
security prices of Example 22.3.1 could not be equilibrium prices under strictly increasing utility
functions.
Notes
As in two-date security markets, the assumption of no arbitrage plays a central role in multidate
markets. Influential papers in which the importance of arbitrage is recognized are Ross [3], Black
and Scholes [1] and Harrison and Kreps [2].
Bibliography
[1] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of
Political Economy, 81:637–654, 1973.
[2] J. Michael Harrison and David M. Kreps. Martingales and arbitrage in multiperiod securities
markets. Journal of Economic Theory, 20:381–408, 1979.
[3] Stephen A. Ross. A simple approach to the valuation of risky streams. Journal of Business,
51:453–475, 1978.
223
224 BIBLIOGRAPHY
Chapter 23
Dynamically Complete Markets
23.1 Introduction
As defined in Chapter 21, security markets are dynamically complete (at prices p) if any consump-
tion plan for future dates can be obtained as a payoff of a portfolio strategy; that is, if M(p) = R
k
.
Security markets are incomplete if M(p) is a proper subspace of R
k
.
In the two-date model of Chapter 1 completeness of security markets requires the existence of
at least as many securities as states. In the multidate model the opportunity to trade securities
at future dates implies that many fewer securities than events are necessary for markets to be
dynamically complete.
In this chapter we provide a characterization of dynamically complete security markets and
show that, for such markets, equilibrium consumption allocations are Pareto optimal.
23.2 Dynamically Complete Markets
An example of securities that result in markets that are dynamically complete at arbitrary prices
are the Arrow securities . The Arrow security for event ξ
t
has a dividend of one in event ξ
t
at date
t and zero in all other events and at all other dates. If all k Arrow securities are traded, then any
consumption plan in R
k
can be generated using a buy-and-hold portfolio strategy.
With Arrow securities, markets are dynamically complete even if trading is limited to date 0.
As noted in Section 23.1, the opportunity to trade at future dates significantly reduces the number
of securities needed for dynamically complete markets. A simple characterization of dynamically
complete markets obtains as an extension of the characterization of complete markets in the two-
date model (see Chapter 1).
The one-period payoff matrix in event ξ
t
at date t, t < T , is a J × k(ξ
t
) matrix with entries
p
j
(ξ
t+1
) + x
j
(ξ
t+1
) for all j and all immediate successors ξ
t+1
of ξ
t
. Here k(ξ
t
) is the number of
immediate successors of event ξ
t
.
23.2.1 Theorem
Markets are dynamically complete iff the one-period payoff matrix in each nonterminal event ξ
t
is
of rank k(ξ
t
).
Proof: Markets are dynamically complete iff, for each nonterminal event ξ
t
and arbitrary
payoffs in immediate successors of ξ
t
, there exists a portfolio that generates those payoffs. Such
portfolio exists iff the one-period payoff matrix in ξ
t
has rank k(ξ
t
). That follows from the charac-
terization of complete security markets for the two-date model as given in Theorem 1.2.1.
2
225
226 CHAPTER 23. DYNAMICALLY COMPLETE MARKETS
It follows that the minimum number of securities required for markets to be dynamically com-
plete equals the maximum number of branches emerging from any node of the event tree. Having
that number of securities is not, however, always sufficient; security prices may be such that one-
period payoffs of securities are redundant in some events, so that markets may be incomplete even
if there exist the necessary number of securities.
23.2.2 Example
In Example 21.2.1 two branches emerge from each nonterminal node, so the necessary condition
for market completeness is that there exist at least two securities.
To see that this condition is not sufficient, suppose that there exist two securities with dividends
x
1
(ξ
g
) = x
1
(ξ
b
) = 0, x
1
(ξ
gg
) = x
1
(ξ
bb
) = 1, x
1
(ξ
gb
) = x
1
(ξ
bg
) = 0, (23.1)
and
x
2
(ξ
g
) = x
2
(ξ
b
) = 0, x
2
(ξ
gg
) = x
2
(ξ
bb
) = 0, x
2
(ξ
gb
) = x
2
(ξ
bg
) = 1. (23.2)
The one-period payoff matrix in each date-1 event is of rank two. However, if the price of each
security in the two date-1 events equals 1/2, then the one-period payoff matrix at date 0 is of rank
one. Thus markets are incomplete. There is no way for agents to trade securities at date 0 so as
to obtain different one-period payoffs in the two date-1 events.
2
23.3 Binomial Security Markets
A binomial event tree is an event tree with an arbitrary number of dates T such that at every
nonterminal date each event has exactly two immediate successors, “up” and “down”. The simplest
example of a binomial event tree was given in Section 21.2.1. Another example follows.
23.3.1 Example
Suppose that there are two securities traded at every date: a discount bond b maturing at date T and
a risky stock a. The dividend of the bond at date T is 1 and its price at date t is p
b
(ξ
t
) = (¯r)
−(T −t)
for every event ξ
t
. The price of the stock at date 0 is p
a0
= 1. In the two possible events at date
1 the price of the stock is u or d (u > d) depending on whether the “up” or “down” event occurs.
Stock prices at subsequent dates are defined similarly; the one-period return on the stock is always
u or d. The stock price at date t is therefore p
a
(ξ
t
) = u
t−l
d
l
in an event ξ
t
such that the number
of “downs” preceding it from date 0 to date t is l where 1 ≤ l ≤ t. The dividend on the stock is
nonzero only at the terminal date T , and is x
a
(ξ
T
) = u
T −l
d
l
in an event ξ
T
such that the number
of “downs” preceding it is l.
Such binomial security markets are dynamically complete. At every date and in every nonter-
minal event, the one-period return matrix is
"
¯r ¯r
u d
#
which has full rank 2 since u > d by assumption. Thus we have dynamically complete markets
with two securities and 2
T
events at terminal date T .
The particular specifications of stock and bond prices in this example are very restrictive. For
instance, there is no reason in general to expect the one-period return on the bond to be the same in
every nonterminal event. The property of dynamic completeness does not require this simplification;
all that is needed is that the one-period payoff matrix be of full rank at each nonterminal event.
2
23.4. EVENT PRICES IN DYNAMICALLY COMPLETE MARKETS 227
23.4 Event Prices in Dynamically Complete Markets
If security markets are dynamically complete, then the payoff pricing functional q is a linear
functional on the space R
k
. It can be identified by its values on the unit vectors in R
k
. The
event-ξ unit vector, denoted by e(ξ), is the dividend of the Arrow security associated with ξ. We
define q(ξ) ≡ q(e(ξ)) and refer to q(ξ) as the event price of ξ.
Since every z ∈ R
k
can be written as z =
P
ξ∈Ξ
z(ξ)e(ξ), we have
q(z) = q(
X
ξ∈Ξ
z(ξ)e(ξ)) =
X
ξ∈Ξ
q(e(ξ))z(ξ) =
X
ξ∈Ξ
q(ξ)z(ξ). (23.3)
Equation 23.3 is the representation of the payoff pricing functional by event prices. Using the same
notation to denote the functional q and the k-dimensional vector of event prices q(ξ) for all ξ ∈ Ξ,
23.3 can be written
q(z) = qz. (23.4)
Event prices are (strictly) positive iff the payoff pricing functional is (strictly) positive. The-
orems 3.4.1 and 3.4.2 allow us to conclude that event prices are strictly positive iff there is no
arbitrage and positive iff there is no strong arbitrage. Thus, calculating event prices and deter-
mining whether they are strictly positive (positive) is a way of verifying whether security prices
exclude arbitrage (strong arbitrage).
The event prices associated with security prices p can be calculated by finding portfolio strategies
with payoffs e(ξ) for all ξ. The event price q(ξ) is then the date-0 price of the portfolio strategy
with payoff e(ξ). It is more convenient to describe event prices as a solution to a system of linear
equations as in two-date security markets (see Chapter 2). The event prices satisfy:
q(ξ
t
) p
j
(ξ
t
) =
X
ξ
t+1
⊂ξ
t
q(ξ
t+1
)(p
j
(ξ
t+1
) + x
j
(ξ
t+1
)), (23.5)
for every event ξ
t
, t ≥ 0, and every security j, with q(ξ
0
) set equal to 1.
To prove this consider the portfolio strategy of buying one share of security j at date t ≥ 1
in event ξ
t
and selling it at the subsequent date t + 1 in every possible successor event ξ
t+1
⊂
ξ
t
(see Example 21.3.1). Denoting this portfolio strategy by
ˆ
h, we have z(
ˆ
h, p)(ξ
t
) = −p
j
(ξ
t
),
z(
ˆ
h, p)(ξ
t+1
) = p
j
(ξ
t+1
) + x
j
(ξ
t+1
) for ξ
t+1
⊂ ξ
t
, and z(
ˆ
h, p)(ς) = 0 in all other events ς. Since
ˆ
h
0
= 0, we have that q(z(
ˆ
h, p)) = p
0
ˆ
h
0
= 0. Using the representation 23.4 of the payoff pricing
functional by event prices, we obtain 23.5.
Eq. 23.5 for t = 0 is derived from the portfolio strategy consisting of buying one share of security
j at date 0 and selling it in all date-1 events. This portfolio strategy has the payoff p
j
(ξ
1
) + x
j
(ξ
1
)
in each date-1 event ξ
1
and zero elsewhere. Its date-0 price is p
j
(ξ
0
), so 23.5 results.
The system of equations 23.5 can be solved for event prices q under given security prices p. One
starts by solving for date-1 event prices. Knowing these, one can solve for date-2 event prices from
appropriate versions of 23.5; and so on. In the case of nonzero event prices, one can alternatively
rewrite equations 23.5 in terms of relative event prices q(ξ
t+1
)/q(ξ
t
), solve for the relative prices,
and then calculate event prices from the relative prices. Note that the satisfaction of the rank
condition of Theorem 23.2.1 assures a unique solution for equations 23.5.
Results of this section will be extended to incomplete markets in Chapter 24.
23.5 Event Prices in Binomial Security Markets
Event prices in the binomial security markets of Example 23.3.1 can easily be found using 23.5.
We have two equations for the two securities in each event ξ
t
:
q(ξ
t
) = uq(ξ
u
t+1
) + dq(ξ
d
t+1
) (23.6)
228 CHAPTER 23. DYNAMICALLY COMPLETE MARKETS
and
q(ξ
t
) = ¯rq(ξ
u
t+1
) + ¯rq(ξ
d
t+1
), (23.7)
where ξ
u
t+1
and ξ
d
t+1
denote the immediate successor events of event ξ
t
.
The solution for relative event prices is
q(ξ
u
t+1
)
q(ξ
t
)
=
¯r − d
¯r(u − d)
(23.8)
q(ξ
d
t+1
)
q(ξ
t
)
=
u − ¯r
¯r(u − d)
(23.9)
for every ξ
t
. The event price of event ξ
t
at date t such that the number of “downs” preceding it is
l is
q(ξ
t
) =
µ
u − ¯r
¯r(u − d)
¶
l
µ
¯r − d
¯r(u − d)
¶
t−l
. (23.10)
Event prices q(ξ
t
) are strictly positive iff u > ¯r > d, that is, if the one-period risk-free return
is between the high and the low one-period returns on the risky security. In that case there is
no arbitrage in the binomial security markets. Event prices are positive and there is no strong
arbitrage if u ≥ ¯r ≥ d.
23.6 Equilibrium in Dynamically Complete Markets
An agent’s consumption-portfolio choice problem in multidate security markets is
max
c,h
u(c) (23.11)
subject to
c
0
= w
0
− p
0
h
0
(23.12)
c
t
= w
t
+ z
t
(h, p), t ≥ 1. (23.13)
Since the price p
0
h
0
of portfolio strategy h at date 0 equals the value of its payoff under the
payoff pricing functional q, the budget constraint 23.12 can be written as
c
0
= w
0
− q(c
1+
− w
1+
), (23.14)
where c
1+
denotes the consumption plan c from date 1 on, that is, c
1+
= (c
1
, . . . , c
T
), so that
c = (c
0
, c
1+
). The budget constraint 23.13 can be rewritten as
c
1+
− w
1+
∈ M(p). (23.15)
Consequently, we can rewrite the optimization problem 23.11 as
max
c
u(c) (23.16)
subject to 23.14 and 23.15. If markets are dynamically complete, then M(p) = R
k
and restriction
23.15 is vacuous. Moreover, the budget constraint 23.14 can be written as
c
0
+ qc
1+
= w
0
+ qw
1+
, (23.17)
where q is the vector of event prices associated with security prices p.
The optimization problem 23.16 becomes utility maximization under the single budget con-
straint 23.17. This latter maximization problem is the consumption choice problem of agent i
facing complete contingent commodity markets. At price q(ξ) the agent can purchase one unit of
23.7. PARETO-OPTIMAL EQUILIBRIA 229
consumption in event ξ. One unit of date-0 consumption has price 1. The first-order condition for
an interior solution to the utility maximization under the budget constraint 23.17 is
q(ξ) =
∂
ξ
u
∂
ξ
0
u
(23.18)
for every event ξ.
The equivalence of the optimization problem 23.11 and utility maximization under the single
budget constraint 23.17 tells us that consumption allocation {c
i
} and security prices p are an
equilibrium in security markets which are dynamically complete (under p) if the same allocation
{c
i
} and prices q are an equilibrium in contingent commodity markets. The equilibrium security
prices p and the contingent commodity prices q are related via 23.5; that is, q are the event prices
associated with p.
23.7 Pareto-Optimal Equilibria
As in the two-date model, a consumption allocation is Pareto optimal if it is impossible to reallocate
the total endowment so as to make some agent strictly better off without making any other agent
strictly worse off. That is, allocation {c
i
} is Pareto optimal if there does not exist an alternative
allocation {c
0
i
} which is feasible
I
X
i=1
c
0
i
=
I
X
i=1
w
i
, (23.19)
weakly preferred by every agent,
u
i
(c
0
i
) ≥ u
i
(c
i
), (23.20)
and strictly preferred by at least one agent (so that 23.20 holds with strict inequality for at least
one i).
The first welfare theorem states that an equilibrium allocation in commodity markets is Pareto
optimal under the same assumptions as those of the two-date model.
23.7.1 Theorem
If security markets are dynamically complete under equilibrium security prices and agents’ utility
functions are strictly increasing, then every equilibrium consumption allocation is Pareto optimal.
Proof: The proof is the same as that for Theorem 15.3.1. If markets are dynamically com-
plete, then each equilibrium consumption allocation is also an equilibrium allocation of complete
contingent commodity markets, see Section 23.6. By the first welfare theorem, the latter allocation
is Pareto optimal.
2
The first order conditions for an interior Pareto-optimal allocation are that marginal rates of
substitution ∂
ξ
u/∂
ξ
0
u are the same for all agents. In an interior equilibrium under dynamically
complete markets, marginal rates of substitution are equal to event prices, see 23.18.
Notes
The concept of dynamically complete markets has its origins in the literature on option pricing; see
Black and Scholes [2], Cox and Ross [3], Rubinstein [9] and Harrison and Kreps [6]. The Pareto
optimality of equilibrium allocations in complete security markets was first pointed out by Arrow
[1] in the two-date model. The analysis was extended by Guesnerie and Jaffray [5] and Kreps [7],
[8] to dynamically complete markets in the multidate model.
Binomial security markets were first studied by Cox, Ross, and Rubinstein [4].
230 CHAPTER 23. DYNAMICALLY COMPLETE MARKETS
Bibliography
[1] Kenneth J. Arrow. The role of securities in the optimal allocation of risk bearing. Review of
Economic Studies, pages 91–96, 1964.
[2] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of
Political Economy, 81:637–654, 1973.
[3] John C. Cox and Stephen A. Ross. The valuation of options for alternative stochastic processes.
Journal of Financial Economics, 3:145–166, 1976.
[4] John C. Cox, Stephen A. Ross, and Mark Rubinstein. Option pricing: A simplified approach.
Journal of Financial Economics, 7:229–263, 1979.
[5] Roger Guesnerie and J.-Y. Jaffray. Optimality of equilibrium of plans, prices, and price expec-
tations. In J. Dr`eze, editor, Allocation Under Uncertainty. MacMillan, London, 1974.
[6] J. Michael Harrison and David M. Kreps. Martingales and arbitrage in multiperiod securities
markets. Journal of Economic Theory, 20:381–408, 1979.
[7] David M. Kreps. Multiperiod securities and the efficient allocation of risk: A comment on the
Black-Scholes option pricing model. In John McCall, editor, The Economics of Uncertainty and
Information. University of Chicago Press, 1982.
[8] David M. Kreps. Three essays on capital markets. Revista Espanola de Economia, 1987.
[9] Mark Rubinstein. The valuation of uncertain income streams and the pricing of options. Bell
Journal of Economics, 7:407–425, 1976.
231
232 BIBLIOGRAPHY
Chapter 24
Valuation
24.1 Introduction
Whether for two-date security markets (see Chapter 5) or for multidate security markets, it is useful
to have valuation defined on the entire contingent claim space R
k
, not just on the asset span M(p).
The valuation functional is a linear functional
Q : R
k
→ R (24.1)
that extends the payoff pricing functional from the asset span M(p) to the contingent claim space
R
k
; that is
Q(z) = q(z) for every z ∈ M(p). (24.2)
The valuation functional assigns a value to every multidate contingent claim. We are interested
in valuation functionals that are strictly positive (positive) since this property reflects the absence
of arbitrage (strong arbitrage). A strictly positive (positive) valuation functional will be used in
Chapter 25 to derive event prices and risk-neutral probabilities in the multidate model.
24.2 The Fundamental Theorem of Finance
The Fundamental Theorem of Finance asserts the existence of a strictly positive (positive) valuation
functional. Since the asset span and the payoff pricing functional of the multidate model have
exactly the same properties as the asset span and the payoff pricing functional of the two-date
model, the existence and properties of the valuation functional are the same as well.
24.2.1 Theorem (Fundamental Theorem of Finance)
Security prices exclude arbitrage iff there exists a strictly positive valuation functional.
24.2.2 Theorem (Fundamental Theorem of Finance, Weak Form)
Security prices exclude strong arbitrage iff there exists a positive valuation functional.
As already noted, the proofs of these theorems given in Chapter 5 for the two-date model carry
over to the multidate model. In the proofs of the necessity parts the payoff pricing functional is
extended one dimension at a time. We choose a contingent claim z
∗
which is not in the asset span
and extend the payoff pricing functional to the subspace spanned by M(p) and z
∗
. The value of
z
∗
is selected from an interval defined by the bounds
q
u
(z
∗
) ≡ min
h
{p
0
h
0
: z(h, p) ≥ z
∗
} (24.3)
233
234 CHAPTER 24. VALUATION
and
q
`
(z
∗
) ≡ max
h
{p
0
h
0
: z(h, p) ≤ z
∗
}. (24.4)
If security prices exclude strong arbitrage, then the bounds define an interval [q
`
(z
∗
), q
u
(z
∗
)]
such that assigning to z
∗
a value drawn from this interval leads to a positive linear extension of the
payoff pricing functional. If security prices exclude arbitrage, the interval has nonempty interior
and each value in the interior leads to a strictly positive extension.
The following example illustrates the bounds:
24.2.3 Example
In Example 21.2.1, suppose that there are two securities, a discount bond maturing at date 1
(security 1) and a discount bond maturing at date 2 (security 2). Thus the dividends of the one-
period bond are x
1
(ξ
g
) = x
1
(ξ
b
) = 1 at date 1 and x
1
(ξ) = 0 for all events ξ ∈ F
2
at date 2. For
the two-period bond the dividends are x
2
(ξ
g
) = x
2
(ξ
b
) = 0 at date 1 and x
2
(ξ) = 1 for all events
ξ ∈ F
2
at date 2. Let the price at date 0 for the one-period bond be p
1
(ξ
0
) = 0.9; and the prices
for the two-period bond be p
2
(ξ
0
) = 0.75, p
2
(ξ
g
) = 0.9, and p
2
(ξ
b
) = 0.8.
Markets are incomplete, for the rank condition of Theorem 23.2.1 fails in both events at date
1. The asset span M(p) is 4-dimensional, whereas the contingent claim space is 6-dimensional. In
fact, the contingent claim
z = (z(ξ
g
), z(ξ
b
), z(ξ
gg
), z(ξ
gb
), z(ξ
bg
), z(ξ
bb
)) (24.5)
can be generated by a portfolio strategy iff z(ξ
gg
) = z(ξ
gb
) and z(ξ
bg
) = z(ξ
bb
).
Consider the contingent claim z
∗
given by z
∗
1
= (0, 0) and z
∗
2
= (2, 1, 1, 0). Clearly, z
∗
6∈ M(p).
The upper bound on the value of z
∗
is determined by solving the minimization problem 24.3. We
have
min
h
p
1
(ξ
0
)h
1
(ξ
0
) + p
2
(ξ
0
)h
2
(ξ
0
) (24.6)
subject to
z(h, p) ≥ z
∗
. (24.7)
Constraint 24.7 implies that
h
2
(ξ
g
) ≥ 2, h
2
(ξ
g
) ≥ 1, h
2
(ξ
b
) ≥ 1, h
2
(ξ
b
) ≥ 0, (24.8)
h
1
(ξ
0
) + 0.9(h
2
(ξ
0
) − h
2
(ξ
g
)) ≥ 0, and h
1
(ξ
0
) + 0.8(h
2
(ξ
0
) − h
2
(ξ
b
)) ≥ 0. (24.9)
The solution to the linear programming problem 24.6 calls for a date-1 holding of 2 two-period
bonds if the first corporate report is good (h
2
(ξ
g
) = 2) and 1 two-period bond if the first report
is bad (h
2
(ξ
b
) = 1). These holdings have to be financed by a date-0 portfolio. Purchasing 10 two-
period bonds (h
2
(ξ
0
) = 10) and selling 7.2 one-period bonds (h
1
(ξ
0
) = −7.2) at date 0, generates a
date-1 payoff of 1.8 if the first report is good and 0.8 if the first report is bad—as needed to finance
the date-1 holdings. The date-0 price of this portfolio strategy is 1.02.
The payoff of this portfolio strategy is (0, 0) at date 1, and (2, 2, 1, 1) at date 2. It is the
smallest contingent claim in the asset span that exceeds z
∗
. Since security prices exclude arbitrage,
the date-0 price of 1.02 of this portfolio strategy must be minimal.
In this example the optimal portfolio strategy could have been determined by simply finding
the smallest contingent claim that lies in the asset span and satisfies 24.7 and then identifying
the portfolio strategy that generates that contingent claim. This solution method does not work
in general since usually the smallest element of the asset span does not exist. In general it is
necessary to solve the linear programming problem explicitly, either as one large linear program or,
using backward induction, as several smaller programs.
24.3. UNIQUENESS OF THE VALUATION FUNCTIONAL 235
The lower bound on the value of z
∗
is determined by solving the maximization problem 24.4.
We have
max
h
p
1
(ξ
0
)h
1
(ξ
0
) + p
2
(ξ
0
)h
2
(ξ
0
) (24.10)
subject to
z(h, p) ≤ z
∗
. (24.11)
The solution to this problem is identical to the minimization problem 24.6, except that 9, not
10, units of the two-period bond are purchased at date 0. The date-0 price of this portfolio strategy
is 0.27. It generates a payoff of (0,0,1,1,0,0), which is the greatest payoff that is less than or equal
to z
∗
.
2
As in two-date security markets, a strictly positive (positive) valuation functional associated
with an equilibrium payoff pricing functional is given by an agent’s marginal rates of substitution
between consumption at date 0 and at future dates. If the agent’s equilibrium consumption is
interior and his utility function is strictly increasing (increasing), then the vector of marginal rates
of substitution {∂
ξ
u/∂
ξ
0
u} defines a strictly positive (positive) valuation functional that assigns the
value
P
ξ∈Ξ
z(ξ)(∂
ξ
u/∂
ξ
0
u) to a contingent claim z ∈ R
k
.
24.3 Uniqueness of the Valuation Functional
Extension of the payoff pricing functional to a valuation functional is in general not unique. When
markets are incomplete there exists a continuum of values for any contingent claim not in the asset
span, and each value defines a strictly positive extension of the payoff pricing functional. When
markets are dynamically complete the asset span M(p) equals the contingent claim space R
k
and
the payoff pricing functional and the valuation functional are one and the same. Thus we have
24.3.1 Theorem
Suppose that security prices exclude arbitrage. Then security markets are dynamically complete iff
there exists a unique strictly positive valuation functional.
We pointed out in Section 24.2 that if security prices are equilibrium prices, then the marginal
rates of substitution of an agent define a valuation functional. If markets are incomplete, those
marginal rates may differ among agents and multiple valuation functionals result. If markets
are dynamically complete, then there is a unique valuation functional given by marginal rates of
substitution, which are the same for all agents.
Notes
The valuation functional was introduced in the setting of multidate security markets (including
continuous-time markets) by Harrison and Kreps [2]. The derivation of the valuation functional in
this chapter follows the method of Chapter 5 and is due to Clark [1].
236 CHAPTER 24. VALUATION
Bibliography
[1] Stephen A. Clark. The valuation problem in arbitrage price theory. Journal of Mathematical
Economics, 22:463–478, 1993.
[2] J. Michael Harrison and David M. Kreps. Martingales and arbitrage in multiperiod securities
markets. Journal of Economic Theory, 20:381–408, 1979.
237
238 BIBLIOGRAPHY
Part VIII
Martingale Property of Security
Prices
239
Chapter 25
Event Prices, Risk-Neutral
Probabilities and the Pricing Kernel
25.1 Introduction
In this chapter we present two closely related representations of the valuation functional—one by
event prices, the other by risk-neutral probabilities—and a representation of the payoff pricing
functional by the pricing kernel. These representations are the analogues of those of the valuation
functional and the payoff pricing functional of the two-date model of Chapters 6 and 17.
Event prices are the multidate counterpart of state prices in the two-date model. The existence
of strictly positive (positive) event prices indicates the absence of arbitrage (strong arbitrage). The
uniqueness of event prices indicates that markets are dynamically complete. Event prices can be
calculated as a solution to linear equations. Once event prices are known, the price of any payoff
can be found without identifying a portfolio strategy that generates that payoff.
Risk-neutral probabilities are event prices rescaled by discount factors. The existence of a
pricing kernel is a consequence of the Riesz Representation Theorem.
25.2 Event Prices
If security markets are dynamically complete, then the payoff pricing functional q is defined on
the entire contingent claim space R
k
and the event price q(ξ) is defined as the price q(e(ξ)) of the
Arrow security e(ξ) (see Chapter 23). If security markets are incomplete, then the asset span is a
proper subspace of the contingent claim space and some Arrow securities cannot be priced using
the payoff pricing functional. The Fundamental Theorem of Finance 24.2.1 (24.2.2) implies that
if security prices exclude arbitrage (strong arbitrage), then the payoff pricing functional can be
extended to a strictly positive (positive) valuation functional defined on the entire contingent claim
space. Event prices can then be defined using a valuation functional.
Let Q be a valuation functional and let
q(ξ) ≡ Q(e(ξ)), (25.1)
for every ξ ∈ Ξ, where e(ξ) is the event-ξ unit vector in R
k
, that is, the dividend of the Arrow
security associated with ξ. The value q(ξ) is the event price of event ξ under the valuation functional
Q. If Q is a strictly positive (positive) functional, then each event price is strictly positive (positive).
Since every contingent claim z ∈ R
k
can be written as z =
P
ξ∈Ξ
z(ξ)e(ξ), we have
Q(z) =
X
ξ∈Ξ
Q(e(ξ))z(ξ) = qz, (25.2)
241
242 CHAPTER 25. EVENT PRICES
where q is now a vector of event prices. The equation
Q(z) = qz (25.3)
is the representation of the valuation functional by event prices. For a payoff z ∈ M(p), we have
q(z) = qz. (25.4)
Thus the price of a payoff can be obtained using event prices without determining a portfolio
strategy that generates that payoff.
As when markets are dynamically complete (Section 23.4), event prices in incomplete markets
can be identified as a positive solution to the linear equations 23.5. To see this, consider a portfolio
strategy of buying one share of security j at date t ≥ 1 in event ξ
t
and selling it in every successor
event ξ
t+1
⊂ ξ
t
at date t + 1. Denoting that portfolio strategy by
ˆ
h, we have z(
ˆ
h, p)(ξ
t
) = −p
j
(ξ
t
),
z(
ˆ
h, p)(ξ
t+1
) = p
j
(ξ
t+1
) + x
j
(ξ
t+1
) for ξ
t+1
⊂ ξ
t
, and z(
ˆ
h, p)(ς) = 0 for all other events ς. Since
ˆ
h(ξ
0
) = 0, we have that q(z(
ˆ
h, p)) = p(ξ
0
)
ˆ
h(ξ
0
) = 0. Applying 25.4 to the payoff z(
ˆ
h, p), we obtain
q(ξ
t
) p
j
(ξ
t
) =
X
ξ
t+1
⊂ξ
t
q(ξ
t+1
)(p
j
(ξ
t+1
) + x
j
(ξ
t+1
)). (25.5)
Eq. 25.5 holds for every t ≥ 1, every ξ
t
∈ F
t
and every security j. A similar argument shows that
25.5 holds also at date 0 with q(ξ
0
) set equal to one.
Eqs. 25.5 are the same as 23.5 for dynamically complete markets. There are now J equations
with k(ξ
t
) unknowns q(ξ
t+1
)/q(ξ
t
). We just argued that event prices associated with a valuation
functional are a solution to 25.5. A positive valuation functional defines a positive solution, and a
strictly positive functional defines a strictly positive solution. If markets are incomplete, there are
many valuation functionals (see Theorem 24.3.1) and equations 25.5 have many solutions.
25.2.1 Theorem
There exists a strictly positive valuation functional iff there exists a strictly positive solution to
equations 25.5. Each strictly positive solution q defines a strictly positive valuation functional Q
by Q(z) = qz.
Proof: Necessity was proved above. Suppose that q is a strictly positive solution to 25.5.
Then the functional Q defined by Q(z) = qz is linear and strictly positive. Applying 25.5, one
can show that if z ∈ M(p) so that z = z(h, p) for some portfolio strategy h, then qz = p
0
h
0
.
Thus Q(z) = p
0
h
0
, i.e., Q coincides with the payoff pricing functional on M(p). Therefore Q is a
valuation functional.
2
Similarly,
25.2.2 Theorem
There exists a positive valuation functional iff there exists a positive solution to equations 25.5.
Each positive solution q defines a positive valuation functional Q by Q(z) = qz.
Theorems 25.2.1 and 25.2.2 say that equations 25.5 provide a complete characterization of event
prices. Thus event prices can be equivalently defined as a positive or strictly positive solution to
those equations. The Fundamental Theorem of Finance can be restated as saying that security
prices exclude arbitrage (strong arbitrage) iff there exists a strictly positive (positive) solution to
the equations 25.5.
If security prices are equilibrium prices, the vector of marginal rates of substitution of each
agent whose consumption is interior defines a (generally distinct) vector of event prices (see Section
24.2).
25.3. RISK-FREE RETURN AND DISCOUNT FACTORS 243
25.2.3 Example
In Example 24.2.3 equations 25.5 take the following form
q(ξ
gg
) + q(ξ
gb
) = 0.9q(ξ
g
) (25.6)
q(ξ
bg
) + q(ξ
bb
) = 0.8q(ξ
b
) (25.7)
q(ξ
g
) + q(ξ
b
) = 0.9 (25.8)
0.9q(ξ
g
) + 0.8q(ξ
b
) = 0.75. (25.9)
These equations uniquely identify date-1 event prices as q(ξ
g
) = 0.3 and q(ξ
b
) = 0.6, but leave
date-2 event prices as an arbitrary positive (or strictly positive) solution to the following equations
obtained from 25.6 and 25.7:
q(ξ
gg
) + q(ξ
gb
) = 0.27, (25.10)
q(ξ
bg
) + q(ξ
bb
) = 0.48. (25.11)
The existence of strictly positive event prices indicates that there exist no arbitrage. Nonunique-
ness of event prices indicates that markets are incomplete.
2
25.3 Risk-Free Return and Discount Factors
The one-period return on security j in event ξ
t+1
is its one-period (cum-dividend) payoff in ξ
t+1
divided by its price in the immediate predecessor event ξ
t
(where ξ
t
= ξ
−
t+1
),
r
j
(ξ
t+1
) ≡
p
j
(ξ
t+1
) + x
j
(ξ
t+1
)
p
j
(ξ
t
)
. (25.12)
We use r
j,t+1
to denote the one-period return on security j at date t + 1.
A one-period return at date t + 1 is risk-free if it takes the same value for any two date-t + 1
events that have a common predecessor at date t. We denote the one-period risk-free return realized
in event ξ
t+1
by ¯r(ξ
t+1
). By definition, the return ¯r(ξ
t+1
) does not depend on the event ξ
t+1
as
long as ξ
t+1
⊂ ξ
t
for some ξ
t
but, of course, may depend on ξ
t
. In other words, ¯r
t+1
as a function
on states is measurable with respect to F
t
.
Examples of securities with one-period risk-free returns at date t + 1 include the one-period
risk-free bond issued at date t and a discount bond issued at date 0 and maturing at date t +1. We
will frequently assume that at every date and in every event there exists a security (or a portfolio)
with a risk-free one-period return.
If at every date and in every event there exists a security (or portfolio) with a strictly positive
risk-free one-period return, then we can define the discount factor in event ξ
t
as the reciprocal of
the cumulated risk-free return:
ρ(ξ
t
) ≡
t
Y
τ=1
[¯r(ξ
τ
)]
−1
, t = 1, . . . , T, (25.13)
where ξ
τ
is the date-τ predecessor event of ξ
t
, that is ξ
τ
⊃ ξ
t
. Note that ρ(ξ
t
) is the same for any
two date-t events that have a common predecessor at date t −1; that is, ρ
t
is F
t−1
-measurable. We
also set ρ(ξ
0
) ≡ 1. For use later, note that 25.13 implies
ρ(ξ
t
) = ¯r(ξ
t+1
)ρ(ξ
t+1
). (25.14)
244 CHAPTER 25. EVENT PRICES
25.4 Risk-Neutral Probabilities
We define the risk-neutral probability of an event ξ
T
at date T as the ratio of its event price and
the discount factor,
π
∗
(ξ
T
) ≡
q(ξ
T
)
ρ(ξ
T
)
, (25.15)
and the risk-neutral probability of an event ξ
t
at date t for t < T by
π
∗
(ξ
t
) ≡
X
ξ
T
⊂ξ
t
π
∗
(ξ
T
). (25.16)
Risk-neutral probabilities are strictly positive (positive) iff event prices are strictly positive (posi-
tive).
The risk-neutral probability of any event ξ
t
satisfies
π
∗
(ξ
t
) =
q(ξ
t
)
ρ(ξ
t
)
. (25.17)
To see this, we note first that 25.17 holds for date-T events by definition 25.15. Next, we substitute
25.15 in the right hand side of 25.16 to obtain
π
∗
(ξ
t
) =
X
ξ
T
⊂ξ
t
q(ξ
T
)
ρ(ξ
T
)
. (25.18)
Eq. 25.5 when applied to the risk-free security in event ξ
t
implies
q(ξ
t
) =
X
ξ
t+1
⊂ξ
t
¯r(ξ
t+1
)q(ξ
t+1
). (25.19)
Substituting ρ(ξ
t
)/ρ(ξ
t+1
) for ¯r(ξ
t+1
) (see 25.14) in 25.19 and using 25.19 recursively we obtain
q(ξ
t
) =
X
ξ
T
⊂ξ
t
ρ(ξ
t
)
ρ(ξ
T
)
q(ξ
T
). (25.20)
Eqs. 25.18 and 25.20 imply 25.17.
For date-0 event ξ
0
, 25.17 says that
π
∗
(ξ
0
) =
q(ξ
0
)
ρ(ξ
0
)
= 1. (25.21)
Since π
∗
(ξ
0
) =
P
ξ
T
⊂ξ
0
π
∗
(ξ
T
), 25.21 implies that π
∗
is indeed a probability measure.
Eq. 25.17 indicates that risk-neutral probabilities are rescaled event prices. The existence
of strictly positive (positive) risk-neutral probabilities is equivalent to security prices excluding
arbitrage (strong arbitrage). These are restatements of the Fundamental Theorems of Finance.
Further, the risk-neutral probabilities are unique iff markets are dynamically complete.
If risk-neutral probabilities are strictly positive, conditional probabilities can be defined as
π
∗
(ξ
t+1
|ξ
t
) ≡
π
∗
(ξ
t+1
)
π
∗
(ξ
t
)
(25.22)
for ξ
t+1
⊂ ξ
t
. It follows from 25.17 and 25.14 that
π
∗
(ξ
t+1
|ξ
t
) =
q(ξ
t+1
)
q(ξ
t
)
¯r(ξ
t+1
). (25.23)
25.5. EXPECTED RETURNS UNDER RISK-NEUTRAL PROBABILITIES 245
Substituting 25.23 in 25.5 yields
p
j
(ξ
t
) = (¯r(ξ
t+1
))
−1
X
ξ
t+1
⊂ξ
t
π
∗
(ξ
t+1
|ξ
t
)(p
j
(ξ
t+1
) + x
j
(ξ
t+1
)) (25.24)
for every nonterminal event ξ
t
and every security j. Eqs. 25.24 provide a complete charac-
terization of risk-neutral probabilities. They can be used to calculate conditional risk-neutral
probabilities. Marginal risk-neutral probabilities can then be obtained recursively from 25.22 as
π
∗
(ξ
t+1
) = π
∗
(ξ
t+1
|ξ
t
) · π
∗
(ξ
t
), with π
∗
(ξ
0
) = 1.
25.5 Expected Returns under Risk-Neutral Probabilities
When equipped with risk-neutral probabilities, the set of states S can be regarded as a probability
space, just as in the two-date case. All measurable functions on S, such as date-t consumption
plans, portfolio strategies, security prices, dividends and so forth (see Section 21.2), can be regarded
as random variables.
The expected value of a random variable, say the one-period return r
jt
on security j at date
t, with respect to the risk-neutral probabilities π
∗
is denoted by E
∗
(r
jt
). The “
∗
” indicates that
the expectation is taken with respect to π
∗
. In the following sections we will also be using E(r
jt
)
to denote the expected value taken with respect to “natural probabilities” π which reflect agents’
subjective beliefs about the states.
We write E
∗
(r
j,t+1
|ξ
t
) to denote the expected value of r
j,t+1
with respect probabilities π
∗
con-
ditional on event ξ
t
. Thus
E
∗
(r
j,t+1
|ξ
t
) ≡
X
ξ
t+1
⊂ξ
t
π
∗
(ξ
t+1
|ξ
t
)r
j
(ξ
t+1
). (25.25)
We use E
∗
t
(r
j,t+1
) to denote the expected value of r
j,t+1
conditional on F
t
, that is, an F
t
-measurable
random variable that takes value E
∗
(r
j,t+1
|ξ
t
) in event ξ
t
.
Using the notation for conditional expectations, 25.24 is written
p
jt
= (¯r
t+1
)
−1
E
∗
t
(p
j,t+1
+ x
j,t+1
). (25.26)
Thus the date-t price of security j equals the conditional expectation of its one-period payoff
discounted by the one-period risk-free return, where the expectation is taken with respect to risk-
neutral probabilities. Eq. 25.26 can be written in terms of returns as
¯r
t+1
= E
∗
t
(r
j,t+1
)<