The Economics of Risk and Time

562 Pages·2001·7.59 MB·English

by  Christian Gollier

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Cover
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Page i
The Economics of Risk and Time
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The Economics of Risk and Time
Christian Gollier
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© 2001 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form by any electronic or
mechanical means (including photocopying, recording, or information storage and retrieval)
without permission in writing from the publisher.
This book was set in Palatino in '3B2' by Asco Typesetters, Hong Kong.
Printed and bound in the United States of America.
Library of Congress Cataloging-in-Publication Data

Gollier, Christian.
The economics of risk and time / Christian Gollier.
p. cm.
Includes bibliographical references and index.
ISBN 0-262-07215-7 (hc. : alk. paper)
1. Finance. 2. Financial engineering. 3. Risk. 4. Risk assessment. 5. Risk
management. I. Title.
HG101 .G65 2001
658.15'5—dc21
2001030213
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Contents
Preface xv
Acknowledgments xix
I General Theory 1
1 The Expected Utility Model 3
3
1.1 Simple and Compound Lotteries
4
1.2 Axioms on Preferences under Uncertainty
6
1.3 The Expected Utility Theorem
9
1.4 Critics of the Expected Utility Model
10
1.4.1 The Allais Paradox
11
1.4.2 The Allais Paradox and Time Consistency
14
1.5 Concluding Remark
14
1.6 Exercises and Extensions
2 Risk Aversion 17
17
2.1 Characterization of Risk Aversion
18
2.2 Comparative Risk Aversion
20
2.3 Certainty Equivalent and Risk Premium
21
2.4 The Arrow-Pratt Approximation
24
2.5 Decreasing Absolute Risk Aversion
25
2.6 Some Classical Utility Functions
29
2.7 Test for Your Own Degree of Risk Aversion
32
2.8 An Application: The Cost of Macroeconomic Risks
34
2.9 Conclusion
35
2.10 Exercises and Extensions
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3 Change in Risk 39
40
3.1 The Extremal Approach
42
3.2 Second-Order Stochastic Dominance
45
3.3 Diversification
46
3.4 First-Order Stochastic Dominance
47
3.5 Concluding Remark
48
3.6 Exercises and Extensions
II The Standard Portfolio Problem 51
4 The Standard Portfolio Problem 53
53
4.1 The Model and Its Basic Properties
55
4.2 The Case of a Small Risk
57
4.3 The Case of HARA Functions
58
4.4 The Impact of Risk Aversion
59
4.5 The Impact of a Change in Risk
61
4.6 Concluding Remark
62
4.7 Exercises and Extensions
5 The Equilibrium Price of Risk 65
65
5.1 A Simple Equilibrium Model for Financial Markets
68
5.2 The Equity Premium Puzzle
71
5.3 The Equity Premium with Limited Participation
5.4 The Equity Premium and the Integration of International Financial
73
Markets
75
5.5 Conclusion
76
5.6 Exercises
III Some Technical Tools and Their Applications 79
6 A Hyperplane Separation Theorem 81
81
6.1 The Diffidence Theorem
88
6.2 Link with the Jensen's Inequality
89
6.3 Applications of the Diffidence Theorem
89
6.3.1 Diffidence
90
6.3.2 Comparative Diffidence
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91
6.3.3 Central Risk Aversion
92
6.3.4 Central Riskiness
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94
6.4 The Covariance Rule
95
6.5 Conclusion
96
6.6 Exercises and Extensions
7 Log-Supermodularity 99
99
7.1 Definition
102
7.2 Log-Supermodularity and Single Crossing
102
7.2.1 A Theoretical Result
103
7.2.2 Applications to the Standard Portfolio Problem
104
7.2.3 Jewitt's Preference Orders
105
7.3 Expectation of a Log-Supermodular Function
105
7.3.1 A Theoretical Result
106
7.3.2 Two Applications
107
7.4 Concluding Remark
107
7.5 Exercises and Extensions
108
7.6 Appendix
IV Multiple Risks 111
8 Risk Aversion with Background Risk 113
114
8.1 Preservation of DARA
117
8.2 The Comparative Risk Aversion Is Not Preserved
119
8.3 Extensions with Dependent Background Risk
119
8.3.1 Affiliated Background Risk
121
8.3.2 The Comparative Risk Aversion in the Sense of Ross
123
8.4 Conclusion
124
8.5 Exercises and Extensions
9 The Tempering Effect of Background Risk 125
126
9.1 Risk Vulnerability
130
9.2 Risk Vulnerability and Increase in Risk
130
9.2.1 Increase in Background Risk
130
9.2.2 Increase in the Endogenous Risk
131
9.3 Risk Vulnerability and the Equity Premium Puzzle
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9.4 Generalized Risk Vulnerability
135
9.5 Standardness
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