ECONOMIC AND FINANCIAL DECISIONS UNDER UNCERTAINTY

[Pages:238]ECONOMIC AND FINANCIAL DECISIONS UNDER UNCERTAINTY

Louis Eeckhoudt Catholic University of Mons

Christian Gollier University of Toulouse

Harris Schlesinger University of Alabama

February 7, 2004

2

Contents

0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

I Decision theory

11

1 Risk aversion

13

1.1 An historical perspective on risk aversion . . . . . . . . . . . . 14

1.2 Definition and characterization of risk aversion . . . . . . . . . 18

1.3 Risk premium and certainty equivalent . . . . . . . . . . . . . 21

1.4 Degree of risk aversion . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Decreasing absolute risk aversion and prudence . . . . . . . . 28

1.6 Relative risk aversion . . . . . . . . . . . . . . . . . . . . . . . 30

1.7 Some classical utility functions . . . . . . . . . . . . . . . . . . 32

1.8 Bibliographical references and extensions . . . . . . . . . . . . 35

2 The measures of risk

39

2.1 Increases in risk . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.1 Adding noise . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.2 Mean-preserving spreads in probability . . . . . . . . . 42

2.1.3 The integral condition and risk-averse preferences . . . 44

2.1.4 Preference for diversification . . . . . . . . . . . . . . . 46

2.1.5 And the variance? . . . . . . . . . . . . . . . . . . . . . 47

2.2 Aversion to downside risk . . . . . . . . . . . . . . . . . . . . 48

2.3 First-degree stochastic dominance . . . . . . . . . . . . . . . . 49

2.4 Bibliographical references and extensions . . . . . . . . . . . . 51

3

4

CONTENTS

II Risk management

53

3 Insurance decisions

55

3.1 Optimal insurance: An illustration . . . . . . . . . . . . . . . 58

3.2 Optimal coinsurance . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Comparative statics in the coinsurance problem . . . . . . . . 64

3.4 The optimality of deductible insurance . . . . . . . . . . . . . 67

3.5 Bibliographical references and extensions . . . . . . . . . . . . 71

4 Static portfolio choices

77

4.1 The one-risky-one-riskfree-asset model . . . . . . . . . . . . . 78

4.1.1 Description of the model . . . . . . . . . . . . . . . . . 78

4.1.2 The equity premium and the demand for stocks . . . . 80

4.2 The effect of background risk . . . . . . . . . . . . . . . . . . 81

4.3 Portfolios of risky assets . . . . . . . . . . . . . . . . . . . . . 83

4.3.1 Diversification in the expected utility model . . . . . . 83

4.3.2 Diversification in the mean-variance model . . . . . . . 85

4.4 Bibliographical references and extensions . . . . . . . . . . . . 87

5 Static portfolio choices in an Arrow-Debreu economy

91

5.1 Arrow-Debreu securities and arbitrage pricing . . . . . . . . . 92

5.2 Optimal portfolios of Arrow-Debreu securities . . . . . . . . . 95

5.3 A simple graphical illustration . . . . . . . . . . . . . . . . . . 97

5.4 Bibliographical references and extensions . . . . . . . . . . . . 99

6 Consumption and saving

101

6.1 Consumption and saving under certainty . . . . . . . . . . . . 101

6.1.1 Aversion to consumption fluctuations over time . . . . 104

6.1.2 Optimal consumption growth under certainty . . . . . 106

6.2 Uncertainty and precautionary savings . . . . . . . . . . . . . 108

6.3 Risky savings and precautionary demand . . . . . . . . . . . . 111

6.4 Time consistency . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5 Bibliographical references and extensions . . . . . . . . . . . . 115

7 Dynamic portfolio management

119

7.1 Backward induction . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 The dynamic investment problem . . . . . . . . . . . . . . . . 121

7.3 Time diversification . . . . . . . . . . . . . . . . . . . . . . . . 126

CONTENTS

5

7.4 Portfolio management with predictable returns . . . . . . . . . 128 7.5 Bibliographical references and extensions . . . . . . . . . . . . 132

8 Risk and information

135

8.1 The value of information . . . . . . . . . . . . . . . . . . . . . 136

8.1.1 An example . . . . . . . . . . . . . . . . . . . . . . . . 136

8.1.2 A general model . . . . . . . . . . . . . . . . . . . . . . 138

8.1.3 Value of information and risk aversion . . . . . . . . . 141

8.2 Comparative statics analysis . . . . . . . . . . . . . . . . . . . 142

8.2.1 Real-option value and irreversibility . . . . . . . . . . . 144

8.2.2 Savings and the early resolution of uncertainty . . . . . 146

8.3 The Hirshleifer effect . . . . . . . . . . . . . . . . . . . . . . . 147

8.4 Bibliographical references and extensions . . . . . . . . . . . . 149

9 Optimal prevention

153

9.1 Prevention under risk neutrality . . . . . . . . . . . . . . . . . 154

9.2 Risk aversion and optimal prevention . . . . . . . . . . . . . . 155

9.3 Prudence and optimal prevention . . . . . . . . . . . . . . . . 157

9.4 Bibliographical references and extensions . . . . . . . . . . . . 158

III Risk sharing

161

10 Efficient allocations of risks

163

10.1 Risk sharing: An illustration . . . . . . . . . . . . . . . . . . . 164

10.2 Description of the economy and definition . . . . . . . . . . . 166

10.3 Characterization of efficient allocations of risk . . . . . . . . . 168

10.3.1 The mutuality principle . . . . . . . . . . . . . . . . . 169

10.3.2 The sharing of the macroeconomic risk . . . . . . . . . 171

10.4 Aggregation of preferences . . . . . . . . . . . . . . . . . . . . 174

10.5 Bibliographical references and extensions . . . . . . . . . . . . 176

11 Asset pricing

179

11.1 Competitive markets for Arrow-Debreu securities . . . . . . . 179

11.2 The first theorem of welfare economics . . . . . . . . . . . . . 181

11.3 The equity premium . . . . . . . . . . . . . . . . . . . . . . . 182

11.4 The capital asset pricing model . . . . . . . . . . . . . . . . . 185

11.5 Two fund separation theorem . . . . . . . . . . . . . . . . . . 188

6

CONTENTS

11.6 Bond pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 11.6.1 The risk-free rate . . . . . . . . . . . . . . . . . . . . . 189 11.6.2 Factors affecting the interest rate . . . . . . . . . . . . 191 11.6.3 The yield curve . . . . . . . . . . . . . . . . . . . . . . 194

11.7 Bibliographical references and extensions . . . . . . . . . . . . 195

IV Extensions

197

12 Asymmetric Information

199

12.1 Adverse selection . . . . . . . . . . . . . . . . . . . . . . . . . 200

12.1.1 Full insurance . . . . . . . . . . . . . . . . . . . . . . . 201

12.1.2 Pooling contracts . . . . . . . . . . . . . . . . . . . . . 203

12.1.3 Separating contracts . . . . . . . . . . . . . . . . . . . 205

12.2 Moral hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

12.3 The principal-agent problem . . . . . . . . . . . . . . . . . . . 211

12.3.1 Binary effort with a risk-neutral principal . . . . . . . 211

12.3.2 Continuous effort with a risk-averse principal . . . . . . 215

12.4 Bibliographical references and extensions . . . . . . . . . . . . 217

13 Alternative decision criteria

221

13.1 The independence axiom and the Allais' paradox . . . . . . . 224

13.2 Rank-dependent expected utility . . . . . . . . . . . . . . . . . 226

13.3 Ambiguity aversion . . . . . . . . . . . . . . . . . . . . . . . . 230

13.4 Prospect theory and loss aversion . . . . . . . . . . . . . . . . 233

13.5 Some Concluding Thoughts . . . . . . . . . . . . . . . . . . . 235

13.6 Bibliographical references and extensions . . . . . . . . . . . . 236

0.1. PREFACE

7

0.1 Preface

Risk is an ever-prevalent challenge to both individuals and society. When you dress yourself every morning, do you not ask what the weather will be like today? And after recalling the latest weather forecast, do you not wonder whether or not the forecast for today will be accurate? The weather forecast itself relies heavily on the rules of probability theory, as does the fact that today's weather might not behave as the forecast predicts. How you react to the uncertain weather ahead says something about your so-called "risk preferences." If the forecast calls for a 10% chance of rain, do you carry your umbrella when you walk to a restaurant for lunch? How about with a 50% chance of rain? Obviously the answer will not be the same for each individual.

Likewise, an individual may react differently to different consequences from the same risk. A person who decides she does not need to carry her umbrella with such a small risk of rain, may decide nonetheless to stop by the parking lot on the way to the restaurant to put the top up on her new cabriolet automobile. To quote from Peter Bernstein, "The ability to define what may happen in the future and to choose among alternatives lies at the heart of contemporary societies." (Bernstein, 1998) An understanding of risk and how to deal with it is an essential part of modern economies. Recognizing risks, quantifying risks, analyzing them, treating them and incorporating risks into our decision-making processes is the focus of this book.

Of course attempting to model human behavior is never easy. People may behave slightly differently from day to day. They also like to experiment in order to learn about their own tastes and preferences. Still, there are many basic principles that hold with much regularity. For the most part, this book models behavior using the expected utility model as developed in its modern form by von Neumann and Morgenstern (1948). While this basic approach is generally well accepted, it is not without its detractors. We discuss many of the major criticisms in the last chapter of this book.

It is important when reading this book to keep in mind that we are deriving models that help us to understand behavior towards risk. It is not assumed that people actually solve the mathematical problems that we present here. Indeed, most readers probably have a relative who cannot solve an optimization problem, yet decide every year to purchase an automobile insurance policy.

We also confine ourselves to risks that involve economic and financial

8

CONTENTS

decisions. Obviously there are many other risks that one must deal with in everyday life, such as whether or not to take a new medication with potential untoward side effects, or which scientific journal provides the best publication outlet for a newly written research paper.

This book is designed for use in advanced undergraduate and beginning doctoral courses. We cover a broad array of topics in enough detail so that the book may be used as a self-contained text. Alternatively, one can use the first two "basics" chapters, together with a selection of later chapters, as a basis for courses in macroeconomics, insurance, portfolio choice and asset pricing. Such courses can easily adapt the book for the intended use, and supplement it with additional readings or projects.

The book starts by introducing the basic concepts of risk and risk aversion that are crucial throughout the rest of the text. Part two of the text applies these basic concepts to a multitude of personal decisions under risk. Part 3 uses the results about personal decision making to show how markets for risk are organized and how risky assets are priced. Our final part introduces two important points of departure: decision making under imperfect information and alternatives to the expected utility framework.

Each chapter of the book concludes with a discussion of the relevant literature, together with some suggestions for readers who would like to read more on the topic. We also provide an appendix that contains many problems related to each of the thirteen chapters.

The only mathematics contained in this book is calculus and simple algebra. We use discrete examples for time and for probabilities throughout the text. Although the mathematics is important, the logic and intuition are more important and this is stressed throughout the book. Many of the concepts that are derived here might not be easy to understand upon a first reading. We urge the readers to take the time to re-read difficult parts of the book and to work on the related problems in the Appendix.

The book's three authors have spent collectively more than 60 years working on research projects related to the topics we present here. We each learned many new things while writing this book. And we continue to be curious, as we still have much to learn. We will feel that this book has been a "success," if some of our curiosity transfers to the reader.

References Bernstein, P. L., (1998), Against the Gods, John Wiley and Sons.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download