Microfoundations of Financial Economics
Microfoundations of Financial Economics An Introduction to General Equilibrium Asset Pricing
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To Brigitte, with love
List of boxes xi
Preface xiii
1 Introduction 1
1.1 What finance theory is about 1
1.2 Some history of thought 2
1.3 The importance of the puzzles 7
1.4 Outline of the book 9
2 Contingent claim economy 10
2.1 The commodity space 10
2.2 Preferences and ordinal utility 14
2.3 Maximization 16
2.4 General equilibrium 23
2.5 The representative agent 32
Notes on the literature 35 Problems 35
3 Asset economy 37
3.1 Financial assets 37
3.2 Pricing by redundancy 41
3.3 Radner economies 46
3.4 Complete markets (and uniqueness of Arrow prices) 53
3.5 Complications arising from market incompleteness 60
Notes on the literature 65 Problems 65
4 Risky decisions 68
4.1 Bernoulli’s St. Petersburg paradox 69
4.2 Using more structure: probabilities and lotteries 71
4.3 The von Neumann–Morgenstern representation 75
4.4 Measures of risk preference 81
4.5 Assumptions and evidence 86
4.6 Often used specifications 91
Notes on the literature 99 Problems 99
5 Static finance economy 102
5.1 An economy with von Neumann–Morgenstern agents 102
5.2 Efficient risk-sharing 107
5.3 A representative NM agent 112
5.4 Who holds what kind of portfolio? 121
5.5 The stochastic discount factor 126
5.6 The equilibrium price of time 130
5.7 The equilibrium price of risk 132
5.8 Some important special cases 134
Notes on the literature 138 Problems 138
6 Dynamic finance economy 141
6.1 A static dynamic model 141
6.2 Dynamic trading 149
6.3 Models of the real interest rate 162
6.4 Portfolio selection 168
Notes on the literature 170 Problems 170
7 Empirics and the puzzles 172
7.1 Collecting the right data 172
7.2 The equity premium puzzle 176
7.3 Alternative interpretations of the data 184
7.4 Excessive volatility 192
7.5 Anomalies 197
Notes on the literature 198
8 Adapting the theory 199
8.1 Assumptions of the mainstream model 199
8.2 Non-standard preferences 201
8.3 Heterogeneity 214
8.4 Efficiency failure 225
Contents ix Notes on the literature 238
9 Epilog 239
9.1 A mystery 240
9.2 A challenge 241
9.3 The party’s over 242
Appendix A Appendix B Bibliography Index
Symbols and notation 245 Solutions to the problem sets 247 269 285
List of boxes
2.1 Definition of a commodity 14
2.2 Equivalent utility functions 15
2.3 Maximality condition 19
2.4 Classical dichotomy 20
2.5 Interest rate as relative price 22
2.6 Insurance premium as relative price 22
2.7 Contingent claim economy 24
2.8 Competitive equilibrium 25
2.9 First welfare theorem 29
2.10 Loss of distributional information 33
3.1 Law of one price 43
3.2 Decomposition 43
3.3 Risk-neutral probabilities 44
3.4 Risk-neutral pricing 45
3.5 Risk-neutral returns 45
3.6 Asset economy 46
3.7 Absence of arbitrage opportunities 49
3.8 Radner equilibrium 51
3.9 Loss of information on composition 53
3.10 Complete markets 54
3.11 Reverse decomposition and uniqueness of Arrow prices 54
3.12 State-wise dichotomy 58
3.13 The one-good one-agent economy 59
3.14 Effects of incomplete markets 63
3.15 Equilibrium in a quasi-complete market 64
4.1 Certainty equivalent and risk premium 73
4.2 Expected utility representation 78
4.3 Demand for full coverage 82
4.4 ARA 84
List of boxes
CARA—DARA—IARA 84 Determine your preferences 89 Likely utilities 91 HARA (Merton, 1971) 92 Approximate mean–variance analysis 99 Mutuality principle 107 Efficient allocation of aggregate risk (Wilson, 1968) 112 A distribution-independent representative (Rubinstein, 1974) 116 Sufficient condition for quasi-completeness (Ross, 1976) 118 Quasi-completeness without options 120
4. 5 4. 6 4. 7 4. 8
5. 2 5. 3 5. 4 5. 5
5.6 SDF 126
5.7 Equilibrium SDF as function of aggregate data 127
5.8 Risk-neutral probabilities are pessimistic 129
5.9 Growth and the risk-free rate 131
5.10 Aggregate risk and the risk-free rate 132
5.11 CCAPM 133
6.1 Fundamental pricing formula 147
6.2 Qualitative features of the term structure of interest rates 164
6.3 The expectations hypothesis of the term structure 166
Dear reader
This book takes you from the level of microeconomics principles through a sequence of carefully elaborated and detailed steps to modern topics in finance. The book is for you if you have been exposed to indifference curves, budget constraints, and maximization but would like to know about the consumption capital asset pricing model, the theory of the term structure of interest rates, the equity premium puzzle, and the social cost of the business cycle. Even if you have not taken a course in basic microeconomics, you can still read this book, because it reviews the part of microeconomics and general equilibrium theory that is relevant for the topics covered in this book.
The book was written with three groups of readers in mind:
graduate students with a focus on macroeconomics, financial eco- nomics, or monetary economics;
MBA students specializing in finance;
professionalsofthefinancialcommunitywithasufficientpriorknow-
ledge of mathematics and economics.
Essentially, it is suitable for everyone who is seriously interested in financial economics and its relation to the macroeconomy, and has an appetite for the formal analysis of these issues.
Dear instructor
The book is geared to the needs of MA/MSc or PhD students specializing in financial economics. It can also be used for undergraduate students
xiv Preface
with a sufficient appetite for formal analysis as an introduction to general equilibrium theory, macroeconomics, or finance—three terms that have begun to overlap increasingly over the last two decades.
The material covered fits comfortably into a two semester course. For students with sufficient prior exposure to economics (knowledge of gen- eral equilibrium theory and expected utility theory), chapters 2–4 can be reviewed quickly and the remainder of the book should then fit into one semester.
Some knowledge of mathematics is required. This becomes quickly ob- vious simply by looking at the density of the equations in the book: there are fewer than a typical mathematics textbook would have, but considerably more than what someone who is not used to mathematics will feel comfort- able with. We use Euclidean spaces (finite-dimensional real vector spaces), basic statistics (mean, variance, covariance), maximization subject to con- straints, and calculus. More precisely, knowledge of mathematics at the level of Bartle (1976), Simon & Blume (1994), Sundaram (1996), Wein- traub (1982), or any other slightly advanced “Mathematics for Economists” text is more than enough.
Use in combination with other books
Depending on the class level you may wish to emphasize or de-emphasize different aspects or topics, and it may make sense to combine this text with another one.
For a more applied audience and for practitioners, I recommend Cornell (1999) or Siegel (1998) as a starting point. Both books contain detailed discussions of the equity premium puzzle, yet both manage to do away almost completely with mathematics and econometrics. Another good place to start is AIMR’s (2002) published forum on the topic. In this forum, several prominent researchers in the field present their ideas in not too technical a fashion. These ideas are then discussed by a panel.
LeRoy & Werner (2001) and Danthine & Donaldson (2002) are compa- rable in style and difficulty to the present book, should you wish to offer an alternative presentation of the material. The main differences are that these two books put less emphasis on aggregation conditions and cover empirical issues to a lesser extent (Danthine & Donaldson) or not at all (LeRoy & Werner).
If you want to focus somewhat more on empirical work and your students are technically well-trained, the textbooks by Cochrane (2001) and Camp- bell, Lo & MacKinlay (1997) should provide nice complements, containing
Preface xv
much more information on econometric issues related to financial market data.
The present book can also be used as a supplement to modern macroe- conomics courses. For instance, Ljungqvist & Sargent (2000) contains two chapters on equilibrium asset pricing; if you want to focus more on this topic that would be a valuable addition, especially since, unlike Ljungqvist & Sargent, we develop the topic at sub-sonic speed.
For the more theory-minded reader, Gollier (2001a) has created an out- standing research book on the theory of decisions under risk and relations to general equilibrium and asset pricing. An older classic in this domain is Duffie (1988). These titles are clearly more advanced than the present.
Finally, Brunnermeier (2001) presents a book that would be very appro- priate as a basis for a follow-on course from this one. He explores the con- sequences of asymmetric information in general equilibrium asset pricing theory, a topic we touch upon here only marginally.
Website and supporting material
The book’s website1 offers supporting material for instructors. First of all, there is a list of all the references made in the book, with links to online sources where available. This should help instructors collect material for a reader accompanying their course, and should help students collect the relevant literature on their own. The same website offers some Excel files in connection with the problem sets. Finally, there is a collection of PowerPoint files that should help instructors prepare their lectures. These files can easily be amended using PowerPoint.
I also maintain a list of errors. If you find any errors or omissions, please let me know (see the email address on the book’s website).
Acknowledgements
The editor of Princeton University Press, , who guided this project, did a great job—never late, always precise and supportive. Among other services, he selected the anonymous referees, who deserve a big thank you. Their comments and suggestions have made this a far better book. I am also grateful to my colleagues , , , , , , , , ̈derlind, , and for many sug-
1 pup.princeton.edu/titles/7724.html
xvi Preface gestions, and for the encouragement they gave me. enhanced
my mediocre English. I am sure her effort will make your life easier.
Throughout I use the male pronoun. This is not intended as a sexist state- ment. I find it clumsy to keep using “she or he.” Alternating between the sexes—“She chooses a portfolio that maximizes expected utility . . . Higher interest rates therefore incite him to save more”—is confusing and keeps re- minding readers of important issues of gender inequality, when they should be focusing their minds on asset pricing.
1 Introduction
1.1 What finance theory is about
Howmuchshouldyousave? andHowmuchriskshouldyoubear? Whenwethink about these questions, it becomes clear pretty quickly that they are of great importance to our overall material well-being. Saving is essential because most of us will retire at some point. From that point onwards, although we will still be consuming, we will receive no more labor income. Moreover, we will all face significant economic risks during our lives—the risk of losing our job, for instance, or—much worse—of becoming unable to work because of illness or other misfortunes. Clearly, the risks we are exposed to can have a huge effect on our future life, and it is therefore essential to make rational decisions about how much risk to bear.
Important as these questions are for each one of us, individuals’ deci- sions about saving and risk-taking also matter for society as a whole. Total saving determines the amount of investment that the economy as a whole can realize and thus affects future production possibilities. The amount of risk that people are willing to bear determines whether risky projects will be undertaken. Individual decisions in the face of future retirement and risk and the capital requirements of more or less risky investment projects are coordinated through financial markets. If markets work well, risk is allocated to those people who are least hurt by it, impatient people get to consume before they earn (by taking out a loan), and capital is allocated to those projects that generate the most attractive risk-return profile. Finance is concerned with the determination of those prices that equalize demand and supply on these markets and with their effect on the allocation of capital and risk across agents in the economy.
Finance theory is also useful in interpreting financial market prices in ways that are of interest for public policy and social welfare issues.
2 1 Introduction
(1987), for instance, has examined the social costs of business cycles. This is obviously important for economic policy making, but it is also important for macroeconomic theory. To learn the answer to this question, we need to know how much people dislike risk, that is, variations in income. More specifically, in order to judge how expensive business cycles are, we need to determine a price that people would be prepared to pay to avoid the income variations caused by business cycles. Modern asset pricing theory allows us—at least in principle—to do just that.
1.2 Some history of thought
General equilibrium theory, macroeconomics, and asset pricing theory are three fields in economics that have converged more and more over the last thirty years or so. In this section we consider how this convergence came about.
1.2.1 General equilibrium theory
General equilibrium theory is an approach to describing the behavior of an economy as a whole by working out the optimal behavior of each member of the set of agents that make up the economy, and looking for a point of mutual compatibility or consistency. The theory assumes that individuals do not interact with each other directly. Interaction occurs only indirectly, through anonymous markets on which prices (exchange rates for different commodities) are posted. A second assumption that goes hand in hand with anonymity is that each individual is small in relation to the market, so that everyone neglects his own influence on market prices. This assumption is called perfect competition. Models that make these two assumptions are called Walrasian, in honor of Léon Walras (1874) who was the first to formulate such a model. We say that the economy is in equilibrium if, at a certain price, each individual buys or sells the optimal quantities (given his tastes and possibilities) of all commodities and the total supply of each commodity equals the total demand for it.1
Modern general equilibrium theory in the tradition of Arrow & Debreu (1954) accommodates a large number of different goods and very diverse preferences of individuals. This research has established conditions that guarantee the existence of an equilibrium. It has also developed properties of equilibrium allocations, such as the welfare theorems. The two welfare
1Note that it is left unspecified in this model who posts the prices, since everyone takes them as given.
1.2 Some history of thought 3
theorems demonstrate that market equilibrium allocations and socially ef- ficient allocations are equivalent, under some conditions. This equivalence will be extremely useful for our purposes.
This theory was significantly advanced by Hirshleifer (1965, 1966) and Radner (1972). These authors built financial markets into the model and thus provided the first crucial ingredient for making general equilibrium theory applicable to finance. Moreover, their work opened up the possibility of analyzing financial markets that are incomplete in the sense that the available financial instruments may not be sufficient to trade all individual risks efficiently. This incompleteness opens the door to various sorts of coordination failures in a market economy.
By the late 1950s, general equilibrium theory had become the cornerstone of microeconomics, and remained so until it was slowly pushed aside by advances in game theory and information economics in the 1970s. General equilibrium theory has, however, received a new lease of life through its applications to the theory of macroeconomic fluctuations and the theory of asset pricing.
1.2.2 Macroeconomics
When Keynes (1936) developed his General Theory, the world was in disarray. Mass unemployment and mass bankruptcy had erupted— first in the U.S.A. but then quickly spreading throughout the capitalist world. Possibly because of these events, Keynes chose a style of model that broke with the tradition of classical economics. His model did not feature indi- vidual agents explicitly, nor did it feature dynamics of any sort. Instead, he focused on the interdependence between different aggregate variables. In that sense, Keynes’s model is a general equilibrium model, yet one in which the aggregate demand and supply functions are not developed from an individual optimization perspective. This became most clear in Hicks’s (1937) version of Keynes’s model, which came to dominate macroeconomic thinking.
This lack of microfoundation led to problems associated with the endoge- nous determination of expectations. Clearly, expectations should affect an individual’s decisions. We would expect that rational decision makers will try to collect information if faulty decisions are costly. Hicks’s version of Keynes’s model really lacked a convincing theory of expectations. This omis- sion led to increasing dissatisfaction with the model on purely theoretical grounds and ultimately to a dramatic empirical failure with the stagflation of the 1970s, which was an impossible event in the Keynes–Hicks orthodoxy.
4 1 Introduction
These developments gave impetus to a new, or rather renewed old, ap- proach,2 namely to construct dynamic models of aggregate economic fluctu- ations based on individual decisions together with shocks of some sort (most prominently to technology). The rational expectations revolution in macroe- conomics is nothing but a simplified version of Radner’s (1972) idea of an “equilibrium of plans, prices, and price expectations.” The early macro versions of this idea were simplified, in the sense that agents were assumed to have an unbiased expectation of the mean of stochastic variables only, whereas in Radner’s model agents have correct state-contingent expecta- tions.
In essence, this is what the and later Real Business Cycle the- ory consist of: computable dynamic stochastic general equilibrium models. Compared with traditional general equilibrium theory, the macroeconomic variants are typically simpler because they feature only one good and one agent, and give scant attention to the conditions for aggregation. They also make much stronger assumptions concerning preferences and technology in order to get easily computable equilibria. Modern New Keynesian theory and the Synthesis models deviate from the Walrasian ortho- doxy by introducing various frictions into the model. But they, too, work within the general equilibrium framework and assume representative goods and agents.3,4
1.2.3 Finance
Finance theory started out as a field of business administration. Sensible decision making about how to finance operations is obviously vital for any firm, and the placement of free reserves into financial assets can have a substantial impact on the profitability of the enterprise. Markowitz’s (1952) mean–variance mechanics was a breakthrough, offering a much more sophis- ticated decision rule than was common at the time, but one that was still simple to apply.
Markowitz’s contribution serves as a tool for decision making; accordingly, his research is silent about the determination of asset prices. Their stochastic properties are taken as given. Subsequently, emphasis shifted away from
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