Asset Pricing
. 12, 2000
Acknowledgments
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This book owes an enormous intellectual debt to and . Most of the ideas in the book developed from long discussions with each of them, and trying to make sense of what each was saying in the language of the other. I am also grateful to all my col- leagues in Finance and Economics at the University of Chicago, and to especially, for many discussions about the ideas in this book. I thank , , , , , , , , , , , , an anony- mous reviewer, and several generations of Ph.D. students at the University of Chicago for many useful comments. I thank the NSF and the Graduate School of Business for research support.
Additional material and both substantive and typographical corrections will be maintained at
http://www-gsb.uchicago.edu/fac/john.cochrane/research/papers
Comments and suggestions are most welcome This book draft is copyright °c . Cochrane 1997, 1998, 1999, 2000
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Acknowledgments 2 Preface 8 Part I. Asset pricing theory 12
1 Consumption-based model and overview 13
1.1 Basic pricing equation 14
1.2 Marginal rate of substitution/stochastic discount factor 16
1.3 Prices, payoffs and notation 17
1.4 Classic issues in finance 20
1.5 Discount factors in continuous time 33
1.6 Problems 38
2 Applying the basic model 41
2.1 Assumptions and applicability 41
2.2 General Equilibrium 43
2.3 Consumption-based model in practice 47
2.4 Alternative asset pricing models: Overview 49
2.5 Problems 51
3 Contingent Claims Markets 54
3.1 Contingent claims 54
3.2 Risk neutral probabilities 55
3.3 Investors again 57
3.4 Risk sharing 59
3.5 State diagram and price function 60
4 The discount factor 64
4.1 Law of one price and existence of a discount factor 64
4.2 No-Arbitrage and positive discount factors 69
4.3 An alternative formula, and x∗ in continuous time 74 4.4 Problems 76
5 Mean-variance frontier and beta representations 77
5.1 Expected return – Beta representations 77
5.2 Mean-variance frontier: Intuition and Lagrangian characterization 80
5.3 An orthogonal characterization of the mean-variance frontier 83
5.4 Spanning the mean-variance frontier 88
5.5 A compilation of properties of R∗, Re∗ and x∗ 89
5.6 Mean-variance frontiers for m: the Hansen-Jagannathan bounds 92
5.7 Problems 97
6 Relation between discount factors, betas, and mean-variance frontiers 98
6.1 From discount factors to beta representations 98
6.2 From mean-variance frontier to a discount factor and beta representation 101
6.3 Factor models and discount factors 104
6.4 Discount factors and beta models to mean – variance frontier 108
6.5 Three riskfree rate analogues 109
6.6 Mean-variance special cases with no riskfree rate 115
6.7 Problems 118
7 Implications of existence and equivalence theorems 120
8 Conditioning information 128
8.1 Scaled payoffs 129
8.2 Sufficiency of adding scaled returns 131
8.3 Conditional and unconditional models 133
8.4 Scaled factors: a partial solution 140
8.5 Summary 141
8.6 Problems 142
9 Factor pricing models 143
9.1 Capital Asset Pricing Model (CAPM) 145 4
9.2 9.3 9.4 9.5 9.6
Intertemporal Capital Asset Pricing Model (ICAPM) 156 Comments on the CAPM and ICAPM 158 Arbitrage Pricing Theory (APT) 162 APT vs. ICAPM 171 Problems 172
Estimating and evaluating asset pricing models 174
10 GMM in explicit discount factor models 177
10.1 The Recipe 177
10.2 Interpreting the GMM procedure 180
10.3 Applying GMM 184
11 GMM: general formulas and applications 188
11.1 General GMM formulas 188
11.2 Testing moments 192
11.3 Standard errors of anything by delta method 193
11.4 Using GMM for regressions 194
11.5 Prespecified weighting matrices and moment conditions 196
11.6 Estimating on one group of moments, testing on another. 205
11.7 Estimating the spectral density matrix 205
11.8 Problems 212
12 Regression-based tests of linear factor models 214
12.1 Time-series regressions 214
12.2 Cross-sectional regressions 219
12.3 Fama-Mac 228
12.4 Problems 234
13 GMM for linear factor models in discount factor form 235
13.1 GMM on the pricing errors gives a cross-sectional regression 235
13.2 The case of excess returns
13.3 Horse Races
13.4 Testing for characteristics 240
13.5 Testing for priced factors: lambdas or b’s? 241
13.6 Problems 245
14 Maximum likelihood 247
14.1 Maximum likelihood 247
14.2 ML is GMM on the scores 249
14.3 When factors are returns, ML prescribes a time-series regression 251
14.4 When factors are not excess returns, ML prescribes a cross-sectional
regression 255 14.5 Problems 256
15 Time series, cross-section, and GMM/DF tests of linear factor models 258
15.1 Three approaches to the CAPM in size portfolios 259
15.2 Monte Carlo and Bootstrap 265
16 Which method? 271
Part III. Bonds and options 284
17 Option pricing 286
17.1 Background 286
17.2 Black-Scholes formula 293
17.3 Problems 299
18 Option pricing without perfect replication 300
18.1 On the edges of arbitrage 300
18.2 One-period good deal bounds 301
18.3 Multiple periods and continuous time 309
18.4 Extensions, other approaches, and bibliography 317
18.5 Problems
19 Term structure of interest rates
19.1 Definitions and notation
19.2 Yield curve and expectations hypothesis 325
19.3 Term structure models – a discrete-time introduction 327
19.4 Continuous time term structure models 332
19.5 Three linear term structure models 337
19.6 Bibliography and comments 348
19.7 Problems 351
Part IV. Empirical survey 352
20 Expected returns in the time-series and cross-section 354
20.1 Time-series predictability 356
20.2 The Cross-section: CAPM and Multifactor Models 396
20.3 Summary and interpretation 409
20.4 Problems 413
21 Equity premium puzzle and consumption-based models 414
21.1 Equity premium puzzles
21.2 New models
21.3 Bibliography
21.4 Problems
22 References
Part V. Appendix
23 Continuous time
23.1 Brownian Motion
23.2 Diffusion model
23.3 Ito’s lemma
23.4 Problems
Asset pricing theory tries to understand the prices or values of claims to uncertain payments. A low price implies a high rate of return, so one can also think of the theory as explaining why some assets pay higher average returns than others.
To value an asset, we have to account for the delay and for the risk of its payments. The effects of time are not too difficult to work out. However, corrections for risk are much more important determinants of an many assets’ values. For example, over the last 50 years U.S. stocks have given a real return of about 9% on average. Of this, only about 1% is due to interest rates; the remaining 8% is a premium earned for holding risk. Uncertainty, or corrections for risk make asset pricing interesting and challenging.
Asset pricing theory shares the positive vs. normative tension present in the rest of eco- nomics. Does it describe the way the world does work or the way the world should work? We observe the prices or returns of many assets. We can use the theory positively, to try to understand why prices or returns are what they are. If the world does not obey a model’s pre- dictions, we can decide that the model needs improvement. However, we can also decide that the world is wrong, that some assets are “mis-priced” and present trading opportunities for the shrewd investor. This latter use of asset pricing theory accounts for much of its popular- ity and practical application. Also, and perhaps most importantly, the prices of many assets or claims to uncertain cash flows are not observed, such as potential public or private invest- ment projects, new financial securities, buyout prospects, and complex derivatives. We can apply the theory to establish what the prices of these claims should be as well; the answers are important guides to public and private decisions.
Asset pricing theory all stems from one simple concept, derived in the first page of the first Chapter of this book: price equals expected discounted payoff. The rest is elaboration, special cases, and a closet full of tricks that make the central equation useful for one or another application.
There are two polar approaches to this elaboration. I will call them absolute pricing and relativepricing. Inabsolutepricing,wepriceeachassetbyreferencetoitsexposuretofun- damental sources of macroeconomic risk. The consumption-based and general equilibrium models described below are the purest examples of this approach. The absolute approach is most common in academic settings, in which we use asset pricing theory positively to give an economic explanation for why prices are what they are, or in order to predict how prices might change if policy or economic structure changed.
In relative pricing, we ask a less ambitious question. We ask what we can learn about an asset’s value given the prices of some other assets. We do not ask where the price of the other set of assets came from, and we use as little information about fundamental risk factors as possible. Black-Scholes option pricing is the classic example of this approach. While limited in scope, this approach offers precision in many applications.
Asset pricing problems are solved by judiciously choosing how much absolute and how much relative pricing one will do, depending on the assets in question and the purpose of the calculation. Almost no problems are solved by the pure extremes. For example, the CAPM and its successor factor models are paradigms of the absolute approach. Yet in applications, they price assets “relative” to the market or other risk factors, without answering what deter- mines the market or factor risk premia and betas. The latter are treated as free parameters. On the other end of the spectrum, most practical financial engineering questions involve as- sumptions beyond pure lack of arbitrage, assumptions about equilibrium “market prices of risk.”
The central and unfinished task of absolute asset pricing is to understand and measure the sources of aggregate or macroeconomic risk that drive asset prices. Of course, this is also the central question of macroeconomics, and this is a particularly exciting time for researchers who want to answer these fundamental questions in macroeconomics and finance. A lot of empirical work has documented tantalizing stylized facts and links between macroeconomics and finance. For example, expected returns vary across time and across assets in ways that are linked to macroeconomic variables, or variables that also forecast macroeconomic events; a wide class of models suggests that a “recession” or “financial distress” factor lies behind many asset prices. Yet theory lags behind; we do not yet have a well-described model that explains these interesting correlations.
In turn, I think that what we are learning about finance must feed back on macroeco- nomics. To take a simple example, we have learned that the risk premium on stocks – the expected stock return less interest rates – is much larger than the interest rate, and varies a good deal more than interest rates. This means that attempts to line investment up with inter- est rates are pretty hopeless – most variation in the cost of capital comes from the varying risk premium. Similarly, we have learned that some measure of risk aversion must be quite high, or people would all borrow like crazy to buy stocks. Most macroeconomics pursues small deviations about perfect foresight equilibria, but the large equity premium means that volatil- ity is a first-order effect, not a second-order effect. Standard macroeconomic models predict that people really don’t care much about business cycles (Lucas 1987). Asset prices are be- ginning to reveal that they do – that they forego substantial return premia to avoid assets that fall in recessions. This fact ought to tell us something about recessions!
This book advocates a discount factor / generalized method of moments view of asset pricing theory and associated empirical procedures. I summarize asset pricing by two equa- tions:
pt = E(mt+1xt+1) mt+1 = f(data, parameters).
where pt = asset price, xt+1 = asset payoff, mt+1 = stochastic discount factor. 9
The major advantage of the discount factor / moment condition approach are its simplicity and universality. Where once there were three apparently different theories for stocks, bonds, and options, now we see each as just special cases of the same theory. The common language also allows us to use insights from each field of application in other fields.
This approach also allows us to conveniently separate the step of specifying economic assumptions of the model (second equation) from the step of deciding which kind of empiri- cal representation to pursue or understand. For a given model – choice of f(·) – we will see how the first equation can lead to predictions stated in terms of returns, price-dividend ra- tios, expected return-beta representations, moment conditions, continuous vs. discrete time implications and so forth. The ability to translate between such representations is also very helpful in digesting the results of empirical work, which uses a number of apparently distinct but fundamentally connected representations.
Thinking in terms of discount factors often turns out to be much simpler than thinking in terms of portfolios. For example, it is easier to insist that there is a positive discount factor than to check that every possible portfolio that dominates every other portfolio has a larger price, and the long arguments over the APT stated in terms of portfolios are easy to digest when stated in terms of discount factors.
The discount factor approach is also associated with a state-space geometry in place of the usual mean-variance geometry, and this book emphasizes the state-space intuition behind many classic results.
For these reasons, the discount factor language and the associated state-space geometry is common in academic research and high-tech practice. It is not yet common in textbooks, and that is the niche that this book tries to fill.
I also diverge from the usual order of presentation. Most books are structured follow- ing the history of thought: portfolio theory, mean-variance frontiers, spanning theorems, CAPM, ICAPM, APT, option pricing, and finally consumption-based model. Contingent claims are an esoteric extension of option-pricing theory. I go the other way around: con- tingent claims and the consumption-based model are the basic and simplest models around; the others are specializations. Just because they were discovered in the opposite order is no reason to present them that way.
I also try to unify the treatment of empirical methods. A wide variety of methods are pop- ular, including time-series and cross-sectional regressions, and methods based on generalized method of moments (GMM) and maximum likelihood. However, in the end all of these ap- parently different approaches do the same thing: they pick free parameters of the model to make it fit best, which usually means to minimize pricing errors; and they evaluate the model by examining how big those pricing errors are.
As with the theory, I do not attempt an encyclopedic compilation of empirical procedures. The literature on econometric methods contains lots of methods and special cases (likelihood ratio analogues of common Wald tests; cases with and without riskfree assets and when factors do and don’t span the mean variance frontier, etc.) that are seldom used in practice. I
try to focus on the basic ideas and on methods that are actually used in practice.
The accent in this book is on understanding statements of theory, and working with that theory to applications, rather than rigorous or general proofs. Also, I skip very lightly over many parts of asset pricing theory that have faded from current applications, although they occupied large amounts of the attention in the past. Some examples are portfolio separation theorems, properties of various distributions, or asymptotic APT. While portfolio theory is still interesting and useful, it is no longer a cornerstone of pricing. Rather than use portfolio theory to find a demand curve for assets, which intersected with a supply curve gives prices, we now go to prices directly. One can then find optimal portfolios, but it is a side issue for the asset pricing question.
My presentation is consciously informal. I like to see an idea in its simplest form and learn to use it before going back and understanding all the foundations of the ideas. I have or- ganized the book for similarly minded readers. If you are hungry for more formal definitions and background, keep going, they usually show up later on in the chapter.
Again, my organizing principle is that everything can be traced back to specializations of the basic pricing equation p = E(mx). Therefore, after reading the first chapter, one can pretty much skip around and read topics in as much depth or order as one likes. Each major subject always starts back at the same pricing equation.
The target audience for this book is economics and finance Ph.D. students, advanced MBA students or professionals with similar background. I hope the book will also be useful to fellow researchers and finance professionals, by clarifying, relating and simplifying the set of tools we have all learned in a hodgepodge manner. I presume some exposure to undergraduate economics and statistics. A reader should have seen a utility function, a random variable, a standard error and a time series, should have some basic linear algebra and calculus and should have solved a maximum problem by setting derivatives to zero. The hurdles in asset pricing are really conceptual rather than mathematical.
PART I Asset pricing theory
Chapter 1. Consumption-based model
and overview
I start by thinking of an investor who thinks about how much to save and consume, and what portfolio of assets to hold. The most basic pricing equation comes from the first-order conditions to that problem, and say that price should be the expected discounted payoff, using the investor’s marginal utility to discount the payoff. The marginal utility loss of consuming a little less today and investing the result should equal the marginal utility gain of selling the investment at some point in the future and eating the proceeds. If the price does not satisfy this relation, the investor should buy more of the asset.
From this simple idea, I can discuss the classic issues in finance. The interest rate is related to the average future marginal utility, and hence to the expected path of consumption. High real interest rates should be associated with an expectation of growing consumption. In a time of high real interest rates, it makes sense to save, buy bonds, and then consume more tomorrow.
Most importantly, risk corrections to asset prices should be driven by the covariance of asset payoffs with consumption or marginal utility. For a given expected payoff of an asset, an asset that does badly in states like a recession, in which the investor feels poor and is consuming little, is less desirable than an asset that does badly in states of nature like a boom when the investor feels wealthy and is consuming a great deal. The former assets will sell for lower prices; their prices will reflect a discount for their riskiness, and this riskiness depends on a co-variance. This is the fundamental point of the whole book.
Of course, the fundamental measure of how you feel is marginal utility; given that assets must pay off well in some states and poorly in others, you want assets that pay off poorly in states of low marginal uti
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