Scand. J. of Economics 116(3), 593–634, 2014 DOI: 10.1111/sjoe.12070
Empirical Asset Pricing: , Hansen, and
. Campbell†,∗
Harvard University, Cambridge, MA 02138, USA
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The Prize in Economic Sciences for 2013 was awarded to , Hansen, and for their contributions to the empirical study of asset pricing. Some observers have found it hard to understand the common elements of the laureates’ research, preferring to highlight areas of disagreement among them. In this paper, I argue that empirical asset pricing is a coherent enterprise, which owes much to the laureates’ influential contributions, and that important themes in the literature can best be understood by considering the laureates in pairs. Specifically, after summarizing modern asset-pricing theory using the stochastic discount factor as an organizing framework, I discuss the following: the joint hypothesis problem in tests of market efficiency, which is as much an opportunity as a problem (Fama and Hansen); patterns of short- and long-term predictability in asset returns (Fama and Shiller); and models of deviations from rational expectations (Hansen and Shiller). I conclude by reviewing the ways in which the laureates have already influenced the practice of finance, and how they might influence future innovations.
Keywords: Behavioral finance; financial innovation; market efficiency; stochastic discount factor
JEL classification: G10; G12 I. Introduction
The 2013 Prize in Economic Sciences in Memory of , awarded for the empirical analysis of asset prices, was un- forgettably exciting for financial economists. The 2013 laureates, , Hansen, and , are giants of finance and ar- chitects of the intellectual structure within which all contemporary research in asset pricing is conducted.
†Also, Research Associate at the National Bureau of Economic Research.
∗This paper has been commissioned by The Scandinavian Journal of Economics for its annual survey of the Prize in Economic Sciences in Memory of . I am grateful to , , , , – nathan, , , , , , Andrei Shleifer, , and the Nobel laureates for helpful comments on an earlier draft. I also acknowledge the inspiration provided by the Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences in their scientific background paper “Under- standing Asset Prices”, available online at http://www.nobelprize.org/nobel_prizes/economic- sciences/laureates/2013/advanced-economicsciences2013.pdf.
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The fame of the laureates extends far beyond financial economics. Eu- gene Fama is one of the world’s most cited economists in any field. Hansen is an immensely distinguished econometrician, so the field of econometrics naturally claims a share of his Nobel glory. is a founder of behavioral economics, a creator of the Case–Shiller house-price indices, and the author of important and widely read books for a general audience.
The 2013 prize attracted attention in the media, and stimulated discussion among economists, for two additional reasons. First, the behavior of asset prices interests every investor, including every individual who is saving for retirement, and it is a core concern for the financial services industry. Second, the laureates have interpreted asset-price movements in strikingly different ways. is famous for his writings on asset-price bubbles, and his public statements that stocks in the late 1990s and houses in the mid-2000s had become overvalued as the result of such bubbles. is skeptical that the term “bubble” is a well-defined or useful term. More broadly, Fama believes that asset-price movements can be understood using economic models with rational investors, whereas Shiller does not.
The purpose of this paper is to celebrate the 2013 Prize in Economic Sciences, and to explain the achievements of the laureates in a way that brings out the connections among them. I hope to be able to communicate the intellectual coherence of the award, notwithstanding the differing views of the laureates on some unsettled questions.
I should say a few words about my own connections with the laureates. changed my life when he became my PhD dissertation adviser at Yale University in the early 1980s. In the course of my career, I have written 12 papers with him, the earliest in 1983 and the most recent (and hopefully not the last) in 2009. , the oldest of the 2013 laureates, was already a legend over 30 years ago, and his research on market efficiency was intensively discussed in , CT, and every other center of academic economics. I first met Hansen when I visited Chicago while seeking my first academic job in 1984. I have never forgotten the first conversation I had with him about financial econometrics, in which I sensed his penetrating insight that would require effort to fully understand but would amply reward the undertaking.
Among financial economists, I am not unusual in these feelings of strong connection with the 2013 Nobel laureates. The 2013 award ceremony in Stockholm was notable for the celebratory atmosphere among the co- authors and students of the laureates who were present, including academics , , , , nathan, , and , central banker , and asset management practitioners , , and Antti
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Ilmanen. Any of them could write a similar paper to this one, although of course the views expressed here are my own and are probably not fully shared by any other economists, including the Nobel laureates themselves.
The organization of this paper is as follows. In Section II, I give a basic explanation of the central concept of modern asset-pricing theory, the stochastic discount factor (SDF). While the basic theory is not due to the laureates, their work has contributed to our understanding of the concept, and many of their empirical contributions can most easily be understood by reference to it.
In Section III, I discuss the concept of market efficiency, formulated by Fama in the 1960s. Fama first stated the “joint hypothesis problem” in testing market efficiency, which Hansen later understood to be as much an opportunity as a problem, leading him to develop an important econometric method for estimating and testing economic models: the generalized method of moments (GMM).
In Section IV, I review empirical research on the predictability of asset returns in the short and long run. Fama’s early work developed econometric methods, still widely used today, for testing the short-run predictability of returns. Typically, these methods find very modest predictability, but both Fama and Shiller later discovered that such predictability can cumulate over time to become an important, and even the dominant, influence on longer-run movements in asset prices. Research in this area continues to be very active, and is distinctive in its tight integration of financial theory with econometrics.
In Section V, I discuss the work of the laureates on asset pricing when some or all market participants have beliefs about the future that do not conform to objective reality. Shiller helped to launch the field of behavioral economics, and its most important subfield of behavioral finance, when he challenged the orthodoxy of the early 1980s that economic models must always assume rational expectations by all economic agents. Later, Hansen approached this topic from the very different perspective of robust optimal control.
In Section VI, I explore the implications of the laureates’ work for the practice of finance. Contemporary methods of portfolio construction owe a great deal to the work of Fama on style portfolios, that is, portfolios of stocks or other assets sorted by characteristics such as value (measures of cheapness that compare accounting valuations to market valuations) or momentum (recent past returns). The quantitative asset management industry uses many ideas from the work of the laureates, and Shiller’s recent work emphasizes the importance of financial innovation for human welfare in modern economies.
In each of these sections, I refer to the work of more than one of the 2013 Nobel laureates. In this way, I hope to foster an appreciation
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for the intellectual dialogue among the laureates and the many researchers
following their lead.
II. SDF: The Framework of Contemporary Finance
SDF in Complete Markets
The modern theory of the SDF originates in the leading theoretical contri- butions of Ross (1978) and Harrison and Kreps (1979). Here, I present a brief summary in an elementary discrete-state model with two periods, the present and the future, and complete markets.
Consider a simple model with S states of nature s = 1,…, S, all of which have strictly positive probability π(s). I assume that markets are complete; that is, for each state s, a contingent claim is available that pays $1 in state s and nothing in any other state. I write the price of this contingent claim as q(s).
I assume that all contingent claim prices are strictly positive. If this were not true, then there would be an arbitrage opportunity in one of two senses. First, if the contingent claim price for some state s were zero, then an investor could buy that contingent claim, paying nothing today, while having some probability of receiving a positive payoff if state s occurs tomorrow, and having no possibility of a negative payoff in any state of the world. Second, if the contingent claim price for state s were negative, then an investor could buy that contingent claim, receiving a positive payoff today, while again having some probability of a positive payoff and no possibility of a negative payoff in the future.
Any asset, whether or not it is a contingent claim, is defined by its state-contingentpayoffs X(s)forstatess=1,…,S.Thelawofoneprice (LOOP) says that two assets with identical payoffs in every state must have the same price. If this were not true, then again there would be an arbitrage opportunity – this time in the sense that an investor could go long the cheap asset and short the expensive one, receiving cash today while having guaranteed zero payoffs in all states in the future. The LOOP implies that we must have
q ( s ) S
π(s)π(s) X(s) =
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q(s)X(s). (1) The next step in the analysis is to multiply and divide equation (1) by
π(s)M(s)X(s) = E[MX], (2)
the objective probability of each state, π(s):
where M(s) = q(s)/π(s) is the ratio of state price to probability for state s, the SDF in state s. Because q(s) and π(s) are strictly positive for all states s, M(s) is also. The last equality in equation (2) uses the definition of an expectation as a probability-weighted average of a random variable to write the asset price as the expected product of the asset’s payoff and the SDF. This equation is sometimes given the rather grand title of the fundamental equation of asset pricing.
Consider a riskless asset with payoff X(s) = 1 in every state. The price
1+Rf = 1 = 1 . (4) Pf E[M]
This tells us that the mean of the SDF must be fairly close to one. A riskless real interest rate of 2 percent, for example, implies a mean SDF of 1/1.02 ≈ 0.98.
Utility Maximization and the SDF. Consider a price-taking investor who chooses initial consumption C0 and consumption in each future state C(s) to maximize the time-separable utility of consumption. Assume, for now, that the investor’s subjective state probabilities coincide with the objective probabilities π(s) (i.e., the investor has rational expectations). The investor’s maximization problem is
Pf = so the riskless interest rate
q(s) = E[M], (3)
subject to
q(s)C(s) = W0, (6)
where W0 is initial wealth (including the present value of future income, discounted using the appropriate contingent claims prices). The first-order conditions of the problem can be written as
u′(C0)q(s) = βπ(s)u′[C(s)] for s = 1, . . . , S. (7) These first-order conditions imply that
M(s) = q(s) = βu′[C(s)]. (8) π(s) u′(C0)
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Max u(C0) +
βπ(s)u[C(s)], (5)
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In other words, the SDF is the discounted ratio of marginal utility tomorrow to marginal utility today. This representation of the SDF is the starting point for the large body of literature on equilibrium asset pricing, which seeks to relate asset prices to the arguments of consumers’ utility, and particularly to their measured consumption of goods and services.
Heterogeneous Beliefs. In the above discussion, I assume that all investors have rational expectations and thus assign the same probabilities to the dif- ferent states of the world. If this is not the case, we must assign investor- specific subscripts to the probabilities, writing πj(s) for investor j’s sub- jective probability of state s. In general, we must also allow for differences in the utility function across investors, adding a j subscript to marginal utility as well. Then, for any state s and investor j,
βπj(s)u′ [Cj(s)]
q(s) = j . (9)
u′ (Cj0) j
The state price is related to the product of the investor’s subjective proba- bility of the state and the investor’s marginal utility in that state. In other words, it is a composite “util-prob” to use the terminology of Samuelson (1969).
A similar observation applies to the SDF, the ratio of state price to
objective probability:
M(s)= = j j . (10)
q(s) π (s)βu′ [Cj(s)] π(s) π(s) u′ (Cj0)
Volatility of the SDF across states can correspond either to volatile devi- ations of investor j’s subjective probabilities from objective probabilities, or to volatile marginal utility across states. The usual assumption that in- vestors have homogeneous beliefs rules out the first of these possibilities, while the behavioral finance literature embraces it.
The SDF and Risk Premia. I now return to the assumption of rational expectations and adapt the notation above to move in the direction of empirical research in finance. I add the subscript t for the initial date at which the asset’s price is determined, and the subscript t + 1 for the next period at which the asset’s payoff is realized. This can easily be embedded in a multiperiod model, in which case the payoff is the next period’s price plus dividend. I add the subscript i to denote an asset. Then, we have
Pit = Et [Mt+1 Xi,t+1] = Et [Mt+1]Et [Xi,t+1] + Covt (Mt+1, Xi,t+1), (11) where the t subscripts on the mean and covariance indicate that these are
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price of the asset at time t is included in the information set at time t, and hence there is no need to take a conditional expectation of this variable. Because the conditional mean of the SDF is the reciprocal of the gross riskless interest rate from equation (4), equation (11) says that the price of any asset is its expected payoff, discounted at the riskless interest rate, plus a correction for the conditional covariance of the payoff with the SDF.
For assets with positive prices, we can divide through by Pit and use (1 + Ri,t+1) = Xi,t+1/Pit to obtain
1 = Et [Mt+1(1 + Ri,t+1)]
= Et [Mt+1]Et [1 + Ri,t+1] + Covt (Mt+1, Ri,t+1). (12)
Rearranging and using the relation between the conditional mean of the SDF and the riskless interest rate, we have
Et [1 + Ri,t+1] = (1 + R f,t+1)[1 − Covt (Mt+1, Ri,t+1)]. (13)
This says that the expected return on any asset is the riskless return times an adjustment factor for the covariance of the return with the SDF.
Subtracting the gross riskless interest rate from both sides, the risk premium on any asset is the gross riskless interest rate times the covariance of the asset’s excess return with the SDF:
Et [Ri,t+1 − R f,t+1] = −(1 + R f,t+1)Covt (Mt+1, Ri,t+1 − R f,t+1). (14) Generalizing and Applying the SDF Framework
In the above discussion, I assume complete markets, but the SDF frame- work is just as useful when markets are incomplete. The work of Hansen and Richard (1987) and Hansen and Jagannathan (1991) is particularly im- portant in characterizing the SDF for incomplete markets. Shiller (1982) is an insightful early contribution. Cochrane (2005) offers a textbook treat- ment.
In incomplete markets, the existence of a strictly positive SDF is guar- anteed by the absence of arbitrage – a result sometimes called the fun- damental theorem of asset pricing – but the SDF is no longer unique as it is in complete markets. Intuitively, an SDF can be calculated from the marginal utility of any investor who can trade assets freely, but with in- complete markets, each investor can have idiosyncratic variation in his or her marginal utility. Hence, there are many possible SDFs.
However, there is a unique SDF that can be written as a linear combi- nation of asset payoffs and that satisfies the fundamental equation of asset pricing (2). This unique random variable is the projection of any SDF onto the space of asset payoffs, and thus any other SDF must have a higher variance.
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Volatility Bounds on the SDF. Shiller (1982), a comment by Hansen (1982a), and Hansen and Jagannathan (1991) used this insight to place lower bounds on the volatility of the SDF, based only on the properties of asset returns.
A simple lower bound of this sort can be calculated from a single risky asset return and the return on a riskless asset, using the fact that the correlation between the SDF and any excess return must be greater than −1. Equation (14) implies that
Et [Ri,t+1 − R f,t+1] = −Covt (Mt+1, Ri,t+1 − R f,t+1) Et Mt+1
= −Corrt (Mt+1, Ri,t+1 − R f,t+1)σt (Mt+1)σt (Ri,t+1 − R f,t+1) Et Mt+1
≤ σt(Mt+1)σt(Ri,t+1 − Rf,t+1). (15) Et Mt+1
Rearranging, we obtain
σt(Mt+1) ≥ Et[Ri,t+1 − Rf,t+1]. (16) Et Mt+1 σt (Ri,t+1 − R f,t+1)
The standard deviation of the SDF, divided by its mean (which is always close to one, given the relation between the mean SDF and the riskless interest rate), must be at least as great as the mean of the risky asset’s excess return divided by its standard deviation (i.e., the Sharpe ratio of the risky asset). The tightest lower bound is achieved by finding the risky asset, or portfolio of assets, with the highest Sharpe ratio.
This is a very simple way to understand the famous equity premium puzzle of Mehra and Prescott (1985). If, for example, the Sharpe ratio of the aggregate stock market is 0.4 and the average riskless interest rate is 1.03, then the mean SDF must be 1/1.03 or about 0.97, and the standard deviation of the SDF must be at least 0.97 × 0.4 = 0.39. This is a sub- stantial volatility for a random variable with a mean close to one, which must always be positive, and it is far greater than the volatilities produced by simple equilibrium models. A consumption-based asset-pricing model with a representative agent with power utility, for example, implies that the volatility of the SDF is the coefficient of relative risk aversion times the standard deviation of consumption growth, which is of the order of 0.01. So, the volatility bound
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