markets,and perfectly rational economic agents.Of course,a great deal of research is devoted to understanding asset prices under market imperfections like information and transactions costs.The perfect markets models reviewed here represent a baseline,and a starting point for understanding these more complex issues. Work in empirical asset pricing over the last few years has provided a markedly improved understanding of the relations among the various asset-pricing models.Bits and pieces of this are scattered across a number of published papers,and some is "common"knowledge,shared by aficionados.This paper provides an integrative discussion,refining the earlier review in Ferson (1995) to reflect what I hope is an improved understanding. Much of our understanding of how asset pricing models'empirical predictions are related flows from representing the models as stochastic discount factors.Section 2 presents the stochastic discount factor approach,briefly illustrates a few examples of stochastic discount factors,and then relates the representation to beta pricing and to mean variance efficiency.These three concepts: stochastic discount factors,beta pricing and mean variance efficiency,are at the core of modern empirical asset pricing.We show the relation among these three concepts,and a "large-markets" interpretation of these relations.The discussion then proceeds to refinements of these issues in the presence of conditioning information.Section 2 ends with a brief discussion of how the risk factors have been identified in the empirical literature,and what the empirical evidence has to say about the selection of factors. Section 3 begins with a fundamental empirical application of the stochastic discount factor approach-the variance bounds originally developed by Hansen and Jagannathan(1991).Unlike the case where a model identifies a particular stochastic discount factor,the question in the Hansen- Jagannathan bounds is:Given a set of asset returns,and some conditioning information,what can be
2 markets, and perfectly rational economic agents. Of course, a great deal of research is devoted to understanding asset prices under market imperfections like information and transactions costs. The perfect markets models reviewed here represent a baseline, and a starting point for understanding these more complex issues. Work in empirical asset pricing over the last few years has provided a markedly improved understanding of the relations among the various asset-pricing models. Bits and pieces of this are scattered across a number of published papers, and some is "common" knowledge, shared by aficionados. This paper provides an integrative discussion, refining the earlier review in Ferson (1995) to reflect what I hope is an improved understanding. Much of our understanding of how asset pricing models' empirical predictions are related flows from representing the models as stochastic discount factors. Section 2 presents the stochastic discount factor approach, briefly illustrates a few examples of stochastic discount factors, and then relates the representation to beta pricing and to mean variance efficiency. These three concepts: stochastic discount factors, beta pricing and mean variance efficiency, are at the core of modern empirical asset pricing. We show the relation among these three concepts, and a "large-markets" interpretation of these relations. The discussion then proceeds to refinements of these issues in the presence of conditioning information. Section 2 ends with a brief discussion of how the risk factors have been identified in the empirical literature, and what the empirical evidence has to say about the selection of factors. Section 3 begins with a fundamental empirical application of the stochastic discount factor approach - the variance bounds originally developed by Hansen and Jagannathan (1991). Unlike the case where a model identifies a particular stochastic discount factor, the question in the HansenJagannathan bounds is: Given a set of asset returns, and some conditioning information, what can be
3 said about the set of stochastic discount factors that could properly "price"the assets?By now,a number of reviews of the original Hansen-Jagannathan bounds are available in the literature.The discussion here is brief,quickly moving on to focus on less well-known refinements of the bounds to incorporate conditioning information. Section 4 discusses empirical methods,starting with Hansen's (1982)Generalized Method of Moments(GMM).This important approach has also been the subject of several review articles and textbook chapters.We briefly review the use of the GMM to estimate stochastic discount factor models.This section is included only to make the latter parts of the paper accessible to a reader who is not already familiar with the GMM.Section 4 then discusses two special cases that remain important in empirical asset pricing.The first is the cross-sectional regression approach,as developed by Fama and MacBeth(1973),and the second is the multivariate regression approach,popularized in the finance literature following Gibbons(1982). Once the mainstay of empirical work on asset pricing,cross-sectional regression continues to be used and useful.Our main focus is on the economic interpretation of the estimates.The discussion attempts to shed light on recent studies that employ the empirical factors advocated by Fama and French (1993,1996),or generalizations of that approach.The multivariate regression approach to testing portfolio efficiency can be motivated by its immunity to the errors-in-variables problem that plagues the two step,cross-sectional regression approach.The multivariate approach is also elegant, and provides a nice intuition for the statistical tests. A regression approach,with a beta pricing formulation,and a GMM approach with a stochastic discount factor formulation,may be considered as competing paradigms for empirical work in asset pricing.However,under the same distributional assumptions,and when the same moments are estimated,the two approaches are essentially equivalent.The present discussion attempts to clarify
3 said about the set of stochastic discount factors that could properly "price" the assets? By now, a number of reviews of the original Hansen-Jagannathan bounds are available in the literature. The discussion here is brief, quickly moving on to focus on less well-known refinements of the bounds to incorporate conditioning information. Section 4 discusses empirical methods, starting with Hansen's (1982) Generalized Method of Moments (GMM). This important approach has also been the subject of several review articles and textbook chapters. We briefly review the use of the GMM to estimate stochastic discount factor models. This section is included only to make the latter parts of the paper accessible to a reader who is not already familiar with the GMM. Section 4 then discusses two special cases that remain important in empirical asset pricing. The first is the cross-sectional regression approach, as developed by Fama and MacBeth (1973), and the second is the multivariate regression approach, popularized in the finance literature following Gibbons (1982). Once the mainstay of empirical work on asset pricing, cross-sectional regression continues to be used and useful. Our main focus is on the economic interpretation of the estimates. The discussion attempts to shed light on recent studies that employ the empirical factors advocated by Fama and French (1993, 1996), or generalizations of that approach. The multivariate regression approach to testing portfolio efficiency can be motivated by its immunity to the errors-in-variables problem that plagues the two step, cross-sectional regression approach. The multivariate approach is also elegant, and provides a nice intuition for the statistical tests. A regression approach, with a beta pricing formulation, and a GMM approach with a stochastic discount factor formulation, may be considered as competing paradigms for empirical work in asset pricing. However, under the same distributional assumptions, and when the same moments are estimated, the two approaches are essentially equivalent. The present discussion attempts to clarify
4 these points,and suggests how to think about the choice of empirical method. Section 5 brings the models and methods together,in a review of the relatively recent literature on conditional performance evaluation.The problem of measuring the performance of managed portfolios has been the subject of research for more than 30 years.Traditional measures use unconditional expected returns,estimated by sample averages,as the baseline.However,if expected returns and risks vary over time,this may confuse common time-variation in fund risk and market risk premiums with average performance.In this way,traditional methods can ascribe abnormal performance to an investment strategy that trades mechanically,based only on public information. Conditional performance evaluation attempts to control these biases,while delivering potentially more powerful performance measures,by using lagged instruments to control for time-varying expectations. Section 5 reviews the main models for conditional performance evaluation,and includes a summary of the empirical evidence.Finally,Section 6 of this paper offers concluding remarks. 2.Multifactor Asset Pricing Models:Review and Integration 2.1 The Stochastic Discount Factor Representation Virtually all asset pricing models are special cases of the fundamental equation: P=Et{m+1(P+1+D+i)}, (1) where Pt is the price of the asset at time t and D is the amount of any dividends,interest or other payments received at time t+1.The market-wide random variable m is the stochastic discount factor (SDF).The prices are obtained by"discounting"the payoffs using the SDF,or multiplying by mso 1 The random variable m is also known as the pricing kernel,benchmark pricing variable,or
4 these points, and suggests how to think about the choice of empirical method. Section 5 brings the models and methods together, in a review of the relatively recent literature on conditional performance evaluation. The problem of measuring the performance of managed portfolios has been the subject of research for more than 30 years. Traditional measures use unconditional expected returns, estimated by sample averages, as the baseline. However, if expected returns and risks vary over time, this may confuse common time-variation in fund risk and market risk premiums with average performance. In this way, traditional methods can ascribe abnormal performance to an investment strategy that trades mechanically, based only on public information. Conditional performance evaluation attempts to control these biases, while delivering potentially more powerful performance measures, by using lagged instruments to control for time-varying expectations. Section 5 reviews the main models for conditional performance evaluation, and includes a summary of the empirical evidence. Finally, Section 6 of this paper offers concluding remarks. 2. Multifactor Asset Pricing Models: Review and Integration 2.1 The Stochastic Discount Factor Representation Virtually all asset pricing models are special cases of the fundamental equation: Pt = Et {mt+1 (Pt+1 + Dt+1)}, (1) where Pt is the price of the asset at time t and Dt+1 is the amount of any dividends, interest or other payments received at time t+1. The market-wide random variable mt+1 is the stochastic discount factor (SDF).1 The prices are obtained by "discounting" the payoffs using the SDF, or multiplying by mt+1, so 1 The random variable mt+1 is also known as the pricing kernel, benchmark pricing variable, or
5 that the expected"present value"of the payoff is equal to the price. The notation Ef.denotes the conditional expectation,given a market-wide information set, Since empiricists don't get to see it will be convenient to consider expectations conditioned on an observable subset of instruments,Zt.These expectations are denoted as E(.Zt).When Zt is the null information set,we have the unconditional expectation,denoted as E(.).Empirical work on asset pricing models like (1)typically relies on rational expectations,interpreted as the assumption that the expectation terms in the model are mathematical conditional expectations.Taking the expected values of equation (1),rational expectations implies that versions of(1)must hold for the expectations E(.Zt) and E(.). Assuming nonzero prices,equation(1)is equivalent to: E(m+1R+1-112)=0, (2) where R+is the N-vector of primitive asset gross returns and 1 is an N-vector of ones.The gross return Rit+i is defined as (Pit++Dit+1)/Pit.We say that a SDF "prices"the assets if equations (1)and (2)are satisfied.Empirical tests of asset pricing models often work directly with equation(2)and the relevant definition of m+1. Without more structure the equations (1-2)have no content because it is almost always possible to find a random variable mti for which the equations hold.There will be some mui that "works,"in this sense,as long as there are no redundant asset returns.?With the restriction that mis intertemporal marginal rate of substitution,depending on the context.The representation (1)goes at least back to Beja(1971),while the term "stochastic discount factor"is usually ascribed to Hansen and Richard(1987). 2 For example,take a sample of assets with a nonsingular second moment matrix and let m be
5 that the expected "present value" of the payoff is equal to the price. The notation Et{.} denotes the conditional expectation, given a market-wide information set, Ωt. Since empiricists don't get to see Ωt, it will be convenient to consider expectations conditioned on an observable subset of instruments, Zt. These expectations are denoted as E(.|Zt). When Zt is the null information set, we have the unconditional expectation, denoted as E(.). Empirical work on asset pricing models like (1) typically relies on rational expectations, interpreted as the assumption that the expectation terms in the model are mathematical conditional expectations. Taking the expected values of equation (1), rational expectations implies that versions of (1) must hold for the expectations E(.|Zt) and E(.). Assuming nonzero prices, equation (1) is equivalent to: E(mt+1 Rt+1 - 1 |Ωt)=0, (2) where Rt+1 is the N-vector of primitive asset gross returns and 1 is an N-vector of ones. The gross return Ri,t+1 is defined as (Pi,t+1 + Di,t+1)/Pi,t. We say that a SDF "prices" the assets if equations (1) and (2) are satisfied. Empirical tests of asset pricing models often work directly with equation (2) and the relevant definition of mt+1. Without more structure the equations (1-2) have no content because it is almost always possible to find a random variable mt+1 for which the equations hold. There will be some mt+1 that "works," in this sense, as long as there are no redundant asset returns.2 With the restriction that mt+1 is intertemporal marginal rate of substitution, depending on the context. The representation (1) goes at least back to Beja (1971), while the term "stochastic discount factor" is usually ascribed to Hansen and Richard (1987). 2 For example, take a sample of assets with a nonsingular second moment matrix and let mt+1 be
6 a strictly positive random variable,equation(1)becomes equivalent to the no arbitrage principle,which says that all portfolios of assets with payoffs that can never be negative,but which are positive with positive probability,must have positive prices [Beja(1971),Rubinstein(1976),Ross(1977),Harrison and Kreps(1979),Hansen and Richard(1987)1. The no arbitrage condition does not uniquely identify mui unless markets are complete.In that case,mi is equal to primitive state prices divided by state probabilities.To see this write equation(1) as Pit =EfmiXitti),where Xit+i Pit+I Ditt1.In a discrete-state setting,Pit EsnsXis Esqs(/qs)Xis,where gs is the probability that state s will occur and ns is the state price,equal to the value at time t of one unit of the numeraire to be paid at time t+l if state s occurs at time t+1.Xis is the total payoff of the security i at time t+1 if state s occurs.Comparing this expression with equation(1) shows that ms=/gs>0 is the value of the SDF in state s. While the no arbitrage principle places some restrictions on mt+1,empirical work typically explores the implications of equilibrium models for the SDF,based on investor optimization.Consider the Bellman equation for a representative consumer-investor's optimization: J(Wtst)=Max E{U(Ct.)+J(Wt+1,St+I), (3) where U(Ct.)is the direct utility of consumption expenditures at time t,and J(.)is the indirect utility of wealth.The notation allows the direct utility of current consumption expenditures to depend on variables such as past consumption expenditures or other state variables,s.The state variables are sufficient statistics,given wealth,for the utility of future wealth in an optimal consumption-investment plan.Thus,changes in the state variables represent future consumption-investment opportunity risk [I(E{R+R+)]R+I
6 a strictly positive random variable, equation (1) becomes equivalent to the no arbitrage principle, which says that all portfolios of assets with payoffs that can never be negative, but which are positive with positive probability, must have positive prices [Beja (1971), Rubinstein (1976), Ross (1977), Harrison and Kreps (1979), Hansen and Richard (1987)]. The no arbitrage condition does not uniquely identify mt+1 unless markets are complete. In that case, mt+1 is equal to primitive state prices divided by state probabilities. To see this write equation (1) as Pi,t = Et{mt+1Xi,t+1}, where Xi,t+1 = Pi,t+1 + Di,t+1. In a discrete-state setting, Pit = ΣsπsXi,s = Σsqs(πs/qs)Xi,s, where qs is the probability that state s will occur and πs is the state price, equal to the value at time t of one unit of the numeraire to be paid at time t+1 if state s occurs at time t+1. Xi,s is the total payoff of the security i at time t+1 if state s occurs. Comparing this expression with equation (1) shows that ms = πs/qs > 0 is the value of the SDF in state s. While the no arbitrage principle places some restrictions on mt+1, empirical work typically explores the implications of equilibrium models for the SDF, based on investor optimization. Consider the Bellman equation for a representative consumer-investor's optimization: J(Wt,st) ≡ Max Et{ U(Ct,.) + J(Wt+1,st+1)}, (3) where U(Ct,.) is the direct utility of consumption expenditures at time t, and J(.) is the indirect utility of wealth. The notation allows the direct utility of current consumption expenditures to depend on variables such as past consumption expenditures or other state variables, st. The state variables are sufficient statistics, given wealth, for the utility of future wealth in an optimal consumption-investment plan. Thus, changes in the state variables represent future consumption-investment opportunity risk. [1' (Et{Rt+1Rt+1'})-1]Rt+1