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Chapter 9

CONDITIONAL ASSET PRICING

WAYNE E. FERSON, Boston College, MA

Abstract

Conditional asset pricing studies predictability in thereturns of financial assets, and the ability of asset pricing models to explain this predictability. The re-lation betweenpredictability andasset pricing modelsis explained and the empirical evidence for predict-ability is summarized. Empirical tests of conditional asset pricing models are then briefly reviewed.

KeyWords: stochastic discount factors; financialasset returns; predictability; rational expectations;conditional expectations; discount rates; stockprices; minimum-variance portfolios; mean vari-ance efficiency; latent variables; capital asset pri-cing models; market price of risk; multiple-betamodels

9.1. Introduction

Conditional Asset Pricing refers to a subset of Asset Pricing research in financial economics.(See Chapter 8.) Conditional Asset Pricing focuseson predictability over time in rates of return onfinancial assets, and the ability of asset pricingmodels to explain this predictability.

Most asset pricing models are special cases of the fundamental equation:

P t E t{mt 1(P t 1 D t 1)}, (9 :1)

where P t is the price of the asset at time t, and D t 1is the amount of any dividends, interest or otherpayments received at time t 1. The market-wide

random variable mt 1 is the stochastic discountfactor (SDF). By recursive substitution in Equa-tion (9.1), the future price may be eliminated to

express the current price as a function of the futurecash flows and SDFs only: P t E t{ P j > 0( Q k1,... , j mt k)D t j }. Prices are obtained by dis-counting the payoffs, or multiplying by SDFs, sothat the expected present value of the payoff isequal to the price. A SDF prices the assets if Equation (9.1) is satisfied, and any particular assetpricing model may be viewed as a specification forthe stochastic discount factor.

The notation E t{:} in Equation (9.1) denotes theconditional expectation, given a market-wide in-formation set, V t . Empiricists dont get to see V t ,so it is convenient to consider expectations condi-tional on an observable subset of instruments, Z t .These expectations are denoted as E (:jZ t ). WhenZ t is the null information set, we have the uncon-ditional expectation, denoted as E (.).

Empirical work on conditional asset pricingmodels typically relies on rational expectations,which is the assumption that the expectation termsin the model are mathematical conditional expect-ations. This carries two important implications.First, it implies that the law of iteratedexpectations can be invoked. This says that theexpectation, given coarser information, of the con-

ditional expectation given finer information, is theconditional expectation given the coarser informa-tion. For example, taking the expected value of Equation (9.1), rational expectations implies thatversions of Equation (9.1) must hold for the ex-

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pectations E (:jZ t ) and E (.). Second, rational ex-pectations implies that the differences between

realizations of the random variables and the ex-pectations in the model, should be unrelated to theinformation that the expectations in the model areconditioned on. This leads to implications for thepredictability of asset returns.

Define the gross asset return, R it 1 (P it 1 D it 1)=P it The return of the asset i maybe predictable. For example, a linear regressionover time of R it 1 on Z t may have a nonzeroslope coefficient. Equation (9.1) implies thatthe conditional expectation of the product of mt 1 and R it 1 is the constant, 1.0. Therefore,

1 mt 1R it 1 should not be predictably differentfrom 0 using any information available at time t. If there is predictability in a return R it 1 using anylagged instruments Z t , the model implies that thepredictability is removed when R it 1 is multipliedby the correct mt 1 . This is the sense inwhich conditional asset pricing models are askedto explain predictable variation in asset returns.

If a conditional asset pricing model fails to ex-plain predictability as described above, there aretwo possibilities (Fama, 1970, 1991). Either thespecification of mt 1 in the model is wrong, orthe use of rational expectations is unjustified. Thefirst instance motivates research on better condi-tional asset pricing models. The second possibilitymotivates research on human departures from ra-tionality, and how these show up in asset marketprices. For a review of this relatively new field,behavioral finance, see Barberis and Shleifer(2003).

Studies of predictability in stock and long-termbond returns typically report regressions that at-tempt to predict the future returns using laggedvariables. These regressions for shorter horizon(monthly, or annual holding period) returns typic-

ally have small R-squares, as the fraction of thevariance in long-term asset returns that can bepredicted with lagged variables over short horizonsis small. The R-squares are larger for longer-hori-zon (two- to five-year) returns, because expected

returns are considered to be more persistent thanreturns themselves. Thus, the variance of the

expected return accumulates with longer horizonsfaster than the variance of the return, and theR-squared increases (Fama and French, 1988).

Because stock returns are very volatile, smallR-squares can mask economically important vari-ation in the expected return. To illustrate, considera special case of Equation (9.1), the simple Gordon(1962) constant-growth model for a stock price:P kE =(R g), where P is the stock price, E isthe earnings per share, k is the dividend payoutratio, g is the future growth rate of earnings, and Ris the discount rate. The discount rate is the re-

quired or expected return of the stock. Stocks arelong duration assets, so a small change in theexpected return can lead to a large fluctuation inthe asset value. Consider an example where theprice-to-earnings ratio, P =E 15, the payoutratio, k 0:6, and the expected growth rate, g 3 percent. The expected return, R, is 7 percent.Suppose there is a shock to the expected return,ceteris paribus . In this example a change of 1 per-cent in R leads to approximately a 20 percentchange in the asset value.

Of course, it is unrealistic to hold everything elsefixed, but the example suggests that small changesin expected returns can produce large and econom-ically significant changes in asset values. Campbell(1991) generalizes the Gordon model to allow forstochastic changes in growth rates, and estimatesthat changes in expected returns through time mayaccount for about half of the variance of equityindex values. Conditional Asset Pricing modelsfocus on these changes in the required or expectedrates of return on financial assets.

9.2. The Conditional Capital Asset Pricing Model

The simplest example of a conditional asset pricingmodel is a conditional version of the Capital AssetPricing Model (CAPM) of Sharpe (1964):

E (R it 1jZ t ) g o(Z t ) b imt g m(Z t ), (9:2)


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where R it 1 is the rate of return of asset i betweentimes t and t 1, and b imt is the market beta at

time t. The market beta is the conditional covar-iance of the return with the market portfolio div-ided by the conditional variance of the marketportfolio; that is, the slope coefficient in a condi-tional regression of the asset return on that of themarket, conditional on the information at time t.Z t is the conditioning information, assumed to bepublicly available at time t. The term g m(Z t)represents the risk premium for market beta, andg o(Z t ) is the expected return of all portfolios withmarket betas equal to zero. If there is a risk-freeasset available at time t, then its rate of return

equalsg

o (Z t ).Sharpe (1964) did not explicitly put the condi-tioning information, Z t , into his derivation of theCAPM. The original development was cast in asingle-period partial equilibrium model. However,it is natural to interpret the expectations in themodel as reflecting a consensus of well-informedanalysts opinion conditional expectations giventheir information and Sharpes subsequent writ-ings indicated this intent (e.g. Sharpe, 1984). Themultiple-beta intertemporal models of Merton(1973) and CoxIngersollRoss (1985) accommo-date conditional expectations explicitly. Merton(1973, 1980) and CoxIngersollRoss also showedhow conditional versions of the CAPM may bederived as special cases of their models.

Roll (1977) and others have shown that a port-folio is minimum variance if and only if a modellike Equation (9.2) fits the expected returns for allthe assets i , using the minimum-variance portfolioas Rmt 1. A portfolio is minimum variance if andonly if no portfolio with the same expected returnhas a smaller variance. According to the CAPM,the market portfolio with return Rmt 1 is minimumvariance. If investors are risk averse, the CAPM

also implies that the market portfolio is mean-variance efficient, which says that g m(Z t ) inEquation (9.2) is positive. In the CAPM, risk-averse investors choose portfolios that have themaximum expected return, given the variance.This implies that there is a positive tradeoff be-

tween market risk, as measured by b imt , and theexpected return on individual assets, when inves-

tors are risk averse. In the conditional CAPM,meanvariance efficiency is defined relative to theconditional expectations and conditional variancesof returns. Hansen and Richard (1987) and Fersonand Siegel (2001) describe theoretical relations be-tween conditional and unconditional versions of meanvariance efficiency.

The conditional CAPM may be expressed in theSDF representation given by Equation (9.1) as:mt 1 c0t c1tRmt 1. In this case, the coefficientsc0t and c1t are specific measurable functions of theinformation set Z t , depending on the first and

second conditional moments of the returns. Toimplement the model empirically, it is necessaryto specify functional forms for c0t and c1t . Shanken(1990) suggests approximating the coefficientsusing linear functions, and this approach is fol-lowed by Cochrane (1996), Jagannathan andWang (1996), and other authors.

9.3. Evidence for Return Predictability

Conditional asset pricing presumes the existence of some return predictability. There should be instru-ments Z t for which E (mt 1jZ t) or E (R t 1jZ t ) varyover time, in order for E (mt 1R t 1 1jZ t ) 0 tohave empirical bite. At one level, this is easy. SinceE (mt 1jZ t ) should be the inverse of a risk-freereturn, all we need for the first condition to biteis observable risk-free rates that vary over time.Indeed, a short-term interest rate is one of themost prominent of the lagged instruments used torepresent Z t in empirical work. Ferson (1977)shows that the behavior of stock returns andshort-term interest rates, as documented by Famaand Schwert (1977), imply that conditional covar-iances of returns with mt 1 must also vary over

time.Interest in predicting security returns is prob-

ably as old as the security markets themselves.Fama (1970) reviews the early evidence andSchwert (2003) reviews anomalies in asset pricingbased on predictability. It is useful to distinguish,


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following Fama (1970), predictability based on theinformation in past returns (weak form) from

predictability based on lagged economic variablesthat are public information, not limited to pastprices and returns (semi-strong form).

A large body of literature studies weak-formpredictability, focusing on serial dependence in re-turns. High-frequency serial dependence, such asdaily or intra-day patterns, are often considered torepresent the effects of market microstructure,such as bidask spreads (e.g. Roll, 1984) and non-synchronous trading of the stocks in an index (e.g.Scholes and Williams, 1977). Serial dependencemay also represent predictable changes in the

expected returns.Conrad and Kaul (1989) report serial depend-ence in weekly returns. Jegadeesh and Titman(1993) find that relatively high-return recentwinner stocks tend to repeat their performanceover three- to nine-month horizons. DeBondt andThaler (1985) find that past high-return stocksperform poorly over the next five years, andFama and French (1988) find negative serial de-pendence over two- to five-year horizons. Theseserial dependence patterns motivate a large num-ber of studies, which attempt to assess the eco-nomic magnitude and statistical robustness of theimplied predictability, or to explain the predictabil-ity as an economic phenomenon. For a summaryof this literature subsequent to Fama (1970), seeCampbell et al. (1997). Research in this area con-tinues, and its fair to say that the jury is still out onthe issue of predictability using lagged returns.

A second body of literature studies semi-strongform predictability using other lagged, publiclyavailable information variables as instruments.Fama and French (1989) assemble a list of vari-ables from studies in the early 1980s, which as of this writing remain the workhorse instruments for

conditional asset pricing models. In addition to thelevel of a short-term interest rate, as mentionedabove, the variables include the lagged dividendyield of a stock market index, a yield spread of long-term government bonds relative to short-termbonds, and a yield spread of low-grade (high-

default risk and low liquidity) corporate bondsover high-grade corporate bonds. In addition,

studies often use the lagged excess return of amedium-term over a short-term Treasury bill(Campbell, 1987; Ferson and Harvey, 1991). Add-itional instruments include an aggregate book-to-market ratio (Pontiff and Schall, 1998) and laggedconsumption-to-wealth ratios (Lettau and Ludvig-son, 2001a). Of course, many other predictor vari-ables have been proposed and more will doubtlessbe proposed in the future.

Predictability using lagged instruments remainscontroversial, and there are some good reasons themeasured predictability could be spurious. Studies

have identified various statistical biases in predict-ive regressions (e.g. Hansen and Hodrick, 1980;Stambaugh, 1999; Ferson et al., 2003), and havequestioned the stability of predictive relationsacross economic regimes (e.g. Kim et al., 1991; orPaye and Timmermann, 2003) and raised the pos-sibility that the lagged instruments arise solelythrough data mining (e.g. Lo and MacKinlay,1990; Foster et al., 1997).

A reasonable response to these concerns is to seeif the predictive relations hold out-of-sample. Thiskind of evidence is also mixed. Some studies findsupport for predictability in step-ahead or out-of-sample exercises (e.g. Fama and French, 1989;Pesaran and Timmerman, 1995). Similar instru-ments show some ability to predict returns outsidethe United States, where they were originally stud-ied (e.g. Harvey, 1991; Solnik, 1993; Ferson andHarvey, 1993, 1999). However, other studies con-clude that predictability using the standard laggedinstruments does not hold in more recent samples(e.g. Goyal and Welch, 2003; Simin, 2002). Itseems that research on the predictability of securityreturns will always be interesting, and conditionalasset pricing models should be useful in framing

many future investigations of these issues.

9.4. Tests of Conditional CAPMs

Empirical studies have rejected versions of theCAPM that ignore lagged variables. This evidence,




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and mounting evidence of predictable variation inthe distribution of security returns led to empirical

work on conditional versions of the CAPM start-ing in the early 1980s. An example from Equation(9.2) illustrates the implications of the conditionalCAPM for predictability in returns. Rational ex-pectations implies that the actual return differsfrom the conditional expected value by an errorterm, uit 1, which is orthogonal to the informationat time t. If the actual returns are predictable usinginformation in Z t , the model implies that either thebetas or the premiums ( g m(Z t ) and g o(Z t )), arechanging as functions of Z t , and the time variationin those functions should track the predictable

components of asset returns. If the time variationin g m(Z t ) and g o (Z t ) can be modeled, the condi-tional CAPM can be tested by examining its abilityto explain the predictability in returns.

The earliest empirical tests along these lines werethe latent variable models, developed by Hansenand Hodrick (1983) and Gibbons and Ferson(1985), and later refined by Campbell (1987) andFerson et al. (1993). These models allow time vary-ing expected returns, but maintain the assumptionthat the conditional betas are fixed parametersover time.

Consider the conditional representation of theCAPM. Let r it 1 R it 1 R0t 1, and similarly forthe market return, where R 0t 1 is the gross, zerobeta return. The conditional CAPM may then bestated for the vector of excess returns r t 1, asE (r t 1jZ t ) b E (rmt 1jZ t ), where b is the vectorof assets betas. Let r1t be any reference assetexcess return with nonzero beta, b 1, so thatE (r1t 1jZ t ) b 1 E (rmt 1jZ t ). Solving this expres-sion for E (rmt 1jZ t ) and substituting, we haveE (r t 1jZ t ) CE (r1t 1jZ t ), where C (b = b 1). and.=denotes element-by-element division. Theexpected market risk premium is now a latent

variable in the model, and C is the N -vector of the model parameters. Gibbons and Ferson (1985)argued that the latent variable model is attractivein view of the difficulties associated with measur-ing the true market portfolio of the CAPM, butWheatley (1989) emphasized that it remains neces-

sary to assume that ratios of the betas measuredwith respect to the unobserved market portfolio,

are constant parameters.Campbell (1987) and Ferson and Foerster

(1994) show that a single-beta latent variablemodel is rejected by the data. This rejects the hy-pothesis that there is a conditional minimum-vari-ance portfolio such that the ratios of conditionalbetas on this portfolio are fixed parameters. There-fore, the empirical evidence suggests that condi-tional asset pricing models should be consistentwith either (1) a time varying beta, or (2) morethan one beta for each asset.

Conditional CAPMs with time varying betas

are examined by Harvey (1989), replacing theconstant beta assumption with the assumptionthat the ratio of the expected market premium tothe conditional market variance is a fixed param-eter: E (rmt 1jZ t )=Var( rmt 1jZ t ) g . Then, theconditional expected returns may be writtenaccording to the conditional CAPM asE (r t 1jZ t ) g Cov( r t 1 , rmt 1jZ t ). Harveys ver-sion of the conditional CAPM is motivated fromMertons (1980) model, in which the ratio g ,called the market price of risk, is equal to therelative risk aversion of a representative investorin equilibrium. Harvey also assumes that the con-ditional expected risk premium on the market(and the conditional market variance, given fixedg ) is a linear function of the instruments:E (rmt 1jZ t ) dm 0Z t , where dm is a coefficient vec-tor. He rejects this version of the conditionalCAPM for monthly data in the United States. InHarvey (1991), the same formulation is rejectedwhen applied to a world market portfolio andmonthly data on the stock markets of 21 devel-oped countries.

Lettau and Ludvigson (2001b) examine a condi-tional CAPM with time varying betas and risk

premiums, using rolling time-series and cross-sec-tional regression methods. They condition themodel on a lagged, consumption-to-wealth ratio,and find that the conditional CAPM works betterfor explaining the cross-section of monthly stockreturns.




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9.5. Multi-beta Conditional Asset Pricing Models

A multi-beta asset pricing model essentially ex-pands Equation (9.2) to allow for multiple sourcesof risk and expected return. Such a model assertsthat the expected return is a linear function of several betas, i.e.

E t (R it 1) l 0t j 1,... ,K b ijt l jt, (9:3)

where the b ijt , j 1, . . . , K , are the conditionalmultiple regression coefficients of the return of asset i on K risk factors, f jt 1, j 1, . . . , K . Thecoefficient l 0t is the expected return on an assetthat has b 0 jt 0, for j 1, . . . , K ; i.e. it is theexpected return on a zero-(multiple-) beta asset. If there is a risk-free asset, then l 0t is the return of this asset. The coefficient l kt , corresponding to thekth factor has the following interpretation: it is theexpected return differential, or premium, for aportfolio that has b ikt 1 and b ijt 0 for all j 6 k, measured in excess of the zero-beta assetsexpected return. In other words, it is the expectedreturn premium per unit of beta risk for the riskfactor, k. Multiple-beta models follow when mt 1can be written as a conditional linear function of the K factors, as shown by Ferson and Jagan-nathan (1996).

Bansal and Viswanathan (1993) developed con-ditional versions of the CAPM and multiple-factormodels in which the stochastic discount factor mt 1is a nonlinear function of the market or factorreturns. Using nonparametric methods, they findevidence to support the nonlinear versions of themodels. Bansal et al. (1993) compare the perform-ance of nonlinear models with linear models, usingdata on international stocks, bonds, and currencyreturns, and they find that the nonlinear modelsperform better. Farnsworth et al. (2002) comparedthe empirical performance of a large set of condi-

tional asset pricing models using the SDF repre-sentation.

Conditional multiple-beta models with constantbetas are examined empirically by Ferson and Har-vey (1991), Evans (1994), and Ferson and Korajc-zyk (1995), who find that while such models are

rejected using the usual statistical tests, they stillcapture a large fraction of the predictability of

stock and bond returns over time. Allowing fortime varying betas, these studies find that thetime variation in betas contributes a relativelysmall amount to the time variation in expectedasset returns, while time variation in the risk pre-mium are relatively more important.

While time variation in conditional betas is notas important as time variation in expected riskpremiums, from the perspective of modeling pre-dictable time variation in asset returns, this doesnot imply that beta variation is empirically unim-portant. From the perspective of modeling the

cross-sectional variation in unconditionalexpected asset returns, beta variation over timemay be empirically very important. This idea wasfirst explored by Chan and Chen (1988). To seehow this works, consider the unconditionalexpected excess return vector, obtained from themodel as E {E (rjZ )} E { l (Z )b (Z )} E ( l )E (b ) Cov( l (Z ),b (Z )). Viewed as a cross-sectional re-lation, the term Cov( l (Z ),b (Z )) may differ signifi-cantly in a cross-section of assets. Therefore theimplications of a conditional version of the CAPMfor the cross-section of unconditional expectedreturns may depend importantly on commontime variation in betas and expected market riskpremiums.

Jagannathan and Wang (1996) used the condi-tional CAPM to derive a particular uncondi-tional 2-factor model. They show thatmt 1 a 0 a 1E (rmt 1jV t ) Rmt 1, where V t de-notes the information set of investors and a0 and a1are fixed parameters, is a validSDF in the sense thatE (R i ,t 1mt 1) 1 for this choice of mt 1 . Assumingthat E (rmt 1jZ t) is a linear function of Z t , they findthat their version of the model explains the cross-section of unconditional expected returns better

than an unconditional version of the CAPM.New empirical tests of the conditional CAPM

and multiple-beta models, using the multi-betarepresentation and SDF representations, continueto appear regularly in the literature. Future studieswill continue to refine the relationships among the




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various empirical specifications. Research on thepredictability of security returns will always be

interesting, and Conditional Asset Pricing Modelsshould be useful in framing many future investiga-tions of these issues.

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