Portfolio Selection and Asset Pricing Models

Finance theory can be used to form informative prior beliefs in financial decision making. This paper approaches portfolio selection in a Bayesian framework that incorporates a prior degree of belief in an asset pricing model. Sample evidence on home bias and value and size effects is evaluated from an asset-allocation perspective. U.S. investors' belief in the domestic CAPM must be very strong to justify the home bias observed in their equity holdings. The same strong prior belief results in large and stable optimal positions in the Fama-French book-to-market portfolio in combination with the market since the 1940s. FINANCE THEORY HAS PRODUCED A VARIETY of models that attempt to provide some insight into the environment in which financial decisions are made. How should these models be used by financial decision makers? Empirical finance typically approaches a theoretical model by testing whether its implications are supported by the data. Based on the result of a hypothesis test, the model is either rejected or not rejected. It is not clear what such an outcome implies about the usefulness of the model for decision making. If the model is not rejected, should it be used as the truth? And if it is rejected, should it be discarded as worthless? Such a simplistic approach, based solely on the result of a hypothesis test, fails to capture many aspects of both the model and the data that could potentially be useful to a decision maker. Instead, it might be reasonable to assume that financial models are neither perfect nor useless. By definition, every model is a simplification of reality. Hence, even if the data fail to reject the model, the decision maker may not necessarily want to use the model as a dogma. At the same time, the notion that models implied by finance theory could be entirely worthless seems rather extreme. Hence, even if the data reject the model, the decision maker may want to use the model at least to some degree. * Graduate School of Business, University of Chicago. The paper is based on my dissertation at the Wharton School, University of Pennsylvania. I am grateful to the members of my dissertation committee, Don Keim, Karen Lewis, Craig MacKinlay, Frank Schorfheide, and especially to the committee chair Rob Stambaugh, for their numerous helpful comments. The comments of Greg Bauer, Michael Brandt, Mark Britten-Jones, Chris G6czy, Chris Jones, Ron Kaniel, Krishna Ramaswamy, Jay Shanken (the referee), Ren6 Stulz (the editor), and seminar participants at Columbia University, Harvard University, MIT, New York University, the University of California at Los Angeles, the University of Chicago, the University of Pennsylvania, the University of Rochester, Yale University, and the 1999 meetings of the American Finance Association are also appreciated. Of course, I am fully responsible for all the weaknesses of the paper. 179 180 The Journal of Finance A natural approach to using financial models in decision making can be developed in a Bayesian framework. A model can be used as a point of reference around which the decision maker can center his prior beliefs. These prior beliefs are combined with the data, which may violate the implications of the model, and the revised beliefs are used to make decisions. The relative importance of the sample evidence versus the model depends on the strength of the violations of the model in the data relative to the strength of the prior belief in the model.' Following such an approach, this paper explores how asset pricing models can be used in portfolio selection. The goal of portfolio selection is to find an optimal allocation of wealth across a number of assets. At least two approaches to portfolio selection are commonly used in finance. A "data-based" approach assumes a functional form for the distribution of asset returns and estimates its parameters from the time series of returns. For example, sample estimates of the mean and covariance matrix of asset returns can be used to compute the optimal weights in a mean-variance framework. This approach ignores the potential usefulness of asset pricing models.2 Asset pricing models imply an alternative approach to portfolio selection. In this "model-based" approach, the optimal portfolio of every investor is a combination of benchmark portfolios that expose the investor only to priced sources of risk.3 For example, under the Capital Asset Pricing Model, the market portfolio is the single benchmark, and is therefore the optimal portfolio of every investor. This approach makes no use of the time series of returns on the nonbenchmark assets. The two typical approaches to portfolio selection essentially reflect two extreme views about the validity of asset pricing models. The first approach regards asset pricing models as useless, and the second approach considers one of these models to be a perfect description of reality. Such polar views could be adopted as a result of a hypothesis test, as described at the outset.4 However, the portfolio literature is silent about what happens in between. For example, what if an investor thinks highly of a certain pricing model, but is concerned that the model may not hold exactly due to mild violations of its assumptions in the real world?5 1 The idea of using financial models to form prior beliefs in decision making is also mentioned in Stambaugh (1998). 2 Examples of asset pricing models include the Capital Asset Pricing Models (CAPM) of Sharpe (1964) and Lintner (1965), the intertemporal CAPM of Merton (1973), and models based on the arbitrage pricing theory of Ross (1976). The precise meaning of the term data-based approach is clarified in Section I.A. 3 Examples of benchmark portfolios are factor-mimicking portfolios, whose returns mimic the realizations of the factors in a factor-based asset pricing model. 4 Frequentist tests of asset pricing models are too numerous to list. Bayesian tests include Shanken (1987), Harvey and Zhou (1990), McCulloch and Rossi (1990, 1991), Kandel, McCulloch, and Stambaugh (1995), and Geweke and Zhou (1996). 5 For instance, Jagannathan and Wang (1996) argue that "We have to keep in mind that the CAPM, like any other model, is only an approximation of reality. Hence, it would be rather surprising if it turns out to be "100 percent accurate"." Portfolio Selection and Asset Pricing Models 181 This paper approaches the portfolio selection problem in a Bayesian framework that incorporates the investor's prior degree of confidence in an asset pricing model. The degree of confidence can range from a dogmatic belief in the model to a belief that the model is useless. As the degree of skepticism about the model grows, the resulting optimal allocation moves away from a combination of benchmark portfolios toward the allocation obtained in the data-based approach. We explore how fast the optimal allocation moves from one extreme to the other in response to sample evidence and what determines the strength of the influence of sample evidence on the optimal allocation. The approach developed in the paper uses both an asset pricing model and the time series of asset returns to find the optimal portfolio. The investor specifies an informative prior distribution on the assets' mispricing a within an asset pricing model. As is typical in Bayesian analysis, the sample mispricing a' is "shrunk" toward the prior mean of a to obtain the posterior mean of a, which is used in portfolio analysis. The most natural choice for the prior mean of a is zero, the value implied by the model. Prior confidence in the model's implication that a = 0 is expressed through 0-a, the prior standard deviation of a. Due to the shrinkage in a, the sample mean is shrunk toward the expected return implied by the pricing model. Shrinking the sample mean reduces the sensitivity of the optimal weights to the sampling error in a'. The weights have less extreme values and are more stable over time than in the data-based approach. The idea of specifying an informative prior on a is proposed in Pastor and Stambaugh (1999), where the authors suggest a Bayesian approach to estimating costs of equity and analyze the sources of uncertainty in the cost of equity estimates for individual firms. The part of our methodology that leads to the posterior distribution can be viewed as a multivariate extension of a part of the methodology in Pastor and Stambaugh. However, the focus of this paper is quite different. Instead of examining an estimation problem, we concentrate on the decision problem of forming the optimal portfolio of multiple risky assets. In this paper, the investor's prior beliefs are centered around an asset pricing model. In a related study, Black and Litterman (1992) suggest using the CAPM as a benchmark toward which the investor can shrink his subjective views about expected returns. The extent of the deviations from the CAPM depends on the investor's degree of confidence in his subjective views. That study makes no direct use of sample information about expected returns. In contrast, our approach shrinks the sample means toward their values implied by the model. The extent of the deviations from the model depends on the strength of the violations of the model in the data as well as on the investor's degree of confidence in the model. The focus of the empirical analysis is to investigate the extent to which optimal holdings depart from the market portfolio, which plays a central role in finance theory. In our mean-variance examples, wealth is allocated between the market portfolio and an asset (or assets) with nonzero sample mispricing a' within the CAPM. It is well known that one should invest 182 The Journal of Finance (disinvest) in any asset whose a is positive (negative), since combining the asset with the market portfolio increases the portfolio's Sharpe ratio.6 However, the true value of a is unknown. How much attention should the investor pay to a nonzero value of the sample estimate of a? The impact of a' on the optimal allocation is investigated for different assets and different prior beliefs about a. The empirical analysis investigates the home bias in the equity holdings of U.S. investors and the issues of investing based on value and size. The home bias puzzle is associated with the observation that investors' equity holdings typically include a substantially larger proportion of domestic equities than is suggested by standard portfolio theory. U.S. investors hold only about eight percent of their equity holdings in foreign equities, although their optimal allocation in foreign equities based on the sample moments of asset returns is more than 40 percent.7 However, several recent studies cannot reject the hypothesis that the global mean-variance efficient portfolio puts zero weights on non-U.S. stocks. These two different ways of looking at the data, one based on point estimates and the other on the result of a hypothesis test, lead to different conclusions about the home bias and about the benefits of international diversification for U.S. investors. This paper assesses the evidence in the data from an asset allocation perspective. A U.S. investor confronted with the data decides how much to invest in foreign stocks. In our framework, the bias toward domestic equities can simply reflect a certain degree of prior confidence in the domestic CAPM. However, we find that U.S. investors' belief in the global mean-variance efficiency of the U.S. market portfolio must be very strong to justify the home bias observed in their equity holdings. Their actual holdings are consistent with the prior belief that the annual mispricing of a foreign stock portfolio within the domestic CAPM is in the tight interval between -2 percent and 2 percent. Surprisingly, even the same strong prior belief in the CAPM is significantly revised by the sample evidence about the Fama and French (1993) book-to-market portfolio (HML, or "high minus low"). Consider an investor who allocates his wealth between HML and the market portfolio, and who believes that the annual mispricing of HML within the CAPM is between -2 percent and 2 percent. As of January 1997, this investor should optimally invest 40 percent of his wealth in HML, despite his strong belief in the 6An asset with a nonzero a that is combined with a passive portfolio is sometimes referred to as an "active portfolio," following Treynor and Black (1973). The portfolio's Sharpe ratio is the ratio of its expected excess return and the standard deviation of its return. Adding an asset with mispricing a to the market portfolio increases the portfolio's squared Sharpe ratio by (a/o-)2, where o(2 is the residual variance from the market model regression. See Gibbons, Ross, and Shanken (1989). 7 See Lewis (1999). The foreign portfolio in her example is Morgan Stanley's EAFE index, and her sample period is January 1970 through December 1996. The home bias puzzle is also present from the perspective of the international CAPM. The weight of non-U.S. equity in the value-weighted world market portfolio is about 60 percent. Portfolio Selection and Asset Pricing Models 183 mean-variance efficiency of the market portfolio. Moreover, this investor's optimal position in HML is large and rather stable, mostly between 20 percent and 40 percent, in every month since the early 1940s. The optimal positions in HML for investors with weaker beliefs in the CAPM are even larger and still fairly stable. For example, an investor who is completely skeptical about the CAPM should have held approximately 50 to 80 percent of his wealth in HML in the last five decades. The robust optimal weights in HML are primarily due to the robust value premium over the last 60 years. The rest of the paper is organized as follows. Section I first describes the data-based approach to portfolio selection and then develops our "model-anddata-based" Bayesian methodology. Section II describes the data used in the empirical analysis. In Sections III through V, optimal combinations of the market portfolio with a number of different assets are explored. Section III examines the home bias puzzle. Section IV looks into investing based on value and size. Section V explores the effect of imposing prior beliefs that depart from the model. Section VI concludes. I. Methodology The methodology section is divided into four subsections. The first subsection lays out the portfolio selection problem and discusses the data-based approach to this problem. The remaining subsections develop a methodology that can be used to compute optimal weights in the presence of a nontrivial prior degree of belief in an asset pricing model. The second subsection specifies the assumptions on the stochastic behavior of returns and the resulting likelihood function. The third subsection describes the prior distribution on the model parameters. The final subsection describes how the predictive distribution of returns is obtained. A. Portfolio Selection Consider a risk-averse investor with a one-period investment horizon who must allocate funds between a riskless asset and a portfolio of (N + K) risky assets, K of which are benchmark portfolios. The returns on the benchmark portfolios replicate the realizations of K priced sources of risk in a certain asset pricing model. The (N + K) risky assets are referred to as "investable assets," and the N risky assets are referred to as "nonbenchmark assets" or simply "assets." The investor is assumed to consider the past to be informative about the future. The allocation decision is made based on the information set 'I containing a finite history of returns on the investable assets and prior information. The investor believes that his portfolio decision has no effect on the probability distribution of asset returns. The markets are assumed to be frictionless, with no transaction costs or taxes. Let W denote the investor's current wealth, and 8 the proportion of the wealth invested in the riskless asset. The optimal value of 6 depends on the degree of the investor's risk aversion and is not investigated in this paper. 184 The Journal of Finance Let w denote the (N + K) x 1 vector of the weights in the portfolio of the investable assets, wherew 'LN?K =1 and CN?K is an (N + K) -vector of ones. The investor's wealth one period later is W+1 = W(1 + rf + (1 8)w'r+?), (1) where rf stands for the rate of return on the riskless asset and r+1 is the (N + K) x 1 vector of the next-period returns on the investable assets in excess of rf. The investor chooses w to maximize the expected utility of the next-period wealth: max fu(W+i)p(r+?1') dr+?, (2) w where u is the investor's utility function and p (r+1 1) is the probability density of r+1 conditional on (P, often referred to as the predictive density.8 Although the predictive density is in general unknown, the density p (r+ 1 0, P) is usually assumed to be known, where 0 denotes the parameters of the statistical model that describes the stochastic behavior of returns. However, 0 is unknown. The simplest way to deal with this challenge is to treat the sample estimates 0 as their true values. However, such an approach ignores the estimation risk in the estimates and hence understates the true level of uncertainty faced by the investor. As shown by Zellner and Chetty (1965), Brown (1976), Klein and Bawa (1976), and others, estimation risk can be accounted for in a Bayesian framework. Instead of using p(r+1 0S,'), the predictive density can be obtained as p (r+l l D) = Jp(r+110S, D)p( (StD) d0 (3) The posterior distribution of 0, p (0 I), is proportional to the product of the prior distribution and the likelihood function, P (O I b) oc p (0) L(0; (D). (4) One approach to specifying the likelihood function is to assume that, in each period, the joint distribution of the excess returns on the investable assets is multivariate normal with parameters E and V. If the prior distribution of 0 -(E,V) is noninformative and all investable assets have return histories of the same length, the resulting predictive density is a multivariate Student t and the tangency portfolio weights are the same as the weights 8 Throughout the paper, p is a generic notation for any probability density function. Portfolio Selection and Asset Pricing Models 185 obtained when E and V are simply replaced by their maximum-likelihood estimates. Such an approach corresponds to our approach when N = 0 and the investor seeks an optimal mix of the benchmark portfolios. Stambaugh (1997) assumes a standard noninformative prior on 0 and derives closed-form expressions for the first two moments of the predictive density when the assets have return histories that differ in length. In our framework, the K benchmarks may have longer histories than the N assets. When our investor is completely skeptical about the pricing model implied by the K benchmarks, the results essentially coincide with the results obtained in Stambaugh's unequal-history framework. The expressions for the first two moments of the predictive density p (r 1 1), E and V, are given in Appendix A. As in the equal-history framework, estimation risk is included in V, which exceeds the maximum-likelihood estimate of the covariance matrix by a positive definite matrix. Unlike in the equal-history framework, the weights in the tangency portfolio are affected by estimation risk and in general differ from the weights produced by the maximum-likelihood estimates of E and V. Although more complicated approaches based solely on the data can be constructed, our designation "data-based approach" refers to the approach that obtains the weights using the moments given in Appendix A. The remainder of Section I proposes an alternative methodology for obtaining E and V. Unlike the data-based approach described above, the new methodology allows the investor to use an asset pricing model and incorporate his prior degree of confidence in the model's pricing abilities. Regardless of the approach used to obtain E and V, the (N + K) x 1 vector of the weights in the portfolio with the maximum Sharpe ratio is V-1k W* ~~~~~~~~~~~~~~(5) LN+KVE The above expression is a standard mean-variance result, which requires the return on the riskless asset to be smaller than the return on the global minimum variance portfolio of risky assets. A risk-averse mean-variance investor optimally chooses a portfolio with the maximum Sharpe ratio. For simplicity, the examples presented in this paper focus on the familiar meanvariance case and calculate the optimal portfolio weights using equation (5). However, our procedure can be used to construct the entire predictive distribution of returns on the investable assets, not only its first two moments. As a result, the portfolio choice problem in equation (2) can also be solved for utility functions that involve higher order moments such as skewness and kurtosis. Note that our investor should not be viewed as a representative investor. The equilibrium in asset markets cannot be supported if all investors have the same perception of the likelihood function and identical nondogmatic prior beliefs. For example, with identical imperfect beliefs in the CAPM, all investors would deviate from the market portfolio in the direction pointed by 186 The Journal of Finance the data. If all investors perceive the same likelihood, the existence of an equilibrium requires heterogeneity of investors' prior beliefs, with some prior beliefs about mispricing centered at nonzero values. B. Likelihood Suppose that L returns on K benchmark portfolios are available and denote the L x K matrix of those returns in excess of rf by FL. Let Ft denote the tth row of FL, t = 1,... ,L. Also, suppose that T c L returns on N risky assets are available in the most recent periods t = L T + 1,... ,L. Let R denote the T x N matrix of those returns in excess of rf. Let Rt denote the (t-L+T)throwofR,t=L-T+1,...,L,andletFTdenotetheTxK submatrix of FL corresponding to the same period as R. The multivariate regression of the asset returns on the benchmark returns can be written as R =XB + U, vec(U) N(O, I IT), (6) where X =[LT FT], "vec" denotes an operator that stacks the columns of a matrix into a vector, "0" denotes the Kronecker product, and