A Mean-Variance Framework for Tests of Asset Pricing Models Shmuel Kandel University of Chicago Tel-Aviv,University Robert F.Stambaugh University of Pennsylvania This article presents a mean-variance framework for likelihood-ratio tests of asset pricing models.A pricing model is tested by examining the position of one or more reference portfolios in sample mean- standard-deviation space.Included are tests of both single-beta and multiple-beta relations,with or with- out a riskless asset,using either a general or a spe- cific alternative hypothesis.Tests with a factor that is not a portfolio return are also included.The mean- variance framework is illustrated by testing the zero- beta CAPM,a two-beta pricing model,and the con- sumption-beta model. Many asset pricing models imply a linear relation between the expected return on an asset and covari- ances between the asset's return and one or more fac- tors.The implications of such models can also be stated in terms of the mean-variance efficiency of a benchmark portfolio.In single-beta pricing relations,the bench- mark portfolio can be identified specifically.For exam- ple,in the capital asset pricing model(CAPM)of Sharpe (1964),(Lintner 1965),and Black (1972),it is well Financial support from the Center for Research in Security Prices is gratefully acknowledged.This research was completed while the first author was a Bat- terymarch Fellow.The authors are grateful for comments by Michael Gibbons, Jack Glen,John Long,and participants in workshops at the University of Alberta, the University of British Columbia,the University of Pennsylvania,and Prince- ton University.Address reprint requests to Prof.Robert Stambaugh,Finance Department,The Wharton School,University of Pennsylvania,Philadelphia,PA 19104-6367 The Review of Finauncial Studies 1989 Volume 2.number 2,pp.125-156 1989 The Review of Financial Studies 0893-9454/89/S1.50
A Mean-Variance Framework for Tests of Asset Pricing Models Shmuel Kandel University of Chicago Tel-Aviv, University Robert F. Stambaugh University of Pennsylvania This article presents a mean-variance framework for likelihood-ratio tests of asset pricing models. A pricing model is tested by examining the position of one or more reference portfolios in sample meanstandard-deviation space. Included are tests of both single-beta and multiple-beta relations, with or without a riskless asset, using either a general or a specific alternative hypothesis. Tests with a factor that is not a portfolio return are also included. The meanvariance framework is illustrated by testing the zerobeta CAPM, a two-beta pricing model, and the consumption-beta model. Many asset pricing models imply a linear relation between the expected return on an asset and covariances between the asset's return and one or more factors. The implications of such models can also be stated in terms of the mean-variance efficiency of a benchmark portfolio. In single-beta pricing relations, the benchmark portfolio can be identified specifically. For example, in the capital asset pricing model (CAPM) of Sharpe (1964), (Lintner 1965), and Black (1972), it is well Financial support from the Center for Research in Security Prices is gratefully acknowledged. This research was completed while the first author was a Batterymarch Fellow. The authors are grateful for comments by Michael Gibbons, Jack Glen, John Long, and participants in workshops at the University of Alberta, the University of British Columbia, the University of Pennsylvania, and Princeton University. Address reprint requests to Prof. Robert Stambaugh, Finance Department, The Wharton School, University of Pennsylvania, Philadelphia,PA 19104-6367. The Review of Finauncial Studies 1989 Volume 2, number 2, pp.125-156 © 1989 The Review of Financial Studies 0893-9454/89/$1.50
Tbe Revlew of Financial Studies /v 2n 2 1989 known that mean-beta linearity is equivalent to mean-variance efficiency of the market portfolio [Fama (1976)Roll (1977),and Ross (1977)].Sim- ilarly,the consumption-beta model implies the mean-variance efficiency of the portfolio having maximal correlation with consumption [Breeden (1979)].In multiple-beta pricing relations,the benchmark portfolio gen- erally cannot be identified specifically but instead is characterized as some combination of a set of reference portfolios.For example,an exact K-factor arbitrage pricing relation is equivalent to the mean-variance efficiency of some portfolio that combines K factor-mimicking portfolios [Grinblatt and Titman (1987)and Huberman,Kandel,and Stambaugh (1987)]. Although the equivalence between linear pricing relations and mean- variance efficiency is well understood at a theoretical level,links between tests of the pricing models and a mean-variance framework are limited to a few special cases.'This article presents a complete framework for the characterization and investigation of likelihood-ratio tests of the pricing restrictions in a mean-variance setting.Our treatment includes tests with either a single beta or multiple betas,with or without a riskless asset,using either a general or a specific alternative hypothesis.We also extend the mean-variance framework to test the pricing relation with a factor that is not a portfolio return.All the tests considered should be viewed as tests of mean-variance efficiency defined in terms of unconditional distributions rather than as tests of conditional mean-variance efficiency. A major virtue of the mean-variance framework presented in this article is that it allows the researcher to represent graphically in two familiar dimensions the outcome of a test of a multidimensional pricing restriction. A pricing model is tested by examining the position of one or more ref- erence portfolios in sample mean-standard-deviation space.In this approach, the likelihood-ratio-test statistic can be viewed not only as the outcome of a numerical procedure but also as a quantity with simple economic and statistical interpretations. One case for which the mean-variance framework has been developed is one in which a pricing model,that includes a riskless asset is tested against a general alternative hypothesis.The likelihood-ratio test in this case can be characterized as comparing the position in sample mean- standard-deviation space of a benchmark portfolio,or a set of reference portfolios,to the position of the sample tangent portfolio [e.g.,Jobson and Korkie (1982)and Gibbons,Ross,and Shanken (1989)].The rejection region in sample mean-standard-deviation space is defined by a pair of lines. We show that,in the absence of a riskless asset,the rejection region for the likelihood-ratio test using a general alternative hypothesis is defined by a hyperbola in sample mean-standard-deviation space.As in the case in which a riskless asset exists,the rejection region depends only on the See Jobson and Kodde(1982,1988),Gibbons,Ross,and Shanken (1989),Kandel (1984,1986),and Roll (1985). 126
known that mean-beta linearity is equivalent to mean-variance efficiency of the market portfolio [Fama (1976) Roll (1977), and Ross (1977)]. Similarly, the consumption-beta model implies the mean-variance efficiency of the portfolio having maximal correlation with consumption [Breeden (1979)]. In multiple-beta pricing relations, the benchmark portfolio generally cannot be identified specifically but instead is characterized as some combination of a set of reference portfolios. For example, an exact K-factor arbitrage pricing relation is equivalent to the mean-variance efficiency of some portfolio that combines K factor-mimicking portfolios [Grinblatt and Titman (1987) and Huberman, Kandel, and Stambaugh (1987)]. Although the equivalence between linear pricing relations and meanvariance efficiency is well understood at a theoretical level, links between tests of the pricing models and a mean-variance framework are limited to a few special cases.’ This article presents a complete framework for the characterization and investigation of likelihood-ratio tests of the pricing restrictions in a mean-variance setting. Our treatment includes tests with either a single beta or multiple betas, with or without a riskless asset, using either a general or a specific alternative hypothesis. We also extend the mean-variance framework to test the pricing relation with a factor that is not a portfolio return. All the tests considered should be viewed as tests of mean-variance efficiency defined in terms of unconditional distributions rather than as tests of conditional ‘mean-variance efficiency. A major virtue of the mean-variance framework presented in this article is that it allows the researcher to represent graphically in two familiar dimensions the outcome of a test of a multidimensional pricing restriction. A pricing model is tested by examining the position of one or more reference portfolios in sample mean-standard-deviation space. In this approach, the likelihood-ratio-test statistic can be viewed not only as the outcome of a numerical procedure but also as a quantity with simple economic and statistical interpretations. One case for which the mean-variance framework has been developed is one in which a pricing model, that includes a riskless asset is tested against a general alternative hypothesis. The likelihood-ratio test in this case can be characterized as comparing the position in sample meanstandard-deviation space of a benchmark portfolio, or a set of reference portfolios, to the position of the sample tangent portfolio [e.g., Jobson and Korkie (1982) and Gibbons, Ross, and Shanken (1989)]. The rejection region in sample mean-standard-deviation space is defined by a pair of lines. We show that, in the absence of a riskless asset, the rejection region for the likelihood-ratio test using a general alternative hypothesis is defined by a hyperbola in sample mean-standard-deviation space. As in the case in which a riskless asset exists, the rejection region depends only on the 1 See Jobson and Kodde (1982, 1988), Gibbons, Ross, and Shanken (1989), Kandel (1984, 1986), and Roll (1985). 126
estimated means and variance-covariance matrix for the observed universe of assets and does not depend on the specified benchmark or reference portfolios.It is not necessary to estimate a zero-beta expected return to conduct the test,With a single benchmark portfolio,the likelihood-ratio test consists of asking whether the position of the benchmark portfolio in sample mean-standard-deviation space lies within the rejection region. With a collection of reference portfolios,the researcher first plots the sample minimum-standard-deviation boundary of all combinations of the reference portfolios.The test then consists of asking whether this entire boundary lies within the rejection region.We illustrate these procedures by testing a zero-beta CAPM and a two-beta pricing model. The mean-variance framework is also used to investigate likelihood-ratio tests of pricing models against specific alternative hypotheses.We consider tests of a K-beta pricing model against a specific K2-beta pricing model. The null hypothesis identifies K,reference portfolios to be used in explain- ing expected returns,and the specific alternative hypothesis identifies an additional set of K2-K reference portfolios.If a riskless asset exists, then a test of a K,-beta model against a K2-beta model is conducted by testing whether the tangent portfolio of the K,portfolios is also the tangent port- folio of the larger set of K,portfolios.The test is identical to the test of a K,-beta model against a general alternative,except that the set of K2 ref- erence portfolios replaces the original universe of n assets.No other infor- mation about the other n-K,assets is used. When a riskless asset is not included,the specific alternative hypothesis is that some combination of the K2 portfolios is efficient with respect to the set of n assets.As in the case with a riskless asset,there is a close correspondence between the mean-variance representations of the tests against the general and specific alternatives,and the critical hyperbolas in sample mean-standard-deviation space are from the same class.Unlike the case with a riskless asset,however,the critical hyperbola in the case without a riskless asset depends on the returns of all n assets,Tests using a specific alternative are illustrated by testing a single-beta model against a two-beta model. We extend the mean-variance framework to tests of a pricing relation with a factor,such as consumption,that is not a portfolio return.In par- ticular,we consider the role of a reference portfolio with weights estimated. within the sample,to approximate those of the portfolio having maximal correlation with the factor.We show that,if a riskless asset exists,then the likelihood-ratio test of a single-beta pricing model,where betas are defined with respect to a factor,is similar to the test of a single-beta model using a reference portfolio with prespecified weights.In both cases,the position of the reference portfolio is compared with the position of the sample tangent portfolio of the observed universe of n assets.With a prespecified reference portfolio,the critical value for this comparison depends only on the sample means and variance-covariance matrix of the n assets.In the test with estimated weights,however,the critical value also depends on 127
estimated means and variance-covariance matrix for the observed universe of assets and does not depend on the specified benchmark or reference portfolios. It is not necessary to estimate a zero-beta expected return to conduct the test, With a single benchmark portfolio, the likelihood-ratio test consists of asking whether the position of the benchmark portfolio in sample mean-standard-deviation space lies within the rejection region. With a collection of reference portfolios, the researcher first plots the sample minimum-standard-deviation boundary of all combinations of the reference portfolios. The test then consists of asking whether this entire boundary lies within the rejection region. We illustrate these procedures by testing a zero-beta CAPM and a two-beta pricing model. The mean-variance framework is also used to investigate likelihood-ratio tests of pricing models against specific alternative hypotheses. We consider tests of a K1 -beta pricing model against a specific K2 -beta pricing model. The null hypothesis identifies K1 reference portfolios to be used in explaining expected returns, and the specific alternative hypothesis identifies an additional set of K2 - K1 reference portfolios. If a riskless asset exists, then a test of a K1 -beta model against a K2 -beta model is conducted by testing whether the tangent portfolio of the K1 portfolios is also the tangent portfolio of the larger set of K2 portfolios. The test is identical to the test of a K1 -beta model against a general alternative, except that the set of K2 reference portfolios replaces the original universe of n assets. No other information about the other n - K2 assets is used. When a riskless asset is not included, the specific alternative hypothesis is that some combination of the K2 portfolios is efficient with respect to the set of n assets. As in the case with a riskless asset, there is a close correspondence between the mean-variance representations of the tests against the general and specific alternatives, and the critical hyperbolas in sample mean-standard-deviation space are from the same class. Unlike the case with a riskless asset, however, the critical hyperbola in the case without a riskless asset depends on the returns of all n assets, Tests using a specific alternative are illustrated by testing a single-beta model against a two-beta model. We extend the mean-variance framework to tests of a pricing relation with a factor, such as consumption, that is not a portfolio return. In particular, we consider the role of a reference portfolio with weights estimated, within the sample, to approximate those of the portfolio having maximal correlation with the factor. We show that, if a riskless asset exists, then the likelihood-ratio test of a single-beta pricing model, where betas are defined with respect to a factor, is similar to the test of a single-beta model using a reference portfolio with prespecified weights. In both cases, the position of the reference portfolio is compared with the position of the sample tangent portfolio of the observed universe of n assets. With a prespecified reference portfolio, the critical value for this comparison depends only on the sample means and variance-covariance matrix of the n assets. In the test with estimated weights, however, the critical value also depends on 127
the sample correlation between the return on the estimated reference portfolio and the factor.We illustrate this procedure by testing the con- sumption-beta model. The mean-variance framework offers directions for future research beyond the scope of this study.For example,the mean-variance framework pre- sented here,coupled with previously developed analysis,allows the researcher to investigate problems associated with measuring accurately the returns on relevant benchmark or reference portfolios.Kandel and Stambaugh (1987)conducted such an investigation for the Sharpe-Lintner form of the CAPM,where a riskless asset is included.They computed the maximum correlation between a given benchmark portfolio and a portfolio that gives a different inference about the model,and they tested the hypoth- esis that the correlation between the benchmark and the ex ante tangent portfolio exceeds a given level.Their analysis combined a mean-variance framework for the likelihood-ratio test with the results of Kandel and Stambaugh (1986),which derived the maximum correlation between a given portfolio and another portfolio with a given location in mean-vari- ance space.Similar analyses can be conducted for other pricing models by combining the mean-variance framework for tests of these models with the results of Kandel and Stambaugh (1986). The article proceeds as follows.Section 1 defines terms and notation used.Section 2 analyzes likelihood-ratio tests using a general alternative hypothesis,and Section 3 presents tests using specific alternative hypoth- eses.As each test is discussed,we include an illustration using weekly returns on stock market indexes and common stock portfolios formed according to firm size.Section 4 extends the framework to models with a factor that is not a portfolio return and provides an illustration using con- sumption data.Section 5 concludes the article. 1.Definitions and Notation We consider a set of n risky assets,which are often portfolios formed from a larger universe of individual assets.An n x matrix G with full column rank contains the weights for portfolios that are combinations of the n assets.A given set of K reference portfolios is represented by the matrix A,a specific choice of G having K columns.A single reference portfolio is denoted by the vector p,a specific choice of G having one column. Let R,denote returns in period t on the K reference portfolios,and let r,denote returns in period t on the remaining n-K assets.If a riskless asset exists,then R,and r,denote excess returns on these assets,that is, returns in excess of the riskless rate reIt is assumed throughout the paper If the riskless rate is changing,the use of excess returns is problematic in terms of defining unconditional mean-variance efficiency.In that our primary goal is to provide a simple framework in which to Interpret previously applied tests,we follow the convention of previous research in our use of excess retums.This issue,along with more general questions about the appropriateness of testing unconditional relations,lie beyond the scope of this study. 128
the sample correlation between the return on the estimated reference portfolio and the factor. We illustrate this procedure by testing the consumption-beta model. The mean-variance framework offers directions for future research beyond the scope of this study. For example, the mean-variance framework presented here, coupled with previously developed analysis, allows the researcher to investigate problems associated with measuring accurately the returns on relevant benchmark or reference portfolios. Kandel and Stambaugh (1987) conducted such an investigation for the Sharpe-Lintner form of the CAPM, where a riskless asset is included. They computed the maximum correlation between a given benchmark portfolio and a portfolio that gives a different inference about the model, and they tested the hypothesis that the correlation between the benchmark and the ex ante tangent portfolio exceeds a given level. Their analysis combined a mean-variance framework for the likelihood-ratio test with the results of Kandel and Stambaugh (1986), which derived the maximum correlation between a given portfolio and another portfolio with a given location in mean-variance space. Similar analyses can be conducted for other pricing models by combining the mean-variance framework for tests of these models with the results of Kandel and Stambaugh (1986). The article proceeds as follows. Section 1 defines terms and notation used. Section 2 analyzes likelihood-ratio tests using a ‘general alternative hypothesis, and Section 3 presents tests using specific alternative hypotheses. As each test is discussed, we include an illustration using weekly returns on stock market indexes and common stock portfolios formed according to firm size. Section 4 extends the framework to models with a factor that is not a portfolio return and provides an illustration using consumption data. Section 5 concludes the article. 1. Definitions and Notation We consider a set of n risky assets, which are often portfolios formed from a larger universe of individual assets. An n × l matrix G with full column rank contains the weights for l portfolios that are combinations of the n assets. A given set of K reference portfolios is represented by the matrix A, a specific choice of G having K columns. A single reference portfolio is denoted by the vector p, a specific choice of G having one column. Let R, denote returns in period t on the K reference portfolios, and let rt denote returns in period t on the remaining n - K assets. If a riskless asset exists, then Rt and rt denote excess returns on these assets, that is, returns in excess of the riskless rate rFt· 2 It is assumed throughout the paper 2 If the riskless rate is changing, the use of excess returns is problematic in terms of defining unconditional mean-variance efficiency. In that our primary goal is to provide a simple framework in which to Interpret previously applied tests, we follow the convention of previous research in our use of excess returns. This issue, along with more general questions about the appropriateness of testing unconditional relations, lie beyond the scope of this study. 128
Tests of Asset Pricing Models that the n-vector of returns(R)'is distributed multivariate normal with a nonsingular variance-covariance matrix.'Let E and denote the popu- lation mean and covariance matrix of (R)'and partition E and V as E=E R E V=COV (1) For a sample of T observations,define The sample means of (rR)' The sample covariance matrix of (rR)' The minimum sample variance of any portfolio with sample mean return m that is constructed from the set of n assets A useful matrix that summarizes the sample feasible set is given by [&E器阊 (2) where:iss an n -vector of ones.The determinant of the matrix in (2)is D=L·N-MP For a given portfolio p.define .(p):The sample mean return of portfolio p .2(p):The sample variance of portfolio p For a set of portfolios represented by the matrix G,define .(m):The minimum sample variance of any portfolio with sample mean return m that is constructed from the set of portfolios represented by the matrix G The following are defined only for the case in which a riskless asset exists: S(p):The sample Sharpe measure of portfolio p.defined as the ratio of the mean excess return on p to the standard deviation of excess return on p.That is, (3) where excess returns are used in computing (p)and (p). ·* The portfolio having the highest absolute value of the sample Sharpe measure of any portfolio constructed from the set of n assets. The simple partitioning of the set of n assets into sets of size K and nK.is for ease of discussion.Both the K reference portfolios and the other n-K assets can be combinations of the n "primitive"assets. 129
that the n-vector of returns is distributed multivariate normal with a nonsingular variance-covariance matrix.’ Let E and V denote the population mean and covariance matrix of and partition E and V as (1) For a sample of T observations, define The sample means of The sample covariance matrix of The minimum sample variance of any portfolio with sample mean return m that is constructed from the set of n assets A useful matrix that summarizes the sample feasible set is given by (2) where is an n -vector of ones. The determinant of the matrix in (2) is For a given portfolio p, define The sample mean return of portfolio p The sample variance of portfolio p For a set of portfolios represented by the matrix G, define The minimum sample variance of any portfolio with sample mean return m that is constructed from the set of portfolios represented by the matrix G The following are defined only for the case in which a riskless asset exists: The sample Sharpe measure of portfolio p, defined as the ratio of the mean excess return on p to the standard deviation of excess return on p. That is, (3) where excess returns are used in computing and The portfolio having the highest absolute value of the sample Sharpe measure of any portfolio constructed from the set of n assets. 3 The simple partitioning of the set of n assets into sets of size K and n - K is for ease of discussion. Both the K reference portfolios and the other n - K assets can be combinations of the n "primitive" assets. 129
The Review of Financtal Studies /v 2 n 2 1989 .pe:The portfolio having the highest absolute value of the sample Sharpe measure of any portfolio constructed from the set of portfolios represented by the matrix G. 2.Likelihood-Ratio Tests Using a General Alternative Hypothesis Numerous studies have developed and applied tests of asset pricing models against a general (unspecified)alternative hypothesis using the multi- variate regression r=a十BR+ (4) A linear mean-beta pricing relation states that,for some scalar Y, E(r)=Yw-K+EE(R)一r (5) where E()is the expectation operator and denotes an (n-K )-vector of ones.Furthermore,if a riskless asset exists (so that and R,are stated as excess returns),then y=0.The pricing relation in (5)implies the following restriction on the parameters in the multivariate regression in (4): a-Y(n-k-B) (6) which simplifies to the restriction a=0 when a riskless asset exists. The pricing restriction in (5)can be viewed as a set of restrictions on E and V.the true (population)mean vector and variance-covariance matrix of the n risky assets.These restrictions can be written as E=Ytn-k Viz(V22)[E-Yh] (7) Let the parameter vector 0 contain the elements of E and V,and let denote the entire parameter space (wherein E can be any real-valued vector and V can be any symmetric positive-definite matrix).The restrictions in (7)are represented asE(A),where denotes the region of defined by the restrictions.The notation "(A)"is chosen to emphasize the fact that this region depends on the choice of the K reference portfolios.The notation "w(p)"is used with a single reference portfolio p.Let Z denote the sample of T observations ofR)',and let(;Z)denote the likelihood function (given by the multivariate normal distribution). The likelihood ratio for testing a K-beta pricing model with the reference portfolios represented by A against a general alternative is given by max f(0;2) A(A)= max f(0;2) (8) a(万 In the absence of a riskless asset and when K1.a test of the restrictionsands equivalent to a test of "mean-variance spanning."that is,that the mean-variance frontier of the K assets coincides with that of the larger set of n assets.See Huberman and Kandel (1987). 130
The portfolio having the highest absolute value of the sample Sharpe measure of any portfolio constructed from the set of portfolios represented by the matrix G. 2. Likelihood-Ratio Tests Using a General Alternative Hypothesis Numerous studies have developed and applied tests of asset pricing models against a general (unspecified) alternative hypothesis using the multivariate regression (4) A linear mean-beta pricing relation states that, for some scalar (5) where E ( ) is the expectation operator and denotes an ( n - K ) -vector of ones. Furthermore, if a riskless asset exists (so that stated as excess returns), then 0. The pricing relation in (5) implies the following restriction on the parameters in the multivariate regression in (4): which simplifies to the restriction a = 0 when a riskless asset exists.4 The pricing restriction in (5) can be viewed as a set of restrictions on E and V, the true (population) mean vector and variance-covariance matrix of the n risky assets. These restrictions can be written as (7) Let the parameter vector contain the elements of E and V, and let denote the entire parameter space (wherein E can be any real-valued vector and V can be any symmetric positive-definite matrix). The restrictions in (7) are represented as denotes the region of defined by the restrictions. The notation is chosen to emphasize the fact that this region depends on the choice of the K reference portfolios. The notation is used with a single reference portfolio p. Let Z denote the sample of T observations of and let denote the likelihood function (given by the multivariate normal distribution). The likelihood ratio for testing a K -beta pricing model with the reference portfolios represented by A against a general alternative is given by (8) 4 In the absence of a riskless asset and when K> 1, a test of the restrictions a - 0 and is equivalent to a test of “mean-variance spanning,” that is, that the mean-variance frontier of the K assets coincides with that of the larger set of n assets. See Huberman and Kandel (1987). 130
Tests of Asset Pricing Models When a single-beta pricing model is tested,the matrix 4 is replaced in (8)by p,representing the tested reference portfolio. This section presents a framework in sample mean-variance space for conducting likelihood-ratio tests of the pricing restrictions.We first sum- marize existing results for models with a riskless asset (Section 2.1);we then present new geometrical interpretations for testing models without a riskless asset (Sections 2.2 and 2.3). 2.1 Tests of models with a riskless asset When a riskless asset exists,efficiency is defined with respect to the set of n risky assets plus the riskless asset.If the pricing model contains a single beta,that is,the matrix B in (4)has one column,then a test of the pricing model is equivalent to a test of the mean-variance efficiency of the specified reference portfolio with return R.If the pricing model contains several betas,that is,B has more than one column,then in general one cannot identify a specific benchmark portfolio that is implied by the pricing model to be mean-variance efficient.The linear pricing relation in (5)is equivalent to the statement that some portfolio of the K reference portfolios is mean- variance efficient [Jobson and Korkie (1985),Grinblatt and Titman (1987) and Huberman,Kandel,and Stambaugh (1987)]. The finite-sample distribution of the likelihood-ratio-test statistic for models with a riskless asset is presented by Gibbons,Ross,and Shanken (1989).Following Anderson (1984),they show that a transformation of the likelihood-ratio statistic for testing a =0 in (4)(when r,and R,are stated in excess of the riskless rate)obeys an F-distribution in finite sam- ples.?The following proposition summarizes the sample mean-variance representation of this test provided by Jobson and Korkie (1982)and Gibbons,Ross,and Shanken (1989). Proposition 1.The likelihood-ratio test with significance level a rejects the hypothesis that some portfolio of the K reference portfolios represented by the matrix A is efficient with respect to the set of n assets plus the riskless asset if and only if IS()川<S (9) wbere S S()2-E.(n-KT-n) 1+vF(n-K T-n) (10) s Jobson and Korkie(1985)and MacKinlay(1987)also present the same result for the single-beta CAPM. A similar result is also presented by Jobson and Kodde (1982),except that they characterize what is in fact the finite-sample distribution as being valid only asymptotically.and they misstate the number of degrees of freedom. 6These results are also summarized in a recent paper by Jobson and Korkie(1988). 131
When a single-beta pricing model is tested, the matrix A is replaced in (8) by p, representing the tested reference portfolio. This section presents a framework in sample mean-variance space for conducting likelihood-ratio tests of the pricing restrictions. We first summarize existing results for models with a riskless asset (Section 2.1); we then present new geometrical interpretations for testing models without a riskless asset (Sections 2.2 and 2.3). 2.1 Tests of models with a riskless asset When a riskless asset exists, efficiency is defined with respect to the set of n risky assets plus the riskless asset. If the pricing model contains a single beta, that is, the matrix B in (4) has one column, then a test of the pricing model is equivalent to a test of the mean-variance efficiency of the specified reference portfolio with return Rt . If the pricing model contains several betas, that is, B has more than one column, then in general one cannot identify a specific benchmark portfolio that is implied by the pricing model to be mean-variance efficient. The linear pricing relation in (5) is equivalent to the statement that some portfolio of the K reference portfolios is meanvariance efficient [Jobson and Korkie (1985), Grinblatt and Titman (1987) and Huberman, Kandel, and Stambaugh (1987)]. The finite-sample distribution of the likelihood-ratio-test statistic for models with a riskless asset is presented by Gibbons, Ross, and Shanken (1989). Following Anderson (1984), they show that a transformation of the likelihood-ratio statistic for testing a = 0 in (4) (when rt and Rt are stated in excess of the riskless rate) obeys an F-distribution in finite samples.5 The following proposition summarizes the sample mean-variance representation of this test provided by Jobson and Korkie (1982) and Gibbons, Ross, and Shanken (1989).6 Proposition 1. The likelihood-ratio test with significance level a rejects the hypothesis that some portfolio of the K reference portfolios represented by the matrix A is efficient with respect to the set of n assets plus the riskless asset if and only if 5 Jobson and Korkie (1985) and MacKinlay (1987) also present the same result for the single-beta CAPM. A similar result is also presented by Jobson and Kodde (1982), except that they characterize what is in fact the finite-sample distribution as being valid only asymptotically. and they misstate the number of degrees of freedom. 6 These results are also summarized in a recent paper by Jobson and Korkie (1988). 131
Tbe Review of Financtal Studtes/v 2 n 2 1989 if the bracketed quantity in (10)is positive,s equals zero otherwise (in which case there is no rejection),F(n-K,T-n)is the critical value for significance level a of the F-distribution with n-K and T-n degrees of freedom,and y=(n-K)(T-n). Proof.See the Appendix. For a given sample of assets and returns,there may exist no specification of the reference portfolio(s)that results in a rejection of the pricing model, This situation,wherein the maximum squared sample Sharpe measure s(p*)2 is less than vF(n-K,T-n)and thus the bracketed quantity in (10)is negative,is more likely to occur as the number of assets (n)grows large relative to the number of time-series observations (T). As the above proposition states,in a test of a single-beta model (K=1) the efficiency of a portfolio can be tested by plotting its position in sample mean-standard-deviation space,where all returns are stated in excess of the riskless rate.The tested portfolio's position is compared to the location of the two critical lines with intercepts of zero and slopes with absolute values equal to s.If the tested portfolio lies between the critical lines, then its efficiency is rejected. Proposition 1 also indicates that in the test of a multiple-beta model (K >1)the portfolio tested is p,the sample tangent portfolio for the set of K assets.The position of portfolio p is compared to the two critical lines in sample mean-standard-deviation space in precisely the same manner as was the single reference portfolio in the case of K=1.(The differences in S between the two cases simply reflect different degrees of freedom.) Note that Is(p)<s,and thus the multiple-beta model is rejected,if and only if the minimum-standard-deviation boundary of the K reference portfolios does not intersect either of the two critical lines. We illustrate here a test of a two-beta pricing model (K =2)with the weekly returns data used by Kandel and Stambaugh (1987)in tests of the Sharpe-Lintner version of the CAPM(K =1).The set of 12 risky assets (n =12)consists of two market proxies-t h e equally weighted and the value- weighted portfolios of stocks on the New York and American Exchanges- and 10 value-weighted portfolios of common stocks formed by ranking all firms on both exchanges by market value at the end of the previous year. The riskless rate is the return on a U.S.Treasury bill with one week to maturity.'A two-beta model is tested using the two market proxies as the two reference portfolios.We choose these proxies simply to illustrate the testing framework rather than to conduct comprehensive new tests of asset pricing models."For the same reason,we use,for Proposition 1 as well as We thank Richard Rogalski for providing the Treasury bill data. 8 The use of the two proxies may be partially motivated by the well-known"size anomaly"of the single- beta CAPM.The value-weighted Index primarily reflects changes In the prices of large firms,whereas the equally weighted index is affected more by the returns on medium-size and small firms. 132
if the bracketed quantity in (10) is positive, equals zero otherwise (in which case there is no rejection), is the critical value for significance level a of the F-distribution with n - K and T - n degrees of freedom, and Proof. See the Appendix. n For a given sample of assets and returns, there may exist no specification of the reference portfolio(s) that results in a rejection of the pricing model, This situation, wherein the maximum squared sample Sharpe measure and thus the bracketed quantity in (10) is negative, is more likely to occur as the number of assets (n) grows large relative to the number of time-series observations (T). As the above proposition states, in a test of a single-beta model ( K = 1) the efficiency of a portfolio can be tested by plotting its position in sample mean-standard-deviation space, where all returns are stated in excess of the riskless rate. The tested portfolio’s position is compared to the location of the two critical lines with intercepts of zero and slopes with absolute values equal to If the tested portfolio lies between the critical lines, then its efficiency is rejected. Proposition 1 also indicates that in the test of a multiple-beta model ( K > 1) the portfolio tested is the sample tangent portfolio for the set of K assets. The position of portfolio is compared to the two critical lines in sample mean-standard-deviation space in precisely the same manner as was the single reference portfolio in the case of K = 1. (The differences in between the two cases simply reflect different degrees of freedom.) Note that and thus the multiple-beta model is rejected, if and only if the minimum-standard-deviation boundary of the K reference portfolios does not intersect either of the two critical lines. We illustrate here a test of a two-beta pricing model ( K = 2) with the weekly returns data used by Kandel and Stambaugh (1987) in tests of the Sharpe-Lintner version of the CAPM ( K = 1). The set of 12 risky assets ( n = 12) consists of two market proxies-the equally weighted and the valueweighted portfolios of stocks on the New York and American Exchangesand 10 value-weighted portfolios of common stocks formed by ranking all firms on both exchanges by market value at the end of the previous year. The riskless rate is the return on a U.S. Treasury bill with one week to maturity.7 A two-beta model is tested using the two market proxies as the two reference portfolios. We choose these proxies simply to illustrate the testing framework rather than to conduct comprehensive new tests of asset pricing models.8 For the same reason, we use, for Proposition 1 as well as 7We thank Richard Rogalski for providing the Treasury bill data. 8 The use of the two proxies may be partially motivated by the well-known “size anomaly" of the singlebeta CAPM. The value-weighted Index primarily reflects changes In the prices of large firms, whereas the equally weighted index is affected more by the returns on medium-size and small firms. 132
Tests of Asset Pricing Models MInimum-standard-devlation 200 boundary Critical ne t50 100 Boundary of the roference portfollos 60 Equally welghted market Q 0 Value-welghted market -50 -100 -150 -200 Crltical ne =250 10 30 60 70 90 110 Standard Devlation (per year) Figurel A likelihood-ratio test of a two-beta pricing model in the presence of a riskless asset The test Is based on weekly retums in excess of a riskless rate.The two reference portfolios are the value- weighted NYSE-AMEX and the equally weighted NYSE-AMEX.The sample minimum-standard-deviation boundary is constructed using 12 assets:10 size-based portfolios plus the two market proxies.The critical lines reflect a 5 percent significance level.The pricing model is not rejected if the boundary of the reference portfolios intersects a critical line. the propositions to follow,only one of the three subperiods examined by Kandel and Stambaugh (1987).The subperiod selected extends from Oct 8,1975,through Dec.23,1981,and includes 324 weekly observations. Figure 1 displays the test at a 5 percent significance level.The hyperbola representing combinations of the two reference portfolios does not Inter- sect either critical line.Thus,the two-beta model is rejected. 2.2 Tests of single-beta models without a riskless asset A likelihood-ratio test of (6)with a single beta,where y is an unknown zero-beta rate,was first proposed by Gibbons(1982).The hypothesis tested is equivalent to the mean-variance efficiency of the benchmark portfolio with respect to the n risky assets.The exact finite-sample distribution of the likelihood-ratio-test statistic has not been obtained for this case,although a lower bound for the distribution is obtained by Shanken (1986).Thus, selection of an appropriate critical value is more difficult than in the case where a riskless asset exists.Once a critical value is specified,however, we show that this test can be conducted in a mean-variance framework. For discussions of finite-sample properties of the likelihood-ratio statistic and other large-sample equiv- alents,see also Stambaugh(1982),Shanken (1985),and Amsler and Schmidt (1985).Shanken (1985) derives an upper bound on the finite-sample distribution of one altemative to the likelihood-ratio statistic. 133
Figure1 A likelihood-ratio test of a two-beta pricing model in the presence of a riskless asset The test Is based on weekly returns in excess of a riskless rate. The two reference portfolios are the valueweighted NYSE-AMEX and the equally weighted NYSE-AMEX. The sample minimum-standard-deviation boundary is constructed using 12 assets: 10 size-based portfolios plus the two market proxies. The critical lines reflect a 5 percent significance level. The pricing model is not rejected if the boundary of the reference portfolios intersects a critical line. the propositions to follow, only one of the three subperiods examined by Kandel and Stambaugh (1987). The subperiod selected extends from Oct. 8, 1975, through Dec. 23, 1981, and includes 324 weekly observations. Figure 1 displays the test at a 5 percent significance level. The hyperbola representing combinations of the two reference portfolios does not Intersect either critical line. Thus, the two-beta model is rejected. 2.2 Tests of single-beta models without a riskless asset A likelihood-ratio test of (6) with a single beta, where g is an unknown zero-beta rate, was first proposed by Gibbons (1982). The hypothesis tested is equivalent to the mean-variance efficiency of the benchmark portfolio with respect to the n risky assets. The exact finite-sample distribution of the likelihood-ratio-test statistic has not been obtained for this case, although a lower bound for the distribution is obtained by Shanken (1986).9 Thus, selection of an appropriate critical value is more difficult than in the case where a riskless asset exists. Once a critical value is specified, however, we show that this test can be conducted in a mean-variance framework. 9 For discussions of finite-sample properties of the likelihood-ratio statistic and other large-sample equivalents, see also Stambaugh (1982), Shanken (1985), and Amsler and Schmidt (1985). Shanken (1985) derives an upper bound on the finite-sample distribution of one alternative to the likelihood-ratio statistic. 133
Tbe Revtew of Financtal Studtes/v 2 n 2 1989 Definition.W(p)=A(p)2/T-1).This is a monotonic transformation of the lkelibood ratio(p)for testing the efficiency of portfolto p witb respect to the set of n assets.T.In[1 W(p)]is asymptotically distributed as x12 witb n-2 degrees of freedom if portfolto p is effictent. Definition.we is tbe critical value for W(p)at tbe chosen significance level.Tbat is,the efficiency of portfolio p is rejected ff w(p)>W. Proposition 2.Tbe likelibood-ratio test rejects tbe effictency ofportfolto p in the absence of a riskless asset,that is w(p)>w,ff and only if 2(p)>i,()+2()·(a(D) (11) ubere the functionsδ，()andi,()are given by 6,(=x+) and 6,(=-Dx+1) (12) Lx-D Lx-D and where L and D are defined in(2). Proof.See the Appendix. Proposition 2 states that the likelihood-ratio test of efficiency can be performed by first constructing a critical parabola in sample mean-variance (2)space given by the equation2=+2).Note that this critical parabola is a linear transformation of the sample minimum-variance bound- ary of the n assets and that neither 6()nor 6()require information about the tested portfolio (other than that the tested portfolio can be constructed from the set of n assets).If the tested portfolio lies inside the convex region defined by this critical parabola,then the efficiency of that portfolio is rejected.The critical parabola becomes a critical hyperbola in sample mean-standard-deviation space,and we use the latter representa- tion in the illustration below. Using the same 12 assets and the same sample period as were used in the previous example,we test the zero-beta CAPM [Black(1972)]with each of the two indexes as the market proxy.Because this formulation of the model does not include a riskless asset,total (not excess)returns are used. The critical value,we,is based on the result by Shanken (1986)that,under the null hypothesis,the lower bound on the distribution of W(p).(T-K -1)is a T-variate with degrees of freedom n-K and T-K-1. Equivalently,the lower bound on the distribution of W(p)(T-n)/n K)is central F with degrees of freedom n-K and T-n.Therefore,for a significance level of 5 percent and for the 324-week sample size the critical value is =1,312)0=0.0641 312 Figure 2 displays the results of this test.Each of the two market proxies lies inside the rejection region defined by the critical hyperbola,and thus 134
Proposition 2 states that the likelihood-ratio test of efficiency can be performed by first constructing a critical parabola in sample mean-variance space given by the equation Note that this critical parabola is a linear transformation of the sample minimum-variance boundary of the n assets and that neither require information about the tested portfolio (other than that the tested portfolio can be constructed from the set of n assets). If the tested portfolio lies inside the convex region defined by this critical parabola, then the efficiency of that portfolio is rejected. The critical parabola becomes a critical hyperbola in sample mean-standard-deviation space, and we use the latter representation in the illustration below. Using the same 12 assets and the same sample period as were used in the previous example, we test the zero-beta CAPM [Black (1972)] with each of the two indexes as the market proxy. Because this formulation of the model does not include a riskless asset, total (not excess) returns are used. The critical value, is based on the result by Shanken (1986) that, under the null hypothesis, the lower bound on the distribution of W(p). (T - K - 1) is a T 2 -variate with degrees of freedom n - K and T - K - 1. Equivalently, the lower bound on the distribution of W(p) · (T - n)/( n - K) is central F with degrees of freedom n - K and T - n. Therefore, for a significance level of 5 percent and for the 324-week sample size the critical value is Figure 2 displays the results of this test. Each of the two market proxies lies inside the rejection region defined by the critical hyperbola, and thus 134
