The Conditional Relation between Beta and Returns TOR Glenn N. Pettengill, Sridhar Sundaram, Ike Mathur The Journal of financial and Quantitative Analysis, Volume 30, Issue 1 (Mar, 1995) 101-116. Stable url: http://links.jstor.org/sici?sici=0022-1090%028199503%02930%3a1%03c101%03atcrbba%3e2.0.c0%3b2-b Your use of the STOR archive indicates your acceptance of JSTOR'S Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the jStOR archive only for your personal, non-commercial use Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Journal of financial and Quantitative Analysis is published by University of Washington School of Business Administration. Please contact the publisher for further permissions regarding the use of this work, Publisher contactinformationmaybeobtainedathttp://www.jstororg/journals/uwash.html The Journal of financial and Quantitative analysi o1995 University of Washington School of Business Administration jSTOR and the jstoR logo are trademarks of jStoR, and are registered in the u.s. Patent and Trademark Office. For more information on JSTOR contact jstor-info @ umich. edu @2002 JSTOR http://wwwjstor.org Thu nov1403:12:192002
JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS OL 30. NO. 1 MARCH 1995 The Conditional relation between Beta and Returns Glenn N. Pettengill, Sridhar Sundaram, and Ike Mathur* Abstract Unlike previous studies, this paper finds a consistent and highly significant relationship between beta and cross-sectional portfolio returns. The key distinction between our tests and previous tests is the recognition that the positive relationship between returns and beta predicted by the Sharpe-Lintner-Black model is based on expected rather than realized returns. In periods where excess market returns are negative, an inverse relationship between beta and portfolio returns should exist. When we adjust for the egative market excess returns, we find a consistent and signifi lationship between beta and returns for the entire sample, for subsample periods, and for data divided by months in a year. Separately, we find support for a positive payment for beta risk 1. Introduction The Sharpe-Lintner-Black(SLB)model, which is predicated on the assump- ion of a positive risk-return tradeoff, asserts that the expected return for any asset is a positive function of only three variables: beta( the covariance of asset return and market return), the risk-free rate, and the expected market return. This as sertion implies that an asset s responsiveness to general market movements is the only variable to cause systematic differences in returns between assets Empirical tests of this assertion, using average realized returns to proxy for expected returns and an index of equity security returns as a proxy for market returns, initially supported the validity of the SLB model(e.g, Fama and Mac- Beth(1973)). The usefulness of beta as the single measure of risk for a security has, however, been challenged by at least three arguments. First, research has challenged the notion that beta is the most efficient measure of systematic risk for individual securities. Thus, some researchers have argued in favor of measuring ystematic responsiveness to several macroeconomic variables(e. g, Chen, Roll, 5087: Mathur, College of Business and Administration, Southern llinois University at Carbondale, Carbondale, IL 62901, respectively. The authors thank Marcia Comett, Dave Davidson, John Doul Roger Huang, Santosh Mohan, Jim Musumeci, Lilian Ng, Edgar Norton, Nanda rangan, Andy Szak mary, participants of the Southwest Finance Symposium at the University of Tulsa, JFQA Managing Editor Jonathan Karpoff, and JFQA Referees Michael Pinegar and Alan Shapiro for he ments on earlier drafts of the paper. The authors also thank Shari Garnett and Pauletta Avery for their sistance in preparing the manuscript
102 Journal of Financial and Quantitative Analysis and Ross(1986)). Second, other researchers have found empirical evidence that security returns are affected by various measures of unsystematic risk(e.g,Lakon ishok and Shapiro(1986)). Finally, some researchers assert that recent empiric evidence indicates the absence of a systematic relationship between beta and se curity returns(e. g, Fama and French(1992). Collectively, the first two criticisms suggest that beta lacks efficiency and completeness as a measure of risk. The third criticism ies either that there is no risk-return tradeoff or that beta does not measure risk Whar Despite this evidence against the SLB model, Fama((1991),p.1593)asserts market professionals(and academics) still think about risk in terms of market 6. This preference for beta presumably results from the convenience of using a single factor to measure risk and the intuitive appeal of beta. Are these advantages sufficient to justify the continued use of beta if the criticisms cited above are valid? The use of beta may be justified as a measure of risk, even if beta is less efficient than alternative measures of systematic risk or is an incomplete measure of risk. However, if there is no systematic relationship between cross-sectional returns and beta, continued reliance on beta as a measure of risk is inappropriate This paper examines the crucial assertion that beta tionship with returns. Unlike previous studies, this study explicitly recognizes the mpact of using realized market returns to proxy for expected market returns. As developed in the next section, when realized market returns fall below the risk-free rate, an inverse relationship is predicted between realized returns and beta. Ac- knowledging this relationship leads to the finding of a significant and systematic relationship between beta and returns. Further, evidence of a positive risk-return tradeoff is found when beta is used to measure risk. These results cannot be taken as direct support of the SLB model, but they are consistent with the implication that beta is a useful measure of risk In the following section, we discuss the predicted relationship between beta and return distributions for both expected returns and realized returns. Section ml reviews previous tests of the relationship between beta and returns. In Section IV, the data and methodology used to test the relationship between beta and realized returns are described. Section V reports empirical results that show a systematic relationship between returns and beta and support for a positive risk-return tradeoff. Section VI concludes the paper ll. Beta and Returns: The SLB Model and Empirical Tests A. Model Implications The SLB model asserts that investors are rewarded only for systematic risk since unsystematic risk can be eliminated through diversification. Thus, the secu rity market line specifies that the expected return to any risky security or portfolio of risky securities is the sum of the risk-free rate and a risk premium determined IRoll and Ross(1994)attribute the observed lack of a systematic relation between risk and retum the possible mean-variance inefficiency of the market portfolio proxies
Pettengill, Sundaram, and Mathur 1( by beta. Tests of this assertion examine portfolios of securities to reduce both estimation error and nonsystematic risk. The relationship tested is represented as (1) E(RP)=R,+B *(E(Rm)-R) where E(Rp) is the expected return for the risky portfolio P, R is the current risk-free rate, Bp is the covariance between the portfolios return and the markets return divided by the variance of the market, and E(Rm)is the expected return to The interrelationship between these variables provides crucial implica for testing the relationship between beta and returns. On the assumption of a positive risk-return tradeoff, the expected return to the market must be greater than the risk-free return(or all investors would hold the risk-free security ) Since the term(E(Rm)-Ry)must be positive, the expected return to any risky portfolio is a positive function of beta. This relationship has prompted researchers to examine the validity of the SLB by testing for a positive relationship between returns and beta. Since these tests use realized returns instead of expected returns, we argue that the validity of the SLB model is not directly examined. Indeed, recognition of a second critical relationship between the predicted market returns and the risk free return suggests that previous tests of the relationship between beta and returns must be modified The need to modify previous tests results from the model's requirement that a portion of the market return distribution be below the risk-free rate. In addition to the expectation that, on average, the market return be greater than the risk-free rate, investors must perceive a nonzero probability that the realized market return will be less than the risk-free return. If investors were certain that the market eturn would always be greater than the risk-free rate, no investor would hold the risk-free security. This second requirement suggests that the relationship between beta and realized returns varies from the relationship between beta and expected return required by Equation(1). However, the model does not provide a direct indication of the relationship between portfolio beta and portfolio returns when the realized market return is less than the risk-free return. a further examination as detailed below, shows that an inverse relationship between beta and returns can be reasonably inferred during such periods In order to draw this inference, it is necessary to provide an analysis of the portfolio return distribution implied by the SLB model. This model shows that the expected return for each portfolio is a function of the risk-free return, the portfolio beta, and the expected return to the market. The expected return for the portfolio is the mean of the distribution for all possible returns for that portfolio in the appropriate return period. Identical with the market return, for all portfolios with a positive beta, the expected value for the return distribution must be greater than the risk-free rate and the return distribution must contain a non-zero probability of realizing a return below the risk-free rate. To arrive at testable implications, we must extend this analysis to examine the differences in the return distributions of portfolios with different betas Portfolios with higher betas have higher expected returns because of higher risks. For high beta portfolios to have higher risk, there must be some level of realized return for which the probability of exceeding that particular return is
104 Journal of Financial and Quantitative Analysis greater for the low beta portfolio than for the high beta portfolio. If this were not the case, no investor would hold the low beta portfolio. Thus, the SLB model not only requires the expectation that realized returns for the market will, with some probability, be lower than the risk-free rate, but also requires the expectation that, with some probability, the realized returns for high beta portfolios will be lower than the realized returns for low beta portfolios. The model does not require a direct link between these two relationships. A reasonable inference may, however, be that returns for high beta portfolios are less than returns for low beta portfolios when the realized market return is less than the risk-free rate. Although previous tests of the SLB model have not recognized these relationships when testing the validity of the SLB model, the market model used to calculate beta does imply this relationship B. Empirical Tests Previous tests of the implications of the SLB model have sought to find a positive relationship between realized portfolio returns and portfolio betas. The tests are conducted in stages, with the estimation of beta as shown below, (Rpt -Ra)= B*( Rmut-RA) followed by the test for a positive risk-return tradeoff, 1+*+ Equation(2)estimates the beta risk for each portfolio using realized returns for both the portfolio and the market, thus providing a proxy for the beta in the SLB model. Under the assumption that betas in the estimation period proxy betas in the test period, a test for a positive risk-return relationship utilizes Equation (3) If the value for f1 is greater than zero, a positive risk-return tradeoff is supported This procedure may test the usefulness of beta as a measure of risk, but it does not directly test the validity of the SLB model The SLB model not only requires a direct and unconditional relationship between beta and expected returns, but also requires the expectation that the re- lationship between realized returns and beta will vary. As argued in the previot section, in order for high beta portfolios to have more risk, there must be condi tions under which high beta portfolios earn lower returns than low beta portfolio The SLB model does not directly provide the conditions under which the above relationship will be observed. In contrast, Equation(2), which has been used in previous empirical tests, provides an exact condition under which the realized re turns to high beta portfolios are expected to be lower than the realized returns for low beta portfolios. According to Equation(2), the relationship between the return to high and low beta portfolios is conditional on the relationship between realized market returns and the risk-free return. If Rm Rf, then B *(Rmt-Ra)is <0. In these cases, the predicted portfolio return includes a negative risk premium that is proportionate to beta. Hence, if the realized market return is less than the risk-free return, an inverse relationship exists between beta and predicted return ( i.e., high beta portfolios have predicted returns that are less than the predicted returns for