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
Pettengill, Sundaram, and Mathur 105 low beta portfolios). This relationship provides important implications concerning empirical tests for a systematic relationship between beta and returns A systematic relationship must exist between beta and returns for beta to be a useful measure of risk. The SLB model prescribes a systematic and positive tradeoff between beta and expected return, but the above discussion refects a segmented relationship between realized returns and beta, i.e., a positive relation during positive market excess return periods and a negative relation during negative market excess return periods. If realized market returns were seldom less than the risk-free rate, this conditional relationship would have a trivial impact on tests of the relationship between beta and returns. This relationship, however, occurs frequently. A month-by-month comparison of the CRSP equally-weighted index(as the proxy for market return) and the 90-day T-bill rate(as the measure for the risk-free return) over the period 1936 through 1990 reveals that the t- bill rate exceeds the market return in 280 out of 660 total observations The existence of a large number of negative market excess return periods suggests that previous studies that test for an unconditional positive correlation between beta and realized returns are biased against finding a systematic relationship. This paper mploys testing procedures that account for the segmented relationship and finds a significant impact of beta on returns Ill. Previous Tests of the Relationship between Beta and Returns Extant literature examining the relationship between beta and returns has primarily tested for a positive linear relationship as prescribed by the SlB model Although the model postulates a positive tradeoff between risk(beta) and expected return, prior studies have examined the realized returns of equity portfolios forme from rankings of betas. They generally find a weak, but positive, relationship between returns and beta over the entire sample period, but these results are often found to be intertemporally inconsistent and weaker than the association between returns and other variables(e. g, size). These results are generally interpreted as evidence against the validity of a positive relationship between beta and returns The findings of the major studies in this area are briefly described here Fama and MacBeth,'s(1973)seminal study on the validity of the SLB model takes a three-step approach. In the first step, portfolios are formed based on esti mated beta for individual securities. The second step involves estimation of each portfolio's beta in a subsequent time period. In the final step using data from a third time period, portfolio returns are regressed on portfolio betas. Since, on average, for the period 1935 through 1968, a positive relationship exists between beta and monthly returns, Fama and MacBeth conclude that the SlB model ad- equately describes the risk-return behavior observed in capital markets. Schwert (1983), however, suggests that this evidence provides surprisingly weak support for a risk-return tradeoff Following Fama and MacBeth, a number of researchers have conducted em- pirical analyses that suggest that beta may not adequately measure a security'srisk Reinganum(1981)finds that".estimated betas are not systematically related to average returns across securities"and concludes".. that the slB model may lack
106 Journal of Financial and Quantitative Analysis significant empirical content"(p. 439). In a sample of daily returns, Reinganum finds a tendency for portfolio returns to decrease as beta increases. In contrast, for a sample of monthly returns, Reinganum finds a positive relationship between returns and beta, but argues that this apparent corroboration is spurious on two counts. First, the difference in returns across portfolios is not significant. Sec ond, the positive relationship between beta risk and return is not consistent across subl Tinic and West (1984)also reject the validity of the slB model based on ntertemporal inconsistencies. Using monthly data, they find a positive and signif- icant slope when regressing portfolio returns on portfolio betas when return data for the entire year are included. Tinic and West are, however, unable to reject the null hypothesis of no difference in returns across portfolios if return data from the month of January are excluded. Further, for several months of the year, nega- across months of the year led them to conclude that their resul the slb model tive slope coefficients are observed. This inconsistent support for doubt on the validity of the two-parameter model.. "and"... to the extent that the risk-return tradeoff shows up only in January, much of what now constitutes the received version of modern finance is brought into question"(Tinic and West (1984),p.573) Several other studies stress that the ability of beta to explain changes in return is weak relative to other variables. Lakonishok and Shapiro(1984),(1986)find an insignificant relationship between beta and returns. Further, Lakonishok and Shapiro find a significant relationship between returns and market capitalization values From these tests, Lakonishok and Shapiro conclude that an dividual securitys return did not appear to be specifically related to its degree of systematic risk”(1984),p.36 Fama and French(1992) study monthly returns and find an insignificant re- lationship between beta and average returns. In contrast, market capitalization and the ratio of book value to market value have significant explanatory power for portfolio returns. Fama and French state: We are forced to conclude that the SlB odel does not describe the last 50 years of average stock returns"(p 464) In summary, Reinganum finds therelationship between beta and cross-section- al returns to vary across subperiods. Tinic and West find the relationship between beta and the returns to vary with months in a year. Lakonishok and Shapiro and Fama and french find the relationship between beta and returns to be weaker than the relationship between returns and other variables. Collectively, these results have been taken as evidence that the slb model provides an inadequate expla nation for the risk-return behavior observed in capital markets. In contrast, the methodology described below accounts for the conditional relationship between beta and realized returns, and finds a systematic relationship between these vari- Although they argue in support of the SLB model, Fama and Mac Beth find similar inconsistency This result is consistent with Rozeff and Kinney(1976), who find the risk premium for Janua
Pettengill, Sundaram and mathur 107 IV. Data and Methodology A. Data The sample period for this study extends from January 1926 through Decem ber 1990. Monthly returns for the securities included in the sample and the CRSP equally-weighted index(as a proxy for the market index)were obtained from the CRSP monthly databases. The three-month Treasury bill rates(a proxy for the risk-free rate) for the period 1936 through 1990 were collected from the Federal Reserve bulletin B. Test of a Systematic Relationship between Beta and Returns A The purpose of this paper is twofold. The first is to test for a systematic, conditional relationship between betas and realized returns. The second is to test for a positive long-run tradeoff between beta risk and return Tests for a systematic relationship utilize a modified version of the three-step portfolio approach first used by Fama and MacBeth(1973). The sample period is first separated into 15-year subperiods, which are further divided into a portfolio formation period, a portfolio beta estimation period, and a test period of five years each. In the portfolio formation period, betas are estimated for each security in the sample by regressing the securitys return against the market return. Based on the relative rankings of the estimated beta, securities are equally divided into 20 portfolios. Securities with lowest betas are placed in the first portfolio, the next lowest in the second portfolio, and so on. Portfolio betas are estimated in the second five-year period within each subsample by regressing portfolio returns ( the equally-weighted return of all securities in the portfolio) against the market eturns The third step, which tests the relationship between portfolio beta and returns is modified to account for the conditional relationship between beta and realized returns. As argued in Section IL, if the realized market return is above the risk-free return, portfolio betas and returns should be positively related, but if the realized market return is below the risk-free return, portfolio betas and returns should be inversely related. Hence, to test for a systematic relationship between beta and returns, the regression coefficients from Equation (4)are examined, (4) Rir =%0r+fir*8*Bi+i2*(1-6)*B;+Er where 8= l, if(Rmt-Ra)>0(i.e, when market excess returns are positive), and 8=0, if(Rmt-Rn)<o(i. e, when market excess returns are negative), The above relationship is examined for each month in the test period by estimating either yI or 12, depending on the sign for market excess returns Since 1 is estimated in periods with positive market excess returns, the expected sign of this coefficient is positive. Hence, the following hypotheses are sted There is no material difference in the results when the value-weighted index is used he entire sample period into up markets and down markets was first performed
108 Journal of Financial and Quantitative Analysis Ho:作=0, Ha:1>0. Since y2 is estimated in periods with negative market excess returns, the expected sign of this coefficient is negative. Hence, the following hypotheses are Ho:2=0, 0. ported if, in both cases, the null hypothesis is rejected in favor of the alternate A systematic conditional relationship between beta and realized returns is st C. Subsample Procedures The sample period of 1926 through 1990 allows for the creation of 1l distinct 15-year subsamples. The first subsample extends from 1926 through 1940, the second from 1931 through 1945, and so on. In the first subsample, the first five- year period (1926-1930)is the portfolio formation period, the second(1931 1935)and third( 1936-1940)five-year periods are the portfolio beta estimation period and the test period, respectively. Each subsample includes all securities available from the CRSP monthly returns file that have at least 45 observations in each of the three periods within the subsample. The number of securities in the 11 subsamples ranged from a low of 366(first subsample)to a high of 1350 (penultimate subsample). The entire three-step procedure is conducted separately for each subperiod Equation(4)is first examined using all 660 monthly observations. In addition, the data are divided into three approximately equal subperiods: 1936 through 1950, 1951 through 1970, and 1971 through 1990. Separately, the data are divided by months in a year. Applying Equation(4)to each of these subperiods tests whether intertemporal inconsistencies and seasonality observed by previous studies result from the conditional nature of the relationship between beta and realized returns D. A Test of the Positive Risk-Return Tradeoff The second goal of the study is to determine if a systematic relationship between beta and return translates into a positive reward for holding risk (i.e, do high beta assets, on average, earn higher returns than low beta assets? ) If a systematic, conditional relationship between beta risk and returns exists, a positive reward for holding beta risk will occur if two conditions are met: i)market excess returns are, on average, positive; and ii)the risk-return relationship is symmetrical etween periods of positive and negative excess market returns. We test each of these conditions and then provide a direct test of a positive risk-return tradeoff. The average market excess return for the total sample period and the various subperiods are calculated to test for the first condition. A standard t-test is used to determine if market excess returns are, on average, positive. The risk premiums during up and down markets, as captured by f1 and f2, are compared to test for symmetry. Since the expected signs of these coefficients differ, a direct comparison of their average values would be inappropriate. To facilitate comparisons, the sign for i2 is reversed and its mean value is reestimated. These adjustments allow a
Pettengill, Sundaram, and Mathur direct comparison while preserving the effects of slope estimates with unexpected signs(i.e, a negative sign for m1 and a positive sign for 12). After the adjustments, he following hypotheses are tested by using a standard two-population t-test, Ho:71-12=0, Finally, a direct test of the risk-return tradeoff is employed by regressing the average portfolio betas against average portfolio return V. Empirical Results A. Beta vs Realized Returns Previous studies, following Fama and MacBeth, test for a positive linear rela- this relation by estimating the slope coefficients for Equation(3). Results for the total sample period reported in Table I reject the hypothesis of no relation between sk and return at the 0.05 level. However. the results are inconsistent across sub- periods. The null hypothesis is rejected in the first subperiod at the 0.05 level but the null cannot be rejected in the second(t=0.18)or third (t=1.30) subperiods Extant literature cites this weak correlation and the intertemporal inconsistency as evidence against a systematic relationship between risk and return TABLE Estimates of Slope Coefficients(CRSP Equally-Weighted Index) 十 Period T-Statistic P-Value Total Sample 0.0050 00129 (1936-1990) 00111 (1936-1950) 04277 (1951-1970) 00005 0.0975 (1971-1990) We argue that the above results are biased due to the aggregation of positive and negative market excess return periods. Given the conditional relation between risk(beta) and realized returns, we test the dual hypothesis of a positive relation between beta and returns during periods of positive market excess returns and egative relation during periods of negative market excess returns. The hypothesis is tested by examining the regression coefficients f1 and f2 of Equation(4). The regression estimates are presented in Table 2 examination of the estimated regression coefficients provides strong support for a systematic but conditional relationship between beta and realized returns f is estimated in each of the 380 months for which the market excess return is
