Beta Is Still Alive! ∗ Yexiao Xu and Yihua Zhao School of Management The University of Texas at Dallas This revision: March 2012 Abstract This paper investigates whether beta can predict the expected return after control- lingforthe beta instabilityresulting fromshiftinthe covariance structure. Such ashiftis driven by idiosyncratic volatility’sclientele effect: speculative investors prefer stocks with high idiosyncratic volatility. Consequently, these stocks tend to have low future returns from overpricing, and high beta because clientele-based trading also contains systematic component. Indeed, we find that the beta estimate of the current period is positively related to the beta estimate and negativelyrelated to the idiosyncratic volatilitymeasure of the last period. More important, different from existing studies, we find that beta estimates of the current period can significantly explain the cross-sectional differences in future returns of individual stocks, when allowing for an interaction between the current idiosyncratic volatilityand the beta estimates. We also show that our simple model can predict the historicalexpected return well. Allresults are robust with respect to different measures of beta and idiosyncratic volatilityand to different subsamples. Key Words: Expected Return, Idiosyncratic Volatility, Beta Instability, and Misspric- ing ∗WeareindebttoJohnY.Campbell,MichaelRebello,HaroldZhang,FengZhaoforhisinsightfulcomments. Wehave also benefited fromhelpful commentsfromseminar participants atCheung Kong Graduate School of Business and the University of Texas at Dallas. The corresponding author’s address is: Yexiao Xu, 800 West Campbell Road, SM31, The University of Texas at Dallas, Richardson, TX 75083; Telephone (972)883-6703; FAX(972)883-6522; Email: [email protected]. i Beta Is Still Alive! Abstract This paper investigates whether beta can predict the expected return after control- lingforthe beta instabilityresulting fromshiftinthe covariance structure. Such ashiftis driven by idiosyncratic volatility’sclientele effect: speculative investors prefer stocks with high idiosyncratic volatility. Consequently, these stocks tend to have low future returns from overpricing, and high beta because clientele-based trading also contains systematic component. Indeed, we find that the beta estimate of the current period is positively related to the beta estimate and negativelyrelated to the idiosyncratic volatilitymeasure of the last period. More important, different from existing studies, we find that beta estimates of the current period can significantly explain the cross-sectional differences in future returns of individual stocks, when allowing for an interaction between the current idiosyncratic volatilityand the beta estimates. We also show that our simple model can predict the historicalexpected return well. Allresults are robust with respect to different measures of beta and idiosyncratic volatilityand to different subsamples. Key Words: Expected Return, Idiosyncratic Volatility,Beta Instability,and Misspricing ii 1 Introduction The Capital asset pricing model (CAPM) of Sharpe (1964),Lintner (1965),and Black (1972) predicts that differences in the expected returns of individual securities are completely deter- mined by the covariance based beta measure of risk. Many empirical studies including Fama and French (1992), however, provide no or weak evidence to support this prediction. In this study, we show that the lack of empirical evidence is largely due to short-run shifts in the co- variancestructurebetween individualstockreturnand themarketreturn. We furtheridentify that such a shift can be predicted by idiosyncraticvolatility. Consequently, we provide strong evidence in supporting a positive relation between the beta measure of risk and the expected return once controlling for the interaction between beta and idiosyncratic volatility. Although the classical CAPM model is an equilibrium model, it is static in nature with a constant beta measure of risk for each security. There are several reasons, however, to believe that the covariance based beta measure of risk might shift in structure over time. Perhaps,Merton(1973)is the firstone topropose amodel thatallowsfortime-varyingriskas a result of changes in the investment opportunities. Using labor income as a proxy for time- varying investment opportunities, Jagannathan and Wang (1996) provide evidence on the 1 validity of the conditional CAPM. Alternatively, one can treat the position held by equity- holders as a call optionon the firm’s totalasset (Blackand Scholes, 1973)since they have the limited liability and the residual claim. Following the idea, Galai and Masulis (1976) (also see Berk, Green, and Naik, 1999) have shown that the equity beta will vary not only with leverage, but also with the volatility of underlying assets even when the asset beta is stable. Bernardo, Chowdhry, and Goyal (2007) and Da, Guo, and Jagannathan (2011) have provide some empirical evidence in supporting this view. 1Lewellen and Nagel (2006) find that the covariance between time-varying beta and time-varying risk pre- miumis too smallto explain deviations fromthe CAPM. 1 In this paper, we propose and test an alternative explanation for the failure of the beta measure of risk to differentiate the cross-sectional return differences of individual securities. Usingdailyreturnswithinamonthtoestimateamonthlybetameasureofanindividualstock, we find that the current and the next month beta estimates are weakly correlated and vary a lot (with an average autocorrelation being less than 25%). This means that even when the CAPMholdsmonthbymonth,itisdifficulttousethepastbetameasuretopredictnextmonth returns. Such a large instability in the beta measure is unlikely to be a result of changing fundamentalrisksofafirmwhichtendtooccuroveralongerperiod. Estimationerrorsarealso implausible to account for such large changes in the beta measure since betas are estimated usinghigh frequency returns. One possibleexplanationisthe speculativeinvestmentbehavior of both institutions and individual investors who tend to chase certain stocks. As a result, not only these stocks tend to be over- or under-priced relative to their rational prices, but the collective activities of these investors will also move the market in the same direction as well. Therefore, the current beta estimates of these stocks will rise temporarily. Moreover, if over-pricing reverses with a lag because of short-sale constraints, returns tend to drop in the subsequent periods. Such a negative relationbetween current beta and future returns will obscure the true CAPM relation. There are ample evidence to support our view. Using mutual fund equity holdings data, Falkenstein (1996) shows that mutual funds have a significant preference towards stocks with highvisibilityandareaversetostockswithlowidiosyncraticvolatility. BasedontheJapanese experience from 1975 to 2003, Chang and Dong (2006) find that institutional herding is positively related to idiosyncratic volatility. For individual investors, Han and Kumar (2008) have shown that retail investors prefer to hold and actively trade high idiosyncratic volatility stocksdue to their propensity tospeculate. Such special preference by investors’willnot only move the prices of these individual stocks but will move the market as well. Using both the 2 TAQ and ISSM data,Barber, Odean, and Zhu (2009)showthat notonly individualinvestors’ tradingtends tobe correlated,but theircoordinated tradingmovethe marketina substantial way that causes significant future return difference between heavily bought and heavily sold stocks. One thingin commonfromallthese studiesisinvestors’preference towardstockswith high idiosyncratic volatility. If this is the case, the possible instability of the beta measure should be related to idiosyncratic volatility. Following this logic, idiosyncratic volatility can predictchangesinbetaestimates,whichmeanswemightbeabletorestoretheCAPMrelation once controlling for the instabilityissue. Indeed, forstockswithlargebetas and highidiosyncraticvolatilitiesinthe current month, future betas tend to be low. At the same time, these stocks also appear to have low future returns due to limited arbitrage (see, Ang, et. al., 1996). In other words, contemporaneously, the CAPM relation seems to hold because small (large) beta seems to be associated with low (high)return. However,the predictive cross-sectionalregressionwillfail. Aftercontrollingfor the interactive effect between beta and idiosyncratic volatility, we show that the rolling beta measureestimatedbasedonthepastmonthlyreturns(seeFamaandFrench,1992)canexplain the cross-sectional return differences of individual stocks. Moreover, if it is these stocks with high current betas and large idiosyncratic volatilitiesbut low future returns that obscure the true beta and return relation, we should expect to see that the CAPM relation holds for the rest of the stocks. After deleting 10% of the stocks with the largest beta and idiosyncratic volatility (accounting for 5% of the market capitalization), the 25 size and book-to-market sorted portfolio returns using the remaining stocks are significantly and positively related to their betas. Our approach not only restore the CAPM relation in a simple way, but also showsthe importanceof accounting for the instabilityin beta estimate. To the very least, the CAPM holds in a first order. 3 Our findings are robust. Both portfolio analysis and Fama-MacBeth regression analysis provides consistent conclusions. While allpreviously documented firm-level variables,such as the book-to-market ratio, Amihud illiquidity measure, momentum, and return reversal, have significant explanatory power for stock returns, they do not subsume the predictive power of the conventional beta measure for expected return once we control for the interactive effect between beta and idiosyncratic volatility. Moreover, our results are also robust to both the NYSE/AMEX market subsample and the NASDAQ market subsample, and to the two evenly split subsample periods from 1963 to 1986 and from 1987 to 2010. In all these cases, wecontinuetofind bothsignificantexplanatorypowerofthebeta variableand theinteraction termbetween beta and idiosyncraticvolatilityforthe cross-sectionalreturn differencesamong individual stocks. In addition, our results are insensitive to differentbeta estimates. For example, Fama and MacBeth (1973)use a two-step procedure to enhance the power of tests by reducing the noise in the beta estimates. In particular, Fama and French (1992) use the post-sorting portfolio beta estimates instead of the pre-sorting firm-level beta estimates. Recently, Ang, Liu and Schwarz (2010) argue that portfolio beta estimates conceal important information contained in the individual stocks’ betas. Therefore, our main results are based on the rolling beta estimates of individual stocks. This choice is also motived by our argument of instable betas. In the robustness section, we also apply the Lewellen and Negal’s (2006) short window beta estimator and the Fama and French’s (1992) portfolio beta in the cross-sectional regressions. In all these cases, we consistently show that our main finding of significant effect of beta on the expected return is not altered when using portfolio betas provided that we continue to control for the interaction between beta and idiosyncratic volatility. This study is also related to several studies on the pricing of idiosyncratic risk. Using the 4 realized idiosyncratic volatility measured estimated from daily returns, Ang, Hodric, Xing, and Zhang (2006) document a negative relation between current idiosyncratic volatility and the next month return. No matter whether such a negative relation is due to return rever- sal (Huang, Liu, Rhee, and Zhang, 2010) or gambling (Bali, Cakici, Whitelaw, 2010), it is consistent with our findings. However, we take a step further to examine how idiosyncratic volatilityaffectsthe roleofbeta, and in turn altersfuture returns. Usingalternativemeasures of idiosyncratic risk, such as the conditional measure (Fu, 2009) or the portfolio measure of idiosyncraticrisk(Malkieland Xu, 2003),others find thatidiosyncraticvolatilityis positively related to future returns, which suggests a pricing effect of idiosyncratic risk. As suggested by Cao and Xu (2009), the priced component of idiosyncratic risk is a relatively small por- tion due to general diversification effect, mispricing might be a first order effect in short-run. Therefore, we primarily focus on the realized idiosyncratic volatility measure. Other recent studies including Paster and Veronesi (2009) also document the importance of idiosyncratic risk in affecting asset prices. In addition, researchers have found that idiosyncratic volatility is related to the growth option of a firm (see Bernardo, Chowdhry, and Goyal, 2007, Cao, Simin, and Zhao, 2008, Da, Guo, and Jagannathan, 2011, and Johnson, 2004). We contribute to the asset pricing literature in several important way. First, we show that individual securities’ betas vary a lot over time. Such instability makes it difficult for the beta variable alone to predict future returns even if the CAPM holds period-by-period. In Jagannathan and Wang (1996), apart from time-varying betas, the risk premia are also required to change significantly over time in order for the covariance between time-varying betaandtime-varyingriskpremiumtobelargeenoughtopatchthedeviationfromtheCAPM. In contrast, we only make an effort to predict possible deviations from the CAPM for some stocks directly. Second, motivated by possible investors’ preferences toward trading volatile stocks, we find that the realized idiosyncratic volatility is capable of predicting variations 5 in beta estimates. This means that we are able to predict and adjust deviation from the CAPM for stocks that fail the empirical tests. Finally, as a practical matter, we demonstrate that the simple CAPM model holds for 90% of the individual stocks (or 95% of the market capitalization). Atthesametime,thebetaandreturnrelationholdswellforthewholesample of stocks once we controlfor the interactionbetween the idiosyncraticvolatilityand the time- varying beta. In fact, the estimated market risk premium from a multivariatecross-sectional regression resembles the historical average excess return of the market portfolio. Also related to this paper is the study by Ang and Chen (2007), where they focus on an econometrics approach that explicitly model the dynamics of market risk premium, market volatility, and asset betas. They find that the time-varying beta estimates explains return differences between valueand growthstocks. In contrast,we rely on a much simplerapproach and be able to show how pervasive is the return and beta relation. The rest of the paper proceeds as follows. In the next section,we describe the data and defines variablesused in the study. In addition, we discuss our framework to implement the cross-sectional tests. Section 3 reports our main results. Robustness study is carried out in section 4. Finally, Section 5 concludes. 6 2 Data and Methodology In thissection,wewillfirstmotiveourtestingstrategy. In ordertobe consistentwithexisting studies, we also provide informationon our sample selection and variable construction in this section. 2.1 Methodology Fama and French’s (1992) results are both surprising and controversial. Some researchers argue thatboth the size and the book-to-marketvariablesare notrobust or subject tocertain 2 bias. Regardless the merits of these arguments, the beta variable continues to be insignifi- cant in explaining the cross-sectional return differences. Others try to patch the CAPM with different elements. One example is the idea of time-varying risk and risk premium of Merton (1976) as a result of changing investment opportunities. Even when a conditional CAPM model holds perfectly, investors require additional compensation for the covariance risk be- tween time-varying risk and time-varying risk premium. (see Jagannathan and Wang, 1996) Despite the fact that the beta estimatedoes vary substantiallyover time, Lewellen and Nagel (2007) find that the covariance between time-varying beta and time-varying risk premium is too small to account for deviations from the CAPM. Lewellen and Nagel (2007) also find that there is a large variation in the beta estimates. Such large changes in the beta estimates from month to month make it difficult for the beta variable to predict future returns even when the CAPM holds period by period. The large instability in the beta estimates is unlikely to be a result of changes in the fundamental risk (time-varying risk) since it is over relatively short time period. To some degree, Fama and 2An incomplet list includes Ang and Chen (2007), Daniel and Titman (1997), Daniel, Titman and Wei (2001), Horowitz,Loughran and Savin (2000), Knez and Ready (1997), Kim (1997), Kothari, Shanken, Sloan (1995), Loughran (1997), Shumway (1997),B arber and Lyon (1997), Dijk(2011). 7 Frenh (1992)recognize the issue as a estimationerrorproblem and offerto use portfoliobetas as a proxy for individual stocks’ betas. We believe that the failure of the beta measure to explain return differences is not mainly an issue of estimation error since we can use high frequency data to estimate beta. Instead, if change in beta has a systematic component and is predictable, we may be able to restore the predictive power of beta as prescribed by the CAPM model. We propose that one possible cause for beta instabilityis related to investors’ speculative trading behavior. When investors are actively chasing certain stocks, their action will not only affect the prices of these individual stocks but will move the market as well. Consequently, the covari- ance based beta estimates for these stocks will systematicallydeviate from their fundamental values. By its nature, such a deviation will not alter the long-term expected return of a firm and is likely to revert in near future. In other words, individual stocks’ return may still be contemporaneously correlatedwith the market return, which makes the market factorremain to be the single most powerful factor in explaining time-series asset returns, while the beta 3 measure is incapable to differentiate the cross-sectional return differences. In addition, the nature of these deviations suggests that they are not necessarily covary with macro factors, which makes it difficult to correct by relying on the idea of time-varying betas. This may be the reason that, even considering the time-varying factor, the conventional measure of beta is insignificant in cross-sectional tests. Note for a particular stock, such a deviation may be temporally. For the market as a whole, there exist such deviations in the beta estimates at anygivenpointoftime,but fordifferentstocks. Therefore, instabilityislikelytobe pervasive. There are ample evidence supporting that speculators tend to focus on stocks with large idiosyncratic volatilities. For example, using retail level data, Han and Kumar (2008) have 3In fact, these stocks tend to have low future returns because of temporally increase in prices despite increases inbetas, which furtherweakens thepossibility of finding a positive relationbetween beta and future return. 8