The beta anomaly? Stock s quality matters!

Transcription

1 The beta anomaly? Stock s quality matters! John M. Geppert a (corresponding author) a University of Nebraska Lincoln College of Business 425P Lincoln, NE, USA, jgeppert1@unl.edu Lei Zhao b b University of Nebraska Lincoln College of Business 423 Lincoln, NE, USA, lei.zhao@huskers.unl.edu Abstract The beta anomaly is the empirical phenomenon that stocks with high (low) betas have negative (positive) alphas. That is, returns of high-beta stocks are too low relative to returns of low-beta stocks. This anomaly has been challenging and puzzling to empirical asset pricing researchers for more than 40 years since 1970s (Bali, Brown, Murray, and Tang, 201; Hong and Sraer, 201; Liu, Stambaugh, and Yuan, 2017). In this paper, we demonstrate that stock s quality is a key driver of the beta anomaly. The beta anomaly disappears when beta-sorted portfolios are neutralized to stock s quality. In addition, we find that the beta anomaly exists only when stock s quality is highly correlated with beta. Moreover, we find that high institutional ownership attenuates, although not eliminate, both the beta anomaly and stock s quality effect. Finally, we show that stock s quality plays an important role in the relation between future stock returns and beta. Keywords: Beta, Beta anomaly, Quality, Stock returns, Institutional ownership JEL classification: G11, G12, G14 1

2 The beta anomaly? Stock s quality matters! Abstract The beta anomaly is the empirical phenomenon that stocks with high (low) betas have negative (positive) alphas. That is, returns of high-beta stocks are too low relative to returns of low-beta stocks. This anomaly has been challenging and puzzling to empirical asset pricing researchers for more than 40 years since 1970s (Bali, Brown, Murray, and Tang, 201; Hong and Sraer, 201; Liu, Stambaugh, and Yuan, 2017). In this paper, we demonstrate that stock s quality is a key driver of the beta anomaly. The beta anomaly disappears when beta-sorted portfolios are neutralized to stock s quality. In addition, we find that the beta anomaly exists only when stock s quality is highly correlated with beta. Moreover, we find that high institutional ownership attenuates, although not eliminate, both the beta anomaly and stock s quality effect. Finally, we show that stock s quality plays an important role in the relation between future stock returns and beta. Keywords: Beta, Beta anomaly, Quality, Stock returns, Institutional ownership JEL classification: G11, G12, G14 2

3 1. Introduction The beta anomaly is the empirical phenomenon that stocks with high (low) betas have negative (positive) alphas. That is, returns of high-beta stocks are too low relative to returns of low-beta stocks. This anomaly has been challenging and puzzling to empirical asset pricing researchers for more than 40 years since 1970s (Bali, Brown, Murray, and Tang, 201; Hong and Sraer, 201; Liu, Stambaugh, and Yuan, 2017). In this paper, we demonstrate that stock s quality is a key driver of the beta anomaly. The beta anomaly disappears when beta-sorted portfolios are neutralized to stock s quality. In addition, we find that the beta anomaly exists only when stock s quality is highly correlated with beta. Moreover, we find that high institutional ownership attenuates, although not eliminate, both the beta anomaly and stock s quality effect. Finally, we show that stock s quality plays an important role in the relation between future stock returns and beta. The earliest study documenting the beta anomaly can be traced back to Black, Jensen, and Scholes (1972). Black et al. (1972) find that the security market line for US stocks is flatter than predicted by CAPM. In other words, returns of high-beta stocks are too low relative to returns of low-beta stocks. Since then, the beta anomaly has been extensively studied, and explanations almost unanimously argue that beta itself is the relevant stock characteristic driving the anomaly (Liu et al., 2017). For instance, Black (1972), Fama (197), and Frazzini and Pedersen (2014) argue that leverage- and/or margin-constrained investors overweight high-beta assets in their portfolios, thereby bidding up high-beta assets and causing those assets to generate low returns. Barber and Odean (2000) and Antoniou, Doukas, and Subrahmanyam (201) hypothesize that optimistic periods induce unsophisticated overconfident traders to invest heavily in high-beta stocks, leading high-beta stocks to be overpriced during such periods. Baker, Bradley, and 3

4 Wurgler (2011) and Christoffersen and Simutin (2017) argue that attempting to beat benchmarks leads fund managers to overweight their portfolios toward high-beta stocks and decrease their holdings of low-beta stocks, which can result in or reinforce the beta anomaly. Hong and Sraer (201) suggest that high-beta stocks are more likely to be mispriced than low-beta stocks when aggregate disagreement among investors about the stock market s prospects is high and that short-sale constraints prevent arbitrageurs from correcting overpricing of high-beta stocks. Despite the abundance of beta-driven explanations of the beta anomaly, more recently, Bali et al. (201) and Liu et al. (2017) find that beta is not the direct stock characteristic driving the anomaly. Instead, both studies show that beta is responsible for the beta anomaly through association. Specifically, Bali et al. (201) suggest and find that there is a strong positive crosssectional correlation between beta and stock s lottery demand characteristic (proxied by MAX) and that it is lottery demand price pressure induced by high-lottery demand stocks that is responsible for the beta anomaly. 1 However, it is not clear what the MAX variable used in Bali et al. (201) truly reflects. For example, Hou and Loh (201) note that MAX is simply a rangebased measure of volatility in essence; Barinov (2017) finds that MAX proxies for volatility rather than stock s lottery characteristic; Chen and Petkova (2012) find that high MAX stocks have higher R&D than low MAX stocks. Similarly, Liu et al. (2017) find that beta is positively correlated with idiosyncratic volatility (IdioVol) in the cross-section and that there is a negative relation between alpha and IdioVol, both of which together produce the negative relation between alpha and beta, i.e. the beta anomaly. One important finding in Liu et al. (2017) is that the beta anomaly exists only among overpriced stocks. 2 Liu et al. (2017) comment that beta- 1 Bali et al. (201) use MAX, defined as the average of the five highest daily returns of the given stock in the given month, to proxy for stock s lottery demand characteristic. 2 As Liu et al. (2017) note, this finding is not surprising. The relation between alpha and IdioVol is positive among underpriced stocks, the relation between alpha and IdioVol is negative among overpriced stocks, and the latter 4

5 driven explanations of the beta anomaly proposed by prior studies are challenged by this finding because it is not clear why their proposed investor preferences for high-beta stocks are only confined to the high-beta stocks that are overpriced. Recently, Asness, Frazzini, and Pedersen (2017) derive a dynamic valuation model that links stock price to firm characteristics. Motivated by this theoretical model, Asness et al. (2017) define quality stocks as those of firms that are profitable, growing, safe, and well managed and define junk stocks as those of firms that are unprofitable, stagnant, risky, and poorly managed. The authors find that quality stocks have positive risk-adjusted returns while junk stocks have negative risk-adjusted returns i.e. stock s quality effect. After ruling out other possible explanations for stock s quality effect, the authors suggest and show that the effect arises from mispricing i.e. quality stocks are underpriced and junk stocks are overpriced. Motivated by Asness et al. (2017) combined with Liu et al. (2017), we ask the following question: does stock s quality play a role in the beta anomaly? We find that the beta anomaly no longer exists after controlling for stock s quality. Specifically, within each quality tercile (i.e. quality, medium, and junk), the alpha of the zero-cost portfolio that is long high-beta stocks (i.e. the beta decile 10) and short low-beta stocks (i.e. the beta decile 1) is extremely small in magnitude and statistically insignificant. Moreover, when beta-sorted portfolios are constructed to be neutral to stock s quality, the alpha of the high-minus-low beta portfolio is also extremely small in magnitude and statistically insignificant. Furthermore, we rule out a spurious relation between beta and stock s quality by conducting reverse-order bivariate portfolio analyses. We negative relation is stronger than the former positive relation, leading the overall relation between alpha and IdioVol to be negative. Since beta is (imperfectly) positively correlated with IdioVol, a negative relation between alpha and beta should happen only among overpriced stocks. Due to the weaker positive relation between alpha and IdioVol among underpriced stocks and the imperfect positive correlation between IdioVol and beta, the overall relation between alpha and beta turns out to be negative. 5

6 find that when quality terciles are neutralized to beta, the alpha of the quality-minus-junk portfolio is not only statistically significant but also economically large, indicating that controlling for beta cannot explain stock s quality effect. Finally, all these results are robust no matter whether alphas are calculated using the 3-factor model, the 4-factor model, the 5-factor model, or the -factor model. 3 High-beta stocks are more risky and thereby more likely to be junk stocks, whereas lowbeta stocks are less risky and thereby more likely to be quality stocks. In other words, there should be a negative relation between beta and stock s quality. We find that the median (mean) cross-sectional correlation between beta and stock s quality is -0.5 (-0.59). Because the crosssectional correlation varies over time, we split the whole sample period into high- and lowcorrelation months. We find that in months when the correlation is high the beta anomaly is strong, whereas in months when the correlation is low the beta anomaly is nonexistent. Especially, even in high-correlation months when the beta anomaly is strong, the anomaly disappears when alphas are calculated controlling for the quality-minus-junk factor (QMJ) developed by Asness et al. (2017). On the other hand, we find that stock s quality effect remains economically strong and statistically significant in both high- and low-correlation months. All these results together further corroborate the evidence that stock s quality is a key driver of the beta anomaly. Nagel (2005) notes that if stocks are underpriced investors can always correct the underpricing by buying and that if stocks are overpriced short-sale constraints can prevent arbitragers from correcting the overpricing. The author also suggests and shows that stocks with 3 The 3-factor model is the Fama and French (1993) three-factor model (FF3 hereafter). The 4-factor model is the FF3 model augmented with the momentum factor (UMD). The 5-factor model is the Fama and French (2015) fivefactor model (FF5 hereafter). The -factor model is the FF5 model augmented with UMD.

7 high institutional ownership are less likely to be subject to short-sale constraints and that mispricing is less concentrated among such stocks. Consistent with Asness et al. s (2017) conclusion that stock s quality effect arises from mispricing, we find that the alphas of the quality-minus-junk portfolios are much smaller in magnitude among stocks that are predominantly owned by institutions. Furthermore, if the beta anomaly is indeed driven by stock s quality effect that arises from mispricing, the beta anomaly should be strong (weak) among stocks with low (high) institutional ownership. The results are consistent with our inference. Finally, we further demonstrate the important role of stock s quality in the beta anomaly by conducting Fama and MacBeth (1973) (FM hereafter) regression analyses. Specifically, we regress future excess stock returns of individual stocks on beta and other well-recognized control variables. We find that in the whole sample the coefficient on beta is extremely small in magnitude and statistically insignificant, indicating a flat relation between future stock returns and beta, a result that has been widely documented in the literature. A similar result is also found in the junk stock subsample. However, when we exclude junk stocks from the whole sample and especially when we only keep quality stocks in the sample, the coefficient on beta is not only statistically significant but also economically large. These results, which are robust in different sample periods, provide further support to the evidence that stock s quality is a key driver of the beta anomaly. The rest of this paper is organized as follows. Section 2 describes our variables and data. The empirical results are presented and discussed in Section 3. Section 4 concludes the paper. 2. Variables and data 7

8 2.1 Variables Returns In our portfolio analyses, stocks are sorted based on their characteristics (e.g. beta, quality, and institutional ownership) calculated as of the end of month t. After portfolios are formed at the end of month t (or equivalently at the beginning of month t+1), value-weighted portfolio returns are calculated at the end of month t+1. Similarly, in our FM regression analyses, the dependent variables are excess stock returns at the end of month t+1, while beta, quality, and other control variables are calculated as of the end of month t Stock characteristics Used as value weights in our portfolio analyses, a stock s size (MKTCAP) is its market capitalization (i.e. price times shares outstanding) at the end of month t. Because the distribution of MKTCAP is highly skewed in the cross-section, consistent with the common practice of the literature, we use the natural logarithm of MKTCAP, denoted as SIZE, in our FM regression analyses. We calculate a stock s momentum (MOM) at the end of month t as the stock s 11-month return from months t-11 to t-1 inclusive. When calculating MOM, we require a stock to have no missing monthly returns during the 11-month period. We follow Amihud s (2002) method to calculate a stock s illiquidity (ILLIQ). Specifically, for each trading day in month t, we first calculate the daily ratio of the absolute value of the stock s daily return to its daily dollar trading volume. 4 We then calculate the stock s average 4 Dollar trading volume is price times share volume. 8

9 daily ratio in month t, i.e. the stock s month t ILLIQ. 5 When calculating ILLIQ, we require a stock to have at least 15 daily observations in month t. Consistent with Ang, Hodrick, Xing, and Zhang (200), we calculate a stock s month t IdioVol as its standard deviation of residuals from the FF3 model using daily returns in month t. At least 15 daily returns in month t are required to calculate a stock s month t IdioVol. Institutional ownership data are available on a quarterly basis. The institutional ownership variable (INST) used in this paper is the ratio of total institutional ownership to shares outstanding. To measure a stock s INST in month t of a certain year, when t is Jan. and Feb., the stock s month t INST is its INST as of the 4 th quarter of the previous year; when t is Mar., Apr., and May, the stock s month t INST is its INST as of the 1 st quarter of the current year; when t is June, July, and Aug., the stock s month t INST is its INST as of the 2 nd quarter of the current year; when t is Sept., Oct., and Nov., the stock s month t INST is its INST as of the 3 rd quarter of the current year; when t is Dec., the stock s month t INST is its INST as of the 4 th quarter of the current year Risk variables Using daily returns in a 12-month period from months t-11 to t inclusive, we calculate a stock s market beta (β) at the end of month t by regressing the stock s daily excess returns on daily excess market returns (i.e. CAPM), and we calculate a stock s size risk (β smb ) and value risk (β hml ) at the end of month t as the corresponding loadings from running the FF3 model. When calculating β, β smb, and β hml, we require a stock to have at least 200 daily returns in the 12-month period. 5 Following Amihud (2002, p. 37), the average daily ratio is then multiplied by 10. 9

10 We follow Ang et al. (200) to calculate a stock s aggregate volatility risk. Specifically, using daily returns in a 12-month period from months t-11 to t inclusive, we run the following model: r i d = β 0 + β i i i 1 MKTRF d + β VIX VIX d + ε d (1) where r d i is stock i s excess return on day d, MKTRF d is the excess market return on day d, and VIX d is the daily change in the CBOE S&P 100 Volatility Index on day d. i β VIX in equation (1) is stock i s aggregate volatility risk at the end of month t. At least 200 daily observations in the 12-month period are required to calculate a stock s β VIX. Our methods to calculate a stock s coskewness risk (COSKEW) and cokurtosis risk (COKURT) are consistent with Harvey and Siddique (2000) and Kostakis, Muhammad, and Siganos (2012). Specifically, using daily returns in a 12-month period from months t-11 to t inclusive, we first run the CAPM regression: r i,d = α i + β i MKTRF d + ε i,d (2) where r i,d is stock i s excess return on day d and MKTRF d is the excess market return on day d, and obtain the residuals ε i,d. Then stock i s COSKEW and COKURT at the end of month t are calculated as follows: 2 ) [ d (ε i,d ε m,d ]/N COSKEW i = [ (ε 2 2 d i,d )]/N {[ d (ε m,d )]/N} 3 ) [ d (ε i,d ε m,d ]/N COKURT i = [ (ε 2 3 d i,d )]/N {[ d (ε m,d )]/N} (3) (4) VIX (VXO) is the original (new) ticker symbol for the CBOE S&P 100 Volatility Index. Consistent with the tradition in the literature, we still use the original ticker symbol VIX to denote the CBOE S&P 100 Volatility Index. 10

11 where N denotes the number of daily returns stock i has during the 12-month period, and ε m,d is the deviation of the daily excess market return on day d from the average daily excess market return over the N days. When calculating COSKEW and COKURT, we require N Quality Asness et al. (2017) derive a dynamic valuation model that links a stock s price-to-book ratio to its quality characteristics. Motivated by this theoretical model, the authors assign each stock a single overall quality score, which is constructed based on various measures of the stock s profitability, growth, safety, and payout. 7 The authors then divide stocks into two size groups (small and big), and within each size group stocks are further sorted on their quality scores into three groups (quality, medium, and junk). Finally, the QMJ factor is constructed as the average return of the two quality portfolios minus the average return of the two junk portfolios. In our study, we measure a stock s quality at the end of month t, which is intended to reflect the stock s expected quality in month t+1, as the loading on QMJ (β qmj ) from a regression of excess stock returns on the QMJ factor returns using daily returns in a 12-month period from months t-11 to t inclusive. When calculating β qmj, we require a stock to have at least 200 daily returns in the 12-month period. 2.2 Data We obtain daily and monthly stock data from Center for Research in Security Prices (CRSP) in Wharton Research Data Services (WRDS). The Fama and French (1993) three factors, the momentum factor, and the one-month Treasury bill rate are collected from Fama- 7 For more details, see 11

12 French Portfolios and Factors in WRDS. The Fama and French (2015) five factors are downloaded from Professor Kenneth French s online data library. 8 We get institutional ownership data from Thomson Reuters Institutional (13f) Holdings in WRDS. The CBOE S&P 100 Volatility Index data are downloaded from Chicago Board Options Exchange (CBOE). 9 The quality-minus-junk factor comes from Applied Quantitative Research (AQR). 10 Consistent with the common practice of the literature, our sample consists of all ordinary common shares (share code = 10 or 11) on the NYSE, AMEX, and NASDAQ (exchange code = 1, 2, or 3). For our main analyses, the sample period for all stock characteristics (β, β qmj, and other control variables, except β VIX and INST, described in Section 2.1), which are calculated as of the end of month t, is from July 193 to November 201 (41 months), and the sample period for stock returns, which are as of the end of month t+1, is therefore from August 193 to December 201 (41 months). Restricted by the availability of the CBOE S&P 100 Volatility Index data, analyses that involve β VIX cover 30 months t (return months t+1) from December 198 (January 1987) to November 201 (December 201). Quarterly institutional ownership data are available starting from 1980Q1, therefore analyses that involve INST cover 441 months t (return months t+1) from March 1980 (April 1980) to November 201 (December 201). 3. Results 3.1 The beta anomaly 8 Professor French s online data library: We thank Professors Fama and French for making their data available Daily quality-minus-junk factor: Monthly quality-minus-junk factor: 12

13 Table 1 shows the beta anomaly. At the end of each month t (or equivalently at the beginning of each month t+1), we sort stocks into 10 β decile portfolios (1 = low β decile; 10 = high β decile). We hold the 10 portfolios formed at the beginning of month t+1 for the whole month, and at the end of month t+1 we calculate each portfolio s value-weighted monthly return. Consistent with Liu et al. (2017) among others, alphas reported in Table 1 are calculated relative to the FF3 model. Robust Newey and West (1987) (NW hereafter) t-statistics with 7 lags, which test the null hypothesis that alpha is equal to zero, are reported in parentheses. 11 [Insert Table 1 about here] As shown in Table 1, the alpha of the zero-cost portfolio that is long the high β decile portfolio and short the low β decile portfolio is -37 basis points (bps) per month with a t-statistic of However, this nontrivial abnormal return is entirely driven by the high β decile portfolio. Specifically, the alpha of the low β decile portfolio is 0 bps per month with a t-statistic of 0.02, whereas the alpha of the high β decile portfolio is -37 bps per month with a t-statistic of These results are consistent with arbitrage asymmetry put forth by Nagel (2005), Stambaugh, Yu, and Yuan (2015), and Hong and Sraer (201). That is, buying (underpriced low β stocks) is easier than shorting (overpriced high β stocks), 12 therefore arbitrage should eliminate more underpricing than overpricing. 3.2 The quality effect 11 In this paper, all t-statistics are robust Newey and West (1987) t-statistics, with the number of lags depending on the number of time periods (T) in the sample. For Table 1, the sample has T = 41 months, thus lag = 4 ( T 2 ) 9 = ( 41 2 ) 9 = To be conservative, if the calculated lag number is not an integer, we always round it up to the 100 nearest integer. 12 Prado, Saffi, and Sturgess (201) note that short-sale constraints include the difficulty and/or inability to find shares to borrow, the costs involved in searching and borrowing shares (e.g. loan fees), and the risk and loss incurred when the borrowed shares are recalled by the lender. 13

14 Table 2 shows the quality effect. At the end of each month t, we sort stocks into 3 quality tercile portfolios (1 = low β qmj (junk) tercile; 3 = high β qmj (quality) tercile). Value-weighted portfolio returns are calculated at the end of each month t+1. The FF3 alphas and NW t-statistics with 7 lags are reported. [Insert Table 2 about here] In line with Asness et al. s (2017) findings, the junk tercile portfolio has a negative riskadjusted return of -41 bps per month (t-statistic = -4.15), the quality tercile portfolio has a positive risk-adjusted return of 15 bps per month (t-statistic = 2.7), and the quality-minus-junk portfolio has a positive risk-adjusted return of 55 bps per month (t-statistic = 3.80). Again, consistent with arbitrage asymmetry, the underpricing of the quality tercile portfolio is weaker, in terms of both magnitude and statistical significance, than the overpricing of the junk tercile portfolio. 3.3 Bivariate portfolio analyses first sorted on quality then on β Tables 3 to present our bivariate portfolio analyses. At the end of each month t, we first sort stocks into quality terciles (1 = low β qmj (junk) tercile; 3 = high β qmj (quality) tercile). Then within each quality tercile we further sort stocks into β deciles (1 = low β decile; 10 = high β decile). This dependent sort generates 30 portfolios. At the end of month t+1, we calculate each portfolio s value-weighted monthly return. Alphas calculated using different asset pricing models and NW t-statistics with 7 lags are reported in Tables 3 to. In Table 3, alphas are calculated using the FF3 model. Consistent with arbitrage asymmetry and the existent evidence that junk (quality) stocks are overpriced (underpriced), the alphas of the 10 β decile portfolios within the junk tercile are negative and statistically significant, while 14

15 the alphas of the 10 β decile portfolios within the quality tercile are positive but much weaker in magnitude and statistical significance. As expected, all the 10 β decile portfolios within the medium tercile are fairly priced. The alphas of the high-minus-low β portfolios within the junk, medium, and quality terciles are -20 bps per month (t-statistic = -0.78), bps per month (t-statistic = 0.30), and -5 bps per month (t-statistic = -0.30) respectively, indicating that after controlling for stock s quality the beta anomaly no longer exists. In the last column of Table 3, for each β decile, we average the 3 β decile portfolios across the junk, medium, and quality terciles to create a β decile portfolio that is neutral to stock s quality. As shown in the bottom right corner of Table 3, the alpha of the high-minus-low β portfolio that is neutral to stock s quality is only -7 bps per month (t-statistic = -0.42), which is only 19% of the -37 bps per month reported in Table 1 in which β decile portfolios are not neutral to stock s quality. [Insert Table 3 about here] In Table 4, alphas are calculated using the FF3+UMD model. The alphas of the high-minuslow β portfolios within the junk, medium, and quality terciles are -8 bps per month (t-statistic = -0.28), -5 bps per month (t-statistic = -0.23), and -1 bps per month (t-statistic = -0.05) respectively. The alpha of the high-minus-low β portfolio that is neutral to stock s quality is only -4 bps per month (t-statistic = -0.28), which is only 11% of the -37 bps per month reported in Table 1 in which β decile portfolios are not neutral to stock s quality. [Insert Table 4 about here] In Table 5, alphas are calculated using the FF5 model. The alphas of the high-minus-low β portfolios within the junk, medium, and quality terciles are 3 bps per month (t-statistic = 0.12), 4 bps per month (t-statistic = 0.22), and -12 bps per month (t-statistic = -0.7) respectively. The 15

16 alpha of the high-minus-low β portfolio that is neutral to stock s quality is only -1 bps per month (t-statistic = -0.09), which is only 3% of the -37 bps per month reported in Table 1 in which β decile portfolios are not neutral to stock s quality. [Insert Table 5 about here] In Table, alphas are calculated using the FF5+UMD model. The alphas of the high-minuslow β portfolios within the junk, medium, and quality terciles are 10 bps per month (t-statistic = 0.38), -5 bps per month (t-statistic = -0.22), and -8 bps per month (t-statistic = -0.4) respectively. The alpha of the high-minus-low β portfolio that is neutral to stock s quality is only -1 bps per month (t-statistic = -0.0), which is only 3% of the -37 bps per month reported in Table 1 in which β decile portfolios are not neutral to stock s quality. [Insert Table about here] In summary, the bivariate portfolio analyses show that within each quality tercile (i.e. quality, medium, and junk) the alpha of the zero-cost portfolio that is long high β stocks and short low β stocks is extremely small in magnitude and statistically insignificant and that when β-sorted portfolios are constructed to be neutral to stock s quality, the alpha of the high-minuslow β portfolio is also extremely small in magnitude and statistically insignificant. These results indicate that stock s quality is a key driver of the beta anomaly. 3.4 Reverse-order bivariate portfolio analyses first sorted on β then on quality To rule out a spurious relation between β and stock s quality, in Tables 7 to 10, we reverse the order of sorts and examine whether stock s quality effect disappears after controlling for β. At the end of each month t, we first sort stocks into β deciles (1 = low β decile; 10 = high β 1

17 decile). Then within each β decile we further sort stocks into quality terciles (1 = low β qmj (junk) tercile; 3 = high β qmj (quality) tercile). This dependent sort generates 30 portfolios. At the end of month t+1, we calculate each portfolio s value-weighted monthly return. In addition, for each quality tercile, we average the 10 quality tercile portfolios across the β deciles to create a quality tercile portfolio that is neutral to β. Alphas calculated using different asset pricing models and NW t-statistics with 7 lags are reported in Tables 7 to 10. In Table 7 in which alphas are calculated using the FF3 model, the alpha of the qualityminus-junk portfolio that is neutral to β is 42 bps per month (t-statistic = 4.08), which is 7% of the 55 bps per month reported in Table 2 in which quality tercile portfolios are not neutral to β. [Insert Table 7 about here] In Table 8 in which alphas are calculated using the FF3+UMD model, the alpha of the quality-minus-junk portfolio that is neutral to β is 37 bps per month (t-statistic = 3.9), which is 7% of the 55 bps per month reported in Table 2 in which quality tercile portfolios are not neutral to β. [Insert Table 8 about here] In Tables 9 and 10 in which alphas are calculated using the FF5 model and the FF5+UMD model respectively, the alphas of the quality-minus-junk portfolios that are neutral to β are 18 bps per month (t-statistic = 2.0) and 17 bps per month (t-statistic = 1.94) respectively, whose magnitudes are nontrivial relative to the 55 bps per month reported in Table 2 in which quality tercile portfolios are not neutral to β. [Insert Table 9 about here] 17

18 [Insert Table 10 about here] In summary, the reverse-order bivariate portfolio analyses show that when quality terciles are neutralized to β, the alpha of the quality-minus-junk portfolio is not only statistically significant but also economically large, indicating that controlling for β cannot explain stock s quality effect. 3.5 Correlation analyses High β stocks are more risky and thereby more likely to be junk stocks, whereas low β stocks are less risky and thereby more likely to be quality stocks. In other words, there should be a negative relation between β and stock s quality. For each month t, we calculate the crosssectional Pearson product-moment correlation between β and stock s quality. The median (mean) cross-sectional correlation between β and stock s quality over the 41-month sample period is -0.5 (-0.59). 13 Because the cross-sectional correlation varies over time, we split the whole sample into high- and low-correlation months, where high-correlation months are those months in which the correlation is less than or equal to the median (i.e. more negative) and lowcorrelation months are those months in which the correlation is greater than the median (i.e. less negative). We then examine the beta anomaly and stock s quality effect in each subsample. Table 11 examines the beta anomaly in high- and low-correlation months. The FF3 alphas (in Panel A), the FF3+QMJ alphas (in Panel B), and NW t-statistics with lags are reported. 14 As shown in Panel A, in high-correlation months the alpha of the high-minus-low β portfolio is -74 bps per month with a t-statistic of -2.54, whereas in low-correlation months the alpha of 13 Only 3 months out of the 41 months have a positive cross-sectional correlation between β and stock s quality. 14 The subsamples of high- and low- correlation months have 321 months and 320 months respectively. Therefore, for both subsamples, NW t-statistics with lags are calculated. 18

19 the high-minus-low β portfolio is 13 bps per month with a t-statistic of 0.4, indicating that the beta anomaly exists only when stock s quality is highly correlated with β. In Panel B in which alphas are calculated controlling for the QMJ factor developed by Asness et al. (2017), the beta anomaly disappears in high-correlation months (-17 bps per month with a t-statistic of -0.5) and remains nonexistent in low-correlation months (-2 bps per month with a t-statistic of -0.07). [Insert Table 11 about here] Table 12 examines stock s quality effect in high- and low-correlation months. The FF3 alphas and NW t-statistics with lags are reported. The results indicate that stock s quality effect remains economically strong and statistically significant in both high- and low-correlation months. Specifically, the alphas of the quality-minus-junk portfolios in high- and low-correlation months are 54 bps per month (t-statistic = 2.3) and 30 bps per month (t-statistic = 1.75) respectively. [Insert Table 12 about here] In summary, stock s quality effect always exists no matter whether the correlation between stock s quality and β is high or low, but the beta anomaly exists only when stock s quality is highly correlated with β. In addition, when abnormal returns are calculated controlling for the QMJ factor, the beta anomaly does not exist at all. All these results together further corroborate the evidence that stock s quality is a key driver of the beta anomaly. 3. The impact of institutional ownership on the beta anomaly and stock s quality effect Nagel (2005) suggests and shows that stocks with high institutional ownership are less likely to be subject to short-sale constraints and that mispricing is less concentrated among such stocks. In this section, we examine the impact of institutional ownership on the beta anomaly and 19

20 stock s quality effect. The sample, which involves INST, covers 441 months t (return months t+1) from March 1980 (April 1980) to November 201 (December 201). At the end of each month t, we first sort stocks into INST deciles based on an ascending ordering of INST. Then within each INST decile we further sort stocks into β deciles (quality terciles) based on an ascending ordering of β (β qmj ). This dependent sort generates 100 (30) portfolios. At the end of month t+1, we calculate each portfolio s value-weighted monthly return. The FF3 alphas and NW t-statistics with lags are reported. 15 Table 13 examines the impact of institutional ownership on stock s quality effect. We first examine the unconditional strength of stock s quality effect in the 441-month sample period. As shown in Panel A of Table 13, the alpha of the quality-minus-junk portfolio is 81 bps per month with a t-statistic of We then examine the strength of stock s quality effect within each INST decile. Consistent with Asness et al. s (2017) conclusion that stock s quality effect arises from mispricing, we find that the quality effect is much smaller in magnitude among stocks that are predominantly owned by institutions. Specifically, as shown in Panel B of Table 13, within INST deciles 1 to almost all the alphas of the quality-minus-junk portfolios are more than 100 bps per month, whereas within INST deciles 7 to 10 the alphas of the quality-minus-junk portfolios are 44 to 74 bps per month. [Insert Table 13 about here] Table 14 examines the impact of institutional ownership on the beta anomaly. We first examine the unconditional strength of the beta anomaly in the 441-month sample period. As shown in Panel A of Table 14, the alpha of the high-minus-low β portfolio is -5 bps per month with a t-statistic of We then examine the strength of the beta anomaly within each INST 15 The sample, which involves INST, has 441 months, thus NW t-statistics with lags are calculated. 20

21 decile. If the beta anomaly is indeed driven by stock s quality effect that arises from mispricing, the beta anomaly should be strong (weak) among stocks with low (high) institutional ownership. The results are consistent with our inference. Specifically, as shown in Panel B of Table 14, within INST deciles 1 to in which stock s quality effect is relatively strong, all the alphas of the high-minus-low β portfolios are negative and highly statistically significant, and almost all of their magnitudes are more than 100 bps per month. On the other hand, within INST deciles 7 to 10 in which stock s quality effect is relatively weak, all the alphas of the high-minus-low β portfolios are negative but much weaker in magnitude (49 to 58 bps per month) and statistical significance. [Insert Table 14 about here] In summary, the results in this section demonstrate that high institutional ownership attenuates, although not eliminate, both the beta anomaly and stock s quality effect. More interestingly, within exactly the same low (high) INST deciles in which stock s quality effect is relatively strong (weak), the beta anomaly is also relatively strong (weak). As we have ruled out a spurious relation between β and stock s quality, these results provide more support to the evidence that stock s quality is a key driver of the beta anomaly. 3.7 Fama-MacBeth regression analyses We run FM regressions for 5 samples: the whole sample and 4 subsamples. At the end of each month t, we sort stocks into quality terciles (i.e. junk (low β qmj ), medium, and quality (high β qmj )). Stocks in the junk, medium, and quality terciles constitute the junk, medium, and quality subsamples respectively, and stocks in the medium and quality terciles constitute the non-junk subsample. Each month t, for each of the 5 samples, we run a cross-sectional regression of future 21

22 excess stock returns (at the end of month t+1) of individual stocks on β and other wellrecognized control variables, all of which are calculated as of the end of month t. The time-series averages of the regression coefficients, NW t-statistics with 7 lags (in Panel A), and NW t-statistics with lags (in Panel B) are reported in Table As shown in Panel A of Table 15, in the whole sample and the junk subsample the coefficients on β are (t-statistic = 0.40) and (t-statistic = -0.37) respectively, whereas in the non-junk subsample and the quality subsample the coefficients on β are (t-statistic = 1.90) and (t-statistic = 2.05) respectively. The signs of coefficients on the control variables are consistent with previous studies. 17 [Insert Table 15 about here] In Panel B of Table 15, we additionally control for β VIX as suggested in Ang et al. (200). Although the sample period is shorter than that in Panel A, the findings are qualitatively the same. Specifically, in the whole sample and the junk subsample the coefficients on β are (t-statistic = 0.53) and (t-statistic = 0.24) respectively, whereas in the non-junk subsample and the quality subsample the coefficients on β are (t-statistic = 1.9) and (t-statistic = 2.19) respectively. In summary, the FM regression analyses show that in the whole sample the coefficient on β is extremely small in magnitude and statistically insignificant, indicating a flat relation between future stock returns and β, a result that has been widely documented in the literature. A similar result is also found in the junk stock subsample. However, when we exclude junk stocks from the 1 In Panel A of Table 15, the sample period covers 41 months, thus NW t-statistics with 7 lags are calculated. In Panel B of Table 15, the sample period covers 30 months, thus NW t-statistics with lags are calculated. 17 SIZE (-), ILLIQ (+), MOM (+), IdioVol (-), β smb (-), β hml (+), COSKEW (-), COKURT (+). 22

23 whole sample and especially when we only keep quality stocks in the sample, the coefficient on β is not only statistically significant but also economically large. These results, which are robust in different sample periods, provide further support to the evidence that stock s quality is a key driver of the beta anomaly and shed new light on the relation between risk and expected returns. 3.8 Persistence of β qmj In our study, we measure a stock s quality at the end of month t, which is intended to reflect the stock s expected quality in month t+1. In this section, we use a portfolio transition matrix to demonstrate the persistence of β qmj. That is, we examine the probability that a stock in a certain quality tercile in month t will be in the same quality tercile in month t+1. Specifically, at the end of each month t, we sort stocks into 3 quality tercile portfolios (1 = low β qmj (junk) tercile; 3 = high β qmj (quality) tercile). For a certain quality tercile in each month t, we calculate the percentages of stocks in the tercile that (conditional on survival) fall in the same tercile and other terciles in month t+1. We then calculate the time-series averages of the percentages. As shown in Table 1, 89.87% of stocks in the junk tercile in month t remain in the junk tercile in month t+1, 79.44% of stocks in the medium tercile in month t remain in the medium tercile in month t+1, and 88.20% of stocks in the quality tercile in month t remain in the quality tercile in month t+1. These results demonstrate that stock s quality measured in month t is persistent to reflect stock s quality in month t+1. [Insert Table 1 about here] 4. Conclusion 23

24 This paper demonstrates that stock s quality is a key driver of the beta anomaly. While stock s quality effect remains economically strong and statistically significant when the qualityminus-junk portfolio is neutralized to β, the beta anomaly disappears when controlling for stock s quality. Moreover, stock s quality effect always exists no matter whether the correlation between stock s quality and β is high or low, but the beta anomaly exists only when stock s quality is highly correlated with β. Even when stock s quality is highly correlated with β, the beta anomaly no longer exists when abnormal returns are calculated controlling for the QMJ factor. Furthermore, within exactly the same low (high) INST deciles in which stock s quality effect is relatively strong (weak), the beta anomaly is also relatively strong (weak). Lastly, after excluding junk stocks from the sample, FM regressions indicate a significantly positive relation between future stock returns and β. This paper sheds new light on the relation between risk and expected returns, a central issue in financial economics. Also, the findings of this paper are important to investors and traders. An investment strategy of shorting high β stocks and buying low β stocks generates a Sharpe ratio that is even higher than those of the famous value and momentum premiums (Baker et al., 2011; Frazzini and Pedersen, 2014; Hong and Sraer, 201). However, as shown in this paper, picking high and low β stocks with the same quality characteristics renders the profits insignificant. 24

25 References Amihud, Y., Illiquidity and stock returns: cross-section and time-series effects. Journal of Financial Markets 5, Ang, A., Hodrick, R., Xing, Y., Zhang, X., 200. The cross-section of volatility and expected returns. Journal of Finance 1, Antoniou, C., Doukas, J., Subrahmanyam, A., 201. Investor sentiment, beta, and the cost of equity capital. Management Science 2, Asness, C., Frazzini, A., Pedersen, L., Quality minus junk. Unpublished working paper. AQR Capital Management. Baker, M., Bradley, B., Wurgler, J., Benchmarks as limits to arbitrage: understanding the low-volatility anomaly. Financial Analysts Journal 7, Bali, T., Brown, S., Murray, S., Tang, Yi., 201. A lottery demand-based explanation of the beta anomaly. Journal of Financial and Quantitative Analysis (forthcoming). Barber, B., Odean, T., Trading is hazardous to your wealth: the common stock investment performance of individual investors. Journal of Finance 55, Barinov, A., Stocks with extreme past returns: lotteries or insurance? Journal of Financial Economics (forthcoming). Black, F., Capital market equilibrium with restricted borrowing. Journal of Business 45, Black, F., Jensen, M., Scholes, M., The capital asset pricing model: some empirical tests. In: Jensen M. C. (Ed.), Studies in the Theory of Capital Markets. Praeger, New York, pp Chen, Z., Petkova, R., Does idiosyncratic volatility proxy for risk exposure? Review of Financial Studies 25, Christoffersen, S., Simutin, M., On the demand for high-beta stocks: evidence from mutual funds. Review of Financial Studies 30, Fama, E., MacBeth, J., Risk, return, and equilibrium: empirical tests. Journal of Political Economy 81, Fama, E., 197. Foundations of Finance. Basic Books, New York. Fama, E., French, K., Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3-5. Fama, E., French, K., A five-factor asset pricing model. Journal of Financial Economics 11, Frazzini, A., Pedersen, L., Betting against beta. Journal of Financial Economics 111, Harvey, C., Siddique, A., Conditional skewness in asset pricing tests. Journal of Finance 55,

26 Hong, H., Sraer, D., 201. Speculative betas. Journal of Finance 71, Hou, K., Loh, R., 201. Have we solved the idiosyncratic volatility puzzle? Journal of Financial Economics 121, Kostakis, A., Muhammad, K., Siganos, A., Higher co-moments and asset pricing on London Stock Exchange. Journal of Banking & Finance 3, Liu, J., Stambaugh, R., Yuan, Y., Absolving beta of volatility s effects. Journal of Financial Economics (forthcoming). Nagel, S., Short sales, institutional investors and the cross-section of stock returns. Journal of Financial Economics 78, Newey, W., West, K., A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, Prado, M., Saffi, P., Sturgess, J., 201. Ownership structure, limits to arbitrage, and stock returns: evidence from equity lending markets. Review of Financial Studies 29, Stambaugh, R., Yu, J., Yuan, Y., Arbitrage asymmetry and the idiosyncratic volatility puzzle. Journal of Finance 70,

27 Table 1: The beta anomaly FF3 alphas β 1 (Low) (0.02) β (1.30) β (1.7) β (0.44) β (0.37) β (0.5) β (-1.1) β (-0.92) β (-2.25) β 10 (High) (-2.41) High - Low (-1.4) At the end of each month t (or equivalently at the beginning of each month t+1), we sort stocks into 10 β decile portfolios (1 = low β decile; 10 = high β decile). We hold the 10 portfolios formed at the beginning of month t+1 for the whole month, and at the end of month t+1 we calculate each portfolio s value-weighted monthly return. High - Low denotes a zero-cost portfolio that is long the high β decile portfolio and short the low β decile portfolio. Alphas are calculated relative to the FF3 model. Newey and West (1987) t-statistics with 7 lags, which test the null hypothesis that alpha is equal to zero, are reported in parentheses. The sample period covers 41 months t (return months t+1) from July 193 (August 193) to November 201 (December 201). 27

28 Table 2: The quality effect FF3 alphas β qmj 1 (Junk) (-4.15) β qmj 2 (Medium) (0.72) β qmj 3 (Quality) (2.7) Quality - Junk (3.80) At the end of each month t (or equivalently at the beginning of each month t+1), we sort stocks into 3 quality tercile portfolios (1 = low β qmj (junk) tercile; 3 = high β qmj (quality) tercile). We hold the 3 portfolios formed at the beginning of month t+1 for the whole month, and at the end of month t+1 we calculate each portfolio s valueweighted monthly return. Quality - Junk denotes a zero-cost portfolio that is long the quality tercile portfolio and short the junk tercile portfolio. Alphas are calculated relative to the FF3 model. Newey and West (1987) t-statistics with 7 lags, which test the null hypothesis that alpha is equal to zero, are reported in parentheses. The sample period covers 41 months t (return months t+1) from July 193 (August 193) to November 201 (December 201). 28

29 Table 3: Portfolios sorted on β controlling for quality FF3 alphas FF3 alphas β qmj 1 (Junk) β qmj 2 (Medium) β qmj 3 (Quality) Neutral to stock s quality β 1 (Low) (-2.41) (-1.17) (1.00) (-1.58) β (-3.53) (0.08) (1.53) (-1.30) β (-3.58) (-0.21) (0.88) (-1.52) β (-3.31) (0.15) (1.45) (-1.24) β (-3.27) (-0.2) (2.39) (-0.84) β (-2.2) (-0.55) (2.44) (-0.89) β (-3.82) (0.90) (1.0) (-1.50) β (-2.88) (1.03) (2.2) (-0.82) β (-3.03) (-0.74) (2.27) (-1.58) β 10 (High) (-3.2) (-0.95) (0.92) (-2.59) High - Low (-0.78) (0.30) (-0.30) (-0.42) At the end of each month t, we first sort stocks into quality terciles (1 = low β qmj (junk) tercile; 3 = high β qmj (quality) tercile). Then within each quality tercile we further sort stocks into β deciles (1 = low β decile; 10 = high β decile). This dependent sort generates 30 portfolios. At the end of month t+1, we calculate each portfolio s valueweighted monthly return. In the last column of the table, for each β decile, we average the 3 β decile portfolios across the junk, medium, and quality terciles to create a β decile portfolio that is neutral to stock s quality. High - Low denotes a zero-cost portfolio that is long the high β decile portfolio and short the low β decile portfolio. Alphas are calculated relative to the FF3 model. Newey and West (1987) t-statistics with 7 lags, which test the null hypothesis that alpha is equal to zero, are reported in parentheses. The sample period covers 41 months t (return months t+1) from July 193 (August 193) to November 201 (December 201). 29