# «The Cross-Section of Volatility and Expected Returns ANDREW ANG, ROBERT J. HODRICK, YUHANG XING, and XIAOYAN ZHANG∗ ABSTRACT We examine the pricing ...»

However, VIX appears to be a stationary series, and using VIX as the innovation in VIX may slightly over-difference. Our finding of low average returns on stocks with high β FVIX is robust to measuring volatility innovations by specifying various models for the conditional mean of VIX. If we fit an AR(1) model to VIX and measure innovations relative to the AR(1) specification, we find that the results of Table I are almost unchanged. Specifically, the mean return of the difference between the first and fifth β VIX portfolios is −1.08% per month, and the FF-3 alpha of the 5-1 difference is −0.90%, both highly statistically significant. Using an optimal BIC choice for the number of autoregressive lags, which is 11, produces a similar result. In this case, the mean of the 5-1 difference is −0.81% and the 5-1 FF-3 alpha is −0.66%; both differences are significant at the 5% level.11 C.2. Robustness to the Portfolio Formation Window In this subsection, we investigate the robustness of our results to the amount of data used to estimate the pre-formation factor loadings β VIX. In Table I, we use a formation period of 1 month, and we emphasize that this window is chosen a priori without pre-tests. The results in Table I become weaker if we extend the formation period of the portfolios. Although the point estimates of the β VIX portfolios have the same qualitative patterns as Table I, statistical significance drops.

The weakening of the β VIX effect as the formation periods increase is due to the time variation of the sensitivities to aggregate market innovations. The turnover in the monthly β VIX portfolios is high (above 70%) and using longer 11 In these exercises, we estimate the AR coefficients only using all data up to time t to compute the innovation for t + 1, so that no forward-looking information is used. We initially estimate the AR models using 1 year of daily data. However, the optimal BIC lag length is chosen using the whole sample.

Cross-Section of Volatility and Expected Returns 275 formation periods causes less turnover; however, using more data provides less precise conditional estimates. The longer the formation window, the less these conditional estimates are relevant at time t, and the lower the spread in the pre-formation β VIX loadings. By using only information over the past month, we obtain an estimate of the conditional factor loading much closer to time t.

C.3. Robustness to Book-to-Market and Size Characteristics Small growth firms are typically firms with option values that would be expected to do well when aggregate volatility increases. The portfolio of small growth firms is also one of the Fama–French (1993) 25 portfolios sorted on size and book-to-market that is hardest to price by standard factor models (see, for example, Hodrick and Zhang (2001)). Could the portfolio of stocks with high aggregate volatility exposure have a disproportionately large number of small growth stocks?

Investigating this conjecture produces mixed results. If we exclude only the portfolio among the 25 Fama–French portfolios with the smallest growth firms and repeat the quintile portfolio sorts in Table I, we find that the 5-1 mean difference in returns is reduced in magnitude from −1.04% for all firms to −0.63% per month, with a t-statistic of −3.30. Excluding small growth firms produces a FF-3 alpha of −0.44% per month for the zero-cost portfolio that goes long portfolio 5 and short portfolio 1, which is no longer significant at the 5% level (t-statistic is −1.79), compared to the value of −0.83% per month with all firms. These results suggest that small growth stocks may play a role in the β VIX quintile sorts of Table I.

However, a more thorough characteristic-matching procedure suggests that size or value characteristics do not completely drive the results. Table III reports mean returns of the β VIX portfolios characteristic matched by size and book-to-market ratios, following the method proposed by Daniel et al. (1997).

Every month, each stock is matched with one of the Fama–French 25 size and book-to-market portfolios according to its size and book-to-market characteristics. The table reports value-weighted simple returns in excess of the characteristic-matched returns. Table III shows that characteristic controls for size and book-to-market decrease the magnitude of the raw 5-1 mean return difference of −1.04% in Table I to −0.90%. If we exclude firms that are members of the smallest growth portfolio of the Fama–French 25 size-value portfolios, the magnitude of the mean 5-1 difference decreases to −0.64% per month. However, the characteristic-controlled differences are still highly significant. Hence, the low returns to high past β VIX stocks are not completely driven by a disproportionate concentration among small growth stocks.

C.4. Robustness to Liquidity Effects ´ Pastor and Stambaugh (2003) demonstrate that stocks with high liquidity betas have high average returns. In order for liquidity to be an explanation behind the spreads in average returns of the β VIX portfolios, high β VIX stocks 276 The Journal of Finance Table III Characteristic Controls for Portfolios Sorted on β ΔVIX The table reports the means and standard deviations of the excess returns on the β VIX quintile portfolios characteristic matched by size and book-to-market ratios. Each month, each stock is matched with one of the Fama and French (1993) 25 size and book-to-market portfolios according to its size and book-to-market characteristics. The table reports value-weighted simple returns in excess of the characteristic-matched returns. The columns labeled “Excluding Small, Growth Firms” exclude the Fama–French portfolio containing the smallest stocks and the firms with the lowest book-to-market ratios. The row “5-1” refers to the difference in monthly returns between portfolio 5 and portfolio 1. The p-values of joint tests for all alphas equal to zero are less than 1% for the panel of all firms and for the panel excluding small, growth firms. Robust Newey–West (1987) t-statistics are reported in square brackets. The sample period is from January 1986 to December 2000.

1 0.32 2.11 0.36 1.90 2 0.04 1.25 0.02 0.94 3 0.04 0.94 0.05 0.89 −0.11 −0.10 4 1.04 1.02 −0.58 −0.29 5 3.39 2.17 −0.90 −0.64 5-1 [−3.59] [−3.75] must have low liquidity betas. To check that the spread in average returns on the β VIX portfolios is not due to liquidity effects, we first sort stocks into five ´ quintiles based on their historical Pastor–Stambaugh liquidity betas. Then, within each quintile, we sort stocks into five quintiles based on their past β VIX coefficient loadings. These portfolios are rebalanced monthly and are value weighted. After forming the 5 × 5 liquidity beta and β VIX portfolios, we average the returns of each β VIX quintile over the five liquidity beta portfolios. Thus, these quintile β VIX portfolios control for differences in liquidity.

´ We report the results of the Pastor–Stambaugh liquidity control in Panel A of Table IV, which shows that controlling for liquidity reduces the magnitude of the 5-1 difference in average returns from −1.04% per month in Table I to −0.68% per month. However, after controlling for liquidity, we still observe the monotonically decreasing pattern of average returns of the β VIX quintile portfolios. We also find that controlling for liquidity, the FF-3 alpha for the 5-1 portfolio remains significantly negative at −0.55% per month. Hence, liquidity effects cannot account for the spread in returns resulting from sensitivity to aggregate volatility risk.

Table IV also reports post-formation β FVIX loadings. Similar to the postformation β FVIX loadings in Table I, we compute the post-formation β FVIX coefficients using a monthly frequency regression with the four-factor model in equation (6) to be comparable to the FF-3 alphas over the same sample period.

Both the pre-formation β VIX and post-formation β FVIX loadings increase from Cross-Section of Volatility and Expected Returns 277 Table IV Portfolios Sorted on β ΔVIX Controlling for Liquidity, Volume and Momentum In Panel A, we first sort stocks into five quintiles based on their historical liquidity beta, following Pastor and Stambaugh (2003). Then, within each quintile, we sort stocks based on their β VIX ´ loadings into five portfolios. All portfolios are rebalanced monthly and are value weighted. The five portfolios sorted on β VIX are then averaged over each of the five liquidity beta portfolios.

Hence, they are β VIX quintile portfolios controlling for liquidity. In Panels B and C, the same approach is used except we control for average trading volume (in dollars) over the past month and past 12-month returns, respectively. The statistics in the columns labeled Mean and Std. Dev. are measured in monthly percentage terms and apply to total, not excess, simple returns. The table also reports alphas from CAPM and Fama–French (1993) regressions. The row “5-1” refers to the difference in monthly returns between portfolio 5 and portfolio 1. The pre-formation betas refer to the value-weighted β VIX within each quintile portfolio at the start of the month. We report the pre-formation β VIX averaged across the whole sample. The last column reports ex post β FVIX factor loadings over the whole sample, where FVIX is the factor mimicking aggregate volatility risk. To correspond with the Fama–French alphas, we compute the ex post betas by running a four-factor regression with the three Fama–French factors together with the FVIX factor, following the regression in equation (6). The row labeled “Joint test p-value” reports a Gibbons et al. (1989) test that the alphas equal zero, and a robust joint test that the factor loadings are equal to zero.

Robust Newey–West (1987) t-statistics are reported in square brackets. The sample period is from January 1986 to December 2000.

negative to positive from portfolio 1 to 5, consistent with a risk story. In particular, the post-formation β FVIX loadings increase from −1.87 for portfolio 1 to

5.38 for portfolio 5. We reject the hypothesis that the ex post β FVIX loadings are jointly equal to zero with a p-value less than 0.001.

returns than stocks with low past trading volume. It could be that the low average returns (and alphas) we find for stocks with high β FVIX loadings are just stocks with low volume. Panel B shows that this is not the case. In Panel B, we control for volume by first sorting stocks into quintiles based on their trading volume over the past month. We then sort stocks into quintiles based on their β FVIX loading and average across the volume quintiles. After controlling for volume, the FF-3 alpha of the 5-1 long–short portfolio remains significant at the 5% level at −0.58% per month. The post-formation β FVIX loadings also monotonically increase from portfolio 1 to 5.

low past returns, or past loser stocks, continue to have low future returns, stocks with high past β VIX loadings may tend to also be loser stocks. Controlling for past 12-month returns reduces the magnitude of the raw −1.04% per month difference between stocks with low and high β FVIX loadings to −0.89%, but the 5-1 difference remains highly significant. The CAPM and FF-3 alphas of the portfolios constructed to control for momentum are also significant at the 1% level. Once again, the post-formation β FVIX loadings are monotonically increasing from portfolio 1 to 5. Hence, momentum cannot account for the low average returns to stocks with high sensitivities to aggregate volatility risk.

D. The Price of Aggregate Volatility Risk Tables III and IV demonstrate that the low average returns to stocks with high past sensitivities to aggregate volatility risk cannot be explained by size, book-to-market, liquidity, volume, or momentum effects. Moreover, Tables III and IV also show strong ex post spreads in the FVIX factor. Since this evidence supports the case that aggregate volatility is a priced risk factor in the crosssection of stock returns, the next step is to estimate the cross-sectional price of volatility risk.

To estimate the factor premium λFVIX on the mimicking volatility factor FVIX, we first construct a set of test assets whose factor loadings on market volatility risk are sufficiently disperse so that the cross-sectional regressions have reasonable power. We construct 25 investible portfolios sorted by β MKT and β VIX as follows. At the end of each month, we sort stocks based on β MKT, computed by a univariate regression of excess stock returns on excess market returns over the past month using daily data. We compute the β VIX loadings using the bivariate regression (3) also using daily data over the past month. Stocks are ranked first into quintiles based on β MKT and then within each β MKT quintile into β VIX quintiles.

Jagannathan and Wang (1996) show that a conditional factor model like equation (1) has the form of a multifactor unconditional model, where the original factors enter as well as additional factors associated with the time-varying information set. In estimating an unconditional cross-sectional price of risk for the aggregate volatility factor FVIX, we recognize that additional factors may also affect the unconditional expected return of a stock. Hence, in our full specification, we estimate the following cross-sectional regression that includes

**FF-3, momentum (UMD), and liquidity (LIQ) factors:**

where the λs represent unconditional prices of risk of the various factors. To check robustness, we also estimate the cross-sectional price of aggregate volatility risk by using the Coval and Shumway (2001) STR factor in place of FVIX in equation (7).