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

To measure ex post exposure to aggregate volatility risk at a monthly frequency, we follow Breeden, Gibbons, and Litzenberger (1989) and construct an ex post factor that mimics aggregate volatility risk. We term this mimicking factor FVIX. We construct the tracking portfolio so that it is the portfolio of asset returns maximally correlated with realized innovations in volatility using a set of basis assets. This allows us to examine the contemporaneous relationship between factor loadings and average returns. The major advantage of using FVIX to measure aggregate volatility risk is that we can construct a good approximation for innovations in market volatility at any frequency. In particular, the factor mimicking aggregate volatility innovations allows us to proxy aggregate volatility risk at the monthly frequency by simply cumulating daily returns over the month on the underlying base assets used to construct the mimicking factor. This is a much simpler method for measuring aggregate volatility innovations at different frequencies, rather than specifying different, and unknown, conditional means for VIX that depend on different sampling frequencies. After constructing the mimicking aggregate volatility factor, we confirm that it is high exposure to aggregate volatility risk that is behind the low average returns to past β VIX loadings.

However, just showing that there is a relation between ex post aggregate volatility risk exposure and average returns does not rule out the explanation that the volatility risk exposure is due to known determinants of expected returns in the cross-section. Hence, our second condition for a risk-based explanation is that the aggregate volatility risk exposure is robust to controlling for various stock characteristics and other factor loadings. Several of these crosssectional effects may be at play in the results of Table I. For example, quintile portfolios 1 and 5 have smaller stocks, and stocks with higher book-to-market ratios, and these are the portfolios with the most extreme returns. Periods of very high volatility also tend to coincide with periods of market illiquidity ´ (see, among others, Jones (2003) and Pastor and Stambaugh (2003)). In Section I.C, we control for size, book-to-market, and momentum effects, and also 270 The Journal of Finance specifically disentangle the exposure to liquidity risk from the exposure to systematic volatility risk.

B.4. A Factor Mimicking Aggregate Volatility Risk Following Breeden et al. (1989) and Lamont (2001), we create the mimicking factor FVIX to track innovations in VIX by estimating the coefficient b in the

**following regression:**

VIX t = c + b X t + ut, (4) where Xt represents the returns on the base assets. Since the base assets are excess returns, the coefficient b has the interpretation of weights in a zerocost portfolio. The return on the portfolio, b Xt, is the factor FVIX that mimics innovations in market volatility. We use the quintile portfolios sorted on past β VIX in Table I as the base assets Xt. These base assets are, by construction, a set of assets that have different sensitivities to past daily innovations in VIX.7 We run the regression in equation (4) at a daily frequency every month and use the estimates of b to construct the mimicking factor for aggregate volatility risk over the same month.

An alternative way to construct a factor that mimics volatility risk is to directly construct a traded asset that ref lects only volatility risk. One way to do this is to consider option returns. Coval and Shumway (2001) construct marketneutral straddle positions using options on the aggregate market (S&P100 options). This strategy provides exposure to aggregate volatility risk. Coval and Shumway approximate daily at-the-money straddle returns by taking a weighted average of zero-beta straddle positions, with strikes immediately above and below each day’s opening level of the S&P100. They cumulate these daily returns each month to form a monthly return, which we denote as STR.8 In Section I.D, we investigate the robustness of our results to using STR in place of FVIX when we estimate the cross-sectional aggregate volatility price of risk.

Once we construct FVIX, then the multifactor model (3) holds, except we can substitute the (unobserved) innovation in volatility with the tracking portfolio that proxies for market volatility risk (see Breeden (1979)). Hence, we can write

**the model in equation (3) as the following cross-sectional regression:**

rti = α i + βMKT MKT t + βFVIX FVIX t + εt, i i i (5) where MKT is the market excess return, FVIX is the mimicking aggregate volatility factor, and βMKT and βFVIX are factor loadings on market risk and i i aggregate volatility risk, respectively.

Our results are unaffected if we use the six Fama–French (1993) 3 × 2 portfolios sorted on size 7 and book-to-market as the base assets. These results are available upon request.

8 The STR returns are available from January 1986 to December 1995, because it is constructed from the Berkeley Option Database, which has reliable data only from the late 1980s and ends in 1995.

Cross-Section of Volatility and Expected Returns 271 To test a factor risk model like equation (5), we must show contemporaneous patterns between factor loadings and average returns. That is, if the price of risk of aggregate volatility is negative, then stocks with high covariation with FVIX should have low returns, on average, over the same period used to compute the β FVIX factor loadings and the average returns. By construction, FVIX allows us to examine the contemporaneous relationship between factor loadings and average returns and it is the factor that is ex post most highly correlated with innovations in aggregate volatility. However, while FVIX is the right factor to test a risk story, FVIX itself is not an investable portfolio because it is formed with future information. Nevertheless, FVIX can be used as guidance for tradeable strategies that would hedge market volatility risk using the cross-section of stocks.

In the second column under the heading “Factor Loadings” of Table I, we report the pre-formation β FVIX loadings that correspond to each of the portfolios sorted on past β VIX loadings. The pre-formation β FVIX loadings are computed by running the regression (5) over daily returns over the past month. The preformation FVIX loadings are very similar to the pre-formation VIX loadings for the portfolios sorted on past β VIX loadings. For example, the pre-formation βFVIX (β VIX ) loading for quintile 1 is −2.00 (−2.09), while the pre-formation βFVIX (β VIX ) loading for quintile 5 is 2.31 (2.18).

B.5. Post-Formation Factor Loadings In the next-to-last column of Table I, we report post-formation β VIX loadings over the next month, which we compute as follows. After the quintile portfolios are formed at time t, we calculate daily returns of each of the quintile portfolios over the next month, from t to t + 1. For each portfolio, we compute the ex post β VIX loadings by running the same regression (3) that is used to form the portfolios using daily data over the next month (t to t + 1). We report the nextmonth β VIX loadings averaged across time. The next-month post-formation β VIX loadings range from −0.033 for portfolio 1 to 0.018 for portfolio 5. Hence, although the ex post β VIX loadings over the next month are monotonically increasing, the spread is disappointingly very small.

Finding large spreads in the next-month post-formation β VIX loadings is a very stringent requirement and one that would be done in direct tests of a conditional factor model such as equation (1). Our goal is more modest. We examine the premium for aggregate volatility using an unconditional factor model approach, which requires that average returns be related to the unconditional covariation between returns and aggregate volatility risk. As Hansen and Richard (1987) note, an unconditional factor model implies the existence of a conditional factor model. However, to form precise estimates of the conditional factor loadings in a full conditional setting like equation (1) requires knowledge of the instruments driving the time variation in the betas, as well as specification of the complete set of factors.

The ex post β VIX loadings over the next month are computed using, on average, only 22 daily observations each month. In contrast, the CAPM and FF-3 272 The Journal of Finance alphas are computed using regressions measuring unconditional factor exposure over the full sample (180 monthly observations) of post-ranking returns.

To demonstrate that exposure to volatility innovations may explain some of the large CAPM and FF-3 alphas, we must show that the quintile portfolios exhibit different post-ranking spreads in aggregate volatility risk sensitivities over the entire sample at the same monthly frequency for which the post-ranking returns are constructed. Averaging a series of ex post conditional 1-month covariances does not provide an estimate of the unconditional covariation between the portfolio returns and aggregate volatility risk.

To examine ex post factor exposure to aggregate volatility risk consistent with a factor model approach, we compute post-ranking FVIX betas over the full sample.9 In particular, since the FF-3 alpha controls for market, size, and value factors, we compute ex post FVIX factor loadings also controlling for these factors in a four-factor post-formation regression,

where the first three factors MKT, SMB, and HML constitute the FF-3 model’s market, size, and value factors. To compute the ex post β FVIX loadings, we run equation (6) using monthly frequency data over the whole sample, where the portfolios on the left-hand side of equation (6) are the quintile portfolios in Table I that are sorted on past loadings of β VIX using equation (3).

The last column of Table I shows that the portfolios sorted on past β VIX exhibit strong patterns of post-formation factor loadings on the volatility risk factor FVIX. The ex post β FVIX factor loadings monotonically increase from −5.06 for portfolio 1 to 8.07 for portfolio 5. We strongly reject the hypothesis that the ex post β FVIX loadings are equal to zero, with a p-value less than 0.001.

Thus, sorting stocks on past β VIX provides strong, significant spreads in ex post aggregate volatility risk sensitivities.10 B.6. Characterizing the Behavior of FVIX Table II reports correlations among the FVIX factor, VIX, and STR, as well as correlations of these variables with other cross-sectional factors. We denote the daily first difference in VIX as VIX, and use m VIX to represent the monthly first difference in the VIX index. The mimicking volatility factor is highly contemporaneously correlated with changes in volatility at a daily 9 The pre-formation betas and the post-formation betas are computed using different criteria ( VIX and FVIX, respectively). However, Table I shows that the pre-formation β FVIX loadings are almost identical to the pre-formation β VIX loadings.

10 When we compute ex post betas using the monthly change in VIX, m VIX, using a four-factor model similar to equation (6) (except using m VIX in place of FVIX), there is less dispersion in the post-formation m VIX betas, ranging from −2.46 for portfolio 1 to 0.76 to portfolio 5, compared to the ex post β FVIX loadings.

Cross-Section of Volatility and Expected Returns 273 Table II Factor Correlations The table reports correlations of first differences in VIX, FVIX, and STR with various factors. The variable VIX ( m VIX) represents the daily (monthly) change in the VIX index, and FVIX is the mimicking aggregate volatility risk factor. The factor STR is constructed by Coval and Shumway (2001) from the returns of zero-beta straddle positions. The factors MKT, SMB, HML are the Fama and French (1993) factors, the momentum factor UMD is constructed by Kenneth French, and ´ LIQ is the Pastor and Stambaugh (2003) liquidity factor. The sample period is January 1986 to December 2000, except for correlations involving STR, which are computed over the sample period January 1986 to December 1995.

frequency, with a correlation of 0.91. At the monthly frequency, the correlation between FVIX and m VIX is lower, at 0.70. The factors FVIX and STR have a high correlation of 0.83, which indicates that FVIX, formed from stock returns, behaves like the STR factor constructed from option returns. Hence, FVIX captures option-like behavior in the cross-section of stocks. The factor FVIX is negatively contemporaneously correlated with the market return (−0.66), ref lecting the fact that when volatility increases, market returns are low. The correlations of FVIX with SMB and HML are −0.14 and 0.26, respectively. The correlation between FVIX and UMD, a factor capturing momentum returns, is also low at −0.25.

In contrast, there is a strong negative correlation between FVIX and the Pastor and Stambaugh (2003) liquidity factor, LIQ, at −0.40. The LIQ factor ´ decreases in times of low liquidity, which tend to also be periods of high volatility. One example of a period of low liquidity with high volatility is the 1987 crash ´ (see, among others, Jones (2003) and Pastor and Stambaugh (2003)). However, the correlation between FVIX and LIQ is far from −1, indicating that volatility risk and liquidity risk may be separate effects, and may be separately priced.

In the next section, we conduct a series of robustness checks designed to disentangle the effects of aggregate volatility risk from other factors, including liquidity risk.

C. Robustness In this section, we conduct a series of robustness checks in which we specify different models for the conditional mean of VIX, we use windows of different estimation periods to form the β VIX portfolios, and we control for potential 274 The Journal of Finance cross-sectional pricing effects due to book-to-market, size, liquidity, volume, and momentum factor loadings or characteristics.

C.1. Robustness to Different Conditional Means of VIX We first investigate the robustness of our results to the method measuring innovations in VIX. We use the change in VIX at a daily frequency to measure the innovation in volatility because VIX is a highly serially correlated series.