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

The main prediction from the factor model setting of equation (1) that we examine is that stocks with different loadings on aggregate volatility risk have different average returns.4 However, the true model in equation (1) is infeasible 4 While an I-CAPM implies joint time-series as well as cross-sectional predictability, we do not examine time-series predictability of asset returns by systematic volatility. Time-varying volatility risk generates intertemporal hedging demands in partial equilibrium asset allocation problems. In a partial equilibrium setting, Liu (2001) and Chacko and Viceira (2003) examine how volatility risk affects the portfolio allocation of stocks and risk-free assets, while Liu and Pan (2003) show how investors can optimally exploit the variation in volatility with options. Guo and Whitelaw (2003) examine the intertemporal components of time-varying systematic volatility in a Campbell (1993,

1996) equilibrium I-CAPM.

264 The Journal of Finance to examine because the true set of factors is unknown and the true conditional factor loadings are unobservable. Hence, we do not attempt to directly use equation (1) in our empirical work. Instead, we simplify the full model of equation (1), which we now detail.

B. The Empirical Framework To investigate how aggregate volatility risk is priced in the cross-section of equity returns we make the following simplifying assumptions to the full specification in equation (1). First, we use observable proxies for the market factor and the factor representing aggregate volatility risk. We use the CRSP valueweighted market index to proxy for the market factor. To proxy innovations in aggregate volatility, (vt+1 − γv,t ), we use changes in the VIX index from the Chicago Board Options Exchange (CBOE).5 Second, we reduce the number of factors in equation (1) to just the market factor and the proxy for aggregate volatility risk. Finally, to capture the conditional nature of the true model, we use short intervals—1 month of daily data—to take into account possible time variation of the factor loadings. We discuss each of these simplifications in turn.

B.1. Innovations in the VIX Index The VIX index is constructed so that it represents the implied volatility of a synthetic at-the-money option contract on the S&P100 index that has a maturity of 1 month. It is constructed from eight S&P100 index puts and calls and takes into account the American features of the option contracts, discrete cash dividends, and microstructure frictions such as bid–ask spreads (see Whaley (2000) for further details).6 Figure 1 plots the VIX index from January 1986 to December 2000. The mean level of the daily VIX series is 20.5%, and its standard deviation is 7.85%.

Because the VIX index is highly serially correlated with a first-order autocorrelation of 0.94, we measure daily innovations in aggregate volatility by using daily changes in VIX, which we denote as VIX. Daily first differences in VIX have an effective mean of zero (less than 0.0001), a standard deviation of 5 In previous versions of this paper, we also consider: Sample volatility, following French et al. (1987); a range-based estimate, following Alizadeh, Brandt, and Diebold (2002); and a highfrequency estimator of volatility from Andersen, Bollerslev, and Diebold (2003). Using these measures to proxy for innovations in aggregate volatility produces little spread in cross-sectional average returns. These tables are available upon request.

6 On September 22, 2003, the CBOE implemented a new formula and methodology to construct its volatility index. The new index is based on the S&P500 (rather than the S&P100) and takes into account a broader range of strike prices rather than using only at-the-money option contracts.

The CBOE now uses VIX to refer to this new index. We use the old index (denoted by the ticker VXO). We do not use the new index because it has been constructed by backfilling only to 1990, whereas the VXO is available in real time from 1986. The CBOE continues to make both volatility indices available. The correlation between the new and the old CBOE volatility series is 98% from 1990 to 2000, but the series that we use has a slightly broader range than the new CBOE volatility series.

Cross-Section of Volatility and Expected Returns 265 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1986 1988 1990 1992 1994 1996 1998 2000 Figure 1. Plot of VIX. The figure shows the VIX index plotted at a daily frequency. The sample period is January 1986 to December 2000.

2.65%, and negligible serial correlation (the first-order autocorrelation of VIX is −0.0001). As part of our robustness checks in Section I.C, we also measure innovations in VIX by specifying a stationary time-series model for the conditional mean of VIX and find our results to be similar to those using simple first differences. While VIX appears to be an ideal proxy for innovations in volatility risk because the VIX index is representative of traded option securities whose prices directly ref lect volatility risk, there are two main caveats with respect to using VIX to represent observable market volatility.

The first concern is that the VIX index is the implied volatility from the Black–Scholes (1973) model, and we know that the Black–Scholes model is an approximation. If the true stochastic environment is characterized by stochastic volatility and jumps, VIX will ref lect total quadratic variation in both diffusion and jump components (see, for example, Pan (2002)). Although Bates (2000) argues that implied volatilities computed taking into account jump risk are very close to original Black–Scholes implied volatilities, jump risk may be priced differently from volatility risk. Our analysis does not separate jump risk from diffusion risk, so our aggregate volatility risk may include jump risk components.

266 The Journal of Finance A more serious reservation about the VIX index is that VIX combines both stochastic volatility and the stochastic volatility risk premium. Only if the risk premium is zero or constant would VIX be a pure proxy for the innovation in aggregate volatility. Decomposing VIX into the true innovation in volatility and the volatility risk premium can only be done by writing down a formal model. The form of the risk premium depends on the parameterization of the price of volatility risk, the number of factors, and the evolution of those factors.

Each different model specification implies a different risk premium. For example, many stochastic volatility option pricing models assume that the volatility risk premium can be parameterized as a linear function of volatility (see, for example, Chernov and Ghysels (2000), Benzoni (2002), and Jones (2003)). This may or may not be a good approximation to the true price of risk. Rather than imposing a structural form, we use an unadulterated VIX series. An advantage of this approach is that our analysis is simple to replicate.

B.2. The Pre-Formation Regression Our goal is to test whether stocks with different sensitivities to aggregate volatility innovations (proxied by VIX) have different average returns. To measure the sensitivity to aggregate volatility innovations, we reduce the number of factors in the full specification in equation (1) to two, namely, the market factor and VIX. A two-factor pricing kernel with the market return and stochastic volatility as factors is also the standard setup commonly assumed by many stochastic option pricing studies (see, for example, Heston (1993)). Hence, the empirical model that we examine is

on market risk and aggregate volatility risk, respectively.

Previous empirical studies suggest that there are other cross-sectional factors that have explanatory power for the cross-section of returns, such as the size and value factors of the Fama and French (1993) three-factor model (hereafter FF-3). We do not directly model these effects in equation (3), because controlling for other factors in constructing portfolios based on equation (3) may add a lot of noise. Although we keep the number of regressors in our pre-formation portfolio regressions to a minimum, we are careful to ensure that we control for the FFfactors and other cross-sectional factors in assessing how volatility risk is priced using post-formation regression tests.

We construct a set of assets that are sufficiently disperse in exposure to aggregate volatility innovations by sorting firms on VIX loadings over the past month using the regression (3) with daily data. We run the regression for all stocks on AMEX, NASDAQ, and the NYSE, with more than 17 daily observations. In a setting in which coefficients potentially vary over time, a 1-month window with daily data is a natural compromise between estimating Cross-Section of Volatility and Expected Returns 267 coefficients with a reasonable degree of precision and pinning down conditional ´ coefficients in an environment with time-varying factor loadings. Pastor and Stambaugh (2003), among others, also use daily data with a 1-month window in similar settings. At the end of each month, we sort stocks into quintiles, based on the value of the realized β VIX coefficients over the past month. Firms in quintile 1 have the lowest coefficients, while firms in quintile 5 have the highest β VIX loadings. Within each quintile portfolio, we value weight the stocks. We link the returns across time to form one series of post-ranking returns for each quintile portfolio.

Table I reports various summary statistics for quintile portfolios sorted by past β VIX over the previous month using equation (3). The first two columns report the mean and standard deviation of monthly total, not excess, simple returns. In the first column under the heading “Factor Loadings,” we report the pre-formation β VIX coefficients, which are computed at the beginning of each month for each portfolio and are value weighted. The column reports the timeseries average of the pre-formation β VIX loadings across the whole sample.

By construction, since the portfolios are formed by ranking on past β VIX, the pre-formation β VIX loadings monotonically increase from −2.09 for portfolio 1 to 2.18 for portfolio 5.

The columns labeled “CAPM Alpha” and “FF-3 Alpha” report the time-series alphas of these portfolios relative to the CAPM and to the FF-3 model, respectively. Consistent with the negative price of systematic volatility risk found by the option pricing studies, we see lower average raw returns, CAPM alphas, and FF-3 alphas with higher past loadings of β VIX. All the differences between quintile portfolios 5 and 1 are significant at the 1% level, and a joint test for the alphas equal to zero rejects at the 5% level for both the CAPM and the FF-3 model. In particular, the 5-1 spread in average returns between the quintile portfolios with the highest and lowest β VIX coefficients is −1.04% per month. Controlling for the MKT factor exacerbates the 5-1 spread to −1.15% per month, while controlling for the FF-3 model decreases the 5-1 spread to −0.83% per month.

Portfolios Sorted by Exposure to Aggregate Volatility Shocks We form value-weighted quintile portfolios every month by regressing excess individual stock returns on VIX, controlling for the MKT factor as in equation (3), using daily data over the previous month. Stocks are sorted into quintiles based on the coefficient β VIX from lowest (quintile 1) to highest (quintile 5). The statistics in the columns labeled Mean and Std. Dev. are measured in monthly percentage terms and apply to total, not excess, simple returns. Size reports the average log market capitalization for firms within the portfolio and B/M reports the average book-to-market ratio.

The row “5-1” refers to the difference in monthly returns between portfolio 5 and portfolio 1. The Alpha columns report Jensen’s alpha with respect to the CAPM or the Fama–French (1993) three-factor model. The pre-formation betas refer to the value-weighted β VIX or β FVIX within each quintile portfolio at the start of the month. We report the pre-formation β VIX and β FVIX averaged across the whole sample. The second to last column reports the β VIX loading computed over the next month with daily data. The column reports the next month β VIX loadings averaged across months. 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 that mimics aggregate volatility risk, following the regression in equation (6). The row labeled “Joint test p-value” reports a Gibbons, Ross and Shanken (1989) test for the alphas equal to 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.

post-ranking factor loadings that are computed over the full sample period.

While the β VIX loadings show very strong patterns of future returns, they represent past covariation with innovations in market volatility. We must show that the portfolios in Table I also exhibit high loadings with volatility risk over the same period used to compute the alphas.

To construct our portfolios, we take VIX to proxy for the innovation in aggregate volatility at a daily frequency. However, at the standard monthly frequency, which is the frequency of the ex post returns for the alphas reported in Table I, using the change in VIX is a poor approximation for innovations in aggregate volatility. This is because at lower frequencies, the effect of the conditional mean of VIX plays an important role in determining the unanticipated change in VIX. In contrast, the high persistence of the VIX series at a daily frequency means that the first difference of VIX is a suitable proxy for the innovation in aggregate volatility. Hence, we should not measure ex post exposure to aggregate volatility risk by looking at how the portfolios in Table I correlate ex post with monthly changes in VIX.