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

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A final possibility is that the idiosyncratic volatility effect is concentrated during the most volatile periods in the market. To test for this possibility, we compute FF-3 alphas of the difference between quintile portfolios 5 and 1 conditioning on periods with the lowest or highest 20% of absolute moves of the market return. These are ex post periods of low or high market volatility. During stable (volatile) periods, the difference in the FF-3 alphas of the fifth and first quintile portfolios is −1.71% (−0.89%) per month. Both the differences in alphas during stable and volatile periods are significant at the 5% level. The 296 The Journal of Finance most negative returns of the high idiosyncratic volatility strategy are earned during periods when the market is stable. Hence, the idiosyncratic volatility effect is remarkably robust across different subsamples.

III. Conclusion

Multifactor models of risk predict that aggregate volatility should be a crosssectional risk factor. Past research in option pricing has found a negative price of risk for systematic volatility. Consistent with this intuition, we find that stocks with high past exposure to innovations in aggregate market volatility earn low future average returns. We use changes in the VIX index constructed by the Chicago Board Options Exchange to proxy for innovations in aggregate volatility.

To find the component of market volatility innovations that is ref lected in equity returns, we construct a factor to mimic innovations in market volatility following Breeden et al. (1989) and Lamont (2001). We first form portfolios on the basis of their past sensitivity to first differences in the VIX index. Then, we project innovations in VIX onto these portfolios to produce a factor that mimics aggregate volatility risk, which we term FVIX. This portfolio of basis assets is maximally correlated with the realized aggregate volatility innovations. Portfolios constructed by ranking on past betas to first differences in VIX also exhibit strong patterns in post-formation FVIX factor loadings. In particular, the ex post increasing patterns in FVIX factor loadings correspond to decreasing Fama–French (1993) alphas over the same period that the alphas are computed.

We estimate a cross-sectional price of volatility risk of approximately −1% per annum, and this estimate is robust to controlling for size, value, momentum, and liquidity effects. Hence, the decreasing average returns to stocks with high past sensitivities to changes in VIX is consistent with the cross-section of returns pricing aggregate volatility risk with a negative sign. However, despite the statistical significance of the negative volatility risk premium, its small size and our relatively small sample mean that we cannot rule out a potential Peso problem explanation. Since the FVIX portfolio does well during periods of market distress, adding another volatility spike like October 1987 or August 1998 to our sample would change the sign of the price of risk of FVIX from negative to positive. Nevertheless, our estimate of a negative price of risk of aggregate volatility is consistent with a multifactor model or Intertemporal CAPM. In these settings, aggregate volatility risk is priced with a negative sign because risk-averse agents reduce current consumption to increase precautionary savings in the presence of higher uncertainty about future market returns. Our results are also consistent with the estimates of a negative price of risk for aggregate volatility estimated by many option pricing studies.

We also examine the returns of a set of test assets that are sorted by idiosyncratic volatility relative to the Fama–French (1993) model. We uncover a very robust result. Stocks with high idiosyncratic volatility have abysmally low average returns. In particular, the quintile portfolio of stocks with the Cross-Section of Volatility and Expected Returns 297 highest idiosyncratic volatility earns total returns of just −0.02% per month in our sample. These low average returns to stocks with high idiosyncratic volatility cannot be explained by exposures to size, book-to-market, leverage, liquidity, volume, turnover, bid-ask spreads, coskewness, or dispersion in analysts’ forecasts characteristics. The effect also persists in bull and bear markets, NBER recessions and expansions, volatile and stable periods, and is robust to considering different formation and holding periods as long as 1 year. Although we argue that aggregate volatility is a new cross-sectional, systematic factor, exposure to aggregate volatility risk accounts for very little of the anomalous low returns of stocks with high idiosyncratic volatility. Hence, the cross-sectional expected return patterns found by sorting on idiosyncratic volatility present something of a puzzle.

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