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

280 The Journal of Finance We use the 25βMKT × β VIX base assets to estimate factor premiums in equation (7) following the two-step procedure of Fama–MacBeth (1973). In the first stage, betas are estimated using the full sample. In the second stage, we use cross-sectional regressions to estimate the factor premia. We are especially interested in ex post factor loadings on the FVIX aggregate volatility factor, and the price of risk of FVIX. Panel A of Table V reports the results. In addition to the standard Fama and French (1993) factors MKT, SMB, and HML, we ´ include the momentum factor UMD and Pastor and Stambaugh’s (2003) nontraded liquidity factor, LIQ. We estimate the cross-sectional risk premium for FVIX together with the Fama–French model in Regression I. In Regression II, we check robustness of our results by using Coval and Shumway’s (2001) STR option factor. Regressions III and IV also include the additional regressors UMD and LIQ.

In general, Panel A shows that the premiums of the standard factors (MKT, SMB, HML) are estimated imprecisely with this set of base assets. The premium on SMB is consistently estimated to be negative because the size strategy performed poorly from the 1980s onward. The value effect also performed poorly during the late 1990s, which accounts for the negative coefficient on HML.

In contrast, the price of volatility risk in Regression I is −0.08% per month, which is statistically significant at the 1% level. Using the Coval and Shumway (2001) STR factor in Regression II, we estimate the cross-sectional price of volatility risk to be −0.19% per month, which is also statistically significant at the 1% level. These results are consistent with the hypothesis that the crosssection of stock returns ref lects exposure to aggregate volatility risk, and the price of market volatility risk is significantly negative.

When we add the UMD and LIQ factors in Regressions III and IV, the estimates of the FVIX coefficient are essentially unchanged. When UMD is added, its coefficient is insignificant, while the coefficient on FVIX barely moves from the −0.080 estimate in Regression I to −0.082. The small effect of adding a momentum control on the FVIX coefficient is consistent with the low correlation between FVIX and UMD in Table II and with the results in Table IV showing that controlling for past returns does not remove the low average returns on stocks with high β FVIX loadings. In the full specification Regression IV, the FVIX coefficient becomes slightly smaller in magnitude at −0.071, but the coefficient remains significant at the 5% level with a robust t-statistic of −2.02.

Moreover, FVIX is the only factor to carry a relatively large absolute t-statistic in the regression, which estimates seven coefficients with only 25 portfolios and 180 time-series observations.

Panel B of Table V reports the first-pass factor loadings on FVIX for each of the 25 base assets from Regression I in Panel A. Panel B confirms that the portfolios formed on past β VIX loadings ref lect exposure to volatility risk measured by FVIX over the full sample. Except for two portfolios (the two lowest β MKT portfolios corresponding to the lowest β VIX quintile), all the FVIX factor loadings increase monotonically from low to high. Examination of the realized FVIX factor loadings demonstrates that the set of base assets, sorted on past β VIX and past β MKT, provides disperse ex post FVIX loadings.

Cross-Section of Volatility and Expected Returns 281 Table V Estimating the Price of Volatility Risk Panel A reports the Fama–MacBeth (1973) factor premiums on 25 portfolios sorted first on β MKT and then on β VIX. MKT is the excess return on the market portfolio, FVIX is the mimicking factor for aggregate volatility innovations, STR is Coval and Shumway’s (2001) zero-beta straddle return, SMB and HML are the Fama–French (1993) size and value factors, UMD is the momentum ´ factor constructed by Kenneth French, and LIQ is the aggregate liquidity measure from Pastor and Stambaugh (2003). In Panel B, we report ex post factor loadings on FVIX, from the regression specification I (Fama–French model plus FVIX). Robust t-statistics that account for the errors-invariables for the first-stage estimation in the factor loadings are reported in square brackets. The sample period is from January 1986 to December 2000, except for the Fama–MacBeth regressions with STR, which are from January 1986 to December 1995.

From the estimated price of volatility risk of −0.08% per month in Table V, we revisit Table I to measure how much exposure to aggregate volatility risk accounts for the large spread in the ex post raw returns of −1.04% per month between the quintile portfolios with the lowest and highest past β VIX coefficients. In Table I, the ex post spread in FVIX betas between portfolios 5 and 1 is 8.07 − (−5.06) = 13.13. The estimate of the price of volatility risk is −0.08% per month. Hence, the ex post 13.13 spread in the FVIX factor loadings accounts for

13.13 × −0.080 = −1.05% of the difference in average returns, which is almost exactly the same as the ex post −1.04% per month number for the raw average return difference between quintile 5 and quintile 1. Hence, virtually all of the large difference in average raw returns in the β VIX portfolios can be attributed to exposure to aggregate volatility risk.

E. A Potential Peso Story?

Despite being statistically significant, the estimates of the price of aggregate volatility risk from Table V are small in magnitude (−0.08% per month, or approximately −1% per annum). Given these small estimates, an alternative explanation behind the low returns to high β VIX stocks is a Peso problem. By construction, FVIX does well when the VIX index jumps upward. The small negative mean of FVIX of −0.08% per month may be due to having observed a smaller number of volatility spikes than the market expected ex ante.

Figure 1 shows that there are two episodes of large volatility spikes in our sample coinciding with large negative moves of the market: October 1987 and August 1998. In 1987, VIX volatility jumped from 22% at the beginning of October to 61% at the end of October. At the end of August 1998, the level of VIX reached 48%. The mimicking factor FVIX returned 134% during October 1987, and 33.6% during August 1998. Since the cross-sectional price of risk of FVIX is −0.08% per month, from Table V, the cumulative return over the 180 months in our sample period is −14.4%. A few more large values could easily change our inference. For example, only one more crash, with an FVIX return of the same order of magnitude as the August 1998 episode, would be enough to generate a positive return on the FVIX factor. Using a power law distribution for extreme events, following Gabaix et al. (2003), we would expect to see approximately three large market crashes below three standard deviations during this period. Hence, the ex ante probability of having observed another large spike in volatility during our sample is quite likely.

Hence, given our short sample, we cannot rule out a potential Peso story and, thus, we are not extremely confident about the long-run price of risk of aggregate volatility. Nevertheless, if volatility is a systematic factor as asset pricing theory implies, market volatility risk should be ref lected in the cross-section of stock returns. The cross-sectional Fama–MacBeth (1973) estimates of the negative price of risk of FVIX are consistent with a risk-based story, and our estimates are highly statistically significant with conventional asymptotic distribution theory that is designed to be robust to conditional heteroskedasticity.

However, since we cannot convincingly rule out a Peso problem explanation, Cross-Section of Volatility and Expected Returns 283 our −1% per annum cross-sectional estimate of the price of risk of aggregate volatility must be interpreted with caution.

II. Pricing Idiosyncratic Volatility in the Cross-Section The previous section examines how systematic volatility risk affects crosssectional average returns by focusing on portfolios of stocks sorted by their sensitivities to innovations in aggregate volatility. In this section, we investigate a second set of assets sorted by idiosyncratic volatility defined relative to the FF-3 model. If market volatility risk is a missing component of systematic risk, standard models of systematic risk, such as the CAPM or the FF-3 model, should misprice portfolios sorted by idiosyncratic volatility because these models do not include factor loadings measuring exposure to market volatility risk.

A. Estimating Idiosyncratic Volatility A.1. Deﬁnition of Idiosyncratic Volatility Given the failure of the CAPM to explain cross-sectional returns and the ubiquity of the FF-3 model in empirical financial applications, we concentrate on idiosyncratic volatility measured relative to the FF-3 model

A.2. A Trading Strategy To examine trading strategies based on idiosyncratic volatility, we describe portfolio formation strategies based on an estimation period of L months, a waiting period of M months, and a holding period of N months. We describe an L/M/N strategy as follows. At month t, we compute idiosyncratic volatilities from the regression (8) on daily data over an L-month period from month t − L − M to month t − M. At time t, we construct value-weighted portfolios based on these idiosyncratic volatilities and hold these portfolios for N months. We concentrate most of our analysis on the 1/0/1 strategy, in which we simply sort stocks into quintile portfolios based on their level of idiosyncratic volatility computed using daily returns over the past month, and we hold these valueweighted portfolios for 1 month. The portfolios are rebalanced each month. We also examine the robustness of our results to various choices of L, M, and N.

The construction of the L/M/N portfolios for L 1 and N 1 follows Jegadeesh and Titman (1993), except our portfolios are value weighted. For example, to construct the 12/1/12 quintile portfolios, each month we construct a value-weighted portfolio based on idiosyncratic volatility computed from daily 284 The Journal of Finance data over the 12 months of returns ending 1 month prior to the formation date.

Similarly, we form a value-weighted portfolio based on 12 months of returns ending 2 months prior, 3 months prior, and so on up to 12 months prior. Each of these portfolios is value weighted. We then take the simple average of these 12 portfolios. Hence, each quintile portfolio changes 1/12th of its composition each month, where each 1/12th part of the portfolio consists of a value-weighted portfolio. The first (fifth) quintile portfolio consists of 1/12th of the lowest valueweighted (highest) idiosyncratic stocks from 1 month ago, 1/12th of the valueweighted lowest (highest) idiosyncratic stocks from 2 months ago, etc.

B. Patterns in Average Returns for Idiosyncratic Volatility Table VI reports average returns of portfolios sorted on total volatility, with no controls for systematic risk, in Panel A and of portfolios sorted on idiosyncratic volatility in Panel B.12 We use a 1/0/1 strategy in both cases. Panel A shows that average returns increase from 1.06% per month going from quintile 1 (low total volatility stocks) to 1.22% per month for quintile 3. Then, average returns drop precipitously. Quintile 5, which contains stocks with the highest total volatility, has an average total return of only 0.09% per month. The FF-3 alpha for quintile 5, reported in the last column, is −1.16% per month, which is highly statistically significant. The difference in the FF-3 alphas between portfolio 5 and portfolio 1 is −1.19% per month, with a robust t-statistic of −5.92.

We obtain similar patterns in Panel B, where the portfolios are sorted on idiosyncratic volatility. The difference in raw average returns between quintile portfolios 5 and 1 is −1.06% per month. The FF-3 model is clearly unable to price these portfolios since the difference in the FF-3 alphas between portfolio 5 and portfolio 1 is −1.31% per month, with a t-statistic of −7.00. The size and book-to-market ratios of the quintile portfolios sorted by idiosyncratic volatility also display distinct patterns. Stocks with low (high) idiosyncratic volatility are generally large (small) stocks with low (high) book-to-market ratios. The risk adjustment of the FF-3 model predicts that quintile 5 stocks should have high, not low, average returns.

The findings in Table VI are provocative, but there are several concerns raised by the anomalously low returns of quintile 5. For example, although quintile 5 contains 20% of the stocks sorted by idiosyncratic volatility, quintile 5 is only a small proportion of the value of the market (only 1.9% on average). Are these patterns repeated if we only consider large stocks, or only stocks traded on the NYSE? The next section examines these questions. We also examine whether the phenomena persist if we control for a large number of cross-sectional effects that the literature has identified either as potential risk factors or anomalies.

In particular, we control for size, book-to-market, leverage, liquidity, volume, 12 If we compute idiosyncratic volatility relative to the CAPM, we obtain almost identical results to Panel B of Table VI. Each quintile portfolio of idiosyncratic volatility relative to the CAPM has a correlation of above 99% with its corresponding quintile counterpart when idiosyncratic volatility is computed relative to the FF-3 model.