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

In Table VIII, we perform a series of robustness tests of the idiosyncratic volatility effect to this possible momentum explanation. In Panel A, we perform 5 × 5 characteristic sorts first over past returns, and then over idiosyncratic volatility. We average over the momentum quintiles to produce quintile portfolios sorted by idiosyncratic risk that control for past returns. We control for momentum over the previous 1 month, 6 months, and 12 months. Table VIII shows that momentum is not driving the results. Controlling for returns over the past month does not remove the very low FF-3 alpha of quintile 5 (−0.59% per month), and the 5-1 difference in alphas is still −0.66% per month, which is statistically significant at the 1% level. When we control for past 6month returns, the FF-3 alpha of the 5-1 portfolio increases in magnitude to −1.10% per month. For past 12-month returns, the 5-1 alpha is even larger in magnitude at −1.22% per month. All these differences are highly statistically significant.

In Panel B, we closely examine the individual 5 × 5 FF-3 alphas of the quintile portfolios sorted on past 12-month returns and idiosyncratic volatility. Note that if we average these portfolios across the past 12-month quintile portfolios, and then compute alphas, we obtain the alphas in the row labeled “Past 12-months” in Panel A of Table VIII. This more detailed view of the 14 We can also reverse the question and ask if the low average returns of stocks with high dispersion of analysts’ forecasts are due to the low returns on stocks with high idiosyncratic volatility by first sorting stocks on idiosyncratic volatility and then by forecast dispersion. Controlling for idiosyncratic volatility, the FF-3 alpha for the quintile portfolio, that is long stocks with the highest forecast dispersion and short stocks in the quintile portfolio with the lowest forecast dispersion, is −0.36% per month, which is insignificant at the 5% level (the robust t-statistic is −1.47).

Cross-Section of Volatility and Expected Returns 291 Table VIII Alphas of Portfolios Sorted on Idiosyncratic Volatility Controlling for Past Returns The table reports Fama and French (1993) alphas, with robust Newey–West (1987) t-statistics in square brackets. All the strategies are 1/0/1 strategies described in Section II.A, but control for past returns. The column “5-1” refers to the difference in FF-3 alphas between portfolio 5 and portfolio

1. In the first three rows labeled “Past 1-month” to “Past 12-months,” we control for the effect of momentum. We first sort all stocks on the basis of past returns, over the appropriate formation period, into quintiles. Then, within each momentum quintile, we sort stocks into five portfolios sorted by idiosyncratic volatility, relative to the FF-3 model. The five idiosyncratic volatility portfolios are then averaged over each of the five characteristic portfolios. Hence, they represent idiosyncratic volatility quintile portfolios controlling for momentum. The second part of the panel lists the FF-3 alphas of idiosyncratic volatility quintile portfolios within each of the past 12-month return quintiles. All portfolios are value weighted. The sample period is July 1963 to December 2000.

interaction between momentum and idiosyncratic volatility reveals several interesting facts.

First, the low returns to high idiosyncratic volatility are most pronounced for loser stocks. The 5-1 differences in alphas range from −2.25% per month for the loser stocks, to −0.48% per month for the winner stocks. Second, the tendency for the momentum effect to be concentrated more among loser, rather than winner, stocks cannot account for all of the low returns to high idiosyncratic volatility stocks. The idiosyncratic volatility effect appears significantly in every past return quintile. Hence, stocks with high idiosyncratic volatility earn low average returns, no matter whether these stocks are losers or winners.

Finally, the momentum effect itself is also asymmetric across the idiosyncratic volatility quintiles. In the first two idiosyncratic volatility quintiles, the 292 The Journal of Finance alphas of losers (winners) are roughly symmetrical. For example, for stocks with the lowest idiosyncratic volatilities, the loser (winner) alpha is −0.41% (0.45%). In the second idiosyncratic volatility quintile, the loser (winner) alpha is −0.83% (0.85%). However, as idiosyncratic volatility becomes very high, the momentum effect becomes highly skewed towards extremely low returns on stocks with high idiosyncratic volatility. Hence, one way to improve the returns to a momentum strategy is to short past losers with high idiosyncratic volatility.

E. Is It Exposure to Aggregate Volatility Risk?

A possible explanation for the large negative returns of high idiosyncratic volatility stocks is that stocks with large idiosyncratic volatilities have large exposure to movements in aggregate volatility. We examine this possibility in Table IX. The first row of Panel A reports the results of quintile sorts on idiosyncratic volatility, controlling for β VIX. This is done by first sorting on β VIX and then on idiosyncratic volatility, and then averaging across the β VIX quintiles. We motivate using past β VIX as a control for aggregate volatility risk because we have shown that stocks with past high β VIX loadings have high future exposure to the FVIX-mimicking volatility factor.

Panel A of Table IX shows that after controlling for aggregate volatility exposure, the 5-1 alpha is −1.19% per month, almost unchanged from the 5-1 quintile idiosyncratic volatility FF-3 alpha of −1.31% in Table VI with no control for systematic volatility exposure. Hence, it seems that β VIX accounts for very little of the low average returns of high idiosyncratic volatility stocks.

Panel B of Table IX reports ex post FVIX factor loadings of the 5 × 5β VIX and idiosyncratic volatility portfolios, where we compute the post-formation FVIX factor loadings using equation (6). We cannot interpret the alphas from this regression, because FVIX is not a tradeable factor, but the FVIX factor loadings give us a picture of how exposure to aggregate volatility risk may account for the spreads in average returns on the idiosyncratic volatility sorted portfolios.

Panel B shows that in the first three β VIX quintiles, we obtain almost monotonically increasing FVIX factor loadings that start with large negative ex post β FVIX loadings for low idiosyncratic volatility portfolios and end with large positive ex post β FVIX loadings. However, for the two highest past β VIX quintiles, the FVIX factor loadings have absolutely no explanatory power. In summary, exposure to aggregate volatility partially explains the puzzling low returns to high idiosyncratic volatility stocks, but only for stocks with very negative and low past loadings to aggregate volatility innovations.

F. Robustness to Different Formation and Holding Periods If risk cannot explain the low returns to high idiosyncratic volatility stocks, are there other explanations? To help disentangle various stories, Table X reports FF-3 alphas of other L/M/N strategies described in Section II.A.

First, we examine possible contemporaneous measurement errors in the 1/0/1 Cross-Section of Volatility and Expected Returns 293 Table IX The Idiosyncratic Volatility Effect Controlling for Aggregate Volatility Risk We control for exposure to aggregate volatility using the β VIX loading at the beginning of the month, computed using daily data over the previous month following equation (3). We first sort all stocks on the basis of β VIX into quintiles. Then, within each β VIX quintile, we sort stocks into five portfolios sorted by idiosyncratic volatility, relative to the FF-3 model. In Panel A, we report FF-3 alphas of these portfolios. We average the five idiosyncratic volatility portfolios over each of the five β VIX portfolios. Hence, these portfolios represent idiosyncratic volatility quintile portfolios controlling for exposure to aggregate volatility risk. The column “5-1” refers to the difference in FF-3 alphas between portfolio 5 and portfolio 1. In Panel B, we report ex post FVIX factor loadings from a regression of each of the 25β VIX × idiosyncratic volatility portfolios onto the Fama–French (1993) model augmented with FVIX as in equation (6). Robust Newey–West (1987) t-statistics are reported in square brackets. All portfolios are value weighted. The sample period is from January 1986 to December 2000.

strategy by setting M = 1. Allowing for a 1-month lag between the measurement of volatility and the formation of the portfolio ensures that the portfolios are formed only with information definitely available at time t. The top row of Table X shows that the 5-1 FF-3 alpha on the 1/1/1 strategy is −0.82% per month, with a t-statistic of −4.63.

A possible behavioral explanation for our results is that higher idiosyncratic volatility does earn higher returns over longer horizons than 1 month, but shortterm overreaction forces returns to be low in the first month. If we hold high idiosyncratic volatility stocks for a long horizon (N = 12 months), we might see a positive relation between past idiosyncratic volatility and future average returns. The second row of Table X shows that this is not the case. For the 1/1/12 294 The Journal of Finance Table X Quintile Portfolios of Idiosyncratic Volatility for L/M/N Strategies The table reports Fama and French (1993) alphas, with robust Newey–West (1987) t-statistics in square brackets. The column “5-1” refers to the difference in FF-3 alphas between portfolio 5 and portfolio 1. We rank stocks into quintile portfolios of idiosyncratic volatility, relative to FF-3, using L/M/N strategies described in Section II.A. At month t, we compute idiosyncratic volatilities from the regression (8) on daily data over an L month period from months 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, following Jegadeesh and Titman (1993), except our portfolios are value weighted. The sample period is July 1963 to December 2000.

−0.18 −0.82 −0.88 1/1/1 0.06 0.04 0.09 [1.47] [0.77] [1.15] [−1.78] [−4.88] [−4.63] −0.02 −0.17 −0.64 −0.67 1/1/12 0.03 0.02 [0.91] [0.43] [−0.37] [−1.79] [−5.27] [−4.71] −0.01 −0.29 −1.08 −1.12 12/1/1 0.04 0.08 [1.15] [1.32] [−0.08] [−2.02] [−5.36] [−5.13] −0.02 −0.35 −0.73 −0.77 12/1/12 0.04 0.04 [1.10] [0.54] [−0.23] [−2.80] [−4.71] [−4.34] strategy, we still see very low FF-3 alphas for quintile 5, and the 5-1 difference in alphas is still −0.67% per month, which is highly significant.

By restricting the formation period to L = 1 month, our previous results may just be capturing various short-term events that affect idiosyncratic volatility.

For example, the portfolio of stocks with high idiosyncratic volatility may be largely composed of stocks that have just made, or are just about to make, earnings announcements. To ensure that we are not capturing specific short-term corporate events, we extend our formation period to L = 12 months. The third row of Table X reports FF-3 alphas for a 12/1/1 strategy. Using one entire year of daily data to compute idiosyncratic volatility does not remove the anomalous high idiosyncratic volatility-low average return pattern: The 5-1 difference in alphas is −1.12% per month. Similarly, the patterns are robust for the 12/1/12 strategy, which has a 5-1 alpha of −0.77% per month.

G. Subsample Analysis Table XI investigates the robustness of the low returns to stocks with high idiosyncratic volatility over different subsamples. First, the effect is observed in every decade from the 1960s to the 1990s. The largest difference in alphas between portfolio 5 and portfolio 1 occurs during the 1980s, with an FF-3 alpha of −2.23% per month, and we observe the smallest magnitude of the FF-3 alpha of the 5-1 portfolio during the 1970s, during which time the FF-3 alpha is −0.77% per month. In every decade, the effect is highly statistically significant.

A possible explanation for the idiosyncratic volatility effect may be asymmetry of return distributions across business cycles. Volatility is asymmetric Cross-Section of Volatility and Expected Returns 295 Table XI The Idiosyncratic Volatility Effect over Different Subsamples The table reports Fama and French (1993) alphas, with robust Newey–West (1987) t-statistics in square brackets. The column “5-1” refers to the difference in FF-3 alphas between portfolio 5 and portfolio 1. We rank stocks into quintile portfolios of idiosyncratic volatility, relative to FF-3, using the 1/0/1 strategy described in Section II.A and examine robustness over different sample periods.

The stable and volatile periods refer to the months with the lowest and highest 20% absolute value of the market return, respectively. The full sample period is July 1963 to December 2000.

(and larger with downward moves), so stocks with high idiosyncratic volatility may have normal average returns during expansionary markets, and their low returns may mainly occur during bear market periods, or recessions. We may have observed too many recessions in the sample relative to what agents expected ex ante. We check this hypothesis by examining the returns of high idiosyncratic volatility stocks conditioning on NBER expansions and recessions.

During NBER expansions (recessions), the FF-3 alpha of the 5-1 portfolio is −1.25% (−1.79%). Both the expansion and recession differences in FF-3 alphas are significant at the 1% level. There are more negative returns to high idiosyncratic volatility stocks during recessions, but the fact that the t-statistic in NBER expansions is −6.55 shows that the low returns from high idiosyncratic volatility also thrive during expansions.