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

Cross-Section of Volatility and Expected Returns 285 Table VI Portfolios Sorted by Volatility We form value-weighted quintile portfolios every month by sorting stocks based on total volatility and idiosyncratic volatility relative to the Fama–French (1993) model. Portfolios are formed every month, based on volatility computed using daily data over the previous month. Portfolio 1 (5) is the portfolio of stocks with the lowest (highest) volatilities. 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 Fama–French (1993) three-factor model. Robust Newey–West (1987) t-statistics are reported in square brackets. Robust joint tests for the alphas equal to zero are all less than 1% for all cases. The sample period is July 1963 to December 2000.

1 1.06 3.71 41.7% 4.66 0.88 0.14 0.03 [1.84] [0.53] 2 1.15 4.48 33.7% 4.70 0.81 0.13 0.08 [2.14] [1.41] 3 1.22 5.63 15.5% 4.10 0.82 0.07 0.12 [0.72] [1.55] −0.28 −0.17 4 0.99 7.15 6.7% 3.47 0.86 [−1.73] [−1.42] −1.21 −1.16 5 0.09 8.30 2.4% 2.57 1.08 [−5.07] [−6.85] −0.97 −1.35 −1.19 5-1 [−2.86] [−4.62] [−5.92] Panel B: Portfolios Sorted by Idiosyncratic Volatility Relative to FF-3 1 1.04 3.83 53.5% 4.86 0.85 0.11 0.04 [1.57] [0.99] 2 1.16 4.74 27.4% 4.72 0.80 0.11 0.09 [1.98] [1.51] 3 1.20 5.85 11.9% 4.07 0.82 0.04 0.08 [0.37] [1.04] −0.38 −0.32 4 0.87 7.13 5.2% 3.42 0.87 [−2.32] [−3.15] −0.02 −1.27 −1.27 5 8.16 1.9% 2.52 1.10 [−5.09] [−7.68] −1.06 −1.38 −1.31 5-1 [−3.10] [−4.56] [−7.00] turnover, bid–ask spreads, coskewness, dispersion in analysts’ forecasts, and momentum effects.

C. Controlling for Various Cross-Sectional Effects Table VII examines the robustness of our results with the 1/0/1 idiosyncratic volatility portfolio formation strategy to various cross-sectional risk factors. The 286 The Journal of Finance Table VII Alphas of Portfolios Sorted on Idiosyncratic Volatility 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 for idiosyncratic volatility computed relative to FF-3, but control for various effects. The column “5-1” refers to the difference in FF-3 alphas between portfolio 5 and portfolio 1. In the panel labeled “NYSE Stocks Only,” we sort stocks into quintile portfolios based on their idiosyncratic volatility, relative to the FF-3 model, using only NYSE stocks. We use daily data over the previous month and rebalance monthly. In the panel labeled “Size Quintiles,” each month we first sort stocks into five quintiles on the basis of size. Then, within each size quintile, we sort stocks into five portfolios sorted by idiosyncratic volatility. In the panels controlling for size, liquidity volume, and momentum, we perform a double sort. Each month, we first sort stocks based on the first characteristic (size, book-to-market, leverage, liquidity, volume, turnover, bid–ask spreads, or dispersion of analysts’ forecasts) and then, within each quintile we sort stocks based on 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 the characteristic. Liquidity represents the Pastor and Stambaugh (2003) historical liquidity beta, leverage is defined as the ratio of total book value of assets to book value of equity, volume represents average dollar volume over the previous month, turnover represents volume divided by the total number of shares outstanding over the past month, and the bid–ask spread control represents the average daily bid–ask spread over the previous month. The coskewness measure is computed following Harvey and Siddique (2000) and the dispersion of analysts’ forecasts is computed by Diether et al. (2002).

The sample period is July 1963 to December 2000 for all controls with the exceptions of liquidity (February 1968 to December 2000), the dispersion of analysts’ forecasts (February 1983 to December 2000), and the control for aggregate volatility risk (January 1986 to December 2000). All portfolios are value weighted.

table reports FF-3 alphas, the difference in FF-3 alphas between the quintile portfolios with the highest and lowest idiosyncratic volatilities, together with t-statistics to test their statistical significance.13 All the portfolios formed on idiosyncratic volatility remain value weighted.

C.1. Using Only NYSE Stocks We examine the interaction of the idiosyncratic volatility effect with firm size in two ways. First, we rank stocks based on idiosyncratic volatility using only NYSE stocks. Excluding NASDAQ and AMEX has little effect on our results. The highest quintile of idiosyncratic volatility stocks has an FF-3 alpha of −0.60% per month. The 5-1 difference in FF-3 alphas is still large in magnitude, at −0.66% per month, with a t-statistic of −4.85. While restricting the universe of stocks to only the NYSE mitigates the concern that the idiosyncratic volatility effect is concentrated among small stocks, it does not completely remove this concern because the NYSE universe still contains small stocks.

C.2. Controlling for Size Our second examination of the interaction of idiosyncratic volatility and size uses all firms. We control for size by first forming quintile portfolios ranked on market capitalization. Then, within each size quintile, we sort stocks into quintile portfolios ranked on idiosyncratic volatility. Thus, within each size quintile, quintile 5 contains the stocks with the highest idiosyncratic volatility.

The second panel of Table VII shows that in each size quintile, the highest idiosyncratic volatility quintile has a dramatically lower FF-3 alpha than the other quintiles. The effect is not most pronounced among the smallest stocks.

Rather, quintiles 2-4 have the largest 5-1 differences in FF-3 alphas, at −1.91%, −1.61%, and −0.86% per month, respectively. The average market capitalization of quintiles 2-4 is, on average, 21% of the market. The t-statistics of these alphas are all above 4.5 in absolute magnitude. In contrast, the 5-1 alphas for the smallest and largest quintiles are actually statistically insignificant at the 5% level. Hence, it is not small stocks that are driving these results.

The row labeled “Controlling for Size” averages across the five size quintiles to produce quintile portfolios with dispersion in idiosyncratic volatility, but which contain all sizes of firms. After controlling for size, the 5-1 difference in FF-3 alphas is still −1.04% per month. Thus, market capitalization does not explain the low returns to high idiosyncratic volatility stocks.

In the remainder of Table VII, we repeat the explicit double-sort characteristic controls, replacing size with other stock characteristics. We first form portfolios based on a particular characteristic, then we sort on idiosyncratic volatility, and finally we average across the characteristic portfolios to create portfolios 13 We emphasize that the difference in mean raw returns between quintile 5 and 1 portfolios is very similar to the difference in the FF-3 alphas, but we focus on FF-3 alphas as they control for the standard set of systematic factors.

288 The Journal of Finance that have dispersion in idiosyncratic volatility but contain all aspects of the characteristic.

C.3. Controlling for Book-to-Market Ratios It is generally thought that high book-to-market firms have high average returns. Thus, in order for the book-to-market effect to be an explanation of the idiosyncratic volatility effect, the high idiosyncratic volatility portfolios must be primarily composed of growth stocks that have lower average returns than value stocks. The row labeled “Controlling for Book-to-Market” shows that this is not the case. When we control for book-to-market ratios, stocks with the lowest idiosyncratic volatility have high FF-3 alphas, and the 5-1 difference in FF-3 alphas is −0.80% per month, with a t-statistic of −2.90.

C.4. Controlling for Leverage Leverage increases expected equity returns, holding asset volatility and asset expected returns constant. Asset volatility also prevents firms from increasing leverage. Hence, firms with high idiosyncratic volatility could have high asset volatility but relatively low equity returns because of low leverage. The next line of Table VII shows that leverage cannot be an explanation of the idiosyncratic volatility effect. We measure leverage as the ratio of total book value of assets to book value of equity. After controlling for leverage, the difference between the 5-1 alphas is −1.23% per month, with a t-statistic of −7.61.

C.5. Controlling for Liquidity Risk ´ Pastor and Stambaugh (2003) argue that liquidity is a systematic risk. If liquidity is to explain the idiosyncratic volatility effect, high idiosyncratic volatility stocks must have low liquidity betas, giving them low returns. We check this ´ explanation by using the historical Pastor–Stambaugh liquidity betas to measure exposure to liquidity risk. Controlling for liquidity does not remove the low average returns of high idiosyncratic volatility stocks. The 5-1 difference in FF-3 alphas remains large at −1.08% per month, with a t-statistic of −7.98.

C.6. Controlling for Volume Gervais et al. (2001) find that stocks with higher volume have higher returns.

Perhaps stocks with high idiosyncratic volatility are merely stocks with low trading volume? When we control for trading volume over the past month, the 5-1 difference in alphas is −1.22% per month, with a t-statistic of −8.04. Hence, the low returns on high idiosyncratic volatility stocks are robust to controlling for volume effects.

Cross-Section of Volatility and Expected Returns 289 C.7. Controlling for Turnover Our next control is turnover, measured as trading volume divided by the total number of shares outstanding over the previous month. Turnover is a noisy proxy for liquidity. Table VII shows that the low alphas on high idiosyncratic volatility stocks are robust to controlling for turnover. The 5-1 difference in FFalphas is −1.19% per month, and it is highly significant with a t-statistic of −8.04. Examination of the individual turnover quintiles (not reported) indicates that the 5-1 differences in alphas are most pronounced in the quintile portfolio with the highest, not the lowest, turnover.

C.8. Controlling for Bid-Ask Spreads An alternative liquidity control is the bid–ask spread, which we measure as the average daily bid–ask spread over the previous month for each stock. In order for bid–ask spreads to be an explanation, high idiosyncratic volatility stocks must have low bid–ask spreads and corresponding low returns. Controlling for bid–ask spreads does little to remove the effect. The FF-3 alpha of the highest idiosyncratic volatility portfolio is −1.26%, while the 5-1 difference in alphas is −1.19% and remains highly statistically significant with a t-statistic of −6.95.

C.9. Controlling for Coskewness Risk Harvey and Siddique (2000) find that stocks with more negative coskewness have higher returns. Stocks with high idiosyncratic volatility may have positive coskewness, giving them low returns. Computing coskewness following Harvey and Siddique (2000), we find that exposure to coskewness risk is not an explanation. The FF-3 alpha for the 5-1 portfolio is −1.38% per month, with a t-statistic of −5.02.

C.10. Controlling for Dispersion in Analysts’ Forecasts Diether, Malloy, and Scherbina (2002) provide evidence that stocks with higher dispersion in analysts’ earnings forecasts have lower average returns than stocks with low dispersion of analysts’ forecasts. They argue that dispersion in analysts’ forecasts measures differences of opinion among investors.

Miller (1977) shows that if there are large differences in stock valuations and short sale constraints, equity prices tend to ref lect the view of the more optimistic agents, which leads to low future returns for stocks with large dispersion in analysts’ forecasts.

If stocks with high dispersion in analysts’ forecasts tend to be more volatile stocks, then we may be finding a similar anomaly to Diether et al. (2002). Over Diether et al.’s sample period, 1983–2000, we test this hypothesis by performing a characteristic control for the dispersion of analysts’ forecasts. We take the quintile portfolios of stocks sorted on increasing dispersion of analysts’ forecasts (Table VI of Diether et al. (2002, p. 2128)) and within each quintile sort stocks 290 The Journal of Finance on idiosyncratic volatility. Note that this universe of stocks contains mostly large firms, where the idiosyncratic volatility effect is weaker, because multiple analysts usually do not make forecasts for small firms.

The last two lines of Table VII present the results for averaging the idiosyncratic volatility portfolios across the forecast dispersion quintiles. The 5-1 difference in alphas is still −0.39% per month, with a robust t-statistic of −2.09.

While the shorter sample period may reduce power, the dispersion of analysts’ forecasts reduces the noncontrolled 5-1 alpha considerably. However, dispersion in analysts’ forecasts cannot account for all of the low returns to stocks with high idiosyncratic volatility.14 D. A Detailed Look at Momentum Hong, Lim, and Stein (2000) argue that the momentum effect documented by Jegadeesh and Titman (1993) is asymmetric and has a stronger negative effect on declining stocks than a positive effect on rising stocks. A potential explanation behind the idiosyncratic volatility results is that stocks with very low returns have very high volatility. Of course, stocks that are past winners also have very high volatility, but loser stocks could be overrepresented in the high idiosyncratic volatility quintile.