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

THE JOURNAL OF FINANCE • VOL. LXI, NO. 1 • FEBRUARY 2006

The Cross-Section of Volatility

and Expected Returns

ANDREW ANG, ROBERT J. HODRICK, YUHANG XING, and XIAOYAN ZHANG∗

## ABSTRACT

We examine the pricing of aggregate volatility risk in the cross-section of stock returns.

Consistent with theory, we find that stocks with high sensitivities to innovations in

aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, Journal of Financial Economics 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility.

IT IS WELL KNOWN THAT THE VOLATILITY OF STOCK RETURNS varies over time. While con- siderable research has examined the time-series relation between the volatility of the market and the expected return on the market (see, among others, Camp- bell and Hentschel (1992) and Glosten, Jagannathan, and Runkle (1993)), the question of how aggregate volatility affects the cross-section of expected stock returns has received less attention. Time-varying market volatility induces changes in the investment opportunity set by changing the expectation of fu- ture market returns, or by changing the risk-return trade-off. If the volatility of the market return is a systematic risk factor, the arbitrage pricing theory or a factor model predicts that aggregate volatility should also be priced in the cross-section of stocks. Hence, stocks with different sensitivities to innovations in aggregate volatility should have different expected returns.

The first goal of this paper is to provide a systematic investigation of how the stochastic volatility of the market is priced in the cross-section of expected stock returns. We want to both determine whether the volatility of the market ∗ Ang is with Columbia University and NBER. Hodrick is with Columbia University and NBER.

Yuhang Xing is at Rice University. Xiaoyan Zhang is at Cornell University. We thank Joe Chen, Mike Chernov, Miguel Ferreira, Jeff Fleming, Chris Lamoureux, Jun Liu, Laurie Hodrick, Paul Hribar, Jun Pan, Matt Rhodes-Kropf, Steve Ross, David Weinbaum, and Lu Zhang for helpful discussions.

We also received valuable comments from seminar participants at an NBER Asset Pricing meeting, Campbell and Company, Columbia University, Cornell University, Hong Kong University, Rice University, UCLA, and the University of Rochester. We thank Tim Bollerslev, Joe Chen, Miguel Ferreira, Kenneth French, Anna Scherbina, and Tyler Shumway for kindly providing data. We especially thank an anonymous referee and Rob Stambaugh, the editor, for helpful suggestions that greatly improved the paper. Andrew Ang and Bob Hodrick both acknowledge support from the National Science Foundation.

is a priced risk factor and estimate the price of aggregate volatility risk. Many option studies have estimated a negative price of risk for market volatility using options on an aggregate market index or options on individual stocks.1 Using the cross-section of stock returns, rather than options on the market, allows us to create portfolios of stocks that have different sensitivities to innovations in market volatility. If the price of aggregate volatility risk is negative, stocks with large, positive sensitivities to volatility risk should have low average returns.

Using the cross-section of stock returns also allows us to easily control for a battery of cross-sectional effects, such as the size and value factors of Fama and French (1993), the momentum effect of Jegadeesh and Titman (1993), and ´ the effect of liquidity risk documented by Pastor and Stambaugh (2003). Option pricing studies do not control for these cross-sectional risk factors.

We find that innovations in aggregate volatility carry a statistically significant negative price of risk of approximately −1% per annum. Economic theory provides several reasons why the price of risk of innovations in market volatility should be negative. For example, Campbell (1993, 1996) and Chen (2002) show that investors want to hedge against changes in market volatility, because increasing volatility represents a deterioration in investment opportunities. Risk-averse agents demand stocks that hedge against this risk. Periods of high volatility also tend to coincide with downward market movements (see French, Schwert, and Stambaugh (1987) and Campbell and Hentschel (1992)).

As Bakshi and Kapadia (2003) comment, assets with high sensitivities to market volatility risk provide hedges against market downside risk. The higher demand for assets with high systematic volatility loadings increases their price and lowers their average return. Finally, stocks that do badly when volatility increases tend to have negatively skewed returns over intermediate horizons, while stocks that do well when volatility rises tend to have positively skewed returns. If investors have preferences over coskewness (see Harvey and Siddique (2000)), stocks that have high sensitivities to innovations in market volatility are attractive and have low returns.2 The second goal of the paper is to examine the cross-sectional relationship between idiosyncratic volatility and expected returns, where idiosyncratic volatility is defined relative to the standard Fama and French (1993) model.3 If the Fama–French model is correct, forming portfolios by sorting on idiosyncratic volatility will obviously provide no difference in average returns. Nevertheless, if the Fama–French model is false, sorting in this way potentially provides a set 1 See, among others, Jackwerth and Rubinstein (1996), Bakshi, Cao and Chen (2000), Chernov and Ghysels (2000), Burashi and Jackwerth (2001), Coval and Shumway (2001), Benzoni (2002), Pan (2002), Bakshi and Kapadia (2003), Eraker, Johannes and Polson (2003), Jones (2003), and Carr and Wu (2003).

2 Bates (2001) and Vayanos (2004) provide recent structural models whose reduced form factor structures have a negative risk premium for volatility risk.

3 Recent studies examining total or idiosyncratic volatility focus on the average level of firmlevel volatility. For example, Campbell et al. (2001) and Xu and Malkiel (2003) document that idiosyncratic volatility has increased over time. Brown and Ferreira (2003) and Goyal and SantaClara (2003) argue that idiosyncratic volatility has positive predictive power for excess market returns, but this is disputed by Bali et al. (2004).

Cross-Section of Volatility and Expected Returns 261 of assets that may have different exposures to aggregate volatility and hence different average returns. Our logic is the following. If aggregate volatility is a risk factor that is orthogonal to existing risk factors, the sensitivity of stocks to aggregate volatility times the movement in aggregate volatility will show up in the residuals of the Fama–French model. Firms with greater sensitivities to aggregate volatility should therefore have larger idiosyncratic volatilities relative to the Fama–French model, everything else being equal. Differences in the volatilities of firms’ true idiosyncratic errors, which are not priced, will make this relation noisy. We should be able to average out this noise by constructing portfolios of stocks to reveal that larger idiosyncratic volatilities relative to the Fama–French model correspond to greater sensitivities to movements in aggregate volatility and thus different average returns, if aggregate volatility risk is priced.

While high exposure to aggregate volatility risk tends to produce low expected returns, some economic theories suggest that idiosyncratic volatility should be positively related to expected returns. If investors demand compensation for not being able to diversify risk (see Malkiel and Xu (2002) and Jones and Rhodes-Kropf (2003)), then agents will demand a premium for holding stocks with high idiosyncratic volatility. Merton (1987) suggests that in an information-segmented market, firms with larger firm-specific variances require higher average returns to compensate investors for holding imperfectly diversified portfolios. Some behavioral models, like Barberis and Huang (2001), also predict that higher idiosyncratic volatility stocks should earn higher expected returns. Our results are directly opposite to these theories. We find that stocks with high idiosyncratic volatility have low average returns. There is a strongly significant difference of −1.06% per month between the average returns of the quintile portfolio with the highest idiosyncratic volatility stocks and the quintile portfolio with the lowest idiosyncratic volatility stocks.

In contrast to our results, earlier researchers either find a significantly positive relation between idiosyncratic volatility and average returns, or they fail to find any statistically significant relation between idiosyncratic volatility and average returns. For example, Lintner (1965) shows that idiosyncratic volatility carries a positive coefficient in cross-sectional regressions. Lehmann (1990) also finds a statistically significant, positive coefficient on idiosyncratic volatility over his full sample period. Similarly, Tinic and West (1986) and Malkiel and Xu (2002) unambiguously find that portfolios with higher idiosyncratic volatility have higher average returns, but they do not report any significance levels for their idiosyncratic volatility premiums. On the other hand, Longstaff (1989) finds that a cross-sectional regression coefficient on total variance for size-sorted portfolios carries an insignificant negative sign.

The difference between our results and the results of past studies is that the past literature either does not examine idiosyncratic volatility at the firm level, or does not directly sort stocks into portfolios ranked on this measure of interest. For example, Tinic and West (1986) work only with 20 portfolios sorted on market beta, while Malkiel and Xu (2002) work only with 100 portfolios sorted on market beta and size. Malkiel and Xu (2002) only use the idiosyncratic 262 The Journal of Finance volatility of one of the 100 beta/size portfolios to which a stock belongs to proxy for that stock’s idiosyncratic risk and, thus, do not examine firm-level idiosyncratic volatility. Hence, by not directly computing differences in average returns between stocks with low and high idiosyncratic volatilities, previous studies miss the strong negative relation between idiosyncratic volatility and average returns that we find.

The low average returns to stocks with high idiosyncratic volatilities could arise because stocks with high idiosyncratic volatilities may have high exposure to aggregate volatility risk, which lowers their average returns. We investigate this conjecture and find that this is not a complete explanation. Our idiosyncratic volatility results are also robust to controlling for value, size, liquidity, volume, dispersion of analysts’ forecasts, and momentum effects. We find the effect robust to different formation periods for computing idiosyncratic volatility and for different holding periods. The effect also persists in bull and bear markets, recessions and expansions, and volatile and stable periods. Hence, our results on idiosyncratic volatility represent a substantive puzzle.

The rest of this paper is organized as follows. In Section I, we examine how aggregate volatility is priced in the cross-section of stock returns. Section II documents that firms with high idiosyncratic volatility have very low average returns. Finally, Section III concludes.

I. Pricing Systematic Volatility in the Cross-Section A. Theoretical Motivation When investment opportunities vary over time, the multifactor models of Merton (1973) and Ross (1976) show that risk premia are associated with the conditional covariances between asset returns and innovations in state variables that describe the time-variation of the investment opportunities. Campbell’s (1993, 1996) version of the Intertemporal Capital Asset Pricing Model (I-CAPM) shows that investors care about risks both from the market return and from changes in forecasts of future market returns. When the representative agent is more risk averse than log utility, assets that covary positively with good news about future expected returns on the market have higher average returns. These assets command a risk premium because they reduce a consumer’s ability to hedge against a deterioration in investment opportunities. The intuition from Campbell’s model is that risk-averse investors want to hedge against changes in aggregate volatility because volatility positively affects future expected market returns, as in Merton (1973).

However, in Campbell’s setup, there is no direct role for f luctuations in market volatility to affect the expected returns of assets because Campbell’s model is premised on homoskedasticity. Chen (2002) extends Campbell’s model to a heteroskedastic environment which allows for both time-varying covariances and stochastic market volatility. Chen shows that risk-averse investors also want to directly hedge against changes in future market volatility. In Chen’s model, an asset’s expected return depends on risk from the market return, Cross-Section of Volatility and Expected Returns 263

for k = 1,..., K represent loadings on other risk factors. In the full conditional setting in equation (1), factor loadings, conditional means of factors, and factor premiums potentially vary over time. The model in equation (1) is written in terms of factor innovations, so rt+1 − γm,t represents the innovation in the m

where λm,t is the price of risk of the market factor, λv,t is the price of aggregate volatility risk, and the λk,t are the prices of risk of the other factors. Note that only if a factor is traded is the conditional mean of a factor equal to its conditional price of risk.