**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 RETURNSvaries 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.

259

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 P ´astor 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 signif- icant negative price of risk of approximately−1% per annum. Economic the- ory 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 opportu- nities. 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 mar-
ket 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 re-
turns. 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 be-
tween idiosyncratic volatility and expected returns, where idiosyncratic volatil-
ity 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

1See, 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).

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

3Recent studies examining total or idiosyncratic volatility focus on the average level of firm- level 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 Santa- Clara (2003) argue that idiosyncratic volatility has positive predictive power for excess market returns, but this is disputed by Bali et al. (2004).

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 rela- tive 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 aggre- gate volatility and thus different average returns, if aggregate volatility risk is priced.

While high exposure to aggregate volatility risk tends to produce low ex- pected returns, some economic theories suggest that idiosyncratic volatility should be positively related to expected returns. If investors demand compen- sation 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 re- quire 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 ex- pected 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 re- turns 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 pos- itive 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 volatil- ity carries a positive coefficient in cross-sectional regressions. Lehmann (1990) also finds a statistically significant, positive coefficient on idiosyncratic volatil- ity 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 inter- est. 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

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 idiosyn- cratic 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 idiosyn- cratic 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 volatil- ity 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 vari- ables that describe the time-variation of the investment opportunities. Camp- bell’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 represen- tative agent is more risk averse than log utility, assets that covary positively with good news about future expected returns on the market have higher av- erage returns. These assets command a risk premium because they reduce a consumer’s ability to hedge against a deterioration in investment opportuni- ties. 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 mar- ket 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,

changes in forecasts of future market returns, and changes in forecasts of fu- ture market volatilities. For an investor more risk averse than log utility, Chen shows that an asset that has a positive covariance between its return and a variable that positively forecasts future market volatilities causes that asset to have a lower expected return. This effect arises because risk-averse investors reduce current consumption to increase precautionary savings in the presence of increased uncertainty about market returns.

Motivated by these multifactor models, we study how exposure to market volatility risk is priced in the cross-section of stock returns. A true conditional multifactor representation of expected returns in the cross-section would take the following form:

*r*_{t}^{i}_{+1}*= a*_{t}^{i}*+ β*_{m,t}^{i}

*r*_{t}^{m}_{+1}*− γ**m,t*

*+ β*_{v,t}* ^{i}* (v

*t*+1

*− γ*

*v,t*)+

^{K}*k*=1

*β*_{k,t}* ^{i}* (

*f*

*k,t*+1

*− γ*

*k,t*), (1)

where*r*_{t}^{i}_{+1}is the excess return on stock*i,β**m,t** ^{i}* is the loading on the excess mar-
ket return,

*β*

_{v,t}*is the asset’s sensitivity to volatility risk, and the*

^{i}*β*

_{k,t}*coefficients for*

^{i}*k= 1, . . . , K represent loadings on other risk factors. In the full conditional*setting in equation (1), factor loadings, conditional means of factors, and fac- tor premiums potentially vary over time. The model in equation (1) is written in terms of factor innovations, so

*r*

_{t}

^{m}_{+1}

*− γ*

*m,t*represents the innovation in the market return,

*v*

_{t+1}*− γ*

*v,t*represents the innovation in the factor ref lecting ag- gregate volatility risk, and innovations to the other factors are represented by

*f*

*k,t*+1

*− γ*

*k,t*. The conditional mean of the market and aggregate volatility are denoted by

*γ*

*m,t*and

*γ*

*v,t*, respectively, while the conditional means of the other factors are denoted by

*γ*

*k,t*. In equilibrium, the conditional mean of stock

*i is*given by

*a*_{t}^{i}*= E**t*

*r*_{t+1}^{i}

*= β**m,t*^{i}*λ**m,t**+ β**v,t*^{i}*λ**v,t*+

*K*
*k*=1

*β**k,t*^{i}*λ**k,t*, (2)

where *λ**m,t* is the price of risk of the market factor, *λ**v,t* is the price of aggre-
gate volatility risk, and the*λ**k,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.

The main prediction from the factor model setting of equation (1) that we
examine is that stocks with different loadings on aggregate volatility risk have
different average returns.^{4}However, the true model in equation (1) is infeasible

4While an I-CAPM implies joint time-series as well as cross-sectional predictability, we do not examine time-series predictability of asset returns by systematic volatility. Time-varying volatility risk generates intertemporal hedging demands in partial equilibrium asset allocation problems. In a partial equilibrium setting, Liu (2001) and Chacko and Viceira (2003) examine how volatility risk affects the portfolio allocation of stocks and risk-free assets, while Liu and Pan (2003) show how investors can optimally exploit the variation in volatility with options. Guo and Whitelaw (2003) examine the intertemporal components of time-varying systematic volatility in a Campbell (1993, 1996) equilibrium I-CAPM.

to examine because the true set of factors is unknown and the true conditional factor loadings are unobservable. Hence, we do not attempt to directly use equa- tion (1) in our empirical work. Instead, we simplify the full model of equation (1), which we now detail.

*B. The Empirical Framework*

To investigate how aggregate volatility risk is priced in the cross-section of
equity returns we make the following simplifying assumptions to the full spec-
ification in equation (1). First, we use observable proxies for the market factor
and the factor representing aggregate volatility risk. We use the CRSP value-
weighted market index to proxy for the market factor. To proxy innovations
in aggregate volatility, (v*t*+1*− γ**v,t*), we use changes in the *VIX index from the*
Chicago Board Options Exchange (CBOE).^{5}Second, we reduce the number of
factors in equation (1) to just the market factor and the proxy for aggregate
volatility risk. Finally, to capture the conditional nature of the true model, we
use short intervals—1 month of daily data—to take into account possible time
variation of the factor loadings. We discuss each of these simplifications in turn.

*B.1. Innovations in the VIX Index*

The*VIX index is constructed so that it represents the implied volatility of a*
synthetic at-the-money option contract on the S&P100 index that has a matu-
rity of 1 month. It is constructed from eight S&P100 index puts and calls and
takes into account the American features of the option contracts, discrete cash
dividends, and microstructure frictions such as bid–ask spreads (see Whaley
(2000) for further details).^{6} Figure 1 plots the*VIX index from January 1986*
to December 2000. The mean level of the daily *VIX series is 20.5%, and its*
standard deviation is 7.85%.

Because the *VIX index is highly serially correlated with a first-order au-*
tocorrelation of 0.94, we measure daily innovations in aggregate volatility by
using daily changes in*VIX, which we denote asVIX. Daily first differences in*
*VIX have an effective mean of zero (less than 0.0001), a standard deviation of*

5In previous versions of this paper, we also consider: Sample volatility, following French et al. (1987); a range-based estimate, following Alizadeh, Brandt, and Diebold (2002); and a high- frequency estimator of volatility from Andersen, Bollerslev, and Diebold (2003). Using these mea- sures to proxy for innovations in aggregate volatility produces little spread in cross-sectional av- erage returns. These tables are available upon request.

6On September 22, 2003, the CBOE implemented a new formula and methodology to construct its volatility index. The new index is based on the S&P500 (rather than the S&P100) and takes into account a broader range of strike prices rather than using only at-the-money option contracts.

The CBOE now uses*VIX to refer to this new index. We use the old index (denoted by the ticker*
*VXO). We do not use the new index because it has been constructed by backfilling only to 1990,*
whereas the*VXO is available in real time from 1986. The CBOE continues to make both volatility*
indices available. The correlation between the new and the old CBOE volatility series is 98% from
1990 to 2000, but the series that we use has a slightly broader range than the new CBOE volatility
series.

19860 1988 1990 1992 1994 1996 1998 2000 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

* Figure 1. Plot of VIX. The figure shows the VIX index plotted at a daily frequency. The sample*
period is January 1986 to December 2000.

2.65%, and negligible serial correlation (the first-order autocorrelation of*VIX*
is−0.0001). As part of our robustness checks in Section I.C, we also measure
innovations in *VIX by specifying a stationary time-series model for the con-*
ditional mean of*VIX and find our results to be similar to those using simple*
first differences. While*VIX appears to be an ideal proxy for innovations in*
volatility risk because the*VIX index is representative of traded option secu-*
rities whose prices directly ref lect volatility risk, there are two main caveats
with respect to using*VIX to represent observable market volatility.*

The first concern is that the *VIX index is the implied volatility from the*
Black–Scholes (1973) model, and we know that the Black–Scholes model is an
approximation. If the true stochastic environment is characterized by stochas-
tic volatility and jumps, *VIX will reflect total quadratic variation in both*
diffusion and jump components (see, for example, Pan (2002)). Although Bates
(2000) argues that implied volatilities computed taking into account jump risk
are very close to original Black–Scholes implied volatilities, jump risk may be
priced differently from volatility risk. Our analysis does not separate jump
risk from diffusion risk, so our aggregate volatility risk may include jump risk
components.

A more serious reservation about the*VIX index is that VIX combines both*
stochastic volatility and the stochastic volatility risk premium. Only if the risk
premium is zero or constant would*VIX be a pure proxy for the innovation in*
aggregate volatility. Decomposing*VIX into the true innovation in volatility*
and the volatility risk premium can only be done by writing down a formal
model. The form of the risk premium depends on the parameterization of the
price of volatility risk, the number of factors, and the evolution of those factors.

Each different model specification implies a different risk premium. For exam-
ple, many stochastic volatility option pricing models assume that the volatility
risk premium can be parameterized as a linear function of volatility (see, for
example, Chernov and Ghysels (2000), Benzoni (2002), and Jones (2003)). This
may or may not be a good approximation to the true price of risk. Rather than
imposing a structural form, we use an unadulterated*VIX series. An advan-*
tage of this approach is that our analysis is simple to replicate.

*B.2. The Pre-Formation Regression*

Our goal is to test whether stocks with different sensitivities to aggregate
volatility innovations (proxied by *VIX) have different average returns. To*
measure the sensitivity to aggregate volatility innovations, we reduce the num-
ber of factors in the full specification in equation (1) to two, namely, the mar-
ket factor and*VIX. A two-factor pricing kernel with the market return and*
stochastic volatility as factors is also the standard setup commonly assumed by
many stochastic option pricing studies (see, for example, Heston (1993)). Hence,
the empirical model that we examine is

*r*_{t}^{i}*= β*0*+ β*_{MKT}^{i}*MKT**t**+ β*_{VIX}^{i}*VIX**t**+ ε*_{t}* ^{i}*, (3)
where

*MKT is the market excess return,VIX is the instrument we use for*innovations in the aggregate volatility factor, and

*β*

_{MKT}*and*

^{i}*β*

_{VIX}*are loadings on market risk and aggregate volatility risk, respectively.*

^{i}Previous empirical studies suggest that there are other cross-sectional factors that have explanatory power for the cross-section of returns, such as the size and value factors of the Fama and French (1993) three-factor model (hereafter FF-3). We do not directly model these effects in equation (3), because controlling for other factors in constructing portfolios based on equation (3) may add a lot of noise. Although we keep the number of regressors in our pre-formation portfolio regressions to a minimum, we are careful to ensure that we control for the FF- 3 factors and other cross-sectional factors in assessing how volatility risk is priced using post-formation regression tests.

We construct a set of assets that are sufficiently disperse in exposure to
aggregate volatility innovations by sorting firms on*VIX loadings over the*
past month using the regression (3) with daily data. We run the regression
for all stocks on AMEX, NASDAQ, and the NYSE, with more than 17 daily
observations. In a setting in which coefficients potentially vary over time, a
1-month window with daily data is a natural compromise between estimating

coefficients with a reasonable degree of precision and pinning down conditional
coefficients in an environment with time-varying factor loadings. P ´astor and
Stambaugh (2003), among others, also use daily data with a 1-month window in
similar settings. At the end of each month, we sort stocks into quintiles, based
on the value of the realized *β**VIX* coefficients over the past month. Firms in
quintile 1 have the lowest coefficients, while firms in quintile 5 have the highest
*β**VIX* loadings. Within each quintile portfolio, we value weight the stocks. We
link the returns across time to form one series of post-ranking returns for each
quintile portfolio.

Table I reports various summary statistics for quintile portfolios sorted by
past*β** _{VIX}* over the previous month using equation (3). The first two columns
report the mean and standard deviation of monthly total, not excess, simple
returns. In the first column under the heading “Factor Loadings,” we report the
pre-formation

*β*

*coefficients, which are computed at the beginning of each month for each portfolio and are value weighted. The column reports the time- series average of the pre-formation*

_{VIX}*β*

*VIX*loadings across the whole sample.

By construction, since the portfolios are formed by ranking on past*β**VIX*, the
pre-formation*β**VIX* loadings monotonically increase from−2.09 for portfolio 1
to 2.18 for portfolio 5.

The columns labeled “CAPM Alpha” and “FF-3 Alpha” report the time-series
alphas of these portfolios relative to the CAPM and to the FF-3 model, respec-
tively. Consistent with the negative price of systematic volatility risk found by
the option pricing studies, we see lower average raw returns, CAPM alphas,
and FF-3 alphas with higher past loadings of *β** _{VIX}*. All the differences be-
tween quintile portfolios 5 and 1 are significant at the 1% level, and a joint
test for the alphas equal to zero rejects at the 5% level for both the CAPM and
the FF-3 model. In particular, the 5-1 spread in average returns between the
quintile portfolios with the highest and lowest

*β*

*coefficients is−1.04% per month. Controlling for the*

_{VIX}*MKT factor exacerbates the 5-1 spread to*−1.15%

per month, while controlling for the FF-3 model decreases the 5-1 spread to

−0.83% per month.

*B.3. Requirements for a Factor Risk Explanation*

While the differences in average returns and alphas corresponding to dif-
ferent*β**VIX* loadings are very impressive, we cannot yet claim that these dif-
ferences are due to systematic volatility risk. We examine the premium for
aggregate volatility within the framework of an unconditional factor model.

There are two requirements that must hold in order to make a case for a fac- tor risk-based explanation. First, a factor model implies that there should be contemporaneous patterns between factor loadings and average returns. For example, in a standard CAPM, stocks that covary strongly with the market factor should, on average, earn high returns over the same period. To test a fac- tor model, Black, Jensen, and Scholes (1972), Fama and French (1992, 1993), Jagannathan and Wang (1996), and P ´astor and Stambaugh (2003), among others, all form portfolios using various pre-formation criteria, but examine

**T****able****I** **PortfoliosSortedbyExposuretoAggregateVolatilityShocks** Weformvalue-weightedquintileportfolioseverymonthbyregressingexcessindividualstockreturnson*VIX*,controllingforthe*MKT*factoras inequation(3),usingdailydataoverthepreviousmonth.Stocksaresortedintoquintilesbasedonthecoefficient*β**VIX*fromlowest(quintile1)to highest(quintile5).ThestatisticsinthecolumnslabeledMeanandStd.Dev.aremeasuredinmonthlypercentagetermsandapplytototal,notexcess, simplereturns.SizereportstheaveragelogmarketcapitalizationforfirmswithintheportfolioandB/Mreportstheaveragebook-to-marketratio. Therow“5-1”referstothedifferenceinmonthlyreturnsbetweenportfolio5andportfolio1.TheAlphacolumnsreportJensen’salphawithrespect totheCAPMortheFama–French(1993)three-factormodel.Thepre-formationbetasrefertothevalue-weighted*β**VIX*or*β**FVIX*withineachquintile portfolioatthestartofthemonth.Wereportthepre-formation*β**VIX*and*β**FVIX*averagedacrossthewholesample.Thesecondtolastcolumnreports the*β**VIX*loadingcomputedoverthenextmonthwithdailydata.Thecolumnreportsthenextmonth*β**VIX*loadingsaveragedacrossmonths.The lastcolumnreportsexpost*β**FVIX*factorloadingsoverthewholesample,where*FVIX*isthefactormimickingaggregatevolatilityrisk.Tocorrespond withtheFama–Frenchalphas,wecomputetheexpostbetasbyrunningafour-factorregressionwiththethreeFama–Frenchfactorstogetherwith the*FVIX*factorthatmimicsaggregatevolatilityrisk,followingtheregressioninequation(6).Therowlabeled“Jointtest*p*-value”reportsaGibbons, RossandShanken(1989)testforthealphasequaltozero,andarobustjointtestthatthefactorloadingsareequaltozero.RobustNewey–West (1987)*t-statistics*arereportedinsquarebrackets.ThesampleperiodisfromJanuary1986toDecember2000. FactorLoadings NextMonthFullSample Std.%MktCAPMFF-3Pre-FormationPre-FormationPost-FormationPost-Formation RankMeanDev.ShareSizeB/MAlphaAlpha*β**VIX**β**FVIX**β**VIX**β**FVIX* 11.645.539.4%3.700.890.270.30−2.09−2.00−0.033−5.06 [1.66][1.77][−4.06] 21.394.4328.7%4.770.730.180.09−0.46−0.42−0.014−2.72 [1.82][1.18][−2.64] 31.364.4030.4%4.770.760.130.080.030.080.005−1.55 [1.32][1.00][−2.86] 41.214.7924.0%4.760.73−0.08−0.060.540.620.0153.62 [−0.87][−0.65][4.53] 50.606.557.4%3.730.89−0.88−0.532.182.310.0188.07 [−3.42][−2.88][5.32] 5-1−1.04−1.15−0.83 [−3.90][−3.54][−2.93] Jointtest*p*-value0.010.030.00

post-ranking factor loadings that are computed over the full sample period.

While the*β**VIX*loadings show very strong patterns of future returns, they rep-
resent past covariation with innovations in market volatility. We must show
that the portfolios in Table I also exhibit high loadings with volatility risk over
the same period used to compute the alphas.

To construct our portfolios, we take *VIX to proxy for the innovation in*
aggregate volatility at a daily frequency. However, at the standard monthly
frequency, which is the frequency of the ex post returns for the alphas reported
in Table I, using the change in *VIX is a poor approximation for innovations*
in aggregate volatility. This is because at lower frequencies, the effect of the
conditional mean of*VIX plays an important role in determining the unantic-*
ipated change in*VIX. In contrast, the high persistence of the VIX series at a*
daily frequency means that the first difference of *VIX is a suitable proxy for*
the innovation in aggregate volatility. Hence, we should not measure ex post
exposure to aggregate volatility risk by looking at how the portfolios in Table I
correlate ex post with monthly changes in*VIX.*

To measure ex post exposure to aggregate volatility risk at a monthly fre-
quency, we follow Breeden, Gibbons, and Litzenberger (1989) and construct
an ex post factor that mimics aggregate volatility risk. We term this mimick-
ing factor*FVIX. We construct the tracking portfolio so that it is the portfolio*
of asset returns maximally correlated with realized innovations in volatility
using a set of basis assets. This allows us to examine the contemporaneous re-
lationship between factor loadings and average returns. The major advantage
of using*FVIX to measure aggregate volatility risk is that we can construct a*
good approximation for innovations in market volatility at any frequency. In
particular, the factor mimicking aggregate volatility innovations allows us to
proxy aggregate volatility risk at the monthly frequency by simply cumulating
daily returns over the month on the underlying base assets used to construct
the mimicking factor. This is a much simpler method for measuring aggregate
volatility innovations at different frequencies, rather than specifying different,
and unknown, conditional means for*VIX that depend on different sampling*
frequencies. After constructing the mimicking aggregate volatility factor, we
confirm that it is high exposure to aggregate volatility risk that is behind the
low average returns to past*β**VIX* loadings.

However, just showing that there is a relation between ex post aggregate volatility risk exposure and average returns does not rule out the explana- tion that the volatility risk exposure is due to known determinants of expected returns in the cross-section. Hence, our second condition for a risk-based expla- nation is that the aggregate volatility risk exposure is robust to controlling for various stock characteristics and other factor loadings. Several of these cross- sectional effects may be at play in the results of Table I. For example, quintile portfolios 1 and 5 have smaller stocks, and stocks with higher book-to-market ratios, and these are the portfolios with the most extreme returns. Periods of very high volatility also tend to coincide with periods of market illiquidity (see, among others, Jones (2003) and P ´astor and Stambaugh (2003)). In Sec- tion I.C, we control for size, book-to-market, and momentum effects, and also

specifically disentangle the exposure to liquidity risk from the exposure to sys- tematic volatility risk.

*B.4. A Factor Mimicking Aggregate Volatility Risk*

Following Breeden et al. (1989) and Lamont (2001), we create the mimicking
factor*FVIX to track innovations in VIX by estimating the coefficient b in the*
following regression:

*VIX**t**= c + b*^{}*X**t**+ u**t*, (4)

where*X**t*represents the returns on the base assets. Since the base assets are
excess returns, the coefficient*b has the interpretation of weights in a zero-*
cost portfolio. The return on the portfolio,*b*^{}*X**t*, is the factor*FVIX that mimics*
innovations in market volatility. We use the quintile portfolios sorted on past
*β**VIX* in Table I as the base assets*X**t*. These base assets are, by construction, a
set of assets that have different sensitivities to past daily innovations in*VIX.*^{7}
We run the regression in equation (4) at a daily frequency every month and use
the estimates of*b to construct the mimicking factor for aggregate volatility risk*
over the same month.

An alternative way to construct a factor that mimics volatility risk is to di-
rectly construct a traded asset that ref lects only volatility risk. One way to do
this is to consider option returns. Coval and Shumway (2001) construct market-
neutral straddle positions using options on the aggregate market (S&P100
options). This strategy provides exposure to aggregate volatility risk. Coval
and Shumway approximate daily at-the-money straddle returns by taking a
weighted average of zero-beta straddle positions, with strikes immediately
above and below each day’s opening level of the S&P100. They cumulate these
daily returns each month to form a monthly return, which we denote as*STR.*^{8}
In Section I.D, we investigate the robustness of our results to using *STR in*
place of*FVIX when we estimate the cross-sectional aggregate volatility price*
of risk.

Once we construct*FVIX, then the multifactor model (3) holds, except we can*
substitute the (unobserved) innovation in volatility with the tracking portfolio
that proxies for market volatility risk (see Breeden (1979)). Hence, we can write
the model in equation (3) as the following cross-sectional regression:

*r*_{t}^{i}*= α*^{i}*+ β*_{MKT}^{i}*MKT**t**+ β*_{FVIX}^{i}*FVIX**t**+ ε*^{i}* _{t}*, (5)
where

*MKT is the market excess return, FVIX is the mimicking aggregate*volatility factor, and

*β*

_{MKT}*and*

^{i}*β*

_{FVIX}*are factor loadings on market risk and aggregate volatility risk, respectively.*

^{i}7Our results are unaffected if we use the six Fama–French (1993) 3× 2 portfolios sorted on size and book-to-market as the base assets. These results are available upon request.

8The*STR returns are available from January 1986 to December 1995, because it is constructed*
from the Berkeley Option Database, which has reliable data only from the late 1980s and ends in
1995.

To test a factor risk model like equation (5), we must show contemporane-
ous patterns between factor loadings and average returns. That is, if the price
of risk of aggregate volatility is negative, then stocks with high covariation
with*FVIX should have low returns, on average, over the same period used to*
compute the *β**FVIX* factor loadings and the average returns. By construction,
*FVIX allows us to examine the contemporaneous relationship between factor*
loadings and average returns and it is the factor that is ex post most highly
correlated with innovations in aggregate volatility. However, while*FVIX is the*
right factor to test a risk story,*FVIX itself is not an investable portfolio because*
it is formed with future information. Nevertheless,*FVIX can be used as guid-*
ance for tradeable strategies that would hedge market volatility risk using the
cross-section of stocks.

In the second column under the heading “Factor Loadings” of Table I, we
report the pre-formation*β**FVIX*loadings that correspond to each of the portfolios
sorted on past*β**VIX* loadings. The pre-formation*β**FVIX*loadings are computed
by running the regression (5) over daily returns over the past month. The pre-
formation*FVIX loadings are very similar to the pre-formationVIX loadings*
for the portfolios sorted on past*β**VIX*loadings. For example, the pre-formation
*β**FVIX*(*β**VIX*) loading for quintile 1 is −2.00 (−2.09), while the pre-formation
*β**FVIX*(*β**VIX*) loading for quintile 5 is 2.31 (2.18).

*B.5. Post-Formation Factor Loadings*

In the next-to-last column of Table I, we report post-formation*β**VIX*loadings
over the next month, which we compute as follows. After the quintile portfolios
are formed at time*t, we calculate daily returns of each of the quintile portfolios*
over the next month, from*t to t*+ 1. For each portfolio, we compute the ex post
*β** _{VIX}* loadings by running the same regression (3) that is used to form the
portfolios using daily data over the next month (t to t+ 1). We report the next-
month

*β*

*loadings averaged across time. The next-month post-formation*

_{VIX}*β*

*loadings range from−0.033 for portfolio 1 to 0.018 for portfolio 5. Hence, although the ex post*

_{VIX}*β*

*loadings over the next month are monotonically increasing, the spread is disappointingly very small.*

_{VIX}Finding large spreads in the next-month post-formation *β**VIX* loadings is
a very stringent requirement and one that would be done in direct tests of a
conditional factor model such as equation (1). Our goal is more modest. We
examine the premium for aggregate volatility using an unconditional factor
model approach, which requires that average returns be related to the uncon-
ditional covariation between returns and aggregate volatility risk. As Hansen
and Richard (1987) note, an unconditional factor model implies the existence
of a conditional factor model. However, to form precise estimates of the con-
ditional factor loadings in a full conditional setting like equation (1) requires
knowledge of the instruments driving the time variation in the betas, as well
as specification of the complete set of factors.

The ex post*β** _{VIX}* loadings over the next month are computed using, on av-
erage, only 22 daily observations each month. In contrast, the CAPM and FF-3

alphas are computed using regressions measuring unconditional factor expo- sure over the full sample (180 monthly observations) of post-ranking returns.

To demonstrate that exposure to volatility innovations may explain some of the large CAPM and FF-3 alphas, we must show that the quintile portfolios exhibit different post-ranking spreads in aggregate volatility risk sensitivities over the entire sample at the same monthly frequency for which the post-ranking re- turns are constructed. Averaging a series of ex post conditional 1-month covari- ances does not provide an estimate of the unconditional covariation between the portfolio returns and aggregate volatility risk.

To examine ex post factor exposure to aggregate volatility risk consistent
with a factor model approach, we compute post-ranking*FVIX betas over the*
full sample.^{9}In particular, since the FF-3 alpha controls for market, size, and
value factors, we compute ex post*FVIX factor loadings also controlling for these*
factors in a four-factor post-formation regression,

*r*_{t}^{i}*= α*^{i}*+ β*_{MKT}^{i}*MKT**t**+ β*_{SMB}^{i}*SMB**t**+ β*_{HML}^{i}*HML**t*

*+ β*_{FVIX}^{i}*FVIX**t**+ ε*^{i}* _{t}*, (6)

where the first three factors*MKT, SMB, and HML constitute the FF-3 model’s*
market, size, and value factors. To compute the ex post*β**FVIX* loadings, we run
equation (6) using monthly frequency data over the whole sample, where the
portfolios on the left-hand side of equation (6) are the quintile portfolios in Table
I that are sorted on past loadings of*β** _{VIX}* using equation (3).

The last column of Table I shows that the portfolios sorted on past*β** _{VIX}*
exhibit strong patterns of post-formation factor loadings on the volatility risk
factor

*FVIX. The ex post*

*β*

*FVIX*factor loadings monotonically increase from

−5.06 for portfolio 1 to 8.07 for portfolio 5. We strongly reject the hypothesis
that the ex post*β**FVIX*loadings are equal to zero, with a*p-value less than 0.001.*

Thus, sorting stocks on past*β** _{VIX}* provides strong, significant spreads in ex
post aggregate volatility risk sensitivities.

^{10}

*B.6. Characterizing the Behavior of FVIX*

Table II reports correlations among the *FVIX factor,* *VIX, and STR, as*
well as correlations of these variables with other cross-sectional factors. We
denote the daily first difference in*VIX asVIX, and use **m**VIX to represent*
the monthly first difference in the*VIX index. The mimicking volatility factor*
is highly contemporaneously correlated with changes in volatility at a daily

9The pre-formation betas and the post-formation betas are computed using different criteria
(VIX and FVIX, respectively). However, Table I shows that the pre-formation β*FVIX*loadings are
almost identical to the pre-formation*β**VIX*loadings.

10When we compute ex post betas using the monthly change in*VIX,**m**VIX, using a four-factor*
model similar to equation (6) (except using*m**VIX in place of FVIX), there is less dispersion in the*
post-formation*m**VIX betas, ranging from*−2.46 for portfolio 1 to 0.76 to portfolio 5, compared to
the ex post*β**FVIX*loadings.

**Table II**

**Factor Correlations**

The table reports correlations of first differences in*VIX, FVIX, and STR with various factors. The*
variable*VIX (**m**VIX) represents the daily (monthly) change in the VIX index, and FVIX is the*
mimicking aggregate volatility risk factor. The factor*STR is constructed by Coval and Shumway*
(2001) from the returns of zero-beta straddle positions. The factors*MKT, SMB, HML are the Fama*
and French (1993) factors, the momentum factor*UMD is constructed by Kenneth French, and*
*LIQ is the P ´astor and Stambaugh (2003) liquidity factor. The sample period is January 1986 to*
December 2000, except for correlations involving*STR, which are computed over the sample period*
January 1986 to December 1995.

Panel A: Daily Correlation

*VIX*

*FVIX* 0.91

Panel B: Monthly Correlations

*FVIX* *m**VIX* *MKT* *SMB* *HML* *UMD* *LIQ*

*m**VIX* 0.70 1.00 −0.58 −0.18 0.22 −0.11 −0.33

*FVIX* 1.00 0.70 −0.66 −0.14 0.26 −0.25 −0.40

*STR* 0.75 0.83 −0.39 −0.39 0.08 −0.26 −0.59

frequency, with a correlation of 0.91. At the monthly frequency, the correlation
between*FVIX and**m**VIX is lower, at 0.70. The factors FVIX and STR have a*
high correlation of 0.83, which indicates that*FVIX, formed from stock returns,*
behaves like the*STR factor constructed from option returns. Hence, FVIX cap-*
tures option-like behavior in the cross-section of stocks. The factor *FVIX is*
negatively contemporaneously correlated with the market return (−0.66), re-
f lecting the fact that when volatility increases, market returns are low. The
correlations of*FVIX with SMB and HML are*−0.14 and 0.26, respectively. The
correlation between*FVIX and UMD, a factor capturing momentum returns, is*
also low at−0.25.

In contrast, there is a strong negative correlation between *FVIX and the*
P ´astor and Stambaugh (2003) liquidity factor,*LIQ, at−0.40. The LIQ factor*
decreases in times of low liquidity, which tend to also be periods of high volatil-
ity. One example of a period of low liquidity with high volatility is the 1987 crash
(see, among others, Jones (2003) and P ´astor and Stambaugh (2003)). However,
the correlation between*FVIX and LIQ is far from*−1, indicating that volatility
risk and liquidity risk may be separate effects, and may be separately priced.

In the next section, we conduct a series of robustness checks designed to dis- entangle the effects of aggregate volatility risk from other factors, including liquidity risk.

*C. Robustness*

In this section, we conduct a series of robustness checks in which we specify
different models for the conditional mean of*VIX, we use windows of different*
estimation periods to form the *β** _{VIX}* portfolios, and we control for potential

cross-sectional pricing effects due to book-to-market, size, liquidity, volume, and momentum factor loadings or characteristics.

*C.1. Robustness to Different Conditional Means of VIX*

We first investigate the robustness of our results to the method measuring
innovations in*VIX. We use the change in VIX at a daily frequency to measure*
the innovation in volatility because*VIX is a highly serially correlated series.*

However,*VIX appears to be a stationary series, and usingVIX as the innova-*
tion in*VIX may slightly over-difference. Our finding of low average returns on*
stocks with high*β**FVIX*is robust to measuring volatility innovations by specify-
ing various models for the conditional mean of*VIX. If we fit an AR(1) model to*
*VIX and measure innovations relative to the AR(1) specification, we find that*
the results of Table I are almost unchanged. Specifically, the mean return of the
difference between the first and fifth*β** _{VIX}*portfolios is−1.08% per month, and
the FF-3 alpha of the 5-1 difference is−0.90%, both highly statistically signifi-
cant. Using an optimal BIC choice for the number of autoregressive lags, which
is 11, produces a similar result. In this case, the mean of the 5-1 difference is

−0.81% and the 5-1 FF-3 alpha is −0.66%; both differences are significant at
the 5% level.^{11}

*C.2. Robustness to the Portfolio Formation Window*

In this subsection, we investigate the robustness of our results to the amount
of data used to estimate the pre-formation factor loadings *β** _{VIX}*. In Table I,
we use a formation period of 1 month, and we emphasize that this window is
chosen a priori without pre-tests. The results in Table I become weaker if we
extend the formation period of the portfolios. Although the point estimates of
the

*β*

*VIX*portfolios have the same qualitative patterns as Table I, statistical significance drops. For example, if we use the past 3 months of daily data on

*VIX to compute volatility betas, the mean return of the 5th quintile portfolio*
with the highest past *β**VIX* stocks is 0.79%, compared with 0.60% with a 1-
month formation period. Using a 3-month formation period, the FF-3 alpha
on the 5th quintile portfolio decreases in magnitude to−0.37%, with a robust
*t-statistic of* *−1.62, compared to −0.53%, with a t-statistic of −2.88, with a*
1-month formation period from Table I. If we use the past 12 months of*VIX*
innovations, the 5th quintile portfolio mean increases to 0.97%, while the FF-3
alpha decreases in magnitude to*−0.24% with a t-statistic of −1.04.*

The weakening of the*β** _{VIX}* effect as the formation periods increase is due
to the time variation of the sensitivities to aggregate market innovations. The
turnover in the monthly

*β*

*VIX*portfolios is high (above 70%) and using longer

11In these exercises, we estimate the AR coefficients only using all data up to time*t to compute*
the innovation for*t*+ 1, so that no forward-looking information is used. We initially estimate the
AR models using 1 year of daily data. However, the optimal BIC lag length is chosen using the
whole sample.

formation periods causes less turnover; however, using more data provides less
precise conditional estimates. The longer the formation window, the less these
conditional estimates are relevant at time*t, and the lower the spread in the*
pre-formation*β**VIX* loadings. By using only information over the past month,
we obtain an estimate of the conditional factor loading much closer to time*t.*

*C.3. Robustness to Book-to-Market and Size Characteristics*

Small growth firms are typically firms with option values that would be expected to do well when aggregate volatility increases. The portfolio of small growth firms is also one of the Fama–French (1993) 25 portfolios sorted on size and book-to-market that is hardest to price by standard factor models (see, for example, Hodrick and Zhang (2001)). Could the portfolio of stocks with high aggregate volatility exposure have a disproportionately large number of small growth stocks?

Investigating this conjecture produces mixed results. If we exclude only the portfolio among the 25 Fama–French portfolios with the smallest growth firms and repeat the quintile portfolio sorts in Table I, we find that the 5-1 mean difference in returns is reduced in magnitude from −1.04% for all firms to

*−0.63% per month, with a t-statistic of −3.30. Excluding small growth firms*
produces a FF-3 alpha of−0.44% per month for the zero-cost portfolio that goes
long portfolio 5 and short portfolio 1, which is no longer significant at the 5%

level (t-statistic is−1.79), compared to the value of −0.83% per month with all
firms. These results suggest that small growth stocks may play a role in the
*β** _{VIX}* quintile sorts of Table I.

However, a more thorough characteristic-matching procedure suggests that
size or value characteristics do not completely drive the results. Table III re-
ports mean returns of the*β** _{VIX}* portfolios characteristic matched by size and
book-to-market ratios, following the method proposed by Daniel et al. (1997).

Every month, each stock is matched with one of the Fama–French 25 size
and book-to-market portfolios according to its size and book-to-market char-
acteristics. The table reports value-weighted simple returns in excess of the
characteristic-matched returns. Table III shows that characteristic controls for
size and book-to-market decrease the magnitude of the raw 5-1 mean return
difference of−1.04% in Table I to −0.90%. If we exclude firms that are members
of the smallest growth portfolio of the Fama–French 25 size-value portfolios, the
magnitude of the mean 5-1 difference decreases to−0.64% per month. However,
the characteristic-controlled differences are still highly significant. Hence, the
low returns to high past*β** _{VIX}* stocks are not completely driven by a dispropor-
tionate concentration among small growth stocks.

*C.4. Robustness to Liquidity Effects*

P ´astor and Stambaugh (2003) demonstrate that stocks with high liquidity
betas have high average returns. In order for liquidity to be an explanation
behind the spreads in average returns of the*β** _{VIX}*portfolios, high

*β*

*stocks*

_{VIX}**Table III**

**Characteristic Controls for Portfolios Sorted on β****ΔVIX**

The table reports the means and standard deviations of the excess returns on the*β**VIX* quintile
portfolios characteristic matched by size and book-to-market ratios. Each month, each stock is
matched with one of the Fama and French (1993) 25 size and book-to-market portfolios according
to its size and book-to-market characteristics. The table reports value-weighted simple returns
in excess of the characteristic-matched returns. The columns labeled “Excluding Small, Growth
Firms” exclude the Fama–French portfolio containing the smallest stocks and the firms with the
lowest book-to-market ratios. The row “5-1” refers to the difference in monthly returns between
portfolio 5 and portfolio 1. The*p-values of joint tests for all alphas equal to zero are less than 1% for*
the panel of all firms and for the panel excluding small, growth firms. Robust Newey–West (1987)
*t-statistics are reported in square brackets. The sample period is from January 1986 to December*
2000.

Excluding Small,

All Firms Growth Firms

Rank Mean Std. Dev. Mean Std. Dev.

1 0.32 2.11 0.36 1.90

2 0.04 1.25 0.02 0.94

3 0.04 0.94 0.05 0.89

4 −0.11 1.04 −0.10 1.02

5 −0.58 3.39 −0.29 2.17

5-1 −0.90 −0.64

[−3.59] [−3.75]

must have low liquidity betas. To check that the spread in average returns on
the*β** _{VIX}* portfolios is not due to liquidity effects, we first sort stocks into five
quintiles based on their historical P ´astor–Stambaugh liquidity betas. Then,
within each quintile, we sort stocks into five quintiles based on their past

*β*

*coefficient loadings. These portfolios are rebalanced monthly and are value weighted. After forming the 5*

_{VIX}*× 5 liquidity beta and β*

*portfolios, we average the returns of each*

_{VIX}*β*

*quintile over the five liquidity beta portfolios. Thus, these quintile*

_{VIX}*β*

*portfolios control for differences in liquidity.*

_{VIX}We report the results of the P ´astor–Stambaugh liquidity control in Panel A of Table IV, which shows that controlling for liquidity reduces the magnitude of the 5-1 difference in average returns from−1.04% per month in Table I to

−0.68% per month. However, after controlling for liquidity, we still observe
the monotonically decreasing pattern of average returns of the*β**VIX* quintile
portfolios. We also find that controlling for liquidity, the FF-3 alpha for the 5-1
portfolio remains significantly negative at−0.55% per month. Hence, liquidity
effects cannot account for the spread in returns resulting from sensitivity to
aggregate volatility risk.

Table IV also reports post-formation *β**FVIX* loadings. Similar to the post-
formation*β**FVIX* loadings in Table I, we compute the post-formation*β**FVIX* co-
efficients using a monthly frequency regression with the four-factor model in
equation (6) to be comparable to the FF-3 alphas over the same sample period.

Both the pre-formation*β** _{VIX}* and post-formation

*β*

*FVIX*loadings increase from