NBER WORKING PAPER SERIES
THE CROSS-SECTION OF VOLATILITY AND EXPECTED RETURNS
Andrew Ang
dynamiteRobert J. Hodrick
好老师Yuhang Xing
Xiaoyan Zhang
Working Paper10852
www.nber/papers/w10852
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachutts Avenue
Cambridge, MA 02138
October 2004
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 minar participants at an NBER Ast 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 article. Andrew Ang and Bob Hodrick both acknowledge support from the NSF. The views expresd herein are tho of the author(s) and not necessarily tho of the National Bureau of Economic Rearch.
©2004 by Andrew Ang, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang. All rights rerved. Short ctions of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.
The Cross-Section of Volatility and Expected Returns
Andrew Ang, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang
citicNBER Working Paper No. 10852
October 2004
JEL No. G12, G13
ABSTRACT
We examine the pricing of aggregate volatility risk in the cross-ction of stock returns. Consistent with theory, we find that stocks with high nsitivities to innovations in aggregate volatility have low average returns. In addition, we find that stocks with high idiosyncratic volatility relative to the Fama and French (1993) 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.
Andrew Ang原始英文
Marshall School of Business, USC
海音寺潮五郎
Department of Finance
701 Exposition Blvd, Room 701
Los Angeles CA 90089-1427
and NBER
aa610@columbia.edu
Robert J. Hodrick
眼霜的用法Graduate School of Business
Columbia University
3022 Broadway
New York, NY 10027
and NBER
rh169@columbia.edu
Yuhang Xing
Rice University
yxing@rice.edu
Xiaoyan Zhang
妇女节快乐英文
Cornell University
xz@cornell.edu
It is well known that the volatility of stock returns varies over time.While considerable rearch has examined the time-ries relation between the volatility of the market and the ex-pected return on the market(e,among others,Campbell and Hentschel(1992),and Glosten, Jagannathan and Runkle(1993)),the question of how aggregate volatility affects the cross-ction of expected stock returns has received less attention.Time-varying market volatility induces changes in the investment opportunity t by changing the expectation of future mar-ket returns,or by changing the risk-return tr
ade-off.If the volatility of the market return is a systematic risk factor,an APT or factor model predicts that aggregate volatility should also be priced in the cross-ction of stocks.Hence,stocks with different nsitivities to innovations in aggregate volatility should have different expected returns.
i adore you
Thefirst goal of this paper is to provide a systematic investigation of how the stochastic volatility of the market is priced in the cross-ction of expected stock returns.We want to de-termine if the volatility of the market 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.1Using the cross-ction of stock returns,rather than options on the market,allows us to create portfolios of stocks that have different nsitivities to innovations in market volatility.If the price of aggre-gate volatility risk is negative,stocks with large,positive nsitivities to volatility risk should have low average returns.Using the cross-ction of stock returns also allows us to easily con-trol for a battery of cross-ctional effects,like 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´a stor and Stambaugh(2003).Option pricing studies do not control for the cross-ctional risk factors.
Wefind that innovations in aggregate volatility carry a statistically significant negative price of risk of
approximately-1%per annum.Economic theory provides veral reasons why the price of risk of innovations in market volatility should be negative.For example,Campbell (1993and1996)and Chen(2002)show that investors want to hedge against changes in mar-ket volatility,becau increasing volatility reprents a deterioration in investment opportuni-ties.Risk aver agents demand stocks that hedge against this risk.Periods of high volatility also tend to coincide with downward market movements(e French,Schwert and Stambaugh (1987),and Campbell and Hentschel(1992)).As Bakshi and Kapadia(2003)comment,asts with high nsitivities to market volatility risk provide hedges against market downside risk. The higher demand for asts with high systematic volatility loadings increas their price and
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lowers their average return.Finally,stocks that do badly when volatility increas tend to have negatively skewed returns over intermediate horizons,while stocks that do well when volatil-ity ris tend to have positively skewed returns.If investors have preferences over coskewness (e Harvey and Siddique(2000)),stocks that have high nsitivities to innovations in market volatility are attractive and have low returns.2
厦门教育
The cond goal of the paper is to examine the cross-ctional relationship between id-iosyncratic volatility and expected returns,where idiosyncratic volatility is defined relative to the standard Fama and French(1993)model.3If 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 fal,sorting in this way potentially provides a t of asts that may have different exposures to aggregate volatility and hence different aver-age returns.Our logic is the following.If aggregate volatility is a risk factor that is orthogonal to existing risk factors,the nsitivity 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 nsitivities to aggregate volatility should therefore have larger idiosyncratic volatilities relative to the Fama-French model,everything el being equal.Differences in the volatilities offirms’true idiosyncratic errors,which are not priced,will make this relation noisy.We should be able to average out this noi by constructing portfolios of stocks to reveal that larger idiosyncratic volatilities relative to the Fama-French model correspond to greater nsitivities 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 econom
ic theories suggest that idiosyncratic volatility should be positively related to expected returns.If investors demand compensation for not being able to diversify risk(e 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-gmented market,firms with largerfirm-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 the theories.Wefind 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.
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In contrast to our results,earlier rearchers either found a significantly positive relation between idiosyncratic volatility and average returns,or they failed tofind any statistically sig-nificant relation between idiosyncratic volatility and average returns.For example,Lintner (1965)shows that idiosyncratic volatility carries a positive coefficient in cross-ctional regres-sions.Lehmann(1990)als
ofinds a statistically significant,positive coefficient on idiosyncratic volatility over his full sample period.Similarly,Tinic and West(1986)and Malkiel and Xu (2002)unambiguouslyfind that portfolios with higher idiosyncratic volatility have higher av-erage returns,but they do not report any significance levels for their idiosyncratic volatility premiums.On the other hand,Longstaff(1989)finds that a cross-ctional regression coeffi-cient 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 thefirm level or does not directly sort stocks into portfolios ranked on this measure of interest.For example,Tinic and West(1986)work only with20portfolios sorted on market beta,while Malkiel and Xu(2002)work only with 100portfolios sorted on market beta and size.Malkiel and Xu(2002)only u the idiosyncratic volatility of one of the100beta/size portfolios to which a stock belongs to proxy for that stock’s idiosyncratic risk and,thus,do not examinefirm-level idiosyncratic volatility.Hence,by not di-rectly 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 wefind.
The low average returns to stocks with high idiosyncratic volatilities could ari becau stocks with
high idiosyncratic volatilities may have high exposure to aggregate volatility risk, which lowers their average returns.We investigate this issue andfind 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.Wefind the effect robust to different formation periods for computing idiosyncratic volatility and for different holding periods.The effect also persists in both bull and bear markets,recessions and expan-sions,and volatile and stable periods.Hence,our results on idiosyncratic volatility reprent 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-ction of stock returns.Section II documents thatfirms with high idiosyncratic volatility have very low average returns.Finally,Section III concludes.
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出纳和会计的区别I.Pricing Systematic Volatility in the Cross-Section
A.Theoretical Motivation
When investment opportunities vary over time,the multi-factor models of Merton(1973)and Ross(197
6)show that risk premia are associated with the conditional covariances between as-t returns and innovations in state variables that describe the time-variation of the investment opportunities.Campbell’s(1993and1996)version of the Intertemporal CAPM(I-CAPM) shows that investors care about risks from the market return and from changes in forecasts of future market returns.When the reprentative agent is more risk aver than log utility,asts that covary positively with good news about future expected returns on the market have higher average returns.The asts command a risk premium becau they reduce a consumer’s abil-ity to hedge against a deterioration in investment opportunities.The intuition from Campbell’s model is that risk-aver investors want to hedge against changes in aggregate volatility becau volatility positively affects future expected market returns,as in Merton(1973).
However,in Campbell’s t-up,there is no direct role forfluctuations in market volatility to affect the expected returns of asts becau Campbell’s model is premid on homoskedastic-ity.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-aver in-vestors also want to directly hedge against changes in future market volatility.In Chen’s model, an ast’s expected return depends on risk from the market return,changes in forecasts of future market return
s,and changes in forecasts of future market volatilities.For an investor more risk aver than log utility,Chen shows that an ast that has a positive covariance between its return and a variable that positively forecasts future market volatilities caus that ast to have a lower expected return.This effect aris becau risk-aver investors reduce current consumption to increa precautionary savings in the prence of incread uncertainty about market returns.
Motivated by the multi-factor models,we study how exposure to market volatility risk is priced in the cross-ction of stock returns.A true conditional multi-factor reprentation of expected returns in the cross-ction would take the following form:
r i t+1=a i
t
+βi
m,t
(r m
t+1
−γm,t)+βi
v,t
(v t+1−γv,t)+
K
k=1
βi
k,t
(f k,t+1−γk,t),(1)
where r i
t+1is the excess return on stock i,βi
m,t
is the loading on the excess market return,βi
v,t
is the ast’s nsitivity to volatility risk,and theβi
k,t
coefficients for K reprent
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