《金融投资学》教学资源（英文文献）Asymmetric Volatility and Risk in Equity Markets

Asymmetric Volatility and Risk in Equity Markets Geert Bekaert Columbia University,Stanford University,and NBER Guojun Wu University of Michigan It appears that volatility in equity markets is asymmetric:returns and conditional volatility are negatively correlated.We provide a unified framework to simultane- ously investigate asymmetric volatility at the firm and the market level and to examine two potential explanations of the asymmetry:leverage effects and volatil- ity feedback.Our empirical application uses the market portfolio and portfolios with different leverage constructed from Nikkei 225 stocks.We reject the pure leverage model of Christie (1982)and find support for a volatility feedback story. Volatility feedback at the firm level is enhanced by strong asymmetries in condi- tional covariances.Conditional betas do not show significant asymmetries.We document the risk premium implications of these findings. There is a long tradition in finance [see,e.g.,Cox and Ross (1976)]that models stock return volatility as negatively correlated with stock re- turns.Influential articles by Black (1976)and Christie (1982)further document and attempt to explain the asymmetric volatility property of individual stock returns in the United States.The explanation put forward in these articles is based on leverage.A drop in the value of the stock (negative return)increases financial leverage,which makes the stock riskier and increases its volatility. Although,to many,"leverage effects"have become synonymous with asymmetric volatility,the asymmetric nature of the volatility response to return shocks could simply reflect the existence of time-varying risk premiums [Pindyck(1984),French,Schwert,and Stambaugh,(1987),and Campbell and Hentschel (1992)].If volatility is priced,an anticipated increase in volatility raises the required return on equity,leading to an We have benefited from discussions with Darrell Duffie,Robert Engle,Heber Farnsworth,Ken Froot,Steve Grenadier,Masao Matsuda,Manju Puri,Hisayoshi Shindo,Ken Singleton,and Jeff Zwiebel.We also thank seminar participants at the American Finance Association Meetings in Chicago,the Nikko Research Center,Stanford University,University of California at San Diego,the Swedish School of Economics in Helsinki,and Tilburg University for useful com- ments.We are especially grateful for the insightful comments of an anonymous referee,which greatly improved the article.Geert Bekaert acknowledges financial support from an NSF grant and Stanford University.Address correspondence to Guojun Wu,School of Business Adminis- tration,University of Michigan,Ann Arbor,MI 48109,or e-mail:giwu@umich.edu. Black(1976)also discusses an operating leverage effect,induced by fixed costs of the firm.but that effect has received little attention in the finance literature. The Review of Financial Studies Spring 2000 Vol.13.No.1,pp.1-42 2000 The Society for Financial Studies

Asymmetric Volatility and Risk in Equity Markets Geert Bekaert Columbia University, Stanford University, and NBER Guojun Wu University of Michigan It appears that volatility in equity markets is asymmetric: returns and conditional volatility are negatively correlated. We provide a unified framework to simultane￾ously investigate asymmetric volatility at the firm and the market level and to examine two potential explanations of the asymmetry: leverage effects and volatil￾ity feedback. Our empirical application uses the market portfolio and portfolios with different leverage constructed from Nikkei 225 stocks. We reject the pure leverage model of Christie 1982 and find support for a volatility feedback story. Ž . Volatility feedback at the firm level is enhanced by strong asymmetries in condi￾tional covariances. Conditional betas do not show significant asymmetries. We document the risk premium implications of these findings. There is a long tradition in finance see, e.g., Cox and Ross 1976 that
Ž . models stock return volatility as negatively correlated with stock re￾turns. Influential articles by Black 1976 and Christie 1982 further Ž. Ž. document and attempt to explain the asymmetric volatility property of individual stock returns in the United States. The explanation put forward in these articles is based on leverage. A drop in the value of the stock negative return increases financial leverage, which makes the Ž . stock riskier and increases its volatility.1 Although, to many, ‘‘leverage effects’’ have become synonymous with asymmetric volatility, the asymmetric nature of the volatility response to return shocks could simply reflect the existence of time-varying risk premiums Pindyck 1984 , French, Schwert, and Stambaugh, 1987 , and
Ž. Ž. Campbell and Hentschel 1992 . If volatility is priced, an anticipated Ž . increase in volatility raises the required return on equity, leading to an We have benefited from discussions with Darrell Duffie, Robert Engle, Heber Farnsworth, Ken Froot, Steve Grenadier, Masao Matsuda, Manju Puri, Hisayoshi Shindo, Ken Singleton, and Jeff Zwiebel. We also thank seminar participants at the American Finance Association Meetings in Chicago, the Nikko Research Center, Stanford University, University of California at San Diego, the Swedish School of Economics in Helsinki, and Tilburg University for useful com￾ments. We are especially grateful for the insightful comments of an anonymous referee, which greatly improved the article. Geert Bekaert acknowledges financial support from an NSF grant and Stanford University. Address correspondence to Guojun Wu, School of Business Adminis￾tration, University of Michigan, Ann Arbor, MI 48109, or e-mail: gjwu@umich.edu. 1 Black 1976 also discusses an operating leverage effect, induced by fixed costs of the firm, but Ž . that effect has received little attention in the finance literature. The Reiew of Financial Studies Spring 2000 Vol. 13, No. 1, pp. 142
2000 The Society for Financial Studies

The Review of Financial Studies/v 13 n 1 2000 immediate stock price decline.Hence the causality is different:the leverage hypothesis claims that return shocks lead to changes in condi- tional volatility,whereas the time-varying risk premium theory contends that return shocks are caused by changes in conditional volatility. Which effect is the main determinant of asymmetric volatility re- mains an open question.Studies focusing on the leverage hypothesis, such as Christie (1982)and Schwert (1989),typically conclude that it cannot account for the full volatility responses.Likewise,the time- varying risk premium theory enjoys only partial success.The volatility feedback story relies first of all on the well-documented fact that volatility is persistent.That is,a large realization of news,positive or negative,increases both current and future volatility.The second basic tenet of this theory is that there exists a positive intertemporal relation between expected return and conditional variance.The increased volatility then raises expected returns and lowers current stock prices, dampening volatility in the case of good news and increasing volatility in the case of bad news.Whereas such a relationship for the market portfolio would be consistent with the capital asset pricing model [CAPM;Sharpe (1964)],it only holds in general equilibrium settings under restrictive assumptions [see Backus and Gregory(1993),Camp- bell (1993),and the discussion in Glosten,Jagannathan,and Runkle (1993)1. Moreover,there are conflicting empirical findings.For example, French,Schwert,and Stambaugh (1987)and Campbell and Hentschel (1992)find the relation between volatility and expected return to be positive,while Turner,Startz,and Nelson (1989),Glosten,Jagannathan, and Runkle (1993),and Nelson (1991)find the relation to be negative. Often the coefficient linking volatility to returns is statistically insignifi- cant.If the relation between market conditional volatility and market expected return is not positive,then the validity of the time-varying risk premium story is in doubt. Furthermore,the time-varying risk premium story does not readily explain the existence of volatility asymmetry at the firm level,since,in the CAPM example,the relevant measure of risk is then the covariance with the market portfolio.For the time-varying risk premium story to explain firm-specific volatility asymmetry,covariances with the market portfolio should respond positively to increases in market volatility. Our first contribution is to develop a general empirical framework to examine volatility asymmetry at the market level and at the firm or portfolio level simultaneously and to differentiate between the two competing explanations.That such an analysis has not been done before reflects the existence of two virtually separate literatures.As the sur- vey of empirical articles in Table 1 shows,studies focusing on the time- varying risk premium story typically use market-level returns,whereas 2

The Reiew of Financial Studies 13 n 1 2000 immediate stock price decline. Hence the causality is different: the leverage hypothesis claims that return shocks lead to changes in condi￾tional volatility, whereas the time-varying risk premium theory contends that return shocks are caused by changes in conditional volatility. Which effect is the main determinant of asymmetric volatility re￾mains an open question. Studies focusing on the leverage hypothesis, such as Christie 1982 and Schwert 1989 , typically conclude that Ž. Ž. it cannot account for the full volatility responses. Likewise, the time￾varying risk premium theory enjoys only partial success. The volatility feedback story relies first of all on the well-documented fact that volatility is persistent. That is, a large realization of news, positive or negative, increases both current and future volatility. The second basic tenet of this theory is that there exists a positive intertemporal relation between expected return and conditional variance. The increased volatility then raises expected returns and lowers current stock prices, dampening volatility in the case of good news and increasing volatility in the case of bad news. Whereas such a relationship for the market portfolio would be consistent with the capital asset pricing model
CAPM; Sharpe 1964 , it only holds in general equilibrium settings Ž . under restrictive assumptions see Backus and Gregory 1993 , Camp-
Ž . bell 1993 , and the discussion in Glosten, Jagannathan, and Runkle Ž . Ž . 1993 . Moreover, there are conflicting empirical findings. For example, French, Schwert, and Stambaugh 1987 and Campbell and Hentschel Ž . Ž . 1992 find the relation between volatility and expected return to be positive, while Turner, Startz, and Nelson 1989 , Glosten, Jagannathan, Ž . and Runkle 1993 , and Nelson 1991 find the relation to be negative. Ž. Ž. Often the coefficient linking volatility to returns is statistically insignifi- cant. If the relation between market conditional volatility and market expected return is not positive, then the validity of the time-varying risk premium story is in doubt. Furthermore, the time-varying risk premium story does not readily explain the existence of volatility asymmetry at the firm level, since, in the CAPM example, the relevant measure of risk is then the covariance with the market portfolio. For the time-varying risk premium story to explain firm-specific volatility asymmetry, covariances with the market portfolio should respond positively to increases in market volatility. Our first contribution is to develop a general empirical framework to examine volatility asymmetry at the market level and at the firm or portfolio level simultaneously and to differentiate between the two competing explanations. That such an analysis has not been done before reflects the existence of two virtually separate literatures. As the sur￾vey of empirical articles in Table 1 shows, studies focusing on the time￾varying risk premium story typically use market-level returns, whereas 2

Asymmetric Volatility and Risk in Equity Markets Table 1 Summary of selected empirical studies on asymmetric volatility Study Volatility measure Presence of asymmetry Explanation B1ack(1976) Gross volatility Stocks.portfolios Leverage hypothesis Christie(1982) Gross volatility Stocks.portfolios Leverage hypothesis French,Schwert and Conditional volatility Index Time-varying risk Stambaugh (1987) premium theory Schwert(1990) Conditional volatility Index Leverage hypothesis Nelson (1991) Conditional volatility Index Unspecified Campbell and Conditional volatility Index Time-varying risk Hentschel (1992) premium theory Cheung and Ng(1992) Conditional volatility Stocks Unspecified Engle and Ng (1993) Conditional volatility Index (Japan Topix) Unspecified Glosten,Jagannathan Conditional volatility Index Unspecified and Runkle (1993) Bae and Karolyi (1994) Conditional volatility Index Unspecified Braun,Nelson and Conditional volatility Index and stocks Unspecified Sunier(199s)】 Duffee (1995) Gross volatility Stocks Leverage hypothesis Ng(1996) Conditional volatility Index Unspecified Bekaert and Conditional volatility Index(Emerging Unspecified Harvey (1997) Markets) This table lists a sample of studies on the relationship between returns and conditional volatility.Conditional volatility studies typically use GARCH models to measure volatility; "gross volatility"typically refers to the standard deviation of daily returns computed over the course of a month.The "unspecified"label in the explanation column means that asymmetry was modeled but the researchers did not specify the exact cause of asymmetry. studies focusing on the leverage hypothesis typically use firm or portfo- lio data.Moreover,the empirical specifications are not entirely compat- ible across the two literatures.Studies focusing on individual firms typically use regression analysis to examine the relation between a measure of volatility during a particular month ("gross"volatility)and the return in the previous month.Studies at the market level have mostly used the GARCH-in-mean framework of Engle,Lilien,and Robbins (1987),focusing on the relation between return innovations and the conditional volatility of the returns (see Table 1).Our model, while using a related framework,nests the riskless debt model for individual firms in Christie (1982). Our second contribution is to document a new phenomenon that helps explain volatility asymmetry at the firm level:covariance asymme- try.When the conditional covariance between market and stock returns responds more to negative than to positive market shocks the volatility feedback effect is particularly strong.Our empirical framework accom- modates this possibility and we find evidence of such covariance asym- metry.Although Kroner and Ng(1998)document covariance asymmetry in the volatility dynamics of portfolios of small and large firms,most previous studies have focused on asymmetric effects in conditional betas [see Ball and Kothari (1989),Braun,Nelson,and Sunier (1995)]with 3

Asymmetric Volatility and Risk in Equity Markets Table 1 Summary of selected empirical studies on asymmetric volatility Study Volatility measure Presence of asymmetry Explanation Black 1976 Gross volatility Stocks, portfolios Leverage hypothesis Ž . Christie 1982 Gross volatility Stocks, portfolios Leverage hypothesis Ž . French, Schwert and Conditional volatility Index Time-varying risk Stambaugh 1987 premium theory Ž . Schwert 1990 Conditional volatility Index Leverage hypothesis Ž . Nelson 1991 Conditional volatility Index Unspecified Ž . Campbell and Conditional volatility Index Time-varying risk Hentschel 1992 premium theory Ž . Cheung and Ng 1992 Conditional volatility Stocks Unspecified Ž . Engle and Ng 1993 Conditional volatility Index Japan Topix Unspecified Ž. Ž . Glosten, Jagannathan Conditional volatility Index Unspecified and Runkle 1993 Ž . Bae and Karolyi 1994 Conditional volatility Index Unspecified Ž . Braun, Nelson and Conditional volatility Index and stocks Unspecified Sunier 1995 Ž . Duffee 1995 Gross volatility Stocks Leverage hypothesis Ž . Ng 1996 Conditional volatility Index Unspecified Ž . Bekaert and Conditional volatility Index Emerging Unspecified Ž Harvey 1997 Markets Ž. . This table lists a sample of studies on the relationship between returns and conditional volatility. Conditional volatility studies typically use GARCH models to measure volatility; ‘‘gross volatility’’ typically refers to the standard deviation of daily returns computed over the course of a month. The ‘‘unspecified’’ label in the explanation column means that asymmetry was modeled but the researchers did not specify the exact cause of asymmetry. studies focusing on the leverage hypothesis typically use firm or portfo￾lio data. Moreover, the empirical specifications are not entirely compat￾ible across the two literatures. Studies focusing on individual firms typically use regression analysis to examine the relation between a measure of volatility during a particular month ‘‘gross’’ volatility and Ž . the return in the previous month. Studies at the market level have mostly used the GARCH-in-mean framework of Engle, Lilien, and Robbins 1987 , focusing on the relation between return innovations Ž . and the conditional volatility of the returns see Table 1 . Our model, Ž . while using a related framework, nests the riskless debt model for individual firms in Christie 1982 . Ž . Our second contribution is to document a new phenomenon that helps explain volatility asymmetry at the firm level: covariance asymme￾try. When the conditional covariance between market and stock returns responds more to negative than to positive market shocks the volatility feedback effect is particularly strong. Our empirical framework accom￾modates this possibility and we find evidence of such covariance asym￾metry. Although Kroner and Ng 1998 document covariance asymmetry Ž . in the volatility dynamics of portfolios of small and large firms, most previous studies have focused on asymmetric effects in conditional betas
see Ball and Kothari 1989 , Braun, Nelson, and Sunier 1995 with Ž. Ž. 3

The Review of Financial Studies/v 13 n 1 2000 conflicting empirical results.We argue below that asymmetry is more likely to be found in conditional covariances and re-examine whether conditional betas display asymmetry for our sample. Third,since our model combines modeling volatility dynamics and risk premiums,we quantify the risk implications of the estimated volatility dynamics.Most applications of GARCH models,with a few exceptions,have not yet embraced asymmetric volatility models.For example,parameterizations of CAPM models that use GARCH [see, e.g.,Engel et al.(1995)1,models of volatility spillover across equity markets [see,e.g.,Hamao,Masulis,and Ng (1990)],and stochastic volatility models for options [Hull and White (1987)]have typically not used asymmetric volatility models.2 This is surprising since a number of sophisticated models have been developed to accommodate asymmetric volatility [see,e.g.,Nelson (1991),Glosten,Jagannathan,and Runkle (1993),and Hentschel (1995)],and the results in Pagan and Schwert (1990)and Engle and Ng (1993)indicate that these volatility models outperform standard GARCH models.If these models yield different conditional volatilities from symmetric GARCH models,their economic implications will be different too.With an asymmetric volatility model, risk and the cost of capital may increase more in response to negative market return shocks than in response to positive shocks.Whereas the economic importance of such effects is indisputable,it is not ex ante clear that statistically significant asymmetric volatility has economically important risk implications. Finally,whereas most of the empirical analysis so far (see Table 1) has focused on U.S.stock returns,our empirical application focuses on the market return and portfolio returns constructed from Japanese stocks in the Nikkei index.As Engle and Ng (1993)conclude for the Japanese Topix index,our results indicate that asymmetry is an impor- tant feature of stock market volatility in the Japanese market as well. The remainder of the article is organized as follows.Section 1 formulates our empirical model,the empirical hypotheses,and explains the role of leverage in generating asymmetric risk and volatility.A set of specification tests is also discussed.Section 2 discusses the data and the empirical results.Section 3 considers the economic implications of our model and Section 4 evaluates the robustness of the empirical results. The final section summarizes the results and outlines directions for further research. 2Exceptions are Koutmos and Booth(1995)and Ng(1996)in the volatility spillover literature and Amin and Ng (1993),Duan (1995),and Wu (1998)in the options literature. 4

The Reiew of Financial Studies 13 n 1 2000 conflicting empirical results. We argue below that asymmetry is more likely to be found in conditional covariances and re-examine whether conditional betas display asymmetry for our sample. Third, since our model combines modeling volatility dynamics and risk premiums, we quantify the risk implications of the estimated volatility dynamics. Most applications of GARCH models, with a few exceptions, have not yet embraced asymmetric volatility models. For example, parameterizations of CAPM models that use GARCH see,
e.g., Engel et al. 1995 , models of volatility spillover across equity Ž . markets see, e.g., Hamao, Masulis, and Ng 1990 , and stochastic
Ž . volatility models for options Hull and White 1987 have typically not
Ž . used asymmetric volatility models.2 This is surprising since a number of sophisticated models have been developed to accommodate asymmetric volatility see, e.g., Nelson 1991 , Glosten, Jagannathan, and Runkle
Ž . Ž. Ž. 1993 , and Hentschel 1995 , and the results in Pagan and Schwert Ž. Ž. 1990 and Engle and Ng 1993 indicate that these volatility models outperform standard GARCH models. If these models yield different conditional volatilities from symmetric GARCH models, their economic implications will be different too. With an asymmetric volatility model, risk and the cost of capital may increase more in response to negative market return shocks than in response to positive shocks. Whereas the economic importance of such effects is indisputable, it is not ex ante clear that statistically significant asymmetric volatility has economically important risk implications. Finally, whereas most of the empirical analysis so far see Table 1 Ž . has focused on U.S. stock returns, our empirical application focuses on the market return and portfolio returns constructed from Japanese stocks in the Nikkei index. As Engle and Ng 1993 conclude for the Ž . Japanese Topix index, our results indicate that asymmetry is an impor￾tant feature of stock market volatility in the Japanese market as well. The remainder of the article is organized as follows. Section 1 formulates our empirical model, the empirical hypotheses, and explains the role of leverage in generating asymmetric risk and volatility. A set of specification tests is also discussed. Section 2 discusses the data and the empirical results. Section 3 considers the economic implications of our model and Section 4 evaluates the robustness of the empirical results. The final section summarizes the results and outlines directions for further research. 2 Exceptions are Koutmos and Booth 1995 and Ng 1996 in the volatility spillover literature and Ž. Ž. Amin and Ng 1993 , Duan 1995 , and Wu 1998 in the options literature. Ž. Ž. Ž. 4

Asymmetric Volatility and Risk in Equity Markets 1.A Model of Asymmetric Volatility and Risk 1.1 Asymmetric volatility and risk at the firm and market level To establish notation,let P.denote the market index,let rM.denote the return on the market portfolio,and let rM.+1=E(rM.+)+ M.,where I,denotes the information set at time t.Similarly,P are the price and return of the stock of firm i,respectively,and ri.+=E(ri.+)++Define conditional variances and covari- ances,.+1=var(rM.+),+1=var(ri.)and iM.+1= cov(r.+M.+). Definition.A return ri,displays asymmetric volatility if var[y.+ilh,∈.var[y.+l山，∈.t>0]-o(1) In other words,negative unanticipated returns result in an upward revision of the conditional volatility,whereas positive unanticipated returns result in a smaller upward or even a downward revision of the conditional volatility.3 One explanation for such asymmetry at the equity level relies on changes in leverage.To illustrate,consider a world where debt is riskless,that is,the return on all debt equals the risk-free rate.We denote the risk-free rate by r1.since it is known at t-1.It is straightforward to show that .,-11=(1+LR.-)(G.-t) (2) where LRi.-1 is the leverage ratio for firm i and .refers to the return on the firm's assets.+Even when the volatility of the return on a firm's assets is constant,the conditional volatility of the equity return should change when leverage changes [see also Christie (1982)and Schwert (1989)].In particular,shocks that increase the value of the firm, reduce leverage,and with it the conditional volatility of the stock's return and vice versa. 3 We will refer to the latter case as"strong asymmetry,"which implies var [r+ill,e >0]-20. 4With D(E)denoting the value of debt (equity),the leverage ratio is the debt:equity ratio: LR=Di./E..The firm return is the value-weighted sum of the return on debt and the DuD retum on equity,D+E E.Multiplying both sides by Di1+E-and dividing through by E.-1,we obtain Equation (2)after rearranging terms. 5

Asymmetric Volatility and Risk in Equity Markets 1. A Model of Asymmetric Volatility and Risk 1.1 Asymmetric volatility and risk at the firm and market level To establish notation, let PM denote the market index, let r denote , t M, t the return on the market portfolio, and let rM, t1  Er I Ž M, t1
t .  M , where I denotes the information set at time t. Similarly, P , r , t1 t i, t i, t are the price and return of the stock of firm i, respectively, and r  Er I Ž
.  . Define conditional variances and covari- i, t1 i, t1 t i, t1 2 Ž
. 2 ances, M, t1  var r I M, t1 t i ,  , t1  varŽr I i, t1
t iM . and  , t1  covŽr , r I
.. i, t1 M, t1 t Definition. A return r displays asymmetric olatility if i, t 2 2 var r I

, 0  var r I , 0   . 1Ž . i, t1 t i, t i, t i, t1 t i, t i, t In other words, negative unanticipated returns result in an upward revision of the conditional volatility, whereas positive unanticipated returns result in a smaller upward or even a downward revision of the conditional volatility.3 One explanation for such asymmetry at the equity level relies on changes in leverage. To illustrate, consider a world where debt is riskless, that is, the return on all debt equals the risk-free rate. We denote the risk-free rate by r f , since it is known at t  1. It is t1, t straightforward to show that f f ri, t t  r 1, t i  Ž . Ž. 1  LR r , t1 Ž . i, t t  r 1, t , 2 where LR is the leverage ratio for firm i and r refers to the i, t1 i, t return on the firm’s assets.4 Even when the volatility of the return on a firm’s assets is constant, the conditional volatility of the equity return should change when leverage changes see also Christie 1982 and
Ž . Schwert 1989 . In particular, shocks that increase the value of the firm, Ž . reduce leverage, and with it the conditional volatility of the stock’s return and vice versa. 3 We will refer to the latter case as ‘‘strong asymmetry,’’ which implies
2 var
r I t1 tt t , 0   0, and 2 var
r I
, 0   0. t1 tt t 4 With D EŽ. Ž . denoting the value of debt equity , the leverage ratio is the debt:equity ratio: i, t i, t LR  D E . The firm return is the value-weighted sum of the return on debt and the i, t i, t i, t D E i, t1 i, t1 f return on equity, r  r  r . Multiplying both sides by i, t t1, t i, t Di, t1  E D i, t1 i, t1  Ei, t1 D  E and dividing through by E , we obtain Equation 2 after rearranging terms. Ž . i, t1 i, t1 i, t1 5

The Review of Financial Studies/v 13 n 1 2000 Our analysis here is premised on two assumptions,which we test below.First,we assume that a conditional version of the CAPM holds that is,the market portfolio's expected excess return is the (constant) price of risk times the conditional variance of the market and the expected excess return on any firm is the price of risk times the conditional covariance between the firm's return and the market.Note that we formulate the volatility feedback effect at the level of the firm's total assets,since it does not at all depend on leverage.Second,we assume that conditional volatility is persistent,which is an empirical fact supported by extensive empirical work [see Bollerslev,Chou,and Kro- ner (1992)].Since the time variation in second moments is not restricted by the CAPM,we explicitly parameterize it in the next subsection.For now,we consider more generally the mechanisms generating asymme- try,including leverage and volatility feedback,at the market level and firm level using the flow chart in Figure 1. We begin by considering news (shocks)at the market level.Bad news at the market level has two effects.First,whereas news is evidence of higher current volatility in the market,investors also likely revise the conditional variance since volatility is persistent.According to the Volatility Feedback Market level Shocks: Leverage Effect dwua Risk Premiu Pu Tu Ewu Persistence Persistence News Leverage Effact→ 3 Persistence Firm Level Shocks: OMJ小 Risk Premium E,) B,h,6 Volatility Feedback Figure 1 News impact at the market level and the firm level This figure shows the impact of market (eM.)and firm (shocks on conditional variances (+and covariances (iM,+).Feedback effects on current prices (P.P.)and returns(ri.rM.)originating from risk premium changes are also shown. 6

The Reiew of Financial Studies 13 n 1 2000 Our analysis here is premised on two assumptions, which we test below. First, we assume that a conditional version of the CAPM holds, that is, the market portfolio’s expected excess return is the constant Ž . price of risk times the conditional variance of the market and the expected excess return on any firm is the price of risk times the conditional covariance between the firm’s return and the market. Note that we formulate the volatility feedback effect at the level of the firm’s total assets, since it does not at all depend on leverage. Second, we assume that conditional volatility is persistent, which is an empirical fact supported by extensive empirical work see Bollerslev, Chou, and Kro-
ner 1992 . Since the time variation in second moments is not restricted Ž . by the CAPM, we explicitly parameterize it in the next subsection. For now, we consider more generally the mechanisms generating asymme￾try, including leverage and volatility feedback, at the market level and firm level using the flow chart in Figure 1. We begin by considering news shocks at the market level. Bad news Ž . at the market level has two effects. First, whereas news is evidence of higher current volatility in the market, investors also likely revise the conditional variance since volatility is persistent. According to the Figure 1 News impact at the market level and the firm level This figure shows the impact of market Ž . Ž. M and firm shocks on conditional variances , t i, t Ž 2 2 M, t1, i, t1. Ž. Ž . and covariances i M, t1 . Feedback effects on current prices Pi, t M , P , t and returns Ž . r , r originating from risk premium changes are also shown. i, t M, t 6

Asymmetric Volatility and Risk in Equity Markets CAPM,this increased conditional volatility at the market level has to be compensated by a higher expected return,leading to an immediate decline in the current value of the market [see also Campbell and Hentschel (1992)].The price decline will not cease until the expected return is sufficiently high.Hence a negative return shock may generate a significant increase in conditional volatility.Second,the marketwide price decline leads to higher leverage at the market level and hence higher stock volatility.That is,the leverage effect reinforces the volatil- ity feedback effect.Note that although the arrows in Figure 1 suggest a sequence of events,the effects described above happen simultaneously, that is,leverage and feedback effects interact. When good news arrives in the market,there are again two effects. First,news brings about higher current period market volatility and an upward revision of the conditional volatility.When volatility increases, prices decline to induce higher expected returns,offsetting the initial price movement.The volatility feedback effect dampens the original volatility response.Second,the resulting market rally (positive return shock)reduces leverage and decreases conditional volatility at the market level.Hence the net impact on stock return volatility is not clear. As Figure 1 shows,for the initial impact of news at the firm level,the reasoning remains largely the same:bad and good news generate opposing leverage effects which reinforce (offset)the volatility embed- ded in the bad (good)news event.What is different is the volatility feedback.A necessary condition for volatility feedback to be observed at the firm level is that the covariance of the firm's return increases in response to market shocks.If the shock is completely idiosyncratic,the covariance between the market return and individual firm return should not change,and no change in the required risk premium occurs.Hence idiosyncratic shocks generate volatility asymmetry purely through a leverage effect.Volatility feedback at the firm level occurs when mar- ketwide shocks increase the covariance of the firm's return with the market.Such covariance behavior would be implied by a CAPM model with constant (positive)firm betas and seems generally plausible.The impact on the conditional covariance is likely to be different across firms.For firms with high systematic risk,marketwide shocks may significantly increase their conditional covariance with the market.The resulting higher required return then leads to a volatility feedback effect on the conditional volatility,which would be absent or weaker for firms less sensitive to market level shocks.From Equation(2),it also follows that any volatility feedback effect at the firm level leads to more pronounced feedback effects at the stock level the more leveraged the firm is. 7

Asymmetric Volatility and Risk in Equity Markets CAPM, this increased conditional volatility at the market level has to be compensated by a higher expected return, leading to an immediate decline in the current value of the market see also Campbell and
Hentschel 1992 . The price decline will not cease until the expected Ž . return is sufficiently high. Hence a negative return shock may generate a significant increase in conditional volatility. Second, the marketwide price decline leads to higher leverage at the market level and hence higher stock volatility. That is, the leverage effect reinforces the volatil￾ity feedback effect. Note that although the arrows in Figure 1 suggest a sequence of events, the effects described above happen simultaneously, that is, leverage and feedback effects interact. When good news arrives in the market, there are again two effects. First, news brings about higher current period market volatility and an upward revision of the conditional volatility. When volatility increases, prices decline to induce higher expected returns, offsetting the initial price movement. The volatility feedback effect dampens the original volatility response. Second, the resulting market rally positive return Ž shock reduces leverage and decreases conditional volatility at the . market level. Hence the net impact on stock return volatility is not clear. As Figure 1 shows, for the initial impact of news at the firm level, the reasoning remains largely the same: bad and good news generate opposing leverage effects which reinforce offset the volatility embed- Ž . ded in the bad good news event. What is different is the volatility Ž . feedback. A necessary condition for volatility feedback to be observed at the firm level is that the covariance of the firm’s return increases in response to market shocks. If the shock is completely idiosyncratic, the covariance between the market return and individual firm return should not change, and no change in the required risk premium occurs. Hence idiosyncratic shocks generate volatility asymmetry purely through a leverage effect. Volatility feedback at the firm level occurs when mar￾ketwide shocks increase the covariance of the firm’s return with the market. Such covariance behavior would be implied by a CAPM model with constant positive firm betas and seems generally plausible. The Ž . impact on the conditional covariance is likely to be different across firms. For firms with high systematic risk, marketwide shocks may significantly increase their conditional covariance with the market. The resulting higher required return then leads to a volatility feedback effect on the conditional volatility, which would be absent or weaker for firms less sensitive to market level shocks. From Equation 2 , it also Ž . follows that any volatility feedback effect at the firm level leads to more pronounced feedback effects at the stock level the more leveraged the firm is. 7

The Review of Financial Studies/v 13 n 1 2000 The volatility feedback effect would be stronger if covariances re- spond asymmetrically to market shocks.We call this phenomenon covariance asymmetry.So far,covariance asymmetry has primarily re- ceived attention in the literature on international stock market linkages, where larger comovements of equity returns in down markets adversely affect the benefits of international diversification [Ang and Bekaert (1998)and Das and Uppal (1996)].Kroner and Ng (1998)document covariance asymmetry in stock returns on U.S.portfolios of small and large firms without providing an explanation. There are two channels through which covariance asymmetry can arise naturally and both channels are embedded in our empirical specification.First,covariance asymmetry in stock returns could be partially explained by a pure leverage effect,without volatility feedback. Using the riskless debt model,it follows that cov-1i.-rf1 M.- =(1+LR,-1)1+LRM,4-) ×cov-i.,-f,iM,-h1, (3) Even with constant covariance at the firm level,the covariance of an individual stock return with the market may exhibit(strong)asymmetry. Conditional stock return betas are somewhat less likely to display pure leverage effects,since 1+LRi.-LB.-1, B.-1=1+LRM-1 (4) where B(B)is the firm (stock)beta.Hence,idiosyncratic shocks should result in asymmetric beta behavior,but the effect of marketwide shocks on betas is ambiguous. Second,at the firm level as well,covariance asymmetry arises more naturally than beta asymmetry.Suppose the conditional beta of a firm is positive but constant over time,still a popular assumption in many asset pricing models.Then the conditional covariance with the market return is proportional to the conditional variance of the market.Hence a market shock that raises the market's conditional variance increases the required risk premium on the firm (unless the price of risk changes)and causes a volatility feedback effect.When the effect of the market shock on market volatility is asymmetric,the firm(and stock)return automati- cally displays covariance asymmetry.Of course,betas do vary over time [see Jagannathan and Wang (1995)and Ghysels (1998)for recent discussions]and may exhibit asymmetry as well,but there is no model we know of that predicts beta asymmetry at the firm level.In the framework set out below,we impose only mild restrictions on the 8

The Reiew of Financial Studies 13 n 1 2000 The volatility feedback effect would be stronger if covariances re￾spond asymmetrically to market shocks. We call this phenomenon coariance asymmetry. So far, covariance asymmetry has primarily re￾ceived attention in the literature on international stock market linkages, where larger comovements of equity returns in down markets adversely affect the benefits of international diversification Ang and Bekaert
Ž. Ž. 1998 and Das and Uppal 1996 . Kroner and Ng 1998 document Ž . covariance asymmetry in stock returns on U.S. portfolios of small and large firms without providing an explanation. There are two channels through which covariance asymmetry can arise naturally and both channels are embedded in our empirical specification. First, covariance asymmetry in stock returns could be partially explained by a pure leverage effect, without volatility feedback. Using the riskless debt model, it follows that f f cov r  r , r  r t1 i, t t1, t M , t t1, t  Ž .Ž . 1  LRi, t1 1  LRM , t1  f f cov r  r , r  r . 3Ž . t1 i, t t1, t M , t t1, t Even with constant covariance at the firm level, the covariance of an individual stock return with the market may exhibit strong asymmetry. Ž . Conditional stock return betas are somewhat less likely to display pure leverage effects, since 1  LRi, t1    , 4Ž . i, t1 i, t1 1  LRM , t1 where  Ž . Ž. is the firm stock beta. Hence, idiosyncratic shocks i, t1 i, t1 should result in asymmetric beta behavior, but the effect of marketwide shocks on betas is ambiguous. Second, at the firm level as well, covariance asymmetry arises more naturally than beta asymmetry. Suppose the conditional beta of a firm is positive but constant over time, still a popular assumption in many asset pricing models. Then the conditional covariance with the market return is proportional to the conditional variance of the market. Hence a market shock that raises the market’s conditional variance increases the required risk premium on the firm unless the price of risk changes and Ž . causes a volatility feedback effect. When the effect of the market shock on market volatility is asymmetric, the firm and stock return automati- Ž . cally displays covariance asymmetry. Of course, betas do vary over time
see Jagannathan and Wang 1995 and Ghysels 1998 for recent Ž. Ž. discussions and may exhibit asymmetry as well, but there is no model we know of that predicts beta asymmetry at the firm level. In the framework set out below, we impose only mild restrictions on the 8

Asymmetric Volatility and Risk in Equity Markets behavior of betas over time and we examine whether they exhibit asymmetry. 1.2 Empirical model specification We use a conditional version of the CAPM to examine the interaction between the means and variances of individual stock returns and the market return.The conditional mean equations are defined as rM.t -r-1.t=Yi-10Mt +EM.t r1.t-{11=Y-1o1M1+e1,4 (5) rn.t -rf1.t Yi-1CnM.t en.t where r is the one-period risk-free interest rate known at time t-1,Y-is the price of risk,M denotes the market portfolio,and n is the number of other portfolios included in the study.Naturally these portfolios are classified by the leverage ratios of the underlying firms, with portfolio 1 having the highest leverage and portfolio n the lowest. We call these portfolios the leverage portfolios. The time variation in the price of risk depends on market leverage: Y Y-1=1+LRM-1 (6) This specification for the price of risk follows from formulating the CAPM at the firm level,not the equity level,with a constant price of risk.That is, y=E-iw小-hu 品： (7) where the bars indicate firm values rather than equity values.Under certain assumptions,Y is the aggregate coefficient of relative risk aversion [see Campbell (1993)].It is critical in this context that the return used in Equation (7)is a good proxy to the return on the aggregate wealth portfolio.Since the stock index we use in the empirical work is highly levered,M.is a better proxy than r.Of course,the specification in Equation (6)relies on the riskless debt model.However, we subject the model to a battery of specification tests,some of which are specifically designed with alternatives to the riskless debt model in mind. 5 Jagannathan,Kubota,and Takehara (1998)argue that a portfolio of listed stocks is unlikely to be a good proxy for the aggregate wealth portfolio in Japan and find that labor income is priced. They ignore leverage effects,however. 9

Asymmetric Volatility and Risk in Equity Markets behavior of betas over time and we examine whether they exhibit asymmetry. 1.2 Empirical model specification We use a conditional version of the CAPM to examine the interaction between the means and variances of individual stock returns and the market return. The conditional mean equations are defined as  f 2 rM  r  Y   , t t1, t t1 M , t M , t f r  r  Y   1, t t1, t t1 1 M , t 1, t ... , 5Ž . ... ... f r  r  Y   n, t t1, t t1 n M , t n, t where r f is the one-period risk-free interest rate known at time t1, t t  1, Y is the price of risk, M denotes the market portfolio, and n is t1 the number of other portfolios included in the study. Naturally these portfolios are classified by the leverage ratios of the underlying firms, with portfolio 1 having the highest leverage and portfolio n the lowest. We call these portfolios the leverage portfolios. The time variation in the price of risk depends on market leverage: Y Y  . 6Ž . t1 1  LRM , t1 This specification for the price of risk follows from formulating the CAPM at the firm level, not the equity level, with a constant price of risk. That is, f E r  r t1 M , t t1, t Y  , 7Ž . 2 M , t where the bars indicate firm values rather than equity values. Under certain assumptions, Y is the aggregate coefficient of relative risk aversion see Campbell 1993 . It is critical in this context that the
Ž . return used in Equation 7 is a good proxy to the return on the Ž . aggregate wealth portfolio. Since the stock index we use in the empirical 5 work is highly levered, rM is a better proxy than r . Of course, the , t M, t specification in Equation 6 relies on the riskless debt model. However, Ž . we subject the model to a battery of specification tests, some of which are specifically designed with alternatives to the riskless debt model in mind. 5 Jagannathan, Kubota, and Takehara 1998 argue that a portfolio of listed stocks is unlikely to Ž . be a good proxy for the aggregate wealth portfolio in Japan and find that labor income is priced. They ignore leverage effects, however. 9

The Review of Financial Studies/v 13 n 1 2000 Since the CAPM does not restrict the time variation in second moments,we employ a multivariate GARCH model.Specifically,the variance-covariance matrix follows an asymmetric version of the BEKK model [Baba et al.(1989),Engle and Kroner (1995),and Kroner and Ng (1998)].This GARCH-in-mean parameterization of the CAPM,incor- porating an equation for the market portfolio,is similar to the interna- tional CAPM parameterization in Bekaert and Harvey (1995)and DeSantis and Gerard (1997),with more general volatility dynamics.In particular,note that the individual shocks need not add up to the market portfolio shock,since we only consider a limited number of leverage-sorted portfolios. To clearly distinguish the leverage effect from volatility feedback,we formulate our GARCH model at the firm level. Define EM.t MM.t e1,1 1,t ,7 .if,<0 7.t i.(8) otherwise en.t 门h,t The bars indicate firm shocks.Of course,they are related to stock return shocks through leverage, e,4=(1+LR,1-i)e. The conditional variance covariance matrix at the firm level is 最： 0M1,t OMn.! E,=E(e,|1,-)= UM1.I 品， 1, (9) 0Mn,1 01n,i which is modeled as E,=20'+BE-1B'+CE,-1(-1C+Dm-1-1D'. (10) In "VEC"notation the model becomes VEC(E)=#+B#VEC(-1) +C*VEC(E-1(-)+D#VEC(--1),(11) with#=VEC(2'),B*=B⑧B,C#=C⑧C,andD*=D⑧D. A,B,C,and D are n+1 by n+1 constant matrices,with elements j and bij,etc.The conditional variance and covariance of each excess 10

The Reiew of Financial Studies 13 n 1 2000 Since the CAPM does not restrict the time variation in second moments, we employ a multivariate GARCH model. Specifically, the variance-covariance matrix follows an asymmetric version of the BEKK model Baba et al. 1989 , Engle and Kroner 1995 , and Kroner and Ng
Ž. Ž. Ž . 1998 . This GARCH-in-mean parameterization of the CAPM, incor- porating an equation for the market portfolio, is similar to the interna￾tional CAPM parameterization in Bekaert and Harvey 1995 and Ž . DeSantis and Gerard 1997 , with more general volatility dynamics. In Ž . particular, note that the individual shocks need not add up to the market portfolio shock, since we only consider a limited number of leverage-sorted portfolios. To clearly distinguish the leverage effect from volatility feedback, we formulate our GARCH model at the firm level. Define  M , t M , t  1, t 1, t i, t i if , t 0  . . ,   ,   i. 8Ž . tt i . . , t ½ 0 otherwise . . 0 0  n, t n, t The bars indicate firm shocks. Of course, they are related to stock return shocks through leverage,  Ž . 1  LR . i, t i, t1 i, t The conditional variance covariance matrix at the firm level is 2   M , t M 1, t Mn   , t 2   M 1, t 1, t  1n, t  Ý  EŽ .
I  . . .. , 9Ž . t tt t1 . . .. . . ..  0 2   M n, t 1n, t n   , t which is modeled as   Ý   BÝ B  C C  D  D . 10 Ž . t t1 t1 t1 t1 t1 In ‘‘VEC’’ notation the model becomes VEC Ž. Ž . Ýt t   B VEC Ý 1    C VEC Ž. Ž.  D VEC   , 11 Ž . t1 t1 t1 t1 Ž . with  VEC , B  B  B, C  C  C, and D  D  D. , B, C, and D are n  1 by n  1 constant matrices, with elements  and b , etc. The conditional variance and covariance of each excess ij ij 10