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 asymmetry, 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 volatility 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 marketwide 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 respond asymmetrically to market shocks. We call this phenomenon coariance asymmetry. So far, covariance asymmetry has primarily received 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 international 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