《金融投资学》教学资源（英文文献）Have Individual Stocks Become More Volatile - An Empirical Exploration of Idiosyncratic Risk.

THE JOURNAL OF FINANCE.VOL.LVI,NO.1.FEBRUARY 2001 Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk JOHN Y.CAMPBELL,MARTIN LETTAU,BURTON G.MALKIEL, and YEXIAO XU* ABSTRACT This paper uses a disaggregated approach to study the volatility of common stocks at the market,industry,and firm levels.Over the period from 1962 to 1997 there has been a noticeable increase in firm-level volatility relative to market volatility. Accordingly,correlations among individual stocks and the explanatory power of the market model for a typical stock have declined,whereas the number of stocks needed to achieve a given level of diversification has increased.All the volatility measures move together countercyclically and help to predict GDP growth.Market volatility tends to lead the other volatility series.Factors that may be responsible for these findings are suggested. IT Is BY NOw A COMMONPLACE OBSERVATION that the volatility of the aggregate stock market is not constant,but changes over time.Economists have built increasingly sophisticated statistical models to capture this time variation in volatility.Simple filters such as the rolling standard deviation used by Officer (1973)have given way to parametric ARCH or stochastic-volatility models.Partial surveys of the enormous literature on these models are given by Bollerslev,Chou,and Kroner (1992),Hentschel (1995),Ghysels,Harvey, and Renault (1996),and Campbell,Lo,and MacKinlay (1997,Chapter 12). Aggregate volatility is,of course,important in almost any theory of risk and return,and it is the volatility experienced by holders of aggregate index funds.But the aggregate market return is only one component of the return to an individual stock.Industry-level and idiosyncratic firm-level shocks are also important components of individual stock returns.There are several reasons to be interested in the volatilities of these components. John Y.Campbell is at Harvard University,Department of Economics and NBER;Lettau is at the Federal Reserve Bank of New York and CEPR;Malkiel is at Princeton University;and Xu is at the University of Texas at Dallas.This paper merges two independent projects,Camp- bell and Lettau (1999)and Malkiel and Xu(1999).Campbell and Lettau are grateful to Sang- joon Kim for his contributions to the first version of their paper,Campbell,Kim,and Lettau (1994).We thank two anonymous referees and Rene Stulz for useful comments and Benjamin Zhang for pointing out an error in a previous draft.Jung-Wook Kim and Matt Van Vlack pro- vided able research assistance.The views are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of New York or the Federal Reserve System.Any errors and omissions are the responsibility of the authors

Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk JOHN Y. CAMPBELL, MARTIN LETTAU, BURTON G. MALKIEL, and YEXIAO XU* ABSTRACT This paper uses a disaggregated approach to study the volatility of common stocks at the market, industry, and firm levels. Over the period from 1962 to 1997 there has been a noticeable increase in firm-level volatility relative to market volatility. Accordingly, correlations among individual stocks and the explanatory power of the market model for a typical stock have declined, whereas the number of stocks needed to achieve a given level of diversification has increased. All the volatility measures move together countercyclically and help to predict GDP growth. Market volatility tends to lead the other volatility series. Factors that may be responsible for these findings are suggested. IT IS BY NOW A COMMONPLACE OBSERVATION that the volatility of the aggregate stock market is not constant, but changes over time. Economists have built increasingly sophisticated statistical models to capture this time variation in volatility. Simple filters such as the rolling standard deviation used by Officer ~1973! have given way to parametric ARCH or stochastic-volatility models. Partial surveys of the enormous literature on these models are given by Bollerslev, Chou, and Kroner ~1992!, Hentschel ~1995!, Ghysels, Harvey, and Renault ~1996!, and Campbell, Lo, and MacKinlay ~1997, Chapter 12!. Aggregate volatility is, of course, important in almost any theory of risk and return, and it is the volatility experienced by holders of aggregate index funds. But the aggregate market return is only one component of the return to an individual stock. Industry-level and idiosyncratic firm-level shocks are also important components of individual stock returns. There are several reasons to be interested in the volatilities of these components. * John Y. Campbell is at Harvard University, Department of Economics and NBER; Lettau is at the Federal Reserve Bank of New York and CEPR; Malkiel is at Princeton University; and Xu is at the University of Texas at Dallas. This paper merges two independent projects, Camp￾bell and Lettau ~1999! and Malkiel and Xu ~1999!. Campbell and Lettau are grateful to Sang￾joon Kim for his contributions to the first version of their paper, Campbell, Kim, and Lettau ~1994!. We thank two anonymous referees and René Stulz for useful comments and Benjamin Zhang for pointing out an error in a previous draft. Jung-Wook Kim and Matt Van Vlack pro￾vided able research assistance. The views are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of New York or the Federal Reserve System. Any errors and omissions are the responsibility of the authors. THE JOURNAL OF FINANCE • VOL. LVI, NO. 1 • FEBRUARY 2001 1

2 The Journal of Finance First,many investors have large holdings of individual stocks;they may fail to diversify in the manner recommended by financial theory,or their holdings may be restricted by corporate compensation policies.These inves- tors are affected by shifts in industry-level and idiosyncratic volatility,just as much as by shifts in market volatility.Second,some investors who do try to diversify do so by holding a portfolio of 20 or 30 stocks.Conventional wisdom holds that such a portfolio closely approximates a well-diversified portfolio in which all idiosyncratic risk is eliminated.However,the adequacy of this approximation depends on the level of idiosyncratic volatility in the stocks making up the portfolio.Third,arbitrageurs who trade to exploit the mispricing of an individual stock(as opposed to a pattern of mispricing across many stocks)face risks that are related to idiosyncratic return volatility,not aggregate market volatility.Larger pricing errors are possible when idiosyn- cratic firm-level volatility is high (Ingersoll (1987),Chapter 7,Shleifer and Vishny (1997)).Fourth,firm-level volatility is important in event studies. Events affect individual stocks,and the statistical significance of abnormal event-related returns is determined by the volatility of individual stock re- turns relative to the market or industry (Campbell et al.(1997),Chapter 4). Finally,the price of an option on an individual stock depends on the total volatility of the stock return,including industry-level and idiosyncratic vol- atility as well as market volatility. Disaggregated volatility measures also have important relations with ag- gregate output in some macroeconomic models.Models of sectoral realloca- tion,following Lilien (1982),imply that an increase in the industry-level volatility of productivity growth may reduce output as resources are di- verted from production to costly reallocation across sectors.Models of"cleans- ing recessions"(Caballero and Hammour(1994),Eden and Jovanovic(1994)) emphasize similar effects at the level of the firm.An exogenous increase in the arrival rate of information about management quality may temporarily reduce output as resources are reallocated from low-quality to high-quality firms;alternatively,a recession that occurs for some other reason may re- veal information about management quality and increase the pace of real- location across firms. There is surprisingly little empirical research on volatility at the level of the industry or firm.A few papers use disaggregated data to study the "le- verage"effect,the tendency for volatility to rise following negative returns (Black (1976),Christie (1982),Duffee (1995)).Engle and Lee (1993)use a factor ARCH model to study the persistence properties of firm-level volatil- ity for a few large stocks.Some researchers have used stock market data to test macroeconomic models of reallocation across industries or firms(Loun- gani,Rush,and Tave(1990),Bernard and Steigerwald(1993),Brainard and Cutler (1993)),or to explore the firm-level relation between volatility and investment (Leahy and Whited (1996)).Roll (1992)and Heston and Rouwen- horst (1994)decompose world market volatility into industry and country- specific effects and study the implications for international diversification. Bekaert and Harvey (1997)construct a measure of individual firm disper- sion to study the volatility in emerging markets

First, many investors have large holdings of individual stocks; they may fail to diversify in the manner recommended by financial theory, or their holdings may be restricted by corporate compensation policies. These inves￾tors are affected by shifts in industry-level and idiosyncratic volatility, just as much as by shifts in market volatility. Second, some investors who do try to diversify do so by holding a portfolio of 20 or 30 stocks. Conventional wisdom holds that such a portfolio closely approximates a well-diversified portfolio in which all idiosyncratic risk is eliminated. However, the adequacy of this approximation depends on the level of idiosyncratic volatility in the stocks making up the portfolio. Third, arbitrageurs who trade to exploit the mispricing of an individual stock ~as opposed to a pattern of mispricing across many stocks! face risks that are related to idiosyncratic return volatility, not aggregate market volatility. Larger pricing errors are possible when idiosyn￾cratic firm-level volatility is high ~Ingersoll ~1987!, Chapter 7, Shleifer and Vishny ~1997!!. Fourth, firm-level volatility is important in event studies. Events affect individual stocks, and the statistical significance of abnormal event-related returns is determined by the volatility of individual stock re￾turns relative to the market or industry ~Campbell et al. ~1997!, Chapter 4!. Finally, the price of an option on an individual stock depends on the total volatility of the stock return, including industry-level and idiosyncratic vol￾atility as well as market volatility. Disaggregated volatility measures also have important relations with ag￾gregate output in some macroeconomic models. Models of sectoral realloca￾tion, following Lilien ~1982!, imply that an increase in the industry-level volatility of productivity growth may reduce output as resources are di￾verted from production to costly reallocation across sectors. Models of “cleans￾ing recessions” ~Caballero and Hammour ~1994!, Eden and Jovanovic ~1994!! emphasize similar effects at the level of the firm. An exogenous increase in the arrival rate of information about management quality may temporarily reduce output as resources are reallocated from low-quality to high-quality firms; alternatively, a recession that occurs for some other reason may re￾veal information about management quality and increase the pace of real￾location across firms. There is surprisingly little empirical research on volatility at the level of the industry or firm. A few papers use disaggregated data to study the “le￾verage” effect, the tendency for volatility to rise following negative returns ~Black ~1976!, Christie ~1982!, Duffee ~1995!!. Engle and Lee ~1993! use a factor ARCH model to study the persistence properties of firm-level volatil￾ity for a few large stocks. Some researchers have used stock market data to test macroeconomic models of reallocation across industries or firms ~Loun￾gani, Rush, and Tave ~1990!, Bernard and Steigerwald ~1993!, Brainard and Cutler ~1993!!, or to explore the firm-level relation between volatility and investment ~Leahy and Whited ~1996!!. Roll ~1992! and Heston and Rouwen￾horst ~1994! decompose world market volatility into industry and country￾specific effects and study the implications for international diversification. Bekaert and Harvey ~1997! construct a measure of individual firm disper￾sion to study the volatility in emerging markets. 2 The Journal of Finance

Have Individual Stocks Become More Volatile? 3 The purpose of this paper is to provide a simple summary of historical movements in market,industry,and firm-level volatility.We provide a de- composition of volatility that does not require the estimation of covariances or betas for industries or firms.In the interest of simplicity we follow Mer- ton(1980),Poterba and Summers(1986),French,Schwert,and Stambaugh (1987),Schwert (1989),and Schwert and Seguin(1990)and use daily data within each month to construct sample variances for that month,without imposing any parametric model to describe the evolution of variances over time.Multivariate volatility models are notoriously complicated and diffi- cult to estimate.Furthermore,although the choice of a parametric model may be essential for volatility forecasting,it is less important for describing historical movements in volatility,because all models tend to produce his- torical fitted volatilities that move closely together.The reason for this was first given by Merton (1980)and was elaborated by Nelson (1992):with sufficiently high-frequency data,volatility can be estimated arbitrarily ac- curately over an arbitrarily short time interval.Recently Andersen et al. (1999)have used a similar approach to produce daily volatilities from intra- daily data on the prices of large individual stocks. We first confirm and update Schwert's(1989)finding that market vola- tility has no significant trend using monthly data from 1926 to 1997.We next estimate market,industry,and firm-level variances using daily CRSP data ranging from 1962 to 1997.We find that market and industry vari- ances have been fairly stable in that sample period also.However,firm-level variance displays a large and significant positive trend,more than doubling between 1962 and 1997.This finding is robust to plausible variations in our methodology,for example,downweighting the influence of the 1987 crash, fixing the number of firms in the sample,or using weekly or monthly re- turns instead of daily returns to estimate volatility.We conclude that,al- though the market as a whole has not become more volatile,uncertainty on the level of individual firms has increased substantially over a 35-year pe- riod.Consistent with this observation,we find declines over time in the correlations among individual stocks and in the explanatory power of the market model for a typical stock. We also study the variations of the volatility measures around their long- term trends.The three volatility measures are positively correlated with each other as well as autocorrelated.Granger-causality tests suggest that market volatility tends to lead the other volatility series.All three volatility measures increase substantially in economic downturns and tend to lead recessions.The volatility measures-particularly industry-level volatility- help to forecast economic activity and reduce the significance of other com- monly used forecasting variables. The paper is organized as follows.In Section I we present the basic de- composition of volatility into market,industry,and idiosyncratic compo- nents.Section II directly measures trends in volatility.In Section III,we provide alternative indirect evidence of increased idiosyncratic volatility.Here we study correlations across individual stocks,the explanatory power of the market model for individual stocks,and the number of stocks needed to

The purpose of this paper is to provide a simple summary of historical movements in market, industry, and firm-level volatility. We provide a de￾composition of volatility that does not require the estimation of covariances or betas for industries or firms. In the interest of simplicity we follow Mer￾ton ~1980!, Poterba and Summers ~1986!, French, Schwert, and Stambaugh ~1987!, Schwert ~1989!, and Schwert and Seguin ~1990! and use daily data within each month to construct sample variances for that month, without imposing any parametric model to describe the evolution of variances over time. Multivariate volatility models are notoriously complicated and diffi￾cult to estimate. Furthermore, although the choice of a parametric model may be essential for volatility forecasting, it is less important for describing historical movements in volatility, because all models tend to produce his￾torical fitted volatilities that move closely together. The reason for this was first given by Merton ~1980! and was elaborated by Nelson ~1992!: with sufficiently high-frequency data, volatility can be estimated arbitrarily ac￾curately over an arbitrarily short time interval. Recently Andersen et al. ~1999! have used a similar approach to produce daily volatilities from intra￾daily data on the prices of large individual stocks. We first confirm and update Schwert’s ~1989! finding that market vola￾tility has no significant trend using monthly data from 1926 to 1997. We next estimate market, industry, and firm-level variances using daily CRSP data ranging from 1962 to 1997. We find that market and industry vari￾ances have been fairly stable in that sample period also. However, firm-level variance displays a large and significant positive trend, more than doubling between 1962 and 1997. This finding is robust to plausible variations in our methodology, for example, downweighting the influence of the 1987 crash, fixing the number of firms in the sample, or using weekly or monthly re￾turns instead of daily returns to estimate volatility. We conclude that, al￾though the market as a whole has not become more volatile, uncertainty on the level of individual firms has increased substantially over a 35-year pe￾riod. Consistent with this observation, we find declines over time in the correlations among individual stocks and in the explanatory power of the market model for a typical stock. We also study the variations of the volatility measures around their long￾term trends. The three volatility measures are positively correlated with each other as well as autocorrelated. Granger-causality tests suggest that market volatility tends to lead the other volatility series. All three volatility measures increase substantially in economic downturns and tend to lead recessions. The volatility measures—particularly industry-level volatility— help to forecast economic activity and reduce the significance of other com￾monly used forecasting variables. The paper is organized as follows. In Section I we present the basic de￾composition of volatility into market, industry, and idiosyncratic compo￾nents. Section II directly measures trends in volatility. In Section III, we provide alternative indirect evidence of increased idiosyncratic volatility. Here we study correlations across individual stocks, the explanatory power of the market model for individual stocks, and the number of stocks needed to Have Individual Stocks Become More Volatile? 3

4 The Journal of Finance achieve a given level of diversification.Section IV studies the lead-lag rela- tions among our volatility measures as well as their cyclical properties.In Section V,we suggest some factors that may have influenced the apparent increase in idiosyncratic volatility.Section VI presents concluding comments. I.Estimation of Volatility Components A.Volatility Decomposition We decompose the return on a "typical"stock into three components:the market-wide return,an industry-specific residual,and a firm-specific resid- ual.Based on this return decomposition,we construct time series of volatil- ity measures of the three components for a typical firm.Our goal is to define volatility measures that sum to the total return volatility of a typical firm, without having to keep track of covariances and without having to estimate betas for firms or industries.In this subsection,we discuss how we can achieve such a representation of volatility.The next subsection presents the estimation procedure and some details of the data sample. Industries are denoted by an i subscript and individual firms are indexed by j.The simple excess return of firm j that belongs to industry i in period t is denoted as Rit.This excess return,like all others in the paper,is mea- sured as an excess return over the Treasury bill rate.Let wiit be the weight of firm j in industry i.Our methodology is valid for any arbitrary weighting scheme provided that we compute the market return using the same weights; in this application we use market value weights.The excess return of in- dustry i in period t is given by Ri=jewRIndustries are aggregated correspondingly.The weight of industry i in the total market is denoted by wit,and the excess market return is Rmt=iwit Rit. The next step is the decomposition of firm and industry returns into the three components.We first write down a decomposition based on the CAPM, and we then modify it for empirical implementation.The CAPM implies that we can set intercepts to zero in the following equations: Rit =Bim Rmt Eit (1) for industry returns and Rm=B:Rt+可m =B元Bim Rmt+B元et+it (2) for individual firm returns.1 In equation (1)Bim denotes the beta for indus- try i with respect to the market return,and e is the industry-specific re- sidual.Similarly,in equation(2)B is the beta of firmjin industry i with We could work with the market model,not imposing the mean restrictions of the CAPM, and allow free intercepts a;and a in equations (1)and (2).However our goal is to avoid estimating firm-specific parameters;despite the well-known empirical deficiencies of the CAPM, we feel that the zero-intercept restriction is reasonable in this context

achieve a given level of diversification. Section IV studies the lead-lag rela￾tions among our volatility measures as well as their cyclical properties. In Section V, we suggest some factors that may have influenced the apparent increase in idiosyncratic volatility. Section VI presents concluding comments. I. Estimation of Volatility Components A. Volatility Decomposition We decompose the return on a “typical” stock into three components: the market-wide return, an industry-specific residual, and a firm-specific resid￾ual. Based on this return decomposition, we construct time series of volatil￾ity measures of the three components for a typical firm. Our goal is to define volatility measures that sum to the total return volatility of a typical firm, without having to keep track of covariances and without having to estimate betas for firms or industries. In this subsection, we discuss how we can achieve such a representation of volatility. The next subsection presents the estimation procedure and some details of the data sample. Industries are denoted by an i subscript and individual firms are indexed by j. The simple excess return of firm j that belongs to industry i in period t is denoted as Rjit. This excess return, like all others in the paper, is mea￾sured as an excess return over the Treasury bill rate. Let wjit be the weight of firm j in industry i. Our methodology is valid for any arbitrary weighting scheme provided that we compute the market return using the same weights; in this application we use market value weights. The excess return of in￾dustry i in period t is given by Rit 5 (j[i wjit Rjit . Industries are aggregated correspondingly. The weight of industry i in the total market is denoted by wit, and the excess market return is Rmt 5 (i wit Rit . The next step is the decomposition of firm and industry returns into the three components. We first write down a decomposition based on the CAPM, and we then modify it for empirical implementation. The CAPM implies that we can set intercepts to zero in the following equations: Rit 5 bim Rmt 1 eI it ~1! for industry returns and Rjit 5 bji Rit 1 hI jit 5 bji bim Rmt 1 bji eI it 1 hI jit ~2! for individual firm returns.1 In equation ~1! bim denotes the beta for indus￾try i with respect to the market return, and eI it is the industry-specific re￾sidual. Similarly, in equation ~2! bji is the beta of firm j in industry i with 1 We could work with the market model, not imposing the mean restrictions of the CAPM, and allow free intercepts ai and aji in equations ~1! and ~2!. However our goal is to avoid estimating firm-specific parameters; despite the well-known empirical deficiencies of the CAPM, we feel that the zero-intercept restriction is reasonable in this context. 4 The Journal of Finance

Have Individual Stocks Become More Volatile? 5 respect to its industry,and is the firm-specific residual.is orthogonal by construction to the industry return Rit;we assume that it is also orthog- onal to the components R and In other words,we assume that the beta of firm jwith respect to the market,Bim,satisfies Bim=BBim The weighted sums of the different betas equal unity: ∑0aBm=1,∑0m月a=1 (3) The CAPM decomposition (1)and (2)guarantees that the different com- ponents of a firm's return are orthogonal to one another.Hence it permits a simple variance decomposition in which all covariance terms are zero: Var(Rit)=Bin Var(Rmt)+Var(it), (4) Var(Rt)-β院mVar(Rmt)+B院Var(et)+Var(m): (5) The problem with this decomposition,however,is that it requires knowledge of firm-specific betas that are difficult to estimate and may well be unstable over time.Therefore we work with a simplified model that does not require any information about betas.We show that this model permits a variance decomposition similar to equations (4)and(5)on an appropriate aggregate level. First,consider the following simplified industry return decomposition that drops the industry beta coefficient Bim from equation (1): Rit Rmt eit. (6) Equation (6)defines eit as the difference between the industry return Rit and the market return Rm.Campbell et al.(1997,Chapter 4,p.156)refer to equation (6)as a "market-adjusted-return model"in contrast to the mar- ket model of equation (1). Comparing equations (1)and(6),we have t=et+（Bm-1)Rmt. (7) The market-adjusted-return residual eit equals the CAPM residual of equa- tion(4)only if the industry beta Bim=1 or the market return Rmt=0. The apparent drawback of the decomposition(6)is that Rmt and eit are not orthogonal,and so one cannot ignore the covariance between them.Com- puting the variance of the industry return yields Var(Rit)=Var(Rmt)+Var(Eit)+2 Cov(Rmt,Eit) Var(Rmt)+Var(Eit)+2(Bim -1)Var(Rmt), (8)

respect to its industry, and hI jit is the firm-specific residual. hI jit is orthogonal by construction to the industry return Rit; we assume that it is also orthog￾onal to the components Rmt and eI it. In other words, we assume that the beta of firm j with respect to the market, bjm, satisfies bjm 5 bjibim. The weighted sums of the different betas equal unity: ( i wit bim 5 1, ( j[i wjit bji 5 1. ~3! The CAPM decomposition ~1! and ~2! guarantees that the different com￾ponents of a firm’s return are orthogonal to one another. Hence it permits a simple variance decomposition in which all covariance terms are zero: Var~Rit ! 5 bim 2 Var~Rmt ! 1 Var~eI it !, ~4! Var~Rjit ! 5 bjm 2 Var~Rmt ! 1 bji 2 Var~eI it ! 1 Var~hI jit !. ~5! The problem with this decomposition, however, is that it requires knowledge of firm-specific betas that are difficult to estimate and may well be unstable over time. Therefore we work with a simplified model that does not require any information about betas. We show that this model permits a variance decomposition similar to equations ~4! and ~5! on an appropriate aggregate level. First, consider the following simplified industry return decomposition that drops the industry beta coefficient bim from equation ~1!: Rit 5 Rmt 1 eit . ~6! Equation ~6! defines eit as the difference between the industry return Rit and the market return Rmt. Campbell et al. ~1997, Chapter 4, p. 156! refer to equation ~6! as a “market-adjusted-return model” in contrast to the mar￾ket model of equation ~1!. Comparing equations ~1! and ~6!, we have eit 5 eI it 1 ~ bim 2 1!Rmt . ~7! The market-adjusted-return residual eit equals the CAPM residual of equa￾tion ~4! only if the industry beta bim 5 1 or the market return Rmt 5 0. The apparent drawback of the decomposition ~6! is that Rmt and eit are not orthogonal, and so one cannot ignore the covariance between them. Com￾puting the variance of the industry return yields Var~Rit ! 5 Var~Rmt ! 1 Var~eit ! 1 2 Cov~Rmt , eit ! 5 Var~Rmt ! 1 Var~eit ! 1 2~ bim 2 1!Var~Rmt !, ~8! Have Individual Stocks Become More Volatile? 5

6 The Journal of Finance where taking account of the covariance term once again introduces the in- dustry beta into the variance decomposition. Note,however,that although the variance of an individual industry re- turn contains covariance terms,the weighted average of variances across industries is free of the individual covariances: ∑u Var(Ru)=Var(R)+∑w Var(e =o品+o2, (9) where omt Var(Rmt)and o=iwt Var(et).The terms involving betas aggregate out because from equation(3)wiBim=1.Therefore we can use the residual eit in equation(6)to construct a measure of average industry- level volatility that does not require any estimation of betas.The weighted average >i wit Var(Ri)can be interpreted as the expected volatility of a ran- domly drawn industry(with the probability of drawing industry i equal to its weight wit). We can proceed in the same fashion for individual firm returns.Consider a firm return decomposition that drops Bi from equation(2): Rt=Rt+门t, (10) where niit is defined as nm=t+(βi-1)Rt (11) The variance of the firm return is Var(Rt)=Var(Rit)+Var(nit)+2 Cov(Rit,nit) Var(Ri)+Var(njit)+2(B:-1)Var(Ri). (12) The weighted average of firm variances in industry i is therefore ∑w Var(Rt)=Var(Rt)+o品t, (13) where=w Var()is the weighted average of firm-level volatility in industry i.Computing the weighted average across industries,using equa- tion (9),yields again a beta-free variance decomposition: ∑wt∑0mVar(Rt)=∑w Var(Rr)+∑wt∑Var((nm) =Var(Rmt)+∑w Var(et)+∑wao品t =o品+o+o品， (14)

where taking account of the covariance term once again introduces the in￾dustry beta into the variance decomposition. Note, however, that although the variance of an individual industry re￾turn contains covariance terms, the weighted average of variances across industries is free of the individual covariances: ( i wit Var~Rit ! 5 Var~Rmt ! 1 ( i wit Var~eit ! 5 smt 2 1 set 2 , ~9! where smt 2 [ Var~Rmt! and set 2 [ (i wit Var~eit !. The terms involving betas aggregate out because from equation ~3! (i wit bim 5 1. Therefore we can use the residual eit in equation ~6! to construct a measure of average industry￾level volatility that does not require any estimation of betas. The weighted average (i wit Var~Rit ! can be interpreted as the expected volatility of a ran￾domly drawn industry ~with the probability of drawing industry i equal to its weight wit!. We can proceed in the same fashion for individual firm returns. Consider a firm return decomposition that drops bji from equation ~2!: Rjit 5 Rit 1 hjit , ~10! where hjit is defined as hjit 5 hI jit 1 ~ bji 2 1!Rit . ~11! The variance of the firm return is Var~Rjit ! 5 Var~Rit ! 1 Var~hjit ! 1 2 Cov~Rit , hjit ! 5 Var~Rit ! 1 Var~hjit ! 1 2~ bji 2 1!Var~Rit !. ~12! The weighted average of firm variances in industry i is therefore ( j[i wjit Var~Rjit ! 5 Var~Rit ! 1 shit 2 , ~13! where shit 2 [ (j[i wjit Var~hjit ! is the weighted average of firm-level volatility in industry i. Computing the weighted average across industries, using equa￾tion ~9!, yields again a beta-free variance decomposition: ( i wit( j[i wjit Var~Rjit ! 5 ( i wit Var~Rit ! 1 ( i wit( j[i wjit Var~hjit ! 5 Var~Rmt ! 1 ( i wit Var~eit ! 1 ( i witshit 2 5 smt 2 1 set 2 1 sht 2 , ~14! 6 The Journal of Finance

Have Individual Stocks Become More Volatile? 7 where o,录=∑iwto品t=∑wt∑jeiWj Var(mt)is the weighted average of firm-level volatility across all firms.As in the case of industry returns,the simplified decomposition of firm returns (10)yields a measure of average firm-level volatility that does not require estimation of betas. We can gain further insight into the relation between our volatility de- composition and that based on the CAPM if we aggregate the latter (equa- tions (4)and(5))across industries and firms.When we do this we find that o2=a2+CSV(Bm)品， (15) where=iw Var()is the average variance of the CAPM industry shock,and CSV,(βim)=∑；wa(βim-1)2 is the cross--sectional variance of industry betas across industries.Similarly, o2=a2+CSV(Bm)o品+CSV,(Br)a品， (16) where2=∑：wt∑jeiw Var(⑦m),CSV,(Bm)=∑iwt∑wm(Bm-1)2is the cross-sectional variance of firm betas on the market across all firms in all industries,and CSV,(βr)=∑：wa∑jwm(βi-1)2 is the cross-sectional variance of firm betas on industry shocks across all firms in all industries. Equations (15)and(16)show that cross-sectional variation in betas can produce common movements in our variance components,and, even if the CAPM variance components and do not move at all with the market variance o.We return to this issue in Section IV.A,where we show that realistic cross-sectional variation in betas has only small effects on the time-series movements of our volatility components. B.Estimation We use firm-level return data in the CRSP data set,including firms traded on the NYSE,the AMEX,and the Nasdaq,to estimate the volatility compo- nents in equation (14)based on the return decomposition (6)and (10).We aggregate individual firms into 49 industries according to the classification scheme in Fama and French (1997).2 We refer to their paper for the SIC classification.Our sample period runs from July 1962 to December 1997. Obviously,the composition of firms in individual industries has changed dramatically over the sample period.The total number of firms covered by the CRSP data set increased from 2,047 in July 1962 to 8,927 in December 1997.The industry with the most firms on average over the sample is Fi- nancial Services with 628 (increasing from 43 to 1,525 over the sample),and the industry with the fewest firms is Defense with 8(increasing from 3 to 12 over the sample).Based on average market capitalization,the three largest 2 They actually use 48 industries,but we group the firms that are not covered in their scheme in an additional industry

where sht 2 [ (i witshit 2 5 (i wit (j[i wjit Var~hjit ! is the weighted average of firm-level volatility across all firms. As in the case of industry returns, the simplified decomposition of firm returns ~10! yields a measure of average firm-level volatility that does not require estimation of betas. We can gain further insight into the relation between our volatility de￾composition and that based on the CAPM if we aggregate the latter ~equa￾tions ~4! and ~5!! across industries and firms. When we do this we find that set 2 5 sI et 2 1 CSVt~ bim!smt 2 , ~15! where sI et 2 [ (i wit Var~eI it ! is the average variance of the CAPM industry shock eI it, and CSVt~ bim! [ (i wit~ bim 2 1! 2 is the cross-sectional variance of industry betas across industries. Similarly, sht 2 5 sI ht 2 1 CSVt~ bjm!smt 2 1 CSVt~ bji!sI et 2 , ~16! where sI ht 2 [ (i wit (j[i wjit Var~hI jit !, CSVt~ bjm! [ (i wit (j wjit~ bjm 2 1! 2 is the cross-sectional variance of firm betas on the market across all firms in all industries, and CSVt~ bji! [ (i wit (j wjit~ bji 2 1! 2 is the cross-sectional variance of firm betas on industry shocks across all firms in all industries. Equations ~15! and ~16! show that cross-sectional variation in betas can produce common movements in our variance components smt 2 , set 2 , and sht 2 , even if the CAPM variance components sI et 2 and sI ht 2 do not move at all with the market variance smt 2 . We return to this issue in Section IV.A, where we show that realistic cross-sectional variation in betas has only small effects on the time-series movements of our volatility components. B. Estimation We use firm-level return data in the CRSP data set, including firms traded on the NYSE, the AMEX, and the Nasdaq, to estimate the volatility compo￾nents in equation ~14! based on the return decomposition ~6! and ~10!. We aggregate individual firms into 49 industries according to the classification scheme in Fama and French ~1997!. 2 We refer to their paper for the SIC classification. Our sample period runs from July 1962 to December 1997. Obviously, the composition of firms in individual industries has changed dramatically over the sample period. The total number of firms covered by the CRSP data set increased from 2,047 in July 1962 to 8,927 in December 1997. The industry with the most firms on average over the sample is Fi￾nancial Services with 628 ~increasing from 43 to 1,525 over the sample!, and the industry with the fewest firms is Defense with 8 ~increasing from 3 to 12 over the sample!. Based on average market capitalization, the three largest 2 They actually use 48 industries, but we group the firms that are not covered in their scheme in an additional industry. Have Individual Stocks Become More Volatile? 7

8 The Journal of Finance industries on average over the sample are Petroleum/Gas (11 percent),Fi- nancial Services (7.8 percent)and Utilities(7.4 percent).Table 4 includes a list of the 10 largest industries.To get daily excess return,we subtract the 30-day T-bill return divided by the number of trading days in a month. We use the following procedure to estimate the three volatility compo- nents in equation(14).Let s denote the interval at which returns are mea- sured.We will use daily returns for most estimates but also consider weekly and monthly returns to check the sensitivity of our results with respect to the return interval.Using returns of interval s,we construct volatility esti- mates at intervals t.Unless otherwise noted,t refers to months.To estimate the variance components in equation(14)we use time-series variation of the individual return components within each period t.The sample volatility of the market return in period t,which we denote from now on as MKT,is computed as MKT=G品=∑(Rm-um)2 (17) sEt where um is defined as the mean of the market return R over the sample.s To be consistent with the methodology presented above,we construct the market returns as the weighted average using all firms in the sample in a given period.The weights are based on market capitalization.Although this market index differs slightly from the value-weighted index provided in the CRSP data set,the correlation is almost perfect at 0.997.For weights in period t we use the market capitalization of a firm in period t-1 and take the weights as constant within period t. For volatility in industry i,we sum the squares of the industry-specific residual in equation(6)within a period t: 品-∑品. (18) s∈t As shown above,we have to average over industries to ensure that the co- variances of individual industries cancel out.This yields the following mea- sure for average industry volatility IND,: ND,=∑wtG品. (19) 3 We also experimented with time-varying means but the results are almost identical.Foster and Nelson(1996)have recently provided a more comprehensive study of rolling regressions to estimate volatility.They show that under quite general conditions a two-sided rolling regres- sion will be optimal.However,such a technique causes serious problems for the study of lead- lag relationships that is one focus of this paper

industries on average over the sample are Petroleum0Gas ~11 percent!, Fi￾nancial Services ~7.8 percent! and Utilities ~7.4 percent!. Table 4 includes a list of the 10 largest industries. To get daily excess return, we subtract the 30-day T-bill return divided by the number of trading days in a month. We use the following procedure to estimate the three volatility compo￾nents in equation ~14!. Let s denote the interval at which returns are mea￾sured. We will use daily returns for most estimates but also consider weekly and monthly returns to check the sensitivity of our results with respect to the return interval. Using returns of interval s, we construct volatility esti￾mates at intervals t. Unless otherwise noted, t refers to months. To estimate the variance components in equation ~14! we use time-series variation of the individual return components within each period t. The sample volatility of the market return in period t, which we denote from now on as MKTt, is computed as MKTt 5 s[ mt 2 5 (s[t ~Rms 2 mm!2 , ~17! where mm is defined as the mean of the market return Rms over the sample.3 To be consistent with the methodology presented above, we construct the market returns as the weighted average using all firms in the sample in a given period. The weights are based on market capitalization. Although this market index differs slightly from the value-weighted index provided in the CRSP data set, the correlation is almost perfect at 0.997. For weights in period t we use the market capitalization of a firm in period t 2 1 and take the weights as constant within period t. For volatility in industry i, we sum the squares of the industry-specific residual in equation ~6! within a period t: s[ eit 2 5 (s[t eis 2 . ~18! As shown above, we have to average over industries to ensure that the co￾variances of individual industries cancel out. This yields the following mea￾sure for average industry volatility INDt: INDt 5 ( i wit s[ eit 2 . ~19! 3 We also experimented with time-varying means but the results are almost identical. Foster and Nelson ~1996! have recently provided a more comprehensive study of rolling regressions to estimate volatility. They show that under quite general conditions a two-sided rolling regres￾sion will be optimal. However, such a technique causes serious problems for the study of lead– lag relationships that is one focus of this paper. 8 The Journal of Finance

Have Individual Stocks Become More Volatile? 9 Estimating firm-specific volatility is done in a similar way.First we sum the squares of the firm-specific residual in equation (10)for each firm in the sample: 品=∑n品 (20) 8后t Next,we compute the weighted average of the firm-specific volatilities within an industry: 品=∑w品t· (21) jEi And lastly we average over industries to obtain a measure of average firm- level volatility FIRM,as FIRM=∑wuG品. (22) As with industry volatility,this procedure ensures that the firm-specific co- variances cancel out. II.Measuring Trends in Volatility A.Graphical Analysis Popular discussions of the stock market often suggest that the volatility of the market has increased over time.At the aggregate level,however,this is simply untrue.The percentage volatility of market index returns shows no systematic tendency to increase over time.To be sure,there have been epi- sodes of increased volatility,but they have not persisted.Schwert (1989) presented a particularly clear and forceful demonstration of this fact,and we begin by updating his analysis. In Figure 1 we plot the volatility of the value weighted NYSE/AMEX/ Nasdag composite index for the period 1926 through 1997.For consistency with Schwert,we compute annual standard deviations based on monthly data.The figure shows the huge spikes in volatility during the late 1920s and 1930s as well as the higher levels of volatility during the oil and food shocks of the 1970s and the stock market crash of 1987.In general,however, there is no discernible trend in market volatility.The average annual stan- dard deviation for the period from 1990 to 1997 is 11 percent,which is actually lower than that for either the 1970s(14 percent)or the 1980s(16 percent). These results raise the question of why the public has such a strong impres- sion of increased volatility.One possibility is that increased index levels have increased the volatility ofabsolute changes,measured in index points,and that the public does not understand the need to measure percentage returns.An-

Estimating firm-specific volatility is done in a similar way. First we sum the squares of the firm-specific residual in equation ~10! for each firm in the sample: s[ hjit 2 5 (s[t hjis 2 . ~20! Next, we compute the weighted average of the firm-specific volatilities within an industry: s[ hit 2 5 ( j[i wjit s[ hjit 2 . ~21! And lastly we average over industries to obtain a measure of average firm￾level volatility FIRMt as FIRMt 5 ( i wit s[ hit 2 . ~22! As with industry volatility, this procedure ensures that the firm-specific co￾variances cancel out. II. Measuring Trends in Volatility A. Graphical Analysis Popular discussions of the stock market often suggest that the volatility of the market has increased over time. At the aggregate level, however, this is simply untrue. The percentage volatility of market index returns shows no systematic tendency to increase over time. To be sure, there have been epi￾sodes of increased volatility, but they have not persisted. Schwert ~1989! presented a particularly clear and forceful demonstration of this fact, and we begin by updating his analysis. In Figure 1 we plot the volatility of the value weighted NYSE0AMEX0 Nasdaq composite index for the period 1926 through 1997. For consistency with Schwert, we compute annual standard deviations based on monthly data. The figure shows the huge spikes in volatility during the late 1920s and 1930s as well as the higher levels of volatility during the oil and food shocks of the 1970s and the stock market crash of 1987. In general, however, there is no discernible trend in market volatility. The average annual stan￾dard deviation for the period from 1990 to 1997 is 11 percent, which is actually lower than that for either the 1970s ~14 percent! or the 1980s ~16 percent!. These results raise the question of why the public has such a strong impres￾sion of increased volatility. One possibility is that increased index levels have increased the volatility of absolute changes, measured in index points, and that the public does not understand the need to measure percentage returns. An￾Have Individual Stocks Become More Volatile? 9

10 The Journal of Finance 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 19201950 1940 1950 1960 1970 1980 1990 2000 Date Figure 1.Standard deviation of value-weighted stock index.The standard deviation of monthly returns within each year is shown for the period from 1926 to 1997. other possibility is that public impressions are formed in part by the behavior ofindividual stocks rather than the market as a whole.Casual empiricism does suggest increasing volatility for individual stocks.On any specific day,the most volatile individual stocks move by extremely large percentages,often 25 per- cent or more.The question remains whether such impressions from casual em- piricism can be documented rigorously and,if so,whether these patterns of volatility for individual stocks are different from those existing in earlier pe- riods.With this motivation,we now present a graphical summary of the three volatility components described in the previous section. Figures 2 to 4 plot the three variance components,estimated monthly, using daily data over the period from 1962 to 1997:market volatility MKT, industry-level volatility IND,and firm-level volatility FIRM.All three series are annualized(multiplied by 12).The top panels show the raw monthly time series and the bottom panels plot a lagged moving average of order 12.Note that the vertical scales differ in each figure and cannot be compared with Fig- ure 1(because we are now plotting variances rather than a standard deviation) Market volatility shows the well-known patterns that have been studied in countless papers on the time variation of index return variances.Com- paring the monthly series with the smoothed version in the bottom panel suggests that market volatility has a slow-moving component along with a

other possibility is that public impressions are formed in part by the behavior of individual stocks rather than the market as a whole. Casual empiricism does suggest increasing volatility for individual stocks. On any specific day, the most volatile individual stocks move by extremely large percentages, often 25 per￾cent or more. The question remains whether such impressions from casual em￾piricism can be documented rigorously and, if so, whether these patterns of volatility for individual stocks are different from those existing in earlier pe￾riods. With this motivation, we now present a graphical summary of the three volatility components described in the previous section. Figures 2 to 4 plot the three variance components, estimated monthly, using daily data over the period from 1962 to 1997: market volatility MKT, industry-level volatility IND, and firm-level volatility FIRM. All three series are annualized ~multiplied by 12!. The top panels show the raw monthly time series and the bottom panels plot a lagged moving average of order 12. Note that the vertical scales differ in each figure and cannot be compared with Fig￾ure 1 ~because we are now plotting variances rather than a standard deviation!. Market volatility shows the well-known patterns that have been studied in countless papers on the time variation of index return variances. Com￾paring the monthly series with the smoothed version in the bottom panel suggests that market volatility has a slow-moving component along with a Figure 1. Standard deviation of value-weighted stock index. The standard deviation of monthly returns within each year is shown for the period from 1926 to 1997. 10 The Journal of Finance