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 industry beta into the variance decomposition. Note, however, that although the variance of an individual industry return 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 industrylevel 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 randomly 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 equation ~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 decomposition and that based on the CAPM if we aggregate the latter ~equations ~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 components 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 Financial 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!, Financial 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 components in equation ~14!. Let s denote the interval at which returns are measured. 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 estimates 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 covariances of individual industries cancel out. This yields the following measure 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 regression 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 firmlevel volatility FIRMt as FIRMt 5 ( i wit s[ hit 2 . ~22! As with industry volatility, this procedure ensures that the firm-specific covariances 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 episodes 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 standard 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 impression 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. AnHave 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 percent or more. The question remains whether such impressions from casual empiricism can be documented rigorously and, if so, whether these patterns of volatility for individual stocks are different from those existing in earlier periods. 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 Figure 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. Comparing 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