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