THE AMERICAN ECONOMIC REVIEW SEPTEMBER /978 rors are distributed independently of the Dividing by o(B*)and assuming that the true values(R, and R), then the last term error 8, is uncorrelated with the values R in the numerator, as well as the last term and B*, we obtain, in the denominator. will tend to zero as the sample size increases indefinitely. Thus, y 8,= COv(R, Rk)/(+ ol/o2)ok. But since 1+σ2(6,)/(2(*) cov(R, Rx)/o2-B* we finally obtain 1+σ2(0,)/σ2(6*) ence 8< Bk This may explain the result of most empiri for all investors k, and hence B i< B where cal studies where y, is below the value pre B* is a weighted average of B*. (I shall de- dicted by the CA PM fine this weighted average later on;see It has also been found in all empirical re equation(35).) search that yo>0, while, according to the Let us now investigate the impact of this CA PM, Yo should equal zero. This bias may bias in measuring the systematic risk, on the be explained as follows: from equation (28) cross-section regression which is essential the estimate of yo is given by o an examination of the validity of the CAPM (see equation (1). Since B,is biased, one can write B as follows where R is the average of the variables R r,and B is the average of the estimates of where p, is an error term. Most empirical However, the true relationship shousets the systematic risks B, of all risky as orks carry out the cross-section regression (from equation(29)) in the following manner(see equation(I')) 13 since according to the above assumptions while the true relationship is given by 1<评B*, R hence we obtain the result yo>Y0=0 Apparently, the em pirI here R, is the average rate of return of the cal result is that y2(see equation (3))is ith asset, r is the riskless interest rate and significantly greater than zero. The latter re s the estimate of the systematic risk ob- sult, however, can be explained by the tained from the time-series regression. Thus model presented in this paper. According to cOv(Ri, B) the CaPM, investors diversify in many se curities. and hence, the residual variance S2 should have no impact on the risk-return equilibrium relationship. The individual σ(8*)+σ2(0)+2cov(8*,日) securitys variance as well should have no cov(R,B↑)+cOv(R,6) impact on this relationship since the con tribution of the individual risk is about 0(81)+0(01)+ 2cov(A:,0)(1/n)02(R )when n is the number of securi- ties available in the market. however if 7In deriving (26)it is assumed that the errors are & Equation (30)is valid even under less restr distributed independently of R, and Rk. However, it is assumptions(see fn. 7) For simplicity's sake we assume that the investor y are distributed independently and that the regres- diversify equally his resources among all securities. sion coefficient of R on Y, is greater than-1 (See Miller and Schole 0m3303038AN652 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1978 rors are distributed independently of the true values (Ri and Rk), then the last term in the numerator, as well as the last term in the denominator, will tend to zero as the sample size increases indefinitely. Thus, fi = cov (Ri, Rk)/(1 ? ,/abo But since cov (Ri, Rk)/aJ = we finally obtain i k (25) - + ? Hence7 (26) /3i < di*k for all investors k, and hence fi < /3* where /3* is a weighted average of /*. (I shall de￾fine this weighted average later on; see equation (35).) Let us now investigate the impact of this bias in measuring the systematic risk, on the cross-section regression which is essential to an examination of the validity of the CAPM (see equation (1')). Since fi is biased, one can write fi as follows, (27) 1=3= 13? +q where , is an error term. Most empirical works carry out the cross-section regression in the following manner (see equation (1')): (28) Ri- r = y0 + y1/3i + ei while the true relationship is given by (29) R-r = y* + y3 + e* where Ri is the average rate of return of the ith asset, r is the riskless interest rate, and fi is the estimate of the systematic risk ob￾tained from the time-series regression. Thus cov (Ri, O3) cov(Ri, i* + 6) a2(/*) + a2(0i) ? 2cov(/:3, 6j) cov(Rj,O* ) + cov(Rj,Oj) 3a2(/*) + +2(6.) ? 2cov(O:3, O1) Dividing by a2(/3*) and assuming that the error 6i is uncorrelated with the values Ri and /3*, we obtain, cov(Ri,f)/a2(ig) =1 + a2(6i)/a2(/*) y* or i = 1 ) I + Uf (ONUV (i*) and hence' (30) *< y This may explain the result of most empiri￾cal studies where j , is below the value pre￾dicted by the CA PM. It has also been found in all empirical re￾search that 'o > 0, while, according to the CA PM, yo should equal zero. This bias may be explained as follows: from equation (28) the estimate of 'yO is given by To = R - where R is the average of the variables Ri - r, and d is the average of the estimates of the systematic risks fi of all risky assets. However, the true relationship should be (from equation (29)) TYo = R - ,Y since according to the above assumptions oy < j* and /i < /3*, also jI, < hence we obtain the result %O > y = 0. Apparently, the most disturbing empiri￾cal result is that 2 (see equation (3)) is significantly greater than zero. The latter re￾sult, however, can be explained by the model presented in this paper. According to the CAPM, investors diversify in many se￾curities, and hence, the residual variance Sei should have no impact on the risk-return equilibrium relationship. The individual security's variance as well should have no impact on this relationship since the con￾tribution of the individual risk is about (I/n)oa2(Ri) when n is the number of securi￾ties available in the market.9 However, if 71n deriving (26) it is assumed that the errors are distributed independently of Ri and Rk. However, it is easy to verify that it is sufficient to require that u and ' are distributed independently and that the regres￾sion coefficient of Rk on T is greater than -1. 8Equation (30) is valid even under less restrictive assumptions (see fn. 7). 9For simplicity's sake we assume that the investor diversify equally his resources among all securities. (See Miller and Scholes.) This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:07:38 AM All use subject to JSTOR Terms and Conditions