VOL, 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM only o of securities included in the kth in- price behavior even better than the esti restors portfolios, are taken into account. mates of the systematic risk(i.e, y, in equa However, if we assume unrealistically that tion (3). I demonstrate below that the fact security i is included in all investors' port. that investors hold portfolios with only folios(equation(17))then for an equilib- few risky assets, rather than the market rium price determination we must take into portfolio, provides a possible explanation account the covariances ou of all securities for the three discrepancies between the ailable in the market since we sum up in theoretical model and the empirical findings equation(17)for all k obtained by various researchers Suppose that an investor holds a port II. The Implication for the Empirical Findin folio k whose random return is the random return on the market portfolio Recent empirical evidence indicates that is Rm. The expected return on R can be empirical data as well as might be expected. of Rm. However, since Rk includes only a Douglas, using annual and quarterly data, few securities while rm consists of all se shows that there is a significant relationship curities available in the market, one would between the mean rate of return of a stock expect that the variance of rm would be d its standard deviation -a fact which smaller than the variance of most selected contradicts the CAPM. Lintner (1965b) portfolios, k. The relationship between Rk regresses annual rates of return of 301 and Rm can be described as follows stocks over the period 1954-63. He esti mates the systematic risk from time and then regresses the mean rate of return (alternatively, one can define this relation- on the systematic risk and on the estimate ship in the form Rm =a+ brk +4, see of the residual variance(see equation(3)). Miller and Scholes), where y is an error His results, too, indicate that the theoretical term. Let us now analyze the impact of the model does not provide a satisfactory de- error in the variables given in(21), on em scription of price behavior. Using a pirical evidence related to the CA PM data, Merton Miller and Myron Scholes In the empirical research, the time-series confirm the basic results of Lintner and regression is formulated as follo suggest possible explanations for the devia tion between the model and the empirical (22) Rat=ai+ B rmt +er evidence. Black, Jensen, and Scholes using where B, derived from(22)is the estimate of monthly data also show that the model the ith security systematic risk. Since the does not provide a satisfactory description investors hold portfolio Rk rather than Rm of price behavior in the stock market he true relationship is given by have investigated the effect of the assumed (23) Ri= a*k+B*Rk+ur investment horizon on the estimates of the where B* is the kth investors true sys stematic risk as well as on the other re- tematic risk We shall see that using (22) sults implied by the CA PM. We have found rather than(23)causes a certain bias in the that the investment horizon plays a crucial estimate of the systematic risk. The estimate role in any econometric research and, par- of B, is given by ticularly, in empirical work which tests the CAPM. However, in analyzing horizons (24) cOk COv ranging from one to twenty-four months ar(Rk+ψ) we have also found that the coefficient of cov(R;,Rk)+cov(R,ψ) the residual variance (y2 in equation (3)) remains significantly +σ+2co(Rk,ψ) nost cases, too, the residual variance explains If we divide by ok and assume that the er- 0m3303038ANVOL. 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM 651 only cij of securities included in the kth in￾vestors' portfolios, are taken into account. However, if we assume unrealistically that security i is included in all investors' port￾folios (equation (17)) then for an equilib￾rium price determination we must take into account the covariances -.i of all securities available in the market since we sum up in equation (17) for all k. II. The Implication for the Empirical Findings Recent empirical evidence indicates that the traditional CA PM does not explain the empirical data as well as might be expected. Douglas, using annual and quarterly data, shows that there is a significant relationship between the mean rate of return of a stock and its standard deviation a fact which contradicts the CA PM. Lintner (1965b) regresses annual rates of return of 301 stocks over the period 1954-63. FIe esti￾mates the systematic risk from time-series and then regresses the mean rate of return on the systematic risk and on the estimate of the residual variance (see equation (3)). His results, too, indicate that the theoretical model does not provide a satisfactory de￾scription of price behavior. Using annual data, Merton Miller and Myron Scholes confirm the basic results of Lintner and suggest possible explanations for the devia￾tion between the model and the empirical evidence. Black, Jensen, and Scholes using monthly data also show that the model does not provide a satisfactory description of price behavior in the stock market. In recent papers David Levhari and I have investigated the effect of the assumed investment horizon on the estimates of the systematic risk as well as on the other re￾sults implied by the CA PM. We have found that the investment horizon plays a crucial role in any econometric research and, par￾ticularly, in empirical work which tests the CA PM. However, in analyzing horizons ranging from one to twenty-four months, we have also found that the coefficient of the residual variance (Y2 in equation (3)) remains significantly positive. In most cases, too, the residual variance explains price behavior even better than the esti￾mates of the systematic risk (i.e., 'Yj in equa￾tion (3)). 1 demonstrate below that the fact that investors hold portfolios with only a few risky assets, rather than the market portfolio, provides a possible explanation for the three discrepancies between the theoretical model and the empirical findings obtained by various researchers. Suppose that an investor holds a port￾folio k whose random return is Rk, while the random return on the market portfolio is Rm. The expected return on Rk can be smaller or greater than the expected return of Rm. However, since Rk includes only a few securities while Rm consists of all se￾curities available in the market, one would expect that the variance of Rm would be smaller than the variance of most selected portfolios, k. The relationship between Rk and Rm can be described as follows: (21) Rm = Rk +VI (alternatively, one can define this relation￾ship in the form Rm = a + bRk + '1, see Miller and Scholes), where ;1 is an error term. Let us now analyze the impact of the error in the variables given in (21), on em￾pirical evidence related to the CA PM. In the empirical research, the time-series regression is formulated as follows: (22) Rit = ai + (iRmt + et where Oi derived from (22) is the estimate of the ith security systematic risk. Since the investors hold portfolio Rk rather than R., the true relationship is given by (23) Rit = aI* Rkt + Ut where d is the kth investor's true sys￾tematic risk. We shall see that using (22) rather than (23) causes a certain bias in the estimate of the systematic risk. The estimate of fi is given by ( cov (Ri, Rm) cov (Ri Rk + f) (24 ' var(Rm) var(Rk + i1) cov (Ri, Rk) + cov (Ri, Vf) 0k + a2 + 2 cov (Rk,y; If we divide by a2 and assume that the er￾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