VOL. 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM 653 one assumes that investors hold undiversified optimization problem with the constraint portfolios which contain stocks of three or on the number of securities nk, and also to four companies(i. e, nk =3, 4)and that the know the amount invested by each investor ith security is not included in all portfolios, in the stock market. To illustrate the diffi en the variance (and hence the residual culties involved in such an empirical test variance)should have a strong impact on let us reexamine equation(6). When we the risk-return relationship. Although we multiply equation (6) by Tk and sum up have already analyzed the role of the only for investors k who hold security i, we variance in price determination(see equa- obtain tion(15)), we can find a more transparen example by looking once again at equation (33)H12Tk-r2T+2TAHx-r)k (6). Rewriting(6)we obtain (31)μ1-r= o? Cov(r, RN) (34)μ1=+∑T4(k-)3k/∑Tk , Assuming, once again, for the sake of By defining 8* as the weighted average, mplicity only, that the typical investor p:=2kTkHk- r)Bk/Xk T(uk- r) ho holds security i will diversify equally and Ek Tk(ur -r/ETk=rt we can re- bet ween three stocks we obtain write(34)as here y i varies from one security to an COMr,iR+iR R;+ other Equation(35) can then be used in order to test empirically the risk-return relation- where i, i-I, and i+ I stand for the three ship as suggested in this paper. However securities included in the portfolio. Thus would like to mention a few characteristic esults as well as difficulties in testing this (32)μ equation empirically:(a) Sinceβ;<β/kfor weighted average of B.(b)n=27(、° lk,β1<*isal I Cov(R,R)+I Cov(R.R,. r)/ET when we sum up only for investors k who hold security i. Thus, rt varies from It is obvious from(32)that variance plays security to security, and any cross-section central role in explaining the risk-return regression will provide an estimate of some relationship. Moreover, one would expect average of all these y t(c) In order to test that the individual variance would have the Ca PM in the present framework, one greater impact on price determination than has to estimate first B*, that is, to have in- the P(as defined in equation (1)) since B, formation, not only on the selected port has very little to do with the stock,'s risk folio by each investor k, but also on the when the portfolios include only a small relative size of his investment, Tk/2TA.(d) number of different securities. Indeed, Finally, it is worth mentioning that if all in Douglas found that the coefficient of the vestors hold security i, yu= EkT(uk-r)/ variance is more important than the coef- > Tk, when we sum up for all investors k ficient of the B in most periods covered in Hence yu=um-r, since in this case >k his empirical research To design a precise empirical study to test where um is the expected rate of return on he model suggested in this paper is not an the market portfolio easy task since equation(6)includes a fac Designing such an empirical research is tor Bu which varies from investor to in yond the scope of this paper. However, if vestor. One has first to find a solution to the the present form of the Ca PM is correct 0m3303038ANVOL. 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM 653 one assumes that investors hold undiversified portfolios which contain stocks of three or four companies (i.e., nk = 3, 4) and that the ith security is not included in all portfolios, then the variance (and hence the residual variance) should have a strong impact on the risk-return relationship. Although we have already analyzed the role of the variance in price determination (see equa￾tion (15)), we can find a more transparent example by looking once again at equation (6). Rewriting (6) we obtain (31) Au - r = lk r Cov(R ,Rk) '7k Assuming, once again, for the sake of simplicity only, that the typical investor who holds security i will diversify equally between three stocks, we obtain k - r r 2 '7k [Cov(Rig - Ri ? Ri_ [ ( ? - i3 i 3 i, 3 i+ ,) where i, i - 1, and i + 1 stand for the three securities included in the portfolio. Thus (32) r = 2 3 [ Ri + 3Cov(Ri.Ri-1) + I Cov (Ri, Ri+ 1) It is obvious from (32) that variance plays a central role in explaining the risk-return relationship. Moreover, one would expect that the individual variance would have greater impact on price determination than the fi (as defined in equation (1)) since f3 has very little to do with the stock's risk when the portfolios include only a small number of different securities. Indeed, Douglas found that the coefficient of the variance is more important than the coef￾ficient of the d in most periods covered in his empirical research. To design a precise empirical study to test the model suggested in this paper is not an easy task since equation (6) includes a fac￾tor Oki which varies from investor to in￾vestor. One has first to find a solution to the optimization problem with the constraint on the number of securities nk, and also to know the amount invested by each investor in the stock market. To illustrate the diffi￾culties involved in such an empirical test, let us reexamine equation (6'). When we multiply equation (6') by Tk and sum up only for investors k who hold security i, we obtain (33) Ai L Tk = r L Tk + E Tk(jk - r)1ki k k k or (34) A i = r + L Tk(juk - r)kil/ ?Tk k k By defining /3* as the weighted average, i= 2kTk(/k r)tki/ 2kTk(Ak r) and ?2 kTk(Ak - r)/l2 kTk = y C we can re￾write (34) as (35) Ai = r + y1/3C where y,C varies from one security to an￾other. Equation (35) can then be used in order to test empirically the risk-return relation￾ship as suggested in this paper. However, I would like to mention a few characteristic results as well as difficulties in testing this equation empirically: (a) Since fi < d3ik for all k, f3i < /3i is also true, since /3* is a weighted averageof Ofik (b) y ji = z Tk(-k k r)/1 Tk when we sum up only for investors k who hold security i. Thus, oy varies from security to security, and any cross-section regression will provide an estimate of some average of all these oY,i- (c) In order to test the CAPM in the present framework, one has to estimate first * that is, to have in￾formation, not only on the selected port￾folio by each investor k, but also on the relative size of his investment, Tk/2 Tk . (d) Finally, it is worth mentioning that if all in￾vestors hold security i, y1i = z2 kTk(/k -r) kTk, when we sum up for all investors k. Hence yji = Im - r, since in this case ?2k￾Tk = To, and2 kTk(/k - r) = To(A m - r) whereurn is the expected rate of return on the market portfolio. Designing such an empirical research is beyond the scope of this paper. However, if the present form of the CA PM is correct, 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