VOL 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM essarily identical to the group of investors Obviously, in such a case, we would expect who hold security j. Thus, the term 2 Tk that the ith security variance will play a r)/>TXoX(price of risk)is a function central role in its equilibrium price deter of the security under consideration, and is mination, quite contrary to the result of the relevant only to investors who decide to traditional CA PM. On the other hand, the hold this security in their portfolio traditional B,(see equation(I) has little to The equilibrium formula given by equa- to with the determination Pio, since 8, in tion(15)has very important implications cludes all the covariances(see equation(7) for the empirical findings of the CA PM. while in the above example we have only To demonstrate, assume that all investors one covariance. Note that few assumptions (namely, omf. urity i hold also security j have been made in order to simplify the investors purchase all the available secu- hold stocks of three or four companies, we rities of these two firms. For simplicity only, still obtain the same result; the ith security and without loss of generality, assume that variance is much more important in price uk-r is a constant (say A)and that determination than one would expect from Tk/2Tk=c for all these investors. Thus the analysis of traditional CA PM. Empiri- (15)reduces to cal support to this theoretical result is given ∑7(k-r) For the specific case in which all inve (15)(1+r)Po=P hold security i, we sum up equation(12) for total aggregate excess dollar return of all TANAσ*2+∑7N n vestors portfolios, which T0(μm-r), whereμ is the expected re turn on the market portfolio and To equal to Too, and hence one does not On the basis of the above simplifying as- have, even in the above specific case, the sumptions, we obtain from(15') ion of the ggregate risk in the market as obtained wh marke is assumed. However, equation(15) can be ∑T(k-l∑No*2+N e Took ET(-) TS (15")(1+r)P0o=Pt T202 ∑7(μ T since 2kNk=Ni, knI&=ni, where N; and N, are the number of outstanding shares of If all investors hold security i, then 2k TK r)and the second term It can readily be seen from(15") that the on the right-hand side is the market price equilibrium price Po is a function of the ith of risk y, when the Ca PM is derived with ecurity variance and of only one covar- out constraint on the number of securities iance, that is, its covariance with security j. in the portfolio(see Lintner 1965a, p. 600) 0m3303038ANVOL. 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM 649 essarily identical to the group of investors who hold security j. Thus, the term I Tk- - r)/2 Tkak (price of risk) is a function of the security under consideration, and is relevant only to investors who decide to hold this security in their portfolio. The equilibrium formula given by equa￾tion (15) has very important implications for the empirical findings of the CA PM. To demonstrate, assume that all investors who hold security i hold also security j (namely, only two risky assets) and these investors purchase all the available secu￾rities of these two firms. For simplicity only, and without loss of generality, assume that Ak- r is a constant (say = A) and that Tk/l Tk = a for all these investors. Thus (15) reduces to Z Tk (Uk - r) ( 15') (1 + r)Pi0 = pi k k TkNik ' + E Tk-Nik rj E Tk k On the basis of the above simplifying as￾sumptions, we obtain from (15') (1 + r)Pio = PiI ETk (k - r)a ik i Njk a* _k k[No k E k k ffk or (15") (1 + r)Pio = Pi, E Tk(8k - r)a[NiCr*2 + Nj,* __ k k Ni, N = Nj, where Ni and Nj are the number of outstanding shares of i andj, respectively. It can readily be seen from (15") that the equilibrium price Pio is a function of the ith security variance and of only one covar￾iance, that is, its covariance with securityj. Obviously, in such a case, we would expect that the ith security variance will play a central role in its equilibrium price deter￾mination, quite contrary to the result of the traditional CA PM. On the other hand, the traditional fi (see equation (1)) has little to to with the determination Pio, since fi in￾cludes all the covariances (see equation (7)) while in the above example we have only one covariance. Note that few assumptions have been made in order to simplify the analysis. However, even when investors hold stocks of three or four companies, we still obtain the same result; the ith security variance is much more important in price determination than one would expect from the analysis of traditional CA PM. Empiri￾cal support to this theoretical result is given in Section IV. For the specific case in which all investors hold security i, we sum up equation (12) for all investors k. Hence ?kTk(gk - r) is the total aggregate excess dollar return of all investors' portfolios, which is equal to To(/um - r), where Am is the expected re￾turn on the market portfolio and To = ?kTk. However, zkTk2k is not necessarily equal to T22 , and hence one does not have, even in the above specific case, the interpretation of the aggregate risk in the market as obtained when a perfect market is assumed. However, equation (15) can be written as (1 + r)P0 = PiI Tk (Ak - r)] T22 T2 2 T2 v 2 2 0 m T.1k (7k k k - r) =T Tk (JUk - r) n Z n t NikoL*2 1:Nj U k I. T(Uk-k r) k If all investors hold security i, then k 7kT (Atk - r) = T0(gm - r) and the second term on the right-hand side is the market price of risk -y, when the CAPM is derived with￾out constraint on the number of securities in the portfolio (see Lintner 1965a, p. 600). 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