THE AMERICAN ECONOMIC REVIEW SEPTEMBER J978 Thus ployed by Lintner (1965a) with only one distinction: Lintner was allowed to sum up of market price per share c,se his equations for all investors. In our model, Ind equation(8)can be written in terms allowed to sum the investors k who hold the security under consideration in their portfolios, since (9)N, Pn-N,Pio( equation (4)(from which we derive equa tion(12)) includes the ith security only fo V, Pox R T+N Po>.1 investors k who hold it. After multiplying for investors k who hold security i e obtain Dividing by N, yields (13){P1-P(1+r)∑T (10)P1-P0(+)=y4-r) 2 7(Hk-7)Na*2+∑Nkσ Poxk02+Po∑xAk0 The equilibrium price of share i, Plo given by Now recall that the proportions invested by the k th investor xuk and xik in the ith and th (14)(1+r)Po= Pi ectively, have been given by ik= Nik Po/Tk, and xik=Nk Pyo/Tk here Nik and Nik stand for the number of shares of firm i and j in the kth iny rtfolio,and Tk is the total amount of In order to derive a more comparable form dollars invested by him in risky assets. Thus, for the equilibrium price as implied by the the substitution of xik and x,k in equation CA PM we multiply and divide by [2kTk (10)yields, (μk-r) to obt (I1)Pil- Pio(I +r) (μk-r) (15)(1+r)Po=Pn r4- ∑T2σ N4a2+∑ Nik Pio Poo By substituting for a* and o*(variance and (k-)Na2+2N0 Inces in terms of one share rat than one dollar), and multiplying and divid ing by Tk, we obtain, [2Tm- (12)P-Po(1+r) where Po is the equilibrium price of stock i as suggested by this model. The price of risk (-m)/ΣT2 relevant only for investors who hold se- curity i. Obviously, investors who do not Equation(12)should apply to the kth in- hold security i are faced by a different price estor, but only for securities which are in- of risk. Moreover, the same investor ma cluded in his portfolio face two(or more) difTerent prices of risk Now, in order to have price equilibrium one appropriate for security i and one for in terms of the aggregate demand for the security j. This may occur since the group th stock we use the same technique as em- of investors who hold security i is not nec 0m3303038AN648 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1978 Thus, 2 = vi2 p,2, (J* = pijp and equation (8) can be rewritten in terms of market price per share, (9) NiPil - NiPio(1 + r) = (k- r) 2 (Jk [N 2NPOIY + NP0 i Piox ik , + N i Pio 1= Xik aij Dividing by Ni yields (10) Pi, - Pio(1 + r) = (Ak - r) a2 (k . PiOXik3i + Pio E xjk ij L X1ol Ju isi Now recall that the proportions invested by the kth investor X,k and Xjk in the ith and jth assets, respectively, have been given by Xik = NikPio/Tk, and Xjk = NjkPj,,O/Tk, where Nik and Nik stand for the number of shares of firm i and j in the kth investor's portfolio, and Tk is the total amount of dollars invested by him in risky assets. Thus, the substitution of Xik and Xjk in equation (10) yields, (11) Pi - Pio(1 + r) (AkI r) *p%2N kai + Njk PioPjo ij By substituting for a* and <* (variance and covariances in terms of one share rather than one dollar), and multiplying and divid￾ing by Tk, we obtain, (12) Pi, - P0o(1 + r) = - 2 Tkkk Likai + E Nj jI Equation (12) should apply to the kth in￾vestor, but only for securities which are in￾cluded in his portfolio. Now, in order to have price equilibrium in terms of the aggregate demand for the ith stock we use the same technique as em￾ployed by Lintner (1965a) with only one distinction: Lintner was allowed to sum up his equations for all investors. In our model, we are allowed to sum them up only for investors k who hold the security under consideration in their portfolios, since equation (4) (from which we derive equa￾tion (12)) includes the ith security only for investors k who hold it. After multiplying equation (12) by Tk24 and summing up only for investors k who hold security i, we obtain (13) [Pi, - P0o(1 + r)] E T = k ET (A - r) NikU*2+ E Nj a k Ljl J = The equilibrium price of share i, PF, is given by (14) (1 + r)Pi = Pi1 - (Tk(Ak - r) [Nik i + L Njk j) +>* Tk k In order to derive a more comparable form for the equilibrium price as implied by the CAPM we multiply and divide by [I2kTk- (k- r)] to obtain [Tk(A k -r)) (15) (1 + r)Pio Pi, [ T2U-2 [ Tk (8k k k k r - r)]+E where P0 is the equilibrium price of stock i as suggested by this model. The price of risk is given by [ ? Tk(gk r)]/k - T k? and is relevant only for investors who hold se￾curity i. Obviously, investors who do not hold security i are faced by a different price of risk. Moreover, the same investor may face two (or more) different prices of risk, one appropriate for security i and one for security j. This may occur since the group of investors who hold security i is not nec￾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