《投资学原理 Investments》课程教学资源（阅读资料）Equilibrium in an Imperfect Market A Constraint on the Number of Securities in the Portfolio

↓OL.68NO.4 EVY: PORTFOLIO EQUILIBRIUM r+ COV(R, Rk) optimal investment. In the rest of the paper we assume that uk and of are the parameters of the optimal unlevered portfolio chosen b: where R, and Rk are the rates of return on the k th investor. This is tantamount to the e ith security and on the portfolio chosen assumption that by the k th investor. Equation(6) can be re- ui=r+(uk- r)Bx In order to examine the impact on equ where Bki is the systematic risk of the ith librium price determination, of not holding asset in the kth investor's optimal portfolio all assets in the portfolio we need to use Ra and is defined as Bk= Cov(R,, Rk)/o2. some algebra. Since Rk=EK,xkR,,equa It is important to note that the equilibrium tion()can be rewritten as relationship given in equations(6) and (6 is independent of the borrowing or lending (7) Vi-Vo=r+(Hs-r) policy of the kth investor, > Thus, without loss of generality, we can assume that xkσ}+∑x,kn when va and vo stand for the expected man and this will not afFect the solution of the ket value of firm i at the end of the period, and for the equilibrium present value, re 'To be more specific suppose that an investor who spectively. Hence, owns TA dollars decides to borrow or lend (2/k xik-I)per each dollar that he owns. Then, if, R, is 8)Vil- Vo(+r)(uk-r) e return (per one dollar)on hi lely from risky assets, the return on his selected port olio(including the borrowing or lending) denoted by Let us denote a*2= the expected variance of the return on one share of the ith firm at the end of the investment perio a*= the expected covariance of the re- turn of a share of firm i and a share of firm j N= the number of outstanding shares of firm i cov(R R Pio= the equilibrium price of a share of firm i Rewriting(6)in terms of RK we obtain Pn- the expected price of a share of firm i at the end of the period Ov(R,RK μ;=F where p and e expected return and 0m3303038AN

VOL. 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM 647 16J ll ~k -- r (6) g1=r + -rcov(RiRk) (F k where Ri and Rk are the rates of return on the ith security and on the portfolio chosen by the kth investor. Equation (6) can be re￾written as (6') Hi = r + (gk -r)ki where Oki iS the systematic risk of the ith asset in the kth investor's optimal portfolio Rk and is defined as flki = Cov(Ri, Rk)/ok It is important to note that the equilibrium relationship given in equations (6) and (6') is independent of the borrowing or lending policy of the kth investor.5 Thus, without loss of generality, we can assume that n k Z Xik = I and this will not affect the solution of the optimal investment. In the rest of the paper we assume that,/k and a' are the parameters of the optimal unlevered portfolio chosen by the kth investor. This is tantamount to the assumption that nk LXik= i = 1 In order to examine the impact on equi￾librium price determination, of not holding all assets in the portfolio we need to use some algebra. Since R k = 2 nk XjkRj, equa￾tion (6) can be rewritten as vil-Vio (Ak - r) (7) - =r+ - V0 ~~~~~~2 [Xik , + jk ji when vil and vio stand for the expected mar￾ket value of firm i at the end of the period, and for the equilibrium present value, re￾spectively. Hence, (8) vil - vio(l + r) = (Ak- r) ck + V nk VioX ik(Y + VO Xjkaij Let us denote (* 2 = the expected variance of the return on one share of the ith firm at the end of the investment period = the expected covariance of the re￾turn of a share of firm i and a share of firm j Ni = the number of outstanding shares of firm i Pio = the equilibrium price of a share of firm I Pi, = the expected price of a share of firm i at the end of the period 5To be more specific suppose that an investor who owns Tk dollars decides to borrow or lend (Lk Xik - 1) per each dollar that he owns. Then, if, Rk is the return (per one dollar) on his optimal portfolio solely from risky assets, the return on his selected port￾folio (including the borrowing or lending) denoted by R* will be /nk \ /' nk \ Rk =KE XLk)Rkj,$ Xikk I)r and hence A k ( Xi k - ( Xik) r + r, nk n 7 2 and, cov* (RiRk) [( ik)] vRik Rewriting (6) in terms of RX we obtain * - r or 2k COv*(RiRk) U7k or nk L xik(Ik-r)+r-r (nk ) f Xik) ak and finally 2i = r + 2 cov(RiRk) Uk where Ik and ak are the expected return and variance of the optimal portfolio of the kth investor when he neither borrows nor lends money. 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

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 0m3303038AN

648 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