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Equilibrium in an Imperfect market: A Constraint on the number of securities in the portfolio he pioneering work of Harry Markowitz obvious that most individuals held a rela- (1952, 1959)and James Tobin in portfolio tively small num ber of stocks in their port theory has led to the development o f a folio. Another source of data which con- uncertainty. This theory, well-known in the Board' s 1967 survey of the Financial Char- literature as the capital asset pricing model acteristics of Consumers. This survey CAPM), was developed independently by covered all households whether or not they William Sharpe, John Lintner(1965a), and filed income tax forms. According to this Jack Treynor. two basic related properties survey, the a verage number of securities implied by the CA PM are: (a) that all in- the portfolio was 3. 41.2 estors hold in their portfolio all the risky The fact that properties(a)and(b)do not securities available in the mar ket, and(b) conform to reality is not a sufficient cause that investors hold the risky assets in the for rejecting the theoretical results of the same proportions, as these assets are avail- Ca PM, One could also accept the Ca PM able in the market, independent of the in- results on positive grounds. If the theoreti vestors preference. This latter property of cal model does indeed explain the price be the CaPM makes it possible to draw many havior of risky assets, one could argue that ing the equilibrium risk- investors behave as if properties(a)and(b) return relationship of risky assets were true, in spite of the fact that these Properties(a)and(b)contradict the mar- properties obviously do not prevail in the ket experience as established in all empirical market. Unfortunately, we can not justify research. First, investors differ in their in- the theoretical results of the CaPM on vestment strategy and do not necessarily ad- positive ground here to the same risky portfolio. Second, the To illustrate the latter difficulty, let us re typical investor usually does not hold many turn in greater detail to the CAPM. Accord- risky assets in his portfolio. Indeed in a re ing to the CaPM, the expected return on ent study, Marshall Blume, Jean Crockett, asset i, E(r,) is related to the expected re and Irwin Friend found that, in the tax year turn on the market portfolio E(Rm)as 1971, individuals held highly undiversified follows: portfolios. The sample, which included ( E(R,)-r=[E(Rm)-rIB 17.056 individual income tax forms,re- vealed that 34. I percent held only one stock where r is the risk-free interest rate, P, is the paying dividends, 50 percent listed no more risk index of the ith security(the"syste- than two, and only 10.7 percent listed more matic risk")and is defined as Cov(r,, rm)/ than ten. Though only firms paying cash var(Rm), and R is the rate of return on a dividends were included in this statistic, it is portfolio which consists of all available risky assets and is called the"market port- *Hebrew University of Jerusalem. I acknowledge folio he helpful comments of Yoram Landskroner, Yoram Kroll and an anonymous referee of this Review. Although the CA PM is formulated in terms of ex ante parameters, it is common disagreement of investors with regards to expected to employ ex post data rather than ex ante parameters. I assume in this model that investors agree values in empirical studies. Thus, we first with regard to future parameters but the model pre sented in this paper can be easily extended to the case 2 For more details of these findings and their analysis, see Blume and Friend (1975) 0m3303038AN

Equilibrium in an Imperfect Market: A Constraint on the Number of Securities in the Portfolio By HAIM LEVY* The pioneering work of Harry Markowitz (1952, 1959) and James Tobin in portfolio theory has led to the development of a theory of the pricing of capital assets under uncertainty. This theory, well-known in the literature as the capital asset pricing model (CAPM), was developed independently by William Sharpe, John Lintner (1965a), and Jack Treynor. Two basic related properties implied by the CA PM are: (a) that all investors hold in their portfolio all the risky securities available in the market, and (b) that investors hold the risky assets in the same proportions, as these assets are available in the market, independent of the investors' preference.' This latter property of the CA PM makes it possible to draw many conclusions regarding the equilibrium riskreturn relationship of risky assets. Properties (a) and (b) contradict the market experience as established in all empirical research. First, investors differ in their investment strategy and do not necessarily adhere to the same risky portfolio. Second, the typical investor usually does not hold many risky assets in his portfolio. Indeed, in a recent study, Marshall Blume, Jean Crockett, and Irwin Friend found that, in the tax year 1971, individuals held highly undiversified portfolios. The sample, which included 17,056 individual income tax forms, revealed that 34.1 percent held only one stock paying dividends, 50 percent listed no more than two, and only 10.7 percent listed more than ten. Though only firms paying cash dividends were included in this statistic, it is obvious that most individuals held a relatively small number of stocks in their portfolio. Another source of data which confirms these findings is the Federal Reserve Board's 1967 survey of the Financial Characteristics of Consumers. This survey covered all households whether or not they filed income tax forms. According to this survey, the average number of securities in the portfolio was 3.41. The fact that properties (a) and (b) do not conform to reality is not a sufficient cause for rejecting the theoretical results of the CA PM. One could also accept the CA PM results on positive grounds. If the theoretical model does indeed explain the price behavior of risky assets, one could argue that investors behave as if properties (a) and (b) were true, in spite of the fact that these properties obviously do not prevail in the market. Unfortunately, we can not justify the theoretical results of the CA PM on positive grounds. To illustrate the latter difficulty, let us return in greater detail to the CA PM. According to the CA PM, the expected return on asset i, E(Ri) is related to the expected return on the market portfolio E(Rm) as follows: (1) E(Ri) - r = [E(Rm)- r- i where r is the risk-free interest rate, fi is the risk index of the ith security (the "systematic risk") and is defined as Cov(Ri, Rm)/ Var(Rm), and Rm is the rate of return on a portfolio which consists of all available risky assets and is called the "market portfolio." Although the CA PM is formulated in terms of ex ante parameters, it is common to employ ex post data rather than ex ante values in empirical studies. Thus, we first *Hebrew University of Jerusalem. I acknowledge the helpful comments of Yoram Landskroner, Yoram Kroll and an anonymous referee of this Review. lLintner (1969) extends the CAPM to the case of disagreement of investors with regards to expected parameters. I assume in this model that investors agree with regard to future parameters but the model presented in this paper can be easily extended to the case of disagreement. 2For more details of these findings and their analysis, see Blume and Friend (1975). 643 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 978 run a time-series regression trary to the expected results from the CAPM Rt=a;+β1Rnt+ea since, if the CA PM is cor find that y2=0. Moreover, in most cases and estimate the systematic risk B, of each the contribution of Se, to the coefficient of asset i(where R and Rmt are the rates of re- correlation is even more important than the urn of the ith asset and the market port- contribution of the systematic risk, B folio, respectively, in year t). In the second In this paper I try to narrow the gap be- step, in order to examine the validity of the tween the theoretical model and the em CA PM, we run a cross-section regression irical findings by deriving a new version of R the Ca PM in which investors are assumed to hold in their portfolios some given num- R, is the average return on the ith ber of securities. obviously, investors' port- risky asset, B, is the estimate of the ith asset folios differ in the proportions of risky as stematic risk take s sets and even in the types of risky assets egression, and u, is a residual term. If the that they hold This, of course, is consistent CA PM is valid one should obtain(see equa- with investors' behavior as established in tion (I))in equilibrium, Yo =0 and y,- previous empirical research. I denote the Rm-r, where Yo and f, are the regression modified model as GCA PM (general capital coefficients estimated by(1), and Rm is the asset pricing model), since the CAPM average observed rate of return on the mar ket portfolio( for example, average rate of return on Standard and Poor's ind these conditions is given in Section Il. In Unfortunately, in virtually all empirical the third section I show that the modified research, ' it emerges that fo is significantly model explains the discrepancy between the positive and y is much below Rm-r. For theoretical results of the CAPM and the rates of return of individual stocks the cor- empirical findings mentioned above. Some elation coefficient of (1)is very low if one empirical results are presented which con employs monthly rates of return, and only firm that the systematic risk 8, plays no role 20-25 percent with annual rates of return in explaining price behavior, once the vari- Finally, in virtually all empirical studies, ance is taken into account, (Section IV) formulation(3)increases the correlation co- Concluding remarks are given in Section V efficient (3)R-r=0+;6,+2S I. Equilibrium in an Imperfect Market The CAPM where i stands for the ith security and s William Sharpe and Lintner(1965a)have e residual variance around the time-series shown that if there is no constraint on the regression(2), 1. e, the variance of the re- number of securities to be included in the siduals eit. In this formulation the estimate investors' portfolio, all investors will hold Y, happens to be significantly positive, con- some combination of m, the market port folio of risky assets, and the riskless asset See Fisher Black. Michael Jensen ron bearing interest rate r(see Figure 1) rton Now, suppose that, as a result of transac- Mill emphasize that the low correlation is obtained tion costs, indivisibility of investment,or and scho when equation(I')is regressed using ck. even the cost of keeping track of the new n order to minimize the measurement errors, it is financial development of all securities, the mon to use in(I')portfolios rather than individual kth investor decides to invest only in nk ks. This portfolio technique increases the correla possible errors, individual stocks show in spite of the securities. Under this constraint he will tion coefficient dramatically. Howe have some interior efficient set (of risky the CAPM defines equilibrium prices of individual sets), say, A'B, and the investor will divide his portfolio between some risky portfolio k 0m3303038AN

644 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1978 run a time-series regression, (2) Rit ai j + 3iRmt + eit and estimate the systematic risk f3 of each asset i (where Rit and Rmt are the rates of return of the ith asset and the market portfolio, respectively, in year t). In the second step, in order to examine the validity of the CA PM, we run a cross-section regression, (1') Ri-r = 'Yo + zlyli + Ui where Ri is the average return on the ith risky asset, fi is the estimate of the ith asset systematic risk, taken from the time-series regression, and ui is a residual term. If the CAPM is valid one should obtain (see equation (1)) in equilibrium, j0 = 0 and j' = Rm - r, where ' and 7 are the regression coefficients estimated by (1'), and Rm is the average observed rate of return on the market portfolio (for example, average rate of return on Standard and Poor's index). Unfortunately, in virtually all empirical research,3 it emerges that 'o is significantly positive and j, is much below Rm - r. For rates of return of individual stocks the correlation coefficient of (1') is very low if one employs monthly rates of return, and only 20-25 percent with annual rates of return.4 Finally, in virtually all empirical studies, formulation (3) increases the correlation coefficient, (3) Ri- r = To+ j lOi+ 72Sei where i stands for the ith security and S2. is the residual variance around the time-series regression (2), i.e., the variance of the residuals eit. In this formulation the estimate y2 happens to be significantly positive, contrary to the expected results from the CA PM since, if the CAPM is correct, one should find that 72 = 0- Moreover, in most cases, the contribution of S2. to the coefficient of correlation is even more important than the contribution of the systematic risk, /3. In this paper I try to narrow the gap between the theoretical model and the empirical findings by deriving a new version of the CA PM in which investors are assumed to hold in their portfolios some given number of securities. Obviously, investors' portfolios differ in the proportions of risky assets and even in the types of risky assets that they hold. This, of course, is consistent with investors' behavior as established in previous empirical research. I denote the modified model as GCA PM (general capital asset pricing model), since the CA PM emerges as a special case. The derivation of the GCAPM under these conditions is given in Section II. In the third section I show that the modified model explains the discrepancy between the theoretical results of the CAPM and the empirical findings mentioned above. Some empirical results are presented which confirm that the systematic risk fi plays no role in explaining price behavior, once the variance is taken into account, (Section IV). Concluding remarks are given in Section V. I. Equilibrium in an Imperfect Market: The GCA PM William Sharpe and Lintner (1965a) have shown that, if there is no constraint on the number of securities to be included in the investors' portfolio, all investors will hold some combination of m, the market portfolio of risky assets, and the riskless asset bearing interest rate r (see Figure 1). Now, suppose that, as a result of transaction costs, indivisibility of investment, or even the cost of keeping track of the new financial development of all securities, the kth investor decides to invest only in nk securities. Under this constraint he will have some interior efficient set (of risky assets), say, A 'B', and the investor will divide his portfolio between some risky portfolio k 3See Fisher Black, Michael Jensen, and Myron Scholes; George Douglas; Lintner (1965b); Merton Miller and Scholes. 4I emphasize that the low correlation is obtained when equation (1') is regressed using individual stock. In order to minimize the measurement errors, it is common to use in (I') portfolios rather than individual stocks. This portfolio technique increases the correlation coefficient dramatically. However, in spite of the possible errors, individual stocks should be used since the CA PM defines equilibrium prices of individual stocks. 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

VOL 68 No. 4 LEVY: PORTFOLIO EQUILIBRIUM K A C Standard deviation Standard deviation FIGURE FIGURE 2 and the riskless asset. Obviously, the in- curities'efficient sets need to be tangent vestor's welfare will decrease if no more the market line rkK. a sufficient condition than nk securities may be included in the for the market to be cleared out, in this ex- portfolio, since for a given expected return, ample, is for two out of three efficient sets he will be exposed to higher risk (see given in Figure 2(i.e, AB, BC, AC) to be Figure 1) tangent to the line rkk. In other words In the specific case in which all investors each of the three assets must be included in hold the same number of risky assets nk in some two-asset portfolio which is tangent to equilibrium, all these interior efficient sets the straight line will be tangent to the same straight line. To In the more realistic case. which will be illustrate, suppose that nk =2 for all k and dealt with below, the kth investor has the that there are n= 3 risky assets available constraint of investing in no more than nk in the market. Figure 2 shows this possibility risky assets when nk varies among investors sing A, B, and C to indicate the three risky securities Without any constraints, all investors E hold portfolio m(i.e, the market portfolio), and all efficient portfolios lie on line rmM that all investors decide to g include only two risky assets in their port- 0 folio. Investors who hold securities A and b a are faced with opportunity line rkK. If all investors decide to include two risky assets in their portfolio, this situation will represent an equilibrium situation, since no one will purchase security C(see Figure 2) Hence the price of securlty C will decline, r and its expected return will increase, until we get a new efficient curve between B and C(or C and A)which will be tangent to line rkk. In this case, however, the market may Standard deviation be cleared out. Note that not all two se- FIGURE 3 0m3303038AN

VOL. 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM 645 a) B mB /, t</A' r Standard deviation FIGURE I and the riskless asset. Obviously, the investor's welfare will decrease if no more than nk securities may be included in the portfolio, since for a given expected return, he will be exposed to higher risk (see Figure 1). In the specific case in which all investors hold the same number of risky assets nk in equilibrium, all these interior efficient sets will be tangent to the same straight line. To illustrate, suppose that nk = 2 for all k and that there are n = 3 risky assets available in the market. Figure 2 shows this possibility using A, B, and C to indicate the three risky securities. Without any constraints, all investors hold portfolio m (i.e., the market portfolio), and all efficient portfolios lie on line rmM. Now suppose that all investors decide to include only two risky assets in their portfolio. Investors who hold securities A and B are faced with opportunity line rkK. If all investors decide to include two risky assets in their portfolio, this situation will not represent an equilibrium situation, since no one will purchase security C (see Figure 2). Hence the price of security C will decline, and its expected return will increase, until we get a new efficient curve between B and C (or C and A) which will be tangent to line rkK. In this case, however, the market may be cleared out. Note that not all two se- . J K Stand ar i V~~ r Standard deviation FIGURE 2 curities' efficient sets need to be tangent to the market line rkK. A sufficient condition for the market to be cleared out, in this example, is for two out of three efficient sets given in Figure 2 (i.e., AB, BC, AC) to be tangent to the line rkK. In other words, each of the three assets must be included in some two-asset portfolio which is tangent to the straight line. In the more realistic case, which will be dealt with below, the kth investor has the constraint of investing in no more than nk risky assets when nk varies among investors m 4-J2 0~ r Standard deviation FIGURE 3 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 /978 mainly as a function of the size of their the covariance between returns of wealth. In this case there are many interio securities i and j efficient sets(see Figure 3), and the existence Hk the portfolio expected return of many market lines does not contradict xik= the proportion invested in the ith the possibility that the market may be in security by the k th investor equilibrium r riskless interest rate In this case, rm is the opportunity line Ak= Lagrange multiplier appropriate without any constraint on the num ber of for the th investor securities in the portfolio: r2 is the marke line with the constraint that no more th an Suppose that the investor selects two securities are included in the portfolio: out of the n available assets to be included r3 is the line with the constraint of no more in his optimal portfolio. Then by differen than three securities in the portfolio, etc. tiating the Lagrangian function we obtain Obviously, the same security may be held in the following nk= I equations, which pro- proportion of 20 percent of one portfolio, vide the optimal diversification strategy 5 percent of a second portfolio, etc. We de- among the nk risky assets rive below the equilibrium prices of risky for the general case in which straint on nk varies from investor to in- (4)xi+22xA=A(1-n vestor. Again, a necessary condition for equilibrium in the stock market is that each x2x402+∑x/02=入4(H2-7) ty be included of the chosen unlevered portfolios from the above efficient sets risk-return relationship under the constraint xnk onk+ 2 X kOnk= Xx(ung- r) that not all risky assets are held in the in vestors portfolio. We assume that there are K investors(or groups of investors), and the =∑x,+ kth investor wealth is Tk dollars. Further- ore, assume that the kth investor invests Thus, the optimal investment strategy of only in nk risky assets while there are in the the k th investor is given by the vector x Ik market n >n, risky assets. thus the kth xnk which solves the above equa- investor minimizes the portfolio's variance tions. We multiply the first equation by xIk subject to the constraint that the number of the second equation by xxk, etc, and then scurities in his portfolio cannot exceed nk. sum up the first nk equations to obtain Mo partially with respect to xik and Ak the Lagrangian function rikai L + +(1-∑x)r-r=A(k-r) +2入k|k He 1 subject to the constraint that no more than (5) nk securities will be included in the optimal portfolio, where where uk and of are the expected return and variance of the k th investors optimal port af= the variance of the ith security re- folio. Using(4)and (5) the kth investor wil turn(per $I of investment) be in equilibrium if and only if 0m3303038AN

646 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1978 mainly as a function of the size of their wealth. In this case there are many interior efficient sets (see Figure 3), and the existence of many market lines does not contradict the possibility that the market may be in equilibrium. In this case, rm is the opportunity line without any constraint on the number of securities in the portfolio; r2 is the market line with the constraint that no more than two securities are included in the portfolio; r3 is the line with the constraint of no more than three securities in the portfolio, etc. Obviously, the same security may be held in proportion of 20 percent of one portfolio, 5 percent of a second portfolio, etc. We derive below the equilibrium prices of risky assets for the general case in which the constraint on nk varies from investor to investor. Again, a necessary condition for equilibrium in the stock market is that each security be included in at least one of the chosen unlevered portfolios from the above efficient sets. Let us turn now to the derivation of the risk-return relationship under the constraint that not all risky assets are held in the investors' portfolio. We assume that there are K investors (or groups of investors), and the kth investor wealth is Tk dollars. Furthermore, assume that the kth investor invests only in nk risky assets while there are in the market n > nk risky assets. Thus, the kth investor minimizes the portfolio's variance subject to the constraint that the number of securities in his portfolio cannot exceed nk. More specifically, one has to differentiate partially with respect to Xik and Xk the Lagrangian function nk nk L -> Ex1kK + 2 E XikXjkO> i= i k=l (i j>i nk tk + 2Xk Ak Xiki - Xik )r subject to the constraint that no more than nk securities will be included in the optimal portfolio, where =-2 = the variance of the ith security return (per $1 of investment) =ij = the covariance between returns of securities i andj =k = the portfolio expected return Xik = the proportion invested in the ith security by the kth investor r = riskless interest rate Xk = Lagrange multiplier appropriate for the kth investor Suppose that the investor selects nk assets out of the n available assets to be included in his optimal portfolio. Then by differentiating the Lagrangian function we obtain the following nk = 1 equations, which provide the optimal diversification strategy among the nk risky assets nk (4) XIkOS + E XjkS l = Xk(MI r) j=2 nk X2k 02 + E XjkU2j = Xk(J2 r) J = I j#2 nk Xnk Unk + E XJvk IT,k; k(n IL = I,L,(-iir j*nk 'tk nk \ Ak -EXik Hi + -EXik r Thus, the optimal investment strategy of the kth investor is given by the vector X,k, X2k,..., Xnk which solves the above equations. We multiply the first equation by Xlk, the second equation by X2k, etc., and then sum up the first nk equations to obtain /nk nk nk a' = Xk(S Xk,Hi - k xkir) Akk = Xik/i ? (i -n~k xik)r r Xk(Mk - r) Hence, 1 /k- r (5) 2 Xk ?k where -k and aS are the expected return and variance of the kth investor's optimal portfolio. Using (4) and (5) the kth investor will be in equilibrium if and only if 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

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 rewritten 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 equilibrium price determination, of not holding all assets in the portfolio we need to use some algebra. Since R k = 2 nk XjkRj, equation (6) can be rewritten as vil-Vio (Ak - r) (7) - =r+ - V0 ~~~~~~2 [Xik , + jk ji when vil and vio stand for the expected market value of firm i at the end of the period, and for the equilibrium present value, respectively. 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 return 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 portfolio (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

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) 0m3303038AN

VOL. 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 equation (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 securities 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 assumptions, 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 covariance, 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 determination, 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 includes 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. Empirical 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 return 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 without constraint on the number of securities in the portfolio (see Lintner 1965a, p. 600). 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

650 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1978 Hence, (16) (1 + r)PiO = Pi, - y To ) L Tk(,k - FN2r) N 2 + Njk Tk - r) k or (17) (1 + r)PiO= Pi, nk L Tk Tk (8k - r)L, Nka,2 + E Nik -ykik JI t Z Tk (jk - r) k where Y mi E T2.jak k Equation (17) is very similar to the classic relationship of the CA PM (see equation (20')). The only two differences are: (a) now the securities' risk is given as the weighted average of the risks of each investor when the weights are Tk (gk - r), so that, the larger the investor's wealth (Tk), the greater his impact on price determination, and (b) the market price of risk y, is defined somewhat differently from the well-known y, as defined by Lintner (1965a). Thus, the classic CA PM may be the approximate equilibrium model for stocks of firms which are held by many investors (for example, AT&T), but not for small firms whose stocks are held by a relatively small group of investors. If we relax the constraint that the kth investor holds only nk securities, then each investor holds the market portfolio and hence6gu - r = /Im - r, and U2 = Sk, where /Im and am are the expected rate of return and variance of the market portfolio, respectively. On the basis of these assumptions we obtain the classic CAPM formula as a special case of the GCA PM suggested in this paper. In this case, equation (16) reduces to T 2 (18) (1 + r)pio =Pi - Y >T k LTk nk But since the relaxation of the imperfection induces all investors to have the same investment strategy in risky assets, (see Sharpe and Lintner, 1965a) all of them hold all the risky assets nk = n and, also Nik/Ni = Tk/ To and hence NTk = NiTk/ To and Njk = NJTk/ T0. By substituting the last reults in equation ( 18) we derive (19) (1 + r)p, ik =pji - = k k or (20) (1 + r)p0o = p,i - -yT k BtSince the =ro,elaxation of20) imerfcst then inuesl-knownvequilirsium haeqution samte ShrpeitindlC M Lintner 11965a, (e l fthmhl F y o ke t k basic diSernce between equation (20) run s the) wEl-nweqiiruequation of),wic avcth,ep traiinalthe CAM (see tneral f , ad ha, p. 600), ~ O . T (20') (1I r)pi = p~i, - [Ni< + Nj bSince difernc bTwee e quations20 (15)ce and (17) wel-nweqiiruequation (1) hc doate,hep tresetintsth most gseeLnnerafoman hence 6Recall that without loss of generality we deal only with the optimal unlevered portfolio. The basic equilibrium equation (equation (6)) and hence all the other results derived from it are unchanged no matter if we deal with the levered or the unlevered portfolio. See fn. 5. 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

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- 0m3303038AN

VOL. 68 NO. 4 LEVY: PORTFOLIO EQUILIBRIUM 651 only cij of securities included in the kth investors' portfolios, are taken into account. However, if we assume unrealistically that security i is included in all investors' portfolios (equation (17)) then for an equilibrium 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 estimates 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 description of price behavior. Using annual data, Merton Miller and Myron Scholes confirm the basic results of Lintner and suggest possible explanations for the deviation 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 results implied by the CA PM. We have found that the investment horizon plays a crucial role in any econometric research and, particularly, 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 estimates of the systematic risk (i.e., 'Yj in equation (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 portfolio 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 securities 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 relationship 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 empirical 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 systematic 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 erThis content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:07:38 AM All use subject to JSTOR Terms and Conditions