American Economic Association Equilibrium in an Imperfect Market: A Constraint on the Number of Securities in the Portfolio Author(s): Haim Levy Source: The American Economic Review, Vol. 68, No. 4(Sep, 1978), pp. 643-658 Published by: American Economic Association StableUrl:http://www.jstor.org/stable/1808932 Accessed:11/09/201303:07 Your use of the JSTOR archive indicates your acceptance of the Terms Conditions of Use, available at JStOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support(@jstor. org American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American economic revie 的d http://www.jstororg This content downloaded from 202. 115.118.13 on Wed, I I Sep 2013 03: 07: 38 AM All use subject to STOR Terms and Conditions
American Economic Association Equilibrium in an Imperfect Market: A Constraint on the Number of Securities in the Portfolio Author(s): Haim Levy Source: The American Economic Review, Vol. 68, No. 4 (Sep., 1978), pp. 643-658 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/1808932 . Accessed: 11/09/2013 03:07 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review. http://www.jstor.org 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
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