W CHICAGO JOURNALS Capital Market Equilibrium with Restricted Borrowing Author(s): Fischer Black Source: The Journal of Business, Vol. 45, No. 3(Jul, 1972), pp. 444-455 Published by: The University of Chicago Press StableurL:http://www.jstor.org/stable/2351499 Accessed:11/09/20130246 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 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 The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Business. 的d http://www.jstororg This content downloaded from 202. 115.118.13 on Wed, I I Sep 2013 02: 46: 59 AM All use subject to STOR Terms and Conditions
Capital Market Equilibrium with Restricted Borrowing Author(s): Fischer Black Source: The Journal of Business, Vol. 45, No. 3 (Jul., 1972), pp. 444-455 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2351499 . Accessed: 11/09/2013 02:46 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. . The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Business. http://www.jstor.org This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
Fischer Black Capital Market Equilibrium with Restricted Borrowing INTRODUCTION Several authors have contributed to the development of a model describ ing the pricing of capital assets under conditions of market equilibrium The model states that under certain assumptions the expected return on any capital asset for a single period will satisfy E(R)=R1+BE(Rm)一R The in equation(1)are defined as follows: R, is the return or le period and is equal to the change in the price of the asset, plu vidends, interest, or other distributions, divided pri of the asset at the start of the period; Rm is the return on the market portfolio of all assets taken together; R, is the return on a riskless asset for the period; B is the"market sensitivity"of asset i and is equal to the slope of the regression line relating R, and Rm The market sensitivity B: of asset i is defined algebraically by B=cov(R, Rm)/var(Rm) (2) are as follows: (a)All investors have the same opinions about the poso: motions that ly used in deriving eq bilities of various end-of-period values for all assets. They have a com- mon joint probability distribution for the returns on the available assets (b) The common probability distribution describing the possible returns on the available assets is joint normal (or joint stable with a single char acteristic exponent).(c) Investors choose portfolios that maximize their expected end-of-period utility of wealth, and all investors are risk averse.( Every investor's utility function on end-of-period wealth in- creases at a decreasing rate as his wealth increases. )(d) An investor may take a long or short position of any size in any asset, including the riskless asset. Any investor may borrow or lend any amount he wants at the riskless rate of interest The length of the period for which the model applies is not specified The assumptions of the model make sense, however, only if the period is taken to be infinitesimal. For any finite period, the distribution of pos- sible returns on an asset is likely to be closer to lognormal than normal Graduate School of Business, University of Chicago Some of the basic ideas in this paper, and many helpful comments, wer rovided ugene Fama, Michael Jensen, John Lintner John Long, Robert Merton, Myron Scholes, william Sharpe, Jack Treynor, and Oldrich Vasicek. This ork was supported in part by Wells Fargo Bank and the Ford Foundation 1. A summary william F. Sharpe, Portfe eory and Capital Markets(New York: McGraw-Hill Book Co,1970) 44 his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
Fischer Black* Capital Market Equilibrium with Restricted Borrowingt I N T R O D U C T I O N Several authors have contributed to the development of a model describing the pricing of capital assets under conditions of market equilibrium.1 The model states that under certain assumptions the expected return on any capital asset for a single period will satisfy E(AJ = Rf + /3JE(Rm) -Rf]. The symbols in equation (1) are defined as follows: Ai is the return on asset i for the period and is equal to the change in the price of the asset, plus any dividends, interest, or other distributions, divided by the price of the asset at the start of the period; Rm is the return on the market portfolio of all assets taken together; Rf is the return on a riskless asset for the period; fli is the "market sensitivity" of asset i and is equal to the slope of the regression line relating Rk and Rm. The market sensitivity /3i of asset i is defined algebraically by A - cov(Ai, Am)/var(Rm). (2) The assumptions that are generally used in deriving equation (1) are as follows: (a) All investors have the same opinions about the possibilities of various end-of-period values for all assets. They have a common joint probability distribution for the returns on the available assets. (b) The common probability distribution describing the possible returns on the available assets is joint normal (or joint stable with a single characteristic exponent). (c) Investors choose portfolios that maximize their expected end-of-period utility of wealth, and all investors are risk averse. (Every investor's utility function on end-of-period wealth increases at a decreasing rate as his wealth increases.) (d) An investor may take a long or short position of any size in any asset, including the riskless asset. Any investor may borrow or lend any amount he wants at the riskless rate of interest. The length of the period for which the model applies is not specified. The assumptions of the model make sense, however, only if the period is taken to be infinitesimal. For any finite period, the distribution of possible returns on an asset is likely to be closer to lognormal than normal; * Graduate School of Business, University of Chicago. t Some of the basic ideas in this paper, and many helpful comments, were provided by Eugene Fama, Michael Jensen, John Lintner, John Long, Robert Merton, Myron Scholes, William Sharpe, Jack Treynor, and Oldrich Vasicek. This work was supported in part by Wells Fargo Bank and the Ford Foundation. 1. A summary of the development of the model may be found in William F. Sharpe, Portfolio Theory and Capital Markets (New York: McGraw-Hill Book Co., 1970). 444 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
45 in particular, if the distribution of returns is normal, then there will be a finite probability that the asset will have a negative value at the end Of these assumptions, the one that has been felt to be the most re- strictive is(d). Lintner has shown that removing assumption (a)does not change the structure of capital asset prices in any significant way, 2 and assumptions(b) and (c) are generally regarded as acceptable ap proximations to reality. Assumption(d), however, is not a very goo approximation for many investors, and one feels that the model would be changed substantially if this assumption were dropped In addition, several recent studies have suggested that the returns on securities do not behave as the simple capital asset pricing model described above predicts they should. Pratt analyzes the relation between risk and return in common stocks in the 1926-60 period and concludes that high-risk stocks do not give the extra returns that the theory predicts they should give. Friend and Blume use a cross-sectional regression be tween risk-adjusted performance and risk for the 1960-68 period and observe that high-risk portfolios seem to have poor performance, while low-risk portfolios have good performance. They note that there is some bias in their test, but claim alternately that the bias is so small that it can be ignored, and that it explains half of the effect they observe.8 In fact, the bias is serious. Miller and Scholes do an extensive analysis of the nature of the bias and make corrections for it. even after their cor- ections, however, there is a negative relation between risk and per formance Black, Jensen, and Scholes analyze the returns on portfolios of stocks at different levels of B, in the 1926-66 period They find that the average returns on these portfolios are not consistent with equation (1) especially in the postwar period 1946-66. Their estimates of the expected eturns on portfolios of stocks at low levels of Bi are consistently higher than predicted by equation(1), and their estimates of the expected re- turns on portfolios of stocks at high levels of B: are consistently lower John Lintner, " The Ags Investors' Diverse Judg Preferences in Perfectl Markets, Journal of fine 3. Shannon P. Pratt, "Relat Levels of Future Returns for Con 4. Irwin Friend and Marshall Blume, "Measurement of Portfolio nce under Uncertainty, " American Economic Review 60( September 19 品 568. Compare the text with n H. Miller and Myron Scholes, Rates of Return in Relation to Risk: A amination of Some Recent Findings, " in Studies in the Theory Capital Markets, ed. Michael C Jensen(New York: Praeger Publishing Co press er Black, Michael C. Jensen, and Myron Scholes,"The Capital Asset Pricing Model: Some Empirica ests, in Studies in the Theory of Capital hael C Jensen(New York: Praeger Publishing Co, in press) his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
445 Capital Market Equilibrium in particular, if the distribution of returns is normal, then there will be a finite probability that the asset will have a negative value at the end of the period. Of these assumptions, the one that has been felt to be the most restrictive is (d). Lintner has shown that removing assumption (a) does not change the structure of capital asset prices in any significant way,2 and assumptions (b) and (c) are generally regarded as acceptable approximations to reality. Assumption (d), however, is not a very good approximation for many investors, and one feels that the model would be changed substantially if this assumption were dropped. In addition, several recent studies have suggested that the returns on securities do not behave as the simple capital asset pricing model described above predicts they should. Pratt analyzes the relation between risk and return in common stocks in the 1926-60 period and concludes that high-risk stocks do not give the extra returns that the theory predicts they should give. Friend and Blume use a cross-sectional regression between risk-adjusted performance and risk for the 1960-68 period and observe that high-risk portfolios seem to have poor performance, while low-risk portfolios have good performance.4 They note that there is some bias in their test, but claim alternately that the bias is so small that it can be ignored, and that it explains half of the effect they observe.5 In fact, the bias is serious. Miller and Scholes do an extensive analysis of the nature of the bias and make corrections for it.6 Even after their corrections, however, there is a negative relation between risk and performance. Black, Jensen, and Scholes analyze the returns on portfolios of stocks at different levels of flb in the 1926-66 period.7 They find that the average returns on these portfolios are not consistent with equation (1), especially in the postwar period 1946-66. Their estimates of the expected returns on portfolios of stocks at low levels of /3i are consistently higher than predicted by equation (1), and their estimates of the expected returns on portfolios of stocks at high levels of /3i are consistently lower than predicted by equation (1). 2. John Lintner, "The Aggregation of Investors' Diverse Judgments and Preferences in Perfectly Competitive Security Markets," Journal of Financial and Quantitative Analysis 4 (December 1969): 347-400. 3. Shannon P. Pratt, "Relationship between Viability of Past Returns and Levels of Future Returns for Common Stocks, 1926-1960," memorandum (April 1967). 4. Irwin Friend and Marshall Blume, "Measurement of Portfolio Performance under Uncertainty," American Economic Review 60 (September 1970): 561-75. 5. Ibid., p. 568. Compare the text with n. 15. 6. Merton H. Miller and Myron Scholes, "Rates of Return in Relation to Risk: A Re-Examination of Some Recent Findings," in Studies in the Theory of Capital Markets, ed. Michael C. Jensen (New York: Praeger Publishing Co., in press). 7. Fischer Black, Michael C. Jensen, and Myron Scholes, "The Capital Asset Pricing Model: Some Empirical Tests," in Studies in the Theory of Capital Markets, ed. Michael C. Jensen (New York: Praeger Publishing Co., in press). This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
446 The Journal of Business Black, Jensen, and Scholes also find that the behavior of well diversified portfolios at different levels of B: is explained to a much greater extent by a two-factor model"than by a single-factor"market model. "8 They show that a model of the following form provides a good fit for the behavior of these portfolios R =a+biRm+(1 -),+E In equation(3), R, is the return on a"second factor"that is independent of the market (its B: is zero), and Ei, i=1, 2,..., N are approxi mately mutually independent residual This model suggests that in periods when R, is positive, the low BL ortfolios all do better than predicted by equation(1), and the high pu portfolios all do worse than predicted by equation(1). In periods whe R is negative, the reverse is true: low B portfolios do worse than ex pected, and high B portfolios do better than expected. In the postwar period, the estimates obtained by Black, Jensen, and Scholes for the mean of R, were significantly greater than zero One possible explanation for these empirical results is that assump- tion(a)of the capital asset pricing model does not hold. What we will show below is that the relaxation of assumption (d)can give models that are consistent with the empirical results obtained by Pratt, Friend and blume, Miller and Scholes, and Black, Jensen and Scholes EQUILIBRIUM WITH NO RISKLESS ASSET Let us start by assuming that investors may take long or short positions of any size in any risky asset, but that there is no riskless asset and that no borrowing or lending at the riskless rate of interest is allowed. This assumption is not realistic, since restrictions on short selling are at least as stringent as restrictions on borrowing. But restrictions on short selling may simply add to the effects that we will show are caused by restric tions on borrowing. Under these assumptions, Sharpe shows that the effi cient set of portfolios may be written as a weighted combination of two basic portfolios, with different weights being used to generate the differ ent portfolios in the efficient set. In his notation, the proportion X, of asset i in the efficient portfolio corresponding to the parameter X satisfies (4), where K and k are constants X=K4+ i=1,2,,,,,N. Thus the weights on the stocks in the two basic portfolios are Ki, i= 1 N, and k, i= 1, 2 The weights satisfy (5), so the sum of the weights Xi is always equal to 1 8. One form of market model is defined in Eugene F. Fama, Risk, Return ibrium, "Journal of Political Economy 79(January/February 1971): 34. his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
446 The Journal of Business Black, Jensen, and Scholes also find that the behavior of welldiversified portfolios at different levels of f83 is explained to a much greater extent by a "two-factor model" than by a single-factor "market model."8 They show that a model of the following form provides a good fit for the behavior of these portfolios: Ri = + bRm + (1 -bi)Rz + e4 (3) In equation (3), R. is the return on a "second factor" that is independent of the market (its jli is zero), and Ei, i - 1, 2, . . , N are approximately mutually independent residuals. This model suggests that in periods when R. is positive, the low f83 portfolios all do better than predicted by equation (1), and the high j3i portfolios all do worse than predicted by equation ( 1 ). In periods when R. is negative, the reverse is true: low 8i portfolios do worse than expected, and high jli portfolios do better than expected. In the postwar period, the estimates obtained by Black, Jensen, and Scholes for the mean of R. were significantly greater than zero. One possible explanation for these empirical results is that assumption (d) of the capital asset pricing model does not hold. What we will show below is that the relaxation of assumption (d) can give models that are consistent with the empirical results obtained by Pratt, Friend and Blume, Miller and Scholes, and Black, Jensen and Scholes. EQUILIBRIUM WITH NO RISKLESS A S S E T Let us start by assuming that investors may take long or short positions of any size in any risky asset, but that there is no riskless asset and that no borrowing or lending at the riskless rate of interest is allowed. This assumption is not realistic, since restrictions on short selling are at least as stringent as restrictions on borrowing. But restrictions on short selling may simply add to the effects that we will show are caused by restrictions on borrowing. Under these assumptions, Sharpe shows that the efficient set of portfolios may be written as a weighted combination of two basic portfolios, with different weights being used to generate the different portfolios in the efficient set.9 In his notation, the proportion Xi of asset i in the efficient portfolio corresponding to the parameter X satisfies (4), where Ki and ki are constants: Xi =Ki + ki i =1, 2,. .N. (4) Thus the weights on the stocks in the two basic portfolios are Ki, i - 1, 2, . . ., N, and ki, i - 1, 2, . . . , N. The weights satisfy (5), so the sum of the weights Xi is always equal to 1. 8. One form of market model is defined in Eugene F. Fama, "Risk, Return, and Equilibrium," Journal of Political Economy 79 (January/February 1971): 34. 9. Sharpe, pp. 59-69. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
Capital Market equilibr (5) Sharpe also shows that the variance of return on an efficient portfolio is a quadratic function of its expected return Similarly, Lintner shows that a number of relations can be derived when there is no riskless asset. 10 His equation(16c) can be interpreted, in the case where all investors agree on the joint distribution of end-of- period values for all assets, as saying that even when there is no riskless asset, every investor holds a linear combination of two basic portfolios And his equation(18)can be interpreted as saying that the prices of assets in equilibrium are related in a relatively simple way even without a riskless asset Cass and Stiglitz show that if the returns on securities are not as sumed to be joint normal, but are allowed to be arbitrary, then the set of efficient portfolios can be written as a weighted combination of tw basic portfolios only for a very special class of utility functions. 1 a notation similar to that used by fama, we can show that every efficient portfolio consists of a weighted combination of two basic portfolios as follows. An efficient portfolio is one that has maximum ex- pected return for given variance, or minimum variance for given expected return. Thus the efficient portfolio held by individual k is obtained by choosing proportions xxi i=1, 2,..., N, invested in the shares of each of the n available assets in order to var (R) 台xyvR,R); (6) ect to E(R)=∑xE(R); xh;=1 (8) Using Lagrange multipliers Sk and Tk, this can be expressed as Minimize kuki cov (R, Ri) E(R)-E(R)」-2T 10. Lintner, Pp. 373-84 11. David and Structure of erences and Asset Returns, and Separability in Contribu tion to the Pure Theory of Mutual Funds, "Journ 2 (June 1970):122-60 his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
447 Capital Market Equilibrium N N Z Ki =1; ki (5) Sharpe also shows that the variance of return on an efficient portfolio is a quadratic function of its expected return. Similarly, Lintner shows that a number of relations can be derived when there is no riskless asset.'0 His equation (16c) can be interpreted, in the case where all investors agree on the joint distribution of end-ofperiod values for all assets, as saying that even when there is no riskless asset, every investor holds a linear combination of two basic portfolios. And his equation (18) can be interpreted as saying that the prices of assets in equilibrium are related in a relatively simple way even without a riskless asset. Cass and Stiglitz show that if the returns on securities are not assumed to be joint normal, but are allowed to be arbitrary, then the set of efficient portfolios can be written as a weighted combination of two basic portfolios only for a very special class of utility functions." Using a notation similar to that used by Fama, we can show that every efficient portfolio consists of a weighted combination of two basic portfolios as follows. An efficient portfolio is one that has maximum expected return for given variance, or minimum variance for given expected return. Thus the efficient portfolio held by individual k is obtained by choosing proportions Xki, i = 1, 2, . . , N, invested in the shares of each of the N available assets, in order to N N Minimize: var(Rk) - XkXkj cov(Ri, Rj); (6) i=l j=l N Subject to: E(Rk) xkjE(Rj); (7) j=1 N Z Xkji1. (8) j=l1 Using Lagrange multipliers Sk and Tk, this can be expressed as N N Minimize: XkiXkj cov(Ri, Rj) i=1 j_-1 (9) NV N - 2SkLZ xkjE(Rj) - E(Rk) - 2Tk Z xkj 1]. j=1 j=1 10. Lintner, pp. 373-84. 11. David Cass and Joseph E. Stiglitz, "The Structure of Investor Preferences and Asset Returns, and Separability in Portfolio Allocation: A Contribution to the Pure Theory of Mutual Funds," Journal of Economic Theory 2 (June 1970): 122-60. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
448 The Journal of Business Taking the derivative of this expression with respect to xki, we have xcov(RR)一SE(R)-T=0 (10) This set of equations, for i=1, 2 n, determines the of xu. If we write Dy for the inverse of the covariance matrix RA), then the solution to this set of equations may be written xM=S8∑DE()+T2D Note that the subscript k, referring to the individual investor, appears on the right-hand side of this equation only in the multipliers Sk and Tk. Thus very investor holds a linear combination of two basic portfolios, and every efficient portfolio is a linear combination of these two basic port folios. In equation(11), there is no guarantee that the weights on the individual assets in the two portfolios sum to one. If we normalize these eights, then equation (11) may be written (12) In equation(12), the symbols are defined as follows w=s∑∑DE(); DiE(R,)/ DEc (Ri) 1; Wkp+ Wig=1 k=1, 2, The last equation in (14)follows from the fact that the xki,s must also sum to one his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
448 The Journal of Business Taking the derivative of this expression with respect to Xki, we have N Z Xkj CoV(Ri, Rj) - SkE(Ri) - Tk (10) j~l This set of equations, for i - 1, 2, , N, determines the values of Xki. If we write Dij for the inverse of the covariance matrix cov(Ri, Rj), then the solution to this set of equations may be written N N XW- Sk DE(Rj) + Tk Di. (11) j=:l j=:l Note that the subscript k, referring to the individual investor, appears on the right-hand side of this equation only in the multipliers Sk and Tk. Thus every investor holds a linear combination of two basic portfolios, and every efficient portfolio is a linear combination of these two basic portfolios. In equation (11), there is no guarantee that the weights on the individual assets in the two portfolios sum to one. If we normalize these weights, then equation (11) may be written Xki WkpXpi + WkqXqi. (12) In equation (12), the symbols are defined as follows: N N Wkp Sk DjE(Rj); i~_i j=:l N N Wkq Tk E D-j i==1 j=1 (13) N N N XPi DjjE(R)/ DijE(Rj); j==1 i==l j=1 N N N Xqi Dtjj 1 Dip. j==1 i==1 j=1 Thus we have N - pi1; (14) N Xqj =1; i= 1 Wkp + Wkq 1 k- 1 2, ..., L. The last equation in (14) follows from the fact that the Xki'S must also sum to one. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
449 Capital Market equilibrium Equation(12), then, shows that the efficient portfolio held by in- vestor k consists of a weighted combination of the basic portfolios p and q. Note, however, that the two basic portfolios are not unique. Supl that we transform the basic portfolios p and q into two different folios u and v, using weights wup, Wug, Wop, and wog. Then we have xui=wuprpi+ wuorgi; (15) Normally, we will be able to solve equations(15) for xps and xge. Let us write the resulting coefficients wpw, Wpu, Wou, and wov. Then we will rud+ (16 xai=wgului+ woori Substituting equations(16) into equation(12), we see that we can write the efficient portfolio k as a linear combination of the new basic port- folios u and v as follows Xut=wkulut+ Whorf. (17) In equation(17), the weights whu and wrw sum to one. Thus the basic portfolios u and v can be any pair of different port folios that can be formed as weighted combinations of the original pair of basic portfolios p and Every efficient portfolio can be expressed as a weighted combination of portfolios u and v, but they need not be efficient themselves Portfolios p and q must have different B's, if it is to be possible generate every efficient portfolio as a weighted combination of these two portfolios. But if they have different B's, then it will be possible to generate new basic portfolios u and v with arbitrary Bs, by choosing ap- propriate weights. In particular, let us choose weights such that n=1;B=0. Multiplying equation (12)by the fraction xmk of total wealth held by investor k, and summing over all investors (k= 1, 2, L),we obtain the weights xmi of each asset in the market portfolio tmt= xmkWip xp+ ImkWkar (19) Since the market portfolio is a weighted combination of portfolios p and q, and since Bm is one, portfolio u must be the market portfolio. Thus we can rename the portfolios u and v specified by(18)portfolios m and z, for the market portfolio and the zero-s basic portfolio. When we write the return on an efficient portfolio k as a weighted combination of the returns on portfolios m and z, the coefficient of the return on portfolio m must be Be. Thus we can write his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
449 Capital Market Equilibrium Equation (12), then, shows that the efficient portfolio held by investor k consists of a weighted combination of the basic portfolios p and q. Note, however, that the two basic portfolios are not unique. Suppose that we transform the basic portfolios p and q into two different portfolios u and v, using weights wu,, Wuq, wa, and Wvq. Then we have XUi- WUVXi + WuqXqi; (15) Xvi WV- X i + WvqXqi. Normally, we will be able to solve equations (15) for xiA and Xqj. Let us write the resulting coefficients wvu, wpv, Wqu, and Wqv. Then we will have x-i WVUX1i + wpVXvi; (16) Xqi WquXui + WqvXvi. Substituting equations ( 16) into equation ( 12), we see that we can write the efficient portfolio k as a linear combination of the new basic portfolios u and v as follows: Xki =WkuXui + WkvXvi. (17) In equation (17), the weights Wku and Wkv sum to one. Thus the basic portfolios u and v can be any pair of different portfolios that can be formed as weighted combinations of the original pair of basic portfolios p and q. Every efficient portfolio can be expressed as a weighted combination of portfolios u and v, but they need not be efficient themselves. Portfolios p and q must have different /3's, if it is to be possible to generate every efficient portfolio as a weighted combination of these two portfolios. But if they have different /3's, then it will be possible to generate new basic portfolios u and v with arbitrary /3's, by choosing appropriate weights. In particular, let us choose weights such that Flu= 1; ,l]V =0. (18) Multiplying equation (12) by the fraction Xmk of total wealth held by investor k, and summing over all investors (k - 1, 2, . . . , L), we obtain the weights xmi of each asset in the market portfolio: L L Xri (ZE XmkWkp) Xi + (Z XmnkWkq )Xqi. (19) k=1 k=1 Since the market portfolio is a weighted combination of portfolios p and q, and since Pm is one, portfolio u must be the market portfolio. Thus we can rename the portfolios u and v specified by (18) portfolios m and z, for the market portfolio and the zero-,8 basic portfolio. When we write the return on an efficient portfolio k as a weighted combination of the returns on portfolios m and z, the coefficient of the return on portfolio m must be /k. Thus we can write This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
450 The journal of business Rk= BuRm+(1-B)R2 Taking expected values of both sides of equation (20), and rewriting ha E(Rx)=E(R:)+BIE(Rm)-E(RD] (21) Equation(21) says that the expected return on an efficient portfolio k is a linear function of its Bk. From(1), we see that the corresponding elationship when there is a riskless asset and riskless borrowing and lending are allowed is )=R+BAE(Rm)一R (22) Thus the relation between the expected return on an efficient portfolio k and its Bk is the same whether or not there is a riskless asset. If there is, then the intercept of the relationship is Ry. If there is not, then the intercept is E(r) We can now show that equation(21)applies to individual securi ties as well as to efficient portfolios. Subtracting equation (10)from itself after permuting the indexes, we get cov(R,R)一cov(R,R)=SAE(R)一E(点)](23) Since the market is an efficient portfolio, we can put m for k, and since i and j can be taken to be portfolios as well as assets, we can put z for Then equation (23)becomes cov(R, Rm)=Sm[E(R)-E(R) (24) Equation (24) may be rewritten as E(R, =E(R)+[var(Rm)/S, 2 var(Rnm)/Sm=E(点n)一E(R) So equation(25) becomes E(RD=E(R,)+B,[E(Rm)-E(R) Thus the expected return on every asset, even when there is no riskless asset and riskless borrowing is not allowed, is a linear function of its B Comparing equation (27)with equation (1), we see that the introduc tion of a riskless asset simply replaces E(R,) with R, ow we can derive another property of portfolio z. Equation (27) holds for any asset and thus for any portfolio Setting B:=0, we see that every portfolio with B equal to zero must have the same expected eturn as portfolio z. Since the return on portfolio z is independent of the return on portfolio m, and since weighted combinations of portfolios m and z must be efficient, portfolio z must be the minimum-variance zero-B portfolio his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
450 The Journal of Business Rk PkRm + (1 -/k)Rz. (20) Taking expected values of both sides of equation (20), and rewriting slightly, we have E(Rk) =E(Rz) +I/8k[E(Rm) -E(Rz)]. (21) Equation (21) says that the expected return on an efficient portfolio k is a linear function of its /3k. From (1), we see that the corresponding relationship when there is a riskless asset and riskless borrowing and lending are allowed is E(Rk) Rf + 1k3[E(Rm) - Rf]. (22) Thus the relation between the expected return on an efficient portfolio k and its /3k is the same whether or not there is a riskless asset. If there is, then the intercept of the relationship is Rf. If there is not, then the intercept is E(R~z). We can now show that equation (21) applies to individual securities as well as to efficient portfolios. Subtracting equation (10) from itself after permuting the indexes, we get cov(Ri, Rk) - cov(Rj, R) = Sk[E(Ri) - E(Rj)]. (23) Since the market is an efficient portfolio, we can put m for k, and since i and j can be taken to be portfolios as well as assets, we can put z for j. Then equation (23) becomes cov(Ai, Am) Sm[E(Ri) - E(Rz)]. (24) Equation (24) may be rewritten as E(Ri) = E(Rz) + [var(Rm)/Sm]/3i. (25) Putting m for i in equation (25), we find var(Rm)/SIn = E(Rm) - E(Rz). (26) So equation (25) becomes E(ftj =E(Rz) + /3i[E(Rm) - E(Rz)]. (27) Thus the expected return on every asset, even when there is no riskless asset and riskless borrowing is not allowed, is a linear function of its /3. Comparing equation (27) with equation (1), we see that the introduction of a riskless asset simply replaces E(RZ) with Rf. Now we can derive another property of portfolio z. Equation (27) holds for any? asset and thus for any portfolio. Setting /i3 0, we see that every portfolio with /8 equal to zero must have the same expected return as portfolio z. Since the return on portfolio z is independent of the return on portfolio m, and since weighted combinations of portfolios m and z must be efficient, portfolio z must be the minimum-variance zero-/3 portfolio. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
451 Capital Market Equilibrium Fama comes close to deriving equation (27). His equation (27) says that the expected return on an asset is a linear function of its risk, measured relative to an efficient portfolio containing the asset. Lintner also derives a linear relationship (eg. [18])between the expected re- turn on an asset and its risk. It is possible to derive equation (27)from either Fama's or Lintner's equations in a relatively small number of Fama, however, goes on to introduce the concept of a new kind of financial intermediary that he calls a"portfolio sharing company In the absence of riskless borrowing or lending opportunities, he says that this fund can purchase units of the market portfolio, and sell shares in its return to different investors. He says that an investor can specify the proportion of the return on this fund that he will receive per unit of his own funds invested. Writing Bx for this proportion, Fama claims that RE= BnR (28) But this is not consistent with market equilibrium. Assuming that E(R) is positive, shares in this fund will be less attractive than direct hold of efficient portfolios with Bk less than one, as given by equation If E(R)is negative, shares in this fund will be less attractive than direct holdings of efficient portfolios with Pk greater than one. So there is no way that the fund can sell all of its shares, except, of course, that it can determine a number R, such that when the return on the holdings of investor k is defined by equation (29), all of the fund's shares can be sold Rk=R,+ Bk(rm-R, (29) Bor unitis. So the concept of portfolio sharing does not cast any light on market equilibrium in the absence of riskless borrowing and lending Starting with equation (23), we can now show one final property of portfolio z. Let p and g be two efficient portfolios and let Wap and Wag be the weights that give portfolio z when applied to portfolios P and q. Putting m for j and p for k to give one equation, and putting m for j for k to give another, we hay cov(R,R2)一coV(Rm,)=SnE(R)一E(Rn)]; (30) Cov(R, R)-cov(Rm, R)=S[E(RS-E(Rm)] Multiplying the equations by Wap and weg, respectively, and adding them ng that cov(Rm, R,)is zero--we have RR)=(wmS2+w2S2)[E(R)一E(点n) Substituting for E(Ri from equation(27), we obtain cov(R,R)=(1-B)(wS2+wS2[E(R)一E(Rm)](32) his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
451 Capital Market Equilibrium Fama comes close to deriving equation (27). His equation (27) says that the expected return on an asset is a linear function of its risk, measured relative to an efficient portfolio containing the asset. Lintner also derives a linear relationship (eq. [18]) between the expected return on an asset and its risk. It is possible to derive equation (27) from either Fama's or Lintner's equations in a relatively small number of steps. Fama, however, goes on to introduce the concept of a new kind of financial intermediary that he calls a "portfolio sharing company." In the absence of riskless borrowing or lending opportunities, he says that this fund can purchase units of the market portfolio, and sell shares in its return to different investors. He says that an investor can specify the proportion of the return on this fund that he will receive per unit of his own funds invested. Writing /k for this proportion, Fama claims that Rkf 8kRm. (28) But this is not consistent with market equilibrium. Assuming that E(R,) is positive, shares in this fund will be less attractive than direct holdings of efficient portfolios with /3k less than one, as given by equation (20). If E(R,) is negative, shares in this fund will be less attractive than direct holdings of efficient portfolios with /3k greater than one. So there is no way that the fund can sell all of its shares, except, of course, that it can determine a number Rf such that when the return on the holdings of investor k is defined by equation (29), all of the fund's shares can be sold: Rk Rf + 3k(Rm - Rf). (29) But this is just an implicit way of creating borrowing and lending opportunities. So the concept of portfolio sharing does not cast any light on market equilibrium in the absence of riskless borrowing and lending opportunities. Starting with equation (23), we can now show one final property of portfolio z. Let p and q be two efficient portfolios and let wz, and Wzq be the weights that give portfolio z when applied to portfolios p and q. Putting m for j and p for k to give one equation, and putting m for I and q for k to give another, we have covCR, A1) - cov-Rm, Ap) Sp[E(Ri) E(Rm)]; (30) cov(&, Aq) cOv(Am, Aq) = Sq[E(Ri) - E(Rm)]. Multiplying the equations by wz, and Wzq, respectively, and adding them -noting that cov(Rm, Rz) is zero-we have cov(Ri, Rz) (wzpSV + WzqSq)[E(Ri) - E(Rm)] (31) Substituting for E (Ri) from equation (27), we obtain cov(Ri, Rz) (1 -ji) (WzpSp + WzqSq)[E(Rz) E(Rm)]. (32) This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
452 The Journal of Business Thus we see that the covariance of the return on any asset i with the return on portfolio z is proportional to 1-Pi. In sum, we have shown that when there is no riskless asset, and no riskless borrowing or lending, every efficient portfolio may be written a weighted combination of the market portfolio m and the minimum variance zero-B portfolio z. The covariance of the return on any asset i with the return on portfolio z is proportional to 1-B The expected return on any asset or portfolio i depends only on pi, and is a linear function of B, Prohibition of borrowing and lending, then, shifts the intercept of the line relating E(R)and B, from R, to E(R). Since this is the effect that complete prohibition would have, it seems likely that partial restric- tions on borrowing and lending, such as margin requirements, would also shift the intercept of the line, but less so. Thus it is possible that restric- tions on borrowing and lending would lead to a market equilibrium con- istent with the empirical model expressed in equation(3)and developed by Black, Jensen, and Scholes QUILIBRIUM WITH NO RISK BORROWING Let us turn now to the case in which there is a riskless asset available such as a short-term government security, but in which investors are not allowed to take short positions in the riskless asset. We will continue to assume that investors may take short positions in risky assets Vasicek has shown that in this case the principal features of the equilibrium with no riskless borrowing or lending are preserved. 12 The expected return on any asset i continues to be a function only of its B The function is still linear. The efficient set of portfolios now has two parts, however. One part consists of weighted combinations of port- folios m and z, and the other part consists of weighted combinations of the riskless asset with a single portfolio of risky assets that we can call portfolio t Je can show this in our notation as follows Since the restriction kinds of efficient portfolios, those that contain the riskless asset and those that do not Let us call the riskless asset number n+ 1 For those efficient portfolios that do not contain the riskless asset equations (6)-(18)of the previous section apply cient portfolio can be expressed as a weighted combination of portfolios u and v, whereβ u is one andβ g Is zero. For those efficient portfolios that do contain the riskless asset, we can extend equation (10) to N+ l assets. The covariance term for N+I vanishes, so we have Borrowing, "memorandum (March 1971); available from the Wells Fargo Bank his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 46: 59 AM All use subject to JSTOR Terms and Conditions
452 The Journal of Business Thus we see that the covariance of the return on any asset i with the return on portfolio z is proportional to 1 - /i. In sum, we have shown that when there is no riskless asset, and no riskless borrowing or lending, every efficient portfolio may be written as a weighted combination of the market portfolio m and the minimumvariance zero-,8 portfolio z. The covariance of the return on any asset i with the return on portfolio z is proportional to 1 - Pi. The expected return on any asset or portfolio i depends only on /Pi, and is a linear function of Pis. Prohibition of borrowing and lending, then, shifts the intercept of the line relating E(RD) and pi from Rf to E(RA). Since this is the effect that complete prohibition would have, it seems likely that partial restrictions on borrowing and lending, such as margin requirements, would also shift the intercept of the line, but less so. Thus it is possible that restrictions on borrowing and lending would lead to a market equilibrium consistent with the empirical model expressed in equation (3) and developed by Black, Jensen, and Scholes. EQUILIBRIUM WITH NO RISKLESS BORROWING Let us turn now to the case in which there is a riskless asset available, such as a short-term government security, but in which investors are not allowed to take short positions in the riskless asset. We will continue to assume that investors may take short positions in risky assets. Vasicek has shown that in this case the principal features of the equilibrium with no riskless borrowing or lending are preserved.'2 The expected return on any asset i continues to be a function only of its /3. The function is still linear. The efficient set of portfolios now has two parts, however. One part consists of weighted combinations of portfolios m and z, and the other part consists of weighted combinations of the riskless asset with a single portfolio of risky assets that we can call portfolio t. We can show this, in our notation, as follows. Since the restriction on borrowing applies only to the riskless asset, there will be only two kinds of efficient portfolios, those that contain the riskless asset and those that do not. Let us call the riskless asset number N + 1. For those efficient portfolios that do not contain the riskless asset, equations (6)-(18) of the previous section apply. Each such efficient portfolio can be expressed as a weighted combination of portfolios u and v, where Pl is one and f, is zero. For those efficient portfolios that do contain the riskless asset, we can extend equation (10) to N + 1 assets. The covariance term for j = N + 1 vanishes, so we have 12. Oldrich A. Vasicek, "Capital Market Equilibrium with No Riskless Borrowing," memorandum (March 1971); available from the Wells Fargo Bank. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:46:59 AM All use subject to JSTOR Terms and Conditions
