《投资学原理 Investments》课程教学资源（阅读资料）capital market equilibrium with restricted borrowing.pdf

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

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