Division of Research Graduate School of Business Administration The University of Michigan CAPITAL MARKET EQUILIBRIUM WITH DIVERGENT BORROWING AND LENDING RATES: COMMENTS AND CORRECTIONS Working Paper No. 118 by Jesus H. Chua August 1975 The University of Michigan FOR DISCUSSION PURPOSES ONLY None of this material is to be quoted or reproduced without the express permission of the Division of Research.

Introduction Brennan (1971) showed that under divergent borrowing and lending rates capital market equilibrium will be achieved in such a way that rates of return on risky assets and the portfolio composition of individual investors may be described by linear equations. This paper shows that his proof is invalid because of an unacceptable implicit assumption. It further shows that his explicit assumptions and procedure should have led to conclusions other than those he presented. Brennan's Proof In the article, Brennan made certain assumptions about the expected utility function, EUi[E(R),o2(R)], of individual i, i=l,...,M: aEUi aB(Ri > 0 (1) aE(R), aEU Ei < 0. (2) aa (R) This constitutes the definition of a risk averter. From this, Brennan proceeded to derive the portfolios that individual investors would hold under different borrowing and lending rate conditions. To simplify the symbols carried throughout, he defined a parameter: aEU. 2 2 aa2 (R) ai = - E ~(3) 3EUi aE(R) He then arrived at equation (9) in his article which was Xi ( (lr-[uiZ+(l-ui)b] I) 4) i~~~

-2 - where X. is the Mx 1 vector of amounts invested in each of the risky securities by investor i; 2 is the Mx M variance-covariance matrix of rates of returns on risky securities; I is the Mx 1 unit vector; u. is a parameter which is 1, if the investor is a net lender, = 0, if the investor is a net borrower, between 0 and 1, if neither; Z is the lending rate; b is the borrowing rate (b>Z); r is the Mx 1 vector of expected returns on risky assets. He interpreted the above equation to mean that every individual investor will hold portfolios consisting of two basic portfolios. The first is the portfolio of risky assets with the minimum variance, the second, the optimal portfolio of risky assets when borrowing and lending rates are both equal to zero. This result was then aggregated to vield the equation determining the rates of return on each risky asset. M I M M 1M r = (i -) X + (I i-) u +(1-ui)b]I (5) 1 ai 1 i 1 ai 1 a On the basis of the preceding equation Brennan concluded that rates of return on risky assets are linear functions of their covariances with the market portfolio.

-3 -It should be pointed out that Brennan's interpretations of equations (4) and (5) are correct only with the implicit assumption that a. is constant. Otherwise, equation (4) could not be the solution for Xi, since the right side of the equation would also contain the unknown X.'s. Likewise, equation (5) would have r terms on the right side of the equation. 2 Constant a. may be represented by utility functions of the 3 following types:3 EU - a + bE(R) + ca (R), (6) K2 2 K2E(R) + - a (R). or:U = Kle a (7) These indifference curves are linear in mean-variance space and parabolic in mean-standard deviation space. They describe investors who demand a constant marginal price for risk when risk is measured by the variance of return. On the other hand, they describe investors who demand increasing v. arginal price for risk when risk is measured by the standard deviatoe s. It is therefore difficult to accept the assumption of constant a.. More crucial is the fact that the utility function implied by a constant a. does not allow preference ordering of the probability distributions of returns consistent in the von Neumann-Morgenstern sense. In the mean-variance space, a preference ordering consistent in the von Neumann.-Morgenstern sense requires a utility function U(R) such that the expected utility,

-4 - EU(E(R),a (R)] = fU(R)dF[R;E(R),2 (R)], (8) -00 can be used to rank probability distributions of return. U(R) must be the same for all the probability distributions being ranked, If EU = a+bE(R) + cE[R-E(R)]2, then U[R,E(R)] = a + bR + c[R-E(R)]2 Ranking with this utility function requires that E(R) of each probability distribution being ranked be used in equation (8) to yield the different variances. Each probability distribution will, in effect, be measured by a different utility function. Consistent preference ordering is therefore nonexistent. bE(R) + cE[R-E(R)], thenIf EU = e bE(R) + cE[R-E(R)]2 then U(R) = e ~.d [bE(R)+co2(R)] dFhis, again, contains E(R) and will vary with the probability dsruThis, again, contains E(R) and will vary with the probability distribu4 tion being ranked. Again, there will not be a consistent ranking of probability distribution of returns when the utility function is used. Brennan has therefore started with a very general utility function but concluded by proving his case for an unacceptable set of conditions. This invalidates the proof provided by Brennan, but it does not invalidate the two main conclusions of his paper, namely: (1) individual investor will hold linear combinations of the two basic portfolios; (2) the rate of return on risky assets is a linear function of its covariance with a market portfolio. Black (1972) arrived at results similar to Brennan's when studying variations on the equal lending and borrowing rate assumption. However, he started with the formulation

-5 -minimize X'iX subject to X'r = E(R) X!I = 1. 5 This formulation bypasses the utility function assumption. Results from the above formulation are generally interpreted as applicable to normally distributed returns or quadratic utility functions. The following section will show that if the problem is solved starting with the maximization of expected utility as Brennan did, but assuming the quadratic utility function explicitly, the same two conclusions will not be arrived at. The conclusion that investors will hold linear combinations of two basic portfolios still holds. However, the conclusion that the rates of return on risky assets are linear functions of their respective covariances with the mnarket portfolio will not hold. Brennan's Procedure with uadratic Utility Functi As shown by Markcwitz, if it is assumed that investors (1) rank probability distributions of returns according to the expected utility maxim, (2) use only expected return and variance as parameters, and (3) make no assumption about the probability distributions, this is equivalent to assuming that investors have quadratic utility functions. The following development will therefore assume that investors have quadratic utility functions: Ui(R) = Mi + n.R + PIR2 i =,..., M (9) n. >0 P. < 0 I

-6 - A. Net lender For net lender i, the expected return on his. portfolio is Ei(R) = Xir + (1-Xil)Q, (10) and the variance on the portfolio return is a (R) XtX. (11) His portfolio selection problem is Maximize EUi = M + n.Ei(R) + PiE (R ) (12) The first order conditions are therefore (nli+2PiE (R))(r- I) + 2PiQX = 0 (13) Simplifying and solving for Xi n. [ -+X'r+(l-XlI)Z](r-ZI) + i 0, (14) n. [K+(r-.rI (r-ZI)' lXi = -(- + Z)(r-.I), (15) n.. X ( &- + g) [6+(r-QI) (r-ZI)' ]-l(r.t). (16) 2P The net lender therefore holds a portfolio which is a linear combination of two basic portfolios, [s+(r-ZIl) (r-I)' ] r and [V (r-.I) (r-ZI) ] I,8 B. Net borrower For the th borrower, the expected return on the portfolio is E(R) = X!r + (1-XjI)b, (17) and the variance of the portfolio return is (R) X a (R) = X!QX.. (18) 3 i

-7 -His portfolio selection problem is - 2 Maximize EUj = Mj + njEj(R) + P E (R ). (19) The first order conditions are therefore {nj+2P E(R)}(r-bI) + 2P Xj = 0 (20) which yield X = -(2P+ b) [Q+(r-bI)(r-bI)'] (r-bI). (21) J 2P This shows that the net borrowers also hold portfolios which are linear combinations of two basic portfolios. C. Non-lender, non-borrower Black (1972) has shown that for non-lender, non-borrowers, the portfolios held are linear combinations of two nonunique basic portfolios. Further, he showed that the basic portfolios may be transformed into any two other portfolios. It is therefore possible to express the portfolios of the non-lender, non-borrowers as linear combinations of the basic lending portfolio and the basic borrowing portfolio. From this it is still correct to conclude that even with divergent lending and borrowing rates, investors still hold portfolios that are linear combinations of two basic portfolios. However, if one examines the equation for the basic lending portfolios, eq. (15), and the equation for the basic borrowing portfolios, eq. (21), one sees that both have quadratic terms in r. Therefore, r cannot, in general, be a linear function of its covariance with the market portfolio.

-8 -Implications The linearity of the dependence of rates of return on their variances and covariances with the market portfolio has been empirically 9 shown. The results above, therefore, are not a proposal for a nonlinear model. Instead, they should be taken as an addition to the list of objections against the use of the quadratic utility function assumption.

-9 - Notes A second order condition is needed to differentiate the diversifier from the plunger. Constant ai is not the same as Lintner's constant Pratt-Arrow risk aversion measure (Lintner, 1969). The second equation is derived by solving aEU 2 2 aa R.EU = ai constant as a variable-separable, aE(R) partial differential equation with constants as boundary conditions. 4Another implication of the above, which is not related to the article, is that consistent ranking by semi-variance cannot be the same as consistent ranking by one-half the variance, unless all the probability distributions being ranked have the same mean. Ranking by semi-variance and expected return may be represented by the following utility function: 2 U(R) = a + bR + c[min(R-h,0)] A consistent ranking by semi-variance requires that h be independent of the probability distributions being ranked. Since ranking by one-half the variance using the above utility function requires that h be equal to each of the means of the probability distributions being ranked, it allows consistent ranking only if all the means are equal. Note that Black's Xi is not the same as Brennan's X.. Brennan's X. is dollar amounts invested while Black's is the proportion of the total amount invested. Therefore, Brennan's X. is equal to Black's X, multiplied by the total amount invested. The next text section will use Black's X. for simpler algebra. H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, pp. 286-88. The three conditions were explicitly or implicitly assumed in Brennan's article.

-10 -Assume that [Q+(r-4I) (r-QI)t'] and [Q+(r-bI)(r-bI)'] are positive definite. When lending rate is zero, the first basic portfolio is the single portfolio.held by the net lending investor who maximizes the quadratic utility function. 9 Black, ns and Scholes, 1972. cf. Black, Jenson, and Scholes, 1972. ~

-11 - References Black, F, "Capital Market with Restricted Borrowing." Journal of Business, July 1972: 444-55, Black, F., M. Jensen, and M. Schol.e. "The. Capital Asset Pricing Model: Some Empirical Tests." Studies in the Theory of Caital Markets. Ed. by M. Jensen. New York: Praeger Publishing Co., 1972. Borch, K. "A Note on Uncertainty and Indifference Curves." Review of Economic Sttldies, January 1969: 1-4. Brennan, M. "Capital Market Equilibrium with Divergent Borrowing and Lending Rates." Journal of Finvancial and Quantitaivve Analyis, December 1971: 1197-1205. Lintner, J. "The Aggregation of Investor's Diverse Judgments and Preferences in Purely Competitive Security Markets." J ournal of Financial and Quantitative Analysis, December 1969: 347-400. Markowvit z, H. Portfolio Selection: Efficient Diversification of Investmicnts. New York: John Wiley and Sons, Inc., 1959. yw