216 THE JOURNAL OF BUSINESS II. SINGLE-PERIOD MODELS points a =0 or a= A; the condition for By a single-period model is meant a the former is that dalu(r)l da is nega theory of the following structure The tive at a=0, which is seen to imply and investor makes his portfolio decision at require E(X).s 0. Thus, the investor the beginning of a period and then waits will hold positive amounts of the risky until the end of the period when the rate asset if and only if its expected rate of of return on his portfolio materializes. return is positive He cannot make any intermediate If the maximum occurs at an interior changes in the composition of his port- value of a, we have at this point folio. The investor makes his decision EU(Y)Ⅺ=0 with the objective of maximizing ex pected utility of wealth at the end of the To see how such an optimal value-of a period (final wealth) depends upon the level of initial wealth we differentiate(2)with respect to A and A. THE SIMPLEST CASE obtain In the simplest possible case there are only two assets, one of which yields a da--eluinxai (3) random rate of return(an interest rate) It is possible to prove that the sign of of X per dollar invested, while the other this derivative is positive, zero, or nega- asset( call it"cash")gives a certain rate tive, according as absolute risk aversion analyzed in some detail in Arrow is decreasing, constant, or increasing. I. one which he invests an amount a in the might consider, preferences over prob risky asset, his final wealth is the random terms of means and variances only. If variable to ar- Y=A+aX (1) bitrary probability distributions,the With a preference ordering U(Y) over utility function must clearly be of the levels of final wealth, the optimal valt f a is the one which maximizes ElU(nI U(n=Y-arz subject to the condition0≤a≤A Then the optimal a is the one which General analysis. The first two de- maximizes rivatives of EU(F]are EIU()]= EA+aX-a(A+ aX)? dElu(r)l-Elu(Y)XI (A-aA)2+(1-2aA)Ea and a(v+e2)a2 dElU(r)I=eu()x. where e and v denote expectation and variance of X, respectively. An interior Assuming general risk aversion(U aximum is then given by 0), the second derivative is negative, so (1-2aA)E that a unique maximum point is guaran (5) teed. This might occur at one of the end Thus, the optimal a depends on the level Arrow, op cit of initial wealth. The same is also true of his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions216 THE JOURNAL OF BUSINESS II. SINGLE-PERIOD MODELS By a single-period model is meant a theory of the following structure: The investor makes his portfolio decision at the beginning of a period and then waits until the end of the period when the rate of return on his portfolio materializes. He cannot make any intermediate changes in the composition of his port￾folio. The investor makes his decision with the objective of maximizing ex￾pected utility of wealth at the end of the period (final wealth). A. THE SIMPLEST CASE In the simplest possible case there are only two assets, one of which yields a random rate of return (an interest rate) of X per dollar invested, while the other asset (call it "cash") gives a certain rate of return of zero. This model has been analyzed in some detail in Arrow.5 If the investor's initial wealth is A, of which he invests an amount a in the risky asset, his final wealth is the random variable Y = A + aX. (1) With a preference ordering U(Y) over levels of final wealth, the optimal value of a is the one which maximizes E[U(Y)], subject to the condition 0 < a < A. General analysis.-The first two de￾rivatives of E[U(Y)] are dE[ U ( -) =E[ U'( Y)X] da and d2E[U(Y)] =E[U"(fY)X . da2 Assuming general risk aversion (U" < 0), the second derivative is negative, so that a unique maximum point is guaran￾teed. This might occur at one of the end points a 0 or a = A; the condition for the former is that dE[U(Y)]/da is nega￾tive at a = 0, which is seen to imply and require E(X) < 0. Thus, the investor will hold positive amounts of the risky asset if and only if its expected rate of return is, positive. If the maximum occurs at an interior value of a, we have at this point E[U'(Y)X] = 0. (2) To see how such an optimal value-of a depends upon the level of initial wealth, we differentiate (2) with respect to A and obtain da E [U"( Y)X] dA E[ U"( Y)X2] (3) It is possible to prove that the sign of this derivative is positive, zero, or nega￾tive, according as absolute risk aversion is decreasing, constant, or increasing. Quadratic utility.-In particular, one might consider preferences over prob￾ability distributions of Y being defined in terms of means and variances only. If such a preference ordering applies to ar￾bitrary probability distributions, the utility function must clearly be of the form U(Y)= Y-aY2. (4) Then the optimal a is the one which maximizes E[U(Y)] = E[A + aX - a(A + aX)2] = (A - aA)2 + (1 - 2aA)Ea - a(V +EI)a2 , where E and V denote expectation and variance of X, respectively. An interior maximum is then given by (1 -2aA)E a 2a(V+E) (5) Thus, the optimal a depends on the level I Arrow, op. cit. of initial wealth. The same is also true of This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions