OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 219 To see that these functions indeed sat- Maximum expected utility is then isfy our requirem when relative risk aversion is constant nax E(n Y)=In A hat is, whe 十E[n(1+1.(14) U(Y)Y Similarly, with U=Y-y, k will be de U'(Y) termined by E(1+kX)X]=0 (YX=-YU(nX and so correspondingly max e(Y-m)=A-vEll+ kx)l-.(15) EU"(Y)XⅥ=-YEU(Y)X] B. MORE GENERAL CASES At an interior maximum point we have almost all the analysis above is easily EU()X]=0 generalized to the case where the yield on he certain asset is non-zero or to the case EU"()X1]=0, where the yields on both dom. Since the analyses are in both cases completely parallel, we shall only give AElU"()X]+aE[U"(n)X2]=0; the results for the more general of two (both yields random). Results for the former case are then obtained simply by replacing the random yield X2 by a non random variable r to represent the inter But from( 3)the left-hand side is da/dA; est on the certain asset hence da/dA a/A, implying a= kA Generalization to an arbitrary number The conclusion is therefore that th of assets would be trivial and add little here m ay exist preferences which can be repre- theoretical interest sented by a utility function in rate of If the random rates of return on the return only, but then it must be of the two assets are X1 and X2, and a is the form In R or Rl-y(n Y and Yl-r are amount in vested in the first asset, ther final wealth and R-1). Other forms, like the quad- Y=(1+X2)A+a(x1-X2) ratic(7)with constant B, are ruled out. By so to say substituting(1+ X,)A for U=In y, the maximum condition b A and X1-X2 for X throughout, most of the conclusions from the discussion of comes the simplest case are readily obtained X Thus, in the general case, an interior maximum point would be one where so that k is determined by the condition EU(Y)(X1-X2)=0,(16) X and the corresponding expression for 十kX da/ dA would be da_B[U"(Y)(x1=x2)(1+x2)] d a EIU"(Y)(X1-X2)2] his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and ConditionsOPTIMAL MULTIPERIOD PORTFOLIO POLICIES 219 To see that these functions indeed sat￾isfy our requirement, we observe that when relative risk aversion is constant, that is, when U"f( Y) Y U'( Y) then U"(Y)YX = -yU'(Y)X, and so E[U"(Y)XY] = --yE[U'(Y)X]. At an interior maximum point we have E[U'(Y)X] = 0, and so E[U"(Y)XY] = 0, or AE[U"(Y)X] + aE[U"(Y)X21=0; thus E[U"(Y)XJ a E[ U"( Y)X2] A' But from (3) the left-hand side is da/dA; hence da/dA = a/A, implying a = kA. The conclusion is therefore that there may exist preferences which can be repre￾sented by a utility function in rate of return only, but then it must be of the form In R or R1- (In Y and Y1Fo are equivalent, as utility functions, to In R and R1zz). Other forms, like the quad￾ratic (7) with constant A, are ruled out. We note for later reference that when U = In Y, the maximum condition be￾comes so that k is determined by the condition E \ Maximum expected utility is then max E (In Y) = InA (14) + E [ln (1 + kX) (4 Similarly, with U = YF1', k will be de￾termined by E[(1 + kX)-YX] = 0, and so correspondingly maxE(Y'-Y) = A-'E[1 + kX)'-z] . (15) B. MORE GENERAL CASES Almost all the analysis above is easily generalized to the case where the yield on the certain asset is non-zero or to the case where the yields on both assets are ran￾dom. Since the analyses are in both cases completely parallel, we shall only give the results for the more general of the two (both yields random). Results for the former case are then obtained simply by replacing the random yield X2 by a non￾random variable r to represent the inter￾est on the certain asset. Generalization to an arbitrary number of assets would be trivial and add little of theoretical interest. If the random rates of return on the two assets are X1 and X2, and a is the amount invested in the first asset, then final wealth is Y= (1 + X2)A + a(Xi - X2) . By so to say substituting (1 + X2)A for A and X1 - X2 for X throughout, most of the conclusions from the discussion of the simplest case are readily obtained. Thus, in the general case, an interior maximum point would be one where E[U'(Y)(X1- X2)] = 0, (16) and the corresponding expression for da/dA would be da E[ U"(Y)(X1-X2)(1 +X2)] d A Et " ( ( X1-X2 ) 2 ](17) 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