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 Conditions
OPTIMAL MULTIPERIOD PORTFOLIO POLICIES 219 To see that these functions indeed satisfy 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 represented 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 quadratic (7) with constant A, are ruled out. We note for later reference that when U = In Y, the maximum condition becomes 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 determined 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 random. 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 nonrandom variable r to represent the interest 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
THE JOURNAL OF BUSINESS It is clear that in general nothing can be said about the sign of -random, the depender the slope of the absolt function is exactly as before, however. In the case where utility is quadratic in wealth(and assuming Xi and X to be independently distributed), the optimal a will be given b E1-E2-2aA[(1+E)(E1-E2)一V2 and correspondingly max ElU(r)1=1(1+B)+2(1+B) (19 4-4是2小++画 With the Tobin formulation, however, the optimal h would be expressed as E1-E2-2B[(1+E2)(E1-E2)-V2] (20 2B[V1+V2+(E1-E2 and, again, if decisions are to be consist- IIL, MULTIPERIOD MODELS ent at different levels of initial wealth, B A. GENERAL METHOD OF SOLUTIOI must be proportional to A The derivation of the utility functions, By a multiperiod model is meant a (13a)and (13b), is clearly independent of theory of the following structure: The the specific setting of the decision prob- investor has determined a certain future lem. The sufficiency part of the proof is point in time(his horizon)at which he also completely analogous. plans to With the utility function U=In y, k then available. He will still make his in- rould now be determined by the condi- vestment decisions with the objective of maximizing expected utility of wealth at X1-X 1+X2+k(X1-X2) that the time between the present and giving his horizon can be subdivided into n peri- ods(not necessarily of the same length E(In[1+ X2+ k(Xi-Xel.(21) at the end of each of which return on the max E(In Y)=In A izes and he can make a new decision Similarly, with U =Yl-y,k is deter- the composition of the portfolio to be mined by held during the next period E(1+X2+k(Xr-X]-(X -X2))=0, This formulation of the problem de- media max E(r-m portfolio decisions are clearly interre (22) lated, and no defense for leaving con- AE{1+X2+k(X1-X2)-? sumption decisions out of the picture can his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions
220 THE JOURNAL OF BUSINESS It is clear that in general nothing can be said about the sign of this derivative; for X2 non-random, the dependence on the slope of the absolute risk-aversion function is exactly as before, however. In the case where utility is quadratic in wealth (and assuming X1 and X2 to be independently distributed), the optimal a will be given by a E1-E2-2aA [ (1 +E2) (E1-E2) -V21 18) 2ca[VI+V2+(El-E2)2( and correspondingly maxE[ U ( Y) V1( 1 +E2) +V2 (1 +E1) VI1+V2 + (El -E2 )2 (19) XF V1 (l1+E2)2?+V2 (l?E1)2 +V1V2A21+ (E1-E2)2 [ V1V( 1 +E2) +V2 (1 +E1) 4a[Vi+V2 + (El-Ei)2] With the Tobin formulation, however, the optimal k would be expressed as E1-E2-2f [ (1 +E2)(E1-E2) -V2( 2fl[V1+V2+(E1-E2)2( and, again, if decisions are to be consistent at different levels of initial wealth, A must be proportional to A. The derivation of the utility functions, (13a) and (13b), is clearly independent of the specific setting of the decision problem. The sufficiency part of the proof is also completely analogous. With the utility function U = In Y, k would now be determined by the condition + 1 -X2 ) giving max E(ln Y) =nA (21) + E{ln [1 + X2+ k(Xi-X2)]} ( Similarly, with U = Y1Fo, k is determined by E{[1 + X2+ k(X -X2)]-'(X1 -X2)}= 0, giving max E(Y'Y) (22) = Al-'E{[l + X2 + k(X1 - X2)'y1I . III. MULTIPERIOD MODELS A. GENERAL METHOD OF SOLUTION By a multiperiod model is meant a theory of the following structure: The investor has determined a certain future point in time (his horizon) at which he plans to consume whatever wealth he has then available. He will still make his investment decisions with the objective of maximizing expected utility of wealth at that time. However, it is now assumed that the time between the present and his horizon can be subdivided into n periods (not necessarily of the same length), at the end of each of which return on the portfolio held during the period materializes and he can make a new decision on the composition of the portfolio to be held during the next period. This formulation of the problem deliberately ignores possibilities for intermediate consumption. Consumption and portfolio decisions are clearly interrelated, and no defense for leaving consumption decisions out of the picture can 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