218 THE JOURNAL OF BUSINESS which is the same solution as(5). For ex- sent the same preference ordering, there ample, if the utility function for final exist constants b and c such that v wealth is Y-(1/400)Y2, it may be per- bU + c. Therefore, if a utility function fectly acceptable to maximize the expec- U determines an ordering of probability tation of R-iR, but only if initial distributions for rate of return and this s It should be kept in mind that when probability distributions for final wealth, ordering is identical with the ordering of we are here speaking of different levels of then U(R)and U(Y)=U(AR)must wealth, this is to be interpreted strictly represent the same ordering. This must in terms of comparative statics; we are mean that U (R) and U(AR only assering that if the investor had transformations of each othe are linear had an initial wealth different from A then his optimal k would have been dif- U(AR)=bU(R)+c ferent from( 1- 2aA)E/2aA(V+E?). Here b and c are independent of R,but When we consider different levels of they may depend upon A health at different points in time (in a se- Differentiation of (9 )with respect to R quence of portfolio decisions), other fac- gives tors may also affect the decisions, as we U(AR)A= bU(R).( shall see later. And it will also become clear that attempting to use a utility Then differentiating (10)with respect to function of the form of equation(7)in A, we have such a setting may easily cause difficul- U"(AR)AR+ U(AR)=bU(R),(11) Uility functions implying constant as- where b denotes derivative with respect set proportions.--If attention is not re- to A. From(10) the right-hand side is stricted to quadratic utility functions, (6A/bU(AR), so that(11)can be writ- however, it may be possible to get invest- ten ment in the risky asset strictly propor- U/(F)Y+U(Y b′A U'(Y) tional to initial wealth Requiring that a/A= k is seen to be or U(Y)Y the same as requiring that choices among U(F)1~b′A (12) portfolios be based upon consideration of the probability distribution for the t, of variations in Y and A, both sides are Since this must hold for independent folio's rate of return independently initial wealth: the choice of a probability constant. This means that relative risk distribution for R=1+kX consists in aversion must be constant, equal to, say, a choice of a value of k, this choice being y. It is easily verified that the only solu- made independently of A. Therefore, the tions to this condition are linear trans- problem of finding the class of utility formations of the function functions with the property that a/A U(Y)= In Y if y=1(13a) k is equivalent to the problem of deter- and mining the class of utility functions with rty that choices among distri- butions for rate of return on the portfolio Thus, utility functions belonging to this are independent of initial wealt class are the only ones permitted if con If two utility functions U and V repre- stant asset proportions are to be optimal his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and Conditions218 THE JOURNAL OF BUSINESS which is the same solution as (5). For ex￾ample, if the utility function for final wealth is Y - (1/400) Y2, it may be per￾fectly acceptable to maximize the expec￾tation of R - 'R2, but only if initial wealth happens to be 100. It should be kept in mind that when we are here speaking of different levels of wealth, this is to be interpreted strictly in terms of comparative statics; we are only asserting that if the investor had had an initial wealth different from A, then his optimal k would have been dif￾ferent from (1 -2 aA)E/2 aA (V + E2). When we consider different levels of wealth at different points in time (in a se￾quence of portfolio decisions), other fac￾tors may also affect the decisions, as we shall see later. And it will also become clear that attempting to use a utility function of the form of equation (7) in such a setting may easily cause difficul￾ties. Utility functions implying constant as￾set proportions.-If attention is not re￾stricted to quadratic utility functions, however, it may be possible to get invest￾ment in the risky asset strictly propor￾tional to initial wealth. Requiring that a/A = k is seen to be the same as requiring that choices among portfolios be based upon consideration of the probability distribution for the port￾folio's rate of return independently of initial wealth: the choice of a probability distribution for R = 1 + kX consists in a choice of a value of k, this choice being made independently of A. Therefore, the problem of finding the class of utility functions with the property that a/A = k is equivalent to the problem of deter￾mining the class of utility functions with the property that choices among distri￾butions for rate of return on the portfolio are independent of initial wealth. If two utility functions U and V repre￾sent the same preference ordering, there exist constants b and c such that V = bU + c. Therefore, if a utility function U determines an ordering of probability distributions for rate of return and this ordering is identical with the ordering of probability distributions for final wealth, then U(R) and U(Y) = U(AR) must represent the same ordering. This must mean that U(R) and U(AR) are linear transformations of each other: U(AR) = bU(R) + c. (9) Here b and c are independent of R, but they may depend upon A. Differentiation of (9) with respect to R gives U'(AR)A = bU'(R). (10) Then differentiating (10) with respect to A, we have U"(AR)AR + U'(AR) = b'U'(R), (1 1) where b' denotes derivative with respect to A. From (10) the right-hand side is (b'A/b)U'(AR), so that (11) can be writ￾ten bA U"(Y)Y+U'(Y)= b U'(Y) or U"(Y)Y bI A U'( = 1---. (12) Since this must hold for independent variations in Y and A, both sides are constant. This means that relative risk aversion must be constant, equal to, say, Ay. It is easily verified that the only solu￾tions to this condition are linear trans￾formations of the functions U(Y) = In Y if '=1 (13a) and U (Y) =y1-- if Py 0 . (1 3b) Thus, utility functions belonging to this class are the only ones permitted if con￾stant asset proportions are to be optimal. 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