STOCHASTIC

ON OPTIMAL MYOPIC PORTFOLIO POLICIES 333
By (31), the expression to be maximized at decision point J — 2, given that the
economy is in state m at that time, becomes
M J-i m
£' pj-^nkj-x .£[(*,_, | mn)w\ . (46)
n=l
Since the constants kj^u„ will in general be different for different n, (46) and, therefore,
the optimal portfolio z$_2,m, depend on the yields in period / — 1. Thus the
optimal investment policy is not myopic at decision point J — 2; neither is it myopic
at decision points 1, ...,/ — 3, which is easily shown by induction.
The existence of positive constants ah . . . , aj-\ and of constants bu, • • • ,
bj^i,Nj_, such that
/,„,(*) = atfJm(x) + bjm all m , j = 1, . . . , J - 1 (47)
are clearly both necessary and sufficient for myopia to be optimal in this model.
As noted, (45) violates (47) for j = J — 1 whenever iV/_i > 1. However, when
N,:= 1 , j=l,...,J, (48)
(47) is satisfied; but (48) also implies that yields are statistically independent in
the various periods. This confirms Mossin's result that the optimal investment
policy is myopic when returns are stochastically independent over time and the
terminal utility function is x 112 .
Let us now assume thatfJm(x) has the form
fjm(x) = log x all m . (49)
Letting Hj-,,m denote the maximum of (34) subject to (35) and (36) when u(x) =
log x, that is,
N J
M J-\.m
ffj-i,»= T,Pj-i.m*E\\og\ £ G3.-,j-i.„» - rj-i,»)»*j-i, + O-i.J [ , (50)
n=.l < L i,2 J )
we obtain from (31)—(33), solving recursively,
fjm(x) = log x + bjm a\\m j = 1, . . . , J — 1 , (51)
(where 6j_i,m = H/_i,m, all m), which is consistent with (47). As a result, the induced
utility functions
/u(*)i • • • ./iw,(*). • • •