STOCHASTIC

W. T. ZIEMBA
One may find the slope of L and the point M (see Fig. 1) by maximizing
(3) *' 3 j h ^ ^ '
s-t ' *- 0, e '*
= L
By letting f, = ^( —10» ' = 1> •••>"> an d utilizing e'x = 1,
We will assume that f'x > 0 for all "interesting" feasible x. This is a very
minor assumption since the fact that the x £ 0 and e'x = 1 implies that there
always exists an x such that |'jc > 0 unless all £, ^ 0 in which case it is optimal
to invest entirely in the risk-free asset. It will now be shown that (3) has a
unique solution.
A differentiable function 0(x): A -> R is said to be strictly pseudo-concave
on A 0 is concave and tj/ > 0 is strictly convex.
Then 0 = x P/i/' is strictly pseudo-concave on A.
Proof V0 = {OAVT-W^W 2 }. Let x e A. Thus
V0(x) (x-x) = r , , ..,, (x — x) < 0, x e A, x # x
[•AW]
{^(x)V^'(3c)- , 5'(*)V^(3c)}'(*-JO ^ 0 (since i/>(x) * 0)
mv*(x)-vm + *(*>DK*)-*M] < ( ^ l s n s d tr it) > o)
i^ (x) ¥ (x) - ¥ (x) i^ (x) which implies that
254 PART III STATIC PORTFOLIO SELECTION MODELS