STOCHASTIC

distributed from the rest. Then an optimal portfolio must involve
X. > 0, with some positive investment in x., as shown in the following.
Theorem III. Let (x,,x2,...,x ) be jointly distributed as
P(x )Q(x_,...,x ) with common mean and finite positive
variances^
E[xt]
and
E[U]
X1dP(x1)dQ(x2
0 < E[(xi - p) ] < »
"9 CX. ,Xj»•..»X )
Xn) = u
ZXjXj dP(X1)dQ(X2 Xn)
where, for U" < 0, 8 is a strictly concave function.
Then if
* * * n
e(X, ,X, X„ ) -Max e(X, X J s.t. IX - 1, X. > 0,
* * l
necessarily X. > 0 and X. < 1.
This will first be proved for n » 2, since the general case can
be reduced down to that case. Denoting dB(\ ,\.)/d\ by 6 (X.,X ), we
need only show the following to be positive
e^o.1) - e2(o,i) - (j^V Jj^V^V
- f x2u'(x2)dp(x2)
- E[X2JETU'(X2)] - EtxjU'Cxj)]
- -E[{X2 - u}{U'(x2) - E[U'(x2)]}]> 0
if U"(xj) < 0, since the Pearsonian correlation coefficient between
any monotone-decreasing function and its argument is negative.
We reduce n > 2 to the n - 2 case by defining
n
X 1 X 1 + ^ X j x j " X 1 X 1 + X II X II
n X.
x » T. —1
11 2 X II
n X.
x , E -r 1 - 1, as definition of X .
3 2 A II
EXISTENCE AND DIVERSIFICATION OF OPTIMAL PORTFOLIO POLICIES