STOCHASTIC

CHOOSING INVESTMENT PORTFOLIOS
Proof (a) Let Sj > 0 be defined by a>} = (\^2)Sf 2 . Then
( m ~) 1/ot m
* (*) = UI (*'"; *) a/2 [ = .15, */,
which was shown to be strictly convex on P in Lemma 1.
(b)
Cm "J l/«
*(*) = i £ (*'«, *) a/2 > = {i(A'Oi *) a/2 } 1/a
b=i J
Hence *P will be strictly convex if /*)" /2 ^ 2$ a } and this set is convex for all q' = 2q"^0
since Press [22] has shown that 27=i(*'fy*) a/2 is convex if 2 ;> a ^ 1.
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 259