STOCHASTIC

CHOOSING INVESTMENT PORTFOLIOS
Tobin's separation theorem [31] and implemented by Ziemba et al. [35]
utilizing Lintner's [15] reformulation of the separation theorem. The analysis
here is analogous and utilizes a mean-dispersion efficient surface.
The following theorem indicates that a point cannot solve (1) unless it lies
on a mean-dispersion efficiency curve.
Theorem 2 Let G and H be two distinct distributions with finite means fit
and p.2 and finite positive dispersions Df and D2 (1 < a g 2), respectively,
such that G(x) = H(y) whenever (x — iii)IDl= (y — fi2)jD2. Let nl^fx2
and G(z)>H(z) for some z. Then $u(w)dG(w)^$u(w)dH(w) for all
concave nondecreasing u if and only if Z^ g Z>2.
Proof Sufficiency: Case (i): \ix = n2 = 0. Let A:s D2\DX ^ 1. Then
/*0O /•oo
u(w) dH(w) - u(w) dG(w)
J — co J — co
/•DO
J ~ 00
[u(kw)-u(wy]dG(w)
(since (7 is symmetric)
r ' {(w(fcw)-w(w)) -(K(-W)-«(-A:W))} dG(w) g 0.
< 0 by concavity
Case (ii): n1 2: \i2. Let 3 and H be the cumulative distribution functions
for x and y, respectively, when their means are translated to zero. Let £ =
J.W «(.) = /.(.+*) *(.) S J.(„+ft)«(.) [b, («,
(since e S; 0 and w is nondecreasing)
g w(w + /* + £) 1 = £>2, a = 2, and u(w) — —e~ w ;
then
-r (e-w/2
-e- w )dG(w) > 0.
The proof is a generalization of the proof given by Hillier [13a] for the
mean-variance case (a = 2). The monotonicity assumption is not crucial,
however; dropping the monotonicity assumption does not seem to add any
apparent generality as free disposal of wealth always seems possible. A proof
1. MEAN-VARIANCE AND SAFETY-FIRST APPROACHES 249