STOCHASTIC

w = Z'x ~ F(?x; l'x, £J-o S,x,',0,a), then
8Ewu(Z'x)
8xt
du(£'xf
8xi
< oo, i — 0,1,...,«,
W. T. ZIEMBA
and the computation of the partials involves only a univariate integration
utilizing a normalized variable w ~ F(w; 0,1,0, a).
(d) The results in (a)-(c) remain valid if the hypotheses are modified
to read lim|W|_00|M(w)|/|w| Pl = vx for some 0 S vY < oo and some /Jj < a,
lim
l
|w|-+oo
lwl-»oo |M'(W)|/| W)" 2 = v2 for some 0 ^ y2 < oo and some /?2 < a, and
tively.
|w'(w)|/|w|^ 3 = *>3 for some 0 ^ v3 < oo and some /?3 < a— 1, respec-
Proo/ See the Appendix.
Remark Most common utility functions have either unbounded expected
utility or they are undefined over portions of the range of w, which is R.
However, one may modify many of these utility functions by adding appropriate
linear segments so that the utility functions are concave, nondecreasing,
defined over the entire range of w, and have bounded expected utility. For the
logarithmic, power, and exponential forms such modified utility functions are
(a) u(w) =
(b) u(w) =
logTW if
(logTW0-l) +-
Wn
w ^ w0,
if w < wn
w0 > 0, z > 0;
if w0,
(1 - 5) w0 d + (dwo ') w if w < w0, w0 > 0, 0 < 8 < 1;
— exp( —TW)
if w 3: w0,
(c) u(w) = < 0 > w0 > -oo, T > 0.
— (1+TW0) exp( — TW0) + texp(—TW0))VV
if w < w0,
Note that all polynomial utility functions of order 2 or more have unbounded
expected utility unless the distribution of w is truncated from above and
below.
It is our purpose to solve (1) using the following two-step procedure:
(i) find an efficient surface independent of the utility function w; and (ii) given
a particular u, find a maximizing point on this surface. Such a procedure for
the case when the random returns are normally distributed was suggested by
248 PART III STATIC PORTFOLIO SELECTION MODELS