STOCHASTIC

The continuous partial derivatives of Z are
(14) ^_J{ 6 + I&(lW.-«.£g>,}
W. T. ZIEMBA
x f(R)dR, i = 0, ...,n.
Now using the Fama-Roll tables, as in Section II, one may obtain a good
approximation to (13) and (14) via
m
(15) Z(x*)s I^HK'^ + M**)} 1 "*,],
and
x j -, i = 0,...,n,
aw
respectively, where P, = Pr{R = R) > 0, TJ=iPj = 1, and /n = 100.
One may then apply any standard nonlinear programming algorithm that
uses function values and/or partial derivatives to solve (1) approximately,
utilizing (15) and/or (16). If one uses an algorithm that utilizes only function
values, such as the generalized programming algorithm (see Ziemba [33]),
then for each evaluation of (15) one merely performs m function evaluations
of the form «(•) and adds them up with weights Pj. The evaluation of the
portfolio is more complicated since for each i (i= l,...,ri) one must perform
m function evaluations of the form {•} du(-)/dw and add these up with the
weights Pj.
One would suspect that it would be economically feasible to solve such
approximate problems when there are say 40-60 investments, the grid m is
say 20-50 points, and u, y, and k are reasonably convenient. It is possible, of
course, to apply this direct solution approach even when the risk-free asset
exists. However, the two-stage decomposition approach appears simpler
because one must solve a fractional program in n variables plus a nonlinear
program in one variable that has m terms. In the direct approach one must
solve one nonlinear program in n variables having m terms that may fail to be
concave or pseudo-concave (because u is concave and y is convex). Some
numerical results are given by Ziemba [34].
262 PART III STATIC PORTFOLIO SELECTION MODELS