STOCHASTIC

SEPARATION IN PORTFOLIO ANALYSIS
II.2 RESTRICTION OF THE UTILITY FUNCTIONS
Cass and Stiglitz [2] pointed out that separation can hold for general
markets only if it holds for a convenient set of special markets. Once the utility
functions have been suitably restricted by the special markets it becomes easier
to examine the general market problem. It is convenient to examine first the
separation property for discrete random returns. Thus, suppose that there
are only finitely many states of nature, 9 = l,...,N, with Pr[0 = z] = pt. The
returns pt{6) are specified completely by an nxN nonnegative matrix
Py, i= \,...,n,j= l,...,N. Assuming an interior solution (or I+ =0), the
following optimality conditions are obtained:
E Pj u '{ Z ykPkAPij = A, i = l,...,w. (16)
; = 1 \* = 1 /
Consider first the case N — n and with nonsingular matrix p. Then (16) can
be solved to give
where
«'(2>P«) = £lJfy, (17)
\ * / Pi J
R = p-K (18)
Since u' is continuously differentiable and strictly decreasing, the inverse
function g(-) = u'~ i (-) is a well-defined, strictly decreasing, continuously
differentiable function on its domain. Furthermore, since u' is piecewise twice
continuously differentiable, so is g. Solving (18) gives
tyjPn=9[~t^\ = 9^d- d9)
From the separation equation (8), it follows that
where
g (Afc;) = a (X) m, + b (A) nt
(20)
wf = £ Xj l pji, A* = £ xj 2 pji. (21)
j j
Clearly, the consideration which yielded Eq. (15) will apply to the vector
g(J.ki), i = 1,...,«, as well as to the vector y(X), so the following necessary
condition obtains:
g(kik) g{kjl) g(k,Z)
ktg'iktX) kjg'ikjX) k,g'(k,X)
ktV(ktX) kj 2 g"(kjX) kfg'X^X)
= 0 (22)
3. SEPARATION THEOREMS 161