STOCHASTIC

R. G. VICKSON
different mutual funds applying in different regions of w. For case (ii),
separation is global for all problems Pk, 0 :g & g n.
Proof (i) w'(£) = a + b£. For problem PO the solution can be written
down explicitly:
where
and
In (39) and (40),
and
x*(w) = k(w)x l + [1 -*(w)] * 2 . (38)
x 1 = a~ l eje I a~ i e, x 2 = a' l pje T (j' 1 n, (38)
A; = 1 +-% T (T~Y (40)
bw
p = (Epu...,Epn) T (41)
a = (EptPj), 1 £ i, ; ^ «. (42)
The solution as written displays global separation explicitly. For problem
Pk with k > 0, (38) will be valid in an interval w e W if JCj*(w) > 0 for z e I+.
Hence, in this case also, local separation will hold. When none of these
conditions is satisfied, local separation can still be proved by noting that the
solution must lie on the Markowitz [9] mean-variance efficiency frontier, and
by noting that any point on this frontier is either a single mutual fund or a
convex combination of two neighbouring mutual funds.
(ii) H'(£) = b£ c . For problem PO the solution x*(w) is identical to the
solution of the following set of nonlinear equations.
and
E\{Pi-p„)
n-l
Pn+ E (Pj-Pn)x* = 0, i= !,...,«- 1, (43)
V = l-lV (44)
>=i
Again, (43) and (44) are also valid for problem Pk with k > 0 if xt*(w) 2: 0
for ie I+. Note that in both these cases x*(w) collapses onto a single mutual
fund which is independent of w. When (43) and (44) are not valid the latter
result is still true, as can be seen most easily by showing that the Kuhn-
Tucker conditions for x*(w) are independent of w. Thus, global separation
holds.
166 PART II QUALITATIVE ECONOMIC RESULTS