STOCHASTIC

SEPARATION IN PORTFOLIO ANALYSIS
It is clear by comparing Theorems 2 and 4, that the question of separation
in general markets without money has been completely solved. However, for
general markets with money, the situation is considerably less clear-cut, as
the following result shows.
Theorem 5 Let p1,...,p„ be a general market with p„ having the certain
(gross) rate of return r > 0, i.e., p„ is money. Then sufficient conditions for
local or global separation of problem Pk are
(i) w'(0 = (fl + bi) c , with either
(ia) c = 1, or
(ib) a > 0, b>0, c 0, c < 0.
(ii) u'(Q = ae H (a>0, b 0 and c < 0. In all cases (i), if
n$I+, global monetary separation obtains, but if n e I+, money may or may
not be one of the mutual funds in an optimal portfolio. However, for case
(ii) local monetary separation holds for large w.
Proof (i) M'(£) = (a + b^) c . For c=\ (quadratic utility), Theorem 4
applies. The fact that global monetary separation* holds if n $ I+ follows
from the work of Lintner [4], reprinted as the first article of this chapter.
Reference to Lintner's paper also shows that monetary separation need not
obtain if n e /+. The general problem may be written
where
and where
max {F(y)\yeY(w)},
l n-l
Y(w) = \yeE n - l \yi^0, iel+-{n} and £ y; S r> + a/w] -1
I ;=i
if n E I+, £ yt unconstrained if n $ 1+ J,
i= 1
+ E
F{y) = \
E\og
1 This is an unsolved problem.
l+b X (Pi-r)h
r "_1
(Pi-r)yt
(c^-1)
(c = -l).
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