STOCHASTIC

SEPARATION IN PORTFOLIO ANALYSIS
Case 2 To derive necessary conditions for separation in general markets
with money, it is only necessary to consider matrices p of the form [2]
L>] =
0 0 0
0 0
0 ••• 0 p„ pna p„a 2
r ••• r r r r
where p is a nonsingular (n — 2) x (n — 2) matrix aud a > 0 is arbitrary. Note
that asset n is money, having the same rate of return r > 0 in all states of
nature. By a simple modification of the proof of Theorem 2, the following
result may be obtained. (See Exercise CR-21.)
Theorem 3 A necessary condition for local or global separation of problem
Pk in general markets with money is that u' have one of the forms
or
for £ 6 Ek.
«'(0 = {a + bt,y
M'(0 = oe bi (constant absolute risk aversion)
Note In the statement of Theorem 3 it is not assumed that money is one
of the mutual funds in an optimal portfolio. The latter property, called
monetary separation is, of course, also covered by the necessary conditions
stated in the theorem.
III. Sufficient Conditions for Separation
In this section, the extent to which the necessary conditions stated in
Theorems 2 and 3 are also sufficient for separation is determined. The first
result is
Theorem 4 The following conditions are sufficient for local or global
separation of problem Pk in general markets without money:
(i) u'(0 = a+bi (a0), or
(ii) u'(Q = bt e (b>0,c