STOCHASTIC

SEPARATION IN PORTFOLIO ANALYSIS
and
e = (l,...,l) T e£" (2)
where T denotes transposition. The investor's problem is
where
and
?k:max{F(y)\yeKk"(w)}, (3)
F(y) = Eu(p T y), (4)
Kk"(w) = {x e E" | e T x = w, xt ^ 0 for i e /+}, (5)
/+ = {/!,..., zj, (6)
The set /+ labels the assets which cannot be sold short. The special case
/+ = 0, problem PO, omits all nonnegativity constraints, and is the problem
considered by Cass and Stiglitz [2]. The contraint set for wealth level unity
will be denoted by Sk":
Sk" = «*"(!)• (7)
Let y*(w) e Kk (w) be the optimal solution to (3), and define x*(w) =
(l/w)y*(w) e Sk". Clearly, x*(w) is the vector of optimal asset proportions.-
Problem Pk is said to exhibit global separation if and only if there exist wealthindependent
vectors x 1 ,* 2 e Sk such that
x*(w) = m(w)x i + [l-w(w)]x 2 (8)
for all w > 0. Problem Pk is said to exhibit local separation if (8) holds only
for w e (a, b) ^ (0, oo). In both of these definitions, the mutual funds x 1 ,x 2
and the proportion m(w) invested in x 1 , may depend on the specific utility
functions used, but this dependence will be suppressed in the notation. 3 The
separation property splits the computation of optimal investment into two
subproblems: (i) the determination of the wealth-independent mutual funds,
and (ii) the determination of investment in each mutual fund as a function
of wealth. Note that if k > 0, global separation of problem Pk might not be
possible, since x*(w) in (8) might become infeasible for certain w. In such a
case, local separation still holds.
3 In the Tobin-Lintner [3, 4] separation, x 1 and x 2 would be independent of the utility
function, but m{w) would not be, since it is governed by mean-variance indifference curves.
3. SEPARATION THEOREMS 159