STOCHASTIC

Separation in Portfolio Analysis
R. G. Vickson
UNIVERSITY OF WATERLOO
I. Introduction
Most modern treatments of portfolio analysis are founded on the expected
utility theory of von Neumann and Morgenstern [1]. When there is a single
risky asset, expected utility maximization leads to a straightforward theory
of demand for risk. In this case such questions as the demand for the risky
asset as a function of wealth, of interest rate, and of taxation policies—in
short, the microeconomics of risk—are all reasonably simple to analyze. This
pleasant state of affairs generally breaks down if many additional risky assets
are introduced into the problem. In the latter case the complexities of multidimensional
nonlinear programming resist not only one's ability to calculate
the optimal portfolio, but also one's ability to conceptualize the problem
effectively. It is therefore important to know if the problem can be recast as
a new problem involving only one or two "composite" assets, for if this is
true the problem becomes effectively one or two dimensional. For example,
if the optimal portfolio as a function of wealth leaves the risky asset proportions
unchanged, the entire set of risky assets is effectively combined into a single
mutual fund. The many-asset case is thus reduced to a single-asset case. In
some circumstances, such a reduction to a single mutual fund might not be
possible, but reduction to two distinct mutual funds might be. The many-asset
case is thus reduced to a two-asset case—a significant reduction in problem
size. A portfolio problem which collapses in this manner down to a problem
involving one or two mutual funds is said to exhibit the separation property.
Such problem behavior clearly has many advantages for both qualitative and
quantitative purposes.
In a recent paper, Cass and Stiglitz [2] have derived restrictions on the
forms of utility functions which lead to separation of the portfolio problem
for general markets. They find that only a very small class of utility functions
allows general separation, these being the functions «(•) such that u'(w) =
{a + bw) c or u'(w) = ae bw . It is well known that general (increasing, concave)
utility functions lead to separation in the case of very special markets, having
joint normal returns, in the presence of a risk-free asset which can be borrowed
or lent without limit at a common interest rate (Tobin [3]; Lintner [4],
3. SEPARATION THEOREMS 157