STOCHASTIC

Sharpe (1964) proved also to be seminal building blocks in the modern theory
of capital asset pricing; see Jensen (1972) for an assessment of the current
st ate of the development and application of this theory.
Exercise CR-9 extends the Tobin-Lintner separation theorem to the case
of homogeneous risk measures more general than variance. Exercise CR-19
examines an equivalent quadratic programming problem for determining the
optimal mutual fund. Exercise CR-20 examines the Lintner separation theorem
when the variance-covariance matrix of the risky assets is not positive definite.
In Exercises CR-22 and 23, optimal portfolios are obtained for a single risky
asset under unlimited and limited borrowing. Exercise ME-18 is concerned
with the existence of solutions to the linear complementary problem arising
from the Lintner paper. Exercise ME-16 examines substitution and complementary
effects in the choice of risky assets, for the case of a quadratic
utility function.
The Vickson paper discusses restrictions on the investor's utility function
which are necessary and sufficient for separation of the general portfolio
problem. The paper is an extension of a recent paper of Cass and Stiglitz
(1970) in which the separation property was examined for the case when
unlimited borrowing and short sales are allowed. The problem is to determine
conditions on the utility function which will ensure that an arbitrary portfolio
problem in many assets (one of which may be riskless) reduces to an equivalent
two-asset problem, involving two mutual funds which are independent of
initial wealth. When such a property holds, determination of the optimal
portfolio is reduced to a two-stage procedure: (1) calculation of the optimal
mutual funds and (2) determination of actual investment in each mutual fund.
If there exists a riskless asset, it may or may not be one of the "mutual funds"
in the optimal portfolio. If it is one of the mutual funds, then the optimal
risky asset proportions are independent of wealth; otherwise the risky asset
proportions will generally be wealth dependent. The type of separation
property discussed in the Vickson paper is totally different from that in the
Lintner paper. In the latter, the optimal mutual fund is determined solely on
the basis of the means, variances, and covariances of the risky assets, and the
rate of return on the riskless asset; it is independent of wealth and the utility
function. In the separation property of the Vickson paper, however, the
mutual funds are independent of wealth but dependent on the parameters of
the utility function. In both types of "separation," the actual investment in
the mutual funds is dependent on both the wealth level and the utility function.
Necessary conditions on marginal utility are derived for separation in
general markets, including markets with or without a risk-free asset. These
conditions on marginal utility are also sufficient for separation in general
markets with unlimited borrowing and short sales. When borrowing constraints
are introduced, these conditions are essentially still sufficient for
86 PART II QUALITATIVE ECONOMIC RESULTS