STOCHASTIC

detailed analysis of the qualitative behavior of portfolios consisting of a blend
of one safe and one risky asset. Several ways to characterize portfolios are
seen to relate to the absolute and relative risk-aversion indices. In Exercise
CR-11, the relevance of absolute risk aversion to the problem of optimal
insurance policies is outlined. Effects of changes in mean and variance of
risky assets are examined in Exercises ME-3 and 4, in connection with the
problem of optimal foreign exchange holdings. Exercise CR-17 presents an
alternative approach to risk aversion, based on mean-variance indifference
curves. In Exercises ME-8 and 9, additional results are presented regarding
the effects of risk-aversion indices on the qualitative behavior of an investor's
insurance premium as a function of mean and variance of risky return (see
also Zeckhauser and Keeler, 1970). In Exercise ME-10 it is shown that a
bounded concave utility function must have a relative risk-aversion index
which increases "on the average," from values less than 1 for small arguments
to values greater than 1 for large arguments. Exercise ME-23 presents a not
commonly assumed utility function that has the desirable properties that it is
increasing, bounded, strictly concave, and has decreasing absolute and
increasing relative risk aversion. See Sankar (1973) for another desirable
utility function. In Exercise ME-1 the reader is asked to devise stochastic
dominance tests which would be appropriate to the class of utility functions
exhibiting decreasing absolute or increasing relative risk aversion.
III. Separation Theorems
The Lintner paper discusses the application of mean-variance and safetyfirst
analysis to portfolio selection. Under the assumptions that there exists a
risk-free asset which can be lent or borrowed in unlimited amounts at a
common positive rate of interest, the optimal portfolio problem becomes
considerably simplified. Lintner proves a very important separation theorem
due to Tobin (1958), which splits the computation into two parts. First, a
single optimal mutual fund, the same for all investors, is calculated by solving
a fractional programming problem. Then the optimal portfolio for any
particular investor is calculated by means of a univariate unconstrained
optimization. The Tobin-Lintner separation theorem implies that all investors
will purchase different amounts of a common mutual fund, and this reduces
the many-risk-asset problem to an effective single-risk-asset problem. If
borrowing of the risk-free asset is limited, the separation theorem need not
hold. In this case the computation of an optimal portfolio becomes considerably
more difficult, and would generally need to be done separately for each
investor's utility function. This is also true if the interest rate of borrowing
exceeds that of lending. Some of the computational aspects of similar problems
are treated in the Ziemba paper in Part III. The Lintner paper along with
INTRODUCTION 85