STOCHASTIC

utility functions dependent on mean and variance parameters when the returns
have normal or two equally likely point distributions.
One may compute the mean-variance efficient surface in several ways; for
example, by solving a parametric qudratic program as discussed in Exercise
CR-13. Paine (1966) illustrates this calculation when the returns follow the
Markowitz-Sharpe (see Sharpe, 1970) diagonal model. When nonnegativity
constraints are not present, one can, as in Exercise CR-14, explicitly solve for
the optimal portfolio allocations. The efficient surface is then an explicit
quadratic function of the stipulated mean portfolio return. See also Hart and
JafFee (1974) for a novel application of mean-variance analysis to the problem
of financial intermediation. Their paper also develops some properties of the
mean-variance efficient set when positive and negative holdings are allowed.
When there is a risk-free asset the efficient surface is a ray in mean-standard
deviation space, and one may find the optimal proportions by solving the
fractional program or linear complementary problem developed in Lintner's
paper. Exercise CR-3 discusses some aspects of the fractional program (see
also Exercises II-CR-19 and 20 and II-ME-18). The continuity properties of
the efficient surface in this case are explored in Exercise ME-3. The number of
efficient portfolios in a given portfolio problem is determined largely by the
constraints on the investment allocations. Exercise CR-11 illustrates how a
useful constraint relaxation increases the number of efficient portfolios. The
analysis is carried out for a numerical problem with one safe and two risky
assets.
As indicated above, if the investment returns have normal distributions the
mean-variance approach and the expected utility approach imply the same
optimal portfolio behavior. That is, all optimal solutions of the expected
utility problem are mean-variance (or equivalently mean-standard deviation)
efficient. In their paper, Pyle and Turnovsky investigate whether or not an
investor who operates with a safety-first criterion also has the same implied
behavior. The basic notion in a safety-first criterion is that the investor wishes
to minimize the chance of obtaining a large loss. There are several ways to
formulate such a notion and Pyle and Turnovsky consider the following three
alternative versions. Choose the investment allocations to: (a) minimize the
probability of obtaining a return below a certain stipulated (aspiration) level;
(2) provide the largest (fractile) return level that is achieved with at least a
given probability level; and (3) provide the largest mean return such that a
stipulated aspiration level is achieved with at least a given probability level.
All of these criteria lead to linear indifference surfaces in mean-standard
deviation space: hence if the mean-standard deviation efficient surface is
convex but not linear there is a unique indifference surface that is tangent to
the efficient surface. Hence there is a unique safety-first investor who will
choose the same portfolio as that chosen by an investor with a given concave
INTRODUCTION 205