STOCHASTIC

the characteristic exponent is 2 (in the multivariate normal distribution case)
the deterministic program reduces to the quadratic program developed in
Exercise CR-1. The reader is asked to verify some results used and stated in
Ziemba's paper in Exercise CR-16. See Ohlson (1972a) for an analysis of the
portfolio problem when the asset returns have log-stable distributions.
The mean-variance and related approaches are discussed in many papers
and books. For additional discussion and results the reader may consult
Archer and D'Ambrosio (1967), Borch (1968), Fama and Miller (1972), Levy
and Sarnat (1972a), Markowitz (1959), Sharpe (1970), Szego and Shell (1972)
Smith (1971), and Tobin (1965).
II. Existence and Diversification of Optimal Portfolio Policies
In financial optimization problems it is important to know if an optimal
decision or policy exists. If the objective function is continuous and the constraint
set is compact, then, by Weierstrass' theorem, a maximizing point
exists. However, it is not always possible or desirable to assume that the
constraint set is compact. In general, of course, an investor may be led to
arbitrarily large investment positions in some investments, and an optimal
allocation does not exist. In his paper Leland considers conditions on the
feasible region, the investment returns, and the utility function that guarantee
that a maximizing point exists. He assumes that the investment choice is among
a finite set of investment alternatives and that there is no arbitrarily large
investment position in any asset or assets which offers a nonnegative net return
with probability 1. It is supposed that the constraint set is closed and convex
and contains the origin. The utility function is strictly increasing and concave
in wealth. Existence of an optimal portfolio is then guaranteed if the utility
function is bounded from above. If the expected returns in all investments are
finite, then it is only necessary to assume that marginal utility converges to
zero as wealth increases without limit. In Exercise CR-5 the reader is asked to
determine whether or not an optimal allocation exists for some numerical
examples. Exercise CR-6 relates the results to boundedness of the utility
function and possible St. Petersburg paradoxes. The reader is asked to investigate
some extensions of Leland's results in Exercise ME-5. Recently, Bertsekas
(1974) has generalized Leland's results. In particular, he develops necessary
and sufficient conditions for the existence of a maximizing point that are
slightly weaker than Leland's sufficiency conditions.
In addition to the existence of optimal portfolio policies one is interested
in how such policies are found numerically. The expected utility function
generally has useful concavity properties and may be equivalently thought of as
a deterministic nonlinear objective function whose variables are the asset
208 PART III STATIC PORTFOLIO SELECTION MODELS