STOCHASTIC

given parameter. Exercise ME-30 develops such results in the special case when
the investor's utility function is a power or logarithmic function and it is
optimal to invest in every available asset. For further results and information
on this delicate topic the reader is referred to the discussion of the maximum
theorem by Berge (1963) and to the work of Danskin (1967) and Hogan (1973).
Not all distributions are "compact" in Samuelson's sense and the reader is
asked to investigate the compactness properties of some common univariate
distributions in Exercise CR-25. Samuelson's presentation does not delve
deeply into many fine and subtle points of the analysis. The reader is therefore
asked to provide some of this material and to verify some statements in the
paper in Exercise CR-26. One can, of course, utilize Taylor series approximations
for very general utility functions and random variables. In Exercise CR-2
the reader is asked to calculate bounds in the maximum error in objective
value terms using such approximations. Taylor series approximations of order
n yield explicit deterministic polynominal functions of order n in terms of the
wealth variable. However, in terms of asset allocation variables (Exercise
CR-12) the approximation yields a sum of n signomial functions (i.e., signed
products of positive variables raised to arbitrary powers) which reduce to
polynomial functions only in very special cases.
Samuelson's paper develops the notion that the mean and variance of wealth
are approximately sufficient parameters for the portfolio selection model when
the probability distribution of wealth is "compact." In the "compact" case,
moments of order 3 and higher are small in magnitude relative to the first
two moments of the portfolio return; hence a limiting approximation indicates
that only the first two moments are relevant for optimal portfolio selection.
Samuelson's presentation is heuristic. The paper by Ohlson presents a rigorous
approach to the study of such approximations. Additionally Ohlson's analysis
generalizes and extends the range of applicability of such asymptotic quadratic
utility approximations. He shows that (essentially) it is sufficient if the third
absolute moment vanishes at a faster rate than the first two moments, even if
moments of order 4 and higher are infinite. Rather general utility functions
also suffice, including the negative exponential, power, and logarithmic cases.
The paper also discusses the specific case when the random returns have
log-normal distributions, assuming that the mean and variance of the growth
rates are linear in time. Such distributions are not "compact," yet the quadratic
approximation is still valid. Exercise ME-31 illustrates a case where not all
moments are finite, yet the asymptotic utility is a quadratic function of the asset
proportions (see also Ohlson, 1974). For an alternative Taylor series justification
for the validity of the mean-variance approximation in portfolio theory,
see Tsiang (1972). Some discussion of Tsiang's paper appears in Borch (1974),
Bierwag (1974), Levy (1974), and Tsiang (1974). See Chipman (1973) and
Klevorick (1973) for a rigorous discussion concerning the existence of expected
204 PART III STATIC PORTFOLIO SELECTION MODELS