STOCHASTIC

utility function. However, many concave utility functions may lead to the
same optimal decision as that obtained by a given safety-first investor. When
there is a risk-free asset the efficient surface is linear and the situation is largely
indeterminate because the indifference surfaces are also linear. In Exercise
CR-24 the reader is invited to show that these results also hold for a reformulated
aspiration objective. The analysis also applies (Exercise CR-23) when the
random returns have appropriate symmetric stable distributions instead of
normal distributions and one utilizes a mean-dispersion analysis as discussed
in Ziemba's paper. See also Exercise CR-21 for a related normal distribution
case. Exercise ME-18 shows that the deterministic equivalent for the aspiration
model that one obtains in the normal distribution case generally does not
generate necessary nor sufficient deterministic equivalent sets when the returns
have nonnormal distributions. Some properties of safety first like optimization
problems are considered in Exercises ME-26 and CR-20. Exercise ME-2
considers a chance-constrained programming problem and the utility function
that it induces. Agnew et al. (1969), Ahsan (1973), Bergthaller (1971),
Dragomirescu (1972), Levy and Sarnat (1972b), Pyle and Turnovsky (1971),
and Telser (1955-1956) consider additional applications of chance-constrained
programming to the portfolio selection problem.
The paper by Ziemba is concerned with the expected utility portfolio model
when the returns have Pareto-Levy distributions. These distributions are also
termed stable because many of their members are closed under addition; they
are of interest for the empirical explanation of asset price changes and other
financial and economic phenomena. An important reason for this is that all
limiting sums (that exist) of independent identically distributed random
variables are stable. Thus it is reasonable to expect that empirical variables
which are sums of random variables conform to stable laws. Stable distributions
represent generalizations of the normal distribution and appropriate stable
families are closed under addition. These distributions have four parameters.
If two of these parameters (namely the skewness and characteristic exponent
coefficients) are constant across investments, then a mean-dispersion analysis
is consistent with the expected utility of wealth approach when preferences
over alternative wealth levels are concave. The mean-dispersion analysis per se
is considered in Exercise CR-7. In particular, if all means exist and are equal
and investment alternatives are independent, then all optimal investment
allocations are positive and are proportional to weighted ratios of the inverses
of the dispersion parameters (as in the normal distribution case). A generalized
risk measure and its properties are explored in Exercise ME-17. In order for
expected utility to be finite it is required that for the large absolute values of
wealth the utility function is not steeper than a power function whose order
is not exceeded by the value of the characteristic exponent. Hence many
common utility functions need to be appropriately modified since their limiting
206 PART III STATIC PORTFOLIO SELECTION MODELS