STOCHASTIC

investment returns are always nonnegative. Several common distributions
might possibly be used to explain such security price changes. One such
distribution that has some attractive analytic and methodological properties
is the log-normal distribution. A variable has a log-normal distribution if its
logarithm has a normal distribution. Many of the important properties of
univariate and multivariate log-normal distributions are discussed in Exercises
ME-20 and 21, respectively. Since log-normal variates are not closed under
addition there does not appear to be an efficient method to solve portfolio
problems that applies for general concave utility functions. However, for
particular utility functions, one can develop approximations which have very
attractive computational and qualitative properties. Such is the case when the
utility function is logarithmic. Exercise ME-22 studies the qualitative properties
of the optimal solution vector for this problem. Exercise ME-23 develops an
approximating problem based on the assumption that the final wealth variate
is log-normal. The approximating problem can be chosen from a surrogate
family so that it has the remarkable property of possessing essentially all of
the qualitative properties of the original problem. Moreover, the approximating
problem is a single quadratic program, which may, as outlined in Exercise
ME-24, be solved in an extremely simple way utilizing the Frank-Wolfe
algorithm. Ohlson and Ziemba (1974) have developed similar results for the
case when the investor's utility is a power function of wealth. Dexter et al.
(1975) present computational results that support the approximation.
III. Effects of Taxes on Risk-Taking
The papers of Stiglitz and Naslund examine the effects of various taxation
policies on an individual's risk-taking behavior. Stiglitz assumes that the
investor is an expected utility maximizer, and analyzes the effects of taxation
on the investor's allocation of wealth among a sure asset with nonnegative
net return and a single risky asset with nonnegative gross return. There are
assumed to be no borrowing or short-sale constraints. Stiglitz emphasizes
two somewhat different measures of risk-taking: (1) the demand for risky
assets, measured by the fraction of wealth devoted to the risky asset, and (2)
"private risk-taking" (PRT), measured by the standard deviation of final
wealth. He shows that a proportional wealth tax increases or decreases the
demand for risky assets according as the Arrow-Pratt relative risk-aversion
index increases or decreases with wealth, and that it increases or decreases
PRT accordingly as the absolute risk-aversion index increases or decreases with
wealth. Of more interest are proportional income taxes with various loss-offset
provisions. A full loss-offset provision implies that the government shares the
risks of loss as well as the potential gains. This reduces the variability of risky
asset after-tax returns. Since the tax reduces expected return from riskless and
212 PART III STATIC PORTFOLIO SELECTION MODELS