STOCHASTIC

isky assets in the same proportion, and reduces the variance of risky asset
returns, it is plausible that such a tax may lead to increased demand for the
risky asset. Stiglitz shows that this is indeed the case if either (i) the return on
the safe asset is zero, (ii) absolute risk aversion is nondecreasing, or (iii)
absolute risk aversion is decreasing and relative risk aversion is nondecreasing.
He shows also that PRT remains unchanged if the risk-free return is zero,
and that otherwise PRT increases or decreases according as absolute risk
aversion is an increasing or decreasing function. The case of an income tax
with no loss offset or with only partial loss offset is also examined. Not surprisingly,
it is found that the demand for the risky asset is always less with no
loss offset or with partial loss offset than with full loss offset. In Exercise
CR-19 the reader is asked to examine the portfolio allocation effects of a
20% income tax with and without loss offset. The Stiglitz paper also considers
in part the question of special provisions for capital gains. In the extreme
case of a tax only on the safe asset, it is shown that the demand for the risky
asset is increased if absolute risk aversion is nondecreasing, or if the relative
risk aversion is not greater than unity. Exercise CR-4 examines the effects on
a portfolio of two securities of a tax on one of the securities, assuming quadratic
utility.
The Naslund paper examines the effects on portfolio allocation of a proportional
income tax with full loss offset. The paper deals mainly with formulations
of the portfolio problem not based on expected utility. In particular, the main
emphasis is on safety-first (see the Pyle and Turnovsky paper) and chanceconstrained
programming models. The basic safety-first model adopted is that
of minimizing the probability of returns below a preassigned "disaster" level d.
As in the Pyle and Turnovsky paper, Naslund assumes that the problem of
minimizing the Tchebychev upper bound on disaster probability will serve as
an equivalent surrogate problem. The relation between these problems is
examined in Exercises CR-20 and 21, where it is shown that the problems
are equivalent for joint-normally distributed assets. For several risky assets
which are not joint-normally distributed, the resulting portfolios may be quite
different in the "true" and "surrogate" problems. This is illustrated in Exercise
ME-18 for a two-asset case. In any case, assuming the validity of the surrogate
problem, Naslund shows (for two assets) that an increase in taxes leads to
increased risk-taking if the asset with higher yield is riskier, and to decreased
risk-taking otherwise. The paper also discusses portfolio allocation in the
so-called £-model of chance-constrained programming, which maximizes
expected portfolio return subject to the constraint of attaining at least a given
level of return B with at least a preassigned probability level a. For jointnormally
distributed asset returns, the corresponding deterministic equivalent
problem has a linear objective function and quadratic constraints. In this case,
the Tobin-Lintner separation theorem is true, thus reducing the many-asset
INTRODUCTION 213