STOCHASTIC

(c) For a single risky asset having only two rates of return, as in the Stiglitz paper,
sketch the before- and after-tax budget lines under various combinations of interest
rate, initial wealth, and risky returns.
(d) Show by either graphical or analytical means that the effects of the tax on risk-taking
are ambiguous, that is, show that risk-taking may increase or decrease depending on the
problem details.
*(e) Try to find conditions under which unambiguous statements can be made regarding
the effects of this tax on risk-taking.
Suppose that B = $2000, w0 = $1000, and t = 20%.
(f) For the investor of Exercise CR-19, formulate the optimality problem if short sales
are not allowed, but borrowing is unlimited. Formulate an algorithm for solving this
problem.
20. (Properties of univariate log-normal distributions) The notion that random investment
returns are subject to limited liability in the sense that the investor can lose no more
than his initial outlay leads to the study of distributions whose outcomes are never negative.
The log-normal distribution is such a distribution and it arises naturally from modified
central limit theorems. If X is an essentially positive variate (0 < x < oo for all outcomes x)
and Y = log X has a normal distibution with mean n and variance a 2 , then X is said to have
the log-normal distribution A(x\n,a 2 ). Hence
and the density of X is
dA(x)
dx
A(*) =
IN (log x)
(o
exp -^Oogx-
= < xo ^2n
nY
lo
if x > 0,
otherwise,
dx if x > 0,
otherwise.
Thus log-normal variates tend to arise in situations where the variate is a productof independent
identically distributed (iid) elementary variates. It is possible to prove that if Xlt X2,...
is a sequence of iid variates, EilogX,} = fi < oo and var[logA r J = a 2 < oo, then the geometric
mean [OJ = I XjY'" is asymptotically distributed as AO,