STOCHASTIC

and a gamble is offered that returns
{
(r+a)
(r—a)
with probability
with probability
p,
(1— p),
where p > i, a > 0, initial wealth B0 = 0, and r is the riskless rate of interest. Show that
there is no optimal policy.
7. Consider a portfolio problem associated with the choice of asset proportions Xj £ 0
of independent Pareto-Levy investments pu ...,p„. Referring to the notation in Ziemba's
paper, let pj have the distribution F(pj; pj, Sj, 0, a), where Sj > 0 and 1 < a ^ 2.
(a) Show that the mean-a-dispersion efficiency frontier may be generated by solving
minf(x) = ^Sjx/, s.t. Xi + ••• + x„ = 1, pt xt + ••• + pnx„ = p.,
xt gO,.-,^gO
for all p.
(b) Show that the = signs in (a) may be replaced by =: signs as long as min Sj > 0 and
minpj < (i < max/5j.
(c) Let a Lagrangian expression for the problem in (a) be
L(x,vuv2) =f(x) + V1(1-'£XJ) + V2(P-'ZPJXJ).
Develop the Kuhn-Tucker conditions for the problem in (a), and show that they have
a unique solution under the assumptions in (b).
(d) Suppose investments / = 1 i»S» have equal means and 0 < Si g ••• g S,„;
then these investments may be optimally blended to form a single composite stable asset
pc. Recall from Samuelson's paper in Part III, Chapter 3, that it is always optimal to allocate
a positive amount of one's resources in each investment whose mean is at least
min^j. Show that such a result applies to this case even if a < 2. Develop the efficiency
problem to form the composite asset pc = Yj=i y*Pj- Show that aSjiyf)"' 1 = vx.
(e) Show that the optimal asset proportions satisfy
yk* \sKJ
and that ys* L m cJfc
Interpret these expressions. Notice that when a = 2, the normal distribution case, the
product of asset proportion and variance is equal for each j.
The results in (d)-(e) indicate that without loss of generality the assets may be assumed to have
different means, say py > p2 > ••• > p„. By Samuelson's results, x/ > 0 implies Xjtt > 0;
hence the only pattern of zeros in (xt*, ...,x„*) must have the form (xS, ...,xr*,0, ...,0).
(f) Develop the Kuhn-Tucker conditions under the assumption that r is known.
(g) Reduce the Kuhn-Tucker conditions to the system
Sjx'f 1 -ajtsAbiP+j^buXj] -aJ2S2\b2p+Y,b2jx\ = 0, 7 = 3,
where the a's and b's are rational functions of pt and p2.
(h) Develop a procedure to find the optimal r using the system of nonlinear equations
in (g). When is r ~ 1 or n?
8. Consider an investor with a utility function u. Suppose his investment alternatives
i = 1,..., n have independent identically distributed Cauchy distributions px ~ F(p ;p,S,0,\),
S > 0 (notation follows Ziemba's paper). Assume that the investor wishes to allocate his
initial wealth of one dollar among these investments in proportions xt so as to maximize
COMPUTATIONAL AND REVIEW EXERCISES 333