STOCHASTIC

(g) Discuss the use of semivariance as a measure of a risky asset particularly in relation
to variance. Illustrate some advantages and disadvantages.
(h) Suppose there are two projects A and B that have the following joint probability
of outcome table per dollar invested in each investment:
A 0
3
4
0
0.2
0.05
0.05
B
2
0.05
0.1
0.2
5
0.05
0
0.2
Perform p-o 1 , p-a, ft-Sh, and p-sh analyses, where it is assumed that h and the investors'
initial wealth is one dollar.
(i) Discuss the results of (h).
9. Suppose an investor's utility function is u(w) = 1 — Ae~ v,v -(1 -X)e~ 6w , where
0g AS \,y> 0, and 0.
(a) Show that u is concave, nondecreasing, bounded from above, and has decreasing
absolute risk aversion.
Suppose that the investment returns have a multivariate stable distribution of Press'
class as discussed in Ziemba's paper.
(b) Show that
Eu(w) = 1 - A exp[-y/5'*+/>•(*)] - (1 — A) exp[ — dp'x + 6"r(xy\ = i//(x),
where x = (xi, ...,x„)' £ 0 are the investment allocations, 1 < «^ 2, /•(*) =
\ YJj^iix'Qjx)" 12 is the dispersion of the portfolio p'x, and p = (pu ...,p„Y is the vector
of mean returns. Investigate the boundedness properties of the expected utility integral.
(c) Show that r is convex.
(d) Show that \ji is concave.
Suppose u has constant absolute risk aversion.
(e) Show that A = 1.
(f) Show that maximizing Eu(w) is equivalent to maximizing the concave function
m
yp'X-(f/2) Z(x'njXy».
(g) Show that the expression in (f) reduces to Freund's result (Exercise 1) when a = 2.
10. Consider Press' class of multivariate stable distributions as discussed in Ziemba's
paper. Suppose there are, pj assets whose prices follow a multivariate law with characteristic
exponent oij, j = \,...,K, and Y,Pj = n. Suppose each of these groupings is independent,
has m=l, and each fi^ is positive definite. Let x J denote the^j x 1 vector of asset allocations
of type otj, p the vector of mean returns, and suppose that short sales are allowed.
(a) Show that final wealth W has the log characteristic function
K
log^CO = itx'p-i Y^'^WSlSx 1 )"' 12 .
(b) Show that Wis not stable unless aj is the same for ally.
(c) Note that the dispersion of w is
r{x) = i i, (x'fii-W 2 .
346 PART III STATIC PORTFOLIO SELECTION MODELS