STOCHASTIC

(d) Suppose n > r2. Formulate the optimization problem.
(e) Suppose rt = 0.02, r2 = 0.01, yt = 1, y2 = 2, and U = log d + log C2. Find x*.
(f) Graph the situation when U = a log Ci +/? log C2 for a, /? > 0.
(g) What must a and /(be for x* to be 0 or 1 ?
4. Suppose an investor holds security A that was purchased last year for one dollar and
has appreciated by the percentage g. Let a and 8 represent the expected percentage (net)
returns over a planning horizon for securities A and B, respectively. Suppose fj > a and that
the rate of taxation on capital gains is 0 < w < 1. Assume that the investor wishes to
maximize the expected value of terminal wealth net of capital gains taxes.
(a) Show that it is optimal to switch from A to B whenever
1+0(1-*)
(b) Interpret this condition. What happens in the special cases w = 0, w = 1, and g = 0?
(c) Suppose the brokerage charge is 0 < b < 1 dollars for switching from A to B. Show
that it is optimal to switch whenever
m> 1+a ' b{2 - b)
(\+g)(\-bY-gw{\-b) (\-b)(\-bw)
(d) Suppose g = 0.10, a = 0.15, fi = 0.20, w = 0.5, and b = 0.02. Is it optimal to switch
from A to Bl
Suppose a and P have the discrete distributions
Pr{« = «,}=/>,> 0, i=l,...,m, Prifi = fij} = qj > 0, j=\,...,n.
(e) Develop the expressions corresponding to those in (a) and (c).
(f) Is the conclusion in (d) altered if