STOCHASTIC

4. Referring to the article by Wilson consider the cash flow pattern for the initial policy
of [d0, {d,}, {dij},...], and the cash flow pattern for an investment project of [c0, {c,}, {cu},...].
(a) Show that if
«I c0 + rf0, S Pi(c, + d,), £PtAc,j + d,j),...\
^ Z PtPuPtjk — uidoydt.dtj,...),
l.J.k,...
or in a more compact notation, u{E(C+D)} £ Eu(D), then exclude the project.
(b) Show that if
then include the project.
X PtPvPtjk"-u(co + do, Ci+di, c,j+d,j,...)
J,k, ...
^ u(d0,J^Pidt> "ZPtjdij, J^Pijkd, Jk,...\
\ l U ijk J
5. Consider an expected utility maximizer with logarithmic utility function for terminal
wealth, u(w) = log w. Assume that he can invest in a risk-free asset returning r £ 1 and a
risky asset returning p S 0 per dollar invested. Suppose that p is log-normally distributed
(i.e., log/? is normal), with Ep = p., varp = a 2 . Recall from Exercise II-CR-18 that if w0 is
initial wealth and a e [O, w0] is the amount invested in the risky asset, then for a 1-period
problem, a = w0 is optimal if p. ^ ril + a 1 ^ 2 ). In an n-period problem, suppose that the
return r on money can vary between periods but always satisfies the inequality above.
(a) Show that the optimal policy is to invest totally in the risky asset in each period.
(b) Calculate the probability distribution of terminal wealth under the optimal policy.
W„ _ W„ W„-i ^ Wi
W0 W„-i W„_2 Wo'
(c) Calculate the expected value and most probable value of terminal wealth w„.
(d) Let p = 1.10, CT = 0.05, and let r = 1.065 for all periods. Evaluate numerically the
expected and most probable wealth ratios H>10/WO for a 10-period problem. Evaluate the
probability that the optimal investment performs better than money, that is,
Pr[w10 § r i0 w»].
6. Referring to Exercise 5, suppose that the investor's utility function for terminal wealth
is u{w) = W, a s (0,1). Suppose that
(a) Show that the optimal 1-period policy is to invest totally in the risky asset.
(b) Show that the optimal n-period policy is to invest totally in the risky asset in each
period.
Exercise Source Notes
Exercises 2 and 3 were adapted from Mossin (1968c), and Exercise A is due to Professor
S. L. Brumelle.
COMPUTATIONAL AND REVIEW EXERCISES
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