STOCHASTIC

(c) Find the long-run average cost of the chain.
(d) Find the expected discounted total cost of the chain, using an interest rate of 25%.
Suppose that the chain starts in state 1, that you win one dollar every time it passes through
state 2 without having returned to state 1, and that the game is over upon returning to 1.
(e) What is the fair price of such a gamble?
16. It is of interest to develop conditions under which an optimal sequence of portfolio
choices is stationary. That is, in each period it is optimal to invest a constant proportion
of total wealth in each asset. Tobin (1965) developed such a result in a simplified dynamic
mean-variance framework. This problem illustrates the result and shows that the result is
not generally true.
Suppose that the investor wishes to make his total consumption during period T. Hence,
only the expected utility of wealth at time T is valued. Assume that asset returns are
statistically independent. Suppose also that expected utility is a function of the mean and
variance of wealth.
Let R,, E,, and a, be the (gross) return, expected value of return, and standard deviation
of return in period t. The efficient set of portfolios is given by a 2 = /(£,). The /"-period
variables are denoted by R, E, and a. Assume that initial wealth is one dollar. Given
reinvestment of intermediate returns R = n E,.
(a) Show that E = exp R = n E, and
o 2 + E 2 = expR 2 = ]ll>' 2 + £< 2 ] = EI [/(£) + E, 2 l
One may explore the properties of efficient sequences of portfolios by minimizing a 2 + E 2
subject to E = constant.
(b) Show that the solution to the first-order conditions is a stationary sequence with
Ei = Ei = ••• = Er.
(c) Show, however, that this sequence does not necessarily satisfy the second-order
conditions. [Hint: Let T= 2 and a, = /"(£,) = (£", — r,) 2 , where r, > 1 is a riskless rate
of interest.]
(d) Show that the second-order conditions are satisfied if a 2 = exp [A (E, _ i)] — E 2 .
*(e) Whether or not the second-order conditions are satisfied depends crucially on the
generalized convexity properties of the objective function and constraints in the problem
minimize Yl Ut (Et)+E, 2 ] EI £ < = const -
Attempt to utilize the results in Mangasarian's second paper in Part I to determine
sufficient conditions on the/, that lead to the satisfaction of the second-order conditions.
17. Suppose an investor has $100 to invest over three investment periods. Investment 3
yields a certain return of 2% per period. Investments 1 and 2 have the following distribution
in each of the three periods
pr{ri = 0.03, r2 = 0.02} = i, pr{ri = 0.02, r2 = 0.04} = i,
pr{ri = 0.01, r2 = -0.02} = ±, pr{ri = 0.12, r2 = 0.08} = }.
Suppose that the goal is to maximize the expected utility of terminal wealth, w, where
u(w) = w 112 .
(a) Formulate the problem as a three-period stochastic program.
(b) Show that the program in (a) is equivalent to a deterministic program that is concave.
(c) Solve the program in (b) by dynamic programming.
(d) Solve the program in (b) by finding a solution to the Kuhn-Tucker conditions.
18. (The Kelly criterion) A gambler makes repeated bets in sequence against an infinitely
rich adversary. At time t his wager is 0, dollars out of his total wealth W,. Suppose that
COMPUTATIONAL AND REVIEW EXERCISES 671