STOCHASTIC

(e) Show that the optimal policy is to borrow $56,000 and to invest $57,000 in the risky
asset.
(f) What is the optimal policy if w0 = $5000 ?
23. For the investor of Exercise 22 having w0 = $1000 initially, find the optimal policy
under the following conditions.
(a) Borrowing is limited to $1000.
(b) Borrowing is unlimited provided there is at least a 0.99 probability of repaying the
loan plus interest.
(c) Borrowing is unlimited provided there is at least a 0.99 probability of repaying the
loan, excluding interest.
(d) The loan, excluding interest, must be repaid with probability 1.
24. A risky investment returns (1 + Xi) > 1 with probability n > 0 and (1+ X2) < 1 with
probability l-n. Let X =nXl+(\-n)X2> 0 and a 2 > 0 be the variance of X. An
investor saves part of his wealth w0 > 0 and invests the balance, say a, in the risky investment.
Let u be the investor's utility functions and suppose that u' > 0 and «" < 0.
(a) Show that the optimal investment, say a*, is the solution of
(b) Suppose u(w) = In w. Show that
Xi nuXwo+Xia) = — Xi (1 - 7r) U'(W0+X2 a).
w0X n J , da*
> 0 and that dw0
-— > 0.
Xi X2
(c) Suppose u(w) = w— w 2 /2a for a & w0. Show that
(a—w0)X _ , . da*
a = -TTrh > ° and that — < 0.
(d) Suppose u{W) = 1 -e~ Xw for A > 0. Show that
1 nXi da*
In , —— > 0 and that — = 0.
Xi.Xi-Xi) {n-\)X2 dw0
(e) Let R(w) denote the investor's risk-aversion function. Show that
da*
— {i0 if R'{g0.
dw0
25. Show that the model and hence the results in Exercise 24 apply to the following three
investment situations.
(a) An investor has initial wealth G > 0. Wealth saved provides a net return of r > 0.
A risky investment has net return of Yi (>r) and Y2 ( n > 0
and 1 -n. [Hint: Let w0 = (1 + r)G, Xl = Y1-r, and X2 = Y2-r.]
(b) An investor has b and L dollars invested in cash and a risky asset, respectively.
With probability 0 < 1— n < 1 the risky asset is worthless; otherwise it has value L.
The investor can insure against loss of any portion y of the risky asset at a cost of py,
where 0