STOCHASTIC

The optimal first-period decision and policy are
Buy for initial portfolio
Second-period decisions:
If event I (rates increase):
Sell
Hold
Buy
If event II (no rate change):
Sell
Hold
Buy
If event III (rates fall):
Sell
Hold
Buy
1-year
maturity ($)
44,900
—
44,900
45,000
—
44,900
—
—
44,900
70,500
20-year
maturity ($)
55,100
55,100
—
—
—
55,100
200
55,100
—
—
(c) Verify that the above is an optimal policy by finding a dual optimal solution to the
problem in (b).
(d) Investigate the dual multipliers on Pt, L, and the P2e.
(e) Attempt to determine why only the extreme maturity bonds have positive weights.
[Hint: Investigate the risk-aversion properties of the program along with the structure
of returns.]
(f) Attempt to prove mathematically that such an optimal policy will always obtain.
(g) Suppose the loss constraint is deleted. Show that the optimum first-period portfolio
is to invest entirely in 20-year bonds. Indicate why intermediate-length bonds are not
chosen when the loss constraint is imposed and is binding.
(h) In additional numerical experiments, it has been found that the percentage invested
in the longest bond tends to increase with increases in the slope of the yield curve, or the
probability of lower rates in the future and decreases in the magnitude of risk aversion and
transaction costs. Explain this behavior.
10. (Interest rates and bond prices: Deterministic case) Let p, be the instantaneous
(one-period) interest at time / = 0,1,..., T. Let R(t) be the long-term interest rate over the
interval [/, T] (i.e., the effective constant one-period interest rate applying in the interval
[t, TJ). Assume that p, is known with certainty.
(a) Using arbitrage arguments in perfect capital markets, show that
[l + i?(0] T -' = fl0+/>x).
(b) In the continuous time limit, show that (a) becomes
R0) = (T-t)- l jt T p(.r)dT.
Suppose a bond of par value $1, time to maturity T, and (continuous) coupon rate r is
issued at time 1 = 0. Let p(t) be market value of the bond at time /.
(c) Show that p(t) satisfies the differential equation
?'{')-Pit) p(t) = -r, and p(T) = 1.
668 PART V DYNAMIC MODELS