FM for Actuaries

288 CHAPTER 8
can also use your savings in your bank account, which offers you no interest
because the principal is too small. Assume that the effective rate of interest
is 8.5%.
(a) You plan to construct a fund consisting of the zero-coupon bond and
bank deposit. At the time of the construction of the fund the surplus of
you position is zero. How much should you invest in the zero-coupon
bond to minimize the interest-rate risk?
(b) Calculate the present value of your portfolio if the effective rate of interest
goes up or down by 50 basis points immediately after you start
up the portfolio and compare it with the present value of the liability.
8.36 Assume that at yield rate i, the cash-flow stream of the asset is structured in
a way such that V A (i) =V L (i) =V . Show that for small ∆i,
S(i +∆i) ≈ V
[−(1 + i) −1 (D A − D L )∆i + 1 2 (C A − C L )(∆i) 2 ]
,
where the durations and convexities are all calculated based on a yield to
maturity of i. Hence, comment on the outcome of the duration-matching
strategy. [Hint: Study the derivation of equation (8.12) in Section 8.3.]
8.37 Bond A is a 3-year annual coupon bond with coupon rate of 4%. Bond B
is a 5-year annual coupon bond with coupon rate of 8%. You are given the
current spot-rate curve i S t as follows:
t 1 2 3 4 5
i S t 6.1% 6.2% 6.3% 6.4% 6.5%
(a) Compute the Fisher-Weil duration of the two bonds, as well as the price
sensitivity measure in equation (8.35).
(b) Compute the prices of the bonds under the following term structure
(with the years to maturity of the bonds remaining unchanged). Comment
on the use of equation (8.35) for the price changes of these two
bonds.
t 1 2 3 4 5
i S t 6.8% 6.9% 7.0% 7.1% 7.2%