FM for Actuaries

286 CHAPTER 8
8.25 Let D S be the Macaulay duration of the surplus S, which is defined by
n∑
D S = t PV(C t)
,
S
t=1
where C t is the net cash flow at time t and S ≠0. Show that
D S = D A − V L
S (D L − D A ).
Hence, show that if D A = D L then D A = D L = D S .
8.26 Corporation X has an obligation to pay $5,000 at the end of each year for 4
years. Given the following 4 non-callable and default-free bonds, construct
a dedicated bond portfolio that eliminates interest-rate risk. Determine the
face value of each of the following bonds purchased:
Bond E: 1-year bond with 4% annual coupon, redeemable at 103%,
Bond F : 2-year zero-coupon bond, redeemable at par,
Bond G: 3-year bond with 5% annual coupon, redeemable at par,
Bond H: 4-year bond with 5.5% annual coupons, redeemable at par.
8.27 The dividend discount model (DDM) for a stock assumes that the owner of
a stock can expect to receive periodic dividends. The constant-growth model
assumes that dividends grow at a constant rate g. The intrinsic value P of
the stock is the present value of all future dividends. Assume that the current
dividend is D 0 and that the yield curve is flat at i>g.
(a) Prove the Gordon formula
P = D 0(1 + g)
.
i − g
(b) Find the modified duration and the convexity of all the future dividends
in terms of i and g. [Hint: Use Exercise 8.12.]
8.28 A company has to pay $100,000 5 years from now. The current market rate
of interest is 6%. The company uses a 8.6% annual coupon bond redeemable
at par after 6 years to fund this liability.
(a) Calculate the face value and the Macaulay duration of the bond.
(b) Is the bond sufficient to meet the liability when there is a one-time
change in interest rate to 5.5% after 2 years?