FM for Actuaries

Bond Management 289
(c) Calculate the new Fisher-Weil duration of the two bonds immediately
after the spot-rate curve shift in (b). Compare this with your results in
(a).
Advanced Problems
8.38 Find the Macaulay duration and convexity of the following cash-flow stream
using a yield rate of 5%:
⎧
⎨ 1 at t =1, 4, 7, ··· ,
C t = 2 at t =2, 5, 8, ··· ,
⎩
3 at t =3, 6, 9, ··· .
8.39 You have to pay $1,000 at the end of year 1, 2 and 4 from now. The only
investments available to you are 1-, 3-, and 4-year zero-coupon bonds.
(a) Can a dedicated bond portfolio be constructed using the three zerocoupon
bonds?
(b) Assuming that the yields to maturity of all bonds are 8%, find the cheapest
bond portfolio that can immunize interest-rate risk according to the
duration matching strategy in Section 8.5. Calculate the face values
of the three zero-coupon bonds in this portfolio. [Hint: There are infinitely
many immunization strategies which have the same price at 8%.
You are only asked to find one.]
8.40 In this exercise we use a probabilistic argument to show Rule 3 in Section 8.4
that Macaulay duration decreases when i increases. Recall that (8.3) states
that D is a weighted average of 1, 2, ··· ,n using weights w 1 ,w 2 , ··· ,w n .
Since w t ≥ 0 for all t and they sum up to 1, we can treat them as probabilities.
(a) Let T be a random variable taking values in 1, 2, ··· ,nwith probabilities
w 1 ,w 2 , ··· ,w n .FindE(T ), the expectation of T .
(b) Show that the variance of T is Var(T )= ∑ n
t=1 t2 w t − D 2 . In financial
economics, Var(T ) is called the M 2 of the cash-flow stream C t .What
does M 2 measure?
(c) Show that
dD
di = −vVar(T )
and complete the proof of Rule 3.