FM for Actuaries

284 CHAPTER 8
(a) Calculate the present value, Macaulay duration and convexity of the
retirement benefit entitled to Dick.
(b) Harry wants to use a $100 par value 5-year coupon bond with annual
coupon rate of 4% and a $100 par value 10-year bond with annual
coupon rate of 5% to construct an immunized portfolio. Find the face
value of each of the two bonds.
8.19 You have a choice of two bonds X and Y . Bond X is a $1,000 par value
30-year 8% semiannual coupon government bond selling at $920. Bond Y is
a $1,000 par value 20-year 10% semiannual coupon corporate bond selling
at $1,057. You believe that the yield curve for government securities will
be flat after 5 years at 7.5% compounded semiannually, while that for the
corporate bond Y will be 8.5% compounded semiannually after 5 years. If
you can re-invest coupons at 3.5% per half-year in the coming 5 years, what
bond would you choose over an investment horizon of 5 years?
8.20 Consider a portfolio of m fixed-income securities, the yield rate of each is
the same. Let w k , D ∗(k) ,andC (k) be the proportion of the value, the modified
duration, and the convexity of the kth security. Show that the modified
duration and convexity of the portfolio are
m∑
w k D ∗(k)
k=1
and
m∑
w k C (k) ,
k=1
respectively.
8.21 You have a liability of $10,000 due in 10 years and you want to construct a
portfolio using two bonds, X and Y , to meet the obligation. The following
table shows the information of X and Y :
Semiannual coupon rate 6% 0%
Face value $100 $100
Redemption value $104 $100
Time to maturity 7 years 14 years
X
Y
The prevailing yield to maturity in the market is 9% compounded semiannually
for all maturities. How much should you invest in Bonds X and Y if
you employ a target-date immunization strategy?