226 Mathematics for Finance
Exercise 10.12
Invest $1,000 in a portfolio of bonds with duration 2 using 1-year zero-
coupon bonds with $100 face value and 4-year bonds with $15 annual
coupons and $100 face value that trade at $102.A portfolio with duration matching the investment lifetime is insensitive to
small changes of interest rates. However in practice we shall have to modify the
portfolio if, for example, the investment is for 3 years and one of the bonds is
a zero-coupon bond expiring after one year. In addition, the duration may, as
we shall see below, go off the target. As a result, it will become necessary to
update the portfolio during the lifetime of the investment. This is the subject
of the next subsection.
10.1.4 Dynamic Hedging
Even if a portfolio is selected with duration matching the desired investment
lifetime, this will only be valid at the initial instant, since duration changes
with time as well as with the interest rate.
Example 10.7
Take a 5-year bond with $10 annual coupons and $100 face value. Ify= 10%,
then the duration will be about 4.16 years. Before the first coupon is paid the
duration decreases in line with time: After 6 months it will be 3.66, and after
9 months 4. 16 − 0 .75 = 3. 31 .If the duration matches our investment’s lifetime
and the interest rates do not change, no action will be necessary until a coupon
becomes payable. As soon as the first coupon is paid after one year, the bond
will become a 4-year one with duration 3.48, no longer consistent with the
investment lifetime.
Exercise 10.13
Assuming that the interest rate does not change, show that before the
first coupon is paid the duration after timetwillD−t,whereDis the
duration computed at time 0.The next example shows the impact of the interest rate on duration.