108156.pdf

10. Variable Interest Rates 223

Duration measures the sensitivity of the bond price to changes in the interest
rate. To see this we compute the derivative of the bond price with respect toy,

d

dy

P(y)=−τn 1 C 1 e−τn^1 y−τn 2 C 2 e−τn^2 y−···−τnN(CN+F)e−τnNy,

which gives
d
dy

P(y)=−D(y)P(y).

The last formula is sometimes taken as the definition of duration.

Example 10.4

A 6-year bond with $10 annual coupons, $100 face value and yield of 6% has a
duration of 4.898 years. A 6-year bond with the same coupons and yield, but
with $500 face value, will have a duration of 5.671 years. The duration of any
zero-coupon bond is equal to its lifetime.

Exercise 10.8

A 2-year bond with $100 face value pays a $6 coupon each quarter and

has 11% yield. Compute the duration.

Exercise 10.9

What should be the face value of a 5-year bond with 10% yield, paying

$10 annual coupons to have duration 4? Find the range of durations that

can be obtained by altering the face value, as long as a coupon cannot

exceed the face value. If the face value is fixed, say $100, find the level of

coupons for the duration to be 4. What durations can be manufactured

in this way?

Exercise 10.10

Show thatPis a convex function ofy.

If we invest in a bond with the intention to close the investment at timet,
then the future value of the money invested in a single bond will beP(y)ety,
provided that the interest rate remains unchanged (being equal to the initial
yieldy(0)). To see how sensitive this amount is to interest rate changes compute
the derivative with respect toy,

d

dy

(P(y)ety)=

(

d

dy

P(y)

)

ety+tP(y)ety=(t−D(y))P(y)ety.