222 Mathematics for Finance
hand, we do not gain in other circumstances. This is explained by the fact,
that a certain parameter of the bond, called duration and defined below, is
exactly equal to the lifetime of our investment. In some sense, the bond behaves
approximately like a zero-coupon bond with prescribed maturity.
Exercise 10.6
Check the numbers given in the above tables.Exercise 10.7
Compute the value after three years of $1,000 invested in a 4-year bond
with $32 annual coupons and $100 face value if the rates in consecutive
years are as follows:
Scenario 1: 12%,11%,12%,12%;
Scenario 2: 12%,13%,12%,12%.
Design a spreadsheet and experiment with various interest rates.10.1.2 Duration
We have seen that variable interest leads to uncertainty as to the future value
of an investment in bonds. This may be undesirable, or even unacceptable, for
example for a pension fund manager. We shall introduce a tool which makes
it possible to immunise such an investment, at least in the special situation of
maturity-independent rates considered in this section.
For notational simplicity we denote the current yieldy(0) byy.Consider a
coupon bond with couponsC 1 ,C 2 ,...,CNpayable at times 0<τn 1 <τn 2 <
... < τnNand face valueF, maturing at timeτnN. Its current price is given
by
P(y)=C 1 e−τn^1 y+C 2 e−τn^2 y+···+(CN+F)e−τnNy. (10.1)
Thedurationof the coupon bond is defined to be
D(y)=τn 1 C 1 e−τn^1 y+τn 2 C 2 e−τn^2 y+···+τnN(CN+F)e−τnNy
P(y). (10.2)
The numbersC 1 e−τn^1 y/P(y),C 2 e−τn^2 y/P(y),...,(CN+F)e−τnNy/P(y)are
non-negative and add up to one, so they may be regarded as weights or proba-
bilities. It can be said that the duration is a weighted average of future payment
times. The duration of any future cash flow can be defined in a similar manner.