224 Mathematics for Finance
If the duration of the bond is exactlyt,then
d
dy(P(y)ety)=0.If the derivative is zero at some point, then the graph of the function is ‘flat’
near this point. This means that small changes of the rate will have little effect
on the future value of the investment.
10.1.3 Portfolios of Bonds .................................
If a bond of desirable duration is not available, it may be possible to create a
synthetic one by investing in a suitable portfolio of bonds of different durations.
Example 10.5
If the initial interest rate is 14%, then a 4-year bond with annual coupons
C= 10 and face valueF= 100 has duration 3.44 years. A zero-coupon bond
withF= 100 andN= 1 has duration 1. A portfolio consisting of two bonds,
one of each kind, can be regarded as a single bond with couponsC 1 = 110,
C 2 =C 3 =C 4 = 10,F= 100.Its duration can be computed using the general
formula (10.2), which gives 2.21 years.
We shall derive a formula for the duration of a portfolio in terms of the
durations of its components. Denote byPA(y)andPB(y) the values of two
bondsAandBwith durationsDA(y)andDB(y). Take a portfolio consisting
ofabondsAandbbondsB,itsvaluebeingaPA(y)+bPB(y). The task of
finding the duration of the portfolio will be divided into two steps:
- Find the duration of a portfolio consisting ofabonds of typeA.Weshall
writeaAto denote such a portfolio. Its price is obviouslyaPA(y). Since
d
dy(aPA(y)) =−DA(y)(aPA(y)),it follows that
DaA(y)=DA(y).
This is clear if we examine the cash flow ofaA. Each coupon and the face value
are multiplied bya,which cancels out in the computation of duration directly
from (10.2).
- Find the duration of a portfolio consisting of one bondAand one bondB,
which will be denoted byA+B. The price of this portfolio isPA(y)+PB(y).