- Variable Interest Rates 225
Differentiating the last expression, we obtain
d
dy(PA(y)+PB(y)) =d
dyPA(y)+d
dyPB(y)
=−DA(y)PA(y)−DB(y)PB(y).The last term can be written as−DA+B(y)(PA(y)+PB(y)) if we put
DA+B(y)=DA(y) PA(y)
PA(y)+PB(y)+DB(y) PB(y)
PA(y)+PB(y).
This means thatDA+B(y) is a linear combination ofDA(y)andDB(y), the
coefficients being the percentage weights of each bond in the portfolio.
From the above considerations we obtain the general formula
DaA+bB(y)=DA(y)wA+DB(y)wB,where
wA=aPA(y)
aPA(y)+bPB(y),wB=bPA(y)
aPA(y)+bPB(y),
are the percentage weights of individual bonds.
If we allow negative values ofaorb(which corresponds to writing a bond
instead of purchasing it, in other words, to borrowing money instead of invest-
ing), then, given two durationsDA=DB, the durationDof the portfolio can
takeanyvaluebecausewB=1−wAand
D=DAwA+DB(1−wA)=DB+wA(DA−DB).The value ofDcan even be negative, which corresponds to a negative cash
flow, that is, sums of money to be paid rather than received.
Example 10.6
LetDA=1andDB =3.Wewishtoinvest$1,000 for 6 months. For the
duration to match the lifetime of the investment we need 0.5=wA+3wB.Since
wA+wB= 1, it follows thatwB=− 0 .25 andwA=1.25. WithPA=0. 92
dollars andPB=1.01 dollars, we invest $1,250 in^12500. 92 ∼= 1 , 358 .70 bondsA
and we issue 1250. 01 ∼= 247 .52 bondsB.
Exercise 10.11
Find the number of bonds of typeAandBto be bought ifDA=2,
DB=3.4,PA=0.98,PB=1.02 and you need a portfolio worth $5, 000
with duration 6.