Advances in Risk Management

MANUEL MORENO 73

Since this bond can be interpreted as a portfolio ofndiscount bonds, we get:

P∗(t,T)=

∑n

i= 1

ciP(t,ti) (4.14)

From (4.12), it is verified that the instantaneous percentage change in the
price of this coupon bond is given by:

dP∗(t,T)

P∗(t,T)

=

1

P∗(t,T)

∑n

i= 1

ciμP(.)dt−Dsσ 1 dw 1 −DLσ 2 dw 2 (4.15)

where

Ds=

1

P∗(t,T)

∑n

i= 1

ci(ti−t)P(t,ti)

∂Y(t,ti)

∂s

(4.16)

DL=

1

P∗(t,T)

∑n

i= 1

ci(ti−t)P(t,ti)

∂Y(t,ti)

∂L

The valuesDsandDLrepresent the generalized duration measures and
reflect the sensitivity of the bond yield to changes in the factorssand
L. In comparison with the conventional duration, we have two duration
measures, one for each factor. Moreover, there is an additional term:

∂Y(t,ti)

∂s

,

∂Y(t,ti)

∂L

, i=1, 2,...,n (4.17)

that reflects the sensitivity of the yield to maturity to changes in each factor.
Formally, we establish the following definition:

Definition (generalized duration) The “generalized durations”Dsand

DLof a bond that paysncouponsciat timesti,i=1, 2,...,nwith respect

to the factorssandLare given by the expressions:

Ds=

1

P∗(t,T)

∑n

i= 1

ci(ti−t)P(t,ti)

∂Y(t,ti)

∂s

(4.18)

DL=

1

P∗(t,T)

∑n

i= 1

ci(ti−t)P(t,ti)

∂Y(t,ti)

∂L

whereP(t,ti) is the price, at timet, of a zero-coupon bond that matures at

timeti(see Proposition 1).

It is easily shown that these measures, for a zero-coupon bond, become:

Ds=B(t,T)=B(τ)

(4.19)

DL=C(t,T)=C(τ)

where we have used equations (4.8) and (4.11).