Exercise 5.8.3 Empirically, long-term rates change less than short-term ones. To incorporate this fact into duration, we may postulate nonproportional shifts as

[ 1+ S(i) ]

1+ S(i)

= Ki−1[ 1+ S(1) ]

1+ S(1)

for some K < 1.

Show that a t-period zero-coupon bond’s price sensitivity satisfies P

P

=−tKt−1[ 1+ S(1) ]

1+ S(1)

under nonproportional shifts.