FM for Actuaries

230 CHAPTER 7
It is assumed that the force of interest δ(t) follows a step function taking constant
values between successive maturities, i.e., δ(t) =δ i for t i−1 ≤ t<t i , i =1, ··· , 4
with t 0 =0. Estimate δ(t) and compute the spot-rate curve for maturity up to time
t 4 .
Solution: As Bond 1 is a zero-coupon bond, its equation of value is
P 1 =exp [ ∫
−t 1 i S ] t1
]
t 1 =exp
[− δ(u) du =exp[−δ 1 t 1 ] ,
0
so that
δ 1 = − 1 t 1
ln P 1 ,
which applies to the interval (0,t 1 ). Now we turn to Bond 2 (which is again a
zero-coupon bond) and write down its equation of value as
[ ∫ t2
]
P 2 =exp − δ(u) du =exp[−δ 1 t 1 − δ 2 (t 2 − t 1 )] .
0
Solving the above equation for δ 2 , we obtain
δ 2 = − 1 [ln P 2 + δ 1 t 1 ]
t 2 − t 1
= − 1 [ ]
P2
ln .
t 2 − t 1 P 1
For Bond 3, there is a coupon payment of amount C 3 at time t ∗ 31 , with t 1 <t ∗ 31 <
t 2 . Thus, the equation of value for Bond 3 is
[ ∫ t3
] [ ∫ ]
t ∗
31
P 3 = (1+C 3 )exp − δ(u) du + C 3 exp − δ(u) du
0
0
= (1+C 3 )exp[−δ 1 t 1 − δ 2 (t 2 − t 1 ) − δ 3 (t 3 − t 2 )]
+ C 3 exp [−δ 1 t 1 − δ 2 (t ∗ 31 − t 1 )] ,
from which we obtain
δ 3 = − 1 [
P3 − C 3 exp [−δ 1 t 1 − δ 2 (t ∗ 31
ln
− t ]
1)]
t 3 − t 2 (1 + C 3 )P 2
= − 1 [
P3 − P 1 C 3 exp [−δ 2 (t ∗ 31
ln
− t ]
1)]
.
t 3 − t 2 (1 + C 3 )P 2