FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 81
From (3.5) we see that the future value at time t of a unit payment at time 0 can
also be written as
a(t) =(1+i S t )t =
t∏
(1 + i F j )=(1+iF 1 )(1 + iF 2 ) ···(1 + iF t ). (3.10)
j=1
Thus, the present value of a unit payment due at time t is
1
a(t) = 1
∏ t
j=1 (1 + (3.11)
iF j
),
and the present value of a n-period unit-payment annuity-immediate can also be
written as
a n⌉ =
=
=
n∑
t=1
n∑
t=1
1
a(t)
1
∏ t
j=1 (1 + iF j )
1
(1 + i F 1 ) + 1
(1 + i F 1 )(1 + iF 2 ) + ···+ 1
(1 + i F 1 ) ···(1 + (3.12)
iF n ).
While the calculation of the present values remains straightforward when interest
rates are varying deterministically, the computation of the future value at time
n of a payment due at time t, where0 <t<n, requires additional assumptions.
We now consider the assumption that a payment occurring in the future earns the
forward rates of interest implicitly determined in (3.4) and (3.5). Therefore, the
future value at time n of a unit payment due at time t is
(1 + i F t,n−t) n−t =(1+i F t+1) ···(1 + i F n ). (3.13)
Hence, the future value at time n of a n-period annuity-immediate is
s n⌉ =
=
[ n−1
] ∑
(1 + i F t,n−t) n−t +1
t=1
⎡ ⎛
⎞⎤
n−1
∑ n∏
⎣ ⎝ (1 + i F j ) ⎠⎦ +1
t=1
j=t+1
=[(1+i F 2 )(1 + iF 3 ) ···(1 + iF n )] + [(1 + iF 3 )(1 + iF 4 ) ···(1 + iF n )] + ···
···+(1+i F n )+1. (3.14)