FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 83
Example 3.5: Suppose the 1-period forward rates of interest for investments due
at time 0, 1, 2 and 3 are, respectively, 4%, 4.8%, 4.8% and 5.2%. Calculate a 4⌉ and
s 4⌉ .
Solution: From (3.12) we have
a 4⌉ = 1
1.04 + 1
1.04 × 1.048 + 1
1.04 × 1.048 × 1.048
1
+
1.04 × 1.048 × 1.048 × 1.052
=3.5867.
As a(4) = 1.04 × 1.048 × 1.048 × 1.052 = 1.2016, we have
Alternatively, from (3.14) we have
s 4⌉ =1.2016 × 3.5867 = 4.3099.
s 4⌉ =1+1.052 + 1.048 × 1.052 + 1.048 × 1.048 × 1.052
=4.3099.
To further understand (3.17), we write (3.13) as (see (3.8))
(1 + i F t,n−t) n−t = (1 + iS n )n a(n)
(1 + i S =
t )t a(t) . (3.19)
Thus, the future value of the annuity is, from (3.14) and (3.19),
Similarly, for annuity-due we have
[ n−1
] ∑
s n⌉ = (1 + i F t,n−t) n−t +1
t=1
n∑ a(n)
=
a(t)
t=1
n∑ 1
= a(n)
a(t)
t=1
= a(n)a n⌉
. (3.20)
¨s n⌉ = a(n)ä n⌉ . (3.21)