FM for Actuaries

Annuities 57
Figure 2.8:
Increasing annuity-immediate
Cash flow
P P + D P +2D P +(n − 1)D
······
Time
0 1 2 3 ······
n
We can see that the annuity can be regarded as the sum of the following annuities:
(a) an n-period annuity-immediate with constant amount P ,and(b)n − 1
deferred annuities, where the jth deferred annuity is a (n − j)-period annuityimmediate
with level amount D to start at time j, forj =1, ··· ,n− 1. Thus, the
present value of the varying annuity is
⎡ ⎤
⎡
⎤
n−1
∑
n−1
∑
Pa n⌉
+ D ⎣ v j a ⎦ n−j⌉
= Pa n⌉
+ D ⎣ v j (1 − vn−j )
⎦
i
j=1
j=1
⎡( ∑n−1
)
⎤
j=1
= Pa n⌉ + D ⎣
vj − (n − 1)v n
⎦
i
⎡( ∑n
) ⎤
j=1
= Pa n⌉ + D ⎣
vj − nv n
⎦
i
[
an⌉ − nv n ]
= Pa n⌉
+ D
. (2.35)
i
For an n-period increasing annuity with P = D =1, we denote its present and
future values by (Ia) n⌉
and (Is) n⌉
, respectively. Readers are invited to show that
(Ia) n⌉ = än⌉ − nv n
(2.36)
i
and
(Is) n⌉
= s n+1⌉ − (n +1)
= ¨s n⌉ − n
. (2.37)
i
i
For an increasing n-payment annuity-due with payments of 1, 2, ··· ,nat time
0, 1, ··· ,n− 1, the present value of the annuity is
(Iä) n⌉ =(1+i)(Ia) n⌉ . (2.38)