FM for Actuaries

44 CHAPTER 2
Figure 2.2:
Time diagram for an n-payment annuity-due
Cash flow
1 1 1 1 1
······
Time
0 1 2 3 ······
n − 1 n
Note that the first payment is made at time 0, and the last payment is made at
time n − 1. We denote the present value of the annuity-due at time 0 by ä n⌉ i (or
ä n⌉
if the rate of interest i per payment period is understood), and the future value
of the annuity at time n by ¨s n⌉ i (or ¨s n⌉ if the rate of interest i per payment period
is understood).
The formula for ä n⌉
can be derived as follows
Also, we have
ä n⌉ = 1+v + ···+ v n−1
= 1 − vn
1 − v
= 1 − vn
. (2.3)
d
¨s n⌉ = ä n⌉ × (1 + i) n
= (1 + i)n − 1
. (2.4)
d
As each payment in an annuity-due is paid one period ahead of the corresponding
payment of an annuity-immediate, the present value of each payment in an
annuity-due is (1 + i) times the present value of the corresponding payment in an
annuity-immediate. Thus, we conclude
ä n⌉
=(1+i) a n⌉
(2.5)
and, similarly,
¨s n⌉ =(1+i) s n⌉ . (2.6)
Equation (2.5) can also be derived from (2.1) and (2.3). Likewise, (2.6) can be
derived from (2.2) and (2.4).