FM for Actuaries

Annuities 41
Figure 2.1 illustrates the time diagram of an annuity-immediate of payments
of 1 unit at the end of each period for n periods. As the payments occur at different
times, their time values are different. We are interested in the value of the annuity
at time 0, called the present value, and the accumulated value of the annuity at time
n, called the future value.
Figure 2.1:
Time diagram for an n-payment annuity-immediate
Cash flow
1 1 1 1 1
······
Time
0 1 2 3 ······
n − 1 n
Suppose the rate of interest per period is i, and we assume the compoundinterest
method applies. Let a n⌉
denote the present value of the annuity, which is
i
sometimes denoted as a n⌉
when the rate of interest is understood. As the present
value of the jth payment is v j ,wherev = 1
1+i
is the discount factor, the present
value of the annuity is (see Appendix A.5 for the sum of a geometric progression)
a n⌉ = v + v 2 + v 3 + ···+ v n
[ 1 − v
n
]
= v ×
1 − v
= 1 − vn
i
1 − (1 + i)−n
= . (2.1)
i
The accumulated value of the annuity at time n is denoted by s n⌉ i or s n⌉. This
is the future value of a n⌉ at time n. Thus, we have
s n⌉ = a n⌉ × (1 + i) n
= (1 + i)n − 1
. (2.2)
i
s n⌉ will be referred to as the future value of the annuity. If the annuity is of
level payments of P , the present and future values of the annuity are Pa n⌉ and
Ps n⌉ , respectively.