FM for Actuaries

Annuities 49
Also, it can be checked that the following result holds:
v m ¨s m+n⌉ =(1+i) n ä m+n⌉ .
2.4 Annuities under Other Accumulation Methods
We have so far discussed the calculations of the present and future values of annuities
assuming compound interest. In this section we shall extend our discussion
to other interest-accumulation methods. We shall consider a general accumulation
function a(·) and assume that the function applies to any cash-flow transactions
in the future. Thus, as stated in Section 1.7, any payment at time t>0 starts to
accumulate interest according to a(·) as a payment made at time 0.
Given the accumulation function a(·), the present value of a unit payment due
1
at time t is
a(t)
. Thus, the present value of a n-period annuity-immediate of unit
payments is
n∑ 1
a n⌉
=
a(t) . (2.17)
t=1
Likewise, the future value at time n of a unit payment at time t<nis a(n − t).
Thus, the future value of a n-period annuity-immediate of unit payments is
s n⌉ =
n∑
a(n − t). (2.18)
t=1
If (1.35) is satisfied so that a(n − t) = a(n)
a(t)
s n⌉ =
n∑
t=1
for n>t>0,then
a(n)
n∑
a(t) = a(n) 1
a(t) = a(n) a n⌉. (2.19)
This result is satisfied for the compound-interest method, but not the simple-interest
method or other accumulation schemes for which equation (1.35) does not hold.
Example 2.7: Suppose δ(t) =0.02t for 0 ≤ t ≤ 5, finda 5⌉
and s 5⌉
.
Solution:
Hence, from (2.17),
t=1
We first calculate a(t), which, from (1.26), is
(∫ t
)
a(t) = exp 0.02s ds
0
= exp(0.01t 2 ).
a 5⌉ = 1
e 0.01 + 1
e 0.04 + 1
e 0.09 + 1
e 0.16 + 1 =4.4957,
e0.25