FM for Actuaries

46 CHAPTER 2
perpetuity, we note that, as v<1, v n → 0 when n →∞. Thus, from (2.1), we
conclude that the present value of a perpetuity of payments of 1 unit when the first
payment is made one period later, is
a ∞⌉ = 1 i . (2.9)
For the case when the first payment is made immediately, we have, from (2.3),
ä ∞⌉
= 1 d . (2.10)
A deferred annuity is one for which the first payment starts some time in the
future. Consider an annuity with n unit payments for which the first payment is
due at time m +1. This can be regarded as an n-period annuity-immediate to start
at time m, and its present value is denoted by m| a n⌉ i (or m|a n⌉
for short). Thus, we
have
m|a n⌉ = v m a n⌉
[ ] 1 − v
= v m n
×
i
= vm − v m+n
i
= (1 − vm+n ) − (1 − v m )
i
= a m+n⌉ − a m⌉ . (2.11)
To understand the above equation, note that the deferred annuity can be regarded
as a (m + n)-period annuity-immediate with the first m payments removed.
It can be seen that the right-hand side of the last line of (2.11) is the present value
of a (m + n)-period annuity-immediate minus the present value of a m-period
annuity-immediate. Figure 2.3 illustrates (2.11).
From (2.11), we have the following results (note that the roles of m and n can
be interchanged)
a m+n⌉
= a m⌉
+ v m a n⌉
= a n⌉ + v n a m⌉ . (2.12)
Multiplying the above equations throughout by 1+i,wehave
ä m+n⌉
= ä m⌉
+ v m ä n⌉
= ä n⌉ + v n ä m⌉ . (2.13)