FM for Actuaries

48 CHAPTER 2
Figure 2.4: Illustration of equation (2.14)
Cash flow
1 1 1 1 1
··· ···
Time
0 1 2 ··· m m +1 ··· m + n
Future value
of annuity
s m+n⌉
s n⌉
s m⌉
(1 + i) n s m⌉
for arbitrary positive integers m and n. This equation expresses the value at time n
of a (m+n)-period annuity-immediate starting at time 0 in two different ways. The
left-hand side discounts the future value at time m + n backwards by m periods,
while the right-hand side brings the present value at time 0 forward by n periods.
Figure 2.5 illustrates (2.16).
Figure 2.5: Illustration of equation (2.16)
Cash flow
1 1 1 1 1
··· ···
Time
0 1 2 ··· n n +1 ··· m + n
Annuity value
at time n
a m+n⌉
(1 + i) n a m+n⌉
s m+n⌉
v m s m+n⌉