FM for Actuaries

54 CHAPTER 2
Figure 2.7: Payments more frequent than interest conversion (m =4)
Cash flow
1 1 1 1
···
Interest-conversion
period
0 1 2 3 4 ··· n − 2 n − 1 n
Payment perioid
(k =2)
k 2k ··· (m − 1)k mk
The above equation parallels (2.22), which can also be written as
a (m)
n⌉i
= i
r (m) a n⌉.
If the mn-payment annuity is due at time 0, 1 m , 2 m , ··· ,n− 1 m
, we denote its
present value at time 0 by ä (m) , which is given by
n⌉
ä (m)
n⌉
=(1+i) 1 m a
(m)
n⌉
Thus, from (1.22) we conclude
ä (m)
n⌉
The future value of this annuity at time n is
¨s (m)
n⌉
For deferred annuities, the following results apply
[
=(1+i) 1 1 − v
n
]
m ×
r (m) . (2.25)
= 1 − vn
d (m) = d
d (m) än⌉. (2.26)
=(1+i) n ä (m)
n⌉
= d
d (m) ¨s n⌉. (2.27)
and
q|a (m)
n⌉
q|ä (m)
n⌉
= v q a (m)
n⌉ , (2.28)
= v q ä (m)
n⌉ . (2.29)