FM for Actuaries

312 CHAPTER 10
The mean of ¨s n⌉
is
E (¨s n⌉
)
⎡
⎤
n∑ t∏
= E⎣
(1 + i n−j+1 ) ⎦
t=1 j=1
n∑ t∏
= E(1 + i n−j+1 )
=
t=1 j=1
n∑
(1 + r s ) t
t=1
= ¨s n⌉ r s
, (10.14)
where ¨s n⌉ r s
is ¨s n⌉ evaluated at the rate r s . Without providing the details of the
proof we state the variance of ¨s n⌉ as
Var (¨s ( ) ( )
) js + r s +2
2js +2
( ) 2
n⌉ = ¨s
j s − r n⌉ j s
−
¨s
s j s − r n⌉ r s
− ¨s n⌉ r s
. (10.15)
s
Next, we consider a n⌉
. Let us define r a such that
This implies
Furthermore, let
(1 + r a ) −1 =E(1+i t ) −1 = e −µ+ 1 2 σ2 .
Similar to (10.14), we can show that the mean of a n⌉ is
r a = e µ− 1 2 σ2 − 1. (10.16)
j a = e 2(µ−σ2) − 1. (10.17)
E ( a n⌉
)
= an⌉
r a
. (10.18)
Along the same argument as in (10.15), the variance of a n⌉
is
Var ( ( ) ( )
) ja + r a +2
2ra +2
( ) 2
a n⌉ = a
r a − j n⌉ j a
−
a
a r a − j n⌉ r a
− a n⌉ r a
. (10.19)
a
For ä n⌉
and s n⌉
, from (2.7) and (2.8), we have
ä n⌉ =1+a n−1⌉ ,
and
s n⌉ =1+¨s n−1⌉ .