FM for Actuaries

Stochastic Interest Rates 311
Next, we examine the distribution of (1 + i t ) −1 under the lognormal model.
Note that
ln [ (1 + i t ) −1] = − ln(1 + i t ).
Thus, from (10.2), we have
ln [ (1 + i t ) −1] ∼ N(−µ, σ 2 ), (10.8)
and (1 + i t ) −1 follows a lognormal distribution with parameters −µ and σ 2 .From
(3.11), we have
1
n
a(n) = ∏
(1 + i t ) −1 .
t=1
Using a similar argument as in (10.6) and (10.7), we obtain
[ ] 1
E = e −nµ+ n 2 σ2 , (10.9)
a(n)
and
[ ] 1
(
Var = e −2nµ+nσ2)( )
e nσ2 − 1 . (10.10)
a(n)
The statistical properties of annuities (say, ¨s n⌉ , a n⌉ , s n⌉ and ä n⌉ ) under the
lognormal interest rate assumption are fairly complex. We shall present and discuss
some applications of their mean and variance formulas without going through the
proof. 3
We begin with ¨s n⌉
. Let us define r s such that
1+r s =E(1+i t )=e µ+ 1 2 σ2 .
This implies
Furthermore, let
where
v 2 s =Var(1+i t )=
r s = e µ+ 1 2 σ2 − 1. (10.11)
j s =2r s + r 2 s + v 2 s, (10.12)
(
e 2µ+σ2)( )
e σ2 − 1 . (10.13)
3 For the theoretical details, interested readers may refer to Kellison, S.G., The Theory of Interest,
3rd Edition, McGraw-Hill, 2008.