FM for Actuaries

310 CHAPTER 10
10.3 Independent Lognormal Model
The Independent Lognormal Model assumes that 1+i t are independently lognormally
distributed with parameters µ and σ 2 . In other words, ln(1 + i t ) follows
a normal distribution with mean µ and variance σ 2 , i.e., (see Appendix A.14)
ln(1 + i t ) ∼ N(µ, σ 2 ). (10.2)
The mean and variance of the lognormal random variable (1 + i t ) are
E(1+i t )=e µ+ 1 2 σ2 , (10.3)
and
Var(1 + i t )=
(
e 2µ+σ2)( )
e σ2 − 1 , (10.4)
respectively. 2
1
We now discuss the mean and variance of a(n),
a(n) , ¨s n⌉, a n⌉ , s n⌉ and ä n⌉
under the independent lognormal interest rate model. Note that we shall assume all
1-period spot rates to be random, including the rate for the first period i 1 . Apart
from theoretical convenience, there are also practical justifications for this assumption
(see Section 10.6 for an empirical example).
First, from (3.10), we have
Thus,
a(n) =
ln [a(n)] =
n∏
(1 + i t ).
t=1
n∑
ln(1 + i t ). (10.5)
t=1
From (10.2), we conclude that the right-hand side of (10.5) is a sum of n independent
normal random variables, each with mean µ and variance σ 2 . Therefore,
ln [a(n)] is normal with mean nµ and variance nσ 2 . This implies a(n) is lognormal
with parameters nµ and nσ 2 . Using (10.3) and (10.4), we have
E[a(n)] = e nµ+ n 2 σ2 , (10.6)
and
Var [a(n)] =
(
e 2nµ+nσ2)( )
e nσ2 − 1 . (10.7)
2 Other statistical properties of the lognormal model can be found in Klugman, S.A., Panjer, H.H.
and Willmot, G.E., Loss Models: From Data to Decisions, 4th Edition, John Wiley & Sons, 2012.