FM for Actuaries

Stochastic Interest Rates 323
10.13 Suggest a reason why it is not appropriate to use an independent normal
model for forward interest rates.
10.14 It is given that 1+i t follows an independent lognormal distribution with
µ =0.06 and σ 2 =0.03 for t =1, 2, ···.
(a) Find E(i t ).
[ 1
(b) Find E
a(5)
]
,and
1
a(5) calculated at rate E(i t). Are they the same?
10.15 It is given that 1+i t follow independent lognormal distributions with µ =
0.04 and σ 2 =0.02 for t =1, 2, ···. Find the mean and variance of
(a) the accumulated value of $1 at t =4,
(b) the present value of $1 payable at t =4.
10.16 It is given that 1+i t follow lognormal distributions with µ = 0.04 and
σ 2 =0.02 for t =1, 2 and 3, and with µ =0.03 and σ 2 =0.03 for t =4
and 5. Furthermore, i t are independent. Find the mean and variance of
(a) the accumulated value of $1 at t =5,
(b) the present value of $1 payable at t =5.
10.17 Given that 1+i t follow independent lognormal distributions with µ =0.04
and σ 2 =0.02 for t =1, 2, ···, find the mean and variance of
(a) a 7⌉
,
(b) ä 7⌉
.
10.18 Given that 1+i t follow independent lognormal distributions with µ =0.01
and σ 2 =0.03 for t =1, 2, ···, find the mean and variance of
(a) s 7⌉ ,
(b) ¨s 7⌉ .
10.19 Given that 1+i t follow independent lognormal distributions with µ =0.05
and σ 2 =0.005 for t =1, 2, ···, find the 95% confidence interval of the
present value of $3 payable at the end of 4 years.
10.20 Describe how the mean and variance of (Ia) n⌉
can be simulated under the
independent lognormal model.