FM for Actuaries

308 CHAPTER 10
are determined by many forces in the economy and they are hardly constant nor
deterministic.
A random-scenario model is a collection of specified plausible interest rate scenarios.
The modeler, however, needs to state the probability distribution of these
scenarios. Both the choice of scenarios and the corresponding probability assignment
require personal judgement, which should reflect the modeler’s view on the
future interest-rate environment of the economy.
Example 10.2: Consider the following random interest rate scenario model.
Scenario Probability i 1 i 2 i 3 i 4 i 5
1 0.1 3.0% 2.0% 2.0% 1.5% 1.0%
2 0.6 3.0% 3.0% 3.0% 3.5% 4.0%
3 0.3 3.0% 4.0% 5.0% 5.0% 5.0%
Find the mean, the variance and the standard deviation of a(5),
1
a(5) , a 5⌉, ä 5⌉ ,s 5⌉
and ¨s 5⌉ under this model.
Solution: Note that
ä n⌉ =
n−1
∑
t=0
1
a(t)
=
n−1
∑ 1
1+
a(t)
t=1
=
n−1
∑ 1
1+ ∏ t
j=1 (1 + i j) . (10.1)
t=1
Using (3.10), (3.11), (3.12), (10.1), (3.14) and (3.15) (with i F t replaced by i t ), we
1
can compute the values of a(5),
a(5) , a 5⌉, ä 5⌉ ,s 5⌉ and ¨s 5⌉ under each scenario. For
example, under the first scenario, we have
a(5) = (1.03)(1.02)(1.02)(1.015)(1.01) = 1.0986,
1
a(5) = 1
1.0986 =0.9103,
a 5⌉
= 1
(1.03) + 1
(1.03)(1.02) + ···+ 1
(1.03)(1.02)(1.02)(1.015)(1.01) =4.6855,
ä 5⌉
=1+ 1
(1.03) + ···+ 1
(1.03)(1.02)(1.02)(1.015) =4.7753,