FM for Actuaries

322 CHAPTER 10
Find the mean and variance of (Is) 4⌉ .
10.8 Give two shortcomings of the random interest rate scenario approach.
10.9 The random walk model (in particular, the binomial-tree model) is widely
used in financial engineering. In essence, the random walk model gives a
convenient way to construct random scenarios. Let i 1 =3%. The 1-period
spot rate over the next period is 105% of the previous 1-period spot rate
with probability 0.55, and is 95% of the previous 1-period spot rate with
probability 0.45.
(a) Construct all scenarios for the 1-period spot rates over the next 3 years
and give the corresponding probability for each scenario.
(b) Find the mean and standard deviation of the accumulated value of $1
after 3 years.
10.10 Assume that i 1 ,i 2 , ··· ,i n are independently and identically distributed interest
rate random variables with mean µ =0.05 and variance σ 2 =0.0001.
Following (3.10), we define
a(n) =
n∏
(1 + i j ).
j=1
Find the mean and standard deviation of a(10).
10.11 Assume that i 1 ,i 2 , ··· ,i n are independently and identically distributed interest
rate random variables with mean µ =0.10 and variance σ 2 =0.0004.
Following (3.15), we define
⎛
⎞
n−1
∑ n∏
¨s n⌉
= ⎝ (1 + i j ) ⎠ .
t=0
j=t+1
Find the mean and standard deviation of ¨s 8⌉ .
10.12 Assume that i 1 ,i 2 , ··· ,i n are independently and identically distributed interest
rate random variables uniformly distributed on the interval [0.06, 0.08].
Following (3.15), we define
⎛
⎞
n−1
∑ n∏
¨s n⌉ = ⎝ (1 + i j ) ⎠ .
t=0
j=t+1
Find the mean and standard deviation of ¨s n⌉ ,forn =11, 12, ··· , 20.