Homework 1
1. Suppose that the probability that a 30-year-old will die before reaching
age 40 is 0.05, the probability that a 40-year-old will die before reaching age
50 is 0.07, and the probability that a 50-year-old will die before reaching age
60 is 0.
1. L IFE I NSURANCE
1.1. Introduction. In this section, we develop formulae for the valuation
of traditional insurance benefits. In particular, we consider whole life, term
and endowment insurance.
Because of the dependence on death or survival, the timin
1. S URVIVAL M ODELS
1.1. Survival function. We are interested in properties of a positive random variable X, which we interpret as age-at-death or age-at-failure.
Definition 1.1. The survival function of a positive random variable X is
SX (t) = P (X > t)
Q UIZ 3
1. You are given the survival function
s(x) = 1 (0.01x)2 , 0 x 100.
Calculate e30:50 , the 50-year temporary complete expectation of life
of (30).
(A) 27
(B) 30
(C) 34
(D) 37
(E) 41
2. For (x):
(i) K is the curtate future lifetime random variable.
1. For a special fully continuous last survivor insurance of 1 on (x) and (y),
you are given:
(i) Tx and Ty are independent, (ii) = 0.06.
(iii) For (x), x+t = 0.08, and for (y), y+t = 0.04, t > 0.
Calculate the actuarial present value of this insurance, A
1. Consider the following life table:
x
52
53
54
lx 89,948 89,089 88,176
Calculate 0.2q52.4 assuming UDD.
(A) 0.001911 (B) 0.001914 (C) 0.001917 (D) 0.001920 (E) 0.001923
2. For a 4-year college, you are given the following probabilities for dropout
from
1. For a whole life insurance of 1 on (41) with death benefit payable at the
end of the year of death, you are given:
(a) i = 0.05; p40 = 0.9972;
(b) A41 A40 = 0.00822;
(c) 2A41 2A40 = 0.00433;
(d) Z is the present-value random variable for this insurance
Q UIZ 1
1. An actuary for a medical device manufacturer initially models the
failure time for a particular device with an exponential distribution
with mean 4 years. This distribution is replaced with a spliced model
whose density function:
(a) is uniform
Homework 2
Exercise 2.1. Suppose that (x) =
Find .
2x
,
2 x2
Exercise 2.2. You are given that t q35 =
and e45 .
for 0 x , and 20 q40 =
50t+t2
7475
5
12
.
for 0 t 65. Find (45)
Exercise 2.3. The future lifetime of a new born has survival function s(x) =
1