Problem 1
(10 points)
1
for a life aged (x), with x > 0, determine the
Given the probability t px = x+2t
force of mortality µx+t .
Problem 2
(10 points)
Given the following distribution for the curtate future lifetime K[80] of a life aged
(80)
k
0
1
2
3
4
≥5
P(K[80] = k)
0.2
0.15
0.1
0.15
0.4
0
Calculate E[K[80] | K[80] ≥ 3].
Problem 3
(15 points)
We want to determine a whole life continuous annuity āx with a payment rate that
is not constant. Further, assume the following:
(a) The force of interest be constant and equal to δ;
(b) For a life aged x assume a constant force of mortality equal to µ.
(c) The payment rate is r(t) = αt with 0 < α < 1.
Determine, if possible, āx , in terms δ, µ and α.
Also formulate a condition for α, so that āx is meaningful.
Problem 4
Given:
(15 points)
(i) lx =100,000(100 − x), 0 ≤ x ≤ 100, and
(ii) i = 0.
Calculate (Iā)95 exactly. Recall that (Iā)x is a continuous annuity with payment
rate r(t) = [t + 1]. Recall also that [t + 1] = 1 for 0 ≤ t < 1, [t + 1] = 2 for
1 ≤ t < 2 and so on.
Note: lx represents the number of survivors after x years of a population of size
100,000.
Problem 5
(10 points)
Assume that the curtate future lifetime K[30] of a life aged (30) is uniformly distributed on {0, 1, 2, . . . , 80}. Assume further that the interest rate i equals 4%.
Calculate the single premium for a 5 year deferred whole life insurance 5| A30 .
Problem 6
Given
(10 points)
(a) The survival function is s(x) = 1 −
x
90 ,
0 ≤ x ≤ 90.
(b) The force of interest is δ = 0.05.
Calculate the single premium of a term insurance that pays $1000 at the moment
of death if an insured aged (40) dies within 25 years.
Note: s(x) represents the probability of a newborn to survive x years.
Problem 7
(15 points)
Let T [57] be the remaining lifetime of a life aged (57). Assume that the force of
mortality of T [57] is constant and equal to 0.04. Further assume that the rate of
interest equals 0.02. Consider a temporary life annuity issued to the life aged (57).
The benefits of the temporary life annuity are paid to the insured at the end of the
year, if the insured is alive. The amounts are given below (from the below scheme
it is clear that the temporary life annuity is deferred). Level premiums P are paid
at the beginning of each year that the insured is alive, but only for the first three
years.
Year
Benefit
0
0
1
0
2
1
3
3
4
5
5
7
Use the equivalence principle to find premium P .
Hint: The table means that at the end of the first year (year 0) there is no benefit
payment, at the end of the second year (year 1) there is no benefit payment, at the
end of the third year (year 2) there is a benefit payment of 1 and so on.
Problem 8: see next page.
Problem 8
(15 points)
Determine A35:20 , ä35:20 , P35:20 , A135:20 and Ā35;20 on the basis of the Illustrative
Life Table (given below, with i = 5%).
x
..
.
Dx
..
.
Nx
..
.
Cx
..
.
Mx
..
.
x
..
.
34
35
36
..
.
1,796,672.2
1,707,865.3
1,623,269.1
..
.
31,531,658.4
29,734,986.2
28,027,170.9
..
.
3,251.079
3,269.268
3,308.445
..
.
295,164.624
291,913.545
288,614.277
..
.
34
35
36
..
.
39
40
41
..
.
1,392,657.4
1,322,891.9
1,256,394.5
..
.
23,395,337.5
22,002,680.1
20,679,788.2
..
.
3,448.443
3,502.576
3,565.756
..
.
278,593.673
275,145.230
271,642.654
..
.
39
40
41
..
.
54
55
56
..
.
625,073.6
590,402.8
557,250.3
..
.
8,544,152.6
7,919,079.0
7,328,676.2
..
.
4,905.357
5,038.129
5,174.462
..
.
218,209.204
213,303.847
208,265.718
..
.
54
55
56
..
.