THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2801/3901 Life Contingencies
Assignment 3
Due Date: December 1, 2016
1. Suppose that the force of mortality of (x) takes the form
(
1
80−t
, 0 ≤ t < 20,
µx (t) = 1
60−t
, 20 ≤ t < 60.
Using constant force of interest δ = 0.06, calculate āx .
¯ x , āx:n| , āx ,
2. If δt = 0.2/(1 + 0.05t) and lx = 100 − x for 0 ≤ x ≤ 100, calculate (Iā)
n| āx .
3. Let Y be the present value random variable for a life annuity that pays 1 at the
end of each year while (x) survives plus a final adjustment payment at the moment
of death. The adjustment payment is t where t is the portion of the year between
the date of the last regular payment and the date of death. Assuming uniform
{6}
distribution of deaths over each year of age, calculate (a) E(Y ) and (b) äx if
δ = 0.06 and ax = 15.
4. Given that lx = 100, 000(100−x) for 0 ≤ x ≤ 100 and i = 0. Calculate the actuarial
present value of a temporary 5-year life annuity issued to (80). The annuity is
paid continuously at an annual rate of $120,000 per year during the first year, and
$240,000 per year for the next four years.
5. On the basis of the AM92 ultimate table with interest at the effective annual rate
(12) (12)
of 6%, calculate ä40:5| , a30:5| , s̈50:6| , s35:6| . Also, calculate ä40:5| and s35:6| assuming
that deaths have a uniform distribution in each year of age.
6. A special deferred annuity provides the following benefits for a life aged 45.
– on survival to age 65 an annuity of 100,000 per annum payable annually in
advance for five years certain and for life thereafter;
– on death before age 55, 300,000 payable at the end of the year of death;
– on death between ages 55 and 65, 600,000 payable at the end of the year of
death.
Determine the actuarial present value of this annuity using the AM92 ultimate table
with 6% interest.
1
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2801/3901 Life Contingencies
Assignment 3
Due Date: December 1, 2016
1. Suppose that the force of mortality of (x) takes the form
(
1
80−t
, 0 ≤ t < 20,
µx (t) = 1
60−t
, 20 ≤ t < 60.
Using constant force of interest δ = 0.06, calculate āx .
¯ x , āx:n| , āx ,
2. If δt = 0.2/(1 + 0.05t) and lx = 100 − x for 0 ≤ x ≤ 100, calculate (Iā)
n| āx .
3. Let Y be the present value random variable for a life annuity that pays 1 at the
end of each year while (x) survives plus a final adjustment payment at the moment
of death. The adjustment payment is t where t is the portion of the year between
the date of the last regular payment and the date of death. Assuming uniform
{6}
distribution of deaths over each year of age, calculate (a) E(Y ) and (b) äx if
δ = 0.06 and ax = 15.
4. Given that lx = 100, 000(100−x) for 0 ≤ x ≤ 100 and i = 0. Calculate the actuarial
present value of a temporary 5-year life annuity issued to (80). The annuity is
paid continuously at an annual rate of $120,000 per year during the first year, and
$240,000 per year for the next four years.
5. On the basis of the AM92 ultimate table with interest at the effective annual rate
(12) (12)
of 6%, calculate ä40:5| , a30:5| , s̈50:6| , s35:6| . Also, calculate ä40:5| and s35:6| assuming
that deaths have a uniform distribution in each year of age.
6. A special deferred annuity provides the following benefits for a life aged 45.
– on survival to age 65 an annuity of 100,000 per annum payable annually in
advance for five years certain and for life thereafter;
– on death before age 55, 300,000 payable at the end of the year of death;
– on death between ages 55 and 65, 600,000 payable at the end of the year of
death.
Determine the actuarial present value of this annuity using the AM92 ultimate table
with 6% interest.
1
