Actuarial Mathematics 2: Assignment 1
Question 1:
(a) Prove that if µ is constant then
+
for all
=
+
, ≥ 0
(b) For a Whole Life Policy with death benefit of $5000, benefit
payable at the end of year of death, determine 1 given
µ = 0. 07 and = 0. 07.
(c) Consider the following information for a given policy :
a. δ = 0. 03
b. Level premium = $3500
c. No expenses.
d. µ = 0. 0007
e. Benefit payment = $150, 000
(i) Using Thiele’s Differential Equation, solve analytically (using
calculus) for the policy value .
(ii) Find the unique solution to this differential equation clearly
showing why this is the case.
(iii) Describe the what happens to the policy value as → ∞.
(d) Consider the following policy basis for a 20 year term life policy
with:
Survival model: De Moivre’s Law ω = 105
Interest: = 0. 06
Expenses: No expenses
Sum Insured: $60,000
By using Microsoft Excel or otherwise, determine 10 using the
numerical approximation (Euler’s Method) for the solution to
Thiele’s Differential Equation. Assume a time step of ℎ = 0. 05.
Total: 20 marks
Question 2:
Consider the following joint density function for (2) lives (x) and (y):
( , ) = ( , ,
2
+
2
) 0 < < 3 0 < < 2
(a) Calculate .
(b) Calculate .
(c) For a whole life policy of a joint status (following the mortality
model above) with sum insured of $250,000, benefit payable
immediately upon death, determine the net single premium.
Total: 15 marks
