## Actuarial Mathematics 2

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