## Actuarial Mathematics 2

Prove that if Β΅ is constant then
π₯+π‘
π π for all
π₯ = π π
π₯+π
π , π πππ π β₯ 0
(b) For a Whole Life Policy with death benefit of \$5000, benefit
payable at the end of year of death, determine 1π given
π₯
Β΅ = 0. 07 and π = 0. 07.
(c) Consider the following information for a given policy :
a. Ξ΄ = 0. 03
b. Level premium π = \$3500
c. No expenses.
d. Β΅ = 0. 0007
e. Benefit payment π = \$150, 000
(i) Using Thieleβs Differential Equation, solve analytically (using
calculus) for the policy value π‘ππ₯.
(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 π‘ β β.
(d) Consider the following policy basis for a 20 year term life policy
with:
Survival model: De Moivreβs Law Ο = 105
Interest: π = 0. 06
Expenses: No expenses
Sum Insured: \$60,000
By using Microsoft Excel or otherwise, determine 10ππ₯ using the
numerical approximation (Eulerβs Method) for the solution to
Thieleβs Differential Equation. Assume a time step of β = 0. 05.
Total: 20 marks
Question 2:
Consider the following joint density function for (2) lives (x) and (y):
π(π₯, π¦) = π(π₯ , ,
2
+ π¦
2
) 0 < π₯ < 3 0 < π¦ < 2
(a) Calculate π‘π .
π₯π¦
(b) Calculate π‘π .
π₯π¦
(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