Actuarial Mathematics 2

Prove that if Β΅ is constant then
π‘₯+𝑑
𝑠𝑝 for all
π‘₯ = 𝑠𝑝
π‘₯+π‘˜
𝑠, 𝑝 π‘Žπ‘›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 1𝑉 given
π‘₯
Β΅ = 0. 07 and 𝑖 = 0. 07.
(c) Consider the following information for a given policy :
a. Ξ΄ = 0. 03
b. Level premium 𝑝 = $3500
c. No expenses.
d. Β΅ = 0. 0007
e. Benefit payment 𝑆 = $150, 000
(i) Using Thiele’s Differential Equation, solve analytically (using
calculus) for the policy value 𝑑𝑉π‘₯.
(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 β†’ ∞.
(d) Consider the following policy basis for a 20 year term life policy
with:
Survival model: De Moivre’s Law Ο‰ = 105
Interest: 𝑖 = 0. 06
Expenses: No expenses
Sum Insured: $60,000
By using Microsoft Excel or otherwise, determine 10𝑉π‘₯ using the
numerical approximation (Euler’s Method) for the solution to
Thiele’s Differential Equation. Assume a time step of β„Ž = 0. 05.
Total: 20 marks
Question 2:
Consider the following joint density function for (2) lives (x) and (y):
𝑓(π‘₯, 𝑦) = 𝑐(π‘₯ , ,
2
+ 𝑦
2
) 0 < π‘₯ < 3 0 < 𝑦 < 2
(a) Calculate 𝑑𝑝 .
π‘₯𝑦
(b) Calculate π‘‘π‘ž .
π‘₯𝑦
(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

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