MATH 403A, fall 98, CHAPTER 4 Section 4.3, Problems

Under UDD assumption

(D Ā)_{x^{1}:\overline{n}|} = \frac{i}{δ} (D A)_{x^{1}:\overline{n}|}.

1. A special continuous whole life insurance issued to a life aged 35 has a death benefit at time t of b_{t} = 1.1^{t}. The forces of mortality and interest are both constant for all ages, and are .08 and .125 respectively. Find the net single premium for this insurance.

2. Find the Variance of the present value random variable for the insurance in problem 1.

3. A special 3–year ensowment insurance issued to a life aged 50 has a death benefit of b_{k+1} payable at the end of the year of death in year k + 1. The endowment benefit is 5000. The following table gives the mortality rates and the death benefits.

k q_{50+k}b_{k+1} 0 .05 1000 1 .08 2000 2 .12 3000

v = .9. Find the actuarial present value of this insurance.

4. Find the variance of the present value random variable for the insurance in problem 3.

5. A special whole life insurance issued to a life aged 45 has death benefit payable at the end of the year of death. The benefit for death in the first year is 100. after that, each year’s benefit is 10% greater than the prior year’s. Mortality follows deMoivre’s law with w = 100. The effective annual rate of interest is 8%. Find the net single premium for this insurance.

6. You are given

q_{x} = q_{x+1} = .1 v = .9 A_{x+2} = .600

Find A_{x}.

7. You are given

A_{x} = .5 A_{x+1} = .52 i = .06.

Find q_{x}.

8. You are given

Ā_{x} = .5 Ā_{x+1} = .52 i = .06.

Deaths are uniformly distributed within each year of age. Find q_{x}.

9. You are given that

(D̄ Ā)_{x^{1}:\overline{20}|} = 2.01 (D Ā)_{x^{1}:\overline{20}|} = 2.06 (D A)_{x^{1}:\overline{20}|} = 2.00 (D A)_{x^{1}:\overline{20}|} = 1.15

Find (ĪĀ)_{x^{1}:\overline{20}|}.

10. A continuously increasing whole life insurance issued to a life aged 55 has a benefit payable at the moment of death of b_{t} = 1000t. Mortality follows deMoivre’s law with w = 100. The force of interest is a constant .06. Find the net single premium for this insurance.