Math 5067 (001) Homework 10 Due 12/2/15 1. A whole life annuity-immediate on (50) pays 1000 in the first year. Subsequent payments increase at a compounded rate of 4%. You are given: a) 50t 0.02, t 0 b) i 0.05 Calculate the expected present value of this annuity. 2. For a 20-year deferred whole life annuity-due of 1 per year on (45), you are given: a. Mortality follows De Moivre's law with 105. b. i 0. Calculate the probability that the sum of the annuity payments actually made will exceed the actuarial present value at issue of the annuity. 3. You are given the following life table data: You are also given that i = 0.06 and Determine a50 . lx qx dx 508 0.00600 91,365 4. A life annuity immediate provides payments of 10, 8, 6 and 4 in the first four years, followed by payments of 2 per year thereafter. You are given: i. Ax 0.2 6 ii. Ax5 0.3 0 iii. (I1 ) x:5| 1 2 a iv. Ax:5| 0.8 v. 5 px 0.9 8 Calculate the actuarial present value of this annuity. a51 11.888 x 5 0 5 1 5 2 5. You are given: i. . ii. iii. ax 24 n1| ax 18 sx:n| 6 Calculate n Ex 6. Smith, age 65, is evaluating three temporary life annuities, each of which provides annual payments. For each annuity the last payment is at age 75. i. The first annuity pays $5,000 at age 66; subsequent payments decrease by $500 per year. The actuarial present value of this annuity is $14,000. ii. The second annuity pays $1,000 at age 65; subsequent payments increase by $1,000 per year through age 75. The actuarial present value of this annuity is $21,000. iii. The third annuity pays $1,000 each year beginning at age 65 and continuing to age 75. The actuarial present value of this annuity is P. Determine P. 7. Jones, aged 50, purchases a life annuity due with annual payments of $1000. The payments for the first 20 years are guaranteed. Find the probability that the present value of the payments which he will receive will exceed the single benefit premium for the contract. Use the Illustrative Life Table with i = 0.06. 8. For a 5-year deferred whole life annuity due of 1 on (x), you are given: a. x+t = 0.01, t 0 b. i = 0.04 c. ax:5| 4.542 d. The random variable S denotes the sum of the annuity payments. Calculate Pr(S5| ax ) 9. You are given: a. = 0.05 b. Mortality follows De Moivre's law with = 100. Calculate Calculate 2 A35 a35 Calculate the Pr(aT | a35 ) 35 Calculate Var(aT ) 35 10. Y is the present value random variable for a continuous 10-year temporary life annuity. You are given: a. = 0.02 for all x x b. = 0.04 Calculate the probability that the total payments on this annuity will exceed E(Y). 11. Y is the present value of a whole life annuity-due on (20). You are given: a. Mortality follows De Moivre's law with = 100. b. i = 0.06 Calculate the probability that the sum of the payments will exceed E(Y). 12. Y is the present value random variable for a continuous whole life annuity on (60). You are given: a. Mortality follows De Moivre's law with = 110. b. = 0.06 Calculate the 90th percentile of Y. 13. You are given: a. for all integral x. ax 8 b. i 0.08 Calculate q . 8 30 14. You are given: a. a19 25.04 b. c. a20 24.85 i 0.03 Calculate p19 . 15. You are given: a. A65 0.325 b. c. d. i 0.06 p64 0.992 p65 0.983 Approximate i. ii. A65 in two ways: Assuming uniform distribution of deaths between integral ages. Using Woolhouse's formulas on annuities. 16. You are given: a. Deaths are uniformly distributed over each year of age. b. 5| ax:10 5.3 5 Ex 0.60 c. d. 15 e. Ex 0.20 i 0.10 Calculate 5| x:10 a(12) . 17. You are given: a. ax 12.5432 b. c. x 0.01 0.05 Estimate ax using Woolhouse's formulas with three terms