PSTAT 172B: ACTUARIAL STATISTICS FINAL EXAM

Transcription

1 PSTAT 172B: ACTUARIAL STATISTICS FINAL EXAM June 10, 2008 This exam is closed to books and notes, but you may use a calculator. You have 3 hours. Your exam contains 7 questions and 11 pages. Please make sure you print your name and sign the honor code below. I acknowledge that I have neither given nor received aid on this examination nor have I concealed any violation of the Honor Code. (Signed) 1

2 1. Two independent lives now age 35 and 50 have mortality based on a constant force of mortality with µ = The force of interest rate is δ = (a) (5 pts.) Calculate the probability that at least one of (35) and (50) will die within 10 years. (b) (5 pts.) Calculate e 35:50, the expected time until both have died. 2

3 1. Two independent lives now age 35 and 50 have mortality based on a constant force of mortality with µ = The force of interest rate is δ = (c) (5 pts.) Calculate A 1 35:50. (d) (5 pts.) A 3-year temporary life annuity-due on (35) and (50) pays $10, 000 if both persons are alive and $2, 000 if exactly one person is alive. Calculate the actuarial present value of this annuity. 3

4 2. Ellen age 30 and Jean age 40 are independent lives and have mortality following de Moivre s law with ω = 100. A 30-year term insurance on Ellen and Jean provides the death benefit payable at the moment of the first death. The interest rate is i = 0. (a) (5 pts.) Calculate the probability that Ellen will die after Jean and within 10 years. (b) (5 pts.) Calculate the probability that the second death occurs between 10 and 20 years. 4

5 2. Ellen age 30 and Jean age 40 are independent lives and have mortality following de Moivre s law with ω = 100. A 30-year term insurance on Ellen and Jean provides the death benefit payable at the moment of the first death. The interest rate is i = 0. (c) (5 pts.) A death benefit of $140, 000 will be paid if Ellen dies before Jean and within 30 years, and a death benefit of $180, 000 will be paid if Jean dies before Ellen and within 30 years. Calculate the actuarial present value of this 30-year term insurance. (d) (5 pts.) Premiums are paid continuously while both are alive, for a maximum of 20 years. Calculate the annual benefit premium. 5

6 3. Two independent lives now age x and y are subject to a constant force of mortality with µ = 0.04, when the effect of common shock is not considered. Both lives are subject to common shock with constant force λ = The force of interest rate is δ = (a) (5 pts.) Calculate the probability of both (x) and (y) dying at the same time. (b) (5 pts.) Calculate the actuarial present value of a last-survivor whole life insurance of $1 on (x) and (y) payable at the moment of death. 6

7 4. ABC Paper Mill company purchases a 5-year term insurance paying a benefit in the event the machine breaks down. If the cause is minor (1), only a repair is needed. If the cause is major (2), the machine must be replaced. The force of decrement by cause (1) is µ (1) (t) = for t > 0, and the force of decrement by cause (2) is µ (2) (t) = for t > 0. The force of interest rate is δ = (a) (5 pts.) Calculate the probability that the machine will not break down for at least three years but will break down within five years. (b) (5 pts.) The benefit for cause (1) is $1, 500 payable at the moment of breakdown and the benefit for cause (2) is $500, 000 payable at the moment of breakdown. Once a benefit is paid, the insurance contract is terminated. Calculate the actuarial present value of this 5-year term insurance. 7

8 5. Johnny Blaze, age 45, is a professional motorcycle stuntman who purchases a 3-year term insurance policy. A double-decrement table is given below, where the index (1) indicates death from stunt accidents and the index (2) indicates death from other causes. The interest rate is i = x l x (τ) d (1) x d (2) x (a) (5 pts.) Each decrement is uniformly distributed over each year of age in the double decrement table. Calculate q 46(1). (b) (5 pts.) The policy pays $500, 000 for death from a stunt accident and nothing for death from other causes. The benefit is paid at the end of the year of death. Calculate the actuarial present value of this 3-year term insurance. 8

9 5. Johnny Blaze, age 45, is a professional motorcycle stuntman who purchases a 3-year term insurance policy. A double-decrement table is given below, where the index (1) indicates death from stunt accidents and the index (2) indicates death from other causes. The interest rate is i = x l x (τ) d (1) x d (2) x (c) (5 pts.) Level annual benefit premiums are payable at the beginning of each year. Calculate the annual benefit premium. (d) (5 pts.) Calculate 2 V, the benefit reserve at time 2. 9

10 6. (10 pts.) 200 new employees plan to participate in a 3-year management-training program. A multiple decrement table is constructed to predict the number of employees who will complete the program, based on the following associated single decrement assumptions: (i) Of 40 employees, the number who fail to make adequate progress in each of the first three years is 10, 6, and 8, respectively. (ii) Of 30 employees, the number who resign from the company in each of the first three years is 6, 8, and 2, respectively. (iii) Of 20 employees, the number who leave the program for other reasons in each of the first three years is 2, 2, and 4, respectively. (iv) Each decrement is uniformly distributed over each year of age in the associated single decrement table. Calculate the expected number of employees who will fail to make adequate progress in the third year. 10

11 7. Ramona, age 50, is an actuarial science professor, whose career is subject to two decrements. Decrement 1 is mortality, and the associated single decrement table follows de Moivre s law with ω = 100. Decrement 2 is leaving academic employment, with µ (2) 50 (t) = 0.04 for t 0. (a) (5 pts.) Calculate 10 q (τ) 50, the probability that decrement occurs during the 11 th year. (b) (5 pts.) Calculate her expected complete future career years as an actuarial science professor. 11