PSTAT 172A: ACTUARIAL STATISTICS FINAL EXAM

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1 PSTAT 172A: ACTUARIAL STATISTICS FINAL EXAM March 19, 2008 This exam is closed to books and notes, but you may use a calculator. You have 3 hours. Your exam contains 9 questions and 13 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)

2 1. Consider a fully discrete whole life insurance of $1, 000 on (45). Let π a denote an annual premium for this policy and L(π a ) denote the loss-at-issue random variable on the basis of the Illustrative Life Table and 6% interest. (a) (5 pts.) Determine the premium π a based on the equivalence principle. (b) (5 pts.) Calculate the variance of L(π a ).

3 2. Consider a fully discrete whole life insurance of $1, 000 on each of 100 lives age 45 whose future lifetimes are independent. Let π b denote an annual premium for this policy and L(π b ) denote the loss-at-issue random variable for one such policy on the basis of the Illustrative Life Table and 6% interest. (a) (5 pts.) Find the expectation and variance of the total loss. (b) (7 pts.) Determine the premium, π b, such that the probability of a positive total loss is 0.05 by the normal approximation.

4 3. (12 pts.) Consider a fully discrete whole life insurance of $1, 000 on independent lives age 45. Let L denote the loss-at-issue random variable for one such policy on the basis of the Illustrative Life Table and 6% interest. The level premium is π = 15. Using the normal approximation, find the minimum number of policies the insurer must issue so that probability of a positive total loss on the policies issued is less than or equal to 0.05.

5 4. Consider a fully discrete whole life insurance of $1, 000 on (45). Let π c denote an annual premium for this policy and L(π c ) denote the loss-at-issue random variable on the basis of the Illustrative Life Table. (a) (7 pts.) For 6% interest rate, approximate the smallest non-negative premium, π c, such that the probability of a positive loss is less than 0.5. (b) (5 pts.) Find π c when the force of interest δ = 0?

6 5. Let T (x) represent the future lifetime random variable. Assume that µ x (t) = 0.02 and δ = 0.03 for all t 0. (a) (5 pts.) Find ā x = E (ā T ) and σ 2 = Var (ā T ). (b) (5 pts.) Evaluate the probability that ā T i.e. Pr(ā T ā x > σ). will exceed ā x by one standard deviation,

7 6. Let Y be the present-value random variable for a special increasing whole life annuity-due on (40) with an initial payment of $10 in the first year, and payments increasing by $10 every 25 years. The payments will be made once every 25 years, beginning immediately. Assume that mortality follows DeMoivre s law with ω = 110 and i = (a) (5 pts.) Calculate E[Y ]. (b) (5 pts.) Calculate Var[Y ].

8 7. (12 pts.) An insurer is planning to issue a special fully discrete 2-payment whole life insurance policy to a life age 70 whose mortality follows the Illustrative Life Table at interest rate 6%. The level benefit premium π for this insurance will be received at the beginning of the first and third years. The death benefit amount, b K+1, payable at the end of the year of death is given by { π 2 b K+1 =, K = 0, 2 100, K = 1, 3, 4, 5, 6, Calculate the level premium π using the equivalence principle.

9 8. A person age 50 wins $10, 000 in the actuarial lottery. Rather than receiving the money at once, the winner is offered the option of receiving an annual payment of $725 at the beginning of each year guaranteed for 10 years and continuing thereafter for life. Assume that mortality follows the Illustrative Life Table and i = (a) (5 pts.) Should the winner accept a single check for $10, 000 today? (b) (5 pts.) What is the probability that the winner is better off if she accepts a single check for $10, 000 today?

10 9. (12 pts.) Your company is planning to issue a special 3-year deferred whole life annuitydue to a life age x at interest 6%. The level benefit premium π for this life annuity will be received at the beginning of each of the first three years. However, there is no death benefit during the three year deferral period. The first annual payment of this life annuity is $1000 and payments in the following years increase by 6% per year. Assume that the curtate expectation of future lifetime for (x) is e x = and mortality follows kp x = k, k = 1, 2, 3 Using the equivalence principle, find the level benefit premium π.

PSTAT 172A: ACTUARIAL STATISTICS FINAL EXAM

PSTAT 172A: ACTUARIAL STATISTICS FINAL EXAM March 17, 2009 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