Two life insurance policies, each with a death benefit of 10,000 and a one-time premium of 500, are sold to a married couple, one for each person. The policies will expire at the end of the tenth year. The probability that only the wife will survive at least ten years is 0.025, the probability that only the husband will survive at least ten years is 0.01, and the probability that both of them will survive at least ten years is 0.96.

Calculate the expected excess of premiums over claims, given that the husband survives at least ten years.

• 350
• 385
• 397
• 870
• 897

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: E

Because the husband has survived, the only possible claim payment is to the wife. So we need the probability that the wife dies within ten years given that the husband survives. The numerator of the conditional probability is the unique event that only the husband survives, with probability 0.01. The denominator is the sum of two events, both survive (0.96) and only the husband survives (0.01). The conditional probability is 0.01/(0.96 + 0.01) = 1/97. The expected claim payment is 10,000/97 = 103 and the expected excess is 1,000 – 103 = 897.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.