A farmer purchases a five-year insurance policy that covers crop destruction due to hail. Over the five-year period, the farmer will receive a benefit of 20 for each year in which hail destroys his crop, subject to a maximum of three benefit payments. The probability that hail will destroy the farmer’s crop in any given year is 0.5, independent of any other year.

Calculate the expected benefit that the farmer will receive over the five-year period.

• 30
• 34
• 40
• 46
• 50

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

Solution: D

Let X be the number of years in which a payment of 20 is received. X has a binomial distribution with n = 5 and p = 0.5. Let p(x) be the probability of x payments. The expected payment is

0 p (0) + 20 p (1) + 40 p (2) + 60[1 − p (0) − p (1) − p (2)] = 0(1/ 32) + 20(5 / 32) + 40(10 / 32) + 60(16 / 32) = 45.625.

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