Consider a market for annuities for 70-year-old men in which people differ in terms of both their expected remaining years of life and their risk preferences. Of the population of 200, half have a life expectancy of 9 years, and the remaining half have a life expectancy of 11 years. We can express risk preference in the following way. The risk people are worried about is that of running out of wealth before they die. The more risk- averse you are, the higher the up-front price you are willing to pay for the annuity. More risk- averse people are willing to pay 1.3 times x times A, where x is the expected years of life remaining and A is the dollar amount paid each year to the annuitant. Less risk-averse people are willing to pay only 1.1 times x times A. Assume that of the 100 people in each health group, half are more risk- averse and half less risk-averse.
The annuity firm sells an annuity of $50,000 per year for as long as the buyer lives, and the price of the annuity is $550,000. Because the annuity firm cannot tell whether any applicant has a short or long life expectancy, it must accept any application for its product. What is the expected profit of the annuity firm? (You may ignore discounting in this example.)