Expected Values

Question

Suppose the same insurance company as in the previous question insures adults ages 25 to 34 in California for the same amount of money per month, but offers a $175,000 policy for that amount of money. The reason for the difference in the payout is that the death rate in California for that age group is 81.6 per 100,000 residents.

a.) Find the expected value per customer for the insurance company at the end of one year for the policy described in California.

b.) If the insurance company has 10,000 customers with these life insurance policies in California, what is its profit at the end of the year?

c.) Which state is more profitable for the insurance company (as compared to Mississippi in the previous problem)?

PREVIOUS PROBLEM:

An insurance company offers Mississippi adults between the ages of 25 and 34 a $100,000 life insurance policy for $18 a month. They use the fact that Mississippi has a yearly death rate of 172.8 per 100,000 residents aged 25-34 years.

Answer 1

I'll first answer the original question, partly because it is referenced in part c, but also because it provides the basis for the variant question.

For the Mississippi part of the problem, the payout is $100,000, and there is an ^172.8/_100,000 probability of a death (requiring a $100,000 payout). The expected payout, then, is the product of these two numbers. It works out really nicely here, in that the 100,000 cancels and just leaves 172.8. Since the insured is paying $18 per month for 12 months, then the company is receiving $216 per year (18 • 12). That makes the net gain for the insurance company, i.e., its profit, $43.20 (216 - 172.8).

a) We do a similar thing for California in part a. The premiums are the same--$216 per year--but the death rate and the payout are different. Here we take ^81.6/₁₀₀₀₀₀ • 175000, which equals 142.8 as the expected payout. Subtract this from the premium revenue--216 - 142.8--and the profit per customer is $73.20.

b) The total expected profit for 10,000 customers will be 73.2 • 10,000 = $732,000.

c) From comparing the original problem to the result from part a, we can see that California is more profitable per customer than Mississippi.