Final answer:
In this scenario, we are given two groups of 50-year-old men with different chances of dying in the next year based on their family cancer history. We calculate the actuarially fair premium for each group separately by considering the probability of dying and the payout amount. If the insurance company cannot differentiate between the two groups, the actuarially fair premium for the whole group can be calculated using a weighted average. Charging the same premium to the entire group may result in an unfair distribution of costs.
Step-by-step explanation:
In this question, we are given information about two groups of 50-year-old men concerning their chances of dying in the next year. Group 1 consists of men with a family history of cancer, which accounts for 20% of the total group of 1,000 men. They have a 1 in 50 chance of dying in the next year. Group 2 consists of men without a family history of cancer and makes up 80% of the total group of 1,000 men. They have a 1 in 200 chance of dying in the next year.
a. To calculate the actuarially fair premium for each group if the insurance company were selling life insurance separately, we need to consider the probabilities of dying in the next year and the payout amount. For Group 1, the chance of dying is 1 in 50, so the probability of surviving is 49/50. The actuarially fair premium would be the probability of surviving multiplied by the payout amount ($100,000), which is (49/50) * $100,000 = $98,000. For Group 2, the chance of dying is 1 in 200, so the probability of surviving is 199/200. The actuarially fair premium would be (199/200) * $100,000 = $99,500.
b. If the insurance company were offering life insurance to the entire group without knowing about their family cancer histories, the actuarially fair premium for the group as a whole can be calculated by taking the weighted average of the probabilities of dying and the payout amount. Since Group 1 makes up 20% of the total group and Group 2 makes up 80% of the total group, we can calculate the actuarially fair premium as (20% * $98,000) + (80% * $99,500) = $99,300.
c. If the insurance company tries to charge the actuarially fair premium to the group as a whole rather than to each group separately, it may face potential issues. Since the actuarially fair premium for Group 1 is $98,000 and for Group 2 is $99,500, charging the same premium to both groups as a whole may result in an unfair distribution of costs. Those in Group 1 may end up subsidizing the premiums for Group 2, which could lead to dissatisfaction and potential legal concerns.