Final answer:
The actuarially fair premium for each group can be calculated by considering the mortality rates and expected payouts for the two groups separately. If the insurance company can't determine the family cancer history, the actuarially fair premium for the group as a whole would be a weighted average of the premiums for each group. Charging the actuarially fair premium to the group as a whole instead of separately could lead to adverse selection.
Step-by-step explanation:
If the insurance company were selling life insurance separately to each group, the actuarially fair premium for each group can be calculated as follows:
For the group with a family history of cancer: 20% * 1000 = 200 men
Mortality rate: 1 in 50
Expected number of deaths: 200 / 50 = 4 deaths
Expected payout: 4 * $100,000 = $400,000
Actuarially fair premium for this group: $400,000 / 200 = $2,000
For the group without a family history of cancer: 80% * 1000 = 800 men
Mortality rate: 1 in 200
Expected number of deaths: 800 / 200 = 4 deaths
Expected payout: 4 * $100,000 = $400,000
Actuarially fair premium for this group: $400,000 / 800 = $500
If the insurance company could not find out about family cancer histories and were offering life insurance to the entire group, the actuarially fair premium for the group as a whole would be the weighted average of the premiums for each group based on their population percentage. In this case, the actuarially fair premium for the group as a whole would be:
(20% * $2,000 + 80% * $500) = $260 + $400 = $660
If the insurance company tries to charge the actuarially fair premium to the group as a whole rather than to each group separately, the company may face adverse selection. Adverse selection occurs when individuals with a higher risk of death are more likely to purchase insurance. In this case, individuals with a family history of cancer would be overpaying for their premiums, while individuals without a family history of cancer would be underpaying. This imbalance in premiums could make the insurance product less attractive to the target market.