Final answer:
To calculate the actuarially fair premium for each group, take the expected payout divided by the probability of death for each group.
Step-by-step explanation:
To calculate the actuarially fair premium for each group, we need to consider the probabilities of death for each group and the payout amount. Let's first calculate the probability of death for each group:
- For the group with a family history of cancer (20% of 1,000 men), the probability of death is 1 in 50, which is 1/50 = 0.02.
- For the group without a family history of cancer (80% of 1,000 men), the probability of death is 1 in 200, which is 1/200 = 0.005.
Now, we can calculate the actuarially fair premium for each group:
- For the group with a family history of cancer, the actuarially fair premium would be the expected payout divided by the probability of death: $100,000 / 0.02 = $5,000,000.
- For the group without a family history of cancer, the actuarially fair premium would be the expected payout divided by the probability of death: $100,000 / 0.005 = $20,000,000.
If the insurance company cannot find out about family cancer histories and has to offer 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 probabilities of death: (0.2 * $5,000,000) + (0.8 * $20,000,000) = $17,000,000.
If the insurance company tries to charge the actuarially fair premium to the group as a whole rather than to each group separately, it would face adverse selection. This means that the healthier individuals without a family history of cancer may find the premium too expensive and decide not to purchase the insurance, leaving a higher proportion of individuals with a family history of cancer who are more likely to claim the policy. As a result, the insurance company may face higher costs and potential losses.