Final answer:
The actuarially fair premium for each group of 50-year-old men with and without a family history of cancer can be calculated using the probability of dying and the payout amount.
If the insurance company cannot find out about family cancer histories, the actuarially fair premium for the entire group as a whole would be a weighted average.
Charging the actuarially fair premium to the group as a whole instead of to each group separately may lead to adverse selection and losses for the insurance company.
Step-by-step explanation:
Actuarially Fair Premium for Life Insurance
a. If the insurance company were selling life insurance separately to each group, the actuarially fair premium for each group would be calculated as follows:
- For the group with a family history of cancer: 20% of 1,000 is 200 men. The chance of dying in the next year is 1 in 50. Therefore, the expected number of deaths is 200/50 = 4. The payout for each death is $100,000. So, the actuarially fair premium for this group would be 4 * $100,000 = $400,000.
- For the group without a family history of cancer: 80% of 1,000 is 800 men. The chance of dying in the next year is 1 in 200. Therefore, the expected number of deaths is 800/200 = 4. The payout for each death is $100,000. So, the actuarially fair premium for this group would also be 4 * $100,000 = $400,000.
b. If the insurance company were offering life insurance to the entire group but could not find out about family cancer histories, the actuarially fair premium for the group as a whole would be the weighted average of the premiums for each group, based on their respective probabilities of dying. Since 20% of men have a family history of cancer and 80% do not, the actuarially fair premium for the group as a whole would be 20% * $400,000 + 80% * $400,000 = $400,000.
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 end up charging too much for the group without a family history of cancer and too little for the group with a family history of cancer.
This can result in adverse selection, where individuals in the higher risk group are more likely to purchase insurance, leading to losses for the insurance company.