Final answer:
If an insurer discovers a misstatement of age after L's death, the insurer will pay a reduced death benefit. The actuarially fair premiums would differ for groups with different risks. Charging the same premium to everyone could result in financial losses due to underpricing.
Step-by-step explanation:
When L takes out a life insurance policy and dies, and the insurer discovers a misstatement of age on the application, the insurer will typically pay a reduced death benefit. This reduction is based on the misstatement of age provision found in most insurance policies, which adjusts the death benefit according to what the premiums would have been if the correct age had been stated initially.
In terms of actuarially fair premiums, if the insurance company were selling life insurance separately to two groups of 50-year-old men, one with a family history of cancer and the other without, the premiums would differ. Group one, with the history, has a higher chance of dying within the next year, so their premium would be higher. For example, 20% out of 1,000 men with a family history of cancer would be 200 men, each with a 1 in 50 chance of dying within a year, resulting in about 4 expected deaths. Thus, an actuarially fair premium would be the expected payout ($100,000) divided by the chance of dying (50), resulting in a premium of $2,000 for this group.
Group two, without a family history, encompasses the remaining 800 men with a 1 in 200 chance of dying. This results in about 4 expected deaths as well. Consequently, the actuarially fair premium for this group would be $500 (which equals $100,000 divided by 200).
If the insurance company were to offer life insurance to the entire group without distinguishing between the subgroups based on cancer history, the actuarially fair premium would be a weighted average of the two separate premiums, which considers both the higher risk of the 200 men with a family history and the lower risk of the 800 others.
The insurance company would face challenges if it tried to charge the same premium to everyone. This is because the higher-risk individuals (those with a family history of cancer) would be more likely to purchase the insurance, leading to an increase in the number of claims and a risk of underpricing the premiums, which could result in financial losses for the company.