Final answer:
Calculating actuarially fair premiums for life insurance requires estimating the risk of payout and assigning it evenly among policyholders. The actuarially fair premium for men with a family history of cancer is higher at $2,000, while it's lower for those without at $500. When the entire group is considered, the premium averages out to $800, but this could lead to adverse selection and potential financial losses for the insurer.
Step-by-step explanation:
Actuarially Fair Premium Calculation
Calculating an actuarially fair premium for life insurance involves estimating the expected payouts and distributing the cost among policyholders. Consider a group of 1,000 50-year-old men.
Separate Group Premiums
Group with Family History of Cancer:
20% of 1,000 men = 200 men
Probability of dying in the next year = 1/50
Expected payout per person = 1/50 * $100,000 = $2,000
Total expected payout for group = 200 men * $2,000 = $400,000
The actuarially fair premium per person = $400,000 / 200 = $2,000.
Group without Family History of Cancer:
80% of 1,000 men = 800 men
Probability of dying in the next year = 1/200
Expected payout per person = 1/200 * $100,000 = $500
Total expected payout for group = 800 men * $500 = $400,000
The actuarially fair premium per person = $400,000 / 800 = $500.
Group as a Whole Premium
When combining both groups without knowledge of cancer history:
Expected payout per person = (200/1,000 * $2,000) + (800/1,000 * $500) = $800
The actuarially fair premium per person for the entire group = $800.
Implications for Insurance Company
If the insurance company charges the actuarially fair premium for the group as a whole, they face the risk of adverse selection. Healthier individuals from the group without a family history of cancer might opt out due to the higher premium, leaving a disproportionate number of higher-risk individuals, which could result in financial losses for the company.