Final answer:
The actuarially fair premium for Mordecai's health insurance would be $11,250, whereas for life insurance, the premium would be $2,000 for those with a family history of cancer and $500 for those without. If an insurer uses a single premium for a diverse group, it risks adverse selection, potentially leading to financial losses.
Step-by-step explanation:
The actuarially fair premium is the premium that, on average, will cover the expected payout for insurance claims. To calculate this, you consider the probability of the insured event occurring and the cost that the insurance company will incur if the event does happen.
Actuarially Fair Premium for Health Insurance
For Mordecai: He faces a 25% chance of becoming ill. If he is ill, his income is $20,000, and if he is healthy, it is $65,000. We don't have direct information about the insurance payout, so we will simplify the problem by calculating the expected income. The expected income is (0.25 * $20,000) + (0.75 * $65,000) = $5,000 + $48,750 = $53,750. The difference between his income when healthy and the expected income is the actuarially fair premium: $65,000 - $53,750 = $11,250.
Actuarially Fair Premium for Life Insurance
a) Selling separately to each group: For those with a family history of cancer (20% of 1,000 men), the risk of dying in the next year is 1 in 50. Therefore, the premium should cover this risk: (1/50) * 200 people * $100,000 = $400,000; per person premium = $400,000 / 200 = $2,000. For those without a family history (80% of 1,000 men), the risk is 1 in 200. The calculation is similar: (1/200) * 800 people * $100,000 = $400,000; per person premium = $400,000 / 800 = $500.
b) For the entire group without considering family history: The total risk is a weighted average of both groups. (1/50 * 200 + 1/200 * 800)/1,000 * $100,000 = ($4,000 + $4,000)/1,000 = $8,000 / 1,000 = $8 per person.
c) Charging the group the actuarially fair premium could lead to adverse selection where those in higher risk groups are more likely to purchase insurance, while lower risk individuals opt out, eventually resulting in financial losses for the insurance company.