To determine the separate actuarially fair premiums, we multiply the probability of death by the benefit amount, resulting in $2,000 for those with a cancer history and $500 without. The combined actuarially fair premium calculates to $800 per person. The insurance company might face a risk of adverse selection if it only charges the combined premium.
Calculating Actuarially Fair Premium
To calculate the actuarially fair premium for each group and for the entire group as a whole, we need to consider the probability of death and the benefit amount which in this example is $100,000.
a. Separate Actuarially Fair Premium
For those with a family history of cancer (20% of 1,000 = 200 men), the chance of dying is 1 in 50. So the expected payout per person is $100,000 / 50 = $2,000. For the 80% without a family history (800 men), the risk is 1 in 200, making the expected payout $100,000 / 200 = $500 per person.
b. Combined Actuarially Fair Premium
Combining the groups, we have (200 men * $2,000) + (800 men * $500) = $400,000 + $400,000 = $800,000 expected total payout. Divided by the whole group (1,000 men), the actuarially fair premium would be $800,000 / 1,000 = $800 per person.
c. Premium Consequences
If the insurance company charges the combined fair premium to everyone, those with a lower risk might not find the premium appealing and choose not to buy the insurance, leading to a pool of primarily high-risk individuals (adverse selection), which could cause losses for the insurance company.