Final answer:
An actuarially fair premium reflects the actual risk of the insured event occurring. For those with a family history of cancer, the fair premium would be $2,000, while for those without it would be $500. If the insurance company charges a combined average premium, they may face adverse selection which could lead to financial losses.
Step-by-step explanation:
The question you've asked pertains to the calculation of actuarially fair premiums for a life insurance policy, specifically for groups with different mortality risks due to family history of cancer. The context provided describes two groups of 50-year-old men: those with and without a family history of cancer, with mortality risks specified as 1 in 50 and 1 in 200 respectively, for a policy that pays out $100,000 upon death.
Actuarially Fair Premiums
To calculate the actuarially fair premium for each subgroup, we take the probability of the event (death in this case) and multiply it by the payout amount. For the group with a family history of cancer (20% of 1,000 men, so 200 men), the probability is 1/50, leading to an expected payout per person of $100,000/50 = $2,000. Therefore, the fair premium for this group would be $2,000.
For the group without a family history of cancer (80% of 1,000 men, so 800 men), the probability is 1/200, resulting in an expected payout per person of $100,000/200 = $500. The fair premium for this group would be $500.
Combined Group Premium
If the insurance company cannot distinguish between the groups, it must consider the average risk across all 1,000 men. The combined actuarial premium can be found by multiplying the relative sizes of each group by their respective risks and summing the results. This combines the risk of both the high-risk and low-risk groups.
Insurance Company's Challenge
When the insurance company charges the combined actuarially fair premium, it may face adverse selection, as the higher-risk individuals will find the policy more attractive compared to those who are at lower risk. If the lower-risk individuals choose not to purchase insurance at the higher combined premium, the insurance company will end up with a pool of higher-risk clients, which may result in financial losses as the actual claims could be higher than the predicted payouts based on the combined risk.