Question

The idea of insurance is that we all face risks that are unlikely but carry high cost. think of a fire destroying your home. so we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. an insurance company looks at the records for millions of home owners and sees that the the mean loss from fire in a year is μ = $500 per house and the standard deviation of the loss is σ = $10000. the distribution of losses is extremely right-skewed: most people have $0 loss, but a few have large losses. the company plans to sell fire insurance for $500 plus enough to cover its costs and profits. (a) explain why it would be unwise to sell only 100 policies. then explain why selling many thousands of such policies is a safe business. (b) suppose the company sells the policies for $600. if the company sells 50000 policies, what is the approximate probability that the average loss in in a year will be greater than $600?

Answer 1

Adverse selection can create financial risks for insurance companies. Selling a larger number of policies helps mitigate the impact of adverse selection. By using the Central Limit Theorem, we can estimate the probability of the average loss exceeding a certain amount in a large sample of policies.

Step-by-step explanation:

Insurance companies face the challenge of adverse selection, which occurs when only high-risk individuals are willing to purchase insurance.

If the insurance company sells only 100 policies, it is likely that the majority of policyholders will be those with higher risks, leading to a higher probability of large payouts.

This can result in financial losses for the insurance company. On the other hand, selling many thousands of policies reduces the impact of adverse selection because it allows for a more diverse pool of policyholders, including those with lower risks.

If the insurance company sells 50,000 policies with a premium of $600, we can estimate the probability of the average loss in a year being greater than $600 using the Central Limit Theorem.

With a large sample size, the distribution of average losses will tend to follow a normal curve. Given the mean loss of $500 and the standard deviation of $10,000, we can calculate the z-score for a loss of $600.

By consulting the z-table, we can find the probability that the z-score is greater than the calculated value, which would give us an estimate of the probability that the average loss is greater than $600.