An individual is facing an outcome of an annual financial loss (in tens of thousands of dollars) whose probability density function is given by
The probability of a loss in the next year is 0.08. If there is a loss, there is only one loss in any given year. An insurance policy is available to protect against the financial loss by paying in full when a loss occurs.
- What is the probability that the insurer’s payment to the insured will exceed $50,000?
- What is the mean payment made by the insurer to the insured?
- What is the variance of the amount of payment made by the insurer?
Suppose that instead of buying a policy that pays the loss in full, the individual buys a policy that has a 80/20 coinsurance provision, i.e., the insurance company pays 80% of the loss and the insured retains the remaining 20% of a loss. Answer the same three questions.
Solution is found below.
Solution to Problem 10a
Let be the loss variable as described in the problem. Then the following is the probability .
One important thing to keep in mind is that the occurrence of a financial loss is not certain. So the answer to question #1 is not . Let be the insurance payment to the insured. Note that is conditional on the occurrence of a loss. If the loss does not happen, . If the loss does happen, . Likewise, in case of no loss and in case of a loss. So we can use the law of total probability to obtain .
The answers to the other two questions can also be obtained by using the law of total probability.
Answers to Problem 10b