Question

Arnold believes he faces health costs in the current year of either $1000 with
probability 0.8 or costs of $5000 with probability 0.2.
(a) Find the mean and variance of the costs that Arnold faces. (For the variance, you could turn this into a sample of 10 individuals. You may also use proportions instead and have the total number of observations be 1).
(b) Given your answer in part (a), should Arnold purchase an insurance policy that
completely covers his losses if the annual premium of the policy is $2000? Explain your
answer.
(c) The company that offered this insurance policy insures 10,000 people with exactly the
same distribution of health losses as that of Arnold. Calculate the standard error that the
insurance company faces in the average claim per individual insured. (Hint: use your
answer in part (a)).
(d) It can be shown that 95 percent of the time the average claim size will equal the
expected average claim plus or minus 1.96 times the standard error of the average
claim. Give this range for the insurance company considered in part (c). Is the insurance
company taking a large risk in selling this insurance policy?

Answer 1

(a) Mean of the cost Arnold faces = $1000 * 0.8 + $ 5000 * 0.2 = $ 1800

Variance of the cost Arnold faces = 0.8 * (1800 - 1000)² + 0.2 * (5000 - 1800)² = 25,60,000

(b) Here the expected loss is less than the premium insurance company desires so here Arnold should not purchase the insurance policy cover of $ 2000

(c) Here the standard deviation of cost faces by a single person = sqrt(2560000) = $ 1600

so if there are 10000 persons

then standard error that the insurance company faces in the average claim per individual insured = 1600/sqrt(10000) = $ 16

(d) Here the 95% confidence interval = 2000 + 1.96 * 16 = ($ 1968.64, $ 2031.36)

given this range the insurance company is not taking large risk as expected health costs would be $ 1800 which is way out of the range.