Question

Suppose an individual has a job that pays $50,000/year. With a 5% probability, next year their wage will be reduced to $20,000/year.

a. What is their expected next year?

b. Suppose that the individual can insure themselves against the risk of reduced consumption next year. What would be the actuarially fair insurance premium?

Answer 1

a) The probability that your income next year will be $50,000 is 0.95;

the probability that your income next year will be $20,000 is .05.

Summing the expected values of the out-comes yields

0.95($50,000) + 0.05($20,000) = $47,500 + $1,000 = $48,500.

This is your expected income next year.

b) An actuarially fair premium would be one that exactly offset the expected value of the loss.

In this case, the expected loss is $50,000 – $48,500 = $1,500,

so an actuarially fair premium would be $1,500.

Another (equivalent) way to determine this premium is to calculate the expected value of the claims the insurance company would pay; here it is the probability of the loss occurring times the dollar value of the loss, or 0.05 ($50,000 – $20,000) = .05($30,000) = $1,500

A third (equivalent) way to determine the premium is to compute the expected profits to the insurance provider for any given premium P. Since the premium is surely paid and the insurance company pays you $30,000 with a probability of 0.05, the expected profits are given by P=0.05($30,000).

Actuarially fair premiums are those that lead to zero expected profits; setting expected profits equal to zero and solving gives P= 0.05($30,000) = $1,500 again