Question

Emma is considering taking a health insurance. The probability that she becomes sick is 0.5. If she keeps being healthy, she will earn $10,000, but if she becomes sick, she will end up with earning only $1,600. With all the information given, answer the following questions:

a. Assume that Emma is risk averse, and her utility has a form of √? where X is her income. The health insurance company would like to offer an unfair insurance to her. Then, what would be the possible maximum premium that the company can offer to her, expecting Emma’s acceptance?

b. Now assume that Emma is risk neutral. Then, what would be the possible maximum premium that the company can offer to her, expecting Emma’s acceptance?

Answer 1

“Fair” insurance is insurance whose premium (policy price) is equal to the expected loss for whatever is being insured. The maximum premium is the premium that’s just high enough to make the customer indifferent between buying and not buying the policy.

(a) Since insurance increases the consumer’s welfare, s/he will be willing to pay some positive price in excess of the actuarially fair premium to defray risk.

Consider a person with an initial endowment consisting of three things: A level of wealth w0 ($10,000); a probability of an accident of p (0.5); and the amount of the loss, L ($10,000 - $1,600 = $8,400) should a loss occur.

Expected utility if uninsured is:

E(U|I = 0) = (1 − p)U(w0) + pU(w0 − L)

E(U|I = 0) = (1 − 0.5)*sqrt(10,000) + 0.5*sqrt(1600)

E(U|I = 0) = 50 + 20 = 70

Premium = p·A where p is the expected probability of a claim, and A is the amount that the insurance company will pay in the event of an accident.

Expected utility if insured is:

E(U|I = 1) = (1 − p)U(w0 − pA) + pU(wo − L + A − pA)

The optimal amount of insurance that the consumer should purchase insurance amount (A) equal to Loss (L)

E(U|I = 1) = (1 − 0.5)*sqrt(10,000 − premium) + 0.5*sqrt(10,000 − premium)

For maximum premium

Expected utility if insured = Expected utility if uninsured

sqrt(10,000 − premium) = 70

Solving for max premium,

Pmax = $5,100

(b) The maximum amount of premium that a risk neutral person will pay is equal to the premium for a "fair policy".

That is,

Expected amount of loss = 0.5*0 + 0.5*($10,000 - $1600) = $4,200

Therefore, a risk-neutral person would be willing to pay a premium of $4200 for this insurance policy.