Question

Suppose Bob has income of $50,000.   There is a 10% chance that Bob will get sick and lose half of his income. Suppose Bob’s income-utility relationship is given by the following equation, where I is Bob’s income.

U(I) = √2I

Use the facts above to calculate the following:

a) The premium and payout for a full and fair insurance contract.

b) The max premium (i.e. the most Bob would ever be willing to pay for insurance)

Answer 1

It is given that Bob’s utility function (U) is (2I)^{1/2 .} His income(I) is $50,000 if he doesn’t get sick and is $25000 is he does get sick, because on getting sick it is mentioned that Bob loses half his income ($50000/2= $25000). Probability of getting sick is 10%= 0.1 and probability of not getting sick is 1-0.1= 0.9

a) Before calculating the premium and payout for a full and fair insurance contract the expected utility is explained below.

U= (2I)^1/2 where I= $50000 with 90% probability of being healthy (p=0.9) and $25000 with 10% probability of falling sick (1-p= 0.1).

Therefore Expected Utility without any insurance is E(U)= p*(utility of being healthy) + (1-p)* (Utility of being sick)

E(U)= 0.9*(2*50000)^1/2 + 0.1* (2*25000)^1/2

E(U) without insurance = 284.605 + 22.361 = 306.966

Now, a fair premium is equal to probability of loss (p=0.1) times the size of loss ( $25,000).

= 0.1*25000= $2500.

Thus, a fair and full premium to cover Bob is worth $2,500, which is the amount that Bob has to pay to the insurance company in order to get a full coverage for his income loss (worth $25000), in case he does get injured. It is a fair premium because the actual value of the premium is equal to the expected value of the payout. Payout refers to the amount paid by the insurance company to Bob when he accepts the contract.

Alternatively, if Bob chooses to sign the contract, this is a fair premium because irrespective of whether gets injured or not, he certainly earns $47500. If healthy, he gets $50,000 but pays $2500 for insurance and is left with $50000-$2500= $47,500. If he falls sick, he gets $25000 (loses half = $25000), pays $2500 for insurance but is reimbursed with $25000 which covers his loss. So he is left with $25000-$2500+$25000=$47,500.

Therefore expected utility from insurance is

E(U)= 0.9*(2*47500)^1/2 + 0.1* (2*47500)^1/2

E(U) = 308.221. This expected utility from insurance is greater than that without insurance which means that Bob is risk averse and going for insurance is a better option.

b) If Bob would not guaranteed himself with the insurance and would have gambled, then, the E(U) without insurance was 306.966 (Calculated in the first half).

Income corresponding to this level of expected utility could be derived from the original income-utility function.

Plugging the values in, 306.966=(2I)^1/2

Squaring both sides and solving for income (I), I= $47114.063 , which is roughly $47114 (rounded off to the nearest whole number).

Thus, Bob’s maximum willingness to pay would be his guaranteed income without falling sick minus the income corresponding to the initial level of expected utility before taking the insurance.

Bob’s maximum willingness to pay for this insurance is thus $50000-$47114 = $2886.