Question

You have 2 million dollars of wealth. You build $1, 500, 000 house at the beginning of the year on the east coast of Florida. During the year there is a 10% chance that the house will be destroyed by a hurricane-related flood. However, you are able to purchase flood insurance for a premium of $10 per $100 dollars of insured house value for the year. In the event that your home is destroyed by a flood, the insurance company makes a payment equal to the insured house value. You are a risk averse agent with utility of end-of-year wealth given by U(W) = −W−1 . Assume that the interest rate for the year is 0% and that the value of the house, if not flooded, remains unchanged at the end of the year.

1. Let x be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the period if the house is not destroyed by the flood?

2. Let x be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the period if the house is destroyed by the flood?

3. What is expected utility in terms of the quantities given in questions 1 and 2?

4. What percentage of the house value did you choose to insure?

5. Given the relationship between the premium and the insured value, is the expected profit for the insurance company positive? Is the insurance contract actuarially fair?

Answer 1

Part (1)

Premium paid = $ 10 per $ 100 of sum insured = 10% = 0.1x

So, wealth at the end of year if no flood, W1 = Your wealth - premium paid = 2,000,000 - 0.1x

Part (2)

So, wealth at the end of year if there is flood, W2 = Your wealth - cost of the house + sum insured - premium paid = 2,000,000 - 1,500,000 + x - 0.1x = 500,000 + 0.9x

Part (3)

U(W1) = -1/W1 = -1/(2,000,000 - 0.1x); p1 = 1 - p2 = 1 - 10% = 90% = 0.9

U(W2) = -1/W2 = -1/(500,000 + 0.9x); p2 = 10% = 0.1

Hence, Expected utility = E(U(W)) = p1 x U(W1) + p2 x U(W2) = -0.9/(2,000,000 - 0.1x) - 0.1/(500,000 + 0.9x)

Part (4)

You will choose to insure that %age of value that will maximize your expected utility.

Hence, we need to find that x at which E(U(W)) is maximum. Without even thinking, differentiate E(U(W)) w.r.t x and equate it to 0

Hence, d[E(U(W))] / dx = 0

Hence, -0.9 x 0.1 / (2,000,000 - 0.1x)² + 0.1 x 0.9 / (500,000 + 0.9x)² = 0

Simplify carefully to get,

Hence, 2,000,000 - 0.1x = 500,000 + 0.9x

Or, x = 2,000,000 - 500,000 = 1,500,000 = 100% of the cost of the house.

Hence, you will choose to insure the entire 100% of the value of the house.

Part (5)

The insurance companies cash flows:

In case of no flood = 0.1x

and in case of flood = 0.1x - x = -0.9x

Hence, expected value = 0.9 x 0.1x + 0.1 x (-0.9x) = 0

So, the expected profit is not positive. It's actually zero.

Hence, the contract is indeed actuarially fair.