2. Suppose you have a house worth $900,000. There are 25% chance that a burglar

Question

2. Suppose you have a house worth $900,000. There are 25% chance that a burglar will vandalize your house so that your house only worth $640,000 if it gets vandalized. To prevent this, you can buy home insurance at $5,000 cost. Your utility is given by vW, where W is your wealth. a) Show that you would prefer to insure your house than not to insure Now, suppose there is deductible of $1,000 for the insurance. However, the insurance only cost $3,000 this time. b) Show that you would prefer insurance with deductible to insurance without deductible

Explanation / Answer

Answer:

In the beginning of the year, owner has $900000 worth of his house. But at the end of the year his wealth would be

W = 900000 with pr = 0.75 or 640000 with pr = 0.25 . Thus his expected wealth would be

W* = 0.75 * 900000 + 0.25 * 640000 = 675000 + 160000 = 835000

Since his utility function is U = W^1/2 , his expected utility is

E[U(W)] = 0.75 * 900000^1/2 + 0.25 * 640000^1/2

= 0.75 * 948.68 + 0.25 * 800

= 711.51 + 200 = 911.51 ..................... (1)

Utility of expected wealth U(W*) = 835000^1/2 = 913.78

Since U(W*) > E[U(W)] , thus the owner will prefer to insure his house

Let P be the amount of insurance premium the owner is willing to pay then expected utility under insurance is

E[U(insurance)] = 0.75 * U(900000 - P) + 0.25 * U(900000 - P)

[Since the owner will get the full amount of loss in case the house is vandalize thus in both the situation he will have $900000 worth of wealth at the end of the year but he has to pay the insurance premium in both the situations]

When the amount of insurance premium is $5000

E[U(P = 5000)] = (900000 - 5000)^1/2 = 895000^1/2 = 946.044 .............. (2)

Now when there is a a deductible of $1000 but the insurance premium is $3000 then his expected utility is given by

E[U(insurance)] = 0.75 * U(900000 - P) + 0.25 * U(899000 - P)

E[U(P = 3000)] = 0.75 * (900000 - 3000)^1/2 + 0.25 * (899000 - 3000)^1/2

= 0.75 * 897000^1/2 + 0.25 * 896000^1/2

= 0.75 * 947.1 + 0.25 * 946.57

= 710.33 + 236.64 = 946.97 .............. (3)

Comparing (2) and (3), the expected utility of owner under insurance is greater when the insurance premium is $3000 with $1000 deductible than when the insurance premium is $5000 with no deductible

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