Question

1. Jill has $200,000 of valuables, faces a probability of .02 of burglary, in which she would lose jewelry worth $50,000. She can buy full insurance for $15,000 that would fully reimburse the $50,000 loss. Her utility function is u(x) = 2*sqrt(x) What is the actuarially fair price for the insurance policy? Should Jill buy this insurance policy? What is the most that Jill is willing to pay for an insurance policy that fully covers her against loss?

Answer 1

Probability of burglary=p=0.02

Wealth in case of burglary=200000-50000=$150000

Utility in case of burglary=U(150000)=2*(150000)^0.5=774.5967 utils

Probability of no burglary=1-p=1-0.02=0.98

Wealth in case of no burglary=$200000

Utility in case of no burglary=U(200000)=2*(200000)^0.5=894.4272 utils

Expected utility=p*U(150000)+(1-p)*U(200000)=0.02*774.5967+0.98*894.4272=892.0306 utils

Actuarially fair price for the insurance for full coverage=Probability of loss*Loss amount

Actuarially fair price for the insurance for full coverage=0.02*50000=$1000

Let he buys the insurance at a price of $1000

Utility in case of burglary=U(200000-50000+50000-1000)=U(199000)=2*199000^0.5=892.1883 utils

Utility in case of no burglary=U(200000-1000)=U(199000)=2*199000^0.5=892.1883 utils

Expected utility in case he buys insurance=p*U(199000)+(1-p)*U(199000)=U(199000)=892.1883

We can observe expected utility has increased from 892.0306 utils to 892.1883 if agent buys insurance at actuarially fair premium. He should buy the insurance at this price.

Let X be Maximum willingness to pay for insurance.

So, U(200000-X)=Expected utility

2*(200000-X)^0.5=892.0306

(200000-X)=(892.0306/2)²=198929.65

X=200000-198929.65=$1070.35

Maximum willingness to pay for insurance=$1070.35