Question

You would like to buy home insurance for a year. The value of your home at the end of the year will be $100; after a fire the value of the remaining lot will be $20. The probability of fire is Pf= 0.25. You are risk-averse and would like to maximize Expected Utility or E(U). Your Utility function is U = M0.5where M = money. Graph your U function and map the prospects and their corresponding utility levels; additionally, compute and map your E(U) as well as the E(M). What is the maximum premium (MP) you would be willing to pay to insure your home for a year? Show that no matter what happens (fire or no fire) you end up with your E(U). Show that your U[E(M)] is greater than your E(U); what does U[E(M)] > E(U) prove?

Answer 1

U=M^0.5

Expected Value of Home=E(M)=Prob of Fire* Value of home after fire+(1-Prob of fire)*Value of home =0.25*20+0.75*100=80

E(M)=80

Expected Utility=E(U)=0.25*sqrt(20)+0.75*sqrt(100)=8.618

Certainty equivalent is inverse 0f E(U)=8.618^2=74.26

Now Risk Premium =E(M)-CE=80-74.26=5.74

Hence actuarily fair premium will be 5.74

U(E(M))=80^0.5=8.9442> E(U)=8.618

As U(E(M)) is greater than E(U) this means E(M)>CE which explains that given consumer is risk averse