Question

In view of the large wave of burglaries, Stephen is keen to take care of damage caused by burglars. Stephen's current consumption is US. 400,000 per month, but if it were violated, it could be assumed that its consumption would fall to US 200,000 a month while he is building back his stuff instead of being stolen. The probability of burglary is 10% and you can describe the expected utility of Stephen with u (x) = x^3/4

a) What is the expected value and what is Stephen's expected utility in these circumstances?

b) How does the answer change if the incidence of burglary increases to 20%?

c) Find certainty equivalent and risk premium for a and b items.

Answer 1

Answer for a)

Expected Value=Prob.of Burglary(Wealth after burglary)+(1-Prob of Burglary)*(Wealth without burglary)

EV(x)=0.1(200,000)+0.9(400,000)=380,000

EU(x)=Prob.of Burglary(Utility after burglary)+(1-Prob of Burglary)*(Utility without burglary)

=0.1*(200,000)^0.75+0.9*(400,000)^0..75=15,260.61

Hence we know Certainty amount=U(CE(x))=EU(x)

U(CE(x))=15260.91

CE(x)=(15260.91)^4/3=$378536.05

Now CE(x)+Actuarily fair Risk Premium= E(x)

378,536.05+=380,000

=2465.95 in case a

Now of Prob of Burglary is 20% then

EV(x)=0.2(200,000)+0.8(400,000)=340,000

EU(x)=Prob.of Burglary(Utility after burglary)+(1-Prob of Burglary)*(Utility without burglary)

=0.2*(200,000)^0.75+0.8*(400,000)^0..75=13343

Hence we know Certainty amount=U(CE(x))=EU(x)

U(CE(x))=13343.38

CE(x)=(13343.38)^4/3=$316,485.98

Now CE(x)+Actuarily fair Risk Premium= E(x)

316,485.98+=340,000

=23,514.02 in case b