Problem 2· Insurance premium. Suppose Joe\'s income is $50,000, and there is a 5

Question

Problem 2· Insurance premium. Suppose Joe's income is $50,000, and there is a 5% chance he has a car accident that will generate $15,000 in loss (1) Express this risky situation in the form of a lottery. (2) What is the lottery if Joe buys full insurance and pays an insurance premium of $x? (3) What is the actuarially fair insurance premium in this context? Suppose Joe's utility is derived according to u(x)- In(x) Reminder of some properties of logs: If Y-In(z), then eY = exply) = x Y = ln(x*) aln(x) Y = In(zz) = ln(x) + ln(z) (4) What is Joe's expected utility without insurance in this context? (5) What is Joe's expected utility if he fully insure for a premium of Sx? (6) What is the maximum insurance premium Sx that Joe is willing to pay in order to fully insure himself against this risk of a car accident? (7) Is this maximum premium more or less than the actuarially fair premium in (3)? (8) Bonus: what about the choice for the utility function (u(x) = ln(x)) explains your answer to (7)? i.e. How could we have know this result just by looking at a specific property of the utility function?

Explanation / Answer

given income : $50000

chances of accident :5% (it means 5/100=1/20 ..so 1 time accident happen in 20 months )

accident loss :$15000.

1) if we go with the risky condition.then it means he have accident .

so he loss $15000

so income left with him is :50000-15000=$35000

2) if he get full insurance then he need to pay $x per month .

so in 20 month he pay only $20x

and according to our above condition 5% chance of accident so it means once in 20 month he stuck with accident .

so it means he loss $15000

so he get profit of =$15000-$20x

4) if joe not take any insurance .so it means 5% chances of accident

if we want monthly risk then its :15000/20=$750

so he have risk of $750 every month ..

and if take insurance then that amount is less then $750

5) so his expected utility is less then$750 insurance per month  

beacause as we calculate in ques 4 .

that if not take insurance then chances is $750.

6)less then$750 he want to pay.

so for calculating premimum we need to know about the formula.how to calculate premimum.

thanks ...if any issue then comment .

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