Question

Question 7

Suppose Alana has personal wealth of $10,000 and there is a probability of 0.2 of losing her car worth $6,400 in an accident.   Her utility (of wealth) function is given by  u(w) =  w^0.5, where  w  is wealth.     

(a) What is Alana's expected wealth, expected utility, and utility of expected wealth? If she can insure "fully", and if this insurance is fair, how much would it cost her?

(b) What is the maximum amount Alana would be prepared to pay for full insurance? What is the certainty equivalent and the risk premium associated with the uncertain situation she is in if she does not have any insurance? What difference would it make if her utility of wealth function were instead  u(w) = 5w?

Answer 1

Question 7) a) Alana’s expected wealth =

= 0.8*10000 + 0.2*3600

= $8720.

Expected utility = 0.8*10000^0.5 + 0.2*3600^0.5

= 0.8*100 + 0.2*60

= 80+12

= 92

Utility of expected wealth = 8720^0.5 = 93.381.

If the car can be insured fully, then the fair premium that Alana would have to pay =

= Probability of accident * Loss from accident

= 0.2*6400

= $1280

b) The maximum amount Alana would be ready to pay for the insurance =

The maximum insurance that would be paid will be the amount for which the wealth after payment of premium should be equal to the expected utility.

So, Expected utility = 92.

The income for which the utility would be 92 =

As utility = wealth^0.5

Wealth = Utility²

= 92² = $8464.

So, the maximum premium that he would be willing to pay = 10000 - 8464 = $1536.

Certainty equivalent if the insurance is not taken =

Expected utility if the insurance is not taken = 92.

So, the certainty equivalent will be a payment for which the utility of the payment is equal to the expected utility = 92² = 8464

Now, if the utility function changes and becomes

u(w) = 5w.

The expected utility, if the insurance is not taken =

= 0.8*5*10000 + 0.2*5*3600

= 40000 + 3600

= 43600.

The amount which would give us a utility of 43600 =

= 5w = 43600

w = $8720.

So, the certainty equivalent amount would be $8720.

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