Question

Betty Bat loves the Vancouver Canucks. She has followed their exploits since she was five years old and believes they will win the Stanley Cup this season. She has $1,000 savings that she has hidden under her bed. She could spend $600 of the $1,000 in making Canucks championship paraphernalia: buttons, cups pens and so on. Then, if the Canucks win, she estimates her total revenue would be $1,500. If the Canucks lose, she won’t be able to sell any of her stock. Betty figures that the Canucks

have a 0.6 chance of winning the Stanley Cup. Her utility function is given by u(M )= M .

a) Will Betty make the $600 investment into Canucks championship gadgets?

b) Calculate the certainty equivalent of Betty’s prospect.

c) Suppose that a friend offer Betty insurance. He says to Betty, “If you pay me F dollars whether or not the Canucks win, then in the event that the Canucks lose, I will pay you $1,500, the amount that you would have earned had the Canucks win the Stanley Cup. If the Canucks win, I will pay you nothing.” What is the maximum value of F that Betty is willing to pay for the insurance policy? If Betty’s friend is risk neutral, will he gain by this venture? Explain.

Answer 1

a) If Vancouver wins Betty's wealth = $400+$1500 = $1900 (400 from the $1000 in savings and $1500 from sales)

If Vancouver loses Betty's wealth = $400 (as she will lose all the money she made from her investment)

Her utitlity function is given by : u(M) = M (where we assume M to be Betty's wealth)

Thus Betty's expected utility if Vancouver has a 0.6 chance of winning is :

EU = 0.6*u(1900) + 0.4*u(400)

= 0.6*1900 + 0.4*400

= 1300

Her utility initially was u($1000) = 1000

Since 1300>1000, her expected utility from making as investment is greater than utility by not maing the investment thus she will make the investment.

b) Certainty equivalent is the amount of money with certainty that Betty would accept such that she would yield the same utility as making the risky investment.

Certainity equivalent that Betty would accept = CE = u-1(1300) = $1300

c) Her expected utility from not having having insurance is EU(no insurance) = EU($1300) = 1300

Let F be the max insurance premium she is willing to pay

Her expected utility if she pays the insurance premium F is:

  EU(insurance) = E($1500-F) = 1500 - F

To find max value Betty will be willing to pay,

EU(insurance) = EU(no insurance)

1500-F = 1300

F = $200

Thus Betty will pay a maximum of $200 for the insurance policy.

For Betty's friend, since he is risk neutral, his utility function can be assumed to be EU(M) = M

His expected utility if he offers this insurance policy will therefore be

= EU = 0.4*F + 0.6*(F-1500)

= 0.4*200 + 0.6*(200-1500)

= -700

Thus he will have a negative expected utility if he is risk neutral and thus he will lose by this venture