Question

You currently have $1,440,000 of wealth, which you keep in the form of two assets: a bank account worth $1,400,000 and $40,000 in cash in your home safe. However, while you are confident that your home safe is secure, you are concerned that there is a probability ?>0 that your bank will fail (i.e., declare bankruptcy), in which case you would lose all the money in your bank account. (Assume this is the only bank available to you.) The von Neumann-Morgenstern utility function you use to evaluate this risk is ?(?)=√?, where ? is the amount of wealth you have.

Since the potential loss would clearly be devastating, an insurance company offers the following insurance contract: in exchange for an insurance premium of $440,000 that would be paid immediately, the insurance company would promise to pay you $960,000 if your bank failed. You would thus have a total wealth of $960,000 + $40,000 = $1 million regardless of whether the bank survived.

a. What is your expected utility (i.) if you buy the policy, and (ii.) if you do not?

b. Find the minimum probability (call it ????) of the bank failing at which you would buy the policy.

c. What is the insurance company’s expected payoff if you buy the policy? Your answer should be a mathematical expression involving the probability ?.

d. If the insurance company is risk neutral, what is the maximum probability (call it ????) of the bank failing at which the insurance company would be willing to offer this insurance policy?

Answer 1

solution:

a) If buy the policy expected utility=√1000000=1000, without insurance the utility will be (1-p)*√1440000+p*√40,000=(1-p)*1200+200p =1200-1000p

b) you would buy the policy at pmin where utility with policy or without policy is same Hence, 1000=1200-1000pmin gives pmin=200/1000=20%

c) If we buy the policy expected pay off for the insurance company = Insurance premium- expected payout payout=440000-p*960000

d) maximum probability pmax can be calculated when expected pay off toninsurance company is 0 i.e.440000=pmax*960000 gives pmax=45.83%