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Risk Averse Investment for a Banking System

Consider two individuals living in the country of Depressia. Depressian banking system is very unstable. On average, 5% of all banks collapse every year. Each of these individuals is considering depositing all their life savings - $10,000 - in a bank for one year. Banks pay fixed interest, so that by the end of the year all would-be bank clients should have their principals back plus interest. If the bank goes bankrupt all money is lost. The only alternative way of making savings is simply by keeping money (cash) in a piggy bank. (There is zero inflation in Depressia).

One of these individuals is risk-neutral, another is risk-averse with the utility of money (wealth) given by U(I) = I0.5.

a) Will either of these individuals be willing to deposit their savings if banks pay 5% annual interest?
b) Will either of these individuals be willing to deposit their savings if banks pay 8% annual interest?
c) Depressian government introduces tougher banking regulations that result in the probability of bankruptcy decreasing to 2.5%. How will this change your answers to (a) and (b)?
d) If the risk-averse individual decides to deposit his savings at 8% interest in these new conditions (i.e. when the risk of bankruptcy is only 2.5% per year), will he be interested in buying an insurance against bankruptcy (deposit insurance)? If yes, how much will he be willing to pay (at most) for such an insurance policy? Will a reasonable private insurance company be willing to offer deposit insurance policies?
e) What problems may arise in the market for deposit insurance? Discuss. Why do you think (based on the preceding discussion) deposit insurance is offered in the US by the government and not by private companies?
f) Explain why deposit insurance was impossible before the introduction of tougher banking regulation, i.e. when the probability of a bankruptcy in Depressia was 5%? (Consider the case of 8 percent interest paid by the banks.)

Solution Summary

This solution provides step by step equations and detailed explanations for various problems regarding risk averse investment.