You are the manager of a small bank; your business model is to raise money by offering depositors an interest rate and the ability to withdraw money any time they want (deposits are fully liquid). You then use this money to lend to small businesses.

The typical small business takes two years to get off the ground. If you lend it money for two years at a 20% annual interest rate, it will return the principal and interest to you with certainty in two years. However, if after one year you ask for your investment back, the business will be forced to liquidate without making any money and you will recover only 85% of your investment. This is because even though it can sell some of its assets it has already incurred costs that cant be recouped. Alternatively, you can invest your money in risk-free government bonds with an annual interest rate of 1%.To finance investment, you raise money from depositors; assume you were able to raise 1M. Your competitor is offering a 4% interest rate on deposits so you cant offer any less. Unfortunately, you do not know for sure when depositors will withdraw their money.

A. Suppose you dont know exactly how many depositors will withdraw after one year, but you estimate that with an 80% probability 40% of depositors will withdraw after one year and with a 20% probability 90% of depositors will withdraw after one year. Calculate your expected profit if you wish to guarantee that you will never need to liquidate any business loans. (Assume there is no limited liability i.e. if the bank loses money you have to pay from your own personal wealth to refund investors). [6 marks]

B. Calculate your expected profit if instead you invest 40% of your money in business loans (Assume there is no limited liability). [7 marks]

C. Suppose limited liability existed, this means that you (the bank manager or equity holder) can walk away with no losses if the bank was insolvent. In this case, when there is no deposit insurance, the depositors may end up losing some of their money. From the aggregate welfare standpoint, is the society better off in (A) or (B)? [7 marks]

D. Suppose you are a depositor in (B) who is planning to withdraw late and you live in a world of limited liability. Given that you know what the banks optimal behavior is (by calculating (A) and (B)), would you still wait to withdraw late? Will it be possible for this economy to exist in (B)? Comment on deposit insurance. [10 marks]