Bank A currently has $100M in assets, with $90 in deposits (only liability) and $10M in equity

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Question:

Bank A currently has $100M in assets, with $90 in deposits (only liability) and $10M in equity capital (all book values). Management has described that $40M of the total assets are liquid and could be quickly sold at their book value (e.g., cash and treasuries). The remaining $60M of total assets consists of illiquid assets such as loans and illiquid securities (e.g., MBS). In case these illiquid assets had to be quickly liquidated, they would be sold at a 50% discount of their book value. The bank has both insured and noninsured deposits, which are fully covered and not covered by deposit insurance, respectively. Insured and noninsured deposits are equal to $30M and $60M, respectively. Kevin has noninsured deposits in the bank and is concerned about a potential run on the bank today due to rumors. There are only two dates (days) in this example.

Today: All depositors will decide to keep or not their deposits in the bank. The bank will allow all depositors to withdraw their deposits as long as it can raise funds to pay them. The bank cannot raise new equity or deposits. It will meet withdraws by first selling the liquid assets and then selling illiquid assets (when there are no liquid assets left).

Tomorrow: Regulators will evaluate the assets and deposits of the bank. If the assets are below deposits the bank will be declared insolvent and liquidated. The funds from the liquidation will be split among all depositors, and the deposit insurance will cover any possible loss on deposits among insured depositors. If the assets are greater or equal than the deposits the bank will not be liquidated and deposits will be worth their full value.

Depositors have an option between running to the bank today to withdraw their deposits and waiting for tomorrow. Running to the bank today is costly (small cost). Note that, in the absence of a liquidation of the bank tomorrow, depositors will prefer to wait (deposits will have full value and avoids cost of running). Note that insured depositors will also never run as they are fully covered tomorrow independently of a possible liquidation of the bank.

Suppose that Kevin is expecting a run by all other (uninsured) depositors today (panic scenario).

(a) Will the bank have enough funds to cover all withdrawals?

(b) What will be the value of the bank’s assets and deposits tomorrow?

(c) Will the bank be liquidated or not tomorrow?

(d) If Kevin expected all other noninsured depositors to run on the bank today, would he run as well? Would this lead to “self-fulfilling” bank run or panic, i.e. a situation where an expectation by all depositors that others would run materializes into an actual run by all noninsured depositors? Explain.

(e) Suppose now that insured and noninsured depositors are equal to $60M and $30M, respectively. All other assumptions remain the same. Show how your answers to each question (a)-(d) would change (if change at all).

(f) In the context of your previous analyses, can partial deposit insurance (i.e., deposit insurance covering only a fraction of deposits, i.e. not all deposits) prevent bank runs? Explain.