What Causes Bank Runs

by Hon Sing Lee, March 19 2002

Banks existed for centuries (by now).  It started from a simple shop which gathers funds from depositors and lends out to borrowers.  This simple model however, has many complications.  In particular, the depositors are allowed to withdraw the funds without much penalty, while the bank cannot so freely withdraw from borrowers.  Thus there is a natural term structure mismatch.  The bank tries to overcome the early withdrawal problem by maintaining a cash reserve, thus only lending out part of the entire deposits.  However a perfectly healthy bank could still suffer a bank run, when more depositors withdraw than what the cash reserves could cover.  If this happens, not all the depositors can get all their funds back.  To make matters worse, the withdrawal works on a first come first serve policy.  Thus only the earliest withdrawals amounting to the cash reserve can be definitely withdrawn in full; beyond the cash reserve level, withdrawals are not guaranteed.  Hence desperate depositors literally run to the bank to be among the earliest, giving this phenomenon the name Bank Run.  If the bank forces borrowers to return immediately, fire sales occur and the total funds retrieved could also be less than the total deposits.  Hence there is no guarantee, in fact it seldom happens, that all depositors get their funds back.

Why do bank runs exits?  Why couldn't the depositors wait till the bank is ready to return the deposits?  There are 3 main reasons.

(I) Individual Liquidity Shock.
The depositor could suffer an individual liquidity shock, such as hospitalization fees, loss of earnings, repair fees for natural disaster damages, etc.  Thus they have no choice but to withdraw their deposits for immediate use.  Now if the individual liquidity shock are totally independent among depositors, and there are a continuum of depositors, we may assume that the aggregate liquidity shock is non-stochastic, that is, for every modeling period, a fixed amount of deposits are withdrawn.  Then the cash reserve policy would solve the bank run problem.  However occurrences like national disasters, world depression, and currency crisis, etc. make the independence assumption non-realistic.  That is, the aggregate liquidity shock is stochastic as well.  Thus unless the cash reserve is the entire deposits, there is always a positive possibility that bank run occurs.

For example, a hyper inflation economy can lead to bank runs.

(II) Bad Loans.
Not all investments are successful.  For the unsuccessful ones, the initial principal may not be recoverable.  When this happens borrowers cannot return what they borrow from the bank.  Thus in the end, the bank has less money than the initial deposits.  The depositors know about this and rushes to withdraw their funds, thus creating a bank run.  Again if we can assume independence among occurrences of bad loans, and that there is a continuum of borrowers, then we may assume that the aggregate amount of bad loans is non-stochastic.  Given this, the bank only needs to increase on the interest amount to cover the losses from bad loans.  Now increasing interest amount do create even more bad loans.  Suppose we have a deterministic function f: interest amount ® losses in bad loans; Â ® Â.  Suppose there is a closed and bounded interval [0,a] such that f:[0,a] ® [0,a] and that f is continuous.  Then by fixed point theorem there exists f(x0) = x0.  Hence its is possible to implement an interest rate that will exactly balance the bad loans.  Then bad loans will not cause bank runs.

However country and world recessions have shown that aggregate bad loans are stochastic too.  Hence for any interest implemented, there is always a positive possibility that the bad loans exceed the interest earned.  Thus in such occurrences, bank runs are induced again.

For example in the currency crisis, lots of businesses go bankrupt and thus their loans turned bad.  This leads to bank runs.

(III)  Banks Losses.
This requires some clarification.  Note that the depositors' fund is not the bank's, as a company, internal fund.  Thus a bank's wealth is independent of its amount of deposits.  A bank, as a company, needs internal funds just as any other companies.  For example the bank pays its salaries, office rents, taxes and maintenance costs through its internal funds.  The internal funds could come from its operating profits, shareholders investment and other borrowed funds.  The bank could borrow from its depositors, that is, the bank becomes a borrower of itself.  Nevertheless, its internal funds is separate from its deposits.  What about the bank's subsidiaries?  For simplicity, and yet without loss of generality, we may exclude these subsidiaries from the depositor-borrower model.  The bank may channel deposits to its subsidiaries as well, but they are properly managed as borrowings, that is the subsidiaries are borrowers.

Thus though a bank may appear cash-rich, its deposits are like its goods inventory, which is entirely separate from its internal funds.  Thus a bank could incur profit and loss just as any other companies.  In essence, a bank incurs loss if its income is not in excess of its cost.  Furthermore, a bank goes bankrupt when it could not meet its financial commitments.

Just as when any company goes bankrupt, it has to cease its business operations, a bankrupt bank has to cease operations as well.  Thus all the loans have to be recalled, and all deposits returned.  With the paradigm of deposits as goods, this is independent of the bank's financial obligation.  Thus the danger comes from the inability of the borrowers from returning the loan.  In particular, if the bank is a borrower of itself, this danger is even greater.  Thus the depositors know this danger, and rushes to withdraw their funds, inducing a bank run. In this case, there is no bad loans or individual liquidity shocks, but a bank run nevertheless occurs.

For example the Nick Leeson case which brought down the Barings Bank London England by a loss of 1.3 billion dollars, this leads to a bank run.