3.4. A STOCHASTIC PROCESS MODEL OF CASH MANAGEMENT 115
- The mathematical expressionδskis theKronecker delta
δsk={
1 ifk=s
0 ifk 6 =s.- IfXis a random variable assuming some values includingk, theindi-
cator random variablewhere
(^1) {X=k}=
{
1 X=k
0 X 6 =k.
The indicator random variable indicates whether a random variable
assumes a value, or is in a set. The expected value of the indicator
random variable is the probability of the event.Mathematical Ideas
Background
Thereserve requirementis a bank regulation that sets the minimum re-
serves of cash a bank must hold on hand for customer deposits. This is also
called theFederal Reserve requirementor the reserve ratio. These
reserves exist so banks can satisfy cash withdrawal demands. The reserves
also help regulate the national money supply. Specifically in 2010 the Fed-
eral Reserve regulations require that the first $10.7 million are exempt from
reserve requirements. A 3 percent reserve ratio is assessed on net transaction
accounts over $10.7 million up to and including $55.2 million. A 10 percent
reserve ratio is assessed on net transaction accounts in excess of $55.2 million.
Of course, bank customers are frequently depositing and withdrawing
money so the amount of money for the reserve requirement is constantly
changing. If customers deposit more money, the cash on hand exceeds the
reserve requirement. The bank would put the excess cash to work, perhaps
by buying Treasury bills. If customers withdraw cash, the available cash can
fall below the required amount to cover the reserve requirement so the bank
gets more cash, perhaps by selling Treasury bills.
The bank has a dilemma: buying and selling the Treasury bills has a
transaction cost, so the bank does not want to buy and sell too often. On
the other hand, excess cash could be put to use by loaning it out, and so the
bank does not want to have too much cash idle. What is the optimal level
of cash that signals a time to sell, and how much should be bought or sold?