Conversations with a banker: ATMs and the Money Supply

I met a friend of mine last weekend who holds a regional manager-type position at a bank.  We got to talking about banking and economics (go figure) and she described a particularly interesting problem they were discussing at work: What is the optimal policy to refill an ATM with cash?

I've never really thought about it before - I guess I always assumed that once the ATM starts to run out of cash, they send a truck over and refill it.  Turns out, that's not the case - at least for this particular bank in this region [for confidentiality reasons I can't specify the region or the bank].

A couple of relevant points:

• Refilling the ATMs is contracted out to a 3rd party.  It turns out that it is far cheaper to refill them on a regular schedule with a planned route than monitor the ATMs and send people out as needed (though this is always an option).  The amount of money that is withdrawn in a day is fairly predictable.   The yearly cost to refill ATMs is proportional to the number of times they are filled up.
• ATMs are rarely filled to capacity, because a dollar in an ATM is a dollar not out earning interest.  If the interest rate is 6% and if on average an ATM has $50,000 in it, the opportunity cost of having the money in the ATM is $3,000 per year.

The ATMs are filled up such that they run out of money around 1% of the time before they are refilled.  This happens when the amount of money taken out is 2.5 standard deviations from the average (where the mean and sd will differ based on seasonality and day-of-the week).  If an ATM typically dispenses $30,000 over a period of time with an s.d. of $10,000, then we'd see it filled up to $55,000.

The bank's existing policy is to fill up each ATM 6 days a week.  However, this policy was enacted when the overnight rate was 5.5%.  The overnight rate is the interest rate considered, since any excess/shortfall is borrowed on the overnight market.  The policy was optimal when rates were so high, but at 1% it turns out there are significant cost savings to dropping down to a 3 days a week schedule, since this cuts the fill-up costs in half.  It does nearly double the amount of money put into ATMs during each fill-up (question for students: Why doesn't it exactly double it?) and thus nearly doubles the opportunity cost from interest rates.  However, since nominal interest rates are so low, this is a worthwhile tradeoff.

My banker friend believed that her competition would come to a similar conclusion and they'd adjust their policies accordingly.  If so, the amount of money held by ATMs should be significantly negatively correlated with interest rates.  Question for our macroeconomists: Is this something we need to consider when implementing monetary policy?

Comments

[spencer]

Wouldn't the money in an ATM count as vault cash, so as far as monetary policy is concerned it does not matter whether the cash is in an ATM or in the bank's vault or a teller's cash drawer.

[Jeffrey Yasskin]

http://en.wikipedia.org/wiki/Automated_teller_machine#Global_use says there are <2M ATMs worldwide. If they were all storing dollars, and all tended to hold on average $50000, then halving the refill rate would mean an extra $50000 (well, slightly less) in 2M ATMs = ~$100B extra demand for dollars). That's significant compared to the total supply of currency, implying that monetary policymakers should take it into account. I'm guessing it's a significant overestimate, though. If only 1M ATMs store dollars, and they each hold on average $10000, then the net change would be $10B, which isn't as significant, and even that may be too high.

If monetary policy targeted inflation or NGDP expectations directly instead of the interest rate, this wouldn't be an issue.

[Evan Harper]

Agree with Spencer. Vault cash counts as bank reserves. Why don't ATMs count as bank vaults?

[Nick Rowe]

lovely example Mike!

I'm still puzzling over your "question for students". I think it's because the withdrawals on two successive days are not perfectly correlated, so if you doubled the amount of cash put into the ATM the probability of running out of cash would fall below 1%? I had to think about that one.

Somewhere in there it ought to be possible to calculate the implied cost to the bank (in terms of pissed-off customers) of running out of cash, if they decide that 1% is their profit-maximising probability. But if interest rates fall, the bank would also re-evaluate that 1% choice, as well as the number of times per week.

Macro-implications: the demand for base money increases, other things equal. But given the very high velocity you report of money in ATMs (once per day, or thereabouts) compared to most estimates of currency demand (about 5% of GDP, or a velocity of once per 18 days), I don't think it's big enough to matter much. Plus, under current operating procedures of the Bank of Canada, it shouldn't have much effect at all.

[Nick Rowe]

What surprises me most is how "sticky" the bank's decision is. The negative interest-elasticity in the demand for currency is there all right -- they are responding exactly as theory predicts, but it has taken the banks how long? a couple of years? to respond as theory says they will. And we estimate money demand equations using monthly or quarterly data. Maybe this explains the lagged dependent variable that always shows up in an estimated money demand equation?

We always assume people will respond instantly to changed incentives. Things happen the way economic theory says they should; but it always seems to take waaay longer than we think it will.

[Just visiting from Macleans]

ATM service fees (their customers as well as competing banks)= opportunity cost.

[Patrick]

Supposing I get what JVFM said, I second the motion (well, supposing I was a banker): Just set the fee to recoup the interest opportunity cost.

[David]

But Just Visiting From Macleans and Patrick, higher fees mean fewer customers. And adjusting fees doesn't change the fact that you still want to minimize the cost of keeping the machines stocked. So adjusting fees while ignoring the cost minimization problem is just a lazy way of approaching the problem.

[Youcef M.]

question for students: Why doesn't it exactly double it?
The amount withdrawn between two days has to be negatively correlated: If I go to the ATM today, I am less likely to also go tomorrow.

But even with independent distributions, we will need less then double the money. Suppose there 100,000$ in the ATM (double). Let x the be the amount withdrawn on day 1. The ATM will run out of money iff on day two, the demand is y>100,000-x. In this event, either x or y is more than 50,000$. Say it's x. The probability of runout is P(x>50,000) times some probability (less then 1). The total probability has to be less than 1%. To get back to 1% you need less then double.

[Just visiting from Macleans]

But Just Visiting From Macleans and Patrick, higher fees mean fewer customers.

No, you and Patrick read more into my point than I intended. You need to take into consideration the opportunity costs of having no money - in terms of service fees, credit card advancements etc. pissed off customers. These are existing fees.

The analysis so far was too simplistic, and did not take this into consideration.

[Just visiting from Macleans]

...and if you switched money to loaves of bread... :)

[Nick Rowe]

Youcef M: so if the correlation between two successive days is anything less than 1, you don't need to double? (I think that's right).

[Youcef M.]

I did not realize that, but you are right. Perfect correlation is the worst case scenario and all you need to do in this case is double the money supply to stay at the 1% threshold. This is a nicer way to see it!

[Just visiting from Macleans]

...and if the same company that fills up the ATMs also collects the deposits (that the bank begins paying interest on the day they're deposited) then other opportunity costs.

[Just visiting from Macleans]

...and if the same company that fills up the ATMs also collects the deposits (that the bank begins paying interest on the day they're deposited) then other opportunity costs.

[Youcef M.]

I did not realize that, but you are right. Perfect correlation is the worst case scenario and all you need to do in this case is double the money supply to stay at the 1% threshold. This is a nicer way to see it!

[Jon]

Nick comments:

Youcef M: so if the correlation between two successive days is anything less than 1, you don't need to double? (I think that's right).

While true, the point is that even if there is no correlation, the joint probability of withdrawing the full amount on two days must necessarily be less than a single day. Ergo, to have a 1% risk of drawing to zero you need less than 2x the cash to cover two days.

[Mike Moffatt]

Answer to question to students: In fact once you account for seasonality & day of the week effects, there's likely to be a slight negative correlation. But assume they're uncorrelated.

We initially had a normal distribution with mean of 30K, s.d. of 10K. We filled up to 2.5 sd's, which meant the ATM was filled to 55K.

The sum of two normal distributions would have a mean of 60K, a s.d. of 14,142. If we fill up to 2.5 s.d's, then the ATM is filled to 95,355 - not 110K.

* Formula is new sd = sqrt (sd1^2 + sd2^2). = sqrt(200,000,000) = 14,142.

[Jim]

This is good news for ATM robbers then............

[Just visiting from Macleans]

Original posting:
ATMs are rarely filled to capacity, because a dollar in an ATM is a dollar not out earning interest. If the interest rate is 6% and if on average an ATM has $50,000 in it, the opportunity cost of having the money in the ATM is $3,000 per year.

So, on a straight line basis ($3,000/365)~ $8/day.

If the guys driving the ATM filling machines are like Brinks, two armed guys, one driver, etc I suspect they charge more than $8 per visit.

So, it seems to me the issue becomes - how much money can you safely cram into an ATM before you need to refill it? The break even point (interest vs service charges) depending upon how much the drivers charge and the interest rate.

[Mike Moffatt]

"So, it seems to me the issue becomes - how much money can you safely cram into an ATM before you need to refill it? "

There are other considerations as well (collecting deposits, etc.), but based on my conversation these things are almost never filled to capacity. Apparently safety has little to do with it - the 'loss' rate from theft on ATMs I'm told is incredibly small.

[Just visiting from Macleans]

But, if your back of the envelope calcs are correct (and the ones I derived from them), then interest opportunity cost would not have been the constraint in the past - and therefore the ops shouldn't change now with lower rates.

[Mike Moffatt]

"But, if your back of the envelope calcs are correct (and the ones I derived from them), then interest opportunity cost would not have been the constraint in the past - and therefore the ops shouldn't change now with lower rates."

Agreed. There are other factors we're not considering (of which theft is a small, but non-negative one). If there weren't, then the optimal policy would have to be always fill up-to-the-max and fill-up as little as possible. Another one that she indicated was there was some value to having people go to the machines and see if they were functioning, were vandalized, etc.

[Mike Moffatt]

Err.. that should be 'non-zero' one.

[Just visiting from Macleans]

The policy was optimal when rates were so high, but at 1% it turns out there are significant cost savings to dropping down to a 3 days a week schedule, since this cuts the fill-up costs in half...My banker friend believed that her competition would come to a similar conclusion and they'd adjust their policies accordingly.

Royal Bank? :)

aside - Reguly wrote a glowing review of RBC head Gord Nixon on the weekend. Turns out he comes from the investment side of banking:

The new boy might have exuded confidence, but it was something of a show because he had a big hole in his knowledge: As a career investment banker, he knew almost nothing about RBC’s retail business, which was not without its problems at the time. “The business I was least comfortable with was [retail] banking, which was our biggest business,” he says. “It probably took a few years to get comfortable with decision making around the retail bank.”
http://tinyurl.com/3ab96ee

Which probably explains why RBC's customer service and IT systems are so atrocious - the downside of oligopolies - a point I have raised in the past.

[bork]

Used to work in the IT dept for a Big 5. Sat in on a number of meetings that were about taking multiple ABMs (places where 2-3 machines were colocated) and spreading them out across the surrounding blocks of the neighbourhood. Getting more granular data about demand (to more optimally stock machines up with cash) was considered valuable enough that it warranted paying extra rent, maintaining physical premises, etc.

[Just visiting from Macleans]

Getting more granular data about demand (to more optimally stock machines up with cash) was considered valuable enough that it warranted paying extra rent, maintaining physical premises, etc.

I suspect the motivation for spreading out ABMs across the neighbourhood was to increase total # of transactions and hence service fees charged (more convenient - thereby more frequent of less amount, more likely to get competitor banks transactions).

This would by necessity require better data per machine - but I suspect not the reason for multiple sites.

[Andrew F]

This sounds like a toy operations research problem.

[david stinson]

"Question for our macroeconomists: Is this something we need to consider when implementing monetary policy?"

As noted by other commenters, ATM cash amounts to a subset of bank reserves. Reserve ratios in Canada are endogenous and determined by the sort of optimization process you laid out above.

I would have thought that whether monetary authorities need to take account of this would be determined by what they are targeting. In the unlikely case that a monetary authroity is targeting a growth path for a monetary aggregate, then yes, they would need to model varying reserve ratios since varying reserve ratios would affect the relationship between base money and the monetary aggregate. If they are targeting a market expectation, in the form of inflation or NGDP a la Scott Sumner, then I would say that no they do not need to take explicit account.

If they are targeting their own forecast, and they forecast inflation based in part on a forecast relationship between money supply and demand, then I suppose that, yes, they would need to take explicit account(?).