Thursday, October 27, 2016

From a purely financial standpoint, overdraft fees make no sense.

Now, let us be clear here, if you throw out everything about finances and view the world through a paradigm best described as "EVIL!" (all caps and exclamation point compulsory) they make perfect sense.  But let us not do that at this moment and instead think of what an overdraft is and what would make sense when one occurs.

You've got a bunch of stuff going on on one day.  Money is coming into your account, money is going out of your account.  When all is said and done, you should have $28.73 left in your account.  That's good because in two days you need to pay $20 even.  This is a tough time for you, things are riding close to the edge of ruin, but it's ok.  The closest you'll come to emptying your account again is having $63.81, and after that you'll never have fewer than 100 dollars in your account.

But things don't go right.

On the day a bunch of stuff happens everything goes out of order.  Payments to you take too long to clear, payments from you clear right away.  Your account goes negative: -$1.98 cents.

By day's end all payments have cleared and everything is done.  What just happened?

You had an overdraft of $1.98 and the bank made a loan to you to cover that $1.98.  Makes sense, banks do loans.  That's how they make money.  Mortgages, car loans, business loans, personal loans, loan this, loan that.

The bank loaned you $1.98 for less than one day.  Let's round up for the sake of ease.  Say the bank loaned you $1.98 for one whole day.  The average rate for a bank loan is 3.something%  Again, for the sake of ease, let's round up.  4%.

4% annual interest rate continually compounding is an APY of a little over 4.08%.  That's a bad deal because of the rounding up we did, but let's get back to the loan that the overdraft was.

It was $1.98 for a day, not for a year.

Well, there's an easy way to tell how much that is after a day.

(For time t after taking out a loan)
[What you owe] = [what you were loaned]*e([interest rate]*t)

So we plug in some numbers:
--You were loaned 1.98
--The interest rate was 4% per year, which is expressed as .04
--You paid back the loan in one day so t = [one over the number of days in a year], which is:
     1/365.2425 (have to remember leap years)

So we plug that in:

[what you owe when you pay back the loan] = 1.98*e.04*(1/365.2425)

And we get this:

[what you owe when you pay back the loan] = 1.98*1.0001095...
[what you owe when you pay back the loan]= 1.980216854109...

A little over $1.98 (interest on a single day is seldom very much) banks round up so that you don't get shit for free, which means that you owe $1.99, right?

So at the end of the day you have $28.72 left in your account, more than enough to cover the coming $20 draw, and at the next tight point you'll hit a bottom of $63.80, and then it's smooth sailing because the next low point is more than $100 and you'll never dip under $100 again.


Well, no.

You get hit for 35 dollars.  You end the day with -$6.27, the next time anything comes out of your account at all, even a penny, you'll get another overdraft because there is less than nothing in there.  So when that $20 is supposed to come out with $8.73 to spare, you get hit for another 35 dollars.  Meaning that that time you were going to have a mere $63.81 in your account?  Yeah, that's $6.19 more than you have in your account.  You get hit for another $35.

Now you have $105 fewer dollars in your account than expected meaning that the whole "never have fewer than $100 in your account" has been transformed into "Your balance will never go lower than negative $5" which isn't very comforting.

All of this because you got charged $35 for a one day loan of $1.98.  Some interest rate.  But how much?

* * *

Well, we flip the equation.  Now the interest rate is unknown but we do know [what you owe when you pay back the loan].  So we plug in what we know and then just solve for [interest rate]

1.98+35 = 1.98*e[interest rate]*(1/365.2425)
36.98 = 1.98*e[interest rate]*(1/365.2425)
divide both sides by 1.98
18.6767... = e[interest rate]*(1/365.2425)
Get rid of the "e" by taking the natural log
2.927280... = [interest rate] * (1/365.2425)
Multiply to isolate interest rate:
1,069.1672... = [interest rate]
convert to percent
106,916.72...% = [interest rate]

Now remember, this is APR not APY.  APY is the one that you really want to look at.  Fortunately the conversion equation is easy.  The APY is . . .

Ok, so here's the thing.  For numbers that start with .0 APR and APY tend to be close, for credit cards they start to diverge, by the time you get to an APR of 1 (100%) you have an APY of 1.72ish (172%ish) and from there it gets, well, exponential.  I mean, it already was, but after you pass one things explode.

An APR of 10 (1,00%) is an APY of 22,025.47ish (2,202,547%)
An APR of 100 (10,00%) is an APY of 26 tredecillion.  26--fourty two zeroes--decimal point

We're talking about something with an APR of great than one hundred thousand percent.  I tried to use Google's built in calculator, it just rounded the number off to infinity.  I shit you not.

So, what is the APY on the overdraft loan?
APY > 2[followed by 464 zeroes]

Technically I'm supposed to subtract one from that answer but the rounding error in the above figure defies description.  The one hundred thousand times the high end estimate of the number of atoms in the universe (that we know of) is smaller than the rounding error.  This is a "1 = 2 for very large numbers" situation.

* * *

Now $1.98 is a value that sticks in my mind because that once set off an overdraft cascade in my life (the other numbers are random) and one could reasonably point out that not every overdraft is quite that small.

So, how large would an overdraft have to be for an instant cost of 35 dollars to be fair?

Pretty easy to plug the simplification of a reasonable interest rate (actually high) from before in and figure out how large the loan would need to be to get 35 dollars in interest in the first day.  You'd need an overdraft of $319,634.71.

Stop, for a moment, to think about that.

Banks do not front over three hundred thousand dollars, collateral free, to the kind of people they hit with overdraft charges.  If you believe otherwise then we're going to have to check you for psychotropic cooties.

Regardless of the strange ways your mind (psyche) may have been turned (tropos) it's clear that looking at the size of the overdrawing isn't going to bring things into a reasonable state.  If it has to be in the hundreds of thousands to even begin to make sense then we're on the wrong track.

One might think that this is a sign that the instant nature of overdraft fees that's the problem.  Maybe if there were some reasonable leniency period before the fee kicked in it would work well.

Let's try that.

* * *

We'll go back to our old equation but this time solve for time.

[What you owe when you pay back the loan] = [what you were loaned]*e(.04*t)
36.98 = 1.98*e(.04*t)
18.6767... = *e(.04*t)
2.927280... = .04 * t
t = 73.182009531313373002331274372038

Seventy three years, two months, five days, eleven hours, twenty seven minutes, forty six seconds.

If the bank let you have a grace period like that, no one would ever be hit with an overdraft charge.

What if someone overdrew by $100 dollars, what would it take for $35 to be reasonable then?

135 = 100*e(.04*t)
1.35 = e(.04*t)
.3ish = .04* t
t = 7.5026148112584520187628033656259

Seven years, six months, no days, twenty two hours, 55 minutes, fifteen seconds, and change.

That's a lot quicker, but if you're a bank you don't want to give someone a seven year long interest free grace period on paying you back.  So a grace period doesn't seem to be the answer.  It's not a bad idea (credit cards charge usury, but even they give you a grace period from purchase to the beginning of the next billing cycle) but it's not the answer.

* * *

The problem comes down to the fact that no matter what you do, $35 doesn't make sense.

You could posit some cut off for overdrafts that are too small (like the $1.98), or some grace period so that you don't have the absurdity of paying so much for a loan that may last mere hours, but 35 dollars doesn't work out no matter how you run the numbers.

When you overdraw the bank either bounces you or spots you.  Overdraft fees are about when they spot you.  Maybe they spot you two dollars, maybe they spot you two hundred.  But the thing is, it's a loan.  They want the money back.

Banks know loans.  They know how to do loans.  They know that fees are a downright stupid way to do loans.  Loans work on interest.

That takes into account how large the loan is and how long it takes to get paid back.  The loan is small, it takes the interest a long time to make a burdensome fee, the loan is large not so much.  The loan is short and there wasn't time to accumulate much interest, it's long and it does.

And, importantly, even at credit card rates it still wouldn't be as fucked up as $35.

Credit card rates are important to us for this simple reason: every time you make a purchase on a credit card (debit cards are different) you're getting a collateral free loan that you didn't consult anyone at the loan providing company about.  Exactly like when the bank spots you the amount you overdrafted.

They do not charge a fixed rate of $35 per transaction.  That would create all kinds of fucked up incentives.

A teller at my bank told me that if I was going to overdraft I should go big.  Set up a withdrawal, or a wire transfer to another account of mine, that covered all of the money I could possibly need.  Then I draw on that for a while and thus only have to pay 35 dollars once instead of it setting a cascade of $35 $35 $35 $35...

And that is the logical thing.  If it's a fixed rate per fuck up, you should fuck up spectacularly once.  Bleed the bank for every penny it will spare before it says, "Fuck overdraft, this is bouncing," and then draw off of that take, rather than your bank account, so that you only get slammed with one fee instead of an unending cycle of fees and debt.

So, from a purely financial standpoint, overdraft fees make no sense.

But, as noted at the beginning, if you throw out everything about finances and view the world through a paradigm best described as "EVIL!" (all caps and exclamation point compulsory) they make perfect sense.

Then the cascading failure that pushes someone from solvency to ruin is a feature, not a bug.  They could never get away with $35 if they called it interest on a loan, because then they'd have to say the interest rate and people would flee in terror.  So instead they call it an "overdraft protection fee".


  1. Those fees don't make much sense from a customer retention standpoint either. Though I suppose that anyone who overdraws their account with any regularity is someone they don't want as a customer. The thing that gets me is that apparently even credit unions charge absurd overdraft fees, though they do seem to be smaller than what banks charge. I decided long ago that it's more about punishing the depositor than it is about anything sensible.

    I think it's perfectly reasonable for banks to charge a fee to cover the cost of processing an overdraft, but with increasing automation, I'd think that those costs have, if anything, gone down, and the fees rather emphatically haven't. Furthermore, I suspect that they probably weren't high enough to justify a %35.00 charge to being with, so that's clearly not why they do it.

    Do you know if anyone's ever studied how much of a drag on the economy those fees are? Because I bet it's not insignificant.

  2. Gah, that percent sign should obviously be a dollar sign.

  3. Well in the particular recent case it's a credit union where the customers who are facing these fees have no other options, so I doubt customer retention bothers them much?

    The accounts themselves don't have fees, so they're a great deal for anyone who is able to avoid spending money they don't have more than twice a year or so.

    The other huge advantage is they let the payment go thru without bouncing, so you don't face jail on top of the horrific money problems.