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My tax refund this year will be $2,043 from the federal government, and $957 from New York State. That comes out to exactly $3000. I figure the chances of that happening are pretty low.

Is there some law I'm running into that is capping my tax return (and I guess thereby making me lose money on my taxes)?

The reason why I believe $3000 is special and has a higher chance of being special than any other random number is that if there was a cap it would likely be at a nice even number, like $3000.

Chris W. Rea
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David says Reinstate Monica
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    The chances of getting a refund of exactly $2,043 from the federal government are pretty low, too. It's just that our brains like round numbers, so they tend to stand out. – Pete Becker Feb 16 '18 at 12:40
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    Actually, the chances of that happening are pretty high, given how many people there are in the US – minseong Feb 16 '18 at 13:51
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    Chance of such a number increase given that he IRS likes to use rounded numbers and prefers you do the same. – RomaH Feb 16 '18 at 14:48
  • Note that in the future (2019 and beyond), there are likely going to be conditional credits (based on what I've heard about the new tax laws), so you'll lose some refund money if you pay too much in taxes up front. You wouldn't get a bigger refund, but also get larger paychecks. – phyrfox Feb 16 '18 at 15:01
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    Our brains tend not to trust round numbers like that. –  Feb 16 '18 at 16:06
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    @phyrfox do you have any references? – stannius Feb 16 '18 at 16:55
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    @PeteBecker But arguably it's more likely that the amount would be capped at a round number, if there were a cap – Ben Aaronson Feb 16 '18 at 17:02
  • @stannius The Child Tax Credit thing, only a portion of it can go towards tax liability instead of a refund. I actually had it backwards, apparently; it's now more refundable than it used to be. – phyrfox Feb 16 '18 at 17:11
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    @phyrfox I think you've misunderstood. Refundable credits are credits that will pay out to you even if it effectively makes your total taxes negative. However, regardless of whether a credit is refundable or not, you're always able to (potentially, i.e. if you have zero tax liability) get all the money back that you had withheld. – stannius Feb 16 '18 at 17:16
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    @BenAaronson, but neither of his refunds were round numbers. He received two refunds that happen to add up to a round number. If OP had filed somewhere other than New York, it likely would have added up to some other random number. – Seth R Feb 16 '18 at 18:40
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    @SethR I guess he's wondering if there's something in one of the tax calculations that takes the other one into account, to put a cap on the total refund. – Barmar Feb 16 '18 at 22:00
  • The probability of it being $3000 is exactly the same as the probability of it being $3762. You wouldn't think $3762 was strange. – Michael Kay Feb 18 '18 at 19:31
  • @MichaelKay I edited to explain to you and multiple others why $3000 is less likely than some random number. – David says Reinstate Monica Feb 18 '18 at 21:13
  • The state and federal tax authorities are totally separate. They don't coordinate with each other to check the total. Also... the amount of your refund is calculated by the return that you filed... it's not a number the government randomly comes up with. You should have written the amount of each refund yourself on each form after calculating it before you even sent them in. – Bart Feb 19 '18 at 05:18
  • @Bart Actually when filing your NY tax return you are asked for federal return amount. There is nothing stopping NY from enacting such a law. – David says Reinstate Monica Feb 19 '18 at 06:17
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    @DavidGrinberg if you had a statistical sample of many tax refunds and multiples of $1000 showed up more frequently than others, then you would be able to assess whether this had any statistical significance. From a sample of one, you can't make any such inference. – Michael Kay Feb 19 '18 at 10:05
  • @SethR True. There would have had to be coordination between the federal and state tax refunds. Not being a resident of the US, I don't know how plausible that is. – Ben Aaronson Feb 20 '18 at 10:53

5 Answers5

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One can overpay their state and federal taxes by tens or hundreds of thousands of dollars. Tax time is when it gets reconciled, a payment due or a refund. No cap either way, your $3000 is just coincidence that it’s a round number.

JTP - Apologise to Monica
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    In fact, it happens to roughly 1/1000 people * 323.1 million people in usa = ~323,000 people every year. – Mooing Duck Feb 16 '18 at 23:37
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    152M tax returns filed. No idea if the owed/overpaid numbers are clustered, e.g. a high number between +/- $900, or if there’s a wider random spread. Still, a large number will end in 000 – JTP - Apologise to Monica Feb 17 '18 at 00:55
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    I had a particularly complex US return one year, and was due a $1 refund. I didn't cash the check; I kept it as a souvenir. Next year the IRS credited me for the check I didn't cash! – Mark Stewart Feb 17 '18 at 03:09
  • @MarkStewart Nicely done. I guess you've set up your tax withholding about as precisely as it can be done. – Daniel Feb 19 '18 at 18:48
  • With the potential for Roth recharactrization after year end, it was easy to get the tax return to show nothing owed, no refund. – JTP - Apologise to Monica Feb 19 '18 at 20:11
  • It was pure luck to get a $1 refund; this was 25+ years ago! – Mark Stewart Feb 21 '18 at 12:59
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Is there some law I'm running into that is capping my tax return

You're filing two returns, not one. Thus, the premise of your question is flawed.

Just as importantly, the two tax returns don't ask each other what the other's current year refund is.

RonJohn
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    Line 10, Form 1040 does in fact ask what the state refund was for the prior year? There may not be a cap, but the number definitely is an input into the calculation of tax. – user662852 Feb 16 '18 at 15:05
  • But only if you itemized deductions the prior year. – jmarkmurphy Feb 16 '18 at 15:18
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    @user662852 Sure but it asks what your state refund was for the prior year. If the question had been "I got $957 back from the state last year and $2043 from the IRS this year," it would still be a coincidence, but at least there would be a connection. – stannius Feb 16 '18 at 16:54
  • Since many of the inputs to the state and federal tax returns are the same, it's theoretically possible that one of them could calculate the refund for the other, and then take that into account when determining its refund. – Barmar Feb 16 '18 at 22:03
  • I disagree that my premise is flawed. New York State can easily have a law that says something like 'your total tax return from NY and the feds is capped at $3000' – David says Reinstate Monica Feb 19 '18 at 02:54
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Roughly, about one in a thousand people will have a tax return in multiples of a thousand. If 100 million people/families file, there could be around 100,000 with a return that’s a multiple of 1,000.

Most deductions are limited to the total amount of taxes you owe, some are not. Therefore, your return is mostly limited by how much taxes you already paid - but not always. Otherwise, there is no cut-off (I once got over 20k$ back).

I think it was pure coincidence that yours was 3,000.

Community
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Aganju
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    The distribution of digits is not uniform... a result of Benford's Law – user71659 Feb 16 '18 at 03:52
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    I appreciate your point is that with a large enough group of people you can expect 'nice' numbers to appear relatively commonly, but your maths is well off the mark here... – Bamboo Feb 16 '18 at 04:54
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    This is so poor mathematically that it will actively hurt understanding the principles. Without an explanation of the probability this response does nothing except say "it's a coincidence". Please fix the maths or delete the answer, and avoid harming future maths knowledge. – Nij Feb 16 '18 at 06:40
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    Aside from Benford's Law, there are other biases to take into account. For example, a tax refund of $0 is probably a lot more common than any other given amount. – BenM Feb 16 '18 at 06:43
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    The Benfords' Law applies for the first digit of a number. Here we are dealing with the last digits, which should be more or less uniformly distributed (of course, other biases not withstanding...). – TonioElGringo Feb 16 '18 at 09:26
  • @TonioElGringo A result of the first digits not being uniformly distributed is that the occurrence of *000 is not equidistant as well. – glglgl Feb 16 '18 at 09:30
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    @user71659 Benford's Law does not cover the digit zero. – pipe Feb 16 '18 at 09:59
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    @pipe: That's a direct result of Benford's Law applying to the first digit, as TonioElGringo already mentioned. Fundamentally, Benford's Law is a statement which says that numbers have an exponential distribution. It's known that this doesn't apply to dates or lottery drawings. It's also a bit flawed for prices set in retail, where $x.99 prices are popular. – MSalters Feb 16 '18 at 10:37
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    The only non-uniformity I see is that a higher than uniform fraction of people will be paying taxes much less $1000. This is of course assuming that we're considering only the people who're paying taxes. – Thirupathi Thangavel Feb 16 '18 at 12:10
  • @TonioElGringo Benford's law applies to other digits. – user71659 Feb 16 '18 at 16:59
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    I agree with @ThirupathiThangavel. Where is the poor math regarding "ending with 000" other than the total number of people in the set? I don't see how Benford's Law or distribution applies here. Are some of you suggesting that a refund of $1256 (or some other number greater than 1000) is more likely than a refund of $1000? – TTT Feb 16 '18 at 17:32
  • @TTT Yes! Benford's law says refund of $10xx occurs 3.6x more often than uniform. (The next two digits are roughly uniform). A refund of $99x is 2.5x less likely than uniform. It is a fallacy when people believe combining a bunch of random variables leads to a uniform distribution, when in fact it is often exponential (Benford's), power law (Pareto) or Gaussian (Central Limit Theorem). – user71659 Feb 16 '18 at 17:52
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    @user71659 - but no where does this answer contradict that $10xx is 3.6x more likely than uniform. This answer is simply saying that in the set of (1000, 1001, 1002 ... 1998, 1999), the chances of hitting 1000 exactly should be about 1 in a thousand. It's irrelevant that 1000 is much more likely than 9000, because in each set of 1000 it should be fairly uniform. – TTT Feb 16 '18 at 17:59
  • @TTT It says right there in the first sentence. The point is more than one in 1,000 will have a refund of $x000, because less than one in 1000 will have a $x999 refund. Benford's law applies to every single digit (with decreasing effect). The last digit alone is reasonably uniform, but not when taken in a whole with the leading digits. This confusion is exactly why it can be used to detect fraud. – user71659 Feb 16 '18 at 18:02
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    @user71659 - How can that be? Are you saying a refund of $1000 is more likely than $999? And that $2000 is more likely than $1999? (I guess my view is that Benford's law probably doesn't apply in this case.) – TTT Feb 16 '18 at 18:11
  • @TTT Yes! I hand-wave that Benford's law applies because income distribution is a power law process (Pareto), and Benford's law applies to Pareto distributions. "I can't see how it applies so it can't." is not a scientific argument. Again, you're confusing uniform distributions in the least significant digits with uniform distribution of the whole number. – user71659 Feb 16 '18 at 18:19
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    Benford's law absolutely does not say that a refund of $1000 is more likely than $999. (If it did, it would also say a refund of Can$1256 was more likely than a refund of Can$1255, which is just silly.) What is says is that a refund beginning with the digits $100... is much more likely than one beginning with the digits $999.... At the scale we're talking about that's equivalent to saying that a refund of $1000, $1001, ..., or $1009 is more likely than a refund of $999 — which doesn't even need a very strong assumption on the refund distribution, just that it's reasonably continuous. – Micah Feb 16 '18 at 20:14
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    In fact, if you restrict yourself to some finite range far away from 0 (say, $500 to $10500), look only at tax refunds within that range, and assume that they obey Benford's law exactly, you will see that the final three digits are very close to uniformly distributed (the most likely probability is at most on the order of 10^-6 more likely than the least likely probability). There may be some nonuniformity worth worrying about here, but Benford's law is in no way the cause of it. – Micah Feb 16 '18 at 20:46
  • Benford's law doesn't apply, but a variant of it does (i.e. derived in the same way). The goal is to minimize the tax return, so you're going to have a non-uniform distribution of tax return amounts. It'll obviously not be a flat line, and may resemble a bell curve of sorts, but it's going to be heavily weighted towards the left. An easy approach in these cases is to go with extremes: instead of an even modulo $1000, what about modulo $2 or modulo $4MM? Certainly not 1 in 4 million people has a $4MM tax return... but it'll be almost certainly 1 in 2 that has an even tax return (rounded to 1$). – Mahmoud Al-Qudsi Feb 16 '18 at 23:33
  • If someone mentions Benford's law, there is roughly one chance in a thousand that they know how to apply it correctly. – jwg Feb 19 '18 at 17:07
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The size of your refund is calculated by a simple formula:

Refund = (amount withheld) + (amount of refundable credits) - (taxes owed)

(This could come out to a negative number, in which case you will have to pay in more money instead of getting a refund, but that's a digression.)

None of those inputs have any influence on each other.** Your refund is just the disparity between the amount you have already given to the IRS (or state agency) for the year, and the amount it turns out you actually owe after going through the complicated steps of the 1040 form.

So, supposed you go through the tax form and find out your tax liability is $8,437 (let's leave credits out of this), and it you actually paid in $11,437 through withholding, your refund will be $3,000. If you had withheld more, your refund would be higher. If you had withheld less, your refund would be lower. Bt your withholding and actual tax liability have nothing to do with each other.

In your case, you actually have 2 refunds which happen to add up to a round number. But these, too, are independent calculations. The federal government doesn't care how much you pay into your state (other than the fact that your state withholding is deductible if you are itemizing, provided you count your state refund as income in the following year), and the state doesn't care what you paid the feds. They each have their own rules for calculating what you owe. The fact that your two refunds combined add to a nice round number is coincidence.

** If you have regular employment, you could argue that the taxes owed influences the amount withheld, as your withholding is supposed to be a projection of what the IRS thinks you will owe according to a calculation using the size of your paycheck and the number of exemptions you claim. It is entirely up to you to declare your exemptions, though, and the actual tax calculation is much more complex than the one used to determine withholding, so it is rare for those numbers to match exactly. But it is to your benefit to get these two numbers as close as possible.

Seth R
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There is no "maximum refund amount". I presume the round number is a coincidence.

How did you calculate your refunds? If you filled out the forms by hand, presumably you know exactly what calculations you did, and you'd know whether you applied some rule about a maximum. I suppose if you used some tax software package you might not know how they calculated your refunds. In that case, you could print out the forms and follow through each step to see where the numbers come from.

If you studied the calculations, it's possible that the number is not strictly a "coincidence", that you come up with a round number because other numbers going into the calculation are round numbers or some such.

Note that while the probability that your tax refund will be an exact multiple of $1000 is, presumably, about 1 in 1000, it is a classical fallacy to make much of this. If either the fed tax alone or the state tax alone had been a round number, you would likely have been equally impressed. Or if either tax, or the total of the taxes, or the total withheld, etc. Just in your taxes there are probably dozens of "interesting" numbers. Then consider your credit card balance, your bank account balance, etc. The probability that one specific number will come out to a round number is small. But the probability that ANY number that you might encounter in your personal finances will turn out to be a round number is not that small.

This is how psychics work. I saw on the news once that a psychic was asked to assist the police in solving a murder. He said things like, "The letter 'R' will be important." And then it turned out that the first name of the man who found the body started with an "R". Amazing prediction! Not really. He would surely also have claimed a hit if "R" had been the first letter of the man's last name, or of the street where the body was found, or the town, or a nearby store, or a river or a hill, or the model car someone involved drove. He never said it would be related to finding the body, so he could claim success if there was an "R" in the name of a witness, if the person was found tied with "rope", etc etc.

Jay
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