Deviations from Benford’s Law over time, in U.S. accounting data

Jialan Wang writes:

So according to Benford’s law, accounting statements are getting less and less representative of what’s really going on inside of companies.  The major reform that was passed after Enron and other major accounting standards barely made a dent.

There is much more at the link.  If you are new to the party, Benford’s Law is that:

…in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on the logarithmic scale.

There is more at the link or more here.  Here are two previous MR posts on Benford’s Law.


Somewhere, Minsky is clapping.

While the increase in deviation is interesting, the fact that accounting data deviates from Benford's law at all is not surprising. Accounting data is based on accruals and not actual cash flows, which mitigates volatility in the data. Thus, you wouldn't expect a semi-artificially smoothed data set to follow Benford's law.

But the deviation was pretty close to zero in the 60's

Let me beat the drum for the financial crisis really being an accounting crisis. Anyone who has worked for a large auditing firm knows that accounting is crooked, what with its swing items and all the other revenue smoothers and enhancements that exist.

Again why? Accountants didn't only recently figure out that accuracy is kind of a big deal. As with peer review, their entire purpose is because people have always been sneaky.

Well, AICPA requires accounts and accounting firms to have peer reviews about every 3 years.


2011 standards:

Note: the documents state "non-SEC issuers" as the SEC has a different set of standards - those apply to publicly traded companies and not these.

Yes . I've said this too. Part of the problem is that accountants are now required to state with certanty numbers that are completely uncertain. This leads to them having to "make shit up."

Big increase during recessions, non?

I bet deviations from Benford's law are correlated to economic hardship. The 90s look pretty good.

Probably a macro-trend related to TGS, too?

BUT: levels matter more than rates! Why do they become entrenched?

"BUT: levels matter more than rates! Why do they become entrenched?"

Regulatory law?

The rise in the deviation seems to me to correlate with the increase in the use of computers for accounting. It's a lot harder to do tricky arithmetic with calculators and slide rules. PCs make it easy to do complex calculations to massage bad numbers into less-bad ones.

The original author (Wang) doesn't even label the graph with enough information to evaluate the claim--is it quarterly or annual? Assets or revenues?

Benford's Law isn't a law for all data sets and I don't know enough about accounting practices to say if this data should follow the law.

"I don’t know enough about accounting practices to say if this data should follow the law."
"Assets or revenues"
I believe he just extracted all the numbers from the public financial statements.
It should. Its actually used by auditors to test for fraud.

Possible alternative explanation: Because there has been an increase in reporting requirements during this time, have some of these numbers been "non-Benford-compliant"?

For example, package sizes in a supermarket are not Benford compliant (and I know this from testing it) because so many packages are the same size: 16 oz, 24 oz, etc. You have a lot of 8 oz packages but not a lot of 7 oz packages. But there's no fraud involved in supermarket package sizes not being Benford-compliant.

Is thee a whole class of numbers being reported now (but not in 1960) which aren't Benford-compliant even if non-fraudulent?

"Is thee a whole class of numbers being reported now (but not in 1960) which aren’t Benford-compliant even if non-fraudulent?"

Possibly. However, I suspect that a lot of it is due to numbers that have to be made up, because their valuation isn't really priced by a market.

Ah, yes. "The man wants a number. I gave him a number." It would come as no surprise if valuations which were not a function of actual working markets would fail to follow Benford's Law.

for noobs

This is amazing! I remember the first time I heard about Benford's Law. It makes sense, intuitively. As all numbers begin with one and rise from there, repeating the same set of digits in the same order, there will naturally be more ones at every order of magnitude than anything else.

I hope someone at OccupyWallSt is doing a teach-in on this stuff, because they will love it.

I think I first heard about Benford's Law and its use by detectives looking into white collar crime on Planet Money. Looks like it's a criminal class running our whole system now.

In science, if reality differs from a "law" more and more and more over time, we conclude that the law must be wrong.

In economics, we conclude that reality must be faked. It's glorious.

The law isn't that it never happens. The law is that for any set of data that's logarithmically distributed, you'll see a preponderance of ones in the leading digit. (In fact, there's even more information, as there's a particular distribution that the values in the leading digit will take, and the same for the second, third, etc. digits.) It's a mathematical result.

The question is, should we expect accounting data to be valid in the domain of Benford's law? It has in the past, apparently.

Deviation from Benford's Law suggests that certain numbers are arbitrarily chosen rather than being directly measured. In recent decades, accounting has become more dependent on management estimates and less dependent on measured historical costs. Moving away from historical costs generally results in more useful financial statements, but it's no surprise that it leads to a greater deviation from Benford's Law. A deviation in certian numbers or a greater deviation in one company as compared to other similar companies can be evidence of fraud. I don't think you can infer fraud by looking at deviations from Benford's Law across industries or over a long period of time.

Everyone I know who has worked in the accounting industry for a long time agrees that accounting fraud has become much less common in recent years. Things that would be considered shockingly fraudulent today were common business practices in the 60s, 70s and 80s.

I have never before seen a Benford's law time series. It is a great way of checking that the law is relevant to the data set. (cf YSK)

Just throwing this out there- could it be due to an increased use of goodwill?

Benford's law is likely only to catch simple frauds. A sophisticated fraud would be aware of the techniques used for investigation, and account for them. Or so I'd hope.

It seems to me that it would be prudent for a fraud to at least look at the results of his work and make some effort to hide tells.

No, no, no. What BL says is that accounting data is becoming less arbitrary. Whether it's less accurate is a different question.

If Benford's distribution is becoming better known, then this graph underestimates the increase in fraud.

Why do we assume that accounting data should follow Benford's law? The fact that it was close in the 60s could be coincidental.

Is the vertical axis of that figure scaled for meaningful comparison of data-points at times with different volumes of economic activity?

Unrelatedly, an alternative explanation. Benford's Law is just an empirical generalization based on historical data of lists of numbers from "many (but not all) real-life sources". There is no theoretical understanding of why the underlying processes of companies' financial activity should generate Benford's distribution. Rather than more companies faking more of their books, it could be that something else about the nature of economic activity has been changing over the past 50 years, in the direction of a new equilibrium where the digits of financial figures are no longer expected to follow Benford's distribution in the absence of cheating.

Very interesting. I am an auditor with one of the Big 4 and I can tell you that we actually use Benfords law to look for fraud. We do it on a transactional basis and not F/S level.

One explanation has nothing to do with cheating
In the period plotted intellectual capital and intangible property have become the prime drivers of profitable companies. But these assets do not appear on the book data presented to Compustat and corporate shareholders - nor on the forms given to the tax authorities.

A company has to keep records that go beyond what accountants can put on the books under GAAP rules in order to understand what is really going on.

I noted that the 10 Oct 2011 report by The President’s Council on Jobs and Competitiveness mentions (financial `capital' 48 times, but cites intellectual property only in the context of protection (twice). Our leaders, advised by Krugman to Laffer, want to fix the country with the 20th century tools of Marx, Smith, Hayek, and Keynes, applying them to in a world where a successful business as Apple can be started in the US with a few smart people and have it all manufactured offshore. No capital required - and the Fed is powerless.

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