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Sunday 16 October 2011

Benford’s Law: Are Euro States and US Stocks Fiddling Their Figures?

Note: the author of the Benford analysis on US stocks has (at least temporarily) recanted: "D'oh, I've discovered an error in my original calculation on Benford's law.  I will post a revised analysis when I'm confident in its accuracy, but in the meantime I'm keeping this post up for the public record only.  The results and conclusions below are not to be trusted."  (  Ahh, the dangers of non-peer reviewed material ...

Scary Statistics

As we’ve previously seen, Benford’s law – one of the odder practical truths revealed by statistics – is a great tool for identifying fraudulent accounting, a good indicator that’s there’s something afoot in the footnotes. Now, though, we have a couple of new examples of forensic analysis using the technique, and they don’t make comfortable reading if you’re long of equities; or indeed anything other than sub-automatic weapons and a bricked-up cave.

So on one hand we have evidence suggesting that US corporations are systematically manipulating their accounts, and on the other that the real depth of the issues in the Eurozone are yet to be revealed. If true, we’re a long way from resolution of this particular, and peculiar bust.

Benford’s Back

To recap: Benford’s law says that in naturally occurring data the leading digit is most often 1 and least often 9, descending in frequency as the numbers rise. This doesn’t work for all data – human height is distributed according to the familiar Bell curve or normal distribution, for instance – but is common in other areas. In particular Benford’s law rules in financial accounting. So, for instance, more companies earn between 100 million and 199 million than between 200 million and 299 million and so on.

Of course, when humans start manipulating and fabricating data they generally try to do so by randomising it. A typical made-up set of numbers will not follow a Benford-style exponential distribution and can therefore be detected. This paper by Durtschi, Hillison and Pacini explains the background and gives a real forensic example:
“When the details of the account were inspected, it was apparent that many more refund checks of just over £1,000 had been written than in the previous period ... A subsequent detailed examination of the account, however, uncovered that the financial officer had created bogus shell insurance companies in her own name and was writing large refund checks to those shell companies”.
The application of Benford’s isn’t always so simple, though. When Paul Kedrosky analysed Bernie Madoff’s returns he determined:
“Bernie Madoff’s results, far from being pulled from his hat, showed every sign of having been generated by a randomizing algorithm. My analysis suggested that – based on Benford’s Law fit – his results would have passed muster in terms of looking sufficiently random to be real.”
Basically, Madoff’s data so perfectly matched Benford that it should have been suspect for the opposite reasons you might have expected. So Benford’s law is a useful tool but it’s especially useful where the perps aren’t really that smart. So you’d probably reckon that EU finance ministers and corporate CFOs were likely to fall foul of it. And it appears you’d be right.

Euro Trillions

Earlier this year Bernhard Rauch, Max Göttsche, Gernot Brähler and Stefan Engel published an eye-opening paper entitled Fact and Fiction in EU-Governmental Economic Data. Unfortunately this is only available from behind a paywall, but it’s a fascinating read, if you’re into horror stories. Unsuprisingly Greece was the biggest culprit.  Of course the deviation of Greece from Benford’s law was pretty much guaranteed: the scale of Greece’s problems is staggering. Michael Lewis in his new book, Boomerang: Travels in the New Third World, puts the problem into context. The Greeks owe:
“About $1.2 trillion, or more than a quarter-million dollars for every working Greek.”
If you read on, preferably from behind the sofa, through your fingers, you get to:
“In just the past twelve years the wage bill of the Greek public sector has doubled in real terms – and that number doesn’t take into account the bribes collected by public officials. The average government job pays almost three times the average private-sector job. The national railroad has annual revenues of 100 million euros against an average wage bill of 400 million, plus 300 million euros in other expenses”.
Birling Pigs

Given that the original research confirmed what was expected about the prime suspect, what looks even more worrying is who else the researchers fingered:
“We established a robust ranking of the EU member states according to the extent of the deviation from Benford’s law. The countries with the greatest deviations are Greece, Romania, Latvia, and Belgium. In the case of Greece, the suspicion of manipulating data has officially been confirmed by the European Commission.”
You’ll note that the remainder of the so-called PIIGS – the main focus of the world’s concerns – Italy, Ireland, Portugal and Spain – don’t appear on that list. Quite what sort of acronym we’ll get if you add Romania, Latvia and Belgium to the list of euro denominated black holes one dreads to think: PIGS BIRL, might be appropriate, though (yeah, I had to look it up, too).

Of course, if this data is correct, the real issue is why would these countries do this? In Greece’s case the answer seems fairly straightforward: as a nation it’s been in default for half of its independent history. For much of the rest of it, it’s been funded by the British as a defence against the Ottoman Empire and the Americans as protection against the Soviets. Like a fading movie star it needs external funding to maintain its lifestyle. Presumably something similar applies to the other countries, although quite what core Eurozone member Belgium is doing on the list is hard to say: probably the fact it’s not had a functioning government for a while isn't helping.

Benford on Firms

Turning away from the travails of the Eurozone, the economist Jialan Wang has created a stir by publishing her analysis of accounting data on firms that file SEC accounts. The results weren’t pretty:
“Deviations from Benford's law have increased substantially over time, such that today the empirical distribution of each digit is about 3 percentage points off from what Benford's law would predict. The deviation increased sharply between 1982-1986 before leveling off, then zoomed up again from 1998 to 2002. Notably, the deviation from Benford dropped off very slightly in 2003-2004 after the enactment of Sarbanes-Oxley accounting reform act in 2002, but this was very tiny and the deviation resumed its increase up to an all-time peak in 2009.”
The analysis suggests that there’s something very peculiar going on with US companies and their finances, and that something has a piscine odour to it:
“While these time series don't prove anything decisively, deviations from Benford's law are compellingly correlated with known financial crises, bubbles, and fraud waves. And overall, the picture looks grim. Accounting data seem to be less and less related to the natural data-generating process that governs everything from rivers to molecules to cities. Since these data form the basis of most of our research in finance, Benford's law casts serious doubt on the reliability of our results. And it's just one more reason for investors to beware.”
Accounting for Benford

Of course, establishing that there might be a problem is one thing; explaining why it’s happening is entirely another. Two things come to mind – firstly, that corporate finances are uniformly stretched and CFO’s are pushing the boundaries to keep their concerns going or, secondly, that this is a social proof effect where as one or two firms push the boundaries of what’s acceptable others are “forced” to follow suit or fall behind in the corporate sack-race that is the quest for quarterly returns. History suggests that, as a defence, accountancy firms are less a solid bulwark and more a permeable membrane against these types of shenanigans.

Without access to the underlying data it’s not possible to tease out exactly how prevalent these problems are or begin to look the behavior behind them; although no doubt hoards of economists are hurrying to analyse the data themselves, as we speak. Nonetheless it’s a worrying sign. Keep an eye on Chapter 11 filings – if the pigs really start birling it won’t matter how good the accounts look if there’s not enough cash in the bank to pay the bills.

Related articles: Forensic Finance, Benford’s Way, The Death of the Accrual Anomaly, Fear and Loathing in the Eurozone


  1. From Wang's blog:

    "Important update 10/16/11 - D'oh, I've discovered an error in my original calculation on Benford's law. I will post a revised analysis when I'm confident in its accuracy, but in the meantime I'm keeping this post up for the public record only. The results and conclusions below are not to be trusted."

  2. Always a risk in non peer-reviewed material. It would have been surprising if this held up under full analysis, because it would have suggested that the whole world had been missing the obvious for a considerable period of time. What's interesting is that the anecdotal evidence seemed to support the analysis so it'll be interesting to see the revision: either there's still something odd going on, or this is another case of illusory correlations.

  3. First contact with this blog, but your example of "Benford's Law" sounds like the well-known Power Law idea that data sets such as corporate size, or size of cities, have lots of little ones, some middling, and a few giants. Called Zipf's law for word frequncies.

  4. Hi Ralph

    You're right, both Benford and Zipf are related, they're both scale invariant. As I understand it they both apply to naturally occurring phenomena, but of different types: so similar, but not equivalent.

  5. Considering the astonishing track record US companies have at consistently beating quarterly earnings forecasts I'd say something smells. They seem a smidgen more truthful with annual reports because those could have legal repercussions if false (as if any regulator would actually react, ha ha haaaaaa).