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…

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