U.S. Markets

Using Benford’s Law to avoid corporate chicanery: James Saft

(Reuters) - Investors wanting to dodge the next Enron, or just outperform the market, might want to pay particular attention to the first digits, and only the first digits, in numbers in company accounts.

Called Benford’s Law, after the physicist who discovered it in 1938, it describes a well observed fact: in naturally occurring sets of numbers of sufficient size the first digits are not evenly distributed. Lower numbers predominate: 1 is the first digit in a number almost 30 percent of the time while 9 begins less than 5 percent of numbers.

No one knows exactly why this happens, but in data sets from rainfall amounts to town population the numbers follow a Benford’s Law distribution.

As the U.S. Internal Revenue Service uses Benford’s Law to sniff out tax cheats, or at least to narrow the field to better channel resources, why not do something similar with the data produced by companies to represent their performance?

Deutsche Bank has done just that, applying Benford’s Law to company financial statements and coming up with results which are, at least in the aggregate, compelling.

A data set which does not conform to Benford’s Law may well indicate that something is not right. This may be fraud, or may be mistakes or misstatements. In any event all of these leave investors operating at a disadvantage when compared to investing in a company with accurate reports.

Deutsche crunched the numbers on Russell 3000 companies and found that a Benford distribution applies to almost every balance sheet and income statement item, from annual sales to accounts receivables to net income. Similar data was found for global firms.

But those companies which don’t show conformity to natural number distribution also show a marked divergence in how they perform for their investors.

“Stocks with potential accounting irregularities underperform the market significantly,” quantitative strategists at Deutsche Bank led by Yin Luo wrote in a March report.

“More importantly, companies with accounting irregularities exhibit more severe drawdowns, higher volatility, and lower risk-adjusted returns compared to the market portfolio.”

Whereas the Russell 3000 has increased 16-fold since 1990, a market cap-weighted basket of those companies whose reports don’t conform to Benford’s Law is only up a bit less than six-fold.

Interestingly the two groups performed about the same from 1990 through 2000, at which point a sharp divergence began. The non-conformers (and remember non-conformity is not proof of fraud) began to sink then, only recently regaining their previous millennium’s peak. The Deutsche study can’t account for this change over time, though it occurs to an outside observer that the past 15 years, with the dotcom bubble followed by the sub-prime and financial stock fiasco, was a period in which fraud and poor accounting often found its comeuppance.


The vast majority of companies’ data adheres to Benford’s Law, with about 5 percent of Russell 3000 companies not conforming based on Deutsche’s calculations. Fewer global large-cap companies throw up suspicious numbers, though it is impossible to say if this indicates that the U.S. suffers from more fraud and sloppy accounting.

To come back to Enron, Deutsche looked at 40 accounting items, captured monthly, over about 20 years. Whereas for Walmart, the gap between what Benford’s Law would predict and what was reported was minimal, for Enron there was a bigger variance, with the numbers 5, 6 and 7 appearing more often than is natural.

In the case of Enron, which went bankrupt in 2001, smoke indicated fire.

The question is then what practically an investor can do with this data. At any given time there are usually only very few companies with accounting irregularities. One possibility suggested by Deutsche is a long/short strategy which bets on the overall index with a smaller short bet against those companies whose numbers look doubtful. Because it can be hard to short smaller stocks, this may work better if done with larger-cap issues.

One obvious problem with this whole approach is that fraudsters too will have heard of Benford’s Law, and can easily do their own analysis to make sure their cooked books conform to a ‘natural’ distribution of numbers. Perhaps the Enrons of tomorrow, or today, will prove to have tightly conforming accounts.

What a Benford’s analysis may do then is to help weed out the incompetent, who misstate accounts based on bad data, or simply get the math wrong.

This group might well be larger than the fraudsters and similarly worth avoiding as investment candidates.

(At the time of publication James Saft did not own any direct investments in securities mentioned in this article. He may bean owner indirectly as an investor in a fund. You can email himat and find more columns at

Editing by James Dalgleish