Using Benford’s Law to avoid corporate chicanery

July 8, 2015

July 8 (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

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

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

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 be
an owner indirectly as an investor in a fund. You can email him
at and find more columns at

(Editing by James Dalgleish)

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