The Simple Mathematical Law That Financial Fraudsters Can't Beat

How hard is it to ferret out securities fraud? It might be as easy as looking for how many times the digit `1’ appears in a company's financial entries instead of ‘9.’
A simple mathematical law that applies to everything from the height of mountains to the population of towns in Tajikistan can also be used to uncover suspicious numbers in public-company accounting, an expert at Columbia Business School says in a new paper.
Benford’s Law states simply that the quantity of most things in the real world is more likely to be described by a number beginning in 1 than any other digit, and the likelihood of it being described by other digits declines at an increasing rate as those digits get higher.
When Columbia’s Dan Amiram and two coauthors applied Benford’s Law to public companies, they found the same rules hold true: Companies whose financial statements were significantly out of compliance with the law were far more likely to get caught by the Securities and Exchange Commission for accounting irregularities. And when companies restated their earnings, a before-and-after comparison showed that the new, real numbers complied with Benford’s Law while the old ones didn’t.
“It was absolutely shocking,” Amiram said of the study with coauthors Zahn Bozanic of Ohio State University and Ethan Rouen of Columbia, which he says already has drawn inquiries from hedge funds, short-sellers annd the SEC. “Every year, every industry fits perfectly to Benford’s Law.”
That result might seem absurd, or at least counterintuitive, given the presumably random nature of things like the number of widgets sold or profit from an aluminum smelter. Why is more likely that a company sold 10 widgets, or 10,000, than 700 or 700,000?
The answer reflects the intersection of statistics and our method of counting things. As a simple matter of math, increasing the quantity of anything from 1 to 2, or 100 to 200, or 1 million to 2 million, means a 100% increase, whereas an increase from 9 to 10, or 9 million to 10 million, only requires an 11% increase. That means any count of something in the real world is far more likely to begin with a low digit than a high digit, including when a company starts the year with revenue of $0 and increases that figure each day.
“In this world, there are more small things than large things,” explained Mark Nigrini, a professor of accounting at West Virginia University and author of Benford's Law, a 100,000-word treatise on the practical application of the law toward fraud, Ponzi schemes, tax evasion and other skullduggery. “Any town or city would have a population with a first digit of 1 far longer than any other digit.”
The actual statistical probability under Benford’s Law is 30% for the digit 1, 17.6% for 2, 12.5% for 3, and so on down to 4.6% for 9. And that’s what Amiram found when he studied more than 40,000 public-company annual reports from 2001 to 2011. Benford's Law described those results in aggregate, whether across the entire dataset or industry by industry. Nearly 86% of the firm-year observations complied with the law.
Things got more interesting when he looked at companies that were busted by the SEC. (He drew his data from the SEC's Accounting and Auditing Enforcement Releases database.)  There, he found companies that were ultimately caught in accounting improprieties had a statistical measure of deviation from the results predicted by Benford’s Law of more than 20 times the average for all firms. Even more fascinating, that measure of deviation plunged in the three years before the companies were caught, supporting the widely accepted theory that the SEC only uncovers frauds after they start falling apart and companies can no longer fudge their numbers.
This method has the same flaw as many other fraud-detection tools: It turns up a lot of false positives, or at least companies that never show up in the SEC's enforcement database. Amiram  suspects many of the companies that deviate significantly from Benford’s Law may actually be cooking the books but will never be caught, either because of the SEC’s limited enforcement budget or because they escape the agency’s five-year statute of limitations.
The reliability of this tool for detecting fraud is supported, however, by direct observations of companies that have restated earnings. Those companies showed a statistically significant shift from non-compliance with Benford's Law to compliance before and after they restated earnings.
“For same firm year, the same operation, the same firm, we have two sets of numbers,” Amiram said. “The corrected numbers are very close to Benford’s law, and the previous ones are absolutely off.”
To give one example, Sprint Nextel’s financials deviated from the distribution of digits predicted by Benford’s Law in the two years before it restated earnings in 2001, and fell into compliance after that. AT&T results, by comparison, followed the law the entire time.
Fraudsters can try and fudge their numbers in a way that complies with Benford’s predictions, but it won’t be easy. In another test, Amiram deliberately altered Alcoa’s financial results for 2011, changing sales, cost of goods sold and taxes randomly in 1,000 simulations. Alcoa’s measure of deviation increased practically every time.
Benford’s Law has been used in a large number of forensic applications, including voter fraud, Greece's effort to hide its debt, and determining whether digital photographs have been altered. It's also been in the toolkit of auditors for years, said Amiram, a former auditor, but only at the level of operating accounts. He said his paper is the first to apply the law to company-level financial reports accessible through databases like Compustat.

Amiram began thinking about a Benford's Law screen on financial results after a friend who teaches at Dartmouth sent him a Facebook link to the `10 Weirdest Math Facts." He detailed Rouen, a Ph.D candidate, to study financial reports and quickly became convinced they had "finally found one of the most useful applications of that law."He's spoken to the SEC about his research, but doesn't know if the agency is using it.
Now he plays a parlor game with his students, separating them into groups of "truth-tellers" and "liars" with the first group compiling a list of the lengths of major world rivers and the second one making the numbers up. Benford's Law identifies the liars every time, he said.
When it comes to financial fraud, he said: "If you manipulate the numbers, 90% of the time you’re going to violate Benford’s Law."
Given the apparently rampant level of securities fraud that plaintiff lawyers claim to find each year, it is surprising they haven't turned to this tool as a first screen to determine whether they're suing the right company. Maybe they could even use it to find real examples of fraud, instead of waiting for the company or the SEC to disclose it first.