Numerology for accountants.

Do financial statements universally "favor" some numbers over others? The idea seems to defy logic. In a random string of numbers pulled from a company's books, each digit, 1-9, would seem to have on chance in nine of starting a given number. But according to a 60-year-old formula making its way into the accounting field, some numbers really are more "popular" than others. A disruption in the pattern may reveal an inefficient process, an honest mistake or outright fraud. Benford's Laws the newest tool in the auditor's arsenal.

Sixty years ago, Frank Benford, a physicist at General Electric, examined disparate sets of data, such as baseball statistics, street numbers of people in listed biographical dictionaries and electricity bills, and concluded some numbers crop up more than others. His law, which is borne out by examination of census data and Dow Jones averages for example, says that the chance that the first digit of a number in a set of data will be 1 is 30.1%, nearly one in three, not one in nine. Each digit, whether it appears in Dow Jones averages or is pulled from the front page of the New York Times, has a different probability of showing up.

Mark J. Nigrini, PhD, started his career as a chartered accountant in South Africa, but, for the last decade, he has been working as a professor--currently on the faculty of thee Cox Schools of Business of Southern Methodist University--researching number patterns and applying them to real-world situations. "Benford's Law gives us the expected patterns of the digits in a list of numbers," he told the Journal. "These patterns should appear in accounts-payable invoices and accounts-receivable numbers, for example." When patterns vary from those set down by Benford, there may be a problem. "Variations may reveal employees are listing expenses as $24 to avoid a $25 voucher limit. Or consider a manager who has a $3,000 signing authority. Benford's Law tells the that, say, only .6% of all numbers should start with 29. If I get more than that, maybe a manager is approving too many $2,900 invoices, right under his limit." Sometimes, said Nigrini, patterns change because of an error. "Invoices entered more than once will change the frequencies."

Benford's Lay may also catch fraud, thanks to psychology, said Nigrini, "People just don't think `Benford-like.' If someone is cutting a $400 check every week for nonexistent janitorial services, those checks will skew the digit distribution. A thorough application of the law will find the fraud."

The big time

Benford's Law is no longer a mathematical curiosity. At least one firm, Ernst & Young LLP, is already applying it. "We're using advanced analytical tools, such as the application of Benford's Law, more and more," said James Searing, a partner and the firm's director if strategic service development. "The future of adults is already here. There's more focus on total business risk, leading to greater use of information technology to detect anomalies, exceptions and errors of misstatement." E&Y has developed proprietary software that applies Benford's Law to a client's data. "The more we use such analytical tools, the more we find." In actual engagements, E&Y has turned up the same problems Nigrini discussed: When "spikes"--digit frequencies that violate Benford's Law--involve numbers corresponding to dollar amounts just under a particular manager's maximum signing authority, Searing suspects trouble. "Maybe managers are breaking down one project into several pieces to circumvent a higher level of oversight for some reason. Clients want to know when employees are going around their control systems."

At other times we see spikes at $15 or $25," said Searing. "Even though these are low-dollar items, that might mean a client is writing a separate check for each express-mail delivery. That is a very expensive way to process invoices and write checks." But advising the client to arrange for monthly billing for express mail, E&Y can make the audit a value-added service.

Searing made a medical comparison: "X-rays give doctors another way of looking at the body, and this analytical tool gives us another way of looking at a company." Neither doctor nor CPA can abandon professional judgement and rely entirely on these tools. In fact, E&Y does not universally apply Benford's Law, nor does U.S. GAAS address it. Nevertheless, Searing said, "In the long term, advanced analytical tools, such as the application of Benford's Law, will be part of the audit. They may very well help us to reduce the overall risk of auditing, increase the reliability of the audit opinion and increase value to the client."