Benford’s Law Among the Russell 3000

Benford’s Law, an arcane rule from the math books, is making something of a splash in the world of financial analysis.

The basic idea is this: in certain sets of data (the lengths of a collection of rivers, the populations of a set of cities and towns, etc.), a predictable pattern occurs. More numbers begin with the digit 1 than begin with 2, more begin with 2 than 3, and so on. In other words, the distribution of the first digit is not random, but instead logarithmic.

Many explanations for this phenomenon have been proposed. Consider the absolute difference between two numbers. Most people can see right away that the difference between 1 and 2 is the same as the difference between 8 and 9, namely 1. On the face of it, these seem like an equal increase. If, however, we think in terms of growth, we see that the “distance” between 1 and 2 (100%) is much larger than that between 8 and 9 (12.5%). This is a characteristic of logarithmic scales and the sorts of datasets that fit well with Benford’s Law.

So, do figures in financial statements correspond to Benford’s Law? The answer is a resounding “yes”. Research has made a more rigorous case, but the following chart and table should suffice to convince all but the extremely skeptical.

We looked at the first-digit distribution of Russell 3000 fiscal 2015 financial statements and found that, in the aggregate, the distribution is almost a perfect match with the predicted distribution.

R3000 Benfords Distribution copy

So what does this mean? Well, if a company’s financial statements deviate from the expected distribution, it could be a sign that something is “wrong” with the numbers. Research along these lines has found some promising correlations.

While the process of using Benford’s Law to detect fraud on an account basis has been around for some time, thanks in part to the work of one Professor Mark Nigrini, a paper entitled Financial Statement Irregularities: Evidence from the Distributional Properties of Financial Statement Numbers by Dan Amiram, Zahn Bozanic and Ethan Rouen explores the use of Benford’s distribution on the face of financial statements. In an overview, the authors found that 1) financial statements, as a whole, conform to the distribution and 2) individual divergences from the distribution “may reflect the informational quality of financial disclosures.” This means that the numbers on the face of financial statements should conform to Benford’s distribution and those financial statements that do not may be of poor quality.