# Benford’s Law and the Decreasing Reliability of Accounting Data"

H/t **Mike Kimel**

Via Economist’s View

This is from Jialan Wang:

Benford’s Law and the Decreasing Reliability of Accounting Data for US Firms, by Jialan Wang: …[T]here are more numbers in the universe that begin with the digit 1 than 2, or 3, or 4, or 5, or 6, or 7, or 8, or 9. And more numbers that begin with 2 than 3, or 4, and so on. This relationship holds for the lengths of rivers, the populations of cities, molecular weights of chemicals, and any number of other categories. …

This numerical regularity is known as Benford’s Law, and specifically, it says that the probability of the first digit from a set of numbers is

dis given byIn fact, Benford’s law has been used in legal cases to detect corporate fraud, because deviations from the law can indicate that a company’s books have been manipulated. Naturally, I was keen to see whether it applies to the large public firms that we commonly study in finance.

downloaded quarterly accounting data for all firms in Compustat,… over 20,000 firms from SEC filings… (revenues, expenses, assets, liabilities, etc.).

And lo, it works! Here are the distribution of first digits vs. Benford’s law’s prediction for total assets…

Next, I looked at how adherence to Benford’s law changed over time, using a measure of the sum of squared deviations of the empirical density from the Benford’s prediction…

Deviations from Benford’s law have increased substantially over time, such that today the empirical distribution of each digit is about 3 percentage points off from what Benford’s law would predict. The deviation increased sharply between 1982-1986 before leveling off, then zoomed up again from 1998 to 2002. Notably, the deviation from Benford dropped off very slightly in 2003-2004 after the enactment of Sarbanes-Oxley accounting reform act in 2002, but this was very tiny and the deviation resumed its increase up to an all-time peak in 2009.

So according to Benford’s law, accounting statements are getting less and less representative of what’s really going on inside of companies.The major reform that was passed after Enron and other major accounting standards barely made a dent.

Next, I looked at Benford’s law for three industries: finance, information technology, and manufacturing. … [shows graphs] … While these time series don’t prove anything decisively, deviations from Benford’s law are compellingly correlated with known financial crises, bubbles, and fraud waves. And overall, the picture looks grim. Accounting data seem to be less and less related to the natural data-generating process that governs everything from rivers to molecules to cities. Since these data form the basis of most of our research in finance, Benford’s law casts serious doubt on the reliability of our results. And it’s just one more reason for investors to beware….

This is facinating. Maybe the next step is to use Benford when someone in congress does budget projections?

Just had my third cup of coffee this morning with a math PhD. I showed him Thoma’s post.

After some unkind words about the use of quant techniques by economists, he told me the author should not be able to make such a conclusion using this technique — interesting but far from conclusive.

STR,

Mathematicians are a strange bunch. Here’s another mathematician saying the exact opposite: http://www.economist.com/comment/1076902#comment-1076902

Hmmm… I guess they’re just like economists!