## Thursday, October 20, 2011

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

In most lists of numbers involving real world observations, the distribution of first digits is governed by Benford's law, whereby the logarithms of the numbers (but not the numbers themselves) are uniformly and randomly distributed. This means that the first digit will be 1 about 30% of the time, and larger numbers will occur as leading digits with lower and lower frequency. The number 9 should be a first digit less than 5% of the time.

Benford's "law" is an empirical statement and not a mathematical theorem. However, it makes sense because a logarithmic distribution of first digits is the only one that would satisfy scale invariance, meaning that the distribution of first digits is independent of whatever measuring unit is used.

Benford's law has implications in investing, because the first digits of numbers on a financial statement should follow this distribution assuming they are real world observations (i.e. not made up). And in fact, there is empirical evidence that deviations from Benford distributions in financial statements indicate misstatements.

I saw a study showing that deviations from Benford distributions have been increasing in U.S. financial statements. More importantly, the patterns of deviations match with industry blow ups - there was a spike in the financial sector in the late 80s (S&L crisis) and in the tech sector in 2000.