Math Theory Offers Way to Detect Cooked Books

Imagine that you deposit $1,000 in a bank at 10 percent compound interest per year. Next year you'll have $1,100, the year after that $1,210, then $1,331, and so on. The first digit of your bank account remains a "1" for a long time.

When your account grows to more than $2,000, the first digit will remain a "2" for a shorter period as your interest increases. And when your deposit finally grows to more than $9,000, the 10 percent growth will result in more than $10,000 in your account the following year and a long return to "1" as the first digit.

If you record the amount in your account each year for a large number of years, these numbers will thus obey Benford's Law.

The law is also "scale-invariant" in that the dimensions of the numbers don't matter. If you expressed your $1,000 in euros or francs or drachmas and watched it grow at 10 percent per year, about 30 percent of the yearly values would begin with a "1," about 18 percent with a "2," and so on.

More generally, Hill showed that such collections of numbers arise whenever we have what he calls a "distribution of distributions," a random collection of random samples of data. Big, motley collections of numbers follow Benford's Law.

Suspiciously High Digits

And this brings us back to Enron, accounting, and Nigrini, who reasoned that the numbers on accounting forms, which come from a variety of company operations, each from a variety of sources, fit the bill and should be governed by Benford's Law.

That is, these numbers should begin disproportionately with the digit "1," and progressively less often with bigger digits, and if they don't, that is a sign that the books have been cooked. When people fake plausible-seeming numbers, they generally use more "5s" and "6s" as initial digits, for example, than would be predicted by Benford's Law.

Nigrini's work has been well-publicized and has no doubt been noted by accountants and by prosecutors. Whether the Enron and Arthur Andersen people have heard of it is unclear, but investigators might want to check if the percentage of leading digits in the Enron documents is what Benford's Law predicts. Such checks are not fool-proof and sometimes lead to false positive results, but they provide an extra tool that might be useful in certain situations.