There Benford’s law helps us to easily unmask the financial fraud. To do this, you know, you need to look at the numbers in the financial statements. Nothing new, one might say. What’s surprising is that, to get started, you might not even need to analyze each item in detail: just look at the first digit of each number. In many collections of real data – such as corporate balance sheets, city populations, or river lengths – the first digit of the numbers follows one very precise schemeknown as Benford’s law. According to this law, derived from empirical observation, the first digit of the numbers is a 1 in approximately 30% of cases, a 2 in approximately 20% of cases, a 3 in approximately 12% and so on, until reaching 9, which should appear less than 5% of the time.
This phenomenon is observed in a very wide range of contexts – from house prices, to financial data, to the lengths of rivers – and, if the numbers present in the company balance sheet do not respect this property, it could be a indicator of a potential scam. It is no coincidence that Benford’s law is today one of the tools used to detect financial fraud.
What is Benford’s law
There Benford’s law it’s aempirical observation on the behavior of digits in real numerical data. To understand it, let’s start with an example: house prices. In Italy, many homes cost between 100,000 and 300,000 euros, some exceed 400,000 and a small percentage reach and exceed 900,000. So, if we look at all the Italian houses, it will be a lot most likely the house price starts with 1 (100,000 euros) or with 2 (200,000 euros), rather than with 8 or 9. In particular, if they respect Benford’s law, approximately 30% (30.1%) will begin with 1, approximately 17% (17.6%) will begin with 2, approximately 12% (12.5%) will begin with 3 and so on, always decreasing, until reaching 4.6% of the prices that will begin with 9.
The incredible thing is that this property is not only found in house prices, but also in river lengths, city populations, newspaper numbers, financial data, molecular weights, election results, even Fibonacci sequence numbers or death rates.
The first person to notice this behavior was Simon Newcomb in 1881, but he was the physicist Frank Benford, in 1938to study it systematically: he collected over 20,000 numbers from 20 different fields and called this strange regularity, now known by his name, the “law of anomalous numbers”.
How Benford’s Law helps expose financial scams
The idea of using Benford’s law to detect financial fraud did not come until many years after it was formulated. The most influential figure in this area is Mark Nigriniprofessor at West Virginia University. Already in 1992, Nigrini proposed using it to analyze the tax returns: in his analysis he noted that the regular ones respected the distribution required by law, while in suspicious cases the frequency of the first digits era altered. Since then, Benford’s law has also become a useful tool for government agencies such as the IRS (Internal Revenue Service, the United States tax agency), which has used it for decades to identify manipulations in accounting data.
An emblematic case of application of Benford’s law is that of Enronthe multinational energy companycollapsed in 2001 due to a massive accounting scam. For years, the company inflated profits and hid debts, rigging its balance sheets to appear more solid than it was. Once the scam was discovered, Nigrini applied Benford’s law to the company’s financial data and found obvious deviations, especially in the revenue reports. Unfortunately, the analysis came too late: if it had been used earlier, it could have triggered controls well before the disaster.
However, one fundamental point needs to be clarified: Benford’s law is not infallible and is not applicable in all contexts. The fact that a data set does not comply with this does not automatically mean that it has been falsified. Likewise, not all scams alter the figures in a way that violates the law. This is a preliminary tool, an indicator of possible anomaly. If the data deviates too much from the expected pattern, it is worth investigating further, but it is not, on its own, sufficient evidence in court.
Some cases where Benford distribution doesn’t work
Benford’s law does not apply to all real data collections. Let’s think, for example, about theadult height expressed in centimeters: almost all of them exceed one metre, so the values almost always start with 1. Nobody is three meters tall, and very few exceed two. In this case, the distribution of the first digits is unbalanced, e it does not reflect Benford’s law.
The same goes for the shoe size: the numbers are concentrated in a narrow range, between 20 and 50. Here too, the initial figures are not distributed according to the percentages required by law.
In general, Benford works best when data they cover several orders of magnitude – for example, values that range from a few tens to hundreds of thousands uniformly, as happens in company balance sheets. The wider the range of orders of magnitude covered by the data, the more likely it is that the law applies accurately.
