There are several different tools and mathematical formulas that can be utilized to determine the strength of a company’s financial statements. One of the most intriguing and useful tools to detect accounting irregularities is the mathematical formula Benford’s Law – a concept developed by physicist Frank Benford in the 1930s.

Benford’s Law states that in many kinds of numerical data, whether financial statements, population numbers, street addresses, length of rivers, etc., a predictable pattern occurs. More numbers begin with the digit 1 than begin with the digit 2, more begin with 2 than 3, and so on. Based on this pattern, the distribution of the first digit in a number is not random, it’s logarithmic, as shown below:

Research has suggested that Benford’s Law can be used to detect anomalies in data, whether from clerical errors, random chance, or outright manipulation. When a set of numbers expected to conform to the distribution do not do so, this can be a sign that there is something wrong with the data. This simple analysis of the first digit of numbers in a data set can be used to help uncover fraud and other data problems in a number of instances, including accounting, scientific and legal cases.

Additionally, research has found that financial statements, as a whole, conform to the distribution and that individual divergences from the pattern “may reflect the informational quality of financial disclosures.”

Based on this theory, if the distribution of numbers in a company’s financial statements doesn’t conform to the distribution expected according to Benford’s Law, it is an indication that those company’s financial statements should be reviewed more thoroughly when evaluating the quality of financial statements.

Digits in financial statements failing to conform to the Benford distribution does not unequivocally mean that fraud has occurred. However, if an individual were to fabricate numbers on financial statements, it’s highly unlikely that the individual would be able to manipulate the numbers in a way that would conform to Benford’s Law.

To demonstrate the usefulness of this statistic, consider the example of Synchronoss Technologies Inc. [SNCR]. In 2017, the Company disclosed an Audit Committee investigation that delayed the filing of their annual report, a non-reliance restatement related to revenue recognition that reduced net income by $112 million, a necessary adjustment to financials related to cost of revenues, and material weaknesses in internal control over financial reporting.

The financial statements for Synchronoss Technologies failed to conform to Benford’s Law for fiscal year ends December 31, 2015 and 2017:

The sharp deviation occurring around the first digits of 5, 7, and particularly 9 should serve as an indicator that those financial statements should be reviewed closely for other anomalies. Individuals assessing financial statements can use a deviation from Benford’s Law as a starting point for determining which companies may warrant additional scrutiny.

There are several methods to measuring a financial statements’ adherence to the Benford distribution. Audit Analytics flags deviations from Benford’s Law in two ways:

**Kolmogorov-Smirnoff**(KS) statistic – Determines if the company’s financial statements deviate from the law in a statistically significant way by comparing the largest single deviation to a critical value**Mean Absolute Deviation**(MAD) statistic – Finds the average deviation of the actual distribution from the company’s financial statements to the ideal Benford distribution

Among the Russell 3000, we looked at companies that failed to conform to Benford’s Law based on financials in their last annual report, and its subsequent correlation with other notable accounting events that are indicators of reporting quality, including ineffective disclosure controls (SOX 302) or internal controls (SOX 404), financials restatements, and late filings.

Companies that failed to conform to the Benford’s Law distribution were more likely to have adverse 302 and 404 opinions, financial restatements and late filings.

This indicates that companies with financial statements that do not conform to Benford’s distribution have a greater chance of having poor internal and disclosure controls – and having poor controls can lead to low quality financials, either due to intentional or unintentional means. This type of environment will often lead to unfavorable outcomes such as non-timely filings and restatements.

While Benford’s Law should not be used as a decision making tool by itself, it may prove to be a useful screening tool to indicate that a set of financial statements deserves a deeper analysis.

This analysis was performed using data from the Accounting Quality + Risk Matrix (AQRM), powered by Audit Analytics. The AQRM is a powerful tool to screen portfolios for risk indicators based on qualitative disclosures, allowing users to quickly highlight indicators of potential earnings management and other accounting quality issues.

For more information on the Accounting Quality + Risk Matrix and how it can be integrated into your current workflow, please contact us.