Pulling at Loose Threads: Unraveling Fraud with Benford's Law

In the intricate tapestry of financial data, anomalies and discrepancies can often hide in plain sight, much like a loose thread on a well-worn garment.

While these irregularities might seem insignificant at first glance, tugging gently at them can sometimes unravel larger issues lurking beneath the surface.

One tool for detecting "loose threads" in financial data is Benford's Law.

Originally observed as a curious pattern in logarithm tables, this mathematical principle has found its niche in the world of forensic accounting, serving as a spool for unwinding financial fraud.

Understanding Benford's Law

The story of Benford's Law began with a curious observation by Frank Benford, a physicist, in the early 20th century.

While working with logarithm tables (a table used to find the logarithm of a specific number to that particular base), Benford noticed that the pages corresponding to numbers starting with the digit 1 were more worn out than those starting with other digits. This led him to hypothesize that in many naturally occurring datasets, the number 1 is more likely to appear as the leading digit than any other number.

Mathematically, Benford's Law predicts the distribution of the first digits in many datasets. The number 1 appears as the leading digit about 30.1% of the time, the number 2 about 17.6% of the time, and so on, with the number 9 appearing as the leading digit only 4.6% of the time.

This distribution is counterintuitive to many people. Someone might expect all digits (from 1 to 9) to appear as the leading digit roughly equally (or about 11.1% of the time) if the distribution were uniform. But that is not the case. In reality, lower digits appear more often.

But why does this matter in the realm of finance and fraud detection?

As it turns out, genuine financial transactions, whether sales figures, expense reports, or stock prices, often exhibit this pattern. When numbers are fabricated or manipulated, they often deviate from this expected distribution, making Benford's Law a powerful tool in the auditor's arsenal.

Why Benford's Law Works for Fraud Detection

The principal underlying Benford’s Law is almost too simple to believe, but its simplicity is its power. Imagine a vast forest where trees of all sizes coexist. The younger, smaller trees are more numerous, while the older, towering giants are fewer but still present.

Trees can die at any time, which means, naturally, that there must be more sprouts than old trees. Just like there are more beginners than experts.

Similarly, in the world of numbers, smaller leading digits like 1s and 2s naturally occur more frequently in many datasets, while larger leading digits like 8s and 9s are less common. This natural occurrence is the essence of Benford's Law.

When humans intervene, especially with the intent to deceive, they often create numbers that feel "random" or "ordinary" to them. For instance, someone fabricating financial data might make up numbers starting with 7 or 8, thinking they're being random, while avoiding numbers starting with 1, thinking they're too common.

The fraudster’s human bias inadvertently disrupts the natural distribution, making the data stand out when analyzed through the lens of Benford's Law.

Moreover, genuine financial data often spans several orders of magnitude. A company might have small expenses like a $12 office supply purchase and large expenses like a $9,000 equipment investment. This wide range ensures that the first digits of these numbers aren't restricted, allowing them to naturally follow Benford's distribution.

While this application showcases the power of Benford's Law, it's essential to remember that it's a tool, not a verdict. It can highlight potential areas of concern, but further investigation is always required to confirm any suspicions.

Setting Up a Dataset for Analysis

Before diving into the intricacies of Benford's Law analysis, it's crucial to ensure that your dataset is primed for scrutiny. Not all datasets are suitable for this type of analysis, so here's what you need to consider:

• Span of Orders of Magnitude: For Benford's Law to be applicable, your dataset should span several orders of magnitude. For instance, a dataset with values ranging from 10 to 10,000 is more likely to fit the bill than one ranging from 100 to 200.

• Naturally Occurring Data: Benford's Law is most effective with datasets that arise naturally, without artificial constraints. For instance, if all your expense reports are capped at $500, the distribution of leading digits might be skewed.

• Sufficient Data Volume: A larger dataset provides a more accurate representation. Analyzing a handful of transactions might not yield meaningful results, but analyzing thousands can.

• Avoid Predetermined Values: Datasets with many predetermined or fixed values (like standard pricing) might not be ideal for Benford's analysis.

Once you've ensured your dataset meets these criteria, it's time to roll up your sleeves and get to work with Excel!

Performing Benford's Analysis in Excel

Excel, with its powerful data analysis capabilities, is a fantastic tool to unravel the threads of potential fraud using Benford's Law.

Data Preparation

Use Excel's data import or Power Query to bring in your dataset, be it from a CSV file, a database, or directly from your accounting software.

Fictitious Expense Report Data

Download this dataset in the Benford's Law Analysis template.

In a new column, use a formula to extract the leading digit from each transaction amount. For an expense amount value in cell B2, the formula would be =LEFT(B2).

Identifying the first digit

Calculating Expected and Observed Frequencies:

In a separate table, list down digits 1 through 9. Assuming cell A2 contains the digit 1, use the formula =LOG10(1/A2+1) to calculate the expected percentage for the digit 1 according to Benford’s Law. Then use relevant references down the range to substitute A2 for each leading digit.

Use Excel's COUNTIF function to tally the occurrences of each leading digit in your dataset.

Divide the count of each digit by the total number of entries to get the actual percentage.

Expected and Actual Percentages

Comparison and Visualization:

Create a combo chart comparing the actual percentages against the expected percentages from Benford's Law. Any significant deviations will stand out visually.

For example in the visualization below, the 1 digits are below the expected frequency, while the 4, 7, and 8 digits are noticeably above the expected frequency.

Benford's Law Visualization

Remember, while Excel provides a visual and mathematical representation, human intuition and judgment are irreplaceable. Use the insights from Excel as a starting point, and then dive deeper into any anomalies you uncover.

Interpreting the Results

The beauty of Benford's Law lies in its simplicity. It provides a clear mathematical expectation against which real-world data can be compared.

However, interpreting the results requires a blend of analytical rigor and judgment.

• Significant Deviations: If your observed frequencies significantly deviate from the expected Benford's distribution, it's a signal to investigate further. However, it's essential to remember that deviation doesn't automatically equate to fraud. There could be legitimate reasons for the discrepancy.

• Context Matters: Always consider the broader context. For instance, if you're analyzing expense reports and notice a deviation, it could be due to a recent company event or policy change rather than fraudulent activity.

• Cross-reference with Other Data: If certain transactions stand out, cross-reference them with other related data. For instance, if a particular employee's expenses seem anomalous, check their travel records, project assignments, or any other relevant data to gain a clearer picture.

• Seek Expertise: If you're unsure about the results or their implications, consult with colleagues or experts in forensic accounting. Collaborative analysis can often shed light on areas you might have overlooked.

Using a Chi-Square Test for Benford's Law Interpretation

The chi-square test is a statistical test used to determine if there's a significant difference between the expected frequencies and the observed frequencies in one or more categories.

Instead of only visually comparing the actual and expected frequencies (e.g., through a combo chart), the chi-square test can determine if the differences between them are statistically significant.

The chi-square test is designed to work with categorical data across multiple categories, making it suitable for the nine leading digit categories (1 through 9) in Benford's Law.

Excel includes the Chi-square test in its function library. We can use the function by entering =CHISQ.TEST([actual counts], [expected counts]) in a cell of our analysis sheet.

The output of Excel’s CHISQ.TEST function will calculate a p-value associated with the chi-square statistic. A small p-value (typically less than or equal to 0.05) indicates that the observed and expected frequencies are significantly different.

Chi-square Test

In this example, the value in D15 is well below 0.05; therefore, the null hypothesis is rejected, and the findings are determined to be significant.

Chi-square Results

Since the Benford’s Law Analysis is just an indicator of fraud, not proof of fraud, the auditor will need to investigate these results further.

Limitations of Benford's Law

While Benford's Law is a powerful tool in the auditor's toolkit, it's not without its limitations. Being aware of these can help ensure that the analysis is both accurate and meaningful:

• Not All Data Fits: As mentioned earlier, not all datasets are suitable for Benford's analysis. Data that doesn't span several orders of magnitude or has artificial constraints might not yield meaningful results.

• Risk of False Positives: Just because a dataset deviates from Benford's distribution doesn't mean there's fraud. It's crucial to avoid jumping to conclusions based solely on the analysis.

• Over-reliance: Benford's Law is just one tool among many. Relying solely on it without considering other methods or tools can lead to an incomplete understanding of the data.

• Human Bias: It's possible for someone knowledgeable about Benford's Law to manipulate data in a way that it adheres to the expected distribution while still being fraudulent. Always approach the analysis with a healthy dose of skepticism.

For these reasons, it's essential to use a Benford’s Law analysis judiciously, in conjunction with other tools and methods, to paint a comprehensive picture of the financial landscape.

Conclusion

In the vast realm of financial data, the subtle threads of inconsistency, though often overlooked, can unravel tales of discrepancies, oversights, and at times, deliberate deception.

Benford's Law, with its mathematical precision and intuitive simplicity, serves as a beacon, guiding auditors and analysts towards these potential anomalies.

As with any tool, its strength lies not just in its application but in the hands of the user. A discerning eye, a questioning mind, and a comprehensive approach are essential to transform raw data and mathematical patterns into actionable insights.

While Benford's Law provides a roadmap, the journey of uncovering, understanding, and addressing financial irregularities is a collaborative effort, requiring diligence, expertise, and often, a touch of intuition.

In the ever-evolving landscape of financial transactions, tools like Benford's Law equip us with the means to provide assurance relevant to transparency, integrity, and validity.