Annual loss to Irish business due to fraud or misappropriation can be as much as 5 per cent of turnover. Even today with sophisticated IT systems, most frauds are detected by chance. Rather than relying on luck, it makes more sense to take a pro active approach to detecting fraudulent transactions in a business. Using Benford’s Law is an effective sample selection method for proactive fraud detection, explains Peter Johnson
Proactive detection of fraud requires the selection of an appropriate sample of transactions for audit. Traditionally there have been two types of sampling: statistical and non-statistical. Whilst these are valid approaches for financial statement auditing, they have their limitations when it comes to detecting fraud. In fact, sampling may not even detect a fraud because fraudulent transactions may not be included in the sample, or if they are, the audit test may not detect a fraud. For the most part, internal and external auditors rarely go into the level of detail necessary to actively detect fraud.
For this reason, using Benford's Law, the sampling and audit risk is reduced because the selection criteria are based on transaction value and probabilities.
SO WHAT IS BENFORD’S LAW?
The law takes its name from Frank Benford, who was employed as an engineer by the General Electric Company in the 1920s. These were the days before computers and calculators so multiplication and division of large numbers was undertaken using slide rules and for more accuracy, logarithms. Logarithms are contained on log tables, for those of us old enough to remember such things. The various log tables were printed in books and all engineers at the GEC research centre would have used them extensively. Benford noticed that the pages at the front of the log tables appeared to be more heavily thumbed (and therefore dirtier at the bottom right hand corner of each page) than those at the back. He formed the view that the engineers were pre disposed to looking up logs of multi-digit numbers starting with low numbers (1, 2 or 3) more frequently than multi-digit numbers starting with digits 4 - 9.
A more modern equivalent of this observation would be the keypad of a well-used adding machine, calculator or computer. The lower numbered keys will generally be more worn than the higher numbered keys.
Benford went on to test this theory by examining over 20,000 sets of numbers including baseball statistics, areas of river, stock market quotes, population of towns etc. The data came from sources that were random and also from sources that followed mathematical rules. However, man-made data such as house numbers, telephone numbers and zip codes were excluded.
The results of the analysis did confirm the empirical observation of thumbed log tables. The probability of a multi-digit number starting with '1' was indeed higher than for the first number to be '9'. In mathematical terms, Benford theorised that the probability (P) of the first digit (n) being a particular multi-digit number is
P(n) = log (1 + 1/n)
By putting various values for n we see that:
For n = 1; the probability is 30.1%
For n = 9; the probability is 4.6%
Intuitively, we might have expected the probability of each number 1 to 9 appearing as the first digit of a multi-digit number to be the same or 11.1%.
Benford went on to conduct the analysis for the second digit of a multi-digit number and found a similar pattern emerged. In further work he looked at the probabilities of the first 2 digits of a multiple digit number occurring. Again a logarithmic rather than the random pattern emerged.
HOW DOES THIS HELP TO DETECT FRAUD?
Consider how you would identify if expense claims being processed by your company were not all bona fide and were being 'padded'. Traditional sampling techniques could not identify likely transactions because:
- Statistical techniques usually depend upon unusual items/errors being randomly distributed.
- Non-statistical methods rely on judgement of the auditor and faced with a large volume of transactions, it is very difficult to know where to start.
This is where Benford’s law can help us. The data for distribution of the first digit of all expenses claims is graphed and this is compared to a Benford distribution produced by the equation given above. This can be done very easily using Microsoft Excel. The mechanics of the process of preparing the source data for analysis is beyond the scope of this article, but readers can get more details from the bibliography.
Having undertaken the analysis, the following might result for the value '4'
Actual Expected (Benford)
21.9% 9.7%
This result does not mean that there is a fraud. It does, however, tell us that the number of multi-digit expense claims starting with the digit '4' is not as we would expect in that it greatly exceeds the probability that Benford predicts. Thus our sample for audit should include all expense claims starting with the digit '4'.
There could, of course, be a perfectly valid reason for the deviation from the expected probability. However, for example, if we also know, that two signatures are required to approve all claims over €500 and only one up to €500. This could mean that controls are being circumvented to avoid the need for two signatures by multiple expense claims being submitted. Clearly this is a circumstance that requires further investigation.
LIMITATIONS OF BENFORD’S LAW
The law does not apply to all sets of numbers and in selecting samples the following need to be considered:
-The sample size must be large enough for the digit pattern to emerge.
-It does not work for numbers that are truly random (e.g. lottery numbers!)
-It will not work where numbers are constrained (e.g. where the data lies in a pre-defined range of values, a limit on the maximum values or the same numbers appear regularly for some reason).
-It does not work for non-naturally occurring numbers (telephone or PPS numbers).
Other work in this area has shown that Benford's Law applies to accounting data. In particular, Nigrini has developed a number of tests based on the law and his book shows how these can applied to detecting fraud. In addition to using Microsoft Excel, there a number of commercially available software packages that will automate the process of analysing the data.
SUMMARY
This article seeks to introduce readers to Benford's' Law and highlight how it can be used to effectively select samples for proactive fraud detection. I would refer readers to a number of articles that go into this subject in more detail and show how to use Microsoft Excel to undertake Benford analyses.
Bibliography
Digital Analysis Using Benford's Law: Test and Statistics for Auditors by Mark Nigrini. Global Audit Publications.
Benford's Law Made Easy by David G. Banks, The White Paper September / October 1999
See also the website of the
Association of Certified Fraud Examiners website has a wealth information.