What Does Benford’s Law Have to Do with Insurance Fraud?
Radost Wenman

What Does Benford’s Law Have to Do with Insurance Fraud?

Benford’s law is not some arcane legal clause reserved only for those who are well-versed in legal jargon and procedurals, though I admit that the “law” reference in the name is a bit misleading. Benford’s law is a little-known mathematical curiosity, but if you mention it to a forensic accountant, you will immediately sense excitement rather than bewilderment. Forensic experts from the financial sector often rely on this technique to chase data anomalies and financial fraud based on the distribution of digits in the numbers they examine. Naturally, this type of digital analysis can also be extended to detecting insurance fraud, or at least red-flagging cases as suspicious and aberrant. The premise is that fraudulent data do not conform to the mathematical patterns of the law, and cases that are isolated as non-conforming should at least warrant further investigation.

What renders Benford’s law interesting are not only its applications in solving real-world problems, but the fact that, at first blush, its concept runs counter to people’s intuition. Try this little experiment: Present your friends with a sequence of numbers for a naturally-occurring, real-life phenomenon, such as the population size of the cities in the U.S., or your credit card charges over time. Then ask them how often they believe the non-zero leading digits 1 through 9 occur in the data (e.g., the leading digit in 95 is 9, and the one in 0.05 is 5). Most of the time, the answer you will hear is: “About the same for each digit, around 11%, or 1 out of 9.” 

We are inclined to think that the leading digits would approximately follow a uniform frequency distribution, but according to Benford’s law, the distribution is actually quite skewed. Indeed, Benford’s law states that the distribution function of the first digit D is roughly proportional to the common logarithm of the digit itself. More precisely, the discrete probability density is given by: 

The graph below illustrates the logarithmic distribution of D. Plugging the digits 1 through 9 into the above formula, we see that 1 is predicted to show up as the leading digit about 30% of the time, 2 about 18% of the time, and so on, while 9 is expected in less than 5% of the cases. Pretty mind-bending, isn’t it?

So, for example, if the most significant digits of the numbers you are investigating follow a pattern that looks something like the pink graph below, you know a funny fish might be swimming in your data. The pink line shows many more 1’s and fewer 9’s than predicted by the law. As a matter of fact, the pink line represents the returns for the Fairfield Sentry Fund, whose investments were placed solely with Bernie Madoff.

The first-digits law has several other variations which describe the distributions of other significant positions (2nd, 3rd and so on), as well as combinations of digits. Even though Benford’s law has some limitations, research suggests that when expertly applied, either alone or in tandem with more complex analytics tools such as neural networks, the law could provide a useful means for insurance companies to mine their data for the presence of fraud. The more fraud detection methods a company has at its disposal, the better equipped it is to spot the funny fish. Because while it is possible to game one method, it is difficult to game them all.

Radost Wenman, FCAS, MAAA, is a Consulting Actuary with Pinnacle Actuarial Resources, Inc. in the San Francisco, California office. She holds a Master of Science degree in Statistics and a Bachelor of Science degree in Mathematics from Stanford University. Radost has over nine years of experience in the capacity of a pricing actuary in the personal lines segment. In this role, she has developed home and auto pricing solutions through the design and implementation of advanced predictive models.
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