Benford’s Lawand

Fraud DetectionOzan Gurel

Certified Fraud Examiner

Bloomberg Quant SeminarSeptember 6, 2016

If you have a large set of numbers, what would you expect the frequencies of the possible first digits of all the numbers to be?

If you have a large set of numbers, what would you expect the frequencies of the possible first digits of all the numbers to be?

With the first digit being one of nine numbers (1-9), would you expect the probability of each to be 0.1111 (1/9, 11.11%)?

If you have a large set of numbers, what would you expect the frequencies of the possible first digits of all the numbers to be?

With the first digit being one of nine numbers (1-9), would you expect the probability of each to be 0.1111 (1/9, 11.11%)?

What became known as Benford’s Law was observed at General Electric by Frank Benford in the 1930s and published as

Benford, Frank, “The Law of Anomalous Numbers,” Proceedings of the American Philosophical Society, 78 (4), Mar. 31, 1938, pp. 551-572, but was first reported in 1881 by Simon Newcomb of the National Almanac Office in

Newcomb, Simon, “Note on the Frequency of Use of the Different Digits in Natural Numbers,” American Journal of Mathematics, 4 (1), 1881, pp. 39-40.

Benford’s Law

The expected proportion of numbers beginning with the leading digit n is:

log10((n + 1)/n)

IntuitionThe percentage increase of a number required to get it to the next first-digit level decreases as the first digit increases from 1 to 9.

100 to 200 - 100%200 to 300 - 50%300 to 400 - 33.33%400 to 500 - 25%500 to 600 - 20%600 to 700 - 16.67%700 to 800 - 14.29%800 to 900 - 12.5%900 to 1000 - 10%

Miller-Nigrini Excel Benford TesterFrance - Populations légales en quartiers de la politique de la ville - 2013

Comparison of First Digits of Returns of Madoff and Apple

Source: http://www.nigrini.com/benfordslaw.htm

Madoff Apple

Benford’s Law and Enron

Disbursement Fraud in Oregon

Unusually high percentage of disbursements beginning with 1 found in audit conducted by Audits Division, Secretary of State, State of Oregon.Further analysis found a large number of $100 transactions, most of which were fraudulent.

Source: https://oregonaudits.org/category/fraud-investigation/

Source: Durtschi, Cindy, William Hillson, and Carl Pacini, “The Effective Use of Benford’s Law to Assist in Detecting Fraud in Accounting Data,” Journal of Forensic Accounting, 5, 2004, pp. 17-34.

Commercial Applications

• ACL – Vancouver, BC – Internal Audit

• AQRM – Accounting Quality + Risk Matrix by Ives Group – Sutton, MA

(Bloomberg Terminal App)

• IDEA by CaseWare Analytics – Ottawa, ON