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It’s All in the Numbers - Benford’s Law

It’s All in the Numbers - Benford’s Law. Ed Tobias, CISA, CIA May 12, 2010. Topics. Expectations Background Why it works Real-world examples How do I use it? Questions. Expectations. How many have heard of it? All over the professional journals J. of Accountancy – 2003, 2007

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It’s All in the Numbers - Benford’s Law

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  1. It’s All in the Numbers -Benford’s Law Ed Tobias, CISA, CIA May 12, 2010

  2. Topics • Expectations • Background • Why it works • Real-world examples • How do I use it? • Questions

  3. Expectations • How many have heard of it? • All over the professional journals • J. of Accountancy – 2003, 2007 • J. of Forensic Accounting – 2004 • Internal Auditor – 2008 • ISACA Journal – 2010 • Fraud Magazine - 2010

  4. Expectations • As of 2004, over 150 articles have been written about Benford’s Law

  5. Background • 1881 – Simon Newcomb, astronomer / mathematician • Noticed that front part of logarithm books was more used • Inferred that scientists were multiplying more #s with lower digits

  6. Background • 1938 – Frank Benford, Physicist at GE Research labs • Front part of the log book was more worn out than the back • Analyzed 20 sets of “random numbers” – 20,299 #s in all

  7. Background • Tested random #s and random categories • Areas of rivers • Baseball stats • #s in magazine articles • Street addresses - first 342 people listed in “American Men of Science” • Utility Bills in Solomon Islands

  8. Background • Benford’s Law: • Random #s are not random • Lower #s (1-3) occur more frequently as a first digit than higher numbers (7-9) • In a sample of random numbers: • #1 occurs 33% • #9 occurs 5%

  9. Background • What are “random numbers”? • Non-manipulated numbers • Population stats, utility bills, • Areas of rivers • NOT human-selected #s • Zip codes, SSN, Employee ID

  10. Background • What’s the practical use? • 1990s – Dr. Mark Nigrini, college professor • Tested insurance costs (reim. claims), sales figures • Performed studies detecting under/overstmts of financial figures • Published results in J. of Accountancy (1990) and ACFE’s The White Paper (1994) • Useful for CFEs and auditors

  11. Background • What about financial txns? • “Random data” = non-manipulated numbers • AP txns, company purchases • NOT human-selected #s • Expense limits (< $25) • Approval limits (No sig < $500) • Hourly wage rates

  12. Background • How will it help me with non-random data? • Aid in detection of unusual patterns • Circumventing controls • Potential fraud

  13. Why it works • You won the lottery – invest $100M in a mutual fund compounding at 10% annually • First digit is “1” • Takes 7.3 yr to double your $ • At $200M, first digit is “2” ...

  14. Why it works • At $500M … First digit is “5” • Takes 1.9 yr to increase $100MM • Although time is decreasing, there are more years that start with lower digits • Eventually, we will reach $1B • First digit is “1”

  15. Why it works • Seems reasonable that the lower digits (1-3) occur more frequently • These 3 digits make up approx. 60% of naturally-occurring digits

  16. Why it works • Scale invariant • 1961-Roger Pinkham • If you multiply the numbers by the same non-zero constant (i.e., 22.04 or 0.323) • New set of #s still follows Benford’s Law • Works with different currencies

  17. Examples • $2M Check Fraud in AZ • $4.8M Procurement fraud in NC

  18. Example #1 • Check fraud in AZ • #s appear random to untrained eyes • Suspicious under Benford’s Law • Counter-intuitive to human nature

  19. Example #1 • Wrote 23 checks (approx. $2M) • Many amts < $100K • Tried to circumvent a control that required a human signature • Mgr tried to conceal fraud • Human choices are not random

  20. Example #1 • Avoided common indicators: • No duplicate amounts • No round #s – all included cents

  21. Example #1 • Mistakes: • Repeated some digits / digit combinations • Tended towards higher digits (7-9) • Count of the leading digit showed high tendency toward larger digits (7-9) • Anyone familiar with Benford’s Law would have recognized the larger digit trend as suspicious

  22. Example #2 • Benford’s Law can be extended to first 2 digits • Allow examiner to focus on specific areas • High-level test of data authenticity

  23. Example #2 • Procurement fraud in NC • 660 invoices from a vendor • Years 2002-2005 • Total of $4.8M submitted for payment • Run the 660 txns through Benford’s Law …

  24. Example #2 See any suspicious areas?

  25. Example #2 Drilling down in the “51” txns

  26. Example #2 • Over a 3-year period, at least $3.8M in fraudulent invoices for school bus and automobile parts were submitted. • The investigation recovered $4.8M from the vendor and former school employees.

  27. How do I use it? • Data Analytics software • ACL / IDEA • Excel • Add-Ons • Built-in Excel Functions

  28. Questions

  29. Summary • Expectations • Background • Why it works • Real-world examples • How do I use it?

  30. Contact Information • Ed Tobias • ed.tobias@hillsclerk.com • LinkedIn • http://www.linkedin.com/in/ed3200

  31. References • Benford’s Law Overview. n.d. Retrieved March 10, 2010 from http://www.acl.com/supportcenter/ol/courses/course.aspx?cid=010&ver=9&mod=1&nodeKey=3 • Browne, M. Following Benford’s Law, or Looking Out for No. 1.n.d. Retrieved March 10, 2010 from http://www.rexswain.com/benford.html • Durtschi, C., Hillison, W., and Pacini, C. The Effective Use of Benford’s Law to Assist in Detecting Fraud in Accounting Data. 2004. Journal of Forensic Accounting. Vol. V. Retrieved March 10, 2010 from http://www.auditnet.org/articles/JFA-V-1-17-34.pdf • Managing the Business Risk of Fraud. EZ-R Stats, LLC. 2009. Retrieved March 10, 2010 from http://www.ezrstats.com/CS/Case_Studies.htm • Kyd, C. Use Benford’s Law with Excel to Improve Business Planning. 2007. Retrieved March 10, 2010 from http://www.exceluser.com/tools/benford_xl11.htm

  32. References • Lehman, M., Weidenmeier, M, and Jones, T. Here’s how to pump up the detective power of Benford’s Law. Journal of Accountancy. 2007. Retrieved March 10, 2010 from http://www.journalofaccountancy.com/Issues/2007/Jun/FlexingYourSuperFinancialSleuthPower.htm • Lynch, A. and Xiaoyuan, Z. Putting Benford’s Law to Work. 2008. Internal Auditor. Retrieved March 10, 2010 from http://www.theiia.org/intAuditor/itaudit/archives/2008/february/putting-benfords-law-to-work/ • Nigrini, M. Adding Value with Digital Analysis. Internal Auditor. 1999. Retrieved March 10, 2010 from http://findarticles.com/p/articles/mi_m4153/is_1_56/ai_54141370/ • Nigrini, M. I’ve Got Your Number. Journal of Accountancy. 1999. Retrieved March 10, 2010 from http://www.journalofaccountancy.com/Issues/1999/May/nigrini.htm • Rose, A. and Rose, J. Turn Excel Into a Financial Sleuth. 2003.Journal of Accountancy. Retrieved March 10, 2010 from http://www.systrust.us/pubs/jofa/aug2003/rose.htm • Simkin, M. Using Spreadsheets and Benford’s Law to Test Accounting Data. ISACA Journal. 2010, Vol. 1. Pp. 47-51.

  33. References • Stalcup, K. Benford’s Law. Fraud Magazine. 2010, Jan/Feb. Pp 57-58.

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