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

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

Ed Tobias, CISA, CIA

May 12, 2010

- Expectations
- Background
- Why it works
- Real-world examples
- How do I use it?
- Questions

- 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

- All over the professional journals

- As of 2004, over 150 articles have been written about Benford’s Law

- 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

- 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

- 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

- 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%

- In a sample of random numbers:

- What are “random numbers”?
- Non-manipulated numbers
- Population stats, utility bills,
- Areas of rivers

- NOT human-selected #s
- Zip codes, SSN, Employee ID

- Non-manipulated numbers

- 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

- 1990s – Dr. Mark Nigrini, college professor

- 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

- “Random data” = non-manipulated numbers

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

- Aid in detection of unusual patterns

- 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” ...

- 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”

- Takes 1.9 yr to increase $100MM

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

- 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

- $2M Check Fraud in AZ
- $4.8M Procurement fraud in NC

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

- 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

- Avoided common indicators:
- No duplicate amounts
- No round #s – all included cents

- 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

- Count of the leading digit showed high tendency toward larger digits (7-9)

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

- 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 …

See any suspicious areas?

Drilling down in the “51” txns

- 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.

- Data Analytics software
- ACL / IDEA

- Excel
- Add-Ons
- Built-in Excel Functions

- Expectations
- Background
- Why it works
- Real-world examples
- How do I use it?

- Ed Tobias
- LinkedIn
- http://www.linkedin.com/in/ed3200

- 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

- 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.

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