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. 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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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

Topics

  • Expectations

  • Background

  • Why it works

  • Real-world examples

  • How do I use it?

  • Questions


Expectations

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


Expectations1

Expectations

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


Background

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


Background1

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


Background2

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


Background3

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%


Background4

Background

  • What are “random numbers”?

    • Non-manipulated numbers

      • Population stats, utility bills,

      • Areas of rivers

    • NOT human-selected #s

      • Zip codes, SSN, Employee ID


Background5

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


Background6

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


Background7

Background

  • How will it help me with non-random data?

    • Aid in detection of unusual patterns

      • Circumventing controls

      • Potential fraud


Why it works

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” ...


Why it works1

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”


Why it works2

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


Why it works3

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


Examples

Examples

  • $2M Check Fraud in AZ

  • $4.8M Procurement fraud in NC


Example 1

Example #1

  • Check fraud in AZ

    • #s appear random to untrained eyes

    • Suspicious under Benford’s Law

    • Counter-intuitive to human nature


Example 11

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


Example 12

Example #1

  • Avoided common indicators:

    • No duplicate amounts

    • No round #s – all included cents


Example 13

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


Example 2

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


Example 21

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 …


Example 22

Example #2

See any suspicious areas?


Example 23

Example #2

Drilling down in the “51” txns


Example 24

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.


How do i use it

How do I use it?

  • Data Analytics software

    • ACL / IDEA

  • Excel

    • Add-Ons

    • Built-in Excel Functions


Questions

Questions


Summary

Summary

  • Expectations

  • Background

  • Why it works

  • Real-world examples

  • How do I use it?


Contact information

Contact Information

  • Ed Tobias

    • [email protected]

  • LinkedIn

    • http://www.linkedin.com/in/ed3200


References

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


References1

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.


References2

References

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


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