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Copyright © 2011
Fraud Detection
using Benford’s Law
James J.W. Lee MBA (Iowa,US), B.Acc (S’pore),
FCPA (S’pore), FCPA (Aust.), CA (M’sia),
CFE, CIA, CISA, CISSP, CGEIT
The Hidden Secrets of Numbers …
Copyright © 2011
Contents
I. History
II. Introduction
III. Real-world Application
IV. Case Studies
V. Caveats
VI. Demo (using ACL)
VII. Demo (using Excel)
Copyright © 2011
History
Benford’s Law
Copyright © 2011
History
In 1881, Professor Simon
Newcomb observed that
the first few pages of his
Logarithm Book (digits 1,
2, 3) looked more worn
and dirtier than the other
pages (digits 7, 8, 9).
Copyright © 2011
History
However, Professor
Simon Newcomb did not
provide any statistical
basis for his observation,
and his discovery did not
have any practical
applications
Copyright © 2011
Don’ t
Believe
?
Copyright © 2011
Introduction
Benford’s Law
Copyright © 2011
Introduction
Who’s Benford?
• In 1938, physicist Frank
Benford, who was unaware of
Newcomb’s observation, also
discovered same phenomena
with his Logarithm Book used
by scientists & engineers.
• Unlike Newcomb, Benford
attempted to test his theory
with empirical data.
Copyright © 2011
Introduction
Frank Benford analyzed 20,229 data sets, eg:
• baseball statistics, areas of rivers, molecular
weights of atoms, electricity bills, stock market
quotes, populations of towns, physical and
mathematical constants.
He discovered that appearance of each digits
(1 – 9) is not equally distributed, instead some
digits appear more frequently than others.
Benford’s Law formulated.
Copyright © 2011
Introduction
So, what’s Benford’s Law?
• Benford’s law states the probability of
obtaining digits 1 through 9 in each
position of a number
• Eg: in number “3879”
• first position: 3
• second position: 8
• third position: 7
• fourth position: 9
What’s likelihood
of getting “3” in
First position ?
Copyright © 2011
Introduction
What’s Benford’s Law? (cont’d)
• Most people assume probability for each digit
1 – 9 in any position of a number is 1/9, i.e.
each digit is equally likely to occur.
• This is not true. Benford’s Law:
in any given position of a number… • probability of 1 is higher than 2 • probability of 2 is higher than 3 • … • probability of 8 is higher than 9
Copyright © 2011
Introduction
Digit 1st Position 2
nd Position 3
rd Position 4
th Position
0 0.11968 0.10178 0.10018
1 0.30103 0.11389 0.10138 0.10014
2 0.17609 0.10882 0.10097 0.1001
3 0.12494 0.10433 0.10057 0.10006
4 0.09691 0.10031 0.10018 0.10002
5 0.07918 0.09668 0.09979 0.09998
6 0.06695 0.09337 0.0994 0.09994
7 0.05799 0.0935 0.09902 0.0999
8 0.05115 0.08757 0.09864 0.09986
9 0.04576 0.085 0.09827 0.09982
Digit “0” is excluded because it’s not first digit. Source: Nigrini, 1996
Copyright © 2011
Introduction
How is Benford’s Law possible?
• If a data entry begins with the digit 1, it has
to double in size (100%) before it begins
with the next digit – digit 2.
• If a data entry begins with the digit 9, it only
has to be increased by only 11% in order
for the first digit to be digit 1 again.
• Hence, chances of digit 1
is more likely than digit 9.
Can’t
Understand ?
See Demo.
Copyright © 2011
Introduction
Mathematical Formula for Benford’s Law
The probability of any number “d” from
1 through 9 being the first digit is….
Log10 (1 + 1/d)
Copyright © 2011
OK, now you
know
Benford’s
Law …
So What?
Copyright © 2011
Real-World
Application
Copyright © 2011
Real-World Application
Both Benford and
Newcomb did not illustrate
any practical use for this
observation.
In 1992, Chartered
Accountant Dr Mark
Nigrini used Benford’s
Law for fraud detection and
assessment of inefficiency.
Copyright © 2011
Real-World Application
Mark Nigrini’s Analysis
What Benford’s Law can do?
To detect …
1. Fraud
2. Non-compliance
3. Inefficiency
Copyright © 2011
Mark Nigrini’s Analysis
Why Benford’s Law can detect fraud?
Because culprits …
1. Attempt to evenly distribute digits, subconsciously
2. Consistently select specific numeric sequences
3. Avoid repetitions of numbers
Copyright © 2011
Mark Nigrini’s Analysis
How to apply Benford’s Law?
• Identify data with frequency of “digits”
deviated from expected pattern
• Investigate deviated data
Copyright © 2011
Mark Nigrini’s Analysis
Where can Benford’s Law be applied on?
• Purchases, Payroll and other payments
• Sales and other income
• Claims and Refunds
• Cash deposits / withdrawal
• Bank balances
• Account Payable and Receivable
• Etc…
Copyright © 2011
Mark Nigrini’s Analysis
Eg: Apply Benford’s Law on Purchases data
• Investigate deviated data sets
• Suppliers
• Purchase value
• Frequency
• Objectives: To uncover …
1. Fraud (duplicate payment / bogus supplier)
2. Non-compliance (avoid threshold)
3. Inefficiency (massive small payment)
Copyright © 2011
Distortion Factor model:
Data conformed with Benford’s Law
Benford Law is
applicable
Examples
Transaction-level data Sales, expenses
Large data sets Full year’s transactions
Data with mean >
median & positive skew
Most accounting
numbers
Data resulted from
mathematical combinatn
of numbers
A/Rec’ble (no. sold x price)
A/Pay’ble (no. bought x pr)
Source: Durtschi, 2004, 24.
Copyright © 2011
Distortion Factor model:
Data not conformed with Benford’s Law
Benford Law not
applicable
Examples
Truly random data Lottery no. (4D, TOTO)
Data with limits
(max/min)
Max ATM withdrawal amount
Artificially created /
assigned data
Telephone no., housing block
no., cheque no., invoice no.,
postal codes
Copyright © 2011
Case
Studies
Copyright © 2011
Case Study #1
State of Arizona v. Wayne James
Nelson (CV92-18841)
• Nelson, a manager in Office of Arizona
State Treasurer, diverted nearly US$2m
funds to a fictitious vendor.
• Amount in defaulted cheques appeared to
be in random, no duplicated amount, no
round numbers (i.e. each amount included
cents).
Copyright © 2011
Case Study #1
• However, subconsciously,
Nelson repeated some
digits & digit combinations,
and tendency toward the
higher digits.
• Nelson was arrested,
despite he argued that he
only demonstrated
absence of safeguards in
new computer system.
Copyright © 2011
Case Study #2
Embezzlement @ SME (Anna M. Rose and Jacob M. Rose, Aug 2003)
• SME Owner expanded his 1-store family-owned
business into a 4-store chain
• He relinquished some control to store managers
• He concerned about bookkeeping errors or
possibility of fraud
• He applied Benford’s Law to analyze each store’s
expenditure data
Copyright © 2011
Case Study #2
Result of his Benford analysis:
• First Digit Test
• Digits 5, 6, & 7 appear
more, while digit 1 appears
less, than expected
• Second Digit Test
• Again, digits 6 & 7 appear
more, while digit 0 didn’t
occur at all
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5 6 7 8 9
Digit
Ra
te Benford
Sample
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6 7 8 9
Digit
Ra
te Benford
Sample
Copyright © 2011
Case Study #2
• First Two-
Digits Test
• 56 & 67
appear
more than
expected
Result of his Benford analysis (cont’d):
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94 98
Two Digit Pair
Rate Benford
Sample
Copyright © 2011
Case Study #2
Action taken after result of his Benford
analysis:
• SME Owner vouched sample of expenditure
starting with the 56 & 67 sequences
• He discovered payments to unfamiliar vendors
• Further investigation revealed vendors did not
exist! … Payments gone to personal account of
culprit store manager!
Copyright © 2011
Caveats
Benford’s Law
Copyright © 2011
Caveats
False positives
• Not necessarily fraud
Certain types of fraud will not be
detected using Benford’s Law analysis
• “My Law” vs Benford’s Law
• Substitute with your own expected
distribution
Copyright © 2011
Demo
(ACL)
Copyright © 2011
Demo
(Excel)
Copyright © 2011
Thank You
