DIGITAL
ANALYSIS IN INTERNAL AUDIT
Digits can do wonders. Analysis of
Digits can do more wonders. It is a methodology to find abnormal duplications of specific
digits, combinations of digits, specific
numbers in corporate data.
Enthusiastic internal auditors and external auditors may likely to use Digital Analysis which
bring value added services to their corporates and clients. The purpose of this paper submission
is to enrich the auditors in the most effective use of digital analysis, based
on Benford’s Law.
Benfords Law :
Similar to the experience of an apple
fell on the head of Sir Isaac Newton, Frank Benford, a physicist at the GE Research
Laboratories also had a unique experience.
In the year, 1920 while at his work
spot, Frank Benford noticed that his log table books were more worn at the
first few pages than at the last few pages. The first pages give the logarithms
of numbers with low first digits. The first digit in a number is the left most
digit. For example, the first digit of 1,18,020 is 1 and the third digit
is 8. The first and second digit of
7,840 is 7 and 8 respectively. Benford hypothesized
that he was looking up the logs of numbers with low first digits more
often, because there were more numbers with low first digits than with high
first digits.
Origin :
In 1881, Simon Newcomb, an astronomer
and mathematician, published the first known article describing what has become
known as Benford’s Law in the American Journal of Mathematics. He observed that
library copies of books of logarithms were considerably more damaged in the starting pages which dealt with low
digits and comparitively less damaged on the pages dealing with higher digits.
He himself inferred from this pattern that fellow scientists used those tables
to look up numbers which started with the numeral one more often than those
starting with two, three and so on. The obvious conclusion was that more
numbers exist which begin with the numeral one than with the large numbers.
Newcomb calculated that the probability that a number has any particular
non-zero first digit is:
P(d) = Log10(1+1/d)
Where d is a number 1..2...3................9... and P is the probability.
Using Newcomb’s formula, the
probability that the first digit of a number is one is about 30% while the
probability of the first digit a nine is only 4.6 percent. The following table
shows the expected frequencies for all digits starting from 0 to 9 in each of
the first two places in any number.
Expected
Frequencies based on Benford’s Law
|
DIGIT
|
FIRST
PLACE
|
SECOND
PLACE
|
|
0
|
---
|
0.11968
|
|
1
|
0.30103
|
0.11389
|
|
2
|
0.17609
|
0.10882
|
|
3
|
0.12494
|
0.10433
|
|
4
|
0.09691
|
0.10031
|
|
5
|
0.07918
|
0.09668
|
|
6
|
0.06695
|
0.09337
|
|
7
|
0.05799
|
0.09035
|
|
8
|
0.05115
|
0.08757
|
|
9
|
0.04576
|
0.08500
|
In a nutshell, we may say that Benford’s
law is based on a peculiar observation
that lower digits appear more frequently than higher digits in data sets. Hitherto, we focused on sample, which is now switched over to population!
Let us
see how this phenomenon applied to Auditing and Accounting.
Benford’s
Law applied to Internal Audit:
Not all data sets are expected to have the
digit frequencies of Benford’s Law: therefore the guidelines for deciding
whether a data set would comply are that:
1. The
numbers in the data set should describe the sizes of the elements in the data
set;
2. There
should be no built-in maximum or minimum to the numbers. A maximum or minimum
that occurs often would cause many numbers to have the digital patterns of the
maximum or minimum;
3. The
numbers should not be assigned. Assigned numbers are those given to objects to
identify them. Examples are social security, bank account and telephone numbers,
and cheque numbers.
Benford’s Law gives auditors the expected frequencies
of the digits in tabulated data.The premise is that we would expect authentic,
and un manipulated data to exhibit these patterns. If a data set does not
follow these patterns, however a few
possible reasons exist to explain this phenomenon.
1. The
data set did not meet the three tests noted above, and/or;
2. The data
set includes invented numbers, biased numbers
or errors.
If
a data set follows Benford Law, auditors might conclude that the data have
passed a reasonableness. This does not mean that all the numbers are correct,
but rather that any errors or manipulations were not significant enough to
distort the digit patterns. Auditors would still have to combine Digital
Analysis with other analytical procedures and possible statistical sampling
procedures.
Audit
tests:
The first and second digit tests are
used as high level tests of reasonableness. Experience has shown that large
accounts payable files have followed Benford’s Law quite closely. Also the
extended values of the Inventory follows Benford’s Law closely. The third, more
focused test is to verify the frequencies of the first two digits. The formula
for the expected first two digit frequencies under Benford’s Law is :
Expected
First Two Digit frequency=log(1+1/FTD)
The fourth, still more focused test is
the number duplication test. Here, the frequencies of the actual numbers in the
audit data set are tabulated. The list begins with the most frequently
occurring number and ends with the least frequently occurring number. Auditors
generally direct their attention to
1. Numbers
that have occurred abnormally often relative to
other numbers;
2. Odd
numbers that have occurred abnormally often;
3. Round
numbers that have occurred abnormally often;
Exemptions
:
Some populations of accounting-
related data do not conform to a Benford
distribution. For example, assigned numbers such as cheque numbers,
purchase order numbers or numbers that are influenced by human thought , for
example, the price of a pair of shoes is 199.99 or ATM withdrawals do not
follow Benford’s Law. Assigned numbers should follow a uniform distribution
rather than a Benford distribution.
In addition to an auditor’s
perception in determining which
populations fit a Benford distribution, there exist some tests that reveal
whether or not Benford’s Law applies to a particular data set. Wallace suggests
that if the mean of a particular set of numbers is larger than the median and
the skewness value is positive, the data set likely follows a Benford
distribution. It follows that the larger the ratio of the mean divided by the
median, the more closely the set will follow Benford’s Law. This is true since
observations from a Benford distribution have a predominance of small values.
The difficulty in relying only on such tests as a screening process, before
applying digital analysis, is that if an account contains sufficient bogus
observations it could fail the tests; thus, digital analysis would not be
applied when, in fact, it should.
Conclusion:
We conclude that Benford’s analysis, when
used correctly, is a useful tool for identifying suspect accounts for further
analysis. Because of its usefulness, digital analysis tools based on Benford’s
Law are now being included in many popular software packages like ACL and
CaseWare 2002.The goal of this paper has been to help auditors more
appropriately apply Benford’s law-based analysis to increase their ability to
detect ubnormal transactions or fraud. Benford analysis is a particularly useful analytical
tool because it does not use aggregated data, rather it is conducted on
specific accounts using all the data available. It can be very useful in
identifying specific accounts for further analysis and investigation.
References :
1. Mark
J Nigrini Ph.D, Institute of Internal
Auditors
247, Maitland Avenue, Altamonte Sprins,
Florida.
2. “ The
effective use of Benford’s Law to assist in detecting fraud in Accounting Data”
– M/s Cindy Durtschi, William Hillison & Carl Pacini.
3. Chartered
Accountant Magazine November 2011 issue.
