Risk Based Audit Approach
Monetary Unit
Sampling
Session Overview
This session is the last of the sessions on
audit sampling. Session 5.1 was on the
basic concepts of sampling and Session 6.1
on the application of attribute sampling for
control procedures. This session will cover
the basics of another type of sampling, the
Monetary Unit Sampling and its application
for substantive test of details.
Actually, Monetary Unit Sampling (MUS)
is used widely both for
 Test of Controls
 Account Balance
In this session, we will discuss MUS for
Substantive Test of Details covering the
following important points:
•
•
•
Monetary Unit Sampling (MUS),
advantages and limitations of MUS and
it’s application to substantive test of
details
Determining the sample size for
substantive test of details; and
Evaluation of results of substantive test
of details.
.
Learning Objectives
At the end of the session, you would be
able to apply MUS for substantive test of
details, to the extent that, the steps are
followed correctly and keeping in mind the
advantages and limitations of MUS.
The session is expected to provide only a
broad overview of MUS and is not meant to
impart expertise on MUS. Participants may
read additional study material suggested in
the bibliography for further knowledge.
Note 7.1
Session 7.1
Basic Concepts
Some of the new concepts that would be
discussed in this session are explained
below:
(a) Monetary Unit Sampling:
Monetary Unit Sampling (MUS) is a
sampling method in which the sampling
unit is not an invoice or any other physical
unit, but an individual rupee. However,
when the individual rupee is selected, the
auditor does not verify just that particular
rupee, but the rupee acts as a hook and
drags the whole invoice with it. For
example, if as a result of sample selection,
Rs.365 is selected for testing and if that
rupee falls in voucher number 14, then that
voucher will be audited and its quality
assigned to the sampling unit.
Let us assume that there are 6 items out of
which 2 items are to be selected. The value
of the 6 items are 100, 200, 300, 400, 500
and 1000. If attribute sampling is used to
select the 2 items, then all the items have
equal chance of selection, as the sampling
unit would be individual item. On the other
hand, if MUS is used, then the total value
of 6 items works out to Rs.2500, i.e., there
are 2500 sampling units. As 2 items are to
be selected, the sampling interval would be
2500/2 (Rs.1250). This means that one
rupee out of every 1250 rupees would be
selected. In such a case, the chances of
1000 rupee item getting picked up is 10
times more than the 100 rupee item getting
picked up. Thus MUS has a bias towards
high value items.
(b) Most Likely Error:
Most Likely Error (MLE) is an estimation
of the error in the population. Initially MLE
will be estimated based on past experience
and used for determining the sample size.
After carrying out substantive test of
details, the MLE will be projected based on
actual sample results and used for drawing
audit conclusions.
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Risk Based Audit Approach
(c) Basic Precision:
Basic Precision (BP) is the allowance for
errors which exist but no evidence of that is
found in the sample. It is dependent on the
confidence level and the size of the sample
and is present even where no errors have
been found in the sample.
(d) Precision Gap Widening:
Precision Gap Widening (PGW) is the
additional allowance that must be made in
the Precision as a result of errors in the
sample.
(e) Planned Precision:
Planned Precision (PP) is the difference
between Upper Error Limit and the Most
Likely Error, i.e., Planned Precision =
Materiality - Most Likely Error = Basic
Precision + Precision Gap Widening.
(f) High value and key items:
The Auditor may decide that all items/
transactions above a particular monetary
value are to be audited 100%. These items
are called high value items. For e.g., the
auditor may decide that all items above
Rs.100,000 are to be audited fully.
Similarly the auditor using his judgement
may decide that some items due to their
nature are prone to error. Such items are
called key items. For e.g., if there is a
complete breakdown of controls in a
particular division in one account area, he
may treat all items relating to that account
area from that division as key items and
check all these items. It is to remembered
that all high and key items are to be
deducted from population to arrive at
representative population for sampling.
(g) Maximum Possible Misstatements or
Upper Error Limit:
The Upper Error Limit (UEL) is the
maximum possible error estimated in the
population as a result of the substantive test
of samples. If the Upper Error Limit is
above the materiality limit, then the auditor
will either perform further substantive tests
Note 7.1
Session 7.1
to check whether there is a material error or
conclude that there is a material error.
(h) Tainting:
Tainting is the percentage of error found in
monetary terms in a sample item. For e.g.,
if the accounts receivable balance of ‘x’ as
per financial statement is Rs.10,000 and its
value as per auditor’s findings is Rs.8,000
then the tainting would be (Rs.10,0008,000)/10,000 = 20%.
MUS – Advantages, Limitations
and Relevance to Substantive Tests
As described previously, substantive tests
are those tests of transactions and balances
that seek to provide evidence as to the
completeness, accuracy and validity of
information in the financial statements. The
objective of substantive testing is to obtain
reasonable assurance that financial
statement assertions individually and
together correspond to established criteria
within limits not exceeding materiality.
Thus, substantive tests are intended to
determine the monetary effect of errors in
the financial statements. For example, the
aim of substantive tests of accounts
receivable could be to check the extent to
which the balance is overstated. For
substantive test of details (audit of
individual transactions), we must use
sampling to select individual items for
examination.
In MUS, as explained earlier, each rupee is
treated as a sampling unit and acts as a
hook for the physical unit in which it
occurs; conclusions on the physical unit in
monetary terms can be reached. The results
from the tests of sample are then used to
project the most likely error and the upper
error limit in the population. As MUS helps
in arriving at audit conclusions in monetary
terms with quantification of the degree of
confidence in the result, it is the preferred
method of sampling for substantive test of
details.
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Risk Based Audit Approach
Advantages of MUS
The important advantages of MUS are:
(i) It normally produces smaller sample
sizes than other substantive sampling
plans.
(ii) There is no difficulty in expressing a
conclusion in monetary terms.
(iii) The application of rupee unit sampling
is not contingent on knowledge of the
population size. This permits sample
selection to be started before the total
value of the final population is known.
As will be explained later in the
session, the average sampling interval
can be worked out without details of
population size.
(iv) No rupee stratification is necessary, as
this
will
be
accomplished
automatically, thus avoiding problems
of determining optimum strata
boundaries and allocation of sample
size among strata.
(v) It is relatively easy to apply compared
to other sampling plans.
(vi) The problem of detecting the large but
infrequent errors is solved, since all
items greater than the sampling
interval will be selected.
Disadvantages of MUS
The main disadvantages of MUS are the
following:
(i) Accounts/items with nil balances will
have no chance of selection.
(ii) The more an item is understated; the
less likely the item has a chance of
selection. Hence, MUS is less useful
for finding understatements.
(iii) A large percentage error in a small
transaction can significantly increase
the computed error limit.
(iv) It is very difficult to use MUS in a
non-computerized environment as
totaling the sample items in the
population for the purpose of finding
Note 7.1
Session 7.1
out the particular item in which the
dollar falls, is an onerous task.
(v) MUS is more time consuming than
other sampling plans, as the number of
sampling units (rupee) is higher than in
attribute sampling (physical unit like
voucher, cheque).
Steps involved in MUS
The stages in MUS are:
(a) determining the sample size;
(b) selecting the sample for performing
substantive test of details; and
(c) evaluation of sample test results.
Steps involved in performing these stages
are explained below:
(a) Determining the Sample Size
The steps involved in determining the
sample size are as follows:
(i) Set Upper Error Limit (UEL) (mostly
equal to materiality).
(ii) Subtract the estimated Most Likely
Error (MLE) (usually based on prior
knowledge, i.e. the results of last
year’s audit; in the absence of sound
prior knowledge approximately 15-20
percent of materiality is a good rule of
thumb)
(iii) Subtract Precision Gap Widening
(PGW) (approximately 1/2 materiality
is a good rule of thumb, but can vary
depending upon the number and
magnitude of errors anticipated and
the assurance level).
(iv) Obtain Basic Precision (BP): BP =
UEL -MLE- PGW.
(v) For the given assurance level, use the
Assurance table to determine the
basic precision factor.
(vi) Calculate the sampling interval by
using the formula: Average Sampling
Interval (ASI) = BP/BP factor.
(vii) Deduct the high value and key value
items from the total population to
arrive at the estimated representative
population.
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Risk Based Audit Approach
(vii)
Session 7.1
Calculate the sample size by using
the formula: Sample size =
representative population/ average
sampling interval.
3. Average sampling interval
(1/3)
Rs.261,000(B)
Calculation of Sample Size
Example 7.1.1
An auditor is performing substantive tests
on the valuation of buses in the company.
The total value of the fleet is
Rs.25,000,000. The materiality limit
established for the audit is Rs.1,000,000.
The auditor anticipates an error of
Rs.250,000 (based on past experience). The
auditor estimates the precision gap
widening of Rs.150,000. The auditor wants
90 percent assurance from the substantive
tests. The auditor has identified high-value
items of Rs.1,000,000 and key-value items
of Rs.500,000 in the population. Using the
table of basic precision and precision gap
widening factors (Appendix - I), the basic
precision, average sampling interval and
representative sample size are calculated as
explained below:
Calculation of Basic Precision
1. Upper Error Limit
(equal to materiality)
Rs.1,000,000
2. Less: anticipated most
likely errors
Rs.250,000
3. Estimated precision
available (1-2)
Rs.750,000
4. Less: estimated precision
gap widening
Rs.150,000
5. Basic precision (3-4)
Rs.600,000 (A)
Calculation of Average Sampling Interval
1. Basic precision for audit
2. Assurance level expected
3. Basic precision factor
(from table for 90%)
Note 7.1
Rs.600,000
90%
2.3
1. Average sampling interval Rs.261,000(B)
as above
2. Total population
Rs.25,000,000
3. Less: High-value items
Rs.1,000,000
4. Less: Key-value items
Rs.1,000,000
5. Representative test
population
Rs.23,500,000
6. Representative sample
size (5/1)
90
Thus the representative sample size for
substantive test of details will be 90.
(b) Selecting the Sample
Out of the four samples selection methods
is described in Session 5.1 Basic
concepts of statistical sampling, the
two methods that are used in MUS are
systematic selection and cell selection.
In systematic selection, one or two
items are selected randomly and the
average sampling interval is added to
arrive at the other items to be selected.
In cell selection, the population is
divided into various cells and one item
is selected from each cell. Systematic
selection ensures that all items whose
value is higher than the average
sampling interval are automatically
selected. In Cell selection all items
whose value is twice more than the
sampling
interval
are
selected
automatically. Thus, chances of highvalue but infrequent error escaping
audit scrutiny are avoided. After
selecting the sample refer to Handout
7.1.3 go through it before we discuss it
in detail.
(c) Evaluating Results of Substantive
Test of Details
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Risk Based Audit Approach
The steps involved in evaluating the results
of substantive test of details are:
(i) Calculate the percentage of tainting
for individual items.
(ii) Add
the
individual
tainting
percentages to arrive at net tainting
percentage.
(iii) Calculate the Most Likely Error
(MLE) for the representative sample
by using the formula: MLE = Net
tainting percentage * Average
Sampling Interval (ASI)
(iv) Add the errors in the high-value and
key-value items to the MLE to arrive
at total most likely errors as follows:
Total most likely error = MLE + error
in high value items + error in key
value items
(v) Arrange the tainting according to the
percentage of tainting, in descending
order,
for
overstatement
and
understatement separately. Only
tainting found out from the
representative sample are to be used
for this purpose and tainting in highand key-value items is to be excluded.
(vi) Calculate the tainting adjusted PGW
factor for each tainting sorted as at
(v).
Tainting adjusted PGW factor =
Tainting percentage * PGW factor.
(vii) Total the tainting adjusted PGW
factors for overstatement and
understatement separately.
(viii) Calculate
overstatements
and
understatements PGW of separately:
PGW = sum of tainting adjusted
PGW factors * ASI.
(ix) Calculate UELs for overstatement and
understatement separately:
UEL(overstatement) = MLE + BP +
PGW
UEL(understatement) = MLE - BP –
PGW
Example 7.1.2
Note 7.1
Session 7.1
In the example 7.1.1 discussed earlier,
assume that the auditor has identified the
following errors:
Item .
No.
Book
value
Representative sample:
14
Rs.5,000
24
Rs.7,500
16
Rs.4,000
High value:
28
Rs.30,000
Audited
value
Rs.3,000
Rs.1,500
Rs.6,000
Rs.21,000
Using the basic precision and precision gap
widening factors table (Appendix 7.1-A),
the total most likely error, the upper error
limits for overstatement and understatement
and audit conclusion can be arrived at as
explained below:
Calculation of Total Most Likely Error
1. Net sample error tainting will be:
Item Book
number value
Audit
value Error Tainting
14 Rs.5,000 Rs.3,000 Rs.2,000
40%
24 Rs.7,500 Rs.1,500 Rs.6,000
80%
16 Rs.4,000 Rs.6,000(Rs.2,000) (50%)
Net tainting
70%
2. Net most likely error for representative
sample will be:
Sum of net tainting percentage % x
Average sampling interval
We know that average sampling interval is
Rs. 261,000 (B) in Example 7.1.1.
70% X Rs.261,000 = Rs.182,700
3. Error in high-value item= Rs.30,000Rs.21,000=Rs.9,000
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Risk Based Audit Approach
Session 7.1
4. Total most likely error 2+3
=Rs.191,700(X)
Upper Error
Limits
Calculation of Upper Error Limits
Audit Conclusions
The most likely error in the population is
Rs.191,700 overstatement. The UELs at 90
percent
confidence
are
that
the
overstatement is at most Rs.958,740 and
the understatement at the most is
Rs.485,295. As the UELs are less than the
materiality limit the auditor can conclude
that there is no material error. If the UEL is
more than the materiality, then the Auditor
will have to conclude either that there is a
material misstatement in the accounts or
increase the quantum of substantive test of
details.
1. Tainting adjusted Precision Gap
Widening factors
(a) Overstatements
Taintings
Ranked
PGW
by size
factors
(from table)
Tainting
adjusted
PGW factors
40%
2nd
0.43
40%X0.43=0.172
80
1st
0.59
80%X0.59=0.472
(b) Understatements
50%
1st
0.59
50%X0.59=0.295
2. Precision Gap Widening
PGW = Sum of tainting adjusted PGW
factors x Average sampling interval
(a) Overstatements = 0.644 X
261,000=Rs.167,040
(Y)
(b) Understatements =0.295 X
261,000=Rs.76,995 (Z)
3. Upper Error Limits
Basic Precision is Rs.600,000 as
worked out at (A) in Example 7.1.1
Overstatements Understatements
Rs.
Rs.
Basic Precision
600,000
PGW
167,040
(Y) as above
(485,295)
Thus MUS is used, both at the stage of
planning the sample size and subsequently
at the stage of evaluation of sample results
of substantive test of details. We will now
discuss
Handout
7.1.3
performing
substantive test of details
Summary
The key points that were discussed in the
session are:
• Monetary Unit Sampling (MUS) –
Advantages, limitations and relevance
of MUS for substantive test of details
• Steps involved in MUS
• Determining the sample size for
substantive test of details
• Selecting the sample for performing
substantive test of details and
• Evaluation of results of substantive
test of details.
(600,000)
(76,995)
(Z) as above
Total Precision
767,040
(676,995)
Add total most
likely error
(X) as above
191,700
191,700
Note 7.1
958,740
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