sampling for internal auditors - ISACA

EXPECTED PRESENTATION OUTCOMES Why Do Auditors Sample? Sampling Policy Statistical & Non-statistical Sampling Statistical Terminologies...

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SAMPLING FOR EFFECTIVE INTERNAL AUDITING By Mwenya P. Chitalu CIA

EXPECTED PRESENTATION OUTCOMES  Why Do Auditors Sample?  Sampling Policy  Statistical & Non-statistical Sampling

 Statistical Terminologies  Statistical Sampling Plans  External Auditing Standards  Sample Selection Methods

 Illustrations

DEMYSTIFYING STATISTICAL SAMPLING  The Principle (or Law) of Parsimony: That things are usually

connected in the simplest or most economical way.  Reducing ideas to small, easy-to-write symbols & saying a lot in a small

area covered by a formula.  Eliminate the Greek, Arabic & Roman language barrier in symbols & Formulae

that mystify Mathematics or Statistics.  Just like any other audit, Probe Statistical Assertions-Life can be made easy

with appropriate sampling.  If it cannot be measured, then it cannot be managed economically,

efficiently, & effectively.  Mathematics or statistics is commitment to logical thinking.  It squeezes the most learning about the population from limited sample

data.

WHY DO AUDITORS SAMPLE?  International Standards for the Professional Practice of Internal Auditing:

 Guides Information should be: Sufficient, Reliable, Relevant & Useful  Acknowledges Sampling Techniques in Evidence Acquisition  Opinions are NOT ABSOLUTE GUARANTEE but REASONABLE

ASSURANCE of Accuracy  Proficiency & Due Professional Care  Cost-Benefit Considerations: The Economy, Efficiency & Effectiveness, …  Corroborating Evidence for Control Processes & Account Balances

SAMPLING POLICY  Written Policy Statement  When to Sample?

 Who Should Sample?  How to Sample?  Inappropriate Uses for Sampling:  When a Total is easily Audited  Inquiry & Observation Procedures  Analytical Procedures

STATISTICAL & NON-STATISTICAL SAMPLING

 Three Characteristics in Common:  Both Require Auditor judgment in Planning, Implementing, &

Evaluating the Sampling Plan  Actual Audit Procedures Performed are the same  Both Non-Statistical & Statistical Techniques are Permitted by the

IPPF

STATISTICAL & NON-STATISTICAL SAMPLING

 Differences between Statistical & Non-statistical Sampling  Sampling Risk is Controlled & Measurable

 Technical Training & Knowledge is Required  Computer Accessibility

STATISTICAL & NON-STATISTICAL SAMPLING  In Summary, the following should be addressed:  What is the Internal Audit Department’s Recommended Policy or

Procedure?  Is a Quantitative measure of Sampling Risk Desired?  What is the relative Cost & Benefit of Statistical versus Nonstatistical Sampling?  Is Technical Expertise Available?  Is Computer Software Accessible or Expertise to Write a Program?

STATISTICAL TERMINOLOGIES  Confidence Level (C): Is the Reliability Level or Degree of Belief in

the Obtained Results.  Measure of Central Tendency:  Mean (µ): The arithmetic average of a set of numbers.

 Median: The halfway value of raw data arranged in numerical order from

lowest to highest.  Mode: The most frequently occurring value.

STATISTICAL TERMINOLOGIES  Standard Deviation (𝝈): The statistical measurement of the variability

of values in a sample (the square root of the variance).

 Range: The difference between the largest and smallest values of any

group.

 Population (N): The total number of items from which the sample is

drawn-It’s the focus of interest comprising sampling units.

 Sampling Unit: Individual items making up a Population.  Sample (n): Collection of sampling units on which audit procedures are

performed.

STATISTICAL TERMINOLOGIES  Logical Unit: Account or transaction selected to be sampled.  Expected Population Deviation Rate (𝝆): Estimate of the actual deviation

rate in the population, usually based on prior experience, inquiries, and observations.

 Precision (P): An assumed amount of possible unknown or the range of

allowable error.

 Tolerable Misstatement: The auditor’s assessment of materiality with

respect to the population.

 Upper Precision Limit: Upper limit on deviations expected in the

population.

STATISTICAL TERMINOLOGIES  Tainting: Percentage of misstatement in a logical unit in a PPS sample.  Upper Misstatement Limit (UML): Estimated maximum misstatement

existing in the population at a specified reliability in PPS sampling.

 Sampling Risk: Conclusions

based on sample differing with conclusions that could be reached if the entire population were examined.

 Non-sampling risk: Drawing incorrect conclusion for reasons other than

sampling due to poor judgment or failure to adhere to professional standards.

STATISTICAL SAMPLING Advantages

Disadvantages

 May yield desired results from

 Can be costly and time-

minimum number of items  Yields quantified data  Includes measures of sampling

risk, confidence level, and precision  Is adaptable to computer testing  Lends credibility to audit

conclusions/recommendations

consuming  May require training and software costs  May preclude experienced auditors’ insights

NON-STATISTICAL SAMPLING Advantages

Disadvantages

 Flexibility

 Results not statistically valid

 Use of internal auditor’s

 No objective measure of

judgment  Allows reasonable reliability at reasonable cost

sampling risk provided  Chance of wrong sample size  Effectiveness depends upon auditor’s skill

STATISTICAL SAMPLING PLANS 1.

ATTRIBUTES SAMPLING (TESTS OF CONTROLS)  Concerns binary, yes/no, or error/non-error populations  It tests the effectiveness of controls.

2.

VARIABLES SAMPLING (SUBSTANTIVE TESTS)  Concerns monetary amounts & other measures.  It assesses materially misstated account balances & ...

3.

THE PPS SAMPLING ( THE CAV SAMPLING)  Concerns primary engagement objective of few overstatements & not understatement.  Difference & Ratio Estimations may not be efficient.

EXTERNAL AUDITING STANDARDS

Internal & External Audit Work Coordination & Recognition:  Statement on Auditing Standards (SA) No. 39: Audit Sampling & SAS No. 47: Audit Risk & Materiality in Conducting an Audit – AICPA.  Audit Risk Model:

Audit Risk

= Inherent Risk x Control Risk x Detection Risk

 Audit Risk: Issuing unmodified opinion on financial statements that are

materially misstated.  Inherent Risk: Material misstatement occurring in the absence of appropriate controls.  Control Risk: Controls ineffective & fails to prevent or detect material misstatement in a timely manner.  Detection Risk: Substantive procedures failing to detect a material misstatement.

EXTERNAL AUDITING STANDARDS  Sampling risk impacts the Efficiency & Effectiveness of an audit

Components of Sampling Risk Audit Test Tests of Controls

Audit Efficiency Risk of Assessing Control Risk Too High (i.e., not depending upon effective controls)

Substantive Tests

Risk of Incorrect Rejection (i.e., Risk of Incorrect Acceptance rejecting a materially correct (i.e., accepting a materially balance) incorrect balance)

Statistical Term Alpha Risk (∝)

Audit Effectiveness Risk of Assessing Control Risk Too Low (i.e., depending upon ineffective controls)

Beta Risk ( 𝛽)

EXTERNAL AUDITING STANDARDS  Non-sampling Risk

“The audit failing to detect an internal control weakness or material misstatement for reasons other than the fact that sampling was used.”  Application of an inappropriate audit procedure  Failure to recognize an error condition  Omission of an essential audit step

 Materiality: Amount of difference tolerated by the auditor & concluding the

assertion tested as reasonable:

 Tolerable deviation rate for tests of control  Tolerable misstatement for substantive testing

 Materiality is inversely related to sample size  Materiality assessment must be a cost versus benefit decision

SAMPLE SELECTION METHODS  Methods Appropriate for Both Statistical & Non-statistical Sampling:  Simple Random Sampling: Items with equal chance of selection.  Systematic Sampling: nth item selection with random start within the n

interval. PPS uses systematic sampling.

 Methods Used Only for Non-statistical Sampling:  Haphazard Selection: Selecting sample items without intentional bias.  Block Selection: Audit of a group of contiguous transactions like delivery

notes for March or invoices in a sequence.  Block Amount: Whole amount is audited.

 Other Considerations in Sample Selection:  Void Items: Select additional sampling units for voided items.  Missing Items: Must be treated as an error condition- In attributes,

control is not effective & in substantive testing, audited value is ZMK 0.00

ATTRIBUTE SAMPLING When to use

. Size of sample (n)

Statistical table specifications

To estimate the number of times a certain characteristic may occur in a population

Based on judgment about probability that errors (or other characteristics) will occur or based on statistical tables 𝐂 𝟐 𝝆𝒒 𝐧= 𝐏𝟐 • Population size (N) • Confidence level (C) • Precision (P) • Expected rate of errors (𝝆) &q=100-𝝆

Attributes Sampling Illustrations

.ITEM 1 2 3 4 5 6

ACCOUNTS RECEIVABLES AS AT 31ST DECEMBER 2013 Population Size of Accounts Receivable Confidence Level Confidence Coefficient Tolerable Deviation Rate (TDR) (Based on Prior Years of Findings or Pilot Sample) Planned Risk of Assessing Control Risk Too Low (Beta Risk) Planned Risk of Assessing Control Risk Too High (Alpha Risk) Desired Precision = Beta x TDR/Alpha

7 8

Sample Size Expected Number of Errors (From Statistical Tables) Assuming Control Procedures Anticipated Deviation Rate = Zero Upper Precision Limit (UPL) from the Statistical Tables (And is Less than Tolerable Deviation Rate=5%)

9

Assuming 2 Actual Control Procedure Errors: Upper Precision Limit (from the Tables)

N C

𝛽 P n

UPL

4,000 Accounts 90% 1.64 5% 5% 10% 2.50% 204 Accounts 5 0% 1.50%

2 3.20%

10

And UPL < ρ

Conclusion???

11

CONCLUSION

Controls are Effective

Attributes Sampling Variations  Stop-or-Go Sampling: The Auditor guards against selecting an

unnecessarily large sample.  Discovery Sampling: The Auditor targets discovering at least one

deviation if the percentage of deviations in the population is at or above a specified level, e.g. Fraud, Substantial mistake or Compliance failure.

VARIABLES SAMPLING

When to use

.

When size matters; e.g., amount of a discrepancy in monetary or weight terms

Size of sample (n)

𝐂 𝟐 𝝈𝟐 𝐧= 𝟐 𝐏 Statistical table specifications

• • • •

Population size (N) Confidence level/Coefficient (C) Precision (P) Standard deviation (𝝈)

Variables Sampling Illustration ITEM ACCOUNTS RECEIVABLES AS AT 31ST DECEMBER 2013

.

1 2 3 4 5 6

8 9

Recorded Amount of Accounts Receivable (N) Tolerable Misstatement Planned Risk of Incorrent Acceptance (Beta Risk) Planned Risk of Incorrect Rejection (Alpha Risk) Number of Accounts Receivable (N) Estimated Population Standard Deviation (Based on Prior Years of Findings or Pilot Sample) Confidence Level Confidence Coefficient Desired Precision = Beta x TM/Alpha Precision per-item basis (Desired Precision/N)

7

Sample Size

7

RM TM 𝛽

C P n

360,000 ZMK 18,000 ZMK 5% 10% 4,000 Accounts 8.68 ZMK 90% 1.64 9,000 ZMK 2.25 ZMK 40 Accounts

Three Types of Variables Sampling  Mean-per-unit Estimation: Estimates the total monetary amount of the

population by calculating a sample mean & multiplying by the number of items in the population.

 Difference Estimation: Estimates the total error in the population.  Useful only if population contains enough errors to generate a reliable sample

estimate & the differences are not proportional to the book values.

 Ratio Estimation: Estimates the total monetary amount of the population

by calculating the ratio between the audited & book values in the sample and using this ratio to make the estimate.  Useful when differences between book & sample values are proportional to book

values.

Variables Sampling: Mean-per-Unit Estimation Case example Population: 4,000 Accounts Total book value: ZMK 360,000.00 Sample size: 40 Accounts Sample book value: ZMK 3,600.00 Sample audit value: ZMK 3,400.00

Step 1: Calculate average audit value (i.e., mean-perunit value for audited samples).

K3,400.00/40 = K85.00 / Account. Step 2: Multiply mean-per-unit value by number of accounts in the population. K85.00  4,000 Accounts = K340,000.00 Over-count = K20,000.00 (K340,000.00 – K360,000.00)

Variables Sampling: Difference Estimation Case example Population: 4,000 Accounts Total book value: ZMK 360,000.00 Sample size: 40 Accounts Sample book value: ZMK 3,600.00 Sample audit value: ZMK 3,400.00

Step 1: Calculate average difference between audit value and book value for the sample. (K3,400.00 – K3,600.00)/40 Accounts = (K5.00) Step 2: Determine the difference estimate for the population. (K5.00)  4,000 accounts = (K20,000.00)

Step 3: Estimate actual value by adding the difference estimate and book value for the population. (K20,000.00) + K360,000.00 = K340,000.00

Book value is Overstated by K20,000.00

Variables Sampling: Ratio Estimation Case example Population: 4,000 Accounts Total book value: ZMK360,000.00 Sample size: 40 Accounts Sample book value: ZMK3,600.00 Sample audit value: ZMK 3,400.00

Step 1: Audit value for sample = K3,400.00 Step 2: Book value for sample = K3,600.00 Step 3: Find ratio of audit value to book value: K3,400.00 / K3,600.00 = 0.94 Step 4: Estimate actual population value by multiplying ratio by population book value:

0.94  K360,000.00 = K338,400.00

Book value is Overstated by K21,600.00

PROBABILITY-PROPORTIONAL-TO-SIZE (PPS) SAMPLING

When to use

Size of sample (n)

When auditing account balances for few . overstated items; e.g., in inventory, receivables, disbursements, etc.

(n1: AM=0, & n2: AM>=1)

Statistical specifications

𝐧𝟏 = • • • • •

𝐑𝐌 𝐱 𝐑𝐅 𝐓𝐌

or

𝐑𝐌 𝐱 𝐑𝐅

𝐧𝟐 = 𝐓𝐌−(𝐀𝐌 𝐱 𝐄𝐅)

Recorded Amount of the Account (RM) Reliability Factor (RF) Tolerable Misstatement (TM) Anticipated Misstatement (AM) Expansion Factor (EF)

PPS ILLUSTRATION ACCOUNTS RECEIVABLE AS AT 31ST DECEMBER 2013

.Recorded Amt of A/C Receivables Tolerable Misstatement Anticipated Misstatement Risk of Incorrect Acceptance AMT A/C No. ZMK ACT0001 ACT0002 ACT0003 ACT0004 ACT0005 ACT0006 . . . ACT4000

CUM AMT

9,450 9,450 480 9,930 2,800 12,730 5,106 17,836 2,100 19,936 8,000 27,050 . . . . . . 6,000 360,000

TOTAL

360,000 18,000 0 5%

Kw acha Sampling Observed Tainting Sampling Projected Selected Unit Amount % Interval Misstatement 9,000

9,450

7,875

*

*

1,575

18,000 27,000 . . . 360,000

2,100 8,000 . . . 6,000

0 8,000

100% 0

9,000 9,000

9,000 0

4,500

25%

9,000

2,250

360,000

Basic Precision(SI x RF = K9,000 x 3) Total Projected Misstatment Allowance for Precision Gap Widening: (4.75-3.00-1.00) x K9,000 (6.30-4.75-1.00) x K2,250 Upper Misstatement Limit (UML)>TM CONCLUSION

RM TM AM

12,825 ZMK ZMK

27,000 12,825

ZMK ZMK ZMK

6,750 1,238 47,813

Accounts Receivable Materially Overstated

CONCLUSION/RECOMMENDATIONS

It is Concluded & Recommended that Internal Auditors comply with the Proficiency & Due Professional Care IIA Standards by Appropriate Application of both Statistical & Non Statistical Sampling to Reasonably Assure that Opinion Evidence is: Sufficient, Reliable, Relevant and Useful.

REFERENCES FOR FURTHER READING

1.

Sampling for Internal Auditors:Text-based Self Study CourseThe Institute of Internal Auditors by Barbara Apostolou, PhD, CPA, DABFA.

2.

Internal Audit Practice-Part 1:The IIA’s CIA Learning System by The Institute of Internal Auditors.

3.

Internal Audit Practice-Part 1: Gleim CIA Review by Professor Irvin N. Gleim, PhD, CPA, CIA, CMA, CFM.

COMMENTS, REMARKS & QUESTIONS

Confidence coefficient, C, Based on the Risk of Incorrect Rejection Risk of Incorrect Rejection 20% 10% 5% 1%

Confidence Level 80% 90% 95% 99%

Confidence Coefficient 1.28 1.64 1.96 2.58

Attributes Sample Size Statistical Tables For Tests of Controls Five Percent (5%) Risk of Assessing Control Risk Too Low (Number of Expected Errors in parentheses)

.Expected Population Deviation Rate (%) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 5.00 6.00 7.00

2% 149(0) 236(1) * * * * * * * * * * * * * * * * * *

3% 99(0) 157(1) 157(1) 208(2) * * * * * * * * * * * * * * * *

4% 74(0) 117(1) 117(1) 117(1) 156(2) 156(2) 192(3) 227(4) * * * * * * * * * * * *

5% 59(0) 93(1) 93(1) 93(1) 93(1) 124(2) 124(2) 153(3) 181(4) 208(5) * * * * * * * * * *

Tolerable Deviation Rate 6% 7% 8% 49(0) 42(0) 36(0) 78(1) 66(1) 58(1) 78(1) 66(1) 58(1) 78(1) 66(1) 58(1) 78(1) 66(1) 58(1) 78(1) 66(1) 58(1) 103(2) 66(1) 58(1) 103(2) 88(2) 77(2) 127(3) 88(2) 77(2) 127(3) 88(2) 77(2) 150(4) 109(3) 77(2) 173(5) 109(3) 95(3) 195(6) 129(4) 95(3) * 148(5) 112(4) * 167(6) 112(4) * 185(7) 129(5) * * 146(6) * * * * * * * * *

9% 32(0) 51(1) 51(1) 51(1) 51(1) 51(1) 51(1) 51(1) 68(2) 68(2) 68(2) 68(2) 84(3) 84(3) 84(3) 100(4) 100(4) 158(8) * *

10% 29(0) 46(1) 46(1) 46(1) 46(1) 46(1) 46(1) 46(1) 46(1) 61(2) 61(2) 61(2) 61(2) 61(2) 76(3) 76(3) 89(4) 116(6) 179(11) *

15% 19(0) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 30(1) 40(2) 40(2) 40(2) 40(2) 50(3) 68(5)

20% 14(0) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 22(1) 30(2) 30(2) 37(3)

Attributes Sample Evaluation Tables For Tests of Controls Upper Limits at Five Percent (5%) Risk of Assessing Control Risk Too Low Sample Size . 25 30 35 40 45 50 55 60 65 70 75 80 90 100 125 150 200

0 11.3 9.5 8.3 7.3 6.5 5.9 5.4 4.9 4.6 4.2 4.0 3.7 3.3 3.0 2.4 2.0 1.5

1 17.6 14.9 12.9 11.4 10.2 9.2 8.4 7.7 7.1 6.6 6.2 5.8 5.2 4.7 3.8 3.2 2.4

2 * 19.6 17.0 15.0 13.4 12.1 11.1 10.2 9.4 8.8 8.2 7.7 6.9 6.2 5.0 4.2 3.2

Actual Number of Deviations Found 3 4 5 6 7 * * * * * * * * * * * * * * * 18.3 * * * * 16.4 19.2 * * * 14.8 17.4 19.9 * * 13.5 15.9 18.2 * * 12.5 14.7 16.8 18.8 * 11.5 13.6 15.5 17.4 19.3 10.8 12.6 14.5 16.3 18.0 10.1 11.8 13.6 15.2 16.9 9.5 11.1 12.7 14.3 15.9 8.4 9.9 11.4 12.8 14.2 7.6 9.0 10.3 11.5 12.8 6.1 7.2 8.3 9.3 10.3 5.1 6.0 6.9 7.8 8.6 3.9 4.6 5.2 5.9 6.5

8 * * * * * * * * * 19.7 18.5 17.4 15.5 14.0 11.3 9.5 7.2

9 * * * * * * * * * * 20.0 18.9 16.8 15.2 12.3 10.3 7.8

10 * * * * * * * * * * * * 18.2 16.4 13.2 11.1 8.4

.

Reliability Factors (RF) for Overstatements

Number of Overstatements

0 1 2

Risk of Incorrect Acceptance

1% 4.61 6.64 8.41

5% 3.00 4.75 6.30

10% 2.31 3.89 5.33

15% 1.90 3.38 4.72

20% 1.61 3.00 4.28

PPS Sampling Expansion Factors For Expected Misstatements .

Risk of Incorrect Acceptance (%) 1 5 10 15 20 25 30 37 50

Expansion Factor 1.90 1.60 1.50 1.40 1.30 1.25 1.20 1.15 1.10