1. Substantive Test Sampling
CAS 500 – Audit evidence
CAS Identifying and Assessing the Risks of Material Misstatement through Understanding the Entity and its Environment
CAS Evaluation of Misstatements Identified during the Audit
CAS 320 – Materiality in planning and performing an audit
CAS 530 – Audit sampling

2. Substantive Procedures for Auditing Account Balances
Substantive tests of details auditing is the performance of procedures to obtain direct evidence about the dollar amounts and disclosures in the financial statements.
Substantive procedures include:

3. The Audit Risk Model
Audit Risk = Inherent Risk x Control Risk x Detection Risk Audit risk Inherent risk Control Risk Detection risk

4. Risk Model Expansion
Detection risk (DR) is actually a combination of two risks:
Analytical procedures risk (APR)
Risk of incorrect acceptance (RIA)

5. Risk Model Expansion AR RIA = IR x CR x APR Expanded risk model:
AR = IR × CR × APR × RIA
Only risk of incorrect acceptance can be controlled by the auditor.
Solve the risk model for RIA:
with AR, IR, and APR held constant RIA varies inversely with CR or control risk.
RIA = AR
IR x CR x APR

6. More About Sampling Risk
In sampling for substantive procedures, the auditor faces two sampling risks.
The risk of incorrect acceptance (RIA)
(effectiveness risk, beta risk)
The risk of incorrect rejection (RIR)
(efficiency risk, alpha risk)

7. Which is considered the more important of the two.
Incorrect acceptance leads to audit failure.
Incorrect rejection leads to additional audit procedures to determine actual misstatement.

8. Sampling Steps for an Account Balance Audit
Specify the audit objectives.
Define the population.
Choose an audit sampling method.
Determine the sample size.
Select the sample.
Perform the substantive-purpose procedures.
Evaluate the evidence.

9. Plan the Procedures – involves the first three steps
The first three steps represent the problem-recognition phase.
So what is being considered in the first three steps?
Remember a key to the audit approach:
Management’s Assertions lead to Audit Objectives lead to Audit Procedures

10. 1. Specify the Audit Objectives
Objective is to decide whether the client’s assertions about existence, rights, and valuation are materially accurate.

11. 2. Define the Population
Definition of population must match objectives.
Examine individually significant items.
Apply stratification to the remaining population.

12. 3. Choose an Audit Sampling Method
Auditor will decide whether to use statistical or judgmental sampling methods.
Steps 4 to 6 represent the evidence-collection phase.
Sample size determination requires consideration of several factors. It is important to get the sample size correct, and to avoid over-auditing or under-auditing.

13. 4. Determine the Sample Size
The sample size is based on the following:
risk of incorrect acceptance (RIA),
risk of incorrect rejection (RIR),
material misstatement amount,
expected dollar misstatement in the population,
variability within the population, and
size of the population.

14. 5. Select the sample
Random or non-random methods, as in sample selection for test of controls.

15. 6. Perform the substantive purpose procedures.
Substantive-purpose audit program produces evidence to support or refute management assertions.
Remember the assertions.

16. 7. Evaluate the Evidence Evaluate the evidence:
Determine amount of known misstatement.
Determine the likely misstatement.
Project the misstatement found in the sample to the population.

17. 7. Evaluate the Evidence Consider sampling risks:
Auditor uses professional judgment and experience to consider these risks.
Qualitative evaluation:
Follow up on all differences uncovered to determine
any misunderstanding of GAAP,
simple mistakes,
intentional irregularities, or
management override of controls.

18. 7. Evaluate the Evidence Evaluate the misstatement:
Known misstatement and likely misstatement are combined and compared to materiality.
Sample risk gives rise to “possible misstatements.”

19. Timing of Substantive Procedures
Account balances can, in part, be audited at an interim date.
Auditor will extend the interim date audit conclusions to balance sheet date.
Audit work is performed at interim dates:
Poor controls, or significant business risk may preclude performing procedures at interim.

20. Monetary Unit Sampling - How Does it Work?
Auditors use monetary unit sampling, also called probability-proportional-to-size or dollar-unit sampling, to determine the accuracy of financial accounts. With monetary unit sampling, each dollar in a transaction is a separate sampling unit. A transaction for $40, for example, contains 40 sampling units. Auditors usually use monetary unit sampling to sample and test accounts receivable.
Here’s an example of how monetary unit sampling works The audit client’s accounts receivable book value is $300,000, and the sample size is set at 96 records.
Figure the sampling interval by dividing book value by sample size
(300,000/96) = 3125
Arrange the client’s accounts receivable in an ordered list using some sort of ordering sequence.
For example, you can arrange them alphabetically by customer name or numerically by customer number.
Pick a random number between 1 and 3,125.
For this method to work correctly, the random number has to be less than the sampling interval and greater than the smallest sampling unit. Auditors usually use a random-number-generator computer program to pick the random number. The sampling unit and sampling interval limits are programmed into the software before the task is run. In this case, say the software selects the random number 556.
Monetary Unit Sampling Table
Customer Name
Customer Balance
Cumulative Balance
Sampling Item
ABC Electric
$435
Best Friend Cat Care
$785
$1,220
(1) $556
Brandy’s Grill
$1,510
$2,730
Buddy’s Gas Station
$5,000
$7,730
(2) $556 + $3,125 = $3,681

21. First, pick the records to test: Take the alphabetically ordered list shown in the Customer Name column, which lists every customer balance by dollar amount, and count each dollar until getting to $556. The random number generator gives the number 556 in Step 3 in the previous slide. The cumulative dollar amount for ABC Electric is under $556.
That shows that the first sampling item is Best Friend Cat Care, which at a cumulative total of $1,220 is the first customer in the list with a cumulative balance over $556. Best Friend Cat Care becomes the first customer in the sample.
Secondly, select the next invoice to sample: Add the sampling interval of $3,125 to the random number of $556. This equals $3,681, which is the next sampled item dollar amount. Brandy’s Grill at $2,730 cumulatively is under $3,681, thus Brandy’s is skipped. Buddy’s Gas Station has the 3,681st dollar.
To pick the next sampling item: Add the sampling interval of $3,125 to the prior sampling item of $3,681, which equals $6,806, and so on until the last name in the customer list is reached. This will give the total sample size of 96.
When sampling, misstatements are being looked for. If a selected customers invoice should have been entered for $986, for example, and it was entered as $896, there is a misstatement. If the total misstatements exceed the tolerable level, there may be a material misstatement.

22. Example for Calculation of Sample Size for MUS
This example uses materiality:
Book Value of Population = 596,566
Acceptable risk of incorrect acceptance (RIA) = 10%
Tolerable misstatement /materiality (a) = 40,000
Expected misstatements in the population (b) = 4,000
Ratio of expected to tolerable misstatements - b/a = .10
Confidence factor (R) – next page (c) = 2.77
Tolerable misstatement (P) as % of population (d) = 0.067
Sample size - c/d (n=R/P) = 42 (round up)
We will use 40 for MUS.

23. MUS Confidence Factors (R Values) (RIA in brackets)
Ratio of Expected to Tolerable Misstatement
Confidence Level (Risk of Incorrect Acceptance)
95%
(5%)
90%
(10%)
85%
(15%)
80%
(20%)
75%
(25%)
70%
(30%)
65%
(35%)
63%
(37%)
50%
(50%)
0.00
3.00
2.31
1.90
1.61
1.39
1.21
1.05
1.00
0.70
0.05
3.31
2.52
2.06
1.74
1.49
1.29
1.12
1.06
0.73
0.10
3.68
2.77
2.25
1.89
1.20
1.13
0.77
0.15
4.11
3.07
2.47
1.28
0.82
0.20
4.63
3.41
2.73
2.26
1.62
1.38
1.30
0.87
0.25
5.24
3.83
3.04
2.49
2.09
1.76
1.50
1.41
0.92
0.30
6.00
4.33
2.30
1.93
1.63
1.53
0.99
0.35
6.92
4.95
3.86
3.12
2.57
2.14
1.79
1.67
0.40
8.09
5.72
4.42
3.54
2.89
2.39
1.99
1.85
1.14
0.45
9.59
6.71
5.13
4.07
3.29
2.70
2.22
1.25
0.50
11.54
7.99
6.04
4.75
3.80
3.08
2.51
2.32
1.37
0.55
14.18
9.70
7.26
5.64
4.47
3.58
2.65
1.52
0.60
17.85
12.07
8.93
6.86
5.37
4.25
3.38
3.09
1.70

24. Problem DC 10-1, Page 547
When Marge Simpson, PA, audited the Candle Company inventory, a random sample of inventory types was chosen for physical observation and price testing. The sample size was 80 different types of candles and candle-making inventory. The entire inventory contained 1,740 types, and the amount in the inventory control account was $166,000. Simpson had already decided that a misstatement of as much as $6,000 in the account would not be material. The audit work revealed the following no errors in the sample of 80. This problem has been changed from the text to show no errors.

25. There are no misstatements in this sample.
Assume the sample was chosen using Monetary Unit Sampling, and the sampling risk is 10%. Simpson also assumes that the Misstatement Assumption for zero misstatements is 50% for both over and under misstatements. Is the inventory materially misstated?
Total Population =
166,000.00
Sample Size = 80
ARACR =
10.00%
Upper Misstatement Unit Error Assumption
50.00%
Lower Misstatement Unit Error Assumption
Materiality
6,000
This part of the population is not in the sample. Thus zero misstatements have been found here because these items have not been examined.
Sample
There are no misstatements in this sample.
The misstatements of the remaining part of the population are not known. Thus a misstatement assumption has to be made. E.g. they are 50% misstated. See next slide.

26. Overstatements Understatements
Number of Misstatements
(1)
Upper Precision Limit Portion
(2)
Recorded
Value
(3)
Misstatement Unit Error Assumption (4)
Bound Portion
2x3x4
Overstatements
0.029
166,000
0.500
$2,407
Upper Precision Limit
Initial Misstatement Bound
Understatements
Number of Misstatements
(1)
Upper Precision Limit Portion
(2)
Recorded
Value
(3)
Misstatement Unit Error Assumption (4)
Bound Portion
2x3x4
Understatements
0.029
166,000
0.500
$2,407
Upper Precision Limit
Initial Misstatement Bound
As the Upper Bound and Lower Bound are less than the materiality of $6,000, there is no material misstatement

27. ACTUAL NUMBER OF DEVIATIONS FOUND
Sample size
ACTUAL NUMBER OF DEVIATIONS FOUND 1 2 3 4 5 6 7 8 9 10
5 PERCENT RISK OF OVER RELIANCE (RIA or Beta Risk) 20
14.0
21.7
28.3
34.4
40.2
45.6
50.8
55.9
60.7
65.4
69.9 25
11.3
17.7
23.2
28.2
33.0
37.6
42.0
46.3
50.4
54.4
58.4 30
9.6
14.9
19.6
23.9
28.0
31.9
35.8
39.4
43.0
46.6
50.0 35
8.3
12.9
17.0
20.7
24.3
27.8
31.1
37.5
40.6
43.7 40
7.3
11.4
15.0
18.3
21.5
24.6
27.5
30.4
33.3
36.0
38.8 45
6.5
10.2
13.4
16.4
19.2
22.0
24.7
27.3
29.8
32.4
34.8 50
5.9
9.2
12.1
14.8
17.4
19.9
22.4
27.1
29.4
31.6 55
5.4
8.4
11.1
13.5
15.9
18.2
20.5
22.6
24.8
26.9
28.9 60
4.9
7.7
12.5
14.7
16.8
18.8
20.8
22.8
26.7 65
4.6
7.1
9.4
11.5
13.6
15.5
17.5
19.3
21.2
23.0 70
4.2
6.6
8.8
10.8
12.7
14.5
16.3
18.0
19.7
21.4
23.1 75
4.0
6.2
8.2
10.1
11.8
15.2
16.9
18.5
20.1
21.6 80
3.7
5.8
9.5
14.3
18.9
20.3 90
3.3
5.2
6.9
9.9
12.8
14.2
100
3.0
4.7
7.6
9.0
10.3
125
2.4
3.8
5.0
6.1
7.2
9.3
12.3
13.2
150
2.0
3.2
5.1
6.0
7.8
8.6
200
1.5
3.9
300
1.0
1.6
2.1
2.6
3.1
3.5
4.4
4.8
5.6
400
0.8
1.2
2.3
2.7
3.6
4.3
500
0.6
1.3
1.9
2.9
3.4

28. ACTUAL NUMBER OF DEVIATIONS FOUND
Sample size
ACTUAL NUMBER OF DEVIATIONS FOUND 1 2 3 4 5 6 7 8 9 10
10 PERCENT RISK OF OVER RELIANCE (RIA or Beta Risk) 20
10.9
18.1
24.5
30.5
36.1
41.5
46.8
51.9
56.8
61.6
66.2 25
8.8
14.7
20.0
24.9
29.5
34.0
38.4
42.6
50.8
54.8 30
7.4
12.4
16.8
21.0
28.8
32.5
36.2
39.7
43.2
46.7 35
6.4
10.7
14.5
18.2
21.6
28.2
31.4
34.5
37.6
40.6 40
5.6
9.4
12.8
16.0
19.0
22.0
27.7
33.2
35.9 45
5.0
8.4
11.4
14.3
17.0
19.7
22.3
24.8
27.3
29.8
32.2 50
4.6
7.6
10.3
12.9
15.4
17.8
20.2
22.5
24.7
27.0
29.2 55
4.2
6.9
11.8
14.1
16.3
18.4
20.5
22.6
24.6
26.7 60
3.8
8.7
10.8
15.0
16.9
18.9
20.8
22.7 65
3.5
5.9
8.0
10.0
12.0
13.9
15.7
17.5
19.3
22.8 70
3.3
5.5
7.5
9.3
11.1
14.6
18.0
19.6
21.2 75
3.1
5.1
7.0
10.4
12.1
13.7
15.2
18.3
19.8 80
2.9
4.8
6.6
8.2
9.8
11.3
15.8
17.2
18.7 90
2.6
4.3
7.3
10.1
11.5
16.7
100
2.3
3.9
5.3
7.9
9.1
12.7
125
1.9
6.3
8.3
10.2
11.2
150
1.6
3.6
4.4
6.1
7.8
8.6
200
1.2
2.0
2.7
3.4
4.0
6.5
7.1
300
0.8
1.3
1.8
4.7
400
0.6
1.0
1.4
1.7
2.4
3.0
500
0.5
1.1
2.1
The Upper and Lower Bounds must be determined. What do these mean?
Obtain the Precision Limits from this table – 10% RIA

29. Problem DC 10-1, Page 547
When Marge Simpson, PA, audited the Candle Company inventory, a random sample of inventory types was chosen for physical observation and price testing. The sample size was 80 different types of candles and candle-making inventory. The entire inventory contained 1,740 types, and the amount in the inventory control account was $166,000. Simpson had already decided that a misstatement of as much as $6,000 in the account would not be material. The audit work revealed the following eight errors in the sample of 80.
Book Value
(a)
Audit value
Error Amount
(b)
% Misstatement
b/a
$600.00
$622.00
$(22.00)
(0.037)
15.50
14.50
1.00
0.065
65.25
31.50
33.75
0.517
83.44
53.45
29.99
0.359
16.78
15.63
1.15
0.069
78.33
12.50
65.83
0.840
13.33
14.22
(0.89)
(0.067)
93.87
39.87
54.00
0.575
$966.50
$803.67
$162.83

30. Assume the sample was chosen using Monetary Unit Sampling, and the sampling risk is 10%. Simpson also assumes that the Misstatement Assumption for zero misstatements is 50% for both over and under misstatements. Is the inventory materially misstated?
Total Population =
166,000.00
Sample Size = 80
ARACR =
10.00%
Upper Misstatement Unit Error Assumption
50.00%
Lower Misstatement Unit Error Assumption
Materiality
6,000
This part of the population is not in the sample. Thus zero misstatements have been found here because these items have not been examined.
Sample
The misstatements of the sample, as percentages, are known . See previous slide 39.
The misstatements of the remaining part of the population are not known. Thus a misstatement assumption has to be made. E.g. they are 50% misstated. See next slide.

31. Overstatements Number of Misstatements (1)
Upper Precision Limit Portion
(2)
Recorded
Value
(3)
Misstatement Unit Error Assumption (4)
Bound Portion
2x3x4
Overstatements
0.029
166,000
0.500
$2,407 1
0.019
0.840
2649 2
0.018
0.575
1718 3
0.016
0.517
1373 4
0.359
954 5
0.015
0.069
172 6
0.065
162
Upper Precision Limit
0.128
Initial Misstatement Bound
$9,435

32. ACTUAL NUMBER OF DEVIATIONS FOUND
Sample size
ACTUAL NUMBER OF DEVIATIONS FOUND 1 2 3 4 5 6 7 8 9 10
10 PERCENT RISK OF OVER RELIANCE (RIA or Beta Risk) 20
10.9
18.1
24.5
30.5
36.1
41.5
46.8
51.9
56.8
61.6
66.2 25
8.8
14.7
20.0
24.9
29.5
34.0
38.4
42.6
50.8
54.8 30
7.4
12.4
16.8
21.0
28.8
32.5
36.2
39.7
43.2
46.7 35
6.4
10.7
14.5
18.2
21.6
28.2
31.4
34.5
37.6
40.6 40
5.6
9.4
12.8
16.0
19.0
22.0
27.7
33.2
35.9 45
5.0
8.4
11.4
14.3
17.0
19.7
22.3
24.8
27.3
29.8
32.2 50
4.6
7.6
10.3
12.9
15.4
17.8
20.2
22.5
24.7
27.0
29.2 55
4.2
6.9
11.8
14.1
16.3
18.4
20.5
22.6
24.6
26.7 60
3.8
8.7
10.8
15.0
16.9
18.9
20.8
22.7 65
3.5
5.9
8.0
10.0
12.0
13.9
15.7
17.5
19.3
22.8 70
3.3
5.5
7.5
9.3
11.1
14.6
18.0
19.6
21.2 75
3.1
5.1
7.0
10.4
12.1
13.7
15.2
18.3
19.8 80
2.9
4.8
6.6
8.2
9.8
11.3
15.8
17.2
18.7 90
2.6
4.3
7.3
10.1
11.5
16.7
100
2.3
3.9
5.3
7.9
9.1
12.7
125
1.9
6.3
8.3
10.2
11.2
150
1.6
3.6
4.4
6.1
7.8
8.6
200
1.2
2.0
2.7
3.4
4.0
6.5
7.1
300
0.8
1.3
1.8
4.7
400
0.6
1.0
1.4
1.7
2.4
3.0
500
0.5
1.1
2.1
1.9
1.8
1.6
1.6
1.5
1.9
The Upper and Lower Bounds must be determined. What do these mean?
Obtain the Precision Limits from this table – 10% RIA

33. Understatements Number of Misstatements (1)
Upper Precision Limit Portion
(2)
Recorded
Value
(3)
Misstatement Unit Error Assumption (4)
Bound Portion
2x3x4
Understatements
0.029
166,000
0.500
$2,407 1
0.019
0.067
211 2
0.018
0.037
111
Upper Precision Limit
0.066
Initial Misstatement Bound
$2,729

34. Offsetting Adjustments
Number of Misstatements
Misstatement Unit Error Assumption
(a)
Sample Size
(b)
Recorded Population
(c)
Point Estimate
a(c/b)
Bounds
Initial Overstatement Bound
$9,435
Understatement Misstatements 1
0.067 80
166,000
139
(139) 2
0.037 77
(77)
Total
0.104
216
(216)
Adjusted Overstatement Bound
9,219
Initial Understatement Bound
2,729
Overstatement Misstatements
0.840
1743
(1743)
0.575
1193
(1193) 3
0.517
1073
(1073) 4
0.359
745
(745) 5
0.069
143
(143) 6
0.065
135
(135)
2.425
5032
(5032)
Adjusted Understatement Bound
-2303
As the Upper Bound is greater than the materiality of $6,000, there may be a material misstatement

35. Tolerable misstatement
The Decision Rule
Reject
Tolerable misstatement
($6,000)
$6,000
$2,303
LMB
$9,219
UMB
Problem
Since materiality is $6,000, there may be material misstatement.

36. Problem EP 10-5, Page 546 – Point Estimate
Assume the results shown below were obtained from a stratified sample. Required : Apply the ratio calculation method (Point Estimate) to each stratum to calculate the projected likely misstatement (PLM). What is the PLM for the entire sample?
SAMPLE RESULTS
Stratum
Population Size
Recorded Amount
Sample
Misstatement Amount
PLM 1 6
$100,000
$(600) 2 80
75,068 23
21,700
(274) 3
168
75,008 22
9,476
(66) 4
342
75,412
4,692
(88) 5
910
74,512
1,973
Total
1,506
$400,000 96
$137,841
$(1,005)

37. Problem DC 10-1, Page 547 – The Difference Method
Sample size is 80
Population consists of 1,740 types
Book Value
(a)
Audit value
Error Amount
(b)
% Misstatement
b/a
$600.00
$622.00
$(22.00)
(0.037)
15.50
14.50
1.00
0.065
65.25
31.50
33.75
0.517
83.44
53.45
29.99
0.359
16.78
15.63
1.15
0.069
78.33
12.50
65.83
0.840
13.33
14.22
(0.89)
(0.067)
93.87
39.87
54.00
0.575
$966.50
$803.67
$162.83