Chapter Nine Audit Sampling:An Application to Substantive Tests of Account Balances
Substantive Tests of Details of Account Balances • The statistical concepts we discussed in the last chapter apply to this chapter as well. Three important determinants of sample size are • Desired confidence level. • Tolerable misstatement. • Estimated misstatement. • Misstatements discovered in the audit sample must be projected to the population, and there must be an allowance for sampling risk.
Substantive Tests of Details of Account Balances Consider the following information about the inventory account balance of an audit client: The ratio of misstatement in the sample is 2%($2,000 ÷ $100,000) Applying the ratio to the entire population produces a bestestimate of misstatement of inventory of $60,000.($3,000,000 × 2%)
Substantive Tests of Details of Account Balances The results of our audit test depend upon the tolerable error associated with the inventory account. If the tolerable error is $50,000, we cannot conclude that the account is fairly stated because our best estimate of the projected error is greater than the tolerable error.
Monetary-Unit Sampling (MUS) MUS uses attribute-sampling theory to express a conclusion in dollar amounts rather than as a rate of occurrence. It is commonly used by auditors to test accounts such as accounts receivable, loans receivable, investment securities, and inventory.
Monetary-Unit Sampling (MUS) MUS uses attribute-sampling theory to estimate the percentage of monetary units in a population that might be misstated and then multiplies this percentage by an estimate of how much the dollars are misstated.
Monetary-Unit Sampling (MUS) Advantages of MUS • When the auditor expects no misstatement, MUS usually results in a smaller sample size than classical variables sampling. • The calculation of the sample size and evaluation of the sample results are not based on the variation between items in the population. • When applied using the probability-proportional-to-size procedure, MUS automatically results in a stratified sample.
Monetary-Unit Sampling (MUS) Disadvantages of MUS • The selection of zero or negative balances generally requires special design consideration. • The general approach to MUS assumes that the audited amount of the sample item is not in error by more than 100%. • When more than one or two misstatements are detected, the sample results calculations may overstate the allowance for sampling risk.
Sampling may be used for substantive testing to: • Test the reasonableness of assertions about a financial statement amount. • Develop an estimate of some amount. Steps in MUS Sampling Application
For MUS the population is defined as the monetary value of an account balance, such as accounts receivable, investment securities, or inventory. Steps in MUS Sampling Application
An individual dollar represents the sampling unit. Steps in MUS Sampling Application
Steps in MUS Sampling Application A misstatement is defined as the difference between monetary amounts in the client’s records and amounts supported by audit evidence.
The auditor selects a sample for MUS by using a systematic selection approach called probability-proportionate-to-size selection. The sampling interval can be determined by dividing the book value of the population by the sample size. Each individual dollar in the population has an equal chance of being selected. Steps in MUS Sampling Application
Steps in MUS Sampling Application Assume a client’s book value of accounts receivable is $2,500,000, and the auditor determined a sample size of 93. The sampling interval will be $26,882 ($2,500,000 ÷ 93). The random number selected is $3,977 the auditor would select the following items for testing:
After the sample items have been selected, the auditor conducts the planned audit procedures on the logical units containing the selected dollar sampling units. Steps in MUS Sampling Application
The misstatements detected in the sample must be projected to the population. Steps in MUS Sampling Application
Steps in MUS Sampling Application Basic PrecisionIf no misstatements are found in the sample, the best estimate of the population misstatement would be zero dollars. $26,882 × 3.0 = $80,646 upper misstatement limit
Steps in MUS Sampling Application Misstatements DetectedIn the sample of 93 items the following misstatements were found: Because the Axa balance of $32,549 is greater than the interval of $26,882, no sampling risk is added. Since all the dollars in the large accounts are audited, there is no sampling risk associated with large accounts. $3,284 ÷ $21,893 = 15%
(0.15 × $26,882 × 1.4 = $5,645) Steps in MUS Sampling Application Compute the Upper Misstatement LimitWe compute the upper misstatement limit by calculating basic precision and ranking the detected misstatements based on the size of the tainting factor from the largest to the smallest.
In our example, the final decision is whether the accounts receivable balance is materially misstated or not. Steps in MUS Sampling Application We compare the tolerable misstatement to the upper misstatement limit. If the upper misstatement limit is less than or equal to the tolerable misstatement, we conclude that the balance is not materially misstated.
Steps in MUS Sampling Application In our example the upper misstatement limit of $150,621 is greater than the tolerable misstatement of $125,000, so the auditor concludes that the accounts receivable balance is materially misstated. • When faced with this situation, the auditor may: • Increase the sample size. • Perform other substantive procedures. • Request the client adjust the accounts receivable balance. • If the client refuses to adjust the account balance, the auditor would consider issuing a qualified or adverse opinion.
Why is Sample Size Not Used in Evaluating MUS Results? Most MUS evaluation approaches use the misstatement factors and increments associated with a sample size of 100, regardless of the actual sample size used by the auditor.
Effect of Understatement Misstatements MUS is not particularly effective at detecting understatements. An understated account is less likely to be selected than an overstated account. The most likely error will be reduced by $2,688(– 0.10 × $26,882)
Nonstatistical Sampling for Tests of Account Balances The sampling unit for nonstatistical sampling is normally a customer account, an individual transaction, or a line item on a transactions. When using nonstatistical sampling, the following items must be considered: • Identifying individually significant items. • Determining the sample size. • Selecting sample items. • Calculating the sample results.
Identifying Individually Significant Items The items to be tested individually are items that may contain potential misstatements that individually exceed the tolerable misstatement. These items are tested 100% because the auditor is not willing to accept any sampling risk.
Population book valueTolerable misstatement SampleSize × Assurance factor = Determining the Sample Size
Selecting Sample Items Auditing standards require that the sample items be selected in such a way that the sample can be expected to represent the population.
If the population total is $200,000, the projected misstatement would be $20,000 ($200,000 × 10%) Calculating the Sample Results One way of projecting the sampling results to the population is to apply the misstatement ratio in the sample to the population. Assume the auditor finds $1,500 in misstatements in a sample of $15,000. The misstatement ratio is 10%.
The projected misstatement would be $30,000, ($300 ÷ 100 = $3 × 10,000). Calculating the Sample Results A second method is the difference estimation. This method projects the average misstatement of each item in the sample to all items in the population. Assume misstatements in a sample of 100 items total $300, and the population contains 10,000 items.
Nonstatistical Sampling Example The auditor’s of Calabro Paging Service have decided to use nonstatistical sampling to examine the accounts receivable balance. Calabro has 11,800 accounts with a balance of $3,717,900. The auditor’s stratify the accounts as follows:
Nonstatistical Sampling Example • The auditor’s decide . . . • There is a low assessment for inherent and control risk. • The tolerable misstatement is $40,000, and the expected misstatement is $15,000. • There is a moderate risk that other auditing procedures will fail to detect material misstatements. • All customer account balances greater than $25,000 are to be audited.
Population book valueTolerable misstatement SampleSize × Assurance factor = $3,717,900 – $550,000 $3,167,900$40,000 SampleSize × 1.2 = 95rounded = Nonstatistical Sampling Example
Nonstatistical Sampling Example The auditor sent positive confirmations to each of the 110 (95 + 15) accounts selected. Either the confirmations were returned or alternative procedures were successfully used. Four customers indicated that their accounts were overstated and the auditors determined that the misstatements were the result of unintentional error by client personnel. Here are the results of the audit testing:
Nonstatistical Sampling Example As a result of the audit procedures, the following projected misstatement was prepared: The total projected misstatement of $10,800 is less than the expected misstatement of $15,000, so the auditors may conclude that there is a low risk that the true misstatement exceeds the tolerable misstatement.
Why Did Statistical Sampling Fall Out Of Favor? • Firms found that some auditors were over relying on statistical sampling techniques to the exclusion of good judgment. • There appears to be poor linkage between the applied audit setting and traditional statistical sampling applications.
Classical Variable Sampling Classical variables sampling uses normal distribution theory to evaluate the characteristics of a population based on sample data. Auditors most commonly use classical variables sampling to estimate the size of misstatement. Sampling distributions are formed by plotting the projected misstatements yielded by an infinite number of audit samples of the same size taken from the same underlying population.
Classical Variables Sampling A sampling distribution is useful because it allows us to estimate the probability of observing any single sample result.
Classical Variables Sampling In classical variables sampling, the sample mean is the best estimate of the population mean.
Classical Variables Sampling • Advantages • When the auditor expects a large number of differences between book and audited values, this method will result in smaller sample size than MUS. • The techniques are effective for both overstatements and understatements. • The selection of zero balances generally does not require special sample design considerations.
Classical Variables Sampling • Disadvantages • To determine sample size, the auditor must estimate the standard deviation of the audited value or differences. • If few misstatements are detected in the sample data, the true variance tends to be underestimated, and the resulting projection of the misstatements to the population is likely not to be reliable.
Applying Classical Variables Sampling Defining the Sampling Unit The sampling unit can be a customer account, an individual transaction, or a line item. In auditing accounts receivable, the auditor can define the sampling unit to be a customer’s account balance or an individual sales invoice included in the account balance.
2 Population size × ZIA × SDTolerable misstatement – Estimated misstatement SampleSize = Applying Classical Variables Sampling Determining the Sample Size where ZIA = One-tailed Z value for the specifiedlevel of the risk of incorrect acceptance. SD = Estimated standard deviation.
Applying Classical Variables Sampling The risk of incorrect acceptance is the risk that the auditor will mistakenly accept a population as fairly stated when the true population misstatement is greater than tolerable misstatement.
2 SampleSize 5,500 × 1.96 × $31$50,000 – $20,000 = 125 = Applying Classical Variables Sampling The year-end balance for accounts receivable contains 5,500 accounts with a book value of $5,500,000. The tolerable misstatement for accounts receivable is set at $50,000. The expected misstatement has been judged to be $20,000. The risk of incorrect acceptance is 2.5%. Based on work completed last year, the auditor estimates the standard deviation at $31. Let’s calculate sample size.
Meanmisstatementper samplingitem Total audit differenceSample size = $330.20125 = $2.65 = Applying Classical Variables Sampling Calculating the Sample Results The sample selection usually relies on random-selection techniques. Upon completion, 30 of the customer accounts selected contained misstatements that totaled $330.20. Our first calculation is the mean misstatement in an individual account which is calculated as follows:
Projectedpopulationmisstatement Population size × Mean misstatement per sampling item = (in sampling units) Applying Classical Variables Sampling The mean misstatement must be projected to the population. = 5,500 × $2.65 = $14,575
Accounts receivablepoint estimate Bookvalue Projected populationmisstatement = – = $5,500,000 – $14,575 = $5,485,425 Applying Classical Variables Sampling Point estimate of accounts receivable balance . . . The sum of the audited differences squared is equal to $36,018.32. We will use this value to calculate the standard deviation.