Using pattern analysis methods to supplement attention directing analytical procedures

Expert Systems With Applications, Vol. 9, No. 4, pp. 513-528, 1995 Copyright© 1995 Elsevier Science Ltd Printed in the USA. All rights reserved 0957-4174/95 $9.50 + .00

Pergamon 0957-4174(95) 00021-6

Using Pattern Analysis Methods to Supplement AttentionDirecting Analytical Procedures JAMES R. COAKLEY Department of Accounting, Finance and Information Management, College of Business, Oregon State University, Bexell Hall 200, Corvallis, OR 97331-2603, U.S.A.

A b s t r a c t - - H o w might the application of analytical procedures be improved given the inherent shortcomings of traditional analytic techniques and the apparent difficulties auditors have in combining all critical cues when evaluating the results of the analytical procedures? This research attempts to improve analytical methods by applying a new technology, Artificial Neural Networks (ANNs), to perform pattern recognition of the investigation signals generated by analytical procedures. ANNs, a type of artificial intelligence technology, are able to recognize patterns in data even when the data is noisy, ambiguous, distorted or variable. Four years of audited financial data from a medium-sized distributor were used to calculate five commonly applied financial ratios. The performance of these ratios, applied independently and in combinations, was evaluated using a presumed lack of actual errors and certain seeded material errors~ The ANN method evaluated the information content of the combinations of financial ratios using an entropy cost function derived from information theory. This exploratory study suggests that the use of an ANN to analyze patterns of related fluctuations across numerous financial ratios provides a more reliable indication of the presence of material errors than either traditional analytic procedures or pattern analysis, as well as providing insight to the plausible causes of the error. Preliminary results suggest that the use of pattern analysis methods as a supplement to traditional analytical procedures will offer improved performance in recognizing material misstatements within the financial accounts.

1. I N T R O D U C T I O N

to known circumstances that affect the financial status of the firm and the relationships among accounts. Analytical procedures are generally governed by pattern recognition tasks: various fluctuations in account balances or their relationships are interpreted as being unusual, thereby signaling the need for additional investigation. However, fluctuations in the balances of financial accounts usually occur from month to month. Analytical procedures may detect these normal fluctuations and signal the need for additional investigation, thereby producing an erroneous alert for the presence of material monetary error. The auditor may not be able to distinguish these erroneous alerts from valid investigation signals that result from unusual fluctuations. Unnecessary investigations add to the overall cost of the audit without providing additional substantive evidence about the reasonableness of financial accounts. Audit research has evaluated the effectiveness of altemative analytical procedures in terms of their ability to direct attention towards financial account balances containing material errors. These analytical procedures have ranged from relatively simple prediction models to complex time-series methods. Much of this research

AUDITORS commonly use analytical procedures to analyze and evaluate financial information by examining relationships among financial and nonfinancial data for plausibility. These analytical procedures play an important role in the preliminary audit process (Cushing & Loebbecke, 1986; Tabor & Willis, 1985). Kreutzfeldt and Wallace (1986) report that 40% of the errors encountered during an audit engagement were detected using analytical procedures. Hylas and Ashton (1982) found that approx. 27% of the audit adjustments made by a major CPA firm were initially signaled by analytical procedures. In addition, Statement on Auditing Standards Number 56 requires that auditors use analytical procedures to help in planning "the nature, timing and extent of other auditing procedures" (American Institute of Certified Public Accountants, 1988, SAS 56, AU 329.04). Analytical procedures are essentially tests of reasonableness that involve a comparison of the client's unaudited book values to the auditor's expectations about what those values should be, giving consideration 513

514

suggests that analytical procedures can be effectively applied to assist the auditor in detecting material misstatements (Albrecht & McKeown, 1978; Coakley, 1982; Coakley & Brown, 1993; Daroca & Holder, 1985; Harper, Strawser, & Tang, 1990; Holder, 1983; Kinney, 1978, 1979, 1987; Kinney & Salamon, 1978, 1982; Knechel, 1986, 1988a,b; Loebbecke & Steinbart, 1987; Neter, 1979; Spires & Yardley, 1989; Wright & Ashton, 1989). However, Loebbecke and Steinbart (1987) caution that analytical procedures do not reliably indicate the absence of errors. Research on the judgmental process used by auditors suggests that recognition of patterns in financial ratios is an important part of an analytical procedure. Libby (1985) argues that the detection of audit errors is a diagnostic problem. Using protocol analysis of four auditors from two firms, Biggs, Mock, and Watkins (1988) find that auditors use financial ratios to investigate audit problems and opportunities in a complex, realistic task setting. Biggs and Wild (1985) note that auditors could identify patterns in financial data; they concluded, "this evidence suggests that pattern recognition is an important part of the auditor's analytical procedures." (p. 631). Bedard and Biggs (1991) found that auditors sometimes fail to include the combination of all crucial cues during pattern recognition tasks. They suggest that the subject's lack of pattern recognition seems plausible given Kinney's (1987) conclusion that analytical procedures as performed in practice do not stress the use of data in combination. This previous research suggests the need for a methodology that can be applied to analyze the complex patterns (or cues) across the financial accounts. This concept was briefly investigated by Kinney (1987) who found that patterns of deviations from expectation over several related financial ratios was useful in identifying the cause of a material monetary error in the accounts. A method which analyzes the pattems of related fluctuations across numerous financial accounts may provide a more reliable indication of the presence of material errors, as well as providing insight to the plausible causes of the error. Artificial neural networks (ANNs), a type of artificial intelligence technology, are able to recognize patterns in data even when the data is noisy, ambiguous, distorted, or variable (Hecht-Nielsen, 1989). The objective of this exploratory research is to determine if an ANN may be applied to perform pattern recognition of the investigation signals generated by analytical procedures. This research extends previous work which evaluated whether an ANN method could produce more appropriate investigation signals for the fluctuating data typically contained in financial statements (Coakley & Brown, 1993). The previous effort focused on the forecasting ability of the ANN and varied many factors which affect the performance of alternative analytical procedures, including materiality levels, statistical levels of con-

J. R. Coakley

fidence placed on analytical procedures, and sources of material errors. This study differs from that previous research because it applies the ANN procedure to analyze patterns across financial ratios, rather than independently predict the future value of each ratio. The remainder of this article is organized as follows: The first section discusses the financial model of the firm and the financial ratios selected for the analysis. The second section develops the financial ratio pattern analysis methodology, while the third section discusses the use of entropy theory to develop the pattern analysis methodology based on the ANN. Next, the simulation procedures that were applied to evaluate the financial ratio and ANN methods are described. The fifth section presents the results, and the last section summarizes the study and discusses implications for future research. 2. FINANCIAL MODEL OF FIRM

Financial statements of any firm contain numerous accounts. For each account; many individual transactions constitute the reported balance (also referred to as the book value) for each annual reporting period. This reported balance can be partitioned into 12 monthly balances. An auditor needs to determine if an account balance is free of material errors that require an adjustment to the financial statements. Let xa, reflect the reported balance in account a for month t, and Y~t reflect the correct but unknown balance of account a for month t. Then the error in the account prior to the audit is represented by e~,. (ea,=xa,-ya,).

(1)

An analytical procedure is applied to estimate the expected balance in the account, 3~a,. This expected balance is compared with the reported balance (xat) to determine if a discrepancy exists. There are two outcomes from this comparison: • If a material discrepancy does exist, then the analytical procedure has signaled the need for further investigation of the financial accounts. • If a material discrepancy is not discovered, then the need for further investigation is not signaled. In addition, there are two states of nature which could Occur"

• The size of the error in the reported balance is less than the materiality threshold; or • the size of the error is greater than or equal to the materiality threshold. The combinations of outcomes and states of nature result in four types of attention-directing decisions (Table 1). Correct decisions result when the size of the monetary error is less than a material amount and an investigation is not signaled, and when the size of the monetary error is material and an investigation is

Attention-Directing Analytical Procedures

515 TABLE 1 Types of Attention-Directing Decisions

Results of Analytical Procedure Size of Error in Reported Account Balance

Investigation Not

Investigation

Signaled

Signaled

Less than materiality threshold

Correct decision

Type I error

Type II error

Correct decision

Greater than or equal to materiality threshold

signaled. A Type I error is an incorrect decision to investigate which occurs if the analytical procedure signals the existence of an error when no material error is present in the reported account balance. The Type I error rate provides a measure of efficiency. A large number o f Type I errors may reduce the efficiency o f the audit since additional accounts m a y be needlessly investigated. A Type II error is an incorrect decision to not investigate which occurs when t h e analytical procedure fails to signal the existence of a material error when one actually exists. The Type II error rate provides a measure of the reliability o f the analytical procedure and is consistent with the audit risk model in SAS 47 (AICPA 1983) that attempts to control the risk of concluding that an account balance is correct when it actually contains a material error. SAS 47 does allow for the concept of a tolerable error, and the total tolerable errors at the individual account level may exceed the materiality level. In this exploratory study, we do not distinguish tolerable errors. W h e n the total error amount exceeds a material amount, then an investigation signal should be signaled or a Type II error results.

Receivables are overstated by the amount of fictitious sales recorded. If a physical count of inventory is taken and the Inventory account is adjusted to match the physical count, then the errors to the Inventory and Cost of Sales accounts are corrected leaving only the error to Receivables and Sales. If a perpetual inventory system is used, then the actual cost o f the items apparently sold would be recorded to the Inventory and Cost of Sales accounts. The error in these accounts, on average, will equal the gross margin percentage times the actual error. As a result of aggregation, Current Assets are also affected by an amount equal to the difference between the sales revenue and the cost. W h e n purchases on an account are not recorded, Payables are understated by the amount of unrecorded purchases. If a physical inventory is taken and the Inventory is adjusted to match the actual count, the error is forced into Cost of Sales as an understatement. As a result of aggregation, Current Liabilities are understated. In a perpetual i n v e n t o r y system, Inventory is also understated by the amount o f the unrecorded purchase. As a result of aggregation, Current Assets and Current Liabilities are understated.

2.1. Definition of Sources of Error

2.2. Financial Ratios Selected

Two sources of errors were used to affect analyses of the financial accounts, namely: (1) unrecorded purchase o f merchandise on account; and (2) recorded fictitious sales on account. Coakley and Loebbecke (1985) report that these sources of error are among the most frequently encountered in practice. Also, affected accounts typically require adjustments (Hylas & Ashton, 1982; Kinney, 1987). Two assumptions were used to analyze the errors, namely, (1) a physical count of inventory was taken, and the Inventory account was adjusted to match the physical count (this assumption was used by Kinney (1987)); (2) no physical inventory was taken so the Inventory account was not corrected (this assumption matches the process used by the firm in this study for interim statements). The impact of the errors on the financial accounts could thereby be predicted (Table 2). W h e n fictitious sales on an account are recorded, both Sales and

Five financial ratios were selected on the basis of the ability of the ratio to reflect changes in the account balances due to sources of error (Table 3). These ratios (or variants) have previously been evaluated to determine their effectiveness as analytical procedures (Coakley, 1982; Kinney, 1987). The first four ratios (Receivables Turnover, Inventory Turnover, Cost of Sales and Accruals) were used in the Kinney 1987 study. The fifth ratio (a Debt-to-Equity variant ~) was added to provide a measure which uniquely reflected the Purchases Not Recorded error under the assumption o f no physical count of inventory.

~During one year in the base period, the reported balances for OwnersEquity included negative and zero amounts. Thus, the common debt-to-equity ratio based on the relationship of Total Liabilities to OwnersEquityproduced some undefined values. The ratio based on the aggregates TotalLiabilities and Total Liabilities and Equities captured the essence of the debt-to-equity relationship and produced ratio values that were more consistent.

516

J. R. Coakley TABLE 2 Impact of Source of Monetary Error on Financial Accounts or Aggregates

Source of Monetary Error Fictitious Sales on Account Recorded

Financial Accounts or Aggregates Receivables

Inventory Adjusted to Physical Count

No Physical Count of Inventory

Overstate by S

Overstate by S

Inventory

Purchase on Account Not Recorded Inventory Adjusted to Physical Count

Understate by C

Payables

Understate Understate

Sales

Overstate by S

Overstate by C Overstate by S

Understate

Overstate by S

Cost of sales Current assets

No Physical Count of Inventory

Understate

Overstate by S - C

Current liabilities

Understate Understate

Understate

Note: The firm uses a perpetual inventory system: S, sales revenue for items sold; C, cost associated with items sold.

2.3. Description of the Firm in the Case Study Actual month-end balances were obtained for a wholesale distributor for four calendar years. Although each of the four years was audited, the monthly balances were not audited individually. It was assumed that these monthly balances were free o f material error. There were no year-end accounting adjustments to the accounts. The firm under study has a single, disaggregated operation which produces revenues of approx. $20 million per year. No changes of accounting methods or

other unusual circumstances occurred in the four-year period. F o r the purposes of this exploratory study, only internal financial data were examined; no external industry or economic indicators or operating data were included. Nine income statement and balance sheet accounts or aggregates affected by the financial ratios were selected for this study. The average monthly balance and coefficient of variation were calculated over the three year base period for these accounts or aggregates (Table 4). The coefficient of variation provides an indication of

TABLE 3 Impact of Source of Monetary Error on Evaluated Financial Ratios

Source of Monetary Error Fictitious Sales on Account Recorded

Inventory Adjusted to PhysicalCount (Kinney)

No Physical Count of Inventory

Up

Down

Up

Down

Down

Down

Accruals ratio Receivables~Payables

Up

Up

Up

Up

Debt/equity ratio Total liabilities~total liabilities and equities

Not evaluated

Not evaluated

Down slightly

Financial Ratios Receivables turnover Sales~Receivables

Inventory Adjusted to Physical Count (Kinney)

No Physical Count of Inventory

Up

Up

Purchase on Account Not Recorded

Inventory turnover Cost of sales/Inventory Cost of sales ratio Cost of sales/Sales

Attention-Directing Analytical Procedures

517 TABLE 4 Financial Accounts Included in the Study

Financial Account Receivables (trade accounts receivable) Inventory *Current assets Payables *Current liabilities *Total liabilities *Total liabilities and equities Sales Cost of sales

Average Monthly Account Balance over Three Year Base Period

of Variation

4,356,329

0.201

1,373,011 5,799,543 2,829,479 6,103,791 6,436,349 6,514,291 1,498,937 1,068,835

0.376 0.211 0.213 0.176

Coefficient

0.165

0A69 0.190 0.204

Note: Aggregatesare annotatedwith an asterisk.

the degree of noise in the absolute balances of the accounts and aggregates. 3. F I N A N C I A L RATIO PATTERN ANALYSIS METHOD Financial ratios were calculated for the four audit periods using ending balances. Ending balances are preferred in auditing over the traditional financial method of using average balances. If a material error is present in the ending balance of an account, averaging the ending balance with the beginning balance will dilute the effect of the error, making it more difficult to detect. Auditor's actions following application of an analytical procedure depends on the decision rule used with the procedure. The decision rule is a model for evaluating the effect of errors on the financial ratios. Many alternative decision rules applied to financial ratios are investigated in the literature (Coakley, 1982; Kinney, 1987; Knechel, 1986, 1988b; Loebbecke & Steinbart, 1987). This study employs a statistical decision rule [proposed by Kinney (1987)] that triggers an investigation when the difference between reported ratio value (calculated from the reported balances for the test month) and the expected ratio value (derived from past audited balances) is so large in relation to past differences that it is unlikely that the difference can be attributed to chance. Let rt represent the reported value of a ratio based on the reported balances in month t. The expected value of the ratio based on the audited balances for month t (denoted rt) is assumed to be the average value of the same ratio over the previous audit period, adjusted for changes in the industry. Finally, let s~ represent the standard deviation of the audited monthly values of ratio r for the previous audit period. Then the test statistic can be derived as the difference between the reported and expected values, divided by the standard deviation [see equation (2)] Test Statistic = ( r , - %)~st.

(2)

The decision rule signals the need for additional investigations of the accounts comprising the ratio if the calculated test statistic exceeds a preset critical value for the deviation. If the distribution of standardized changes over the past audit periods is normal (which is assumed), a " Z " value based on a risk level specified by the auditor may be used as the critical value. Thus, the auditor is alerted to investigate the accounts if the test statistic in equation (2) is greater than Zl-~/2 or is less than -Z1-~/2, where a is the probability of a Type I error. Similar to the Kinney (1977) study, an a level of 0.33 was used.

3.1. Pattern Analysis Procedure The financial ratio procedures attempt to identify unusual fluctuations in the financial ratios and alert the auditor that additional investigations may be warranted. The purpose of the pattern analysis procedure is to determine whether the signaled fluctuations within each stream of ratios are consistent with the expected combination of fluctuations that would occur if the types of errors considered plausible for the engagement are present. The model for evaluating the pattern analysis of the financial ratios is based on the Guidelines for Selection of Analytical Procedures presented by Coakley (1982). The combination of investigation signals from the financial ratios is compared with the expected combination of signals that would occur if either of the two specified types of error is present. The comparison is based on an analysis that predicts the direction of the change in the financial ratio if a monetary error from the specified financial transaction (source of error) is present in the accounts (see Table 3). The calculated financial ratios (depicted in Figure 1) demonstrate the value of employing pattern analysis as an analytical procedure. At month 37 (during the fourth audit period depicted), there is a large increase in the Inventory Turnover and Debt-to-Equity ratios. At month 42 there is a large decrease in the Inventory Turnover and

518

J. R. Coak!ey

I Accruals I

r

Inventory Turnove{ N

I

, Debt/

Equity

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O 0.5

Cost of 5

L Receivables TL I

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'"'"'"'"{'"2 Audit Year (Period)

I I 1,41

I [

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Potential {I Errors

FIGURE 1. Financial ratios applied to case study balances.

Receivables Turnover ratios, and a large increase in the Accruals ratio. Analytical procedures which evaluate each financial ratio independently should signal that additional investigations are warranted for these large fluctuations. Since the reported account balances are supposedly free of material error, these signaled investigations would result in Type I errors. Applying pattern analysis across these financial ratios yields a different result. According to the impact analysis in Table 3, a decrease in the Debt-to-Equity ratio accompanied by a significant increase in the Inventory Turnover ratio should alert the auditor that there is a high likelihood that a material error exists as a result of a Purchases Not Recorded transaction. Since the fluctuations in month 37 were in the same direction, they were not caused by either source of error considered plausible for this engagement. Thus, the pattern analysis approach would not signal the need for additional investigations. In a similar manner, the fluctuations identified in the ratios during month 42 can be analyzed with respect to the two error sources considered plausible for this engagement. If the fluctuations in all three ratios (Receivables Turnover, Inventory Turnover and Accruals) were positive, the auditor should be alerted for the potential presence of a material error resulting from the

Fictitious Sales on Account transaction. If the Inventory Turnover and Accruals ratio both had positive fluctuations and the Receivables Turnover was as expected, then the auditor should be alerted for the potential presence of a material error resulting from the Purchase Not Recorded transaction. But, since the fluctuations in the Accruals and turnover ratios are in different directions, they could not be caused by either of the two error sources. Thus, the pattern analysis approach would not signal the need for additional investigations. The pattern analysis procedure requires investigation signals from multiple analytical procedures to generate an alert for a potential monetary error. Thus, an investigation signal from a single source will not generate any alerts for potential errors. In his study, Kinney (1987) noted that the Accruals ratio provided redundant information to the Cost of Sales ratio since each changes in the same direction for both sources of error. As a result, Kinney did not include the Accruals ratio in the pattern analysis since it had a lower signal-tonoise ratio, thus providing less information about possible accounting errors. In this study, there are five financial ratios being investigated. Use of a pattern analysis procedure which alerts for potential misstatements if two or more of the

Attention-Directing Analytical Procedures

519

TABLE 5 Decision Table for Investigation Alerts from Combinations of Financial Ratios that Detect Monetary Errors from Two Plausible Transactions Investigation Signal Generated by Financial Ratio Analytical Procedure? Debt/equity ratio No Accruals ratio

No

Cost of sales ratio

Yes

No

Inventory turnover ratio

No

Yes Yes

Receivables turnover ratio

N

Y

N

No alerts

X

X

X

Alert for error from Fictitious Sales transaction

No

Y

N

No Yes

Y

N

Y

X

No N

Yes Yes

Y

N

No

Y

Yes

N

Y

X

X

N

Y

X

X

X

X

X

X

X

X

Alert for error from Purchase Not Recorded transaction Alert for error from Combination of transactions

X

X

Investigation Signal Generated by Financial Ratio Analytical Procedure? Debt/equity ratio Yes Accruals ratio

No

Cost of sales ratio

Yes

No

Inventory turnover ratio

No

Yes Yes

Receivables turnover ratio

N

Y

No alerts

X

X

N

Y

No N

Yes Y

N

Y

No N

Yes Yes

No

Yes

Y

N

Y

N

Y

N

Y

X

X

X

X

X

X

X

X

Alert for error from Fictitious Sales transaction Alert for error from Purchase Not Recorded transaction

No

X

X

X

Alert for error from Combination of transactions

financial ratios signal investigation results in five sources of investigation signals (one from each ratio). This produces 32 combinations of decision results which must be analyzed (Table 5). The pattern analysis approach as presented in Table 5 provides an equal weight to each of the investigation signals generated by each of the ratios. However, for a given error source, some ratios may be providing complementary redundant information while other ratios may be providing conflicting information. A methodology needs to resolve conflicting investigation signals and provide more weight to those signals that effectively discriminate between alternative sources of error. One

X

X

X

approach to derive these weights is to apply ANNs to recognize the patterns across financial ratios that signal potential sources of monetary error. 4. A R T I F I C I A L NEURAL N E T W O R K PATTERN ANALYSIS M E T H O D Determining the appropriate ANN model is not a straight-forward task. Within the literature, ANNs are broadly categorized using three criteria: the network architecture; the activation function; and the training algorithm. This study is limited to investigating variations of the multi-layer, feed-forward, perceptron

520

(architecture) employing a hyperbolic tangent activation function and the back-propagation training algorithm. This is not the only model that could be applied to the data set, nor is it necessarily the best m o d e l for this purpose. But comparing the performance of alternative models is beyond the scope of this paper. Other published descriptions provide insights into various networks (Anderson & Rosenfeld, 1988; Hecht-Nielsen, 1990; Hertz, Krogh, & Palmer, 1991; Hoptroff, Hall, & Bramson, 1991; Lawrence, 1991; Rumelhart & McClelland, 1986; Waite & Hardenbergh, 1989; Wasserman, 1989).

4.1. The Artificial Neural Network Architecture The architecture represents the nature and degree of connectivity among the processing nodes in the ANN. Within this study, the five calculated monthly financial ratios were the input data stream to the ANN. Three output nodes were used: a separate output node to signal the presence of material error from each of two potential sources of error investigated in the study, and a third node to signal the presence of material error from either or both sources. Specifying the internal architecture requires determining the number of hidden layers, and the number of processing nodes in each hidden layer. Although there has been much research on deriving the 'proper' internal architecture [see Cybenko (1989); Lapedes & Farber (1988); Hecht-Nielsen (1990); Hertz et al. (1991); Hornik, Stinchcombe, & White (1989)], experiments must be conducted to determine the performance differences between alternative network configurations. The best results for the data within this study were obtained with an architecture using 2 hidden layers and 11 nodes in each hidden layer.

4.2. The Activation Function Four non-linear activation functions are commonly used with the back-propagation training algorithm: sigmoid, half-sigmoid, arc-tangent and hyperbolic tangent (Anderson & Rosenfeld, 1988; Hecht-Nielsen, 1990; Hertz et al., 1991; Lawrence, 1991; Rumelhart & McClelland, 1986; Stometta & Huberman, 1987; Waite & Hardenbergh, 1989; Wasserman, 1989). The hyperbolic tangent activation function is used in this study since it provided the easiest derivation of the entropy cost function (see discussion below). For the hyperbolic tangent activation function, the input data streams were scaled to a range of (_+ 1) to center the data on the midpoint of the activation function and limit the initial feed-forward calculations to the near-linear portion of the function. The range of the target values was also adjusted to lie within the output range of activation function. A 'better' fit was obtained with the target

J. R. Coakley

values scaled to 90% of the ---1 output range of the activation function.

4.3. The Training Algorithm The back-propagation algorithm has emerged as the dominant paradigm for training multi-layered perceptrons. A complete description of the algorithm and its derivations can be found in numerous sources, including Hertz et al. (1991) and Rumelhart et al. (1986). Backpropagation models typically use a quadratic cost function which minimizes the sum of the squared errors between the calculated output value (O) and the specified target value (T). This sum of squared error (SSE) cost function uniformly weights each training trial error in accordance with the square of magnitude of the error vector ( T - O ) . This error measurement scheme ensures that large errors receive much greater attention than small errors, and it also sensitive to errors made on commonly encountered inputs than it is to errors on rare inputs (Hecht-Nielsen, 1990). However, the cost function should reflect the statistical nature of the original data as well as any assumptions about measurement errors. This study employs an alternative cost function, proposed by Hertz et al. (1991), which maximizes the relative entropy between the input data streams. This cost function, based on information theory, involves learning the correct probabilities of a set of hypotheses represented by the output unit. Information theory is applied to derive the expected information content of an investigation signal, which is expressed as a conditional likelihood function. The resulting entropy cost function results from maximizing the conditional likelihood function, and corresponds to the maximum conditional likelihood function commonly used in Logit analysis [see Loh (1991) and Palepu (!986)]. The maximum conditional likelihood function is appropriate for dichotomous output variables, which in the case of this study, corresponds to the likelihood of the presence or absence of an investigation signal. As a result of using the entropy cost function, the ANN will place more weight on those signals that produce complementary information, and less weight on those signals that provide redundant or conflicting information.

5. S I M U L A T I O N P R O C E D U R E S

5.1. Seeding of Monetary Errors A simulation was performed to evaluate the effectiveness of the financial ratio and ANN pattern analysis methods in signaling the absence and presence of material errors. To simulate unaudited values to which the auditor would apply analytical procedures, the audited account balances for each month were seeded with a monthly cutoff error. This error then propagated through the various accounts

Attention-Directing Analytical Procedures 0.45

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M o n t h in p r e d i c t i o n p e r i o d

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~ ) Type I error

i {

FIGURE 2. Effect of alpha risk level on Type I error rate.

and aggregates due to the aggregation process of generating monthly financial reports. Since cutoff errors are self correcting, the errors were only reflected in the seeded months. The simulation process produces three distinct cases regarding the seeding of monetary errors into the financial accounts and aggregates: • No monetary errors are seeded. Since we have assumed that the reported balances are free of material error, they can be used to evaluate the efficiency of the procedures. The procedures produce correct decisions if they do not signal the need for additional investigations when monetary errors do not occur in the financial balances. W h e n additional investigations are signaled in this case, they are assessed as Type I errors. (This case corresponds to the first 'state of nature' in Table 1 where the size of the error in the account balance is less than the materiality threshold.) • Monetary errors less than a material amount are seeded. If the seeded monetary error was less than a material amount, an efficient analytical procedure would not signal further investigation. When additional investigations are signaled in this case, they are assessed as Type I errors. (This case also corresponds to the first 'state of nature' in Table 1 where the size o f the error in the account balance is less than the materiality threshold.) • Material monetary errors are seeded. When material monetary errors are seeded into the accounts, a reliable analytical procedure should signal the need for further investigation. When additional investigations are not signaled in this case, they are assessed as Type II errors. (This case corresponds to the second 'state o f

nature' in Table 1 where the size of the error is greater than or equal to the materiality threshold.)

5.2.

Performance

Measure

The combined Type I and Type II error rates should be used to evaluate analytical procedures which rely on statistical processes since changes in the specified levels of ct risk (used to establish the confidence regions) will result in tradeoffs between the Type I and Type II error rates. This tradeoff in error rates is demonstrated using Figure 2 which displays four sets of data for the 12 month prediction period: the actual account balances, the estimated point-values for the account balances, a 90% confidence interval around the estimates, and a 67% confidence interval around the estimates. Assume that the graph in Figure 2 represents the first case described above where no material errors are seeded into the accounts. With the a risk set at 0.10 (90% confidence region), there are false signals to investigate in months 4 and 12, resulting in a Type I error rate of 0.167. W h e n the a risk is changed to 0.33 (67% confidence region), two additional months (1 and 11) are now signaled for false investigation, resulting in a Type I error rate of 0.333. Thus, the Type I error rate may be increased solely by increasing the specified a risk for the analytical procedure. A similar situation occurs when material monetary errors are seeded into the account balances. In this situation, an increase in the specified alpha risk for the analytical procedure will decrease the Type II error rate. As a result, the overall effectiveness of an analytical procedure should be assessed by evaluating the combina-

522

J. R. Coakley

tion of the Type I and Type II error rates. This will compensate for those situations which result in a tradeoff between the Type I and Type II error rates. Since each analytical procedure is evaluated under the same conditions, differences in the overall effectiveness measure reflect comparative differences in the ability of the analytical procedure to signal the absence and presence of material errors. If a purely random process was used to signal investigations, then, on average, half of the months evaluated would be signaled for investigation. If no material errors were present in the accounts, the resulting Type I error rate would be 0.5. That is, half of the accounts evaluated were signaled for further investigation when no material errors were present. If material errors were present in the accounts, the resulting Type II error rate would also be 0.5. In this case, only half of the accounts containing material errors were correctly signaled for further investigation. The expected value of the sum of the error rates would equal 1.0. Using a method proposed by Loebbecke and Steinbart (1987), the overall effectiveness of the analytical procedure can be assessed by comparing the sum of the Type I and Type II error rates to a benchmark value of 1.0.

5.3. Experimental Factors Two factors were varied during the simulation: the transaction sources for the material errors and the materiality level (size of the error). The two transactions sources of monetary error were: (1) unrecorded purchase of merchandise on account; and (2) recorded fictitious sales on account (discussed earlier). The amount defined as material is based on the functional relationship developed by Warren and Elliot (1986) [see equation (3)]. Icerman and Hillison (1991) compare material amounts derived from the formula in equation (3) with empirical data collected from 49 manufacturing firms. Their analysis reports that the formula produced values consistent with the size of individual errors that resulted in adjustments to the financial statements. Relying on their study, the formula in equation (3) is used to provide a reasonable approximation for materiality. Similar to the Kinney (1987) study, the materiality levels were varied from 0, 0.5, 1.0 and 2.0 times the material amount. Materiality (M)= 0.038657 * (Revenues) °'8672°3.

(3)

5.4. ANN Training The A N N pattern analysis procedure was trained using the evaluated financial ratios in the base period. The ratios were first computed under the assumed absence of material error and provided as input to the ANN. The target values for all three output nodes was set to - 1 , indicating that the probability of a material error should be 0. The monetary errors associated with fictitious sales

on accounts were then seeded into each period, varying the size of the monetary error from 0.5, 1.0 and 2.0 times a material amount. When the error size was less than a material amount, the target value for the output node associated with the fictitious sales error was set to - I (probability of a material error= 0). When the error seeded was equal to or greater than a material amount, the target value for the output node was set to + 1 (probability of a material error = 1.0). The same process was used to derive the input data streams associated with monetary errors from the purchases not recorded transactions. The approximation accuracy of the A N N must be evaluated using a separate set of data (in this case, the forecast period). What is desired is for the A N N to generalize from the limited set of training data to the entire problem environment. However, training with repeated applications of the same data set results in a phenomenon of overtraining. The A N N attempts to exactly fit the limited set of points and loses its ability to interpolate between those points. (Hecht-Nelson, 1990). The issue associated with overtraining is determining when sufficient iterations have been accomplished to assure the 'best' prediction accuracy in the forecast period. Training iterations beyond that point may actually degrade predictive performance. As a result, three sets of data are required: one set to train the network, one set to evaluate when the 'best' predictive performance is attained, and a third set which contains the forecast period. Within this study, three years of data were used for training set and the fourth year of data was used as the forecast period to evaluate the performance of the ANN. A bootstrap method was used to randomly select eight months of data from the training set as a holdout sample to test for overfitting. The remaining 28 months of data in the training set were used to train the network. 6. R E S U L T S Results in this study have been divided into findings based on a comparison of financial ratio results with the Kinney (1987) study, and a comparison of pattern analysis methods.

6.1. Comparison of Financial Ratio Methods Results from applying financial ratios to the 48 months of data in the current study were compared to the results from Kinney's research (1987). Table 6 presents the means and standard deviations of monthly ratios for firms evaluated in both studies. These data suggest that the financial ratios for the two firms have similar distributions. Hence, one might anticipate that the analytical procedures based on these financial ratios would exhibit similar performance across the two firms. Table 7 illustrates the error rates from the Kinney study and Table 8 illustrates the error rates for the firm in

523

Attention-Directing Analytical Procedures TABLE 6 Comparison of Means and Standard Deviations of the Monthly Ratios of Firms Evaluated in Kinney (1987) and this Study

Kinney (1987)

This Study

Financial Ratios

Mean

SD

Mean

SD

Receivables turnover Sales~Receivables

0.35 0.03 0.86 0.17 0.72 0.02 1.62 0.12 Not investigated

0.35 0.84 0.71 1.55 0.92

0.03 0.20 0.02 0.10 0.05

Inventory turnover Cost of Sales~Inventory Cost of sales ratio Cost of sales/Sales Accruals ratio Receivables~Payables Debt/equity ratio Total liabilities~Total liabilities and equities

this study. Type I error rates were derived for each ratio when no error and an error one-half materiality was seeded into the accounts. Type II error rates were derived when a material error and an error twice materiality was seeded into accounts. In the Kinney (1987) study, the Fictitious Sales on Account error source does not impact the Inventory Turnover ratio (see Table 3 under the assumption that a physical adjustment to inventory is made). Thus, investigations signaled b y Inventory Turnover ratio when material Fictitious Sales errors are seeded are false signals to investigate. This is depicted as a Type I error in Table 7. Note that the Type I error rate reported in the Fictitious Sales column is 0.354 which reflects the Type I error rate when no monetary errors are seeded into the financial accounts or aggregates composing the ratio. The Type I error rate reported in the 'Less than Material Error Seeded' column is 0.431. This value reflects two conditions: (1) no monetary errors seeded; and ( 2 ) t h e seeding of monetary errors that are half o f a material amount. As the monetary errors that are half of a material amount are seeded into the financial accounts and aggregates, the analytical procedures sense the unusual fluctuation and signal the need for additional investigations. Since these seeded monetary errors are less than

materiality, an investigation should not be signaled, so it is assessed as a Type I error. As a result, the Type I error rate reported in the 'Less than Material Error Seeded' column should be higher than rate reported in the Fictitious Sales colunm for the Inventory Turnover ratio, A similar situation occurs for the Receivables Turnover ratio. The material Purchases Not Recorded errors do not affect the financial accounts or aggregates composing this ratio. Thus, any signal to investigate would be a false signal, assessed as a Type I error. The Type I error rate reported in the 'Less than Material Error Seeded' column is higher because it includes those conditions where monetary errors that were less than a material amount were seeded into the financial accounts and aggregates. Under the assumption that no physical adjustments to inventory are made (financial data in this study, summarized in Table 8), Table 3 indicates that error from the Fictitious Sales transaction should not impact the Debt to Equity ratio, a n d errors from the Purchases Not Recorded transaction should n o t impact the Receivables Turnover or Cost of Sales ratios. Investigation signals generated under these conditions were assessed as Type I errors. Note again that the Type I error rates generated under the

TAB LE 7 Financial Ratio Results from Kinney (1987) Study

Less than Material Error Seeded Financial Ratios

Type I

Receivables turnover

0.403

Material Fictitious Sales on Account Error Seeded Typel

Material Purchase on Accounts not Recorded Error Seeded

Typell

Combined

Typel

0.281

0.689

0.375

Type II

Combined

0.281

0.712

Sales~Receivables Inventory turnover

0.431

0.354

Cost of sales~Inventory Cost of sales

0.653

0.021

0.674

0.021

0.674

0.347

0.542

0:889

0.438

0.785

Cost of Sales/Sales Accruals

Receivables~Payables

524

J. R. Coakley TABLE 8 Financial Ratio Results from this Study Material Purchase on Less than Material Error Seeded

Financial Ratios

Type I

Receivables turnover

Sales~Receivables Inventory turnover Cost of sales/Inventory Cost of sales Cost of Sales/Sales Accruals Receivables/Payables Debt/equity Total Liabilities~Total Liabilities and Equities

Material Fictitious Sales on Account Error Seeded

Type II

Combined

Typel

0.269

0.564

0.833

0.231

0.192

0.308

0.500

0.231

0.718

0.949

0.385

0.538

0.923

0.346

Type I

Accounts not Recorded Error Seeded

0.308

conditions when no monetary errors are seeded that impact the financial accounts or aggregates tend to be smaller than the Type I error rates which occur when less than material monetary errors are seeded. Applying the financial ratio based analytical procedures to the data within this study resulted in lower Type I and higher Type II error rates compared with Kinney's results for the Receivables Turnover, Inventory Turnover and Cost o f Sales ratios. Note however, that with the exception of the Inventory Turnover ratio, the combined error rates for the data within this study were higher. Thus, the analytical procedures based on these financial ratios tended to be more effective when applied to the financial data in Kinney's study. Even though the financial ratios had similar variation across the two studies, it was much more difficult to distinguish the fluctuations caused by seeded errors from those that normally occurred in the financial data of the firm used in this study. One possible explanation is that a different formula was used to derive a material amount. Kinney (1987) used 0.5% of last year's audited sales, adjusted for the sales growth rate for the industry. The resulting value for a material error was approximately twice as large as the value derived from materiality formula [equation (3)] used in this study. The higher materiality threshold used in the Kinney (1987) study explains the lower Type II error rates since larger errors should be easier to detect. However, changes in the materiality threshold would not explain the differences in the Type I error rates since materiality does not affect the decision roles used with these analytical procedures. The differences in the Type I error rates are most likely due to the variability in the monthly account balances. Recall that the decision rule compares the value of the financial ratio in the current month with the average of the financial ratio values from the previous audit period, adjusting for the variation in the previous audit period. High variability in the calculated Inventory

Typell

Combined

0.436

0.628

0.436

0.821

0.615

0.962

0.231

Turnover ratio values (see Figure 1 and Table 4) results in fewer signals to investigate (lower Type I and higher Type II errors). The values for the Cost of Sales ratio appear consistent across the audit periods with relatively low variation. However, a material error affects both the numerator and denominator of the ratio. The small fluctuations are more difficult to detect, resulting in fewer signals to investigate. This, in turn, produces lower Type I and higher Type II error rates. The Kinney (1987) study found the four financial ratios to be relatively effective analytical procedures. The results from this study are more consistent with the conclusions of Loebbecke and Steinbart (1987) which found that financial ratios may not reliably indicate the absence of errors. The implication is that financial ratios applied as analytical procedures may produce inconsistent results even when applied across similar firms.

6.2. Comparison of Pattern Analysis Methods The results of the pattern analysis using financial ratio procedures from the Kinney study are presented in Table 9. The two sources of error produce four combinations of investigation signals: neither error source signaled, Fictitious Sales on Account signaled, Purchase on Account Not Recorded signaled, or both error sources signaled. Kinney's (1987) reported simulation results for no errors seeded and errors of one-half materiality were combined to produce the first column: Less than Material Error Seeded. Simulation results for errors of a material and twice-material amount were combined to derive the Type II error rates for each error source. The error rates for the four possible outcomes are summed to provide an overall measure of the effectiveness of the pattern analysis procedure. When material errors resulting from the Fictitious Sales on Account transaction were seeded into the financial accounts, no investigation signals were gen-

525

Attention-Directing Analytical Procedures TABLE 9 Pattern Analysis Results from Kinney (1987) Study

Investigation Signals for:

Material Fictitious Material Purchase on Less than Material Error Seeded Sales on Account Error Seeded Accounts not Recorded Error Seeded Type I Type II Type II 0.125

Neither error source

0.115 0.167

Fictitious sales on account error source

0.167

Purchase on account not recorded error source

0.181

Both sources of error

0.097

Combined error rates to detect presence of material error

0.445

0.570

0.560

Combined error rates to detect source of material error

0.445

0.747

0.727

erated in 12.5% of the audit periods evaluated. This equates to a Type II error rate o f 0.125. Thus, the combined error rate to detect the presence of a material monetary error is 0.57. Note, however, that in 17.7% of the audit periods evaluated, an investigation signal was generated, but the error source was erroneously identified as the Purchase on Account Not Recorded transaction. This is a false signal to investigate this error source, or a Type I error. It is also a Type II error for the Fictitious Sales error source since a material error does exist yet the audit period was not signaled for investigation. This Type II error rate must be included in the combined error rate to detect the source of the material monetary error. The combined error rates from applying pattem analysis within the Kinney (1987) study for the Purchase

0.177

on Account Not Recorded error source are 0.56 to detect the presence of an error and 0.727 to detect the source of an error. These combined rates to detect the source of an error are actually higher than the combined rates for the analytical procedures based on the Receivables Turnover, Inventory Turnover and Cost of Sales ratios (depicted in Table 8). Thus, the equally weighted pattern analysis approach does not provide a significant improvement over the independent analytical procedures derived from the financial ratio method. The results from the equally weighted pattern analysis method applied to the financial ratio procedures within this study are presented in Table 10. Except for the Inventory Turnover ratio, the pattern analysis method provides improved performance in detecting both the

TABLE 10 Pattern Analysis Results Using Financial Ratio Procedure (this Study) Investigation Signals for:

Less than Material Fictitious Material Purchase on Material Error Seeded Sales on Account Error Seeded Accounts not Recorded Error Seeded Type I Type II Type II

Neither error source

0.282

0.436

Fictitious sales on account error source

0.096

Purchase on account not recorded error source

0.154

Both sources of error

0.115

Combined error rates to detect presence of material error

0.365

0°647

0.801

Combined error rates to detect source of material error

0.365

0°673

0.929

0.128 0,026

526

J. R. Coakley TABLE 11 Pattern Analysis Results from ANN Procedure

Investigation Signals for:

Less than Material Fictitious Material Purchase on Material Error Seeded Sales on Account Error Seeded Accounts not Recorded Error Seeded Type I Type II Type II

Neither error source

0.103

0.128

Fictitious sales on account error source

0.103

Purchase on account not recorded error source

0.103

0.077

Both sources of error

0.077

0.026

0.103

Combined error rates to detect presence of material error

0.283

0.386

0.411

Combined error rates to detect source of material error

0.283

0.489

0.796

presence and the source of material monetary errors from the Fictitious Sales on Account transactions. Results for the Purchase on Account Not Recorded error source are mixed. The results from the pattern analysis method based on the A N N procedure are presented in Table 11, The combined error rates to detect the presence of an error are 0.386 for the Fictitious Sales on Account transactions and 0.411 for the Purchase on Account Not Recorded transactions. The combined error rates to detect the source of an error are 0.489 for the Fictitious Sales on Account transactions and 0.796 for the Purchase on Account Not Recorded transactions. The A N N pattern analysis procedure is noticeably more effective in detecting errors resulting from either source than the independent financial ratio analytical procedures and the equally weighted pattern analysis methods. Note that the majority of the Type II errors from the Purchase on Accounts Not Recorded transactions are false signals to investigate the Fictitious Sales transactions. The combination of the first four financial ratios (Receivables Turnover, Inventory Turnover, Cost o f Sales and Accruals) have a very limited ability to discriminate between the two sources of error (refer to Table 3). Whenever significant fluctuations are detected in both the Inventory Turnover and Accruals ratios, the cause could be attributed to either, or both, of the error sources. When the error source is Purchases on Account Not Recorded, the Debt to Equity ratio becomes the discriminating analytical procedure. If the fluctuation in this ratio is not significant, then the Fictitious Sales error source may be falsely identified as the cause of the fluctuations. Results from the financial ratio analysis suggest that the Debt to Equity ratio is not very sensitive to material fluctuations (indicated by the high Type II error rate). Alternative financial ratios that are more

0.282

sensitive to material fluctuations from these error sources may further increase the effectiveness of the pattern analysis procedure. 7. S U M M A R Y AND I M P L I C A T I O N S FOR FUTURE RESEARCH This study continued the development of a methodology that can be applied to analyze the complex patterns of related fluctuations across numerous financial accounts and identify the presence and plausible source of a material monetary error in the accounts. One pattern analysis method combined investigation signals from five commonly applied financial ratios, and compared these signals with the expected combination of signals that would occur if either of two specified types of error were present. The comparison is based on an analysis that predicts the direction of the change in the financial ratios if a monetary error from the specified financial transaction (source of error) is present in the accounts. The pattern analysis method was evaluated using four years of audited financial data from a medium-sized distributor. These results were compared with a pattern analysis method presented in Kinney (1987). The general conclusion from these two studies is that the equallyweighted pattern analysis method does not provide an overall improvement in performance compared to applying these same financial ratios independently as analytical procedures. An alternative pattern analysis method was developed which used an A N N model to weight the investigation signals from financial ratios based on their information content. The A N N method evaluated the information content of the combinations of financial ratios using an entropy cost function derived from information theory. The pattern analysis method based on weights derived

Attention-Directing Analytical Procedures

from an A N N provided a more reliable indication of the presence of material errors, as well as providing insight into the plausible sources of the error. Results suggest that the use of pattern analysis methods as a supplement to traditional analytical procedures will offer improved performance in recognizing material misstatements within the financial accounts. As a preliminary study on the use of A N N in auditing, this study was limited to the analysis of a single firm over a 48 month period under ideal conditions. It was a single, disaggregated operation, there were no changes in the accounting method during the four year test period, there were no unusual circumstances during the four year test period, the monthly data was reasonable stable (with the exception of Inventory), and the monthly balances were presumed to be error free. Although performance improvements were obvious, the overall performance of the analytical techniques continues to be disappointing. This limited analysis needs to be extended in the following manner: • Expand scope to include multiple firms. • Modify method to use quarterly financial data that is readily available from existing sources (Compustat, 10-K reports, etc). • Expand sources of error. The analytical procedure should include error sources that are frequently encountered within a specified industry. Coakley and Loebbecke (1982) provide an assessment of the likelihood of material errors for medium-sized manufacturing firms. Additional research is needed to identify like sources of material errors for other industries. Alternative neural network architecture may further enhance the effectiveness of the pattern analysis procedure. • For a set of likely error sources, a set of financial ratios must be developed that are sensitive to material fluctuations and can discriminate between the alternative sources of error. Really meaningful breakthroughs may not occur until a set of financial ratios with significantly better discriminatory power are developed.

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