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The Cost of Fraud Prediction Errors

Messod D. Beneish

Indiana University

Kelley School of Business

Patrick Vorst

Maastricht University

School of Business and Economics

September 2019

Abstract

We estimate the costs of fraud prediction errors from the perspective of auditors, investors, and regulators

using an extensive sample of 140,233 observations (of which 779 are fraud firm-years) over the period 1982-

2016. We propose total costs of misclassification, which incorporates the costs of both false negatives (missed

detections) and false positives (incorrectly identifying non-fraud firms), as a solid basis for comparing the

economic value of fraud prediction models and we describe conditions under which these models are useful

screening devices. We estimate auditors’ false negative costs as the sum of litigation costs, reputation costs,

and client losses and find the mean (median) cost to be $14.1 ($1.94) million in 2016 dollars. We estimate

auditors’ false positive costs absent resignation as the unbilled incremental audit investment and conditional

on resignation as the lost audit fee in the year following the resignation and we find the mean (median) cost

to be $0.258 ($0.084) million. Although false negative costs appear to dwarf false positive costs, with

average and median cost ratios of 54.5 and 23.1, these costs ratios are themselves dwarfed by the proportion

of non-fraud to fraud observations in the sample (179:1). Indeed, the fact that all prediction models have

large numbers of false positives, and that even the best models trade false positives for negatives at the rate

of 92:1 explains why auditors would avoid using fraud prediction models in practice. Investors’ false

negatives costs (averaging $446.6 to $845.6 million) are 9 to 40 times larger than their false positive costs,

the latter predominantly consisting of the abnormal return foregone by not investing in firms incorrectly

identified as fraudulent. However, we show that as the number of false positives increases, investors’ use

of fraud prediction models becomes a value-destroying proposition. For example, at a cut-off point of 1.0,

the F-Score has a false positive rate of nearly 40% and results in a net loss to investors of over $50 billion.

We estimate regulators’ false negative costs on average to range from $5.4 to $20.1 million, but we cannot

estimate their false positive costs unless we assume that regulators systematically announce public

investigations of flagged firms, in which case these costs become prohibitive. Overall, we document

considerable variation in the cost of misclassification, both across models and across user groups.

Furthermore, our evidence suggests that future researchers should focus on lowering the false positive rate of

their models, because increasing the true positive rate appears like a futile pursuit.

JEL classification: G31; G32; G34; M40

Keywords: Financial statement fraud, false positive, false negative, cost of errors

We thank Colleen Honisberg for her help in implementing the litigation and settlement prediction models in

Honisberg, Rajgopal, and Srinivasan (2018). We also thank Dan Amiram, Zahn Bozanic, and Ethan Rouen for giving

us access to the code to compute the FSD measure.

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1. Introduction

Costs are central to many predictions in accounting, auditing, and financial economics as economic

agents trade off costs and benefits in making decisions.1 Yet, despite their important role, empirical

documentation of the magnitude of many costs has been scarce. This scarcity is even more acute

in the context of fraud prediction models because such models have a cost multiplier effect: the

costs vary by decision-maker, and predictions generate two types of error costs, one of which has

thus far not been studied. That is, while we have evidence of the cost of failing to detect an instance

of fraud (e.g., a false negative), the cost of incorrectly classifying a firm as a fraud (e.g., a false

positive) has not been investigated before. Consequently, the rates at which auditors, investors,

and regulators trade off the costs of Type I and II errors has remained in the domain of assumptions.

This paper attempts to fill this gap by investigating two questions related to the costs of fraud

prediction errors: (1) what is the nature and magnitude of such costs from the perspective of

auditors, investors, and regulators?; and (2) how do existing prediction models perform when the

comparisons take into account the cost such errors for the various decision-makers? Although we

cannot observe their objective functions, we conjecture that auditors seek to maximize their profits,

that investors seek to maximize their wealth, and that regulators (e.g., SEC, PCAOB) seek to

minimize wealth losses related to asymmetric and/or misleading information (e.g., incomplete or

inaccurate reports and disclosures). Our investigation of costs assumes that decision-makers

commit to taking actions consistent with the model’s predictions.

1 Among others, researchers invoke agency, disclosure, information, litigation, monitoring, political, proprietary,

regulatory, and reputation costs to develop hypotheses about the incentives and behavior of economic agents in various

contexts. Indeed, the three papers that first developed many of these costs concepts (Stigler 1971; Alchian and Demsetz

1972; Jensen and Meckling 1976) have been cited over 100,000 times in aggregate, according to a July 2019 Google

Scholar search.

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The motivation for our analysis is twofold. First, model predictions are based on

classification cut-offs, which depend on the relative magnitude of the costs of false negatives and

the costs of false positives. Although most studies assume that false negatives are more costly than

false positives, there is considerable disagreement as to how much more costly they are, and to

whom. Indeed, prior studies propose decision rules based on assumed cost ratios of false negatives

to false positives as disparate as 20:1 and 142:1, which makes choosing among a model’s cut-offs

and comparing models difficult (e.g., Beneish 1997; Beneish 1999; Cecchini et al. 2010; Dechow

et al. 2011; Perols et al. 2017).

Second, our analysis is of interest because we propose a cost-based method for comparing

the performance of fraud classifiers that addresses the deficiencies in two methods that have been

frequently used in previous literature to compare model performance (area under the curve,

hereafter AUC, and expected costs of misclassification, hereafter ECM). To explain, although the

measure based on AUC used in Cecchini et al. (2010) and Perols et al. (2017) is not threshold

dependent, it treats false positives and false negatives as equally costly (e.g., Adams and Hand

1999, Lobo et al. 2008). This leads to inaccurate conclusions about the relative performance of

extant fraud prediction models because, for all decision-makers, we systematically reject the null

hypothesis that false positives and false negatives are equally costly. In addition, when researchers

use expected costs of misclassification (ECM) to assess model performance, they assume that all

classification errors of a given type are equally costly. However, we document that there is

considerable variation in the costs of fraud across firms. Indeed, this is the case even among the

top 20 instances of fraud cases in Beneish, Lee, and Nichols (2013). For example, the loss on the

market—a component of investors’ cost of missed detection—is 15 to 20 times larger for frauds

at Enron and Waste Management compared to frauds at Sunbeam and Vivendi.

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Our results are based on a sample of 140,233 firm-years, of which 779 firm-years represent

firms that are subject to SEC enforcement actions (hereafter fraud firms) and 139,454 non-fraud

firm-years. The 779 fraud firm-years consist of 313 unique non-financial firms that were charged

with violations other than violations of the Foreign Corrupt Practices Act and for which data are

available on CRSP and COMPUSTAT. We use this sample to evaluate the ability of five models

to predict fraud over the period 1982-2016: the M-Score (Beneish 1999), an accrual model based

on Kothari et al. (2005), the unexplained audit fee model (Hribar et al. 2014), the F-Score (Dechow

et al. 2011), and the measure of financial statement divergence based on how the distribution of

first digits differs from Benford’s Law (Amiram et al. 2015).2

We estimate that auditors’ costs of false negatives amount to $4.382 billion in aggregate (all

amounts are in 2016 dollars) for the 313 instances of fraud in our sample. We rely on prior research

to estimate three components of auditors’ costs of false negatives: litigation costs, reputation costs,

and the foregone profits associated with any unusual client losses following the public revelation

of undetected accounting fraud at one of the auditor’s clients.

We find that the auditors of 87 fraud firms were sued (27.8% of the sample). Of those 87

lawsuits, 23 (26.4%) were dismissed, and a further 15 (17.2%) were settled for zero damages,

leaving 49 cases with an aggregate settlement of $3.128 billion, a finding that is in line with

Honisberg, Rajgopal, and Srinivasan (2018). In terms of reputational losses, we follow Gleason et

al. (2008) and Weber et al. (2008) and estimate reputation cost as the price reaction around fraud

revelation for the other clients of the auditor in the same industry (i.e., the contagion effect). We

find that they represent $102.7 million in aggregate. Finally, we follow Lyon and Maher (2005)

2 Our comparison focuses on studies for which model parameters, data, and decision rules are readily available. Thus,

we do not compare models that rely on textual or tonal analyses such as Brown et al. (2018) who propose a model that

combines financial data with textual information, or models that use data mining techniques such as Cecchini et al.

(2010) who propose a 23-variable financial kernel based on support vector machines.

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and estimate the profits foregone due to abnormal client losses in the two years following the

revelation of the fraud and find that they represent in aggregate a cost of $1.151 billion.3

In terms of auditors’ false positives costs, our cost estimates are based on the actions we

expect auditors to take when prompted by a flag from a fraud prediction model. Specifically, we

expect that with 97% probability, the auditor would bear the cost of increasing its audit investment

and with 3% probability, the auditor would lose the annual audit fee by resigning from the audit

engagement. Indeed, previous literature has shown that auditors increase their audit investment

when they perceive a higher risk of misstatement (e.g., Glover et al. 2003; Hammersley et al. 2011;

Boritz et al. 2015) and has documented that risk factors such as a control weakness or the payment

of bribes result in higher fees (Hogan and Wilkins 2008; Lyon and Maher 2005; Munsif et al.

2011). Based on this research and on our analysis of audit fees in periods subsequent to remediation

of an internal control weakness, we use an estimate of 23% more audit work (with a range of 20%

to 28%) and calculate that the auditor bears average (median) costs of $0.258 ($0.084) million for

each false positive.

Collectively, our findings suggest that auditors’ false negative costs dwarf their false positive

costs, as the average cost ratios range from 26.8 to 54.5, and the median cost ratios range from 2.0

to 23.1.4 However, these costs ratios are themselves dwarfed by the proportion of non-fraud to

fraud observations, which we calculate as 179 to 1 (139,454/779). This could explain why auditors,

3 We conduct sensitivity analyses on client losses and litigation costs. On client losses, we vary the length of the period

over which we measure client loss (e.g., +1 v. +2 years), and in terms of litigations costs, we estimate an expected

settlement amount from the model proposed by Honisberg, et al. (2018). We find that these costs remain significant

but their magnitudes are lower ($828.2 million for client losses and $902.7 million for estimated litigation costs). 4 These ratios are likely overstated because we are unable to measure the effect of an increase in workload (23% for

all false positives) on either the productivity of the audit staff, or the quality of the audits. This increased workload

could create significant time pressure on all audits because prediction models generate a large number of false

positives, and, in the limit, could make it impossible for audit firms to make the extra audit investment necessary to

investigate all of the model’s alerts.

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on the advice of general counsel apprehensive about the risk of discovery of fraud alerts in auditor

working papers, would avoid the use of fraud prediction models with high false positive rates.

For investors, we estimate the costs false negatives as the abnormal market value loss from

day -1 relative to the fraud-revelation announcement to periods ending as early as day +1 and as

late as day +252 (12 months). Using market-adjusted returns, the average loss ranges from $446.6

million (over 3 days) to $845.6 million (over 12 months) and the corresponding medians are $20.6

million and $49.09 million. We estimate investors’ false positive costs as the sum of two

components: (1) the incremental audit fee which we assume is billed by the incumbent or required

by the newly hired auditor, and (2) the profit foregone (or minus the loss avoided) by not investing

over the next year in firms flagged by the prediction model as potentially fraudulent. The average

ratio of false negative to false positive costs ranges from 21.4 to 40.52 using market-adjusted

returns and from 4.73 to 8.95 when using returns that are risk-adjusted using the Fama-French

four-factor model.

For regulators, we argue that the cost of false negatives is the cost of losing investors’ trust

in the institution, which we estimate as the market-wide dollar abnormal returns in three periods

centered on the fraud-revelation announcement (day 0, days -1 to +1, and days -2, +2). On average,

we estimate that regulators false negative costs range between $5.4 and $20.1 million. In terms of

false positives, regulators’ costs depend on whether they internally or publicly investigate firms

flagged by the model. We cannot estimate regulators’ costs of internal investigations, which we

surmise include work overload and the investment of scarce resources in the pursuit of false targets.

However, if regulators’ make the investigations public, the costs are very large under the

assumption that falsely identified firms experience a market value loss in the range of 3% to 10%,

which is the typical market reaction to comment letters and revelations of SEC investigations and

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charges. In that case, the average ratio of false negative to false positive costs is systematically less

than one, thereby limiting the usefulness of fraud prediction models for regulators.

To evaluate the relative performance of the different models, we assess the true and false

positive performance by decile over the period 1982-2016 and find that the F-Score ranks first

with a top quintile (top four deciles) success rate of 47.5% (72.3%). This is followed by the M-

Score with a success rate of 40.4% (67.5%), and by current accruals with a success rate of 37.2%

(57.9%). Our analysis also reveals that all models have systematically large percentages of false

positives: for example, the false positive rate ranges from 19.94% to 19.99% if the top quintile of

the various models is used to classify firms as frauds and it ranges from 39.93% to 39.99% if firms

in the top four deciles are classified as frauds. In effect, the fact that all prediction models have

large numbers of false positives, and that even the best models trade false positives and false

negatives at a rate of approximately 92:1, explains why auditors avoid using fraud prediction

models in practice. As the number of false positives increases, also investors’ use of fraud

prediction models becomes a value-destroying proposition. For example, at a cut-off point of 1.0,

the F-Score has a false positive rate of nearly 40% and results in a net loss to investors of over $50

billion.

We further show that there is considerable variation in the cost of false negatives (an

untenable assumption of ECM analyses) and that the cost of false negatives and false positives

differs both within and across decision-makers (the former being an untenable assumption of most

AUC analyses), as well as across models. We propose total costs of misclassification as a basis for

comparing the economic value of fraud prediction models and describe conditions under which

these models are useful screening devices. In particular, our evidence suggests that researchers

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should focus on lowering the false positive rate of their models, because increasing the true positive

rate appears like a futile pursuit.

The remainder of the paper is structured as follows: Section 2 describes the fraud prediction

models we consider, Section 3 presents the sample and data, Section 4 describes and illustrates the

process by which we estimate costs for auditors, investors, and regulators, Section 5 compares the

predictive ability of various models, Section 6 discussed additional analyses, and Section 6

concludes.

2. Fraud Prediction Models

We evaluate the ability of six models to identify firms that are subsequently subject to SEC

accounting and enforcement actions: measures of raw accruals and abnormal accruals, the M-Score

(Beneish 1999), the unexplained audit fee model (Hribar et al. 2014), the F-Score (Dechow et al.

2011), and a measure of financial statement divergence based on how the distribution of first digits

differs from Benford’s Law (Amiram et al. 2015).

2.1 Accruals and Performance-Matched Accruals

We report three measures of accruals in our analyses. Total accruals, calculated from the statement

of cash flows as the difference between IBC and OANCF deflated by lagged assets, current

accruals calculated as total accruals plus depreciation deflated by lagged assets, and performance-

matched total abnormal accruals, estimated based on the Jones (1991) model as implemented by

Kothari et al. (2005). In addition to these three measures, we obtain qualitatively similar results

when we consider alternatives measures of raw accruals (drawn from Sloan (1996) and Hribar and

Collins (2002)) and alternative measures of abnormal accruals based on Dechow et al. (1995),

Beneish (1998), and Dechow and Dichev (2002).

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2.2 The Beneish M-Score

Beneish (1997; 1999) profiles firms that manipulate earnings (firms either charged with

manipulation by the SEC or that admit to manipulation in the public press) and develops a

statistical model to discriminate manipulators from non-manipulators. In this paper, we use the

unweighted probit model presented in Beneish (1999) which relies exclusively on financial

statement data and whose usefulness in assessing fraud potential out-of-sample has been shown by

academics and professionals.5 The M-Score below classifies a firm as a potential manipulator if

the M-Score exceeds -1.78, assuming that false negatives are 20 times more costly than false

positives (variables are defined in the Appendix):

M-SCORE = – 4.840 + 0.920DSRIit + 0.528GMIit + 0.404AQIit + 0.892SGIit

+0.115DEPIit – 0.172SGAIit + 4.679TATAit – 0.327LGVIit (1)

2.3 The Hribar et al. (2014) unexplained audit fee model

Hribar et al. (2014) draw on prior audit research to argue that audit fees reflect the

expected costs of misreporting (e.g., litigation, reputation) and that the portion of the audit fee

that is not explained by known determinants reflects in part an auditor’s expectation of the cost

of misreporting. They propose the residual from the following equation as a measure of

unexplained audit fees:

LogFeest = Industry Indicators + β1BigNt + β2log(Assetst) + β3Inventoryt/Avg Assetst

+β4Receivablest/Avg Assetst + β5LTDt/Avg Assetst + β6Earnt/Avg Assetst

+β7Losst + β8Qualifiedt + β9Auditor Tenuret + ɛt (2)

5 Since the publication of the original study (which used data from 1982 to February 1993), the model has been

featured in financial statement analysis textbooks (e.g., Fridson and Alvarez 2011) and in articles directed at auditors,

certified fraud examiners, and investment professionals (e.g., Ciesielski 1998, Merrill Lynch 2000, Wells 2001, DKW

2003, Harrington 2005). The model gained notoriety when a group of MBA students at Cornell University posted the

earliest warning about Enron’s accounting manipulation score using the Beneish (1999) model a full year before the

first professional analyst reports (Morris 2009). This episode in American financial history is preserved in the Enron

exhibit at Museum of American Finance, New York (www.moaf.org) and is also recounted in Gladwell (2009).

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In this model, higher unexplained audit fees are associated with poorer accounting quality.

However, Hribar et al. (2014) do not directly study classification in a sample of fraud cases (and

thus do not provide a threshold with which to classify firms). Consequently, we sort on

unexplained audit fees and evaluate the model’s detective performance by focusing on the top

quintile.

2.4 The Dechow et al. (2011) F-Score

Dechow et al. (2011) follow a methodology similar to Beneish (1997; 1999) in developing a score

to detect accounting fraud and estimate three alternative models that differ in whether the models

include returns and/or non-financial statement data (e.g., number of employees, security issuance).

The results are similar across models, and we use the following version of F-Score (variables

defined in the Appendix):

F-SCORE = - 6.789 + 0.817RSST + 3.230∆REC + 2.436∆INV + 0.122∆Cash Sales

-0.992∆Earnings + 0.972ACT Issuance (3)

Dechow et al. (2011) suggest three potential cut-offs for classifying firms as frauds depending on

whether the F-score exceeds either 1.0, 1.85, and 2.45, which correspond to assumed costs ratios

of false negatives to false positives of 143:1, 86:1, and 82:1, respectively.

2.5 The Amiram et al. (2015) FSD Score

Amiram et al. (2015) construct an FSD Score by comparing the distribution of first digits in over

100 financial statement items relative to Benford’s law, as research in different disciplines has

used deviations from Benford’s distribution to detect errors or manipulation in data. The FSD

Score is based on the mean absolute deviation statistic (MAD) for the financial items considered

and is calculated as MAD = (∑|AD-ED|)/K, where AD (ED) is the actual (expected) proportion of

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leading digits and K is the number of leading digits being analyzed.6 Amiram et al. (2015) suggest

that the FSD score is a useful tool for detecting financial statement errors, as they document that

lagged FSD scores positively correlate with material misstatements while contemporaneous FSD

scores are negatively correlated with material misstatements. Although they suggest that the FSD

score can be used as a leading indicator of material misstatements, they do not report an FSD score

threshold with which to classify firms. As a result, we sort on FSD score and evaluate the model’s

detective performance by focusing on the top quintile.

3. Data

3.1 Sample

Our evaluation of the performance of extant fraud classifiers is based on a comprehensive sample

drawn from Accounting and Auditing Enforcement Actions (AAER) by the Securities and

Exchange Commission (SEC) between April 1982 and July 2016 (AAER#1 to AAER #3793). We

identify accounting enforcement actions against 574 firms after eliminating multiple and

unassigned AAERs (2351), those related to financial institutions (319), auditing actions against

independent CPAs (280), enforcement actions for the payment of bribes under the Foreign Corrupt

Practices Act (112), and those related to related to violations in 10-Qs resolved within the fiscal

year (131).

Table 1 reports the selection of the final sample. The main sample consists of 574 fraud cases

over the period 1982-2016, of which we are able to match 492 cases to the Compustat-CRSP

merged database.7 Those 492 cases relate to 1,185 firm-years with misstated financial statements.

6 We are thankful to Dan Amiram, Zahn Bozanic, and Ethan Rouen for giving us access to the code to extract leading

digits and to compute the FSD Score. 7 Many of those cases can be matched to Compustat, but do not have CRSP return data available to calculate the stock

market reaction to the revelation of the fraud. As we use the price reaction as one of the key measure of fraud costliness,

we drop these observations from the sample and continue with firms that are matched to both Compustat and CRSP.

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We then drop firms with missing returns around the revelation date of the fraud, for example,

because the firm delisted prior to the date on which the fraud is revealed, leaving a sample of 413

fraud cases (1,041 firm-years). As we are interested in determining the usefulness of fraud

prediction models, we further drop observations for which we do not have data to compute the

fraud prediction models we compare. Overall, the sample restrictions lead to a final sample of 313

fraud cases involving 779 misstatement years.

3.2 Backfilling Missing Audit Fees

As we later discuss, audit fees are important to estimate the costs of fraud prediction errors from

an auditors’ perspective. Given that Audit Analytics reports audit fee data from fiscal year 2000

onwards, we estimate an audit fee model in order to backfill audit fees for the earlier years in our

sample. The regression specification we estimate follows Hribar et al. (2014) and uses a wide set

of audit fee determinants, capturing the size and complexity of the client and the audit engagement.

We estimate the regression over the period 2000-2001 as those years precede the enactment of the

Sarbanes Oxley Act, which likely makes them more relevant to predicting audit fees in the pre-

2000 period.

The results of the estimation, which we report in Table 2, Panel A, are in line with

expectations and closely resemble those in Hribar et al. (2014). That is, BigN auditors are able to

charge a fee premium and bigger and more complex clients; those with a higher asset base, a

greater number of business segments, or more foreign sales, command higher audit fees. In

contrast, less risky firms with higher current ratios, lower market-to-book ratios, and better

performance (higher ROA, no losses) pay lower audit fees. Finally, firms that receive a modified

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audit opinion also pay higher fees. We use these coefficient estimates in conjunction with actual

data in Compustat to predict and backfill (missing) audit fees in those earlier years.8

Although the regression results are in line with expectations and the 75.9% explanatory

power suggests that we can obtain reasonable predictions of audit fees, we conduct one further test

to assess the reliability of our audit fee prediction model. Specifically, we compare predicted fees

over the period 2002-2017 with actual fees as reported in Audit Analytics over the same period.

Although ultimately we are interested in predicting fees over the pre-2000 period for which no

Audit Analytics data are available, we can use the period 2002-2017 to estimate the out-of-sample

accuracy of our fee prediction model. The Pearson (Spearman) correlation between (the natural

logarithm of) estimated fees and actual fees is 0.87 (0.86), suggesting that our model performs

well in predicting the (relative) magnitude of audit fees.9

4. The Cost of Classification Errors

We estimate the cost of classification errors for three decision-makers that have incentives to rely

on screening models to detect fraud: auditors, investors, and regulators. The costs that we are

interested in estimating are the costs of false negatives, e.g., the costs of missed fraud detections

or actual frauds not flagged by a model, and the costs of false positives, e.g., the costs of incorrectly

flagging a non-fraud firm as a fraud firm. These costs differ across decision-makers and although

we cannot observe their objective functions, we conjecture that their overarching objectives are as

8 We rely on Compustat data only, including for measuring BigN and OPIN, as we need Compustat data to fill in

missing audit fees over the period for which Audit Analytics was not yet available. 9 However, we find that our estimate of audit fees exhibits a negative bias over the period 2002-2017 (the natural

logarithm of actual fees over the period 2002-2017 is 13.66, while predicted fees are on average 0.87 lower at 12.79).

This is not surprising as prior research has shown a significant increase in average fees following SOX (Ettredge,

Scholz, and Li 2007; Ghosh and Pawlewicz 2009). Although these results suggest that we underestimate audit fees

over the later part of our sample, these results do not necessarily translate to the pre-2000 period, which is also pre-

SOX.

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follows: auditors seek to maximize their profits, investors seek to maximize their wealth, and

regulators seek to minimize wealth losses relating to asymmetric and misleading information (e.g.,

incomplete or inaccurate disclosures and reports). Figure 1 summarizes the major components of

misclassification costs that we estimate for auditors, investors, and regulators.

4.1 Auditors’ Costs of False Positives

To the extent that auditors view a flag from a fraud screening model as an indication of fraud risk,

they have two possible courses of action. First, auditors can increase the nature and extent of the

audit investment to ascertain and report on whether fraud has indeed occurred. Second, they can

resign from the client and forego their typical profit margin on the lost audit fee. Thus, for any

given model, we can estimate auditor i’s expected false positive costs (EAUD [FP COST]) for a

given client j that is falsely flagged in year t as:

EAUDi[FP COST] = p (AUDFEEjt * pm%) + (1-p) (INCRAUDFEEjt * (1-pm%)) (4)

Where p is the probability of resignation, and resignation results in the auditor losing its usual

profit margin (pm%) on the audit fee billed to client j in year t (AUDFEEjt). In the absence of

resignation, the auditor undertakes additional work costing INCRAUDFEEjt * (1-pm%), which we

assume that they cannot pass on to the client in year t. Equation (4) does not take into account the

potential negative effects of time pressure on the quality of the auditor’s output (e.g., Bills et al.

2016). Time pressure could increase because of the additional audit investment and auditors’

limited ability to increase hiring and training in the short run.

4.1.1 Incremental Audit Fees

At least since the early 20th Century, auditors have been trained to increase the amount of audit

work they undertake when they perceive that their client’s financial statements have a higher

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likelihood of containing material misstatements (e.g., see Montgomery 1916). There is extensive

evidence in audit research that auditors respond to a high risk of fraud primarily by changing the

nature and/or extent of audit procedures, increasing the overall audit investment, and charging

higher fees for prospective clients with an element of fraud risk (see Hammersley et al. 2011 for a

recent review of this literature).

We argue that auditors who observe a flag from a fraud-screening model for one of their

clients, undertake an extra investment as part of the audit to ascertain the risk of material

misstatement, and thus reduce litigation and discovery risk.10 If auditors can bill for the additional

work, they bear no direct financial costs from false positives. In that case, the costs of the additional

audit effort are borne by investors as residual claimants. Alternatively, if clients do not accept to

pay for the extra audit effort, the cost is borne by auditors in the first year in which a firm is flagged.

We draw on prior experimental audit judgment research and on empirical audit fee research

to estimate the incremental audit costs that are borne either by the auditor or by investors. For

example, after the issuance of SAS 82 in early 1997, data in Table 3, page 245 of Glover et al.

(2003) suggest that auditors budget 21.8% more audit hours in high fraud risk engagements. More

recently, experimental evidence in Boritz et al. (2015) indicates that auditors increase audit budgets

by 20.4% in the presence of fraud risks. Because it is not possible to directly observe what auditors

would do in terms of audit effort when they perceive an increased likelihood of fraud, we follow

10 Discovery is the risk of having audit documentation contain a record of a fraud alert for a continuing client that is

subsequently found to have committed fraud. Plaintiffs’ attorneys can then argue on discovery that auditors had

advance knowledge of the fraud and were grossly negligent in conducting their audit. This risk likely seemed

significant as Arthur Andersen undertook to shred paper documents and destroy emails related to Enron, which led to

its indictment for obstruction of justice and its subsequent demise. On January 25, 2002, the Wall Street Journal

reported that in recovering deleted e-mails at Arthur Andersen, Congress found evidence that the Chicago office of

Arthur Andersen had issued two “alerts” to the Houston office in the spring of 2001 concerning earnings manipulation

at Enron. The alerts came from a tailored version of the M-Score model that Beneish had estimated under a consulting

relationship with Andersen. (“Andersen Knew of `Fraud' Risk at Enron --- October E-Mail Shows Firm Anticipated

Problems Before Company's Fall”, 01/25/2002, A3).

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prior work on audit fees and view the payment of bribes to foreign government officials and the

existence of internal control weaknesses as fraud risk factors that would lead auditors to increase

their audit investment. The evidence in Lyon and Maher (2005) suggests that audit fees are higher

by 43% for firms reporting paying bribes to foreign government officials. Lyon and Maher

conclude that when auditors assess higher business risks, they pass those costs to their clients.

In terms of empirical work on internal control weaknesses (ICW) on audit fees, we follow

Hogan and Wilkins (2008), Raghunandan and Rama (2006), and Munsif et al. (2011) to estimate

the behavior of audit fees before revelation of an ICW and after the revelation of the ICW,

conditional on whether the latter was remediated or not. Specifically, we augment the audit fee

model by incorporating indicators capturing the existence of an ICW and its remediation status.

N1Y_ICW is an indicator variable that is equal to one if the firm reports an internal control

weakness in the following fiscal year, and zero otherwise. To investigate the impact of

remediation, we include L1Y_ICW_RMD (L1Y_ICW_NRMD), which is an indicator variable that

is equal to one if the firm reported an internal control weakness two years prior to the current fiscal

year that was (was not) remediated in the year prior to the current fiscal year, and zero otherwise.

L1Y_ICW_NEW is an indicator variable that is equal to one if the firm reported a new internal

control weakness in the year prior to the current fiscal year. ICW_RMD, ICW_NRMD, and

ICW_NEW are defined analogously, but are shifted one year ahead.11 We further add the level of

the SEC enforcement budget as a measure capturing the likelihood that fraud, if committed and

going undetected by the auditor, is subsequently uncovered by the SEC. We use two variables

capturing enforcement intensiveness; (1) the natural logarithm of the SEC budget in constant

11 ICW_RMD (ICW_NRMD) is an indicator variable that is equal to one if the firm reported an internal control

weakness in the year prior to the current fiscal year that was (was not) remediated in the current fiscal year, and zero

otherwise. ICW_NEW is an indicator variable that is equal to one if the firm reports a new internal control weakness

in the current fiscal year.

17

dollars, and (2) the SEC budget divided by the number of firms in Compustat. The second measure

takes into account that the SEC budget has to be divided over a constantly changing number of

firms and measures the resources of the SEC relative to the number of firms they have to monitor.

The results are reported in Table 3. We estimate that firms with an ICW pay 18% to 24%

higher fees in the year prior to the reporting of an ICW, which is slightly less than in Hogan and

Wilkins (2008) who report 41% higher audit fees. In the year in which the ICW is first reported,

we find 31% higher fees (ICW_NEW, columns 10-12), while non-remediated internal control

weaknesses lead to 59% higher fees (ICW_NRMD, columns 10-12). Interestingly, from the

coefficient on ICW_RMD (L1Y_ICW_RMD) we find that even in the year of (following)

remediation, fees are 34% (24%-27%) higher. All results are similar after including proxies for

SEC monitoring. Moreover, the positive and significant coefficients on L1Y_LN_SECBUD and

L1Y_SC_SECBUD in all specifications, indicate that increased SEC enforcement activity, and thus

higher fraud detection risk, is associated with higher audit fees. Collectively this evidence suggests

that if auditors increase their investment in the audit after observing a flag from a fraud screening

model, investors bear a cost that ranges from 20% to 30% of lagged audit fees. Alternatively, if

auditors cannot pass on those costs to their clients, they bear costs that are equal to those amounts

multiplied by 1 less their typical profit margin.

In Table 4, Panel A we present statistics on audit fees and on the auditors’ expected false

positive costs for 139,455 non-fraud firm-years in the sample. In 2016 dollars, the average and

median audit fees over the period 1980-2016 are equal to $1.282 and $0.415 million, and the

average (median) false positive costs is $0.259 ($0.084) million. We compute these estimates using

audit fees drawn from Audit Analytics for the period 2000-2016, and audit fees estimated using

the backfilling model we describe in Table 2 for observations in the period 1980-1999. In addition,

18

we assume that the likelihood of resignation is 3%, that incremental fees amount to 23% [range

20-28%] of the contemporaneous audit fee, and, relying on recent reports by the BIG4 in the U.K.,

that audit firms’ profit margins equal 23% [range 20-30%].

4.2 Auditors Costs of False Negatives

4.2.1 Auditor Litigation Costs

Prior research has long recognized that the costs of potential litigation influence audit pricing and

the design of financial reporting (e.g., Simunic 1980, Simunic and Stein 1996, Palmrose 1998, Lys

and Watts 1994). We undertake a comprehensive search of litigation against firms, investment

bankers, and auditors for the 313 fraud firms in the sample. We match firms to litigation cases by

reading lawsuits and resolution notices on Audit Analytics for the post-1999 period and we rely

on Westlaw, Lexis-Nexis, and Factiva searches in the pre-2000 period.12

We tabulate the results of our analysis in Table 4, Panel B. There are lawsuits against 212

fraud firms (67.7%), of which 32 (15.1%) are dismissed and 14 (6.6%) are settled with no damages.

In aggregate, fraud firms have paid $25.9 billion to settle the remaining 166 suits. Investment

bankers paid $20.8 billion to settle 16 out of the 19 lawsuits filed against them in connection with

fraud firms. We also find that the auditors of 87 fraud firms were sued (27.8% of the sample). Of

those 87 lawsuits, 23 (26.4%) were dismissed, and a further 15 (17.2%) were settled for zero

damages, leaving 49 cases with an aggregate settlement of $3.128 billion. Our settlement total for

auditors is in line with that of Honisberg, Rajgopal, and Srinivasan (2018) who identify 540

lawsuits naming auditors over the period 1996-2016 and estimate the aggregate value of auditor

settlement payments at $3.53 billion. As we later report, we also estimate lawsuit settlements

12 Our searches begin with the firm name at the time the fraud is discovered, but we also consider name changes as

applicable, and specify a search period beginning two years before and ending two years after the fraud period

identified in AAERs, and use search terms such as “securities”, “class action”, litigation”, “lawsuit” and “settlement.”

19

against auditors using Honisberg, Rajgopal, and Srinivasan (2018) model that explains settlement

amounts against auditors (Ln(1+Settle)):13

Ln(1 + Settle) = −8.174 + 0.750 ∗ 𝐶𝑙𝑎𝑠𝑠_𝑃𝑒𝑟𝑖𝑜𝑑_𝐿𝑒𝑛𝑔𝑡ℎ − 0.189 ∗ 𝑆ℎ𝑎𝑟𝑒_𝑇𝑢𝑟𝑛𝑜𝑣𝑒𝑟 + 0.040 ∗𝑃𝑟𝑖𝑐𝑒_𝐷𝑟𝑜𝑝 + 0.867 ∗ 𝐺𝑟𝑜𝑤𝑡ℎ + 0.448 ∗ 𝐿𝑛(𝐴𝑠𝑠𝑒𝑡𝑠) + 2.265 ∗ 𝑅𝑂𝐴 + 2.519 ∗ 𝐻𝑖𝑔ℎ_𝐿𝑖𝑡𝑖𝑔𝑎𝑡𝑖𝑜𝑛 +1.782 ∗ 𝐸𝑞𝑢𝑖𝑡𝑦_𝐼𝑠𝑠𝑢𝑎𝑛𝑐𝑒 − 4.044 ∗ 𝐼𝑃𝑂 + 3.356 ∗ 𝑅𝑒𝑠𝑡𝑎𝑡𝑒𝑑 − 1.921 ∗ 𝑃𝑜𝑠𝑡_𝑁𝑖𝑛𝑒_𝐸𝑙𝑒𝑣𝑒𝑛 − 5.165 ∗𝑃𝑜𝑠𝑡_𝑇𝑤𝑜_𝑇ℎ𝑟𝑒𝑒 + 2.135 ∗ 𝑁𝑖𝑛𝑒_𝐸𝑙𝑒𝑣𝑒𝑛 + 4.879 ∗ 𝑇𝑤𝑜_𝑇ℎ𝑟𝑒𝑒 + 2.712 ∗ 𝑃𝑜𝑠𝑡_𝑇𝑒𝑙𝑙𝑎𝑏𝑠 (5)

4.2.2 Auditor Reputation Losses

Franz et al. (1998), Gleason et al. (2008), and Weber et al. (2008) provide evidence that

announcements that reveal either litigation against auditors or the existence of financial statement

fraud at a firm are associated with negative spillover effects on other clients of the auditor. For

example, Weber et al. (2008) find that other clients of KPMG Germany suffered abnormal returns

of -3% around the three events revealing the accounting scandal at another KPMG auditee in

Germany (ComROAD AG). Similarly, Gleason et al. (2008) document a contagion effect and

suggest investors re-assess their reliance on the financial statements of non-restating firms in the

same industry.

13 Honisberg et al. document a declining role of Section 10(b) in class actions against auditors, and provide evidence

on how two Supreme Court rulings in 2007 and 2011 altered the likelihood of a successful lawsuit outcome as a

function of the circuit court in which the litigation proceeds. We assume that the auditors of AAER firms are sued

with probability one and estimate litigation costs as the expected settlement amount using coefficients estimates in

Honisberg, et al. In addition, as an alternative estimation of equation (5) Honisberg et al. use 2008-2012 rather than

2002-2012 as an analysis period. Their goal is to provide evidence on the effect on the 2011 Supreme Court ruling in

Janus v. First Derivative, which made it more difficult to bring cases against auditors in circuit courts in district four

and nine. We also consider the model below as an alternative estimate of settlement costs for our post-2007 sample

(All the variables are described in Honigsberg et al.’s Appendix).

𝐿𝑛(𝑆𝑒𝑡𝑡𝑙𝑒 + 1) = 13.280 − 0.699 ∗ 𝐶𝑙𝑎𝑠𝑠_𝑝𝑒𝑟𝑖𝑜𝑑_𝑙𝑒𝑛𝑔𝑡ℎ − 0.148 ∗ 𝑆ℎ𝑎𝑟𝑒_𝑇𝑢𝑟𝑛𝑜𝑣𝑒𝑟 − 0.067 ∗ 𝑃𝑟𝑖𝑐𝑒_𝐷𝑟𝑜𝑝

+ 2.186 ∗ 𝐺𝑟𝑜𝑤𝑡ℎ − 0.267 ∗ 𝐿𝑛(𝐴𝑠𝑠𝑒𝑡𝑠) + 6.665 ∗ 𝑅𝑂𝐴 ∗ 11.089 ∗ 𝐻𝑖𝑔ℎ_𝐿𝑖𝑡𝑖𝑔𝑎𝑡𝑖𝑜𝑛 − 5.609

∗ 𝐸𝑞𝑢𝑖𝑡𝑦_𝐼𝑠𝑠𝑢𝑎𝑛𝑐𝑒 − 6.424 ∗ 𝐼𝑃𝑂 + 1.662 ∗ 𝑅𝑒𝑠𝑡𝑎𝑡𝑒𝑑 − 6.808 ∗ 𝑃𝑜𝑠𝑡_𝐽𝑎𝑛𝑢𝑠_𝐹𝑜𝑢𝑟_𝑁𝑖𝑛𝑒

− 3.743 ∗ 𝐹𝑜𝑢𝑟_𝑁𝑖𝑛𝑒 + 5.371 ∗ 𝑃𝑜𝑠𝑡_𝐽𝑎𝑛𝑢𝑠

20

In Table 4, Panel C, we use the contagion effect to measure auditor reputational losses.

Specifically, when using the average price reaction in the three days surrounding the fraud-

revealing announcement of the other clients of the auditor in the same industry as the fraud firm

(6-digit GICS), we find that the average reputation loss is equal to $0.37 million in 2016 dollars.

Whereas data from COMPUSTAT provide us with the largest sample coverage among AAER

firms (N=274), we also conduct our analyses relying on the Audit Analytics auditor switching file

and find the average reputation loss for the 139 observations with sufficient data is equal to $2.76

million in 2016 dollars.

4.2.3 Auditor Client Losses

We follow Lyon and Maher (2005) and measure auditor client losses as the difference between an

auditor’s rate of attrition at the office level in the year following the discovery of a missed detection

and (i) the rate of attrition for the same ‘undetecting’ office in the year prior to the public revelation

of the fraud, or (ii) the average rate of attrition for all other of the auditor’s offices. We estimate

the cost of client losses by multiplying the abnormal number of clients lost and the average audit

fee per client. Relying on Compustat auditor data, we find that the average loss due to client losses

is equal to $2.65 million for comparisons in the year after the fraud revelation and $3.68 million

for comparisons in the two year-period after revelation. Our estimates of costs are lower when

relying on Audit Analytics, with average losses of $1.38 million for the one-year comparison and

$2.12 million for the two-year comparison.

4.3 Investors’ Costs

We estimate investors’ false positive costs as the sum of two components: (1) the incremental audit

fee which we assume is billed by the incumbent auditor or required by the newly hired auditor,

and (2) the profit foregone (or minus the loss avoided) by not investing over the next year in firms

21

flagged by the prediction model as potentially fraudulent. We report estimates of these costs in

Table 5, Panel A. The average incremental fee for all non-fraud firms amounts to $0.295 million

in 2016 dollars, whereas non-fraud firms’ dollar abnormal returns range from $15.6 million (size-

adjusted abnormal returns) to $94.2 million (Fama-French 4-factor model).

We posit that investors’ cost of missed detections (false negatives) is the investment loss

associated with the discovery of the misreporting. This follows prior research that has documented

that the revelation of restatements and fraud is typically associated with highly adverse abnormal

price reactions. In terms of AAERs, Beneish (1999) and Karpoff et al. (2008) document three-day

losses of -20% and -25%. For firms that announce restatements due to irregularities, the loss

amounts to between -15% and -25% in the three to six month period after the restatement becomes

public (Badertscher, Collins, and Lys 2008; Hennes, Leone, and Miller 2008). In Table 5, Panel B

we report estimates of the costs of missed detection (false negatives) as the abnormal market value

loss from day -1 relative to the fraud-revealing announcement to periods ending as early as day +1

and as late as day +252 (12 months). Using market-adjusted returns we find that the average loss

ranges from $446.6 million (over 3 days) to $845.6 million (over 12 months) and the corresponding

medians are $20.6 million and $49.09 million.

Collectively, the results on the costs of false positives and false negatives to investors suggest

an average ratio of false negative to false positive costs that ranges from 21.4 to 40.52. This implies

that, ceteris paribus, as the number of false positives increases, investors’ use of fraud prediction

models becomes a value-destroying proposition.14

14Although prior studies commonly implement short and/or hedge strategies, the availability of stocks for borrowing

in order to short sell makes the implementation of short selling on a large scale difficult, if not prohibitively costly

(See Beneish, Lee and Nichols 2015). For this reason, we do not consider the possibility of short selling as a means

of potentially reducing investors’ false positive costs.

22

4.4 Regulator’s Costs

In terms of false positives, regulators’ costs depend on whether they internally or publicly

investigate firms flagged by the model. We cannot estimate regulators’ costs of internal

investigations, which we surmise include work overload and the investment of scarce resources in

pursuit of false targets. However, if regulators’ make the investigations public, the costs are very

large under the assumption that falsely identified firms experience a market value loss in the range

3 to 10%, which is the typical market reaction to comment letters and revelations of SEC

investigations and charges. As we report in Table 6, Panel A, these costs are substantial, averaging

between $84.6 and $281.9 million.

We argue that regulators’ cost of false negatives is the cost of losing investors’ trust in the

institution, which we estimate as the market-wide dollar abnormal returns in three periods centered

on the fraud-revealing announcement (day 0, days -1 to +1, and days -2, +2). Specifically, we

estimate a regression of CRSP daily value-weighted market returns on an indicator variable that is

equal to one on the fraud revelation date and zero on days on which there is no fraud revealed. The

results are reported in Table 6, Panel B. Whereas we find significant results only in the regressions

with the one-day event window, the negative and significant coefficient on EVENT indicates that

market-wide returns are on average lower on fraud revelation dates. This result is consistent with

investors being more pessimistic about stocks in general, an effect that we attribute to a loss of

trust in the regulator’s ability to uncover fraud. In terms of economic significance, we estimate

that regulators’ false negative costs range from $5.4 million to $20.1 million. The average ratio of

false negative to false positive costs is systematically less than one, limiting the usefulness of fraud

prediction models if regulators publicly announce investigations based on alerts from the fraud

prediction models.

23

5. Model Comparisons

5.1 Predictive Ability

In Table 7, we assess the performance of fraud predictions based on total and current accruals,

performance-matched abnormal current accruals, M-Score, F-Score, and the FSD measure over

the period 1982-2016.15 Specifically, we create decile ranks of each of the measures and investigate

the true positive and false positive rates in the top deciles. The table reveals that models have

systematically a large percentage of false positives: for example, the false positive rate ranges from

19.94% to 19.99% if the top quintile of the various models is used to classify firms as frauds and

it ranges from 39.93% to 39.99% if firms in the top four deciles are classified as frauds. In terms

of true positive rates, the F-Score ranks first with a top quintile (top four deciles) success rate of

47.5% (72.3%), followed by the M-Score with a success rate of 40.4% (67.5%), and current

accruals with a success rate of 37.2% (57.9%).

We focus our remaining analyses on comparing the M-Score and F-Score, because it is

possible for decision-makers to implement these models by relying on the cut-offs indicated in the

published studies. Figure 2 relates the rates of false positives and false negatives for two M-Score

and three F-Score cut-offs reported by Beneish (1999) and Dechow et al. (2011). With a threshold

of -1.78, the M-Score identifies 35.7% of frauds with a false positive rate of 16.7%, while with a

threshold of -2.00, the M-Score identifies 43.9% of the frauds, but the false positive rate increases

to 21.8%.16 Dechow et al. (2011) identify three cut-offs for the F-Score (2.45, 1.85, and 1.00) that

are associated with increasing true positive rates (17.8%, 30.2%, and 72.0%) and correspondingly

15 We omit the unexplained audit fee from this comparison because we do not have actual audit fees before 2000. We

re-estimate Table 7 on the post-2000 period for which we have unexplained audit fee data. Whereas the results on the

other measures are quantitatively and qualitatively similar, we find limited predictive ability of unexplained audit fees,

with a true positive rate of 23% in the top quintile. 16 Note that the -1.78 M-Score is the cut-off that minimizes costs of misclassification in the estimation sample in

Beneish (1999). The -2.00 cut-off is sometimes used for illustration purposes to delineate a ‘grey’ area, but is not

based on minimizing misclassification costs.

24

increasing false positive rates (4.0%, 9.3%, and 39.6%). What is noteworthy is that the five points

seem nearly perfectly aligned, and indeed, the R2 of a line fitted through these points equals 98.8%,

which we interpret as indicating that both models trade off true and false positives at a similar rate.

The tradeoff rate is approximately 92 to 1 suggesting that on average for every correctly identified

fraud, the models identify 92 false positives. If frauds were all equally costly and the costs of false

alerts were the same for all non-fraud firms, an approximately constant tradeoff would imply

discretionary model and cut-off choices. However, costs do differ both within and across fraud

and non-fraud groups. For this reason, we turn to evaluating the costs and benefits of evaluating

each model at their suggested cut-offs.

5.2 Costs and Benefits of Implementing Decision Rules Based on F-Score and M-Score

We analyze the costs and benefits of implementing decision rules based on the F-Score and the M-

Score in Table 8. With 1.00 as a cut-off for the F-Score, auditors avoid costs of $3.235 billion from

correctly identifying 226 frauds, but incur false positive costs of $16.297 billion from the 55,212

falsely identified non-fraud observations. Auditors’ net costs of over $13 billion suggest this

particular cut-off is costly. The results are similar for investors who can avoid $96 billion in fraud

related losses, but who forego profits of nearly $157 billion by not investing in flagged firms for a

net cost of nearly $51 billion. For regulators, the results systematically suggest the avoidance of

fraud prediction models, a finding we attribute to the fact that we cannot estimate regulators false

positive costs unless we assume that they go public with an investigation for any model-based

alert.

With 1.85 as a cut-off for the F-Score, auditors avoid costs of $1.876 billion from correctly

identifying 95 frauds but incur false positive costs of $2.077 billion from the 12,940 falsely

identified non-fraud observations. Auditors’ net costs of over $202 million suggesting this cut-off

25

is also costly. On the other hand, investors can avoid $70 billion in fraud related loss and $877

billion of losses by not investing in falsely flagged firms for a net benefit of nearly $947 billion.

Finally, with a cut-off of 2.45 for the F-Score, auditors avoid costs of $0.797 billion from correctly

identifying 56 frauds and incur false positive costs of $0.668 billion from the 5,505 falsely

identified non-fraud observations, resulting in a net benefit of $0.202 billion. As well, investors

can avoid $12.7 billion in fraud-related loss and $892 billion losses by not investing in falsely

flagged firms for a net benefit of nearly $904 billion.

With regard to the M-Score, using the cut-off that minimizes misclassification costs in

Beneish (1999) (e.g., -1.78), auditors avoid costs of $1.516 billion from correctly identifying 112

frauds but incur false positive costs of $2.681 billion from the 23,320 falsely identified non-fraud

observations. Auditors’ net costs of 1.165 billion suggest that using the M-Score is too costly. On

the other hand, investors can avoid $18 billion in fraud related loss, and $1,692 billion losses by

not investing in falsely flagged firms for a net benefit of over $1.71 trillion.

In sum, auditors likely have a strong aversion to using these fraud prediction models because

the models’ false positive rates are too high. The only instance where auditors stand to benefit is

when the false positive rate is 3.95%, but that implies a success rate of only 17.83%. On the other

hand, investors can benefit from using these models so long as they avoid the low cut-off for the

F-Score. Investors’ largest benefit occurs from using the M-Score to avoid firms facing headwinds

as documented in Beneish, Lee and Nichols (2013). 17

6. Model Comparisons

6.1 Conditional Performance of Models

17 Indeed, the gains from the false positives suggest that, although the falsely flagged firms did not engage in fraud

that was uncovered by the SEC, the models do seem to capture potentially underperforming firms.

26

In this section, we discuss the results of additional analyses in which we investigate the

performance of M-Score and F-Score at different cutoffs, conditional on variables that are likely

associated with committing fraud and the probability of detecting fraud. Specifically, we

investigate the models’ performance conditional on quintiles of firm size, firm age, and signed

total accruals, as well as firm life cycle.18 The results can be found in Table 9. We report the total

number of firms in a quintile and/or life cycle stage, as well as the number of firms that are (falsely)

flagged. We then use these numbers to determine the hit rates (the percentage of fraud firms

correctly flagged), false positive rates (the percentage of non-fraud firms incorrectly flagged as

fraudulent), the number of fraud firms as a percentage of all firms that are flagged, and the implied

cost ratio with which models trade off true and false positives.

With respect to firm size, we find that hit rates are generally highest for smaller firms.

However, false positive rates are also highest for smaller firms, such that on a net basis relatively

few fraud firms are correctly flagged in the quintiles containing the smallest firms. Not

surprisingly, the implied cost ratios show a monotonic decrease across quintiles of firm size.

Hence, although hit rates are lower for larger firms, the fact that the number of false positives is

decreasing more substantially in the larger firm quintiles, makes that both M-Score and F-Score

show better performance for larger firms. Surprisingly, we find opposite results for firm age. M-

Score and F-Score perform better for younger firms, both when it comes to hit rates and the rate

at which they trade off true and false positives.

With respect to accruals, both models perform best in the quintiles with extreme (both

positive and negative) accruals. Hit rates are generally highest in the lowest and highest accrual

18 The sample for the life cycle analyses is smaller as the cash flow-based life cycle proxy from Dickinson (2011) is

not available prior to 1987.

27

deciles. However, false positive rates are also much higher in quintiles of extreme accruals, such

that the implied cost ratios show relatively limited variation across quintiles of total accruals.

Interesting results obtain when looking at the results conditional on firm life cycle as both

models perform worst for stable, mature, firms. Hit rates are extremely small for mature firms and

even though false positive rates are small as well, the rate at which the models trade off true and

false positives is worst for mature stage firms. Whereas hit rates are highest for the “extreme”

introduction and decline stage firms, the relative performance of the models is highest for growth

firms, as the implied cost ratio is lowest for firms in these stages.

In short, the results reported in this section present evidence of considerable variation in

model performance conditional on firm size, firm age, accruals, and firm life cycle. Models seem

to work best for larger and younger firms, firms with extreme accruals, and firms in the growth

stage. As costs are also likely to vary across these conditions, these results can provide important

insights into the costs and benefits of fraud prediction models across different subsamples.

7. Conclusion

We estimate the costs of fraud detection errors from the perspective of auditors, investors, and

regulators using an extensive sample of 140,233 observations (of which 779 are fraud firm-years)

over the period 1982-2016. Auditors’ false negative costs (costs of missed detections), estimated

as the sum of litigation costs, reputation costs, and client losses are on average (median) $14.1

($1.94) million in 2016 dollars. Auditors’ false positive costs (costs of incorrectly identifying non-

frauds firms) estimated absent resignation as unbilled incremental audit investment or, conditional

on resignation, as the lost audit fee for one-year, equal on average (median) $0.258 ($0.084)

million. Although false negative costs appear to dwarf false positive costs, with average and

median cost ratios of 54.5 and 23.1, these costs ratios are themselves dwarfed by the proportion of

28

non-fraud to fraud observations in the sample (179:1). Indeed, the fact that all prediction models

have large numbers of false positives and that even the best models trade false positives and false

negatives at a rate of 92:1 explains why auditors would avoid using fraud prediction models in

practice.

Investors’ false negative costs (averaging $446.6 to $845.6 million) are 9 to 40 times larger

than their false positive costs, which predominantly consists of the profit foregone by not investing

over the next year in firms incorrectly identified as fraudulent. As the number of false positives

increases, investors’ use of fraud prediction models becomes a value-destroying proposition. For

example, at a cut-off point of 1.0, the F-Score has a false positive rate of nearly 40% and results

in a net loss to investors of over $50 billion. We estimate regulators’ false negative costs on average

to range from $5.4 to $20.1 million, but we cannot estimate their false positive costs unless we

assume that regulators systematically announce public investigations of flagged firms, in which

case regulators’ false positive costs become prohibitive.

We show that there is considerable variation in the cost of false negatives (an untenable

assumption of ECM analyses) and that the cost of false negatives and false positives differ both

within and across decision-makers (the former being an untenable assumption of most AUC

analyses). We propose the total cost of misclassification, which incorporates the differential

costliness of false positives and false negatives across models, as a basis for comparing the

economic value of fraud prediction models and we describe conditions under which these models

are useful screening devices. In particular, our evidence suggests that researchers should focus on

lowering the false positive rate of their models because increasing the true positive rate appears

like a futile pursuit.

29

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32

Appendix

Variable Definitions

M-Score Variables

M-SCORE = – 4.840 + 0.920DSRIit + 0.528GMIit + 0.404AQIit + 0.892SGIit

+0.115DEPIit – 0.172SGAIit + 4.679TATAit – 0.327LGVIit

DSRI = day’s sales receivable index = (ARt/REVt)/(ARt-1/REV t-1);

GMI = gross margin index

= [(REVt-1- Cost of Goods Soldt-1)/REVt-1]/[(REVt - Cost of Goods Soldt)/REVt];

AQI = asset quality index

= (1 - [Current Assetst + PPEt]/ATt)/(1 - [Current Assets t-1 +PPEt-1]/ATt-1);

SGI = sales growth index = REVt /REVt-1;

DEPI = depreciation index

= (Depreciationt-1/[Depreciationt-1 + PPEt-1])/(Depreciationt/[Depreciation + PPE]);

SGAI = sales, general, and administrative expenses index

= (SGA Expenset/REVt)/(SGAt-1/REVt-1);

TATA = total accruals to total assets = (IBCt-CFOt)/ATt; and

LGVI = leverage index

= ([Long-Term Debtt+Cur. Liabt]/ATt)/([Long-Term Debt t-1+Cur.Liabt-1]/ATt-1).

Unexplained audit fee model

LogFeest = Industry Indicators + β1BigNt + β2log(Assetst) + β3Inventoryt/Avg Assetst

+β4Receivablest /Avg Assetst + β5LTDt /Avg Assetst + β6Earnt /Avg Assetst

+β7Losst + β8Qualifiedt + β9Auditor Tenuret + ɛ

LogFeest = natural logarithm of total audit fees; Industry Indicators = separate indicators for each two-digit SIC code; BigNt = dichotomous variable equal to 1 if a Big N auditor is used, and 0

otherwise; Earnt = operating income after depreciation; Losst = dichotomous variable equal to 1 if a loss occurred within the current or

previous two fiscal years, and 0 otherwise; Qualifiedt = dichotomous variable equal to 1 if the audit opinion was anything

other than a standard unqualified opinion, and 0 if the opinion was a standard unqualified opinion; and

Auditor Tenuret = number of years the auditor has been auditing a company.

F-Score Variables

FSCORE = - 6.789 + 0.817RSST + 3.230∆REC + 2.436∆INV + 0.122∆Cash Sales

-0.992∆Earnings + 0.972 ACT Issuance

33

RSST = (∆WC + ∆NCO + ∆FIN)/Average Total Assets, where WC = [Current Assets - Cash and

Short-Term Investments] - [Current Liabilities - Debt in Current Liabilities]; NCO = [Total

Assets - Current Assets - Investments and Advances - [Total Liabilities - Current Liabilities -

Long-Term Debt]; and FIN = [Short-Term Investments + Long-Term Investments] - [Long-Term

Debt + Debt in Current Liabilities + Preferred Stock]; following Richardson et al. (2005);

∆REC = ∆Accounts Receivables/Average Total Assets;

∆INV = ∆Inventory/Average Total Assets;

∆CASH SALES = percentage change in cash sales [Sales - ∆Accounts Receivables];

∆EARNINGS = [Earningst/Average Total Assetst] - [Earningst-1/Average Total Assetst-1]; and

ACT Issuance = indicator variable coded 1 if the firm issued securities during year t.

34

Figure 1: Estimating the Cost of Fraud Classification Errors

Auditors Investors Regulators Incremental audit work assuming it cannot be

billed to audit client--AT COST. This follows

experimental evidence that auditors increase the audit

investment when thy perceive a higher risk of

mistatement (e.g., Glover et al. 2003, Hammersley et

al. 2011 for a review, Boritz et al. 2015), and

empirical evidence that a risk factor such as a control

weakness or the payment of bribes results in higher

fees (Hogan and Wilkins 2008, Lyon and Maher

2008, and Raghunandan et al. (2011).

Incremental audit work assuming it is billed to

audit client--AT MARKET. This follows

experimental evidence that auditors increase the audit

investment when thy perceive a higher risk of

mistatement (e.g., Glover et al. 2003, Hammersley et

al. 2011 for a review, Boritz et al. 2015), and

empirical evidence that a risk factor such as a control

weakness or the payment of bribes results in higher

fees (Hogan and Wilkins 2008, Lyon and Maher

2008, and Raghunandan et al. (2011).

If investigations into false positives

are made public: Costs estimates

range from 3 to 10% of the market

value of equity at the end of month

three subsequent to firms fiscal year

ends. Percentages encompass three-

day average losses associated with

initial announcements of Wells

notices, restatements involving

irregularities, and SEC

investigations.

Alternatively, one year of lost audit fees from

resigning from the audit client (or profit or such fees).

Loss avoided (profit foregone) by not investing in

false positive firms over a 12-month period beginning

three months after fiscal year end.

If investigations into false positives are

for internal investigations, the costs of

use and alloaction of the regulators'

resources cannot be estimated.

Ligitation Costs: Predicted Settlement Costs

Conditional on the likelihood of auditor litigation

following Honisberg, Rajgopal, and Srinivasan (2018)

Reputation Losses: Market value loss at fraud

revelation due to contagion effect on other clients of

the auditor in the same industry (e.g., Gleason et al.

2008; Weber et al. 2008)

Client Loss: Profits foregone on Abnormal Client

Loss in the two years following the revelation of the

fraud (e.g., Lyon and Maher 2004)

False Positive Costs

(Incorrectly Flagged Non-

Fraud FIrms)

Figure 1--Estimating the Cost of FraudClassification Errors

False Negatives Costs

(Missed Fraud Detection)

Loss on the market: Abnornal Returns in period

varying from days -1 to +1 to days -1 to +252

relative to the first public revelation of the fraud times

the stock's market value on day -2.

The difference in value-weighted market

returns on revelation days over the

period 1982-2016 times the value of the

stock market in aggregate two days prior

to the fraud-revealing announcements.

35

Figure 2: True and False Positive Rates by Prediction Model

36

Figure 3: False Positive versus True Positive Rates

37

TABLE 1

Sample Selection

# AAERs # Firm-Years

Total Sample of Fraud Cases 574 Less: Fraud Cases Not Matched to Compustat-CRSP (82)

Fraud Cases Matched to Compustat-CRSP 494 1185

Less: Fraud Cases with Missing Announcement Returns (79) ( 144)

Fraud Cases with Announcement Returns 415 1041

Less: Fraud Cases with Missing Mscore or Fscore (100) (262)

Final Sample of Fraud Cases 313 779

38

TABLE 2

Audit Fee Model used for Backfilling

Panel A: Audit Fee Model

Depvar: LnFees

Variables Coef. T-stat p-value

Intercept 9.915 265.84 <.001

BigN 0.095 4.62 <.001

LnAssets 0.365 57.48 <.001

BUSSEG 0.089 7.31 <.001

FGN 0.696 23.20 <.001

INV 0.105 2.52 0.012

REC 0.369 7.47 <.001

CR -0.040 -19.74 <.001

BTM -0.035 -5.36 <.001

LEV -0.159 -6.06 <.001

EMPLS 0.097 17.40 <.001

MERGER 0.015 0.86 0.388

DEC_YE -0.037 -2.72 0.007

ROA -0.233 -7.29 <.001

LOSS 0.170 11.45 <.001

OPIN 0.107 6.51 <.001

Litigation -0.017 -1.19 0.235

Adj R2 0.759

N 7311

This table reports the results of the audit fee prediction regression used to backfill missing audit fees in the pre-2000

/ pre-audit analytics period. We estimate the regression over the period 2000-2001 and use the coefficients from this

regression and actuals reported in Compustat to create a predicted fee measure for the earlier years. T-statistics and p-

values are based on standard errors clustered at firm level. The dependent variable LnFees is the natural logarithm of

the audit fee as reported by audit analytics. BigN is an indicator variable that is equal to one if the firm is audited by a

BigN auditor (Compustat AU). LnAssets is the natural logarithm of total assets (AT). BUSSEG is equal to the number

of business segments as reported in Compustat or zero otherwise. FGN is foreign sales as a percentage of total sales

(SALE). INV is inventory as a percentage of total assets (INVT/AT). REC is accounts receivable as a percentage of

total assets (RECT/AT). CR is the current ratio (ACT/LCT). BTM is the book-to-market ratio (CEQ/PRCC_F*CSHO).

LEV is total debt to total assets (DLTT+DLC/AT). EMPLS is the square root of the total number of employees

(EMP^0.5). MERGER is an indicator variable that is equal to one if the Compustat footnote disclosures show that the

firm was involved in a merger, and zero otherwise. DEC_YE is equal to one if the firm's fiscal year-end is not in

December. ROA is operating income after depreciation divided by beginning-of-year total assets (OIADP/AT). LOSS

is an indicator variable that is equal to one if income before extraordinary items was negative in the current year or

any of the previous two years (IB). OPIN is an indicator variable that is equal to one if the firm does not receive an

"unqualified" audit opinion, zero otherwise. Litigation is an indicator variable that is equal to one if the firm is active

in a litigious industry (sic codes 2833:2836, 8731:8734, 3570:3577, 7370:7374, 3600:3674, 5200:5961).

39

TABLE 3

Internal Control Weaknesses, SEC Budgets, and Audit Fees

Variables LnFees LnFees LnFees LnFees LnFees LnFees LnFees LnFees LnFees LnFees LnFees LnFees

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Intercept 8.950*** 9.663*** 5.508*** 10.574*** 10.193*** 9.535*** 10.556*** 10.169*** 9.578*** 10.150*** 10.136*** 9.539***

(48.93) (80.53) (29.56) (20.60) (72.09) (42.70) (20.82) (71.21) (43.03) (23.62) (75.60) (46.54)

N1Y_ICW 0.237*** 0.182*** 0.194***

(11.67) (8.91) (9.51) L1Y_ICW_RMD 0.237*** 0.246*** 0.245*** 0.263*** 0.270*** 0.269***

(14.53) (15.00) (14.96) (15.48) (15.93) (15.89) L1Y_ICW_NRMD 0.485*** 0.496*** 0.494***

(9.91) (10.11) (10.10) L1Y_ICW_NEW 0.310*** 0.321*** 0.320***

(15.46) (16.07) (16.03) ICW_RMD 0.335*** 0.342*** 0.341***

(20.73) (21.34) (21.31)

ICW_NRMD 0.592*** 0.591*** 0.591***

(12.67) (12.74) (12.73)

ICW_NEW 0.313*** 0.307*** 0.306***

(15.41) (15.26) (15.23)

L1Y_SC_SECBUD 2.540*** 0.427*** 0.411*** 0.421***

(20.50) (3.53) (3.45) (3.81) L1Y_LN_SECBUD 0.676*** 0.106*** 0.096*** 0.097***

(29.95) (4.03) (3.69) (4.07)

BigN 0.325*** 0.289*** 0.311*** 0.334*** 0.333*** 0.333*** 0.334*** 0.332*** 0.332*** 0.346*** 0.343*** 0.343***

(17.19) (15.27) (16.47) (15.79) (15.76) (15.79) (15.92) (15.87) (15.88) (17.82) (17.73) (17.75)

LnAssets 0.475*** 0.477*** 0.473*** 0.466*** 0.465*** 0.465*** 0.463*** 0.463*** 0.463*** 0.464*** 0.464*** 0.464***

(67.31) (67.62) (67.08) (61.20) (61.22) (61.28) (61.29) (61.31) (61.38) (64.71) (64.78) (64.85)

BUSSEG 0.029*** 0.028*** 0.029*** 0.019* 0.015 0.015 0.018* 0.014 0.014 0.019* 0.015 0.015

(2.83) (2.69) (2.81) (1.84) (1.47) (1.47) (1.76) (1.41) (1.40) (1.92) (1.55) (1.55)

FGN 0.185*** 0.184*** 0.180*** 0.183*** 0.173*** 0.173*** 0.175*** 0.166*** 0.167*** 0.166*** 0.158*** 0.158***

(7.04) (6.99) (6.84) (6.59) (6.28) (6.28) (6.38) (6.10) (6.11) (6.41) (6.14) (6.15)

INV 0.204*** 0.184*** 0.184*** 0.184** 0.175** 0.174** 0.178** 0.168** 0.168** 0.189*** 0.178*** 0.178***

(3.22) (2.88) (2.88) (2.56) (2.43) (2.42) (2.51) (2.36) (2.35) (2.84) (2.67) (2.66)

40

REC 0.316*** 0.316*** 0.315*** 0.348*** 0.339*** 0.338*** 0.343*** 0.332*** 0.331*** 0.318*** 0.306*** 0.305***

(5.35) (5.34) (5.32) (5.08) (4.94) (4.93) (5.04) (4.88) (4.87) (5.09) (4.90) (4.89)

CR -0.027*** -0.028*** -0.028*** -0.025*** -0.026*** -0.026*** -0.025*** -0.025*** -0.025*** -0.025*** -0.025*** -0.025***

(-11.51) (-11.64) (-11.76) (-8.94) (-9.10) (-9.10) (-8.77) (-8.92) (-8.93) (-9.76) (-9.90) (-9.91)

BTM -0.116*** -0.098*** -0.097*** -0.104*** -0.098*** -0.097*** -0.104*** -0.098*** -0.097*** -0.107*** -0.099*** -0.099***

(-13.44) (-11.43) (-11.29) (-12.83) (-12.22) (-12.15) (-12.95) (-12.34) (-12.29) (-13.76) (-13.06) (-13.01)

LEV -0.132*** -0.141*** -0.144*** -0.116*** -0.106*** -0.107*** -0.110*** -0.101*** -0.102*** -0.118*** -0.111*** -0.111***

(-5.27) (-5.59) (-5.69) (-4.21) (-3.86) (-3.88) (-4.02) (-3.71) (-3.72) (-4.69) (-4.38) (-4.39)

EMPLS 0.052*** 0.051*** 0.052*** 0.052*** 0.053*** 0.053*** 0.054*** 0.055*** 0.055*** 0.054*** 0.055*** 0.055***

(10.10) (9.77) (10.09) (9.87) (9.99) (10.00) (10.20) (10.30) (10.30) (10.47) (10.55) (10.54)

MERGER 0.032*** 0.039*** 0.036*** 0.040*** 0.040*** 0.040*** 0.045*** 0.045*** 0.045*** 0.043*** 0.043*** 0.043***

(3.34) (3.89) (3.64) (3.80) (3.80) (3.80) (4.24) (4.25) (4.26) (4.34) (4.35) (4.35)

DEC_YE -0.016 0.004 0.000 0.009 0.011 0.011 0.008 0.010 0.010 0.006 0.009 0.009

(-0.99) (0.27) (0.03) (0.55) (0.66) (0.68) (0.48) (0.61) (0.63) (0.40) (0.55) (0.55)

ROA -0.299*** -0.284*** -0.272*** -0.307*** -0.306*** -0.305*** -0.304*** -0.303*** -0.303*** -0.281*** -0.280*** -0.280***

(-9.67) (-9.17) (-8.82) (-8.03) (-8.03) (-8.01) (-8.02) (-8.01) (-8.01) (-8.39) (-8.39) (-8.39)

LOSS 0.192*** 0.183*** 0.184*** 0.185*** 0.181*** 0.181*** 0.173*** 0.170*** 0.170*** 0.171*** 0.169*** 0.169***

(18.18) (17.16) (17.37) (15.40) (15.13) (15.17) (14.55) (14.36) (14.40) (15.33) (15.19) (15.23)

OPIN 0.121*** 0.160*** 0.160*** 0.102*** 0.125*** 0.123*** 0.097*** 0.115*** 0.112*** 0.099*** 0.119*** 0.116***

(13.47) (20.29) (20.91) (10.14) (14.18) (14.46) (9.84) (13.23) (13.41) (10.91) (15.34) (15.43)

CLIENT 0.020** 0.035*** 0.020*** 0.017** 0.017** 0.017** 0.025*** 0.026*** 0.027*** 0.019** 0.021*** 0.021***

(2.50) (4.54) (2.63) (1.99) (2.02) (2.06) (3.03) (3.19) (3.28) (2.44) (2.71) (2.82)

LITIGATION -0.060** -0.056** -0.057** -0.071*** -0.070*** -0.070*** -0.070*** -0.070*** -0.070*** -0.062** -0.061** -0.062**

(-2.41) (-2.27) (-2.30) (-2.67) (-2.65) (-2.65) (-2.66) (-2.65) (-2.65) (-2.49) (-2.46) (-2.46)

8.950*** 9.663*** 5.508*** 10.574*** 10.193*** 9.535*** 10.556*** 10.169*** 9.578*** 10.150*** 10.136*** 9.539***

Industry FE Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Year FE Yes No No Yes No No Yes No No Yes No No

Observations 43,762 43,762 43,762 33,837 33,837 33,837 33,837 33,837 33,837 39,921 39,921 39,921

R-squared 0.830 0.817 0.822 0.841 0.840 0.840 0.844 0.843 0.843 0.839 0.838 0.838

41

TABLE 4

Costs of Prediction Errors to Auditors

Panel A: Auditors' False Positive Costs N Mean Std Dev Median Sum

AUDFEE 139,454 1.2822 3.2897 0.4147 178,810

AUD_FPC_Audit_Investment (at Cost) 139,454 0.2587 0.6638 0.0837 36,082

Panel B: Litigation Costs

Incidence of Litigation Lawsuits Dismissed Lawsuits Settled Lawsuits

No Damages Damages

N Percent N Percent N Percent N Percent

v. Firm and Directors 212 67.7% 32 15.09% 14 6.60% 166 78.30%

v. Investment Bankers 19 4.8% 3 15.79% 0 0.00% 16 84.21%

v. Auditors 87 27.8% 23 26.44% 15 17.24% 49 56.32%

Settlement in millions of 2016 $

Paid by N Mean Sum MAX

Firm and Directors 166 82.8 25922.6 4874.3

Investment Bankers 16 66.5 20821.6 15407.5

Auditors 49 10.0 3132.8 512.4

Panel C: Auditors' False Negative Costs N Mean Std Dev Median Sum

AUD_FN_LIT (actual) 311 10.06 45.78 0.00 3128.5

AUD_FN_LIT (Estimated based on Honisberg et al.) 313 2.88 3.58 1.77 902.7

AUD_FN_REP (Clients Same Industry-COMPUSTAT-C) 274 0.37 18.49 0.17 102.7

AUD_FN_CLIENT LOSS (Year+1 v. Year -1)-C 313 2.65 29.62 0.00 828.2

AUD_FN_CLIENT LOSS (Years [+1, +2] v. [-1, -2])-C 313 3.68 30.15 0.00 1150.6

AUD_FN_REP (Clients Same Industry-Aud. Analytics-AA) 139 2.76 30.35 0.26 383.6

AUD_FN_CLIENT LOSS (Year+1 v. -1)-AA 180 1.38 14.35 0.00 248.0

AUD_FN_CLIENT LOSS (Years [+1, +2] v. [-1, -2])-AA 159 2.12 6.65 0.20 336.6

AUD_FN_CLIENT LOSS (Years [+1, +3] v. Y [-1, -3])-AA 126 2.06 6.28 0.00 260.1

Panel D: Ratio of Average False Negative to False Positive Costs

Mean Median False Positive Costs 0.2587 0.0837 False Negative Costs Ratio Ratio

Lit(Actual)+Rep+Client Loss in (-1,+1)-COMPUSTAT 13.08 50.6 0.17 2.0

Lit(Estimated)+Rep+Client Loss in (-1,+1)-COMPUSTAT 5.90 22.8 1.94 23.1

Lit(Actual)+Rep+Client Loss in (-2,+2)-COMPUSTAT 14.11 54.5 0.17 2.0

Lit(Estimated)+Rep+Client Loss in (-2,+2)-COMPUSTAT 6.93 26.8 1.94 23.1 In Panel A, auditors' false positive costs are either a lost audit fee (AUDFEE) in the event of resignation, or in the

absence of resignation, the incremental audit investment--at cost--that a fraud warning flag would generate.

Alternatively, auditor’s false positive costs are calculated as EAUDi[FP COST] = p (AUDFEEjt) + (1-p)

(INCRAUDFEEjt * (1-pm%)), where p is the probability of resignation, AUDFEEjt is the fee lost by the resigning

auditor in year t, INCRAUDFEEjt (Inv_FPC_Auditors (at cost)) is the value of the additional work undertaken by

the auditor as a result of the warning flag (which we assume that they cannot pass on to the client in year t), and where

42

pm% represents the auditor’s usual profit margin. Panel B, reports, for each of the 313 fraud firms in the sample,

lawsuits filed against the firm, its investment bankers, and its auditors, as well as the lawsuit’s resolution. Data are

obtained from extensive searches on Audit Analytics, Westlaw, Lexis Nexis, and Factiva. Our searches begin with the

firm name at the time the fraud is discovered, but also consider name changes as applicable, specify a search period

beginning two years before and ending two years after the fraud period identified in AAERs, and uses search terms

such as “securities”, “class action”, litigation”, “lawsuit” and “settlement.” In Panel C, auditors' false negative costs

have three components. Auditor litigation costs are measured either as AUD_FN_LIT-Actual or as AUD_FN_LIT-

Estimated: AUD_FN_LIT-Actual is based on identifying lawsuits against auditors and their resolution using Audit

Analytics, Westlaw, Lexis Nexis, and Factiva. AUD_FN_LIT-Estimated is the expected settlement amount from the

model proposed by Honisberg, et al. (2018). Auditors’ reputation loss is the abnormal dollar value loss (market value

of equity on day -2 multiplied by abnormal return on day -1 to 1 relative the fraud revelation announcement. for all

clients of the auditor (AUD_FN_REP), and for whether the clients are in the same 6-digit GICS industry relying on

COMPUSTAT ((AUD_FN_REP (Clients Same Industry-COMPUSTAT-C), or AUDIT ANALYTICS

(AUD_FN_REP (Clients Same Industry-Aud. Analyt.-AA)). We measure auditor client losses as the difference

between an auditor’s rate of attrition at the office level in the years following the discovery of a missed detection and

either (i) the rate of attrition for the same ‘undetecting’ office in the year prior to the public revelation of the fraud, or

(ii) the average rate of attrition for all other of the auditor’s offices. When using COMPUSTAT data, we transform

the abnormal attrition rate into a number of clients lost and multiply this estimate by the average audit fee per client.

When using AUDIT ANALYTICS, we calculate the fee lost specifically by assuming it equals last year’s fee. In Panel

D, we report estimates of the ratio of the average cost of false negatives to false positives. Data are in millions of

2016 dollars.

43

TABLE 5

Costs of Prediction Errors to Investors

Panel A: Investors False Positive Costs N Mean Std Dev Median Sum

INV_FPC_Audit_Extracost 139,454 0.2949 0.7566 0.0954 41,126

Inv_FPC_LOSS [PROFIT]-Market 139,454 20.57 3882.41 -4.11 2,869,008

Inv_FPC_LOSS [PROFIT]-Size 132,604 15.60 3873.14 -4.42 2,068,905

Inv_FPC_LOSS [PROFIT]-FF3 138,864 91.02 3750.55 -0.70 12,639,792

Inv_FPC_LOSS [PROFIT]-FF4 138,719 94.23 4086.44 0.13 13,071,245

Panel B: Investors False Negative Costs N Mean Std Dev Median Sum

INV_FN_MKT_Loss-3 day 313 446.6 1975.5 20.6 139,795

INV_FN_MKT_Loss-5 day 311 411.0 2074.4 24.2 127,825

INV_FN_MKT_Loss-1 month 313 414.1 4334.0 34.2 129,627

INV_FN_MKT_Loss-3 month 313 580.4 5344.8 38.6 181,664

INV_FN_MKT_Loss-6 month 313 627.5 7037.0 50.0 196,404

INV_FN_MKT_Loss-12 month 313 845.6 6318.0 49.9 264,665

Panel C: Ratio of Average False Negative to False Positive Costs

Audit_Extracost + Profit Foregone Revelation Period Losses MKT SIZE FF3 FF4 Loss 3-day 21.40 28.10 4.89 4.73 Loss 5-day 19.70 25.85 4.50 4.35 Loss 1-month 19.85 26.05 4.54 4.38 Loss 3-month 27.81 36.51 6.36 6.14 Loss 6-month 30.07 39.47 6.87 6.64 Loss 12-month 40.52 53.19 9.26 8.95

In Panel A, investors’ false positive costs have two potential components: (i) as residual claimants, investors bear the

costs of additional audit fees imposed by auditors as a result of classifying the firm as a potential fraud

(INCRAUDFEEjt), (ii) the second component adds or reduces to the costs of the incremental audit fees depending on

whether investors forego gains or avoid losses by not investing in the firms on which there is a warning flag

(Inv_FPC_PROFIT/LOSS is the one-year dollar abnormal return beginning in month 4 after the end of the firms fiscal

year). In Panel B, investors’ false negative costs as the abnormal dollar value loss (market value of equity on day -2

multiplied by abnormal return over different periods from days -1 to 1, to days -1, to +252 relative the fraud revelation

announcement. In Panel C, we report estimates of the ratio of the average cost of false negatives to false positives.

Data are in millions of 2016 dollars.

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TABLE 6

Costs of Prediction Errors to Regulators

Panel A: Regulators False Positive Costs N Mean Std Dev Median Sum

REG_FPC_3 139,454 84.58 431.25 6.97 11,795,679

REG_FPC_5 139,454 140.97 718.75 11.62 19,659,466

REG_FPC_10 139,454 281.95 1437.50 23.24 39,318,931

Panel B: Regulators False Negative Costs N Mean Std Dev Median Sum

One-day Loss 313 20.05 8.51 22.35 6,276

Three-day Loss 313 5.62 2.39 6.27 1,760

Five-day Loss 313 5.44 2.31 6.06 1,703

Panel C: Ratio of Average False Negative to False Positive Costs

REG_FPC_3 REG_FPC_5 REG_FPC_10

One-day Loss 0.24 0.14 0.07

Three-day Loss 0.07 0.04 0.02

Five-day Loss 0.06 0.04 0.02

Panel D: Estimation of Regulators False Negative Costs: AAER Announcements and Market-wide returns

1-day AAER Event-window:

Value-weighted Coeff. SE T-stat p-value

Intercept 0.00054 0.00011 4.89 <.0001

Event -0.00119 0.00051 -2.36 0.0185

Equal-weighted

Intercept 0.00084 0.00009 9.18 <.0001

Event -0.00096 0.00042 -2.30 0.0216

3-day AAER Event-window:

Value-weighted

Intercept 0.00053 0.00012 4.56 <.0001

Event -0.00033 0.00032 -1.05 0.2932

Equal-weighted

Intercept 0.00084 0.00010 8.70 <.0001

Event -0.00031 0.00026 -1.16 0.2454

5-day AAER Event-window:

Value-weighted

Intercept 0.00055 0.00012 4.54 <.0001

Event -0.00032 0.00027 -1.21 0.2244

Equal-weighted

Intercept 0.00084 0.00010 8.31 <.0001

Event -0.00019 0.00022 -0.88 0.3804

N= 9584 daily market-index returns in the period 1980-2017 In Panel A, we assume regulators publicly announce investigations of firms flagged and estimate regulators’ false

positive costs to range between 3% (REG_FPC_3) and 10% (REG_FPC_10) of a firm’s the market value of equity in

45

month four subsequent to the firm’s fiscal year-end. In Panel B, we use the results of regressions in Panel D of the

value-weighted return on the market from CRSP on indicator variables capturing 1, 3 and 5 days surrounding the

revelation date. Specifically, we estimate regulators’ false negative costs as the regression estimates of the slope times

the value of the market on days -2 relative to the day of revelation. In Panel C, we report estimates of the ratio of the

average cost of false negatives to false positives. Data are in millions of 2016 dollars.

46

TABLE 7

Model Predictive Performance

Total Accruals (Raw) Current Accruals (Raw) Current Accruals (Abnormal)

Decile

True

Positive

Rate (TP)

Cum. TP

Rate

Cum. False

Positive Rate

True Positive

Rate (TP) Cum. TP Rate

Cum. False

Positive

Rate

True

Positive

Rate (TP)

Cum. TP

Rate

Cum. False

Positive

Rate

10 19.9% 19.9% 9.98% 20.7% 20.7% 9.97% 18.0% 18.0% 9.98%

9 16.5% 36.3% 19.96% 16.5% 37.2% 19.96% 11.3% 29.3% 19.98%

8 12.4% 48.7% 29.96% 12.8% 50.0% 29.95% 12.4% 41.7% 29.97%

7 7.9% 56.6% 39.96% 7.9% 57.9% 39.96% 9.4% 51.1% 39.97%

6 8.6% 65.2% 49.96% 6.0% 63.9% 49.97% 7.9% 59.0% 49.98%

5 7.1% 72.3% 59.97% 6.4% 70.3% 59.98% 9.4% 68.4% 59.98%

4 4.9% 77.2% 69.98% 8.3% 78.6% 69.98% 7.1% 75.6% 69.99%

3 5.2% 82.4% 79.99% 6.4% 85.0% 79.99% 6.0% 81.6% 80.00%

2 6.7% 89.1% 90.00% 6.4% 91.4% 90.00% 7.1% 88.7% 90.00%

1 10.9% 100.0% 100.00% 8.6% 100.0% 100.00% 11.3% 100.0% 100.00%

M-Score F-Score FSD

Decile

True

Positive

Rate (TP)

Cum. TP

Rate

Cum. False

Positive Rate

True Positive

Rate (TP) Cum. TP Rate

Cum. False

Positive

Rate

True

Positive

Rate (TP)

Cum. TP

Rate

Cum. False

Positive

Rate

10 22.6% 22.6% 9.97% 30.9% 30.9% 9.95% 13.10% 13.10% 9.99%

9 17.8% 40.4% 19.95% 16.6% 47.5% 19.94% 11.80% 24.90% 19.99%

8 15.3% 55.7% 29.94% 14.3% 61.8% 29.93% 11.50% 36.40% 29.99%

7 11.8% 67.5% 39.94% 10.5% 72.3% 39.93% 10.80% 47.20% 39.99%

6 7.3% 74.8% 49.94% 7.3% 79.6% 49.93% 10.20% 57.40% 49.99%

5 6.7% 81.5% 59.95% 6.4% 86.0% 59.94% 10.20% 67.60% 59.99%

4 5.1% 86.6% 69.96% 5.1% 91.1% 69.95% 9.90% 77.50% 69.99%

3 5.4% 92.0% 79.97% 2.9% 93.9% 79.97% 8.60% 86.10% 79.99%

2 2.9% 94.9% 89.99% 4.1% 98.1% 89.98% 8.60% 94.70% 90.00%

1 5.1% 100.0% 100.00% 1.9% 100.0% 100.00% 5.30% 100.00% 100.00%

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TABLE 8

Costs and Benefits of Implementing Decision Rules Based on F- and M-Score cut-offs.

Net Cost

False Positives True Positives (Benefit)

Decision Rules Number Cost Incurred Number Cost Avoided Fscore>1 Auditors 55,212 16,297 226 3,235 13,062

Investors 55,212 146,921 226 96,165 50,756

Regulators 55,212 4,830,494 226 1,271 4,829,223

Fscore>1.85 Auditors 12,940 2,077 95 1,876 202

Investors 12,940 (876,650) 95 69,784 (946,434)

Regulators 12,940 569,277 95 479 568,798

Fscore>2.45 Auditors 5,505 668 56 797 (129)

Investors 5,505 (891,590) 56 12,678 (904,268)

Regulators 5,505 4,830,494 56 276 4,830,218

Mscore>-1.78 Auditors 23,320 2,681 112 1,516 1,165

Investors 23,320 (1,692,536) 112 18,125 (1,710,661)

Regulators 23,320 787,567 112 458 787,109

This table reports the aggregate costs of false positives are compared to the costs avoided by identifying true positives

for auditors, investors, and regulators, which are described in detail in Tables 4, 5, and 6 respectively. Data are in

millions of 2016 dollars.

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TABLE 9

Conditional Performance of F- and M-Score cut-offs Panel A: Performance conditional on Firm Size

NumFirm NumFraud Flag NoFraud Flag Fraud Hitrate FP Rate Pct Fraud Costratio

Mscore

0 27,327 46 5,082 16 34.78 18.60 0.31 317.63

1 27,353 91 5,621 31 34.07 20.55 0.55 181.32

2 27,354 164 5,407 43 26.22 19.77 0.80 125.74

3 27,352 211 4,261 52 24.64 15.58 1.22 81.94

4 27,337 253 2,365 31 12.25 8.65 1.31 76.29

Fscore1

0 27,327 46 2,600 12 26.09 9.51 0.46 216.67

1 27,353 91 3,073 18 19.78 11.23 0.59 170.72

2 27,354 164 3,082 38 23.17 11.27 1.23 81.11

3 27,352 211 2,567 58 27.49 9.39 2.26 44.26

4 27,337 253 1,345 34 13.44 4.92 2.53 39.56

Fscore2

0 27,327 46 1,097 8 17.39 4.01 0.73 137.13

1 27,353 91 1,430 11 12.09 5.23 0.77 130.00

2 27,354 164 1,394 21 12.80 5.10 1.51 66.38

3 27,352 211 1,032 32 15.17 3.77 3.10 32.25

4 27,337 253 476 14 5.53 1.74 2.94 34.00

Panel B: Performance conditional on Firm Age

NumFirm NumFraud Flag NoFraud Flag Fraud Hitrate FP Rate Pct Fraud Costratio

Mscore

0 25,787 127 7,781 66 51.97 30.17 0.85 117.89

1 26,714 188 5,097 56 29.79 19.08 1.10 91.02

2 28,435 166 4,467 28 16.87 15.71 0.63 159.54

3 28,277 112 3,445 11 9.82 12.18 0.32 313.18

4 27,542 176 1,952 12 6.82 7.09 0.61 162.67

Fscore1

0 25,787 127 3,881 59 46.46 15.05 1.52 65.78

1 26,714 188 2,904 49 26.06 10.87 1.69 59.27

2 28,435 166 2,550 24 14.46 8.97 0.94 106.25

3 28,277 112 2,064 18 16.07 7.30 0.87 114.67

4 27,542 176 1,271 11 6.25 4.61 0.87 115.55

Fscore2

0 25,787 127 2,125 40 31.50 8.24 1.88 53.13

1 26,714 188 1,255 26 13.83 4.70 2.07 48.27

2 28,435 166 995 9 5.42 3.50 0.90 110.56

3 28,277 112 709 8 7.14 2.51 1.13 88.63

4 27,542 176 348 4 2.27 1.26 1.15 87.00

49

Panel C: Performance conditional on Total Accruals

NumFirm NumFraud Flag NoFraud Flag Fraud Hitrate FP Rate Pct Fraud Costratio

Mscore

0 26,931 130 3,011 28 21.54 11.18 0.93 107.54

1 26,932 120 2,220 7 5.83 8.24 0.32 317.14

2 26,931 140 2,078 10 7.14 7.72 0.48 207.80

3 26,932 171 2,992 17 9.94 11.11 0.57 176.00

4 26,931 195 12,107 109 55.90 44.96 0.90 111.07

Fscore1

0 26,931 130 1,770 25 19.23 6.57 1.41 70.80

1 26,932 120 946 10 8.33 3.51 1.06 94.60

2 26,931 140 1,032 14 10.00 3.83 1.36 73.71

3 26,932 171 1,751 20 11.70 6.50 1.14 87.55

4 26,931 195 7,050 91 46.67 26.18 1.29 77.47

Fscore2

0 26,931 130 971 16 12.31 3.61 1.65 60.69

1 26,932 120 321 5 4.17 1.19 1.56 64.20

2 26,931 140 265 8 5.71 0.98 3.02 33.13

3 26,932 171 445 9 5.26 1.65 2.02 49.44

4 26,931 195 3,385 48 24.62 12.57 1.42 70.52

Panel D: Performance conditional on Firm Life Cycle

NumFirm NumFraud Flag NoFraud Flag Fraud Hitrate FP Rate Pct Fraud Costratio

Mscore

0 15,516 110 5,781 57 51.82 37.26 0.99 101.42

1 32,574 286 5,386 52 18.18 16.53 0.97 103.58

2 44,554 205 2,489 12 5.85 5.59 0.48 207.42

3 9,907 53 1,480 6 11.32 14.94 0.41 246.67

4 7,565 36 2,352 13 36.11 31.09 0.55 180.92

Fscore1

0 15,516 110 3,693 43 39.09 23.80 1.16 85.88

1 32,574 286 3,693 65 22.73 11.34 1.76 56.82

2 44,554 205 1,857 16 7.80 4.17 0.86 116.06

3 9,907 53 453 5 9.43 4.57 1.10 90.60

4 7,565 36 584 8 22.22 7.72 1.37 73.00

Fscore2

0 15,516 110 2,035 30 27.27 13.12 1.47 67.83

1 32,574 286 1,422 30 10.49 4.37 2.11 47.40

2 44,554 205 417 5 2.44 0.94 1.20 83.40

3 9,907 53 196 3 5.66 1.98 1.53 65.33

4 7,565 36 304 4 11.11 4.02 1.32 76.00

This table reports the performance of M-Score and F-Score at different cutoffs. Fscore1 reports performance for the

1.85 cutoff, while Fscore2 reports results for the 2.45 cutoff. NumFirm is the total number of firm-years in a quintile

and/or life cycle stage. NumFraud is the number of fraud-years in a quintile and/or life cycle stage. Flag Nofraud and

Flag Fraud are equal to the number of falsely (correctly) flagged firm-years. We then use these numbers to determine

the hit rates (the percentage of fraud firms correctly flagged; Flag Fraud / NumFraud), false positive rates (the

percentage of non-fraud firms incorrectly flagged as fraudulent; Flag NoFraud / NumFirm-NumFraud), the number

50

of fraud firms as a percentage of all firms that are flagged (Flag Fraud / Flag Fraud + Flag NoFraud), and the implied

cost ratio with which models trade off true and false positives (1/pct Fraud).