Materiality uncertainty and earnings misstatement.

ABSTRACT: The concept of materiality provides a basis for auditors to ignore small misstatements, but the definition of "small" in this context is ambiguous. The issue of "materiality-standard-setting" has been raised recently by Arthur Levitt, former chairman of the Securities and Exchange

Commission. We examine how materiality uncertainty affects the auditor's evaluation of audit evidence and a manager's choice of earnings overstatement in a strategic auditing model where earnings misstatements also include unintentional system error. We find that when the expected cost of accepting financial statements that are materially misstated, which we refer to as an audit failure, is relatively large, an increase in materiality uncertainty results in a more conservative auditor evaluation of the audit evidence and a decrease in the amount of intentional overstatement. Alternatively, if the auditor's expected cost of extending audit procedures is relatively high, then an increase in materiality uncertainty results in a less conservative auditor evaluation of the audit evidence and an increase in the manager's earnings overstatement. The auditor also becomes increasingly conservative as the report increases when the information system is sufficiently noisy. Finally, when the expected cost of audit failure is large, the equilibrium audit risk can increase or decrease in materiality uncertainty despite the corresponding increase in auditor conservatism and decrease in intentional overstatement. Audit risk is the average probability of audit failure across all possible evidence outcomes.

Keywords: strategic audit; materiality; intentional overstatement; system of internal control

I. INTRODUCTION

The issue of materiality has recently garnered renewed attention from the Securities and Exchange Commission (SEC) and the accounting profession. In August 1999 the SEC issued Staff Accounting Bulletin No. 99 (SAB No. 99), Materiality. The purpose of this bulletin is not to adopt precise materiality standards, but to provide "guidance in applying materiality thresholds to the preparation of financial statements filed with the Commission and the performance of audits of those financial statements." Prior to 1999, auditors primarily referred to Statement of Financial Accounting Concepts (SFAC) No. 2, which defines a material misstatement as one such that "the judgment of a reasonable person relying on the information would have been changed or influenced by the ... misstatement" (paragraph 10). (1) The SEC has been concerned about public companies "abusing materiality guidelines in order to manipulate their reported earnings" (Journal of Accountancy 2000, 17-18)--that is, the influence of materiality on intentional earnings misstatement.

We examine, in a stylized, game-theoretic model, how uncertainty regarding the materiality threshold affects the interaction between an auditor and manager when the manager has the opportunity and incentive to intentionally overstate earnings. The manager chooses an amount of intentional overstatement. The auditor evaluates the audit evidence by accepting or rejecting reported earnings as materially correct. (2) In addition to intentional overstatements, reported earnings misstatements contain unintentional system errors. The manager has no private information about earnings and overstates earnings by biasing the information system report on earnings. The auditor observes reported earnings before evaluating the audit evidence. The report allows the auditor to update his expectation of materiality and unintentional system error.

While the SEC has identified materiality as a major concern, it is reluctant to provide precise guidance on the issue. Former SEC Chairman Arthur Levitt states that "materiality is not a bright-line cutoff of 3% or 5%. It requires consideration of all relevant factors that could impact an investor's decision." (See WSJ 1998.) (3) Neither the professional literature, common law, nor statutory law provides a precise materiality standard. However, the impact of a vague definition of materiality is unknown, particularly when intentional misstatements are possible. Previous studies in statistical auditing (see for example, Johnstone 1995), decision theory (see for example, Kinney 1975) and strategic auditing (see for example, Newman and Noel 1989) have assumed that a precise materiality standard exists.

Uncertainty about the materiality threshold (materiality uncertainty) arises from the judgmental nature of materiality and potentially creates additional tensions in the strategic interaction between the auditor and manager. While both the auditor and manager have expectations about the materiality threshold, the actual threshold is affected by qualitative factors such as the quality of accounting estimates. (4)

We find that the effect of materiality uncertainty on the equilibrium strategies depends on the auditor's expected cost of accepting materially misstated financial statements, which we refer to as an audit failure, relative to the expected cost of rejection. The expected cost of rejection includes the cost of extending audit procedures and the resulting loss of client goodwill or lucrative consulting contracts. Materiality uncertainty affects the auditor's evidence evaluation because it affects the updated probability of material misstatement (for a fixed earnings report and audit evidence). When the cost of audit failure is high relative to the cost of rejection, the auditor evaluates the evidence more conservatively because he has a greater incentive to avoid audit failure. (5) Greater uncertainty about the materiality threshold increases the updated probability of material misstatement, exposing the auditor to a greater risk of audit failure. Consequently, the auditor evaluates the evidence more conservatively for higher levels of materiality uncertainty, which, in turn, induces the manager to decrease the amount of intentional overstatement. The opposite effects occur when the cost of audit failure is relatively lower than the cost of rejection. In this case, the manager increases the amount of intentional overstatement and the auditor evaluates the evidence less conservatively.

Anecdotal evidence suggests that the auditor's expected cost of audit failure is typically large relative to the expected cost of extending procedures. Historically, the audit profession has focused on developing professional standards that promote the avoidance of audit failure. For example, the independence rules in the Code of Professional Ethics prohibit auditors from holding direct investments (even one share of stock) in their client firms (AICPA 2001). This reduces the auditor's incentive to evaluate the evidence less conservatively. In addition, damage awards against auditors for undetected material misstatements have been large. For example, Coopers & Lybrand settled its case with debenture holders of Miniscribe for $140 million and claims against Arthur Andersen for Enron are estimated to be in the billions of dollars. (6) Furthermore, legislative action has addressed the possibility of large expected rejection costs due to lucrative consulting contracts. The Sarbanes-Oxley Act, signed into law on July 30, 2002, prohibits corporations from hiring the same accounting firms for audits and most types of consulting services. (7) To the extent that the expected cost of audit failure is large relative to the expected cost of extending audit procedures, our model suggests that a fixed materiality standard would be detrimental in promoting the avoidance of audit failure.

Though increasing materiality uncertainty induces more conservative evidence evaluation and less intentional overstatement by the manager when the expected cost of audit failure is large, equilibrium audit risk can increase or decrease. Audit risk is the average probability of audit failure across all possible evidence outcomes. Increased auditor conservatism has an indirect negative effect on audit risk. However, this indirect effect may be overwhelmed by the direct positive effect of materiality uncertainty on audit risk for lower levels of materiality uncertainty. For lower levels of materiality uncertainty, the indirect marginal effect of a change in materiality uncertainty on audit risk is relatively small.

We also find that if the information system is sufficiently noisy, the auditor's evidence evaluation strategy becomes increasingly more conservative as reported earnings increases. This result holds despite the fact that higher earnings reports also imply that the expected materiality threshold has increased. If the system is sufficiently noisy, then increases in the report imply that the updated probability of an audit failure also increases, so the auditor evaluates the evidence more conservatively.

Our model most resembles that of Newman and Noel (1989) because we consider two strategic choices very similar to theirs. The auditor chooses an evidence evaluation strategy that is based upon sample information, and the manager chooses an amount of intentional overstatement that affects the distribution of audit evidence. There are two key differences between our model and Newman and Noel (1989). First, the manager in our setting is not limited to a binary choice of "no fraud" and "fraud" that the auditor finds "acceptable" and "unacceptable," respectively. As a result, an uncertain materiality threshold is not possible in their model. Second, we introduce an additional source of reported earnings misstatement--unintentional system error. Neither the auditor nor the manager knows the amount of misstatement due to unintentional system error. Further, the auditor in our model is concerned about total earnings misstatement consisting of intentional and unintentional error.

In Section II, we describe the players' strategies as well as the economic consequences of those strategies. In Section III we discuss our solution technique and determine the equilibrium strategies, while Section IV provides a comparative analysis of equilibrium strategies and audit risk. We present concluding remarks in Section V.

II. MODEL DESCRIPTION

When formulating an audit plan, the auditor considers the motivation and opportunity of the manager to commit fraud and whether potential misstatements materially affect financial statement fairness. After observing an earnings report and obtaining audit evidence, the auditor either accepts the earnings report as materially correct or rejects, extending audit procedures. As in previous strategic audit settings, the manager in our setting obtains benefits from intentional misstatements that are not identified by audit procedures. Consistent with this literature, we also assume that larger intentional misstatements increase both the likelihood that the misstatements are identified and the manager's expected penalty for misstatements if the auditor rejects. The auditor, on the other hand, must balance the expected cost of audit failure against the expected cost of extending audit procedures.

When we add materiality to this interaction, the tensions within the model change because the manager chooses intentional misstatement knowing that the auditor considers some of them too small to affect financial statement fairness. To explore the strategic effects of materiality, we examine a stylized model in which the manager is limited to overstating earnings by biasing an accounting information system report, while the auditor employs an evidence evaluation strategy. We describe specific elements of our stylized model in the following subsections.

The Accounting Information System and Earnings Misstatement

The manager chooses an amount of overstatement bias, denoted [omega], in anticipation of an earnings report and an audit, but before either event occurs. The manager knows that the distribution of true earnings is y ~ N([[mu].sub.y], [[sigma].sub.y.sup.2]). He also knows that the accounting information system, in the absence of any intentional misstatement, would generate an earnings report, z, that is equal to true earnings plus noise, e where e ~ N(0, [[sigma].sub.e.sup.2]). We assume that earnings and system error are independent random variables. Thus the earnings report is:

(1) z = y + e + [omega]

where z ~ N([[mu].sub.y], + [omega], [[sigma].sub.y.sup.2] + [[sigma].sub.e.sup.2]). We denote the density of z as [lambda](z).

The manager adds intentional overstatement bias, [omega], to the earnings report by misreporting selected transactions. This strategy is advantageous to the manager for two reasons. First, he can inflate earnings without obvious large or unusual year-end adjustments. Second, he can incorporate transaction errors that normal audit procedures do not characterize as intentional or unintentional. For example, suppose two types of errors can occur in the recording of sales: (1) goods are shipped that have not been billed--creating an understatement error or (2) goods are billed that have not been shipped--creating an overstatement error. To bias revenue upward, the manager increases the frequency of the latter error type. The manager could sell goods to some customers as "bill and hold," claiming that the goods are sold but are being held for the customer. He could also send more goods than the customer requested or bill sales based on customer orders, before the shipment of goods occur. (8) Confirmations might uncover these as recording errors, but the manager could claim that they were unintentional system errors. (9)

As a result of the earnings overstatement bias and the system error, the total earnings misstatement included in an earnings report is:

(2) [theta] = z - y = e + [omega].

Because [omega] is not stochastic, the total misstatement is distributed, [theta] ~ N([omega], [[sigma].sub.e.sup.2]).

The auditor observes the report z before he performs audit procedures. From the auditor's point of view, the earnings misstatement is:

(3) [[theta].sub.z] = z - [y.sub.z] = [e.sub.z] + [omega],

where the subscript, z, denotes that the random variable is conditioned on the observed earnings report. Applying Bayes theorem, we find that [y.sub.z] ~ N([[mu].sub.y|z], [[sigma].sub.y|z.sup.2]) where [[mu].sub.y|z] = [sigma](z - [omega]) + (1 - [delta])[[mu].sub.y], [delta] = [[sigma].sub.y.sup.2]/ [[sigma].sub.y.sup.2] + [[sigma].sub.e.sup.2] and [[sigma].sub.y|z.sup.2] = [[sigma].sub.y.sup.2][[sigma].sub.e.sup.2]/ [[sigma].sub.y.sup.2] + [[sigma].sub.e.sup.2]. (10) In other words, once the auditor observes the report z, he revises his expectation of earnings by weighting the unbiased system report, z - [omega], and the unconditional expectation of earnings, [[mu].sub.y], using their relative variances. Consequently, the total misstatement conditioned on the report is [[theta].sub.z] ~ N(z - [[mu].sub.y|z],[[sigma].sub.y|z.sup.2]) where z - [[mu].sub.y|z] = [omega] + (1 - [delta])(z - [omega] - [[mu].sub.y]).

Materiality of an Earnings Misstatement

According to AU[section]312, AICPA (2001), "Audit Risk and Materiality in Conducting an Audit," the auditor should take into account both quantitative and qualitative aspects of misstatements when considering whether they are material. For example, a $300,000 misstatement that is .001 percent of net assets and 5 percent of earnings may be material because users rely on earnings in making investment decisions. Qualitative factors also include items such as the regulatory requirements of the firm. According to SAB No. 99, auditors should not rely entirely on quantitative measures. However, quantitative measures (such as 5 percent of earnings) are important characteristics in judging the materiality of misstatements (see AU[section]9312.11 [AICPA 2001] for discussion). Moreover, percentage-of-earnings is often used by auditors as a quantitative measure of materiality (see Ricchiute 2002, 52).

To incorporate the judgmental characteristics (both quantitative and qualitative) of materiality, we assume that the materiality threshold of reported earnings is a normally distributed random variable [m.sub.z] ~ N([M.sub.z], [[sigma].sub.m.sup.2]). We denote this density t([m.sub.z]). (11) The subscript, z, indicates that the materiality measure depends on the magnitude of reported earnings, z. (12) To simplify the model, we assume that the earnings report affects only the mean of the materiality distribution. In other words, [M.sub.z] = [pi]z where [pi] is an exogenous percentage parameter. For example, in an earnings report of $20,000 and given [pi] = 5%, the auditor's materiality expectation, [M.sub.z], equals $1,000. As the earnings report increases, the expected materiality threshold increases.

Our setting includes the possibility of both intentional and unintentional earnings misstatements. Regarding the distinct nature of these two types of misstatements, the professional literature does not suggest that the auditor plan procedures to separately determine material misstatements of each type. Rather AU[section]312.05, AICPA (2001), points out that "[t]he auditor has no responsibility to plan and perform the audit to obtain reasonable assurance that misstatements, whether caused by errors or fraud, that are not material to the financial statements are detected" (emphasis added). Furthermore, AU[section]312.08, AICPA (2001), states, "When considering the auditor's responsibility to obtain reasonable assurance that the financial statements are free of material misstatement, there is no important distinction between errors and fraud."

In our model there is a single distribution over materiality that does not distinguish between (unintentional) errors and fraud (intentional errors). The auditor considers all misstatements, [[theta].sub.z], whether due to error or fraud, greater than or equal to [m.sub.z] as material.

Audit Evidence

We assume that the auditor observes a noisy evidence signal:

(4) [x.sub.z] = [[theta].sub.z] + [xi]

of the total misstatement [[theta].sub.z] where [xi] ~ N(0,[[sigma].sub.x.sup.2]) and [[theta].sub.z] and [xi] are independent. Thus, [x.sub.z] ~ N(z - [[mu].sub.y|z], [[sigma].sub.y|z.sup.2] + [[sigma].sub.x.sup.2]). We denote the density of [x.sub.z] as f([x.sub.z]) and the distribution function as F([x.sub.z]). Evidence, [x.sub.z], is the outcome of an audit procedure such as a detail test of a sample of transactions or analytical review. The evidence cannot distinguish between intentional and unintentional misstatements but as [x.sub.z] increases, the auditor infers that material misstatement is more likely. Normal, non-forensic-type audit procedures are often limited in their ability to detect a specific source of misstatement. As discussed above, the auditor in our model is concerned about whether, in total, a material misstatement exists, "due to error or fraud." (13)

Once the auditor observes [x.sub.z], he updates the probability of misstatement. The random variable for misstatement, conditional on both [x.sub.z] and z, is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [alpha] = [[sigma].sub.y|z.sup.2]/[[sigma].sub.y|z.sup.2] + [[sigma].sub.x.sup.2]. Based on the updated misstatement the auditor decides whether to accept the earnings report as materially correct or to extend audit procedures. Extended procedures may consist of additional standard audit procedures such as confirmation or forensic-type tests such as authenticating documents. (14)

Finally, we assume that the system is sufficiently noisy relative to the mean materiality percentage [pi] and the precision of the audit evidence [[sigma].sub.x.sup.2]. Specifically, we assume [[sigma].sub.e.sup.2] > [pi][[sigma].sub.x.sup.2][[sigma].sub.y.sup.2]/(1-[pi])([[sigma].sub.x.sup.2]+ [[sigma].sub.x.sup.2]), so that our game has an interior solution. While we make this assumption for tractability reasons, the assumption also makes economic sense for several reasons. First, the accuracy of information systems is limited due to cost. Second, the mean materiality percentage [pi] is likely to be small (less than 10 percent, see Ricchiute [2002, 50-55]). Third, the cost of audit evidence is also likely to be sufficiently small ([[sigma].sub.x.sup.2] is sufficiently small) so that this assumption is met.

The Players' Payoffs

The Auditor

There are two tensions guiding the auditor's choice of accepting the earnings report or extending audit procedures. If the auditor accepts an earnings report that is materially overstated, ([[theta].sub.z] - [m.sub.z] [greater than or equal to] 0), then he incurs a cost of audit failure equal to L. L incorporates all expected costs of a type II error, including potential litigation, sanctions, and reputation loss. After observing the audit evidence and the earnings report, the auditor updates the probability that the earnings misstatement is material. Let H(*|[x.sub.z]) denote the cumulative distribution for the earnings misstatement less materiality, [[theta].sub.z] - [m.sub.z], conditional on the observed evidence, [x.sub.z]. The auditor's expected payoff for accepting earnings as materially correct, given the audit evidence [x.sub.z] and an earnings report z is:

(5) - (1 - H(0|[x.sub.z))L

and H(*|[x.sub.z]) is the distribution function of the random variable [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The term, (1 - H(0|[x.sub.z])) represents the probability that the financial statements are materially misstated given the report z and the audit evidence, [x.sub.z]. That is, (1 - H(0|[x.sub.z])) is the updated, ex post measure of audit risk given the auditor accepts. (15)

If the auditor chooses to reject the report in favor of extending audit procedures, then he incurs a cost, R, that reflects the expected costs of the additional work and the potential bad will that is generated with the client. (16) Typically, L is considered to be large compared to R because auditors expect to incur large litigation costs for undiscovered material misstatements. (17) Most auditors are relatively more concerned with avoiding L relative to R.

The auditor extends audit procedures when:

(6) L(1 - H(0|[x.sub.z])) [greater than or equal to] R.

As a result, the updated probability of audit failure never exceeds R/L. (18)

Based on the Neyman-Pearson Lemma, the auditor's evaluation strategy can be represented as a cut-off value [c.sub.z] such that [x.sub.z] < [c.sub.z] results in acceptance and [x.sub.z] [greater than or equal to] [c.sub.z] results in extending procedures. (19) Consequently, the auditor's expected payoff based on his choice of [c.sub.z] for a given z is:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is his expected payoff from acceptance and -R(1 - F([c.sub.z])) is his expected payoff from extending procedures.

The Manager

The manager's expected payoff depends on reported earnings, which includes both intentional and unintentional misstatements. Because the manager chooses overstatement bias, [omega], before the earnings report, z, is realized, he calculates his expected payoff by anticipating how the earnings report affects his payoff for each report z and by anticipating how the auditor will evaluate the evidence for each observed [x.sub.z].

The manager receives a payoff proportional to the earnings report z but incurs an expected penalty of p[omega] if the auditor extends procedures. The penalty multiplier, p, captures the relative weightings on the positive impact of the earnings report and the negative impact of the penalty. (20) As a result, the manager anticipates an expected penalty that is proportional to to whether the amount of intentional misstatement is material or immaterial. This is consistent with SAS No. 82 (AICPA 1997, para. 38) that says evidence of "fraud involving senior management ... should be reported directly to the audit committee." Finally, we assume that the manager does not benefit from corrected misstatements arising from auditor rejection. (21) Once the auditor rejects, the manager's compensation (based on z) is reduced by p[omega], which represents fines, sanctions, or incarceration for criminal behavior. (22)

The manager chooses system bias prior to the realization of z and [x.sub.z]. Consequently, he averages his expected payoff over all possible reports. For each earnings report z, the manager considers the probability that [x.sub.z] [greater than or equal] [c.sub.z] in assessing the likelihood that audit procedures are extended. (23) The manager's expected payoff is:

(8) U = [[integral].sup.[infinity].sub.-[infinity]] -p[omega](1 - F([c.sub.z]))}[delta](z)dz.

We assume that if the manager chooses [omega] < 0, then he is either penalized zero or some positive amount. As a result, the manager chooses a non-negative system bias that potentially overstates reported earnings.

III. EQUILIBRIUM STRATEGIES

The solution concept for our model is a Bayesian-Nash equilibrium (see Fudenberg and Tirole 1995). Each player chooses his Nash strategy, given his beliefs about the earnings report and the equilibrium choice

of the other player. We solve for the players' equilibrium strategies by first considering the auditor's strategy as a function of the earnings report z and the observed evidence [x.sub.z], fixing the manager's overstatement bias, [omega].

Both [x.sub.z] and z provide the auditor with information about the updated expected earnings misstatement. The auditor maximizes his expected payoff by extending procedures when the expected cost of acceptance exceeds the cost of extending audit procedures as shown in Expression (6).

Expression (6) implies that the auditor extends audit procedures whenever:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[PHI].sup.-1](*) is the inverse of the standard normal distribution. The auditor compares the updated expected earnings misstatement (likely misstatement) to expected materiality less a constant that depends on the variance of the earnings misstatement, the variance of materiality and the auditor's relative costs for type I and type II errors. If L = 2R, then the auditor decides to extend procedures if and only if likely misstatement exceeds expected materiality. However, the expected cost of audit failure is typically large (L > 2R). In this case, the auditor extends audit procedures if likely misstatement exceeds an amount that is less than the expected materiality threshold. The effect of the manager's strategy and expected materiality on the likelihood of extending procedures is intuitive. As the amount of manager bias increases (which increases the mean of total misstatement [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or expected materiality decreases, the auditor is more likely to extend procedures.

When the auditor observes both [c.sub.z] and z, his decision criterion can be expressed as a relation between [x.sub.z] and a function of z denoted [c.sub.z] where the auditor accepts if [x.sub.z] < [c.sub.z] and rejects if [x.sub.z] [greater than or equal to] [c.sub.z]. We find the evaluation strategy [c.sub.z] by solving for the [x.sub.z] that results in auditor indifference between accepting and extending procedures. This [c.sub.z] maximizes the auditor's expected payoff in Expression (7) for each report z. Thus in equilibrium, the auditor's strategy for evaluating the audit evidence [x.sub.z] is equal to:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Expression (10) shows how the auditor's updated information about earnings, [[mu].sub.y|z] affects his evaluation strategy. A higher earnings report indicates a higher expected conditional earnings [[mu].sub.x|y]. However, a higher earnings report also increases the expected misstatement since (z - [[mu].sub.y|z]) [omega] + (1 - [delta])(z - [omega] - [[mu].sub.y]) is increasing in z. (24)

We derive the equilibrium strategies of the auditor and manager by expressing the auditor's strategy in a linear form and then solving for the manager's first order condition. Proposition 1 provides the unique equilibrium strategies of the auditor and manager.

Proposition 1: The unique interior equilibrium strategies of the auditor and manager are characterized as follows.

1) The auditor chooses [c.sub.z] = az + b with:

a = [pi] - (1 - [alpha])(1 - [delta])/[alpha]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2) Given a and b defined in part (1), the manager chooses overstatement bias, [omega], that satisfies:

(11) 1 - p(1 - F(c)) - (1 - a)p[omega]f(c) = 0

where (1 - a) > 0, c = a([omega] + [[mu].sub.y]) + b and f(*) is the density and F(*) is the distribution function of the unconditional evidence signal, x ~ N([omega], [(1 - a).sup.2] [[sigma].sub.e] + [a.sup.2][[sigma].sup.2.sub.y] + [[sigma].sup.2.sub.x]).

All proofs are in the Appendix.

Proposition 1 part (1) shows that the auditor's evaluation strategy is linear in z. For a given z, he responds to the manager's choice of overstatement bias by adjusting [c.sub.z] proportionate to [omega]. This is consistent with the fact that the manager chooses overstatement bias before the realization of y and his choice cannot be affected by y.

We obtain the manager's equilibrium condition in Expression (11) by substituting the auditor's evaluation strategy [c.sub.z] in its general form, az + b, into the manager's payoff function, integrating over z and then differentiating with respect to [omega]. The manager's choice of overstatement bias takes into account the marginal (ex ante) effect on his payoff (1 - p(1 - F(c))) plus the marginal effect of [omega] on the probability of rejection (-(1 - a)p[omega](c)).

IV. COMPARATIVE ANALYSIS

In this section, we provide a comparative analysis in order to explore the interactions of the manager and the auditor. Specifically, we consider how materiality uncertainty and the earnings report affect the equilibrium strategies and audit risk.

The Effect of Materiality Uncertainty on the Equilibrium Strategies

Materiality measures how misstatements affect the decisions of users and, as a result, the size of misstatement the auditor is willing to accept as inconsequential to financial statement fairness. Given that materiality is an inherently uncertain quantity, we perform a comparative analysis of how uncertainty in the materiality threshold affects the players' equilibrium strategies.

Proposition 2 shows how the auditor's choice of evaluation strategy and the manager's choice of overstatement bias change with the degree of materiality uncertainty. (25)

Proposition 2: In equilibrium:

1) the auditor's evaluation strategy, [c.sub.z], and the manager's choice of overstatement bias, to, decrease in materiality uncertainty, [[sigma].sub.m] if and only if 2R < L.

2) the auditor's evaluation strategy, [c.sub.z], and the manager's choice of overstatement bias, [omega], increase in materiality uncertainty, [[sigma].sub.m] if and only if 2R > L.

Some proponents of a well-defined materiality standard argue that materiality ambiguity (i.e., uncertainty) results in non-uniform financial statements, allowing the auditor a great deal of latitude in determining what is material. (26) Thus, some auditors provide more leeway in determining the amount of misstatement that is material. A more lenient evaluation strategy in turn influences the manager to increase the amount of overstatement bias. However, Proposition 2 shows that when the cost of audit failure is large relative to the cost of extending audit procedures (L > 2R), an increase in materiality uncertainty induces a more conservative evaluation rule.

Based on the manager's equilibrium condition in Proposition 1, we find that materiality uncertainty [[sigma].sub.m] affects the manager's decision only through its impact on [c.sub.z]. If the auditor's evidence evaluation [c.sub.z] decreases, then the probability of the auditor finding evidence of fraud increases. This induces the manager to decrease the amount of overstatement bias. Thus:

(12) d[omega]/d[[sigma].sub.m] < 0.

On the other hand, a change in materiality uncertainty, [[sigma].sub.m] affects the auditor's choice of evaluation strategy, [c.sub.z], in two ways, directly and indirectly. The total change in the auditor's evaluation strategy is:

(13) d[c.sub.z]/d[[sigma].sub.m] = [differential][c.sub.z]/[differential][[sigma].sub.m] + ([differential][c.sub.z]/[differential][omega])(d[omega]/d[[sigma].sub.m]) < 0.

We obtain the direct effect from the auditor's equilibrium strategy in Proposition 1. The direct effect is negative if and only if [[PHI].sup.-1](L - R)/L) > 0, where [PHI] is the standard normal cumulative distribution function. This condition is satisfied if and only if L > 2R. Hence, the direct impact of materiality uncertainty [[sigma].sub.m] on evidence evaluation [c.sub.z] is negative if and only if L > 2R.

The indirect effect is the impact that materiality uncertainty [[sigma].sub.m] has on evidence evaluation [c.sub.z] through its impact on overstatement bias [omega]. As discussed above (see Expression (12)), we know that overstatement bias [omega] decreases in [[sigma].sub.m]. From Proposition 1 we know that evidence evaluation [c.sub.z] decreases in intentional misstatement [omega]. Therefore, the indirect effect is positive. However the indirect effect is always overwhelmed by the direct effect because the auditor's reaction to a change in overstatement bias [omega] is not particularly strong relative to his reaction to [[sigma].sub.m]. This is because the auditor is concerned with unintentional and intentional misstatements and the direct effect incorporates both types of misstatement, whereas the indirect effect incorporates only intentional misstatements. Thus the direct effect, arising from [differential][c.sub.z]/[differential][[sigma].sub.m] drives the result of Proposition 2.

The change in sign at L = 2R that we observe in Proposition 2 originates from the auditor's payoff structure. (27) The auditor incurs zero cost for accepting an immaterial misstatement, but as soon as the misstatement reaches the materiality threshold he immediately incurs an expected cost of L. Thus in choosing [c.sub.z] (which also determines the threshold of acceptable updated misstatement, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the auditor weighs the relative costs of accepting and extending procedures based on L and R. The auditor chooses an [x.sub.z] = [c.sub.z] such that the conditional expected loss from a type II error (the constant cost, L, multiplied by the updated probability of a material misstatement) is exactly the same as the certain cost of extending audit procedures, R. For any [x.sub.z] smaller than this critical value, the auditor would strictly be better off by not extending procedures and for any larger [x.sub.z], the auditor would be strictly better off by extending procedures. Consequently, at [x.sub.z] = [c.sub.z] the updated probability of material misstatement is equal to the constant R/L and for every observed z, the maximum updated probability of audit failure is R/L.

Suppose L = 2R or R/L = 1/2. In this case, the auditor is indifferent between acceptance and extending procedures if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This implies that the auditor chooses a critical value [c.sub.z] that equates the threshold of updated misstatement to expected materiality, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], because 1 - [PHI](0) = 1/2.

When the cost of audit failure is high relative to the cost of rejection (L > 2R), the auditor evaluates the evidence more conservatively because he has a greater incentive to avoid audit failure. Greater uncertainty about the materiality threshold increases the updated probability of material misstatement, exposing the auditor to a greater risk of audit failure. Consequently, the auditor evaluates the evidence more conservatively for higher levels of materiality uncertainty. If L > 2R, then R/L < 1/2 and the auditor is indifferent between accepting and extending procedures if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The updated probability of material misstatement is increasing in [c.sub.z] and [[sigma].sub.m]. In order to achieve a constant updated probability of material misstatement, R/L, for a given amount of overstatement bias [omega] at [x.sub.z] = [c.sub.z], an increase in the exogenous parameter, [[sigma].sub.m], must be offset by a commensurate decrease in the auditor's strategic choice, [c.sub.z]. This is the direct effect of materiality uncertainty [[sigma].sub.m], on the auditor's evaluation strategy [x.sub.z].

On the other hand, when L < 2R, the auditor chooses an evaluation strategy [c.sub.z] such that at [x.sub.z] = [c.sub.z] the updated expected misstatement is greater than expected materiality. As a result, the updated probability of material misstatement is decreasing in materiality uncertainty for a given amount of overstatement bias. The effects of materiality uncertainty on the auditor's strategy in turn influences the manager to decrease overstatement bias when L > 2R or increase overstatement bias when L < 2R.

Proposition 2 shows that the effect of materiality uncertainty depends on the relation between L and R. However, we anticipate that L > 2R because the expected economic consequences of audit failure are usually thought to be large relative to the expected cost of extending procedures. The auditor faces a large expected cost from audit failure due to the potential losses from investors and reputation losses. As a result, increased materiality uncertainty produces a more conservative auditor evaluation strategy and a lesser amount of overstatement bias.

Materiality Effects on Audit Risk

According to SAS No. 47 (AICPA 1984, para..02), audit risk is defined as "the risk that the auditor may unknowingly fail to appropriately modify his opinion on financial statements that are materially misstated." Audit risk in our setting is simply the average probability that the auditor accepts a materially misstated earnings report. (28) Proposition 3 describes the effects of materiality uncertainty on audit risk.

Proposition 3: When L > 2R, audit risk, defined as:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has three additive components resulting from an increase in materiality uncertainty.

1) The first component is the direct effect of an increase in materiality uncertainty, holding [c.sub.z] and [omega] constant. This component increases in materiality uncertainty.

2) The second component is the indirect effect of an increase in materiality uncertainty due to its effect on [c.sub.z]. This component decreases in materiality uncertainty.

3) The third component is the indirect effect of an increase in materiality uncertainty due to its effect on [omega]. This component either increases or decreases in materiality uncertainty.

In standard normal form, audit risk is:

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When L > 2R, the auditor's evaluation strategy and the manager's overstatement bias decrease in materiality uncertainty. The decrease in the auditor's evaluation strategy corresponds to a decrease in audit risk, while a decrease in the manager's overstatement bias may increase or decrease audit risk. On the other hand, for given values of [c.sub.z] and [omega], the direct effect of increasing materiality uncertainty, [[sigma].sub.m], is to increase audit risk. Because of these three distinct effects of materiality uncertainty on audit risk, we cannot predict its net affect on the overall change in audit risk. (29)

Figure 1 illustrates this result with a numerical example for two different earnings reports. (30) In this example, audit risk eventually decreases in [[sigma].sub.m], because for larger values of [[sigma].sub.m] the rate of decrease in the auditor's evaluation strategy is higher. As a result, audit risk is decreasing at a faster rate from the indirect effects of materiality uncertainty than from the direct effects. Figure 2, Panel A shows how [c.sub.z] changes as materiality uncertainty increases. Both the size of the earnings report and materiality uncertainty affect the changes in audit risk. When materiality uncertainty is high, the auditor is relatively more cautious in his evidence evaluation for subsequent increases in materiality uncertainty. This effect is more pronounced for higher earnings reports because they signal the possibility of a higher level of system error.

[FIGURES 1-2 OMITTED]

Proposition 3 also provides intuition about the effect that materiality uncertainty has on the auditor's expected payoff (see Expression (7)). The auditor's expected payoff is equal to negative L times audit risk minus R times the probability of extending procedures. The indirect effects of materiality uncertainty affect both audit risk and the probability of extending procedures, often in opposite ways. For example the marginal decrease in audit risk times L exactly offsets the marginal increase in the probability of extending procedures times R when the auditor's evaluation rule decreases. However, the direct affect of materiality uncertainty only changes audit risk and we know that the direct effect of materiality uncertainty increases audit risk (reduces the auditor's expected payoff). While we cannot determine how the auditor's expected payoff changes due to a change in materiality uncertainty, numerical examples suggest that the auditor's expected payoff decreases in materiality uncertainty.

The Effect of the Earnings Report on the Equilibrium Strategies

The earnings report provides the auditor with information about unintentional system error and the expected dollar amount of materiality, [M.sub.z] = [pi]z. As the amount of reported earnings increases, expected materiality increases. The professional literature suggests that as the dollar amount of materiality increases, the auditor increases his evaluation strategy. (31) However, Proposition 4 shows that in our strategic setting just the opposite may occur.

Proposition 4: When the earnings report, z, increases:

1) the auditor's evaluation strategy, [c.sub.z], decreases if and only if

[[sigma].sub.e.sup.2] > [pi][[sigma].sub.x.sup.2][[sigma].sub.y.sup.2]/(1 - [pi])[[sigma].sub.x.sup.2] - [pi] [[sigma].sub.y.sup.2]

and (1 - [pi])[[sigma].sub.x.sup.2] > [pi][[sigma].sub.y.sup.2], and

2) the manager's overstatement bias, [omega], does not change.

The condition in Proposition 4 part (1) depends on the system variance [[sigma].sub.e.sup.2] relative to the materiality percentage [pi] and the variances of both audit evidence and true earnings. If the system variance is large and the materiality percentage [pi] is small so that (1 - [pi])[[sigma].sub.x.sup.2] > [pi][[sigma].sub.y.sup.2], then an increase in the earnings report increases the probability of material error. Consequently, the auditor's evaluation strategy, [c.sub.z], decreases.

Figure 2, Panel B illustrates how the evaluation strategy, [c.sub.z], changes for an increase in the earnings report, z, when the condition of Proposition 4 part (1) holds. Figure 2, Panel B also illustrates the second part of Proposition 4. The amount of the manager's overstatement bias is unaffected by the earnings report, z, because he does not observe earnings prior to his strategy choice.

V. CONCLUSION

Recently, the accounting profession and the SEC have focused attention on the issue of materiality that allows managers to overstate earnings by an immaterial amount. The SEC is concerned that the amount managers manipulate their reports depends on the materiality threshold. If misstatements fall below the materiality threshold, then the auditor is willing to accept those misstatements because, theoretically, they have no impact on the decisions of reasonable users. Furthermore, some controversy exists as to whether materiality should be quantitatively standardized. However, in considering this issue, careful attention should be given to how strategic players would react to the uncertainty of a materiality threshold.

We investigate, in a game-theoretic model, how uncertainty about the materiality threshold affects the conservatism of the auditor's evidence evaluation choice, the extent of the manager's overstatement bias, and the risk of materially misstated financial statements. We find that the auditor's conservatism increases in the uncertainty of materiality when the expected cost of audit failure is large relative to the expected cost of extending audit procedures. Auditor conservatism, in turn, induces the manager to decrease the extent of overstatement bias. Just the opposite occurs when the expected cost of extending procedures is large relative to the expected cost of audit failure. However, because the expected cost of audit failure is likely to be large relative to the expected cost of extending audit procedures, our model suggests that a fixed materiality standard would be detrimental in promoting auditor avoidance of audit failure.

Surprisingly, though, audit risk or the average probability that audit failure occurs, can actually increase as the uncertain materiality threshold increases even though the auditor evaluates the evidence more conservatively and the extent of intentional overstatement decreases. The change in the auditor's equilibrium strategy decreases audit risk indirectly. However, the variance of the materiality threshold directly increases audit risk and this increase can have a dominant effect on the overall change in audit risk.

Of course, our study is limited by our stylized setting. Typically decisions about materiality are more complex, auditors' payoffs are affected by additional legal concerns, and managers consider a vast array of motivational factors. However, adding more complexity to the model is unlikely to negate the result that materiality uncertainty can decrease the extent of overstatement bias while simultaneously increasing or decreasing audit risk.

[FIGURE 1 OMITTED]

APPENDIX

Proof of Proposition 1

Part (1)

In equilibrium, [c.sub.z] = [x.sub.z] equates the left-hand and right-hand sides of Expression (9) so that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Or:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus [c.sub.z] = az + b where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Part (2)

Recall from Expression (8) that the manager's expected payoff is:

(A1) U = [[integral].sup.[infinity].sub.-[infinity]] {z - p[omega](1 - F([c.sub.z]))}[lambda](z) dz. = [[mu].sub.y] + [omega] - p[omega] + p[omega] [[integral].sup.[infinity].sub.-[infinity]] F([c.sub.z])[lambda](z) dz

where:

[[integral].sup.[infinity].sub.-[infinity]] F([c.sub.z])[lambda](z) dz = [[integral].sup.[infinity].sub.-[infinity]] Prob([x.sub.z] < [c.sub.z]|z)[lambda](z) dz.

We also know from general probability theory (see Ash [1972, 240-243] for discussion) that Prob({z [member of] A, x [member of] B}) = [[integral].sup.[infinity].sub.-[infinity]] Prob([x.sub.z] < [c.sub.z]|z)[lambda](z) dz where event B is [x.sub.z] < [c.sub.z] and event A is z [member of] {-[infinity],[infinity]}. Furthermore, because all auditor equilibrium evaluation roles [c.sub.z] have the general form [c.sub.z] = az + b, [x.sub.z] < [c.sub.z] is equivalent to [x.sub.z] < az + b and Prob({z [member of] A, x [member of] B}) = Prob(x - az < b).

Therefore, because the random variable x - az is normal:

Prob(x - az < b) = [PHI] (b - (E[x - az])/[square root of (Var[x - az]))]

where:

E[x - az] = E[e + [omega] + [zi] - a(y + e + [omega])] = [omega] - a([[mu].sub.y] + [omega]) and

Var[x - az] = Var[(1 - a)e - ay + [zi]] = [(1 - a).sup.2][[sigma].sub.e] + [a.sup.2][[sigma].sub.y.sup.2] + [[sigma].sub.x.sup.2].

Thus:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the manager's expected payoff in Expression (A1) can be written as:

(A2) U = [[mu].sub.y] + [omega] - p[omega](1 - F(c))

where c = a([[mu].sub.y] + [omega]) + b and F(*) is the unconditional distribution function of x with mean [omega] and variance [(1 - a).sup.2][[sigma].sub.e.sup.2] + [a.sup.2][[sigma].sub.y.sup.2] + [[sigma].sub.x.sup.2].

Consequently, using the chain role to differentiate U in Expression (A2), we obtain:

(A3) dU/d[omega] = 1 - p(1 - F(c)) - (1 - a)p[omega][??](c) = 0

as the manager's equilibrium condition where a and b are defined as in part (1).

Proof that [omega] Satisfying Expression (A3) Is a Payoff Maximum

Let [U.sub.[omega]] = dU/d[omega] and [u.sub.[omega]j] = [d.sup.2]U/d[omega] dj.

Recall that c = a([omega] + [[micro].sub.y]) + b. Thus, c - [omega] = (a - 1)[omega] + a[[micro].sub.y] + b.

Denote [phi](*) and [PHI](*) as the standard normal density and cumulative distribution, respectively. Using the chain rule where:

[??](c) = 1/[(1 - a).sup.2][[sigma].sub.e.sup.2] + [a.sup.2][[sigma].sub.y.sup.2] + [[sigma].sub.x.sup.2] [phi] (c - [omega]/[(1 - a).sup.2][[sigma].sub.e.sup.2] + [a.sup.2][[sigma].sub.y.sup.2] + [[sigma].sub.x.sup.2])

and

F(c) = [PHI](c - [omega]/[(1 - a).sup.2][[sigma].sub.e.sup.2] + [a.sup.2][[sigma].sub.y.sup.2] + [[sigma].sub.x.sup.2]),

we derive the manager's second order condition:

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, fix [omega] at [omega] that solves [U.sub.[omega]] = 0 and consider the limiting behavior of [U.sub.[omega]](b) as b decreases and increases. Our objective is to determine the sign of [U.sub.[omega][omega]] when [U.sub.[omega]] = 0 and [omega] = [omega]. We determine this sign by first determining the sign of [U.sub.[omega]b] and then exploit the fact (that we will show later) that [U.sub.[omega][omega]] = -(1 - a){[U.sub.[omega]b], + p[??]c)}. In other words, if [U.sub.[omega]b], is positive, then [U.sub.[omega][omega]] is negative if and only if 1 > a.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, as b increases, [U.sub.[omega]], is increasing in b the first time it crosses zero. Furthermore:

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now the term in braces in (A5) remains positive if (1 - a) > 0 because it increases in b. If (1 - a) < 0, then the tern in brackets in (A5) can be zero for only one value of b and we know that at b = + [infinity], [U.sub.[omega]b] > 0. Thus [U.sub.[omega]b] can cross zero only once and must be positive when [U.sub.[omega]] = 0. As a result, [U.sub.[omega]b] > 0 whenever [U.sub.[omega]] = 0.

We substitute [U.sub.[omega]b] (Expression (A5)) into [U.sub.[omega][omega]] (Expression (A4)) to obtain:

(A6) [U.sub.[omega][omega]] = -(1 - a){[U.sub.[omega]b] + p[??](c)} when [U.sub.[omega]] = 0.

Thus:

[U.sub.[omega][omega]] < 0 [??] (1 - a) > 0 when [U.sub.[omega]] = 0.

Finally:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is tree based on our assumption of a noisy information system.

Thus, [omega] is a unique maximizing strategy for the manager.

Proof of Equilibrium Uniqueness

In order to prove the equilibrium {[omega], [c.sub.z]} is unique, we must show three things.

(1) There exists a unique strategy that maximizes the auditor's expected payoff given any choice of [omega] by the manager.

This is satisfied based on part (1) of Proposition 1, which shows that [c.sub.z] is linear in [omega].

(2) There exists a unique strategy that maximizes the manager's expected payoff given any choice of [c.sub.z] = az + b by the auditor.

This is satisfied by the proof above that shows [U.sub.[omega][omega]] < 0, whenever [U.sub.[omega]] = 0.

(3) There is only one set of strategies ({[omega], [c.sub.z]}) that satisfy (1) and (2). To show this we consider the manager's equilibrium condition. Let EqU = 1 - p(1 - F(c)) - (1 - a)p[omega][??](c) = 0 where a and b are defined as in part (1) of Proposition 1. Thus, dEqU/d[omega] = [U.sub.[omega][omega]] + [U.sub.[omega]b] db/d[omega]. Because db/d[omega] = -(1 - [alpha])[delta]/[alpha] = < 0, [U.sub.[omega][omega]] < 0 and [U.sub.[omega]b] > 0 for the equilibrium choice of [omega], dEqU/d[omega] < 0. Thus the equilibrium of our game is unique.

Proof of Proposition 2

By implicit differentiation:

d[omega]/d[[sigma].sub.m] = [dEqU/d[[sigma].sub.m]]/[dEqU/d[omega]]

and in Proposition 1 we show that dEqU/d[omega] = < 0. Using the chain role we obtain:

d[c.sub.z]/d[[sigma].sub.m] = [differential][c.sub.z]/[differential][[sigma].sub.m] + [differential][c.sub.z]/[differential][omega] d[omega]/d[[sigma].sub.m].

Part (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, d[omega]/d[[sigma].sub.m] < 0 [??] 2R < L.

Next [differential][c.sub.z]/[differential][[sigma].sub.m] = -[[PHI].sup.1](L - R/L)[[sigma].sub.m]/ [alpha][square root of ((1 - [alpha])[[sigma].sub.y|z.sup.2] + [[sigma].sub.m.sup.2]]), [differential][c.sub.z]/[differential][omega] = -(1 - [alpha])[differential]/[alpha] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when L > 2R. This implies that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To show that [dc.sub.z]/d[[sigma].sub.m] < 0 implies L > 2R, suppose by way of contradiction that L [less than or equal to] 2R. Then by the proofs of parts (2) and (3) we arrive at a contradiction. Thus d[c.sub.z]/d[[sigma].sub.m] < 0 [??] L > 2R.

Part (2)

Also:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, d[omega]/d[[sigma].sub.m] > 0 [??] 2R > L.

The proof of the sign of d[c.sub.z]/d[[sigma].sub.m] when L < 2R is similar to part (1) except:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when L < 2R.

Thus d[c.sub.z]/d[[sigma].sub.m] = [differential][c.sub.z]/[differential][[sigma].sub.m] + [differential][c.sub.z]/[differential][omega] d[omega]/d[[sigma].sub.m] > -[[PHI].sup.-1](L - R/L)[[sigma].sub.m]/[square root of ((1 - [alpha])[[sigma].sub.y|z.sup.2] + [[sigma].sub.m.sup.2])] (1 - a)/(1 - [pi]) > 0 when L < 2R.

To show that d[c.sub.z]/d[[sigma].sub.m] > 0 implies L < 2R, suppose by way of contradiction that L [greater than or equal to] 2R. Then by the proofs of parts (1) and (3) we arrive at a contradiction. Thus d[c.sub.z]/d[[sigma].sub.m] > 0 [??] L < 2R.

Part (3)

The Knife-Edge Case, L = 2R.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, d[omega]/d[[sigma].sub.m] = 0 [??] 2R = L.

The proof for the sign of d[c.sub.z]/d[[sigma].sub.m] = 0 when L = 2R is similar to part (1) except:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when L = 2R.

Thus d[c.sub.z]/d[[sigma].sub.m] = [differential][c.sub.z]/[differential][[sigma].sub.m] + [differential][c.sub.z]/[differential][omega] d[omega]/d[[sigma].sub.m] = [differential][c.sub.z]/ [differential][[sigma].sub.m] = -[[PHI].sup.-1](L - R/L)[[sigma].sub.m]/[alpha] [square root of ((1 - [alpha])[[sigma].sub.y|z.sup.2] + [[sigma].sub.m.sup.2])] = 0 when L = 2R.

To show that d[c.sub.z]/d[[sigma].sub.m] = 0 implies L = 2R, suppose by way of contradiction that L [not equal to] 2R. Then by the proofs of parts (1) and (2) we arrive at a contradiction. Thus d[c.sub.z]/d[[sigma].sub.m] = 0 [??] L = 2R.

Proof of Proposition 3

Let Audit Risk be denoted as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The overall change in audit risk due to a change in materiality uncertainty is:

dAR/d[[sigma].sub.m] = [differential]AR/[differential][c.sub.z] {[differential][c.sub.z]/[differential][[sigma].sub.m] + [differential][c.sub.z]/[differential][omega] d[omega]/d[[sigma].sub.m]} + [differential]AR/[differential][omega] d[omega]/d[[sigma].sub.m] + [differential]AR/[differential][[sigma].sub.m]

where [differential]AR/[differential][c.sub.z] {[differential][c.sub.z] /[differential][[sigma].sub.m] + [differential][c.sub.z]/[differential] [omega] d[omega]/d[[sigma].sub.m]} is an indirect change arising from the corresponding change in [c.sub.z], [differential]AR/[differential][omega] d[omega]/d[[sigma].sub.m] is an indirect change arising from a corresponding change in [omega], and [differential]AR/[differential][[sigma].sub.m] is the direct change in audit risk.

Part (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

because when L > 2R, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 0 for all [x.sub.z] [less than or equal to] [c.sub.z].

Part (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

because in Proposition 2 we show that {[differential][c.sub.z] /[differential][[sigma].sub.m] + [differential][c.sub.z]/[differential] [omega] d[omega]/d[[sigma].sub.m]} < 0 when L > 2R and because at [x.sub.z] = [c.sub.z], Expression (6) is an equality.

Part (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Proposition 4

Part (1) [Part (2) follows directly from Proposition 1]

Recall from Proposition 1 part (1) that [c.sub.z] = az + b. Thus d[c.sub.z]/dz] = a.

From Proposition 1 we know that:

a = [pi]([[sigma].sub.y.sup.2][[sigma].sub.e.sup.2] + [[sigma].sub.x.sup.2] [[sigma].sub.e.sup.2] + [[sigma].sub.x.sup.2][[sigma].sub.y.sup.2]) - [[sigma].sub.x.sup.2][[sigma].sub.e.sup.2]/[[sigma].sub.y.sup.2] [[sigma].sub.e.sup.2].

The numerator of a equals:

[pi][[sigma].sub.y.sup.2][[sigma].sub.e.sup.2] + [pi][[sigma].sub.x.sup.2] [[sigma].sub.y.sup.2] - (1 - [pi]) [[sigma].sub.x.sup.2][[sigma].sub.e.sup.2] = [[sigma].sub.e.sup.2] - (1 - [pi]) [[sigma].sub.x.sup.2]) [[sigma].sub.x.sup.2][[sigma].sub.y.sup.2].

Consequently:

a < 0 [??] [[sigma].sub.e.sup.2] > [pi][[sigma].sub.x.sup.2] [[sigma].sub.y.sup.2]/(1 - [pi])[[sigma].sub.x.sup.2] - [pi] [[sigma].sub.y.sup.2]

and (1 - [pi])[[sigma].sub.x.sup.2] - [pi][[sigma].sub.y.sup.2] > 0.

The authors appreciate the helpful comments of Paul Newman, Rajib Doogar, Bill Heninger, Bill Kinney, Dale Morse, Kenny Reynolds, Mike Stein, and participants at the American Accounting Association 2001 Midyear Auditing Conference. We give special thanks to Richard Sansing and to two anonymous reviewers for their helpful suggestions.

Editor's note: This paper was accepted by Richard C. Sansing, Special Editor.

(1) This is the definition adopted by the Auditing Standards Board in Statement on Auditing Standards (SAS) No. 47, Audit Risk and Materiality in Conducting an Audit (AICPA 1984).

(2) As we discuss in Section II, rejection in our setting implies that the auditor extends audit procedures rather than actually issuing a modified report.

(3) Proponents of a precise materiality standard argue that the lack of a precise definition induces non-uniformity in the presentation of financial information and makes it impossible for companies to know what is expected of them. Jennings et al. (1985) argue that "[t]he accounting profession should be strongly encouraged to revive their efforts to establish explicit standards in the materiality area, and the legal profession should actively contribute to that process." In addition, ambiguity not only creates a problem for preparers and users, but also for enforcement and for the courts. John Fedders (1998), the former Director of Enforcement for the SEC, asserts that "[t]he tendency of the law must always be to narrow the field of uncertainty, so that citizens may know by what standards their conduct will be judged." He argues that materiality has been impossible to implement from an enforcement perspective because of its ambiguity.

(4) See Section II for a more detailed discussion of the qualitative aspects of materiality.

(5) When the auditor evaluates the evidence more conservatively, he rejects earnings as materially correct for evidence observations that previously would have resulted in acceptance.

(6) See Kiernan and Lewin (1994) and Trager (2002).

(7) This is also referred to as the Anti-Fraud bill. See Engineering News-Record (2002).

(8) See Konrath (2001, 719-721) for an example of an information system with this feature. Anecdotally, turnkey software packages often generate a sales invoice (i.e., record a sale) prior to evidence of the shipment of goods. Thus, the system relies on a control that purges a sale if no shipment occurs.

(9) Despite the choice of overstatement bias by the manager prior to the realization of earnings, the manager is likely to have private information about the information system or the earnings distribution. For simplicity, we do not model this aspect of the manager's information.

(10) See Berger (1985, 127-128) for details on deriving the posterior normal distribution.

(11) Our assumption that the materiality threshold is normally distributed is made for technical convenience. We have no reason to assume that the uncertainty about the threshold is represented by any specific distribution. The assumption of normality allows us to examine how the variance (or degree of uncertainty) affects the interaction between the auditor and manager in our model. In addition, we recognize that this assumption admits some unrealistic values for the materiality threshold, such as negative amounts. This limitation of our assumption does not affect our results qualitatively.

(12) In determining materiality, the auditor typically uses reported values as a base amount (see Ricchiute [2002, 50-58 and 67] for discussion). Of course, materiality based on expected earnings is also reasonable and our results are not affected by considering this measure of materiality.

(13) This is typically how an audit is performed. For example, if the evidence is obtained from analytical review of sales, then the auditor is only concerned with the difference between total predicted sales and total recorded sales. Transaction misstatements that net to zero do not overstate recorded sales nor do they provide the manager with incremental benefits in our model. Our model is not intended to address illegal acts such as illegal payments where individual small amounts can affect the financial statements in a material way. Similarly, our model does not consider the audit of multiple accounts (a multilocation problem) within earnings where the auditor is concerned about the materiality of account classification. Newman et al. (1996), for example, examine a mul tilocation audit setting but do not include materiality in their analysis. We leave the study of materiality in a multilocation setting for future research.

(14) The Public Oversight Board (POB) recently issued "The Panel on Audit Effectiveness: Report and Recommendations'' in 2000 that suggests auditors incorporate forensic-type procedures into the audit plan. If the auditor is not trained to perform forensic-type tests that are warranted, then the auditor should consider engaging a Certified Fraud Examiner.

(15) See footnote 28 for a discussion of ex ante and ex post audit risk measures.

(16) We assume that all identified misstatements are corrected. This is quite common when audit procedures are extended. Extending audit procedures is a form of "announcement" that audit problems likely exist. Moreover, because the client incurs costs for the extra work, the client is usually compelled to make corrections for misstatements the auditor finds. Of course, the remote possibility exists that the auditor will find misstatements so negligible that no adjustments are made. We do not model the technical details of this stage of the game and leave it for future research.

(17) See Kiernan and Lewin (1994) and Trager (2002).

(18) Note that if R > L, then extending audit procedures is a dominated strategy and the auditor always accepts. Because we do not observe this behavior in practice, R is likely to be less than L.

(19) See Berger (1985, 523-530). We technically solve our game as a simultaneous move game in which the auditor chooses [c.sub.z] and the manager chooses [omega]. We could alternatively state that the auditor chooses [c.sub.z] at the beginning of the game, observes z, and then compares the stochastic signal to [c.sub.z] in deciding whether to"accept or reject.

(20) For example, the manager may receive a 10 percent bonus based on reported earnings and be penalized twice the overstatement if caught. The net effect of these two multiples is captured in p. The value of p also captures the possibility that the penalty may not be levied with certainty.

(21) Our model does not include the potential negotiation between the auditor and manager involved in determining whether identified misstatements result in required adjustments by the auditor. Our setting focuses on the identification of fraud when it exists and procedures are extended. Nevertheless, our model captures the concerns of Arthur Levitt and the practitioner literature based on the manager's anticipation of the size of likely misstatement that the auditor finds acceptable. In discussing the evaluation of audit findings, AU[section]312.27, AICPA (2001) notes that "[I]n evaluating whether the financial statements are presented fairly, in all material respects ... the auditor should aggregate misstatements that the entity has not corrected ... that enables him to consider whether ... they materially misstate the financial statements." AU[section]312.28, AICPA (2001) goes on to say that "It]he aggregation of misstatements should include the auditor's best estimate of the total misstatements ... (hereafter referred to as likely misstatement), not just the amount of misstatements he specifically identifies." Our model captures this portion of the evaluation process in which the auditor projects likely misstatement and must decide the likelihood that the misstatement is material to reported earnings.

(22) For simplicity, we assume that all managers are prone to fraud. As a result, the auditor can compute the amount of fraud in equilibrium. This assumption is made without loss of generality though. As discussed in Newman et al. (1996), even though the auditor may be able to infer the amount of fraud no manager penalties can be assessed unless the auditor obtains evidence of fraud. Furthermore, our results remain qualitatively unchanged if we add an inherently honest manager to the game.

(23) Clearly, p must be greater than 1. Otherwise, the manager has no incentive to limit overstatement bias.

(24) We examine the equilibrium effect of the earnings report on [c.sub.z] in Proposition 4.

(25) In the knife-edge case where L = 2R, the auditor's evaluation strategy, [c.sub.z], and the manager's choice of overstatement bias, [omega], are unaffected by materiality uncertainty, [[sigma].sub.m].

(26) See Jennings et al. (1985).

(27) Alternative auditor payoffs lead to similar results. For example, if the auditor's expected cost of audit failure is ([[theta].sub.z] - [m.sub.z])L or [[theta].sub.z]L, then the relative weightings of L and R also affect the choice of [c.sub.z] where [c.sub.z] decreases in L. The cost ([[theta].sub.z] - [m.sub.z])L can be described as Max[0, [[theta].sub.z] - [m.sub.z]], which is continuous in [[theta].sub.z]. In this case, the auditor decreases [c.sub.z] and the manager decreases [omega] for increases in materiality uncertainty regardless of the relation between L and R. However, when the auditor's cost of audit failure is [[theta].sub.z], then, as in Proposition 2, the effect of materiality uncertainty depends on the relative sizes of L and R. (Details of these alternative analyses are available upon request.) This suggests that the discontinuity of the auditor's cost function at ([[theta].sub.z] - [m.sub.z]) influences whether the size of L plays a role in how materiality uncertainty affects the equilibrium behavior of the auditor and manager.

(28) There are two potential measures of audit risk, ex ante and ex post audit risk. Most often studies (see for example, Newman and Noel 1989; Patterson 1993) use ex ante audit risk (risk prior to evidence observation) because it coincides with SAS No. 47's audit risk formula that is "inherent risk times control risk times detection risk." Ex post audit risk depends on evidence observation and is equal to the probability in the braces of Expression (15). In our model this never exceeds R/L when the auditor accepts. To be consistent with prior studies and SAS No. 47, we use the ex ante measure of audit risk.

(29) When L < 2R, the analytical results for changes in audit risk are also ambiguous. In this case, we cannot determine how audit risk changes due to the direct effect, but are unable to find a numerical example in which audit risk decreases in materiality uncertainty when L < 2R.

(30) We calculated all of our numerical examples using the following parameters: rrx = 8.41, ry = 164, re = 280, [[mu].sub.y] = 20,000, [pi] = .05, p = 1.3, L = 1,000, R = 200.

(31) See Konrath (2001).

REFERENCES

American Institute of Certified Public Accountants (AICPA). 1984. Materiality and Audit Risk. SAS No. 47. New York, NY: AICPA.

--. 1997. Considering Fraud in a Financial Statement Audit. SAS No. 82. New York, NY: AICPA.

--. 2001. AICPA Professional Standards. Volume 1. New York, NY: AICPA.

Ash, R. 1972. Real Analysis and Probability. New York, NY: Academic Press Inc.

Berger, J. O. 1985. Statistical Decision Theory and Bayesian Analysis. New York, NY: Springer-Verlag.

Engineering News-Record (ENR). 2002. Anti-fraud bill signed into law. Volume 249 (6).

Fedders, J. 1998. Qualitative materiality: The birth, struggles, and demise of an unworkable standard. The Catholic University Law Review (Fall): 41-91.

Fudenberg, D., and J. Tirole. 1995. Game Theory. Cambridge, MA: MIT Press.

Jennings, M., P. Reckers, and D. Kneer. 1985. A source of insecurity: A discussion and an empirical examination of standards of disclosure and levels of materiality in financial statements. The Journal of Corporation Law 10 (3): 639-688.

Johnstone, D. 1985. Statistically incoherent hypothesis tests in auditing. Auditing: A Journal of Practice & Theory (Fall): 156-175.

Journal of Accountancy. 2000. New SASs address communications and adjustments. Volume 189 (March): 17-18. Available at: http://www.aicpa.org/pubs/jofa/mar2000/newsl.htm.

Kiernan, J. S., and L. L. Lewin. 1994. Those monstrous damage awards against accountants. The CPA Journal (April): 36-41.

Kinney, W. 1975. A decision theory approach to the sampling problem in auditing. Journal of Accounting Research 13 (1): 117-132.

Konrath, L. F. 2001. Auditing, A Risk Analysis Approach. Cincinnati, OH: South-Western college Publishing.

Newman, P., and J. Noel. 1989. Error rates, detection rates, and payoff functions in auditing. Auditing: A Journal of Practice & Theory (Supplement): 50-63.

Newman, P., S. Rhoades, and R. Smith. 1996. Allocating audit resources to detect fraud. Review of Accounting Studies 1 (2): 161-182.

Patterson, E. R. 1993. Strategic sample size choice in auditing. Journal of Accounting Research 31 (2): 272-293.

Public Oversight Board (POB). 2000. The Panel on Audit Effectiveness: Report and Recommendations. Stamford, CT: POB.

Ricchiute, D. 2002. Auditing & Assurance Services. Cincinnati, OH: South-Western College Publishing.

Securities and Exchange Commission (SEC). 1999. Materiality. SEC Staff Accounting Bulletin No. 99. Washington, D.C.: Government Printing Office.

Trager, M. H. 2002. The Enron affair: From an accountant's point of view. The Secured Lender (58): 72-74.

Wall Street Journal. 1998. SEC readies new companies about what is "material" for disclosure. Dow Jones, Inc. (November 3): A2.

Submitted July 2001

Accepted February 2003

Evelyn R. Patterson

State University of New York at Buffalo

Reed Smith

Indiana University