Accepted Manuscript
A PROBABILITY TREE MODEL OF AUDIT QUALITY Erkki K. Laitinen Professor of Accounting and Business Finance , Teija Laitinen Professor of Accounting and Business Finance PII: DOI: Reference:
S0377-2217(14)01022-4 10.1016/j.ejor.2014.12.021 EOR 12680
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
12 January 2014 8 December 2014 11 December 2014
Please cite this article as: Erkki K. Laitinen Professor of Accounting and Business Finance , Teija Laitinen Professor of Accounting and Business Finance , A PROBABILITY TREE MODEL OF AUDIT QUALITY, European Journal of Operational Research (2014), doi: 10.1016/j.ejor.2014.12.021
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Highlights We develop a novel probabilistic model of audit quality. We describe an audit engagement as a random tree model. We show how the fundamental characteristics of audit quality interplay. We introduce a new measure of audit quality. We increase theoretical understanding of how audit quality is generated.
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A PROBABILITY TREE MODEL OF AUDIT QUALITY
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Erkki K. Laitinen, Professor of Accounting and Business Finance* Teija Laitinen, Professor of Accounting and Business Finance Unit of Accounting & Finance ACA research Group University of Vaasa POB 700 FIN-65101 Vaasa FINLAND *Corresponding author Phone: +358 29 449 8443 Email:
[email protected]
There remains little consensus about how to define and formulate audit quality. It does not have
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a consistent definition and operationalization across studies and this has troubled theorists for many years. This study contributes to this discussion by introducing a probability tree model of audit quality. This model is built up of characteristics to create an association with the four sets of audit quality indicators; inputs, process, context, and outcomes (Knechel, Krishnan, Pevzner, Shefchik and Velury 2012). The purpose is to show how these indicators interplay in the context
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of audit quality. The model describes the audit program of an audit engagement as a random tree model based on a stochastic process. Following Simon's (1956; 1957) description of
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adaptive behavior the model describes an audit program as an organic procedure where an auditor does not maximize but is seeking for material misstatements (inadvertent errors) in a random environment. If the auditor under a budget constraint does not (in spite of positive
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inherent risk) detect any misstatement, the audit program will erroneously end with an unqualified report (false negative outcome). In this context, we measure subjective audit quality
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as the probability of the complement of this event (probability of detecting one or more misstatements). We also introduce a concept of perfect auditor with optimal characteristics. Finally, we measure objective audit quality as the relation of the complement event probabilities
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between the auditor and the perfect auditor. The analytical results are demonstrated by numerical examples.
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A PROBABILITY TREE MODEL OF AUDIT QUALITY 1. INTRODUCTION
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The purpose of an audit is according to International Standards of Auditing (ISA) to enhance the degree of confidence of intended users in the financial statements. This is achieved by the expression of an opinion by the auditor on whether the financial statements are prepared, in all material respects, in accordance with an applicable financial reporting framework. This opinion is on whether the financial statements give a true and fair view. It should be based on a reasonable assurance about whether the financial statements as a whole are free from material misstatement. ISA 200 (5) requires that reasonable assurance is a high level of assurance. It is obtained when the auditor has obtained sufficient appropriate audit evidence to reduce audit risk (the risk that the auditor expresses an inappropriate opinion when the financial statements are materially misstated) to an acceptably low level. ISA 200 also continues that reasonable assurance is not an absolute level of assurance, because there are inherent limitations of an audit which result in most of the audit evidence. These standards show that the main task of an auditor is to design an audit plan that provides reasonable assurance of detecting material misstatements. This question is of critical importance to the efficient functioning of capital markets (Dechow, Ge, Larson & Sloan 2011). In this study, we concentrate on this question and introduce a probabilistic tree framework to analyze the quality of an auditor in performing his or her task.
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The quality of an audit is an important but still a controversial issue in both practice and theory. Thus, research on audit area has recently focused heavily on audit quality (Knechel, Krishnan, Pevzner, and Shefchik & Velury 2012). It seems that audit quality studies as many other attempts to measure quality in other contexts, are ending up with the conclusion that there exist different measures of audit quality for different purposes. Two main streams in research defining audit quality are based on 1) attributes of the audit and the audit process and 2) attributes of the auditor and the ex-post evaluation of outcomes. The former stream is based on auditor effort during the audit process and auditor’s ability to detect a possible material misstatement in financial statements. The latter stream connects audit quality to certain attributes of the auditors (size of the audit firm, BigN vs. non-BigN, industry specialization or nonexistence of non-audit services) or ex-post-evaluations of outcomes (adverse outcomes, restatements, litigations), financial reporting quality (discretionary accruals, accounting conservatism), audit reports (missing Going Concern reporting in bankruptcy), and peer review or PCAOB (Public Company Accounting Oversight Board) inspections.
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Despite of the problems in defining audit quality unanimously several factors potentially affecting audit quality have been studied. These factors are classified or categorized in several studies (e.g. Francis 2011; Knechel et al. 2012; Watkins, Hillison, and Morecroft 2004; Duff 2009). These quality concept studies have common characteristics in that they recognize similar patterns in audit quality analysis even though they describe the relationships between patterns in a dissimilar way. These similarities can be seen comparing Table 1, where units of analysis in audit research are presented by Francis (2011), and Figure 1, where indicators of audit quality are presented by Knechel et al. (2012). Francis (2011) argues that audit quality is influenced by six levels of analysis (audit inputs, audit processes, accounting firms, audit industry and audit markets, institutions, and economic consequences of audit outcomes). Knechel et al. (2012) emphasize that audit quality depends on how the fundamental characteristics in Figure 1 (inputs, context, process, and outcomes) manifest in any given engagement. They conclude that a good quality audit is one where there is execution of a well-designed audit process by properly motivated and
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trained auditors who understand the inherent uncertainty of the audit and appropriately adjust to the unique conditions of the client. (Insert Table 1 here) (Insert Figure 1 here)
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Knechel et al. (2012) conclude that audit quality is much debated but however little understood. There remains little consensus about how to define audit quality. Measures of quality depend on whose eyes one looks through (users, auditors, regulators, society). Consequently, audit quality does not have a consistent definition and operationalization across studies (Duff 2009). This has troubled theorists for many years (Herrbach 2001).Therefore, audit quality is a controversial issue, and it is difficult to get a clear overall view of audit quality and how the fundamental characteristics are interplaying in this quality. The purpose of this study is to introduce a comprehensive probabilistic (probability tree) model of audit engagement and show how the characteristics interplay contributing to an overall indicator of audit quality. This model is purposed to increase our theoretical understanding of how audit quality is generated through the fundamental characteristics in an engagement. The objective of the probabilistic model is to describe the audit engagement (program) as an analytical process of the most important characteristics to end with a proposed overall measure of audit quality.
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The issues of identification of material misstatements in financial reporting are complex and many probabilistic approaches have been proposed in the literature of business economics (Matsumura & Tucker 1992; Fairchild 2008; Anastasopoulos & Anastasopoulos 2012). Early models used statistical decision and sampling theories focusing on evidence gathering for detecting misstatements (Kinney 1975; Menzefricke 1984). In these theories, the auditor plays against the state of nature, rather than a strategic opponent (Fairchild 2008). The next stage was to model the audit task in the context of agency theory (Antle 1982; Baiman, Evans & Noel 1987). The last stage of research began to recognize the strategic interactions between auditor and client, and employed game theory. These game-theoretical studies have also analyzed the auditor’s monitoring of the client (Morton 1993) and audit pricing (DeAngelo 1981) but they are strongly concentrated on auditor/manager reporting issues and the clientauditor relationships (Matsumura & Tucker 1992; Matsumura, Subramanyam & Tucker 1997). In the game-theoretical context, the latest research has specifically considered auditing for fraud (Fairchild 2008; Anastasopoulos & Anastasopoulos 2012). Fraud is a form of misstatement where financial reports are manipulated in order to gain an illegal advantage (ISA 240). However, misstatement can also take a form of an inadvertent error due to unintentional mistakes or misinterpretations in gathering or processing of data. In this study, our focus is limited to detect inadvertent errors.
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The common feature of the previous probabilistic approaches is that they explain the behavior of an auditor using optimization (usually, utility maximization) and trying to find best strategies in different contexts of auditing. Thus, these approaches potentially lead to normative results. The present approach is however based on describing the behavior of an auditor under the satisficing principle that is a typical characteristic of adaptive behavior. This kind of approach does not include optimization and only gives descriptive results. This modeling framework can be traced back to Simon (1956; 1957) introducing a model of rational choice and the structure of environment associated with a concept of bounded rationality. When applied in auditing it potentially provides us with a comprehensive description of the fundamental characteristics of auditing and improves our understanding of the interplay between these characteristics. Table 1 and Figure 1 help us to locate the approach in the field of audit research. In Table
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1, this approach refers to audit input and audit process as the unit of analysis. It describes an audit engagement (process) in a probabilistic context and how an auditor equipped with a set of characteristics (input) carries out the audit process. In Figure 1, this model refers to all main fundamental characteristics in the audit quality chart by Knechel et al. (2012). It describes characteristics of the individual auditor (input), features of the client engagement (context), the audit process (process), and finally introduces a probabilistic overall measure of audit quality (outcome). The purpose of this approach is to describe an adaptive audit program or process as an organic procedure where an auditor is searching for material misstatements (inadvertent errors) in a random environment. If the auditor under the budget constraint does not, in spite of given inherent risk, detect any misstatement, the audit program will erroneously end with an unqualified report (false negative outcome).
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The structure of this study is as follows. In this introductory section, the motivation and objective of the study were briefly discussed. It is clear that we need more theoretical research on the interplay of the fundamental characteristics of auditing to increase our understanding of audit quality. The second section presents a short review of prior research on audit research based upon probabilistic approaches. In this section, also the principles of adaptive behavior are briefly discussed in the context of auditing. The third section describes adaptive behavior more carefully and introduces the analytical formulation of the present approach in the form of a probability (random) tree. This section also introduces a concept of subjective audit quality as the probability of detecting one or more misstatements in this random tree. We also introduce a concept of perfect auditor with optimal characteristics and measure objective audit quality as the relation of the probabilities between the auditor and the perfect auditor. In this analytical section, also the effect of learning is shortly discussed. The fourth section presents numerical examples to demonstrate the analytical results. Finally, in the last section we will present concluding remarks of the study, discuss its main limitations and give outlines for further research.
2. PRIOR RESEARCH
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The purpose of this section is briefly to review different probabilistic approaches to auditing especially when they are related to the concepts adopted in this study. We classify these approaches in three categories of studies: 1) decision-theoretical or statistical sampling-theoretical approaches, 2) agencytheoretical approaches, and 3) game-theoretical studies. Because of multitude of different approaches, we only present a sample of relevant classic and fresh approaches in these fields and refer to Matsumura & Tucker (1992), Fairchild (2008), and Anastasopoulos & Anastasopoulos (2012) for further reviews. The common feature of prior studies in these three categories is their basis on optimization theory usually in terms of utility maximization leading to normative results of auditor behavior. In this section, we will also discuss Simon’s (1956; 1957) framework for adaptive behavior in the context of auditing. It does not include optimization but is based on the satisficing behavior under bounded rationality. Therefore, it leads to descriptive results of auditor behavior. Early decision-theoretical or statistical sampling-theoretical studies can be traced to 1970’s. Elliott & Rogers (1972) formulated the fundamental questions of the auditor (are financial statements correct or do they include a material error) as statistical hypotheses based on the uncertainty from partial examination of accounting material. The purpose of Elliott & Rogers (1972) was to suggest an approach to relating statistical (test and sampling) techniques to audit objectives in testing these hypotheses.
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Kinney (1975) continued this line of research and developed a decision-theoretic formulation and solution of the auditor’s decision problem. He also analyzed the sensitivity of the approach to errors in parameter estimates and compared the results especially with those of Elliott & Rogers (1972) and with the traditional confidence interval approach. Menzefricke (1984) further developed statistical analysis in auditing using dollar unit sampling (DUS) methodology in formal decision-theoretic approach instead of standard normal theory approaches. DUS (or Monetary Unit Sampling, MUS) is a sampling strategy in auditing, in which all units are to be randomly selected with probabilities proportional to the book value (Carrizosa 2012). Menzefricke used two different loss functions in sampling, a step and a quadratic function, to obtain optimal sample sizes.
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The next step in the development of research in the field was the introduction of agency theory. While in the decision-theoretical approaches the auditor mainly plays against the state of nature, in agency theory he or she is regarded as an economic agent. Antle (1982) criticized decision-theoretical approaches because the audit objectives were exogenously specified. He calls for a model in which the auditor’s objectives are endogenous and emphasizes that viewing the auditor as an economic agent is the first step in generating such model. In his model, the auditor's primary role is to produce stewardship information used by an owner and manager for contracting purposes. Therefore, he built an agency model and a model of a firm in which there is financial reporting for stewardship purposes, and in which the auditor's product (information) is valuable. The model was initially similar to previous principal-agent models for the demand for auditing. However, the critical and distinguishing feature of Antle’s analysis was the subsequent incorporation of the auditor as an economic agent. This led to the formulation of a two-agent agency model: one agent is the manager and the other the auditor. Such a formulation necessitated careful consideration of the game-theoretic foundations of agency theory. Baiman, Evans & Noel (1987) expanded this kind of approach and developed a principal-agent model for contractual relationships among three individuals. They characterized an optimal pair of contracts (principal-agent and principal-auditor) to analyze how hiring the auditor improves the principal-agent contractual relationship and how the principal overcomes the moral hazard problem with the auditor. They also identified conditions which are sufficient to ensure that hiring a utility-maximizing auditor improves efficiency.
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The third important step in audit research was the inclusion of game theory. The pioneering study by Fellingham & Newman (1985) criticized the single-person decision theory viewing audit risk as an output of the audit plan rather than as an input. They argue that this approach is seriously deficient because it does not allow the audit to influence the behavior of the auditee (client), a fact recognized by the auditor which in turn influences the planning of the audit. Therefore, they formulated the problem in a game-theoretic framework with rational players encompassing strategic factors for both the auditor and auditee. This framework was consistent with behavioral hypotheses regarding the effect of an audit and with randomized audit strategies. They showed that the auditor’s strategy depends on the interaction between the accounting control system and the auditee’s actions. Since this pioneering research a large number of game-theoretic variants of the auditor’s behavior have been modeled. Shibano (1990) derived expressions for the three risks involved in audit (the inherent risk, the control risk, and the detection risk) using game-theoretic models of hidden action and hidden information by the auditee. In hidden action model the auditee can choose high or low effort which affects the audit evidence upon which the auditor has to decide whether to accept or reject the proposed level of profit. In the hidden information model the auditee has to decide whether to report the true profit or not and the auditor has to investigate whether this reported profit is correct. The results are given in terms of
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the conditional probabilities p(z│t) where z is the audit evidence and t is the true level of profit. Patterson (1993) extended the hidden action game model. The audit evidence comes from a normal distribution whose parameters depend on the auditee's decisions on what asset values to report and the auditor's decisions on what size of sample to use.
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Cook, Hatherly, Nadeau & Thomas (1997) extended audit research including cooperative features in a game-theoretic model of audit. They developed a game-theoretical model of the process which seeks to mimic both the internal control investigation and substantive testing involved in auditing. The nonoperative version of the game shows that the solution where both parties work with the maximum effort and the audit report is in accord with the substantive test results, cannot occur unless there is some change to the conditions of the game. They compared the modified results with a cooperative game version of the model and showed that there is a region of parameters where both cooperative and non-cooperative versions of the game lead to this socially desirable outcome. These results indicated that whilst society expects an independent auditor not to cooperate with the auditee, the practical realities of auditing require a considerable degree of cooperation. This leads to an expectation gap between what society expects and what actually happens, except in those parameter regions where both the cooperative and non-cooperative versions of the audit game lead to the same solution.
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Anastasopoulos & Anastasopoulos (2012) criticized classic game theory in failing to determine the way the game reaches equilibrium, whether this equilibrium is stable or not and what is the long term impact of the auditor’s tenure on the quality of auditing. Therefore, they modeled the auditing/fraud detection problem using evolutionary game theory. They showed that the game is stable but not asymptotically stable if the players have accurate information for the parameters involved in the problem. They also analyzed the case of the auditor being partially informed about the auditee firm and concluded that if the auditor is partially informed about the auditee firm, a more comprehensive audit is necessary to guarantee quality of audit. They analyzed how and why the auditors/auditees strategies evolve over time. They provided analytical solutions to a series of problems in auditing including optimal auditor replacement (or auditor tenure), and assessment and prediction of fraud.
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The game-theoretic approach of audit has provided us with a rich set of findings that has remarkably improved our understanding of the fundamental questions of the audit. However, also this approach has limitations. Antle (1982: 526) already in 1980’s touched these limitations introducing two complexities. First, the mathematical program formulated may yield solutions that are not reasonable, because the program may call for the auditor and manager to play dominated Nash equilibria in subgames. Second, the nontrivial nature of the subgames implies that randomized strategies by the auditor and manager may be of crucial importance. However, Fellingham & Newman (1985: 647) concluded that the gametheoretic approach is necessary to deal with strategic considerations in auditing where the audit has behavioral implications for the client. For example, the fundamental assessments of audit risks (are financial statements correct or do they include a material error) are generally incorrect when based on a decision-theoretic analysis. If the client is not influenced by the audit in any way, decision theory is perfectly appropriate for the auditor’s problem. If the audit influences the behavior of the client, there are two alternatives. First, if the client has a pure strategy and the auditor has correct conjectures about the client’s equilibrium strategy, the decision-theoretic approach will result in the same (optimal) decision as the game theory. Second, if the client has an equilibrium randomized strategy, the decision theory generally does not provide guidance for the auditor.
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The analysis of prior studies in auditing has arisen three points which are critical with respect to the choice of the framework for the present study. First, from the viewpoint of the game theory it is important to classify the material misstatement in fraud and inadvertent error. If the material misstatements are caused unintentionally (inadvertent error), the basic audit procedures should be effectively used for error detection (ISA 240). However, if the material misstatements are caused by fraud, a set of more costly risk assessment procedures should be applied to obtain an understanding of the entity and its environment (ISA 315). If we exclude the possibility of fraud and concentrate on the inadvertent error and the basic audit procedures, it is potentially acceptable to expect that the auditor’s behavior does not influence the behavior of the client. The client in the case of unintentional errors has not any incentive for similar behavior as in the case of fraud. Therefore, focusing on the inadvertent errors only makes the game-theoretical approach unnecessary (Fellingham & Newman 1985: 647).
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Second, the characteristics of individual auditors are important determinants (inputs) of audit quality as Figure 1 pointed out (Knechel et al. 2012). Gul, Wu & Yang (2013) have showed that the effects that individual auditors have on audit quality are both economically and statistically significant. They also found that the individual auditor effects on audit quality can be partially explained by auditor characteristics, such as educational background, Big N audit firm experience, rank in the audit firm, and political affiliation. However, Osborne & Rubinstein (1994: 6) conclude that these kinds of differences between individuals which are critical in practice are missing from the game theory in its current form. For this reason, they summarize that: “modeling asymmetries in abilities and in perceptions of a situation by different players is a fascinating challenge for future research, which models of “bounded rationality" have begun to tackle”. Third, the concept of optimality in the game-theoretic models is problematic with respect to audit function. Humphrey (2008: 178) states that audit is recognized as being socially constructed in that one person´s optimality is unlikely to be adequate for others. He concludes that there is no one best method of planning and undertaking an audit. The real priority has to be one of understanding what auditors do and do not do. However, optimization techniques can be very useful as a support for auditing (Dionne, Giuliano & Picard 2009).
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The discussion of these critical points led us to use the adaptive behavior as described by Simon (1956; 1957) as the framework of our analysis. When we limit our analysis to inadvertent errors and exclude frauds, consider the effect a large set of fundamental characteristics on audit quality (Figure 1), and use descriptive analysis rather than normative optimization, this kind of framework is a natural choice. Simon (1956: 129) concludes that the decision-making theory has become a natural meeting ground for psychological and economic theory because of the focus given to rational choice. He continues that a comparative examination of the models of adaptive behavior employed in psychology (learning theories), and of the models of rational behavior employed in economics, shows that in almost all respects the latter postulate a much greater complexity in the choice mechanisms, and a much larger capacity in the organism for obtaining information and performing computations, than do the former. Moreover, in the situations where the predictions of the two theories have been compared, the learning theories appear to account for the observed behavior better than do the theories of rational behavior. The adaptive behavior falls far short of the ideal of "maximizing" postulated in economic theory. Evidently, organisms adapt well enough to "satisfice" and they do not, in general, "optimize." We believe that the chosen framework is appropriate to describe the fundamental characteristics of audit quality presented in Figure 1. Knechel et al. (2012) summarize the potential characteristics that could influence audit quality as follows:
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1. An audit is an economically motivated response to risk, i.e., incentives matter. 2. The output of an audit is a report but the outcome is uncertain and unobservable. It is not possible to ‘‘know’’ the residual risk of an engagement (achieved assurance level), i.e., uncertainty matters. 3. Each engagement is different. The idiosyncratic nature of an audit arises due to variations in client characteristics, audit teams, timing of work, and assessed risk and procedures used, i.e., uniqueness matters. 4. The audit is a systematic activity, i.e., process matters. 5. The execution of the audit process depends on appropriately leveraging the knowledge and skills of experts, i.e., professional judgment matters. The purpose of the present approach is to include many of these fundamental characteristics in our probabilistic tree model of audit project.
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3. PROBABILITY TREE MODEL OF AUDIT PROJECT 3.1. Adaptive organism
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The present probabilistic model of audit quality includes organic features in order to link the identified motivation patterns to an audit project. Miller (1983) offered a good description of the organic organization (referring in this context to audit project, program or engagement). The planning behavior of this kind of organization (engagement) follows the adaptive mode. These organizations tend to exist in a dynamic environment, which changes unpredictably. The organic organization strives to be adaptive to its environment, emphasizing expertise-based power and open communication. It makes great use of well-educated technocrats and engineers, but not of mechanistic planning systems. Such organizations have no time for lengthy planning processes, and because of the turbulence of the environment, planning horizons are not very long. Plans are often not formalized, since they must be able to change rapidly; great emphasis is placed on organizational and strategic flexibility and on maintaining a fastresponse capability to environmental opportunities or threats. These characteristics can obviously be associated with audit projects or programs. Asare & Wright (2004) discuss the evidence on audit programs concluding that formal (standard) audit programs promote mechanistic processing and inflexibility, and inhibit an auditor’s ability to reason strategically. In contrast, informal programs encourage greater cognitive processing, and may facilitate more creative and strategic thinking.
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The organic organization is related to the adaptive planning philosophy (Simon 1957); it also closely resembles Lindblom’s (1968) “Logical Incrementalist” approach to strategy formulation and Mintzberg’s (1989) concept of “Emergent strategies”. These organizations do not plan as formally as the planning organizations. The organic organization makes great use of human intelligence rather than of mechanistic planning models. However, it does not plan as intuitively as the simple organization either. The organic organization is close to the ”adhocracy” and the innovative organization, described by Mintzberg (1973 and 1989). The structure of the innovative organization is fluid, organic, selectively decentralized and “adhocracy” (Mintzberg 1989: 198). Its context is characterized by a complex and dynamic environment, including high technology, frequent product change and development due to intense competition, and large projects of short duration. These kinds of characteristics are also typical for audit engagements which are largely based on human intelligence and cognitive processing but however follow audit standards. ISA 300 requires that the planning stage of the audit should be used to
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establish an overall strategy for the audit, develop an audit plan, and reduce audit risk to an acceptably low level.
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It is impossible to describe all fundamental characteristics of audit quality (Figure 1) in a tractable mathematical model. Therefore the present approach concentrates on some of the most important characteristics that can provide a mathematical description of the audit project. The suggested model is based on a framework set forth by Simon (1957), describing rational choice and the structure of the environment. Laitinen (2001) has applied this kind of framework to describe the growth of small technology firms showing characteristics of adhocracies. Simon (1957: 261) himself comments on the idea of his model as follows: “If an organism is confronted with the problem of behaving approximately rationally, or adaptively, in a particular environment, the kinds of simplifications that are suitable may depend not only on the characteristics—sensory, neural, and other—of the organism, but equally upon the structure of the environment. Hence, we might hope to discover, by a careful examination of some of the fundamental structural characteristics of the environment, some further clues as to the nature of the approximating mechanisms used in decision making.”
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This characterization is valid, as well, for the suggested model, which also describes an adaptive organism and the structure of its environment (an auditor and the audit project). However, Simon’s original model is here extended and applied to describe an audit engagement that has organic or adhocracy characteristics. Many of the organic characteristics are found in adaptive audit projects (Mock and Wright 1999; Asare & Wright 2004; Knechel et al. 2012). Audit projects are typically multistage and adaptive within a complex environment. Projects are idiosyncratic and tailored to each client within the structure of the basic audit methodology. The objectives of the project are not based on maximization but on a need for reasonable assurance based on a knowledge-based professional service producing an uncertain and unobservable outcome. The quality of the project is largely based on the knowledge and expertise of the auditor but also on his investigative intuition based on professional skepticism.
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The organism has only a single goal, survival; It has no problem of maximization; The nature of its perceptions and its environment limit sharply its planning horizon; The nature of its needs and environment create a natural separation between means and ends.
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1. 2. 3. 4.
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Simon (1957: 263) notes that his “satisficing” model differs sharply from the more sophisticated models of human rationality that have been proposed by economists and others. His framework is based on four postulates about the adaptive organism, presented below, which are also valid for the model proposed here:
Simon’s model was developed to describe the behavior of a simple adaptive organism, like a turtle. For purposes of the present framework the simplicity of Simon’s model is retained to describe the characteristics of audit project in simple mathematical terms. However, additional features emerge that alter Simon’s original model. Specifically, a simple mechanism of learning by doing is incorporated in the model to introduce a typical characteristic of adaptive behavior. 3.2. Parameters of the model Following the postulates of an organic organization, let us assume that the audit project or program is dynamic and can be divided into successive stages in the form of a tree. When the auditor is starting the
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first stage of the project, there are assumed g paths or routes of actions for him, to follow in auditing process. Then, after the first stage, the auditor finds himself again in a branch point of g routes for the second stage. To continue the audit project, he is confronted by g paths to follow to the next branch point, and so on. Thus, in this kind of multi-stage audit project, g measures the objective complexity of the audit case or engagement: the more potential paths are available for the auditor to follow; the more complex is the audit engagement. Coenen (2005) describes in a professional journal forensic investigation using this kind of tree: “I liken a forensic investigation to a family tree. The investigator starts going down one path, and there are several branches and directions in which to move. A path is selected for the investigation, and again there are many more paths branching off that one.” This description is nearly identical with our tree model of audit investigation. In audit research, our concept of complexity is closely related to the amount of processing element that varies with the amount of input alternatives and the number of steps or procedures that have to be executed sequentially (Bonner 1994: 217).
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The behavior of the auditor is however constrained by bounded rationality. First, imperfect (industry) experience makes the auditor aware only of 100∙o per cent of total number of existing potential paths: the more experienced the auditor is; the higher is o or the fraction of paths identified. Second, imperfect operational expertise makes the auditor able to follow only 100∙e per cent of the identified paths. The higher the expertise; the higher is e or the fraction of identified paths possible to follow. Thus, the number of paths available for an auditor in practice (d) reflects the perceived complexity of the audit engagement and is defined as follows: (1)
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where g refers to the objective complexity of the auditing case, o to the experience, and e to the operational expertise of the auditor.
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It is further assumed that 100∙p per cent of the d paths have randomly distributed branch points, at which a material misstatement (in this framework, inadvertent errors) can be found. Therefore, p is linked to the riskiness of the audit project in terms of risk for material misstatements. The risk of material misstatement refers to the inherent risk that the accounting material audited is materially misstated. Let r be the probability of detecting the existing misstatement in a branch point. Thus, r* = 1r refers to the detection risk that the procedures performed by the auditor will not detect (and report) a material misstatement that exists in a branch point. This risk is assumed to be a function of the effectiveness of the substantive audit procedures c and their application by the auditor referring to whether the procedures were performed with due professional care, measured by f. This definition of detection risk is consistent with audit standards (PCAOB AS 8). Both c and f vary between 0 and 1 and are assumed independent so that the detection risk r* is:
r* 1 r 1 cf
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where c measures the effectiveness of audit procedures and f the performance of the auditor. If c or f is equal to 0 (effectiveness or performance is zero), then r* (the detection risk) equals 1. When c and f are both equal to 1 (perfect effectiveness and performance), then r* is equal to 0. If a misstatement is detected, it is assumed that it is also reported (so that the probability to report a detected misstatement, say h equals 1). Thus, r factually refers to the joint propensity to detect and report a misstatement while r* refers to the probability of false negative event. The audit risk is a function of p
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and r (IAS 200). In this framework, false positive event is not considered. It refers to an event where the auditor erroneously detects a material error when it does not exist.
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When the auditor is inspecting the audit material in a branch point of paths, it is assumed that he or she can get a gut feel for a potential material misstatement and detect it over v stages of the audit project ahead. Thus, v reflects investigative intuition referring to the range of vision measured as a multiple of the stage of the audit project. This multiple shows the number of stages ahead in which the auditor will be able to forecast a material misstatement. Therefore, v measures the visionary expertise of the auditor. Kinney (1979: 149) describes this kind of auditor characteristic as follows: “the auditor forms an expectation about what the unaudited book value should be by determining a range of acceptable values (i.e., forms a noninvestigation region). The unaudited value is then compared with the noninvestigation region, and depending on the outcome of this comparison, various actions would be taken.” In practice, auditors can develop expected values for unaudited accounts in various ways, including the use of formal statistical models or intuitive judgment (Biggs & Wild 1985: 610). Formal statistical methods such as Zipf analysis and data mining techniques (Multilayer Feed Forward Neural Network (MLFF), Support Vector Machines (SVM), Genetic Programming (GP), Group Method of Data Handling (GMDH), Logistic Regression (LR), and Probabilistic Neural Network (PNN)) can assist auditors also to locate the source of suspicion and further enhance the resulting audit processes (Huang, Yen, Yang & Hua 2008; Ravisankar, Ravi, Raghava Rao & Bose 2011).
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There may be situations where auditors prefer intuitive judgment because time, cost, or data constraints can prevent the use of statistical methods (Biggs & Wild 1985). The AICPA 2009 survey showed that accountants rate investigative intuition in the top 5 characteristics of a forensic accountant (Davis, Farrell & Ogilby 2009). They see the value of both analytical abilities and the much more vague ability to develop productive hunches based on information filtered through their experience. Coenen (2005) concludes that investigative intuition is a good way to describe a necessary element of the forensic accounting equation. Coenen continues that: “I don’t think that the “gut” feeling of a good investigator is teachable. If one possesses that intuition, even in its rawest form, then that ability can be developed and honed. But if a person doesn’t have the basic instinct, it will be hard to make her or him into a good forensic accountant.” It is difficult or even impossible to describe exactly this kind of vague mixture of basic instinct, gut, hunch, and analytical abilities using a concept of a mathematical model. Therefore, in this approach it is only approximated by v describing the ability of an auditor to forecast an error for a multiple of stages ahead.
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The operational limit for the audit engagement is set by the budget allowed to the project. It is assumed that the project is resourced for investigating H stages. Thus, H measures the constraint for the resources of the audit project as a multiple of the stage of the project. Budget limit is an important constraint for audit and this is explicitly mentioned also in accounting standards. In establishing the overall audit strategy, the auditor should take into account the nature, timing, and extent of resources necessary to perform the engagement (PCAOB AS 9). (However, the matter of difficulty, time, or cost involved is not in itself a valid basis for the auditor to omit an audit procedure for which there is no alternative or to be satisfied with audit evidence that is less than persuasive.) The auditor should make an appropriate plan in making sufficient time and resources available for the conduct of the audit (ISA 200). Typically, accounting firms employ constraints in the form of time budgets to contain the cost of audit (Asare, Trompeter & Wright 2000: 546; Yim 2009). Time budgets can, however, induce stress and affect the extent of testing, and auditors can exceed them if necessary. In this framework, the constraint
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is set in the form of the number of stages (H) to be audited. Thus, it mainly refers to a constraint for the depth of audit process. 3.3. Structure of the model
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The audit standards require the auditor to obtain reasonable assurance about whether the financial statements as a whole are free from material misstatement (ISA 200: 5). This reasonable assurance is a high level of assurance. It is obtained when the auditor has obtained sufficient appropriate audit evidence to reduce audit risk (that is, the risk that the auditor expresses an in appropriate opinion when the financial statements are materially misstated) to an acceptably low level. In this framework we know that the client material includes an inherent risk of p. If the probability of detecting one or more misstatements (P) is however very low, it may give to the auditor a reasonable assurance that the material is free from material statements even if p differs from zero. Therefore, in this framework P and its complement Q = 1-P refer to the subjective quality of the audit. Thus, Q measures the probability that the auditor does not detect any material misstatements (inadvertent errors) during the audit process. Therefore, the principal idea of the model is to derive the probability P for the audit project. In each audit stage (branch point of the tree), the auditor can discover d branch points (in the tree) at the distance of one stage, d2 branch points at the distance of two stages and, in general, dk branch points at the distance of k stages. Taking account of investigative intuition v, the auditor is able to investigate material misstatements in (3)
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branch points. Each branch point of the tree is identical with respect to the number of sub-branches (d) and probability of misstatement (p). Thus, the auditor is not faced by the problem of optimization being indifferent to what path to choose and continue. Without option to return back to previous stages, each monotonic investigation strategy (through H-v stages) leads to the same P.
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When the auditor proceeds and continues the project, in the next stage (branch point) he or she is able to see dv new branch points. Thus, after m periods, mdv new branch points will appear. Since the audit project has resources to investigate for H stages ahead, the auditor can under this constraint continue (after the initial stage) for a maximum of H-v stages to search for a material misstatement. Thus, the probability, Q = 1-P, that the auditor is not able to detect any material misstatement (and the audit project will be failed), is equal to the probability that each investigated branch point either 1) does not include a material misstatement or 2) does include a material misstatement but the author is not able to detect it (false negative event). The probability that a branch point does not include a material misstatement is 1-p. However, the probability that a branch point includes a material misstatement but the auditor does not detect it, is p(1-r). Thus, the probability that a branch point does not include a misstatement or it does include a misstatement but the auditor is unable to detect it is q = 1-p + p(1-r) = 1-rp. Probability Q depends on q and the number of branch points investigated (N). When the project starts, the auditor is first able to investigate D branch points. Then, the auditor under the budget constraint can proceed H-v stages and investigate (H-v)dv new branch points and thus N = D + (H-v)dv. For this random tree, Q is simply:
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Q (1 rp) d ( d
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(1 cfp ) goe(( goe)
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which presents the probability that the audit project fails. Equation (4) shows the probability that the auditor potentially gives an unqualified statement, even if the inherent risk is positive. Its complement P = 1-Q directly reflects the subjective quality of audit.
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In this context, we do not consider the event where the auditor erroneously interprets an error-free branch point as including material misstatement (false positive event). This kind of misinterpretation will erroneously decrease the level of assurance. The probability for false positive event can easily be described using the concepts of our model. If the auditor has a probability b to erroneously interpret an error-free branch point as a misstatement, the probability for a false positive event for a branch point is b(1-p). Therefore, the probability that the auditor gets none false positive events for N branch points is simply ((1-b(1-p))N. We will not consider here the false positive events explicitly. Theoretically, this limitation means that we assume that b = 0 and the probability for an auditor to get none false positive events is set equal to 1.
N! (rp) k (1 rp) N k k!( N k )!
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If we define a binary random variable where the probability for the event (outcome) “No detection” is (1-rp) and for the event “Detection” respectively rp, it follows a binomial distribution, because the audit process is here consisted of a sequence of independent experiments. Thus, we can define the probability that the auditor can detect k branch points including a material misstatement in the following way:
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which equals (4) when k = 0. For this binomial distribution it holds that the expected number of branch points to be investigated before the first point including material misstatement is detected, is simply 1/(rp) while the expected value of the total number of such branch points is Nrp.
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Table 2 shows the parameters of the probabilistic audit model. Equation (4) shows the probability Q that the auditor cannot detect any branch point including a material misstatement. This measure is an indicator of subjective audit quality. Panel 1 of Appendix A presents the partial derivatives of Q with respect to p, r, d, H, and v. For values p > 0, r > 0, d > 0, H > 0, and H > v > 0, all these partial derivatives are negative. Thus, Q is a decreasing function of the probability that a branch point includes a material misstatement p, the probability that an existing material misstatement is detected r, the number of paths identified and able to follow d, the number of stages resourced H, and the range of vision v. If v = 0, then the auditor is “myopic” and (4) is simply
Q (1 rp) H and the auditor is able to investigate only H branch points. (Insert Table 2 here)
(6)
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The parameters influencing Q can be classified in two categories on the basis of the object. First, p (inherent risk of the audit case) and r (probability of detecting an existing misstatement) influence the probability to detect a material misstatement in a branch point. Thus, r in the form of (1-r) reflects the detection risk and it is defined as the product of c (effectiveness of audit procedures) and f (performance of the auditor). The higher c and f are, the higher is r, and the lower is Q. Second, d (number of paths identified and able to follow), v (range of vision), and H (number of stages budgeted) affect the number of branch points N investigated by the author. In addition, d is defined as the product of g (total number of paths), o (fraction of paths identified, and e (fraction of paths able to follow). Thus, the higher d, g, and o are, the higher is N and the lower is Q. In practice, N is associated with the number of tests to conduct (extent), the number of hypotheses to test (breadth), the number of tests for each hypothesis (depth) and the number of potential error and non-error hypotheses to test (focus) (Asare, Trompeter & Wright 2000). In summary, Q is affected by eight parameters and their interplay. These parameters indicate that the subjective quality of audit P is influenced by the inherent risk, detection risk, complexity of the audit case, auditor experience, auditor operative expertise, auditor visionary expertise, resources allocated to the audit project, and their interplay. Figure 2 shows that these influences partly cover all fundamental characteristics outlined by Knechel et al. (2012). (Insert Figure 2 here)
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3.4. Perfect auditor
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Probability Q in (4) and its complement 1-P refer to the subjective quality of audit. However, audit standards do not regard reasonable assurance as an absolute level of assurance, because there are inherent limitations of an audit which result in most of the audit evidence on which the auditor draws conclusions and bases the auditor’s opinion being persuasive rather than conclusive (ISA 200: 5). Thus, we suggest a relative measure for the objective quality of audit. The probability drawn for an auditor depends on his or her individual characteristics but also on the context. The author cannot influence the context. Therefore, we introduce the concept of perfect auditor. For this auditor, the individual (input and process) characteristics are perfect but the context is unchanged. We get the following probability Q* for the perfect auditor:
Q* (1 p) g ( g
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when o = 1, e = 1, c = 1, f = 1, and v = H. For the perfect auditor, Q* only depends on inherent risk (p), the objective complexity of the audit case (g), and the budget constraint (H). The partial derivatives in Panel 2 of Appendix A show the higher are p, g, and H, the lower is Q*. The complements of Q and Q* can be used to get a relative measure of audit quality when comparing the probability of detecting at least one misstatement for an ordinary auditor and for the perfect auditor. Thus, we get the following relative measure for audit quality: PO
1 Q P 1 Q * P*
(8)
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where P* is used as a benchmark for P to get P O, an objective measure for audit quality. If the audit case does not include any inherent risk for misstatements so that p = 0, both P and P* will get a value of 0, which leads to 0/0 in (8) leading to an indeterminate form. This implies that in that audit case it is not possible to assess quality (with respect to false negative outcome), since even a perfect auditor cannot detect any misstatement. Equation (4) refers to the audit quality constrained by the bounded rationality of an auditor. This kind of imperfect auditor (“boundedly rational agent”) experiences limits, in formulating and solving complex problems and in processing (receiving, storing, retrieving, transmitting) information (Simon 1957). In summary, we suggest the probability of detecting at least one material misstatement in relation to the probability drawn for the perfect auditor (in that context) as for the overall relative (objective) measure of audit quality. 3.5. Learning by doing
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The present model of audit quality assumes that the eight parameters stay constant in time so that the individual characteristics of the auditor do not change. However, learning is natural for adaptive processes where organisms adapt well enough to satisfice but not in general to optimize (Simon 1956: 129). Beck & Wu (2006) conclude that the auditor can accumulate client-specific knowledge as he or she performs audit engagement updating beliefs about the client and becoming more precise over time. In stable environments, the auditor continuously learns and audit-judgment errors will be eliminated resulting “in the highest possible level of audit quality” (Beck & Wu 2006: 4). Thus, the learning effect has a favorable impact on audit quality as it is also confirmed by empirical evidence (Solomon, Shields, and Whittington 1999; Low 2004). Therefore, we include a simple learning-by-doing mechanism to our model to describe the effect of adaption on audit quality.
(1-rp)u0 (1-rp)u1 … (1-rp)uH-v
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Let is assume that learning improves the ability of the auditor to detect (r) material misstatements in the passage of the audit project, when the auditor improves his or her performance (f) and/or ability to use audit procedures (c). Let us also assume that learning is a steady process by a constant learning coefficient u with respect to the stages of the engagement (i) so that (1-rp) develops as follows: (9)
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where H-v > 0 and u < 1 for positive effects of learning.
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This steady learning process leads to the following probability of detecting none material misstatements during the audit project:
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L u (( H v )(1 H v ) / 2) d
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where L is the learning multiplier. The present framework thus makes it possible to consider the effect of learning L on audit quality as a separate multiplier. For the perfect auditor, L = 1 since H = v. She or he has nothing to learn. Panel 3 of Appendix A presents the partial derivatives of L which show that for u <
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1 L is decreasing in d and H but increasing in u. However, the effect of v on L is not monotonic (see Appendix A). For small H and H-v, L is increasing in v but otherwise decreasing in v. This effect is based on the twofold influence of v on L: if v is higher, the author has fewer stages for learning but at the same time she or he is able to investigate more branch points in each stage.
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The learning process does not affect the inherent risk of audit (p). Therefore the minimum value of the learning coefficient u fulfils the following condition: (11)
1 /( H v )
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which gives the value of u that leads to r = 1 after H-v stages.
4. NUMERICAL EXPERIMENTS
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Equations (4) and (7) give an insight how different fundamental characteristics theoretically influence the audit quality of an ordinary auditor constrained by bounded rationality and the perfect auditor, respectively. However, the influence of these characteristics and their interplay can be better understood performing numerical experiments. Table 3 presents a sample of hypothetical audit cases showing the influence of different parameters on the probability for an ordinary auditor to detect none material misstatement in the limits of the budget (Q). The numerical cases in this table are constructed inserting 2 alternative values for each of the 6 parameters and changing only one at a time (ceteris paribus) to show separate effects of characteristics. However, the number of paths identified and able to follow (d) and the total number of paths (g) are changed together leading to 25 = 32 cases. First, the cases are divided into the simple cases (d =2, g = 3) and the complicated cases (d = 3, g = 4) to show the effect of complexity. The components of d (g, o, and e) are not considered since their influence can be assessed from that of d (d = goe). The simple case is a special case of binary tree. Second, the probability that a branch point includes a misstatement (p, inherent risk) gets the values either 0.01 or 0.02. Third, the rate of detection (r, referring to detection risk 1-r) can have a value 0.90 or 0.95. The components of r are not considered separately (r = cf). Fourth, the range of vision v can take values of 3 or 4 and, fifth, the budget constraint H can be either 6 or 8 stages.
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(Insert Table 3 here)
Table 3 shows that the variation in Q between cases is large. The highest Q is 0.7092 (case 1) and the lowest Q 0.0002 (case 32). If d is 2 (a binary tree), p = 0.01, r = 0.90, v = 3, and H = 6, there is a probability of 0.7092 that none misstatement is detected (case 1). In this simple case, only 38 branch points (N) are investigated when the expected value of the number investigated when the first misstatement is found, is 111. It is expected that only 0.34 misstatements will be detected. When d is 3 (a complicated case), p = 0.02, r = 0.95, v = 4, and H = 8, the number of investigated branch points is 444 leading to Q = 0.0002 (case 32). The expected value for the number of material misstatements detected in this case is 8.44. If d = 3 (case is complicated instead of being simple) but all other parameters are
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identical, Q is 0.1648 (case 16). If d = 2 but v = 3 instead of 4, Q is 0.0355 (case 30). If H = 6 instead of 8 (all other parameters are identical with case 32), Q is 0.0045 (case 31). Thus, the examples indicate that the number of available paths d, the range of vision v, and the budget constraint H have a strong influence on Q. Table 3 also shows the numerical values for the elasticity of Q with respect to d, p, r, H, and v (see Appendix A). These elasticities are very strongly correlated with each other (Pearson correlations are about 0.99) and behave in the same way in different cases. They are identical for r and p. The absolute elasticities are very large for the complicated cases (cases 17-32). The figures indicate that Q is most sensitive to d, v, and H, in this order. For low Q, the absolute elasticities are exceptionally high exceeding 30 for d and v in case 32.
(Insert Table 4 here)
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Table 4 shows the values of Q* (perfect auditor) for the same 32 cases as in Table 3. This probability is very close to zero in 28 cases. The only exceptions are found in the simple cases (g = 3) when p = 0.01 and H = 6 (cases 1, 3, 5, and 7). In these cases, Q* is 0.00017. For the perfect auditor, the number of branch points investigated is very high leading to low Q*. When the ordinary auditor investigates in case 1 only 38 branch points, the number investigated is 1092 for the perfect auditor. The number of branch points investigated rises quickly with the complexity and the budget constraint. The perfect auditor investigates in case 32 (g = 4, H = 8) 87380 branch points when the ordinary auditor investigates only 444. Table 4 also presents the relative measure of audit quality (PO) expressed in (8). This measure varies from 0.2908 (case 1) to 0.9998 (case 32). The experiments indicate that the complexity of the audit case significantly influences this quality measure. Since Q* is in these experiments generally close to zero, PO is close to 1-Q. Table 4 also shows the elasticities of Q* with respect to p, H, and g. The absolute values of these elasticities are very high especially when Q* is close to zero (complicated cases). The elasticities are again highly correlated with each other (Pearson correlation coefficients are about 0.99) behaving in the similar way. The results clearly indicate that Q* is very sensitive to its parameters but in particular to the budget constraint H and the number of paths g, in this order.
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Table 5 presents the values of the learning multiplier L in (10) for the 32 cases considered in previous tables. In these experiments, the learning coefficient u is constant and set very close to unity (u = 0.9998). Its value approximately corresponds to the value which makes (1-rp) equal to (1-p) when r = 0.95, p = 0.02, and H-v = 5. The values of L significantly vary in different cases from 0.8504 to 0.9904. The lower is L, the stronger is the influence of learning on Q. The effect of learning is strongest in the complicated cases (cases 17-32). The minimum value of L is found when d = 3, H = 8, and v = 4 (cases 20, 24, 28, and 32) leading clearly to lower value than similar cases where however v = 3 (L = 0.9222). In these cases, longer range of vision v thus makes the effect of learning stronger. Table 5 also shows the elasticities of L with respect to its parameters. The learning multiplier is extremely sensitive to the learning coefficient u especially in the complicated cases 17-32 where the number of branch points is high. It is however quite insensitive to H, v, and d in particular in simple cases. The elasticity of L with respect to the range of vision v is negative for all complicated cases. For the simple cases 1-16 this elasticity is very close to zero and in several cases approximately zero or slightly positive. These results show that effect of the range of vision v on L is not monotonic as was theoretically shown. (Insert Table 5 here)
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5. CONCLUDING REMARKS
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Knechel et al. (2012) conclude that audit quality is much debated but however little understood and that there remains little consensus about how to define audit quality. In fact, audit quality does not have a consistent definition and operationalization across studies and this has troubled theorists for many years (Herrbach 2001; Duff 2009). The purpose of this study was to introduce a probabilistic model of audit engagement and show how the fundamental characteristics of audit quality interplay contributing to an overall indicator of audit quality. The present model described an audit engagement as a random tree model formed by a stochastic process. The model describes an adaptive audit project as an organic procedure where an imperfect auditor under bounded rationality is seeking for material misstatements in a random environment. If the auditor is not able under budget constraint (measured by multiple of stages) to detect a misstatement when there is positive inherent risk, the audit project will end with a false negative outcome and fail. If the auditor is able to discover at least one misstatement, the project will lead to a true positive outcome and success. We call the probability of success drawn under bounded rationality as subjective audit quality of an imperfect auditor. This probability is in this study related to the probability of success drawn for the perfect auditor with optimal characteristics in the same context, to get a relative indicator of quality. We suggest this kind of relative indicator for measuring the objective audit quality.
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The proposed model included in all eight parameters which measure the fundamental characteristics of an audit project. Six of the eight parameters reflected either audit engagement (context: complexity, inherent risk, and budget constraint) or auditor (input: industrial experience, operative expertise, investigative intuition) while two of them were associated with process (process: effectiveness of audit procedures, performance of auditor). The outcome of the audit process (as a derived measure) was realized as the probability that the audit case includes at least one material misstatement separately derived for a rationally bounded auditor and the perfect auditor. Thus, our model included variables from each of the four categories of fundamental audit quality characteristics presented by Knechel et al. (2012). We built the model following two principal targets. First, we wanted to keep the structure of the model as simple as possible and constructed it without optimization procedures which are rare in auditing practice that is based on social constructs. Instead, we emphasized satisficing and adapting (learning). Second, we selected the parameters of the model to be as relevant as possible for auditing practice and audit standards.
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We hope that the probabilistic model introduced in this study helps us to understand the effects of fundamental characteristics of audit, and their interplay, on audit quality. The effects were analyzed both mathematically and using numerical experiments. The findings indicated that the complexity of the audit case (context), visionary expertise (input), and the budget constraint (context) have a strong influence on subjective audit quality (outcome). The sensitivity of audit quality (as measured by elasticity) to these characteristics behaves in similar way in different audit cases making the elasticities strongly correlated with each other. These elasticities are higher in more complicated audit cases. The findings also indicated that quality of the perfect auditor is very sensitive in particular to the budget constraint and the (objective) complexity of the audit case. These characteristics determine the number of branch points investigated. The findings further indicated that learning is an essential element in audit quality, especially in complicated audit cases. The effect of learning on quality is quite insensitive to budget constraint, visionary expertise, and complexity but very sensitive to learning rate.
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The present study suffers from a number of limitations. First, to simplify the analysis we have assumed that audit process does not affect the behavior of the auditee (client). This simplification justified us to use a descriptive analysis of adaptive behavior. If this assumption is relaxed, it would lead to complicated game-theoretical analyses (Fairchild 2008; Anastasopoulos & Anastasopoulos 2012). This assumption means that the analysis is concentrated merely on detecting unintentional mistakes (inadvertent errors) in accounting rather than on frauds which are based on fraudulent behavior. The PSW Global Economic Crime 2014 Survey indicated that 37% of organizations report being hit by fraud from which only 22% are accounting frauds (see PSW 2014). Thus, although being quite common, the majority of organizations are not hit by a fraud. Second, our model does not consider the false positive event in auditing which is left for future research. Third, the present approach is only based on mathematical analysis and numerical experiments. Although the concepts are largely drawn from previous studies and audit standards, the model and its findings should be empirically assessed. Therefore, we urge further case, experimental, and survey studies to deal with our concepts of audit quality. Especially, we will emphasize future empirical research on the role of investigative intuition.
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Acknowledgements. The authors are grateful to the three anonymous referees and the editor of the journal for excellent comments on the initial manuscript.
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REFERENCES
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Anastasopoulos, N.P. & Anastasopoulos, M.P. (2012). The evolutionary dynamics of audit. European Journal of Operational Research. 216: 469–476. Antle, R. (1982). The Auditor as an Economic Agent. Journal of Accounting Research. 20(2): 503-527. Asare, S.K. & Wright, A.M. (2004). The effectiveness of alternative risk assessment and program planning tools in a fraud setting. Contemporary Accounting Research. 21(2): 325-352. Asare, S.K, Trompeter, G.M. & Wright, A.M. (2000). The Effect of Accountability and Time Budgets on Auditors' Testing Strategies. Contemporary Accounting Research. 17(4): 539-60. Baiman, S., Evans III, J.H. & Noel, J. (1987). Optimal Contract with a Utility Maximizing Auditor. Journal of Accounting Research. 25(2): 217-244. Beck, P.J. & Wu, M.G.H. (2006). Learning by Doing and Audit Quality. Contemporary Accounting Research. 23(1): 1-30. Biggs, S. & Wild, J. (1985). An Investigation of Auditor Judgment in Analytical Review. The Accounting Review. 60(3): 607-633. Bonner, S. E. (1994). A model of the effects of audit task complexity. Accounting, Organizations and Society. 19 (3): 213–234. Carrizosa, E. (2012). On approximate Monetary Unit Sampling. European Journal of Operational research. 217(2): 479-482. Coenen, T.L. (2005). Forensic accounting: A new twist on bean counting. Wisconsin Law Journal. By: dmc-admin November 30, 2005 1:00 am.
[email protected]. Cook, J., Hatherly, D., Nadeau, L. & Thomas, L.C. (1997). Does cooperation in auditing matter? A comparison of a non-cooperative and a cooperative game model of auditing. European Journal of Operational Research 103(3): 470-482. Davis, C., Farrell, R. & Ogilby, S. (2009). Characteristics and skills of the forensic accountant. 2009 survey commissioned by the AICPA. Retrieved October 06, 2014 from: http://www.aicpa.org/. DeAngelo, L. (1981). Auditor size and audit quality. Journal of Accounting and Economics. 3 (December): 183-199. Dechow, P.M., Ge, W., Larson, C.R. & Sloan, R.G. (2011). Predicting Material Accounting Misstatements. Contemporary Accounting Research. 28(1): 17–82. Dionne, G., Giuliano, F., & Picard, P. (2009). Optimal Auditing with Scoring: Theory and Application to Insurance Fraud. Management Science. 55(1): 58-70. Duff, A. 2009. Measuring audit quality in an era of change. An empirical investigation of UK audit market stakeholders in 2002 and 2005. Managerial Auditing Journal 24(5): 400–422. Elliott, R.K. & Rogers, J.R. (1972). Relating statistical sampling to audit objectives.The Journal of Accountancy. 68(July): 46-66; Fairchild, R. (2008). Auditor tenure, managerial fraud and report qualification: a behavioural gametheoretic approach. Journal of Behavioural Accounting and Finance. 1(1): 23 – 37. Fellingham, J.C. & Newman, D.P. (1985). Strategic considerations in auditing. The Accounting Review 60: 634-650. Francis, J.R. (2011). A framework for understanding and researching audit quality. Auditing: A Journal of Practice & Theory. 30(2): 125–152. Gul, F.A., Wu, D. & Yang, Z. (2013). Do Individual Auditors Affect Audit Quality? Evidence from Archival Data. The Accounting Review. 88(6): 1993–2023. Huang, S-M., Yen, D.C., Yang, L-W. & Hua, J.S. (2008). An investigation of Zipf's Law for fraud detection. Decision Support Systems. 46(1): 70-83.
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Herrbach, O. (2001). Audit quality, auditor behaviour and the psychological contract. The European Accounting Review 10(4): 787–802. Humphrey, C. (2008). Auditing research: a review across the disciplinary divide. Accounting, Auditing & Accountability Journal. 21(2): 170-203. International Standard of Auditing (ISA) 200. Overall Objectives of the Independent Auditor and the Conduct of an Audit in Accordance with International Standards on Auditing . Retrieved October, 02 2014, from: http://www.ifac.org/auditing-assurance/clarity-center/clarified-standards International Standard of Auditing (ISA) 240. The auditor’s responsibilities relating to fraud in an audit of financial statements. Retrieved October, 02 2014, from: http://www.ifac.org/auditing-assurance/clarity-center/clarified-standards International Standard of Auditing (ISA) 300. Planning an Audit of Financial Statements. Retrieved October, 02 2014, from: http://www.ifac.org/auditing-assurance/clarity-center/clarified-standards International Standard of Auditing (ISA) 315. Identifying and Assessing the Risks of Material Misstatement through Understanding the Entity and Its Environment. Retrieved October, 02 2014, from: http://www.ifac.org/auditing-assurance/clarity-center/clarified-standards Kinney, W.R. (1975). A Decision-Theory Approach to the Sampling Problem in Auditing. Journal of Accounting Research. (Spring): 117-132. Kinney, W. R. (1979). The predictive power of limited information in preliminary analytical review: An empirical study. Journal of Accounting Research. (Supplement): 148-165. Knechel, W.R, G.V. Krishnan, M. Pevzner, L.B. Shefchik & Velury, U. (2012). Audit Quality: Insights from the Academic Literature. Auditing: A Journal of Practice & Theory. 32(1): 385-421. Laitinen, E.K. (2001): Management Accounting Change in Small Technology Companies: Towards a Mathematical Model of a Technology Firm. Management Accounting Research. 12(4): 507-541. Lindblom, L. E. (1968). The Policy-Making Process. Prentice-Hall. Englewood cliffs, NJ. Low, K-Y. (2004). The effects of industry specialization on audit risk assessments and audit-planning decisions. The Accounting Review. 79(1): 201-219. Matsumura, E.M., Tucker, R.R. (1992). Fraud detection: A theoretical foundation. The Accounting Review 67: 753-782. Matsumura, E.M., Subramanyam, K.R. & Tucker, R. (1997). Strategic Auditor Behavior and Going Concern Decisions. Journal of Business, Finance and Accounting. 24(6): 727-758. Menzefricke, U. (1984). Using Decision Theory for Planning Audit Sample Size with Dollar Unit Sampling. Journal of Accounting Research. 22(2): 570-587. Miller, D. (1983). The correlates of entrepreneurship in three types of firms. Management Science 29: 770–791. Mintzberg, H. (1973). Strategy-making in three modes. California Management Review. 16: 44–53. Mintzberg, H. (1989). Mintzberg on Management. Inside our Strange World of Organizations. The Free Press. MacMillan, Inc. New York, NY. Mock, T. J. & Wright, A. (1999). Are audit program plans risk-adjusted? Auditing: A Journal of Practice & Theory. 18(1): 55–74. Morton, S. (1993). Strategic auditing for fraud. The Accounting Review. 68(October): 825–839. Osborne, M.J. & Rubinstein, A. (1994). A Course in Game Theory. Electronic version of A Course in Game Theory by Martin J. Osborne and Ariel Rubinstein (ISBN 0-262-65040-1). Copyright Massachusetts Institute of Technology. Patterson, E.R. (1993). Strategic sample size choice in auditing, Journal of Accounting Research. 31: 272293.
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PCAOB (AS 8). Audit Risk. Retrieved October, 02 2014, from: http://pcaobus.org/standards/auditing/pages/auditing_standard_8.aspx PCAOB (AS 8). Audit Planning. Retrieved October, 02 2014, from: http://pcaobus.org/standards/auditing/pages/auditing_standard_8.aspx PSW (2014). The 2014 Economic Crime Survey. Retrieved October, 02 2014, from: http://www.pwc.com/gx/en/economic-crime-survey/about-the-survey.jhtml. Ravisankar, P., Ravi, V., Raghava Rao, G. & Bosese, I. (2011). Detection of financial statement fraud and feature selection using data mining techniques. Decision Support Systems. 50(2): 491–500. Shibano, T. (1990). Assessing audit risk from errors and irregularities. Journal of Accounting Research. 28: 110–140. Simon, H.A. (1956). Rational choice and the structure of the environment. Psychological Review. 63(2): 129-138. Simon, H. A. (1957). Models of Man. Social and Rational. John Wiley & Sons. New York. Solomon, I., Shields, M.D. & Whittington, O.R. (1999). What Do Industry-Specialist Auditors Know? Journal of Accounting Research. 37(1): 191-208. Watkins, A.L., Hillison, W. & Morecroft, S.E. (2004). Audit quality: a synthesis of theory and empirical evidence. Journal of Accounting Literature. 23: 153-193. Yim, A. (2009). Efficient Committed Budget for Implementing Target Audit Probability for Many Inspectees. Management Science. 55(12): 2000-2018.
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TABLES Table 1. Units of analysis in audit research (Francis 2011).
Possible research topics
Audit inputs
Relation between engagement team characteristics and financial reporting quality
Audit process
Relation between audit hours, categories of staff and financial reporting quality
Accounting firms
Identifying the source of industry expertise
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Unit of analysis
Audit industry & markets
Role of industry structure in shaping audit quality The role of standards vs. monitoring and regulation in achieving audit quality
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Economic consequences of audit outcomes
Relation between drivers of audit quality and market valuation
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Table 2. The parameters of the probabilistic audit model.
Description
d = g∙o∙e
Number of audit paths identified and able to follow (perceived complexity of the audit case)
g
Total number of audit paths (objective complexity of the audit case)
o
Fraction of paths identified (experience of the auditor)
e
Fraction of paths able to follow (operative expertise of the auditor)
p
Probability that a branch point includes a material misstatement (inherent risk of the audit case)
r=c∙f
Probability to detect the misstatement when existing (1-detection risk r*)
c
Effectiveness of the audit procedures (0-1)
f
Performance of the auditor (0-1)
(h)
(Probability to report a material misstatement when detected)
v
Range of vision (visionary expertise of the auditor, investigative intuition)
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Parameter
Number of stages resourced (budget of the audit project)
Q
Probability to detect none material misstatement under budget constraint (false positive event, failure when p > 0)
P = 1-Q
Probability to detect at least one misstatement under budget constraint (true positive event, success when p > 0)
Table 3. Numerical experiments for the subjective audit quality (Q).
6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8
0,7092 0,6137 0,5709 0,4275 0,6958 0,5972 0,5533 0,4077 0,5015 0,3750 0,3243 0,1813 0,4824 0,3549 0,3044 0,1648 0,3379 0,2074 0,0781 0,0181 0,3181 0,1900 0,0678 0,0144 0,1131 0,0424 0,0060 0,0003 0,1001 0,0355 0,0045 0,0002
M
3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4
Q
Elasticity of Q with respect to: d p r H v -0,96 -0,35 -0,35 -0,43 -0,54 -1,39 -0,49 -0,49 -0,58 -0,84 -2,04 -0,56 -0,56 -0,87 -1,03 -3,20 -0,85 -0,85 -1,16 -1,83 -1,01 -0,36 -0,36 -0,46 -0,56 -1,47 -0,52 -0,52 -0,61 -0,88 -2,16 -0,59 -0,59 -0,92 -1,08 -3,38 -0,90 -0,90 -1,22 -1,93 -1,93 -0,70 -0,70 -0,87 -1,07 -2,80 -0,99 -0,99 -1,16 -1,68 -4,11 -1,14 -1,14 -1,74 -2,06 -6,43 -1,72 -1,72 -2,32 -3,67 -2,03 -0,74 -0,74 -0,92 -1,14 -2,95 -1,05 -1,05 -1,23 -1,77 -4,34 -1,20 -1,20 -1,84 -2,18 -6,79 -1,82 -1,82 -2,46 -3,88 -3,12 -1,09 -1,09 -1,46 -2,89 -4,58 -1,58 -1,58 -1,95 -4,50 -9,71 -2,56 -2,56 -4,39 -8,33 -15,57 -4,03 -4,03 -5,86 -14,77 -3,29 -1,15 -1,15 -1,55 -3,05 -4,84 -1,67 -1,67 -2,06 -4,75 -10,25 -2,70 -2,70 -4,64 -8,80 -16,44 -4,26 -4,26 -6,19 -15,59 -6,27 -2,20 -2,20 -2,94 -5,80 -9,21 -3,19 -3,19 -3,92 -9,04 -19,51 -5,17 -5,17 -8,83 -16,74 -31,28 -8,14 -8,14 -11,77 -29,67 -6,62 -2,32 -2,32 -3,11 -6,13 -9,73 -3,37 -3,37 -4,14 -9,54 -20,60 -5,46 -5,46 -9,32 -17,68 -33,03 -8,60 -8,60 -12,43 -31,34
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
d=g∙o∙e g p r=c∙f v 2 3 0,010 0,90 2 3 0,010 0,90 2 3 0,010 0,90 2 3 0,010 0,90 2 3 0,010 0,95 2 3 0,010 0,95 2 3 0,010 0,95 2 3 0,010 0,95 2 3 0,020 0,90 2 3 0,020 0,90 2 3 0,020 0,90 2 3 0,020 0,90 2 3 0,020 0,95 2 3 0,020 0,95 2 3 0,020 0,95 2 3 0,020 0,95 3 4 0,010 0,90 3 4 0,010 0,90 3 4 0,010 0,90 3 4 0,010 0,90 3 4 0,010 0,95 3 4 0,010 0,95 3 4 0,010 0,95 3 4 0,010 0,95 3 4 0,020 0,90 3 4 0,020 0,90 3 4 0,020 0,90 3 4 0,020 0,90 3 4 0,020 0,95 3 4 0,020 0,95 3 4 0,020 0,95 3 4 0,020 0,95
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Table 4. Numerical experiments for the quality of the perfect auditor (Q*).
3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4
Q 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8
0,7092 0,6137 0,5709 0,4275 0,6958 0,5972 0,5533 0,4077 0,5015 0,3750 0,3243 0,1813 0,4824 0,3549 0,3044 0,1648 0,3379 0,2074 0,0781 0,0181 0,3181 0,1900 0,0678 0,0144 0,1131 0,0424 0,0060 0,0003 0,1001 0,0355 0,0045 0,0002
Elasticity of Q* with respect to: Q* (1-Q)/(1-Q*) p H g 0,000017 0,2908 -11,03 -72,44 -60,45 0,000000 0,3863 -99,39 -869,31 -741,84 0,000017 0,4291 -11,03 -72,44 -60,45 0,000000 0,5725 -99,39 -869,31 -741,84 0,000017 0,3042 -11,03 -72,44 -60,45 0,000000 0,4028 -99,39 -869,31 -741,84 0,000017 0,4467 -11,03 -72,44 -60,45 0,000000 0,5923 -99,39 -869,31 -741,84 0,000000 0,4985 -22,29 -145,62 -121,52 0,000000 0,6250 -200,82 -1747,45 -1491,20 0,000000 0,6757 -22,29 -145,62 -121,52 0,000000 0,8187 -200,82 -1747,45 -1491,20 0,000000 0,5176 -22,29 -145,62 -121,52 0,000000 0,6451 -200,82 -1747,45 -1491,20 0,000000 0,6956 -22,29 -145,62 -121,52 0,000000 0,8352 -200,82 -1747,45 -1491,20 0,000000 0,6621 -55,15 -456,55 -311,04 0,000000 0,7926 -882,63 -9739,68 -6732,96 0,000000 0,9219 -55,15 -456,55 -311,04 0,000000 0,9819 -882,63 -9739,68 -6732,96 0,000000 0,6819 -55,15 -456,55 -311,04 0,000000 0,8100 -882,63 -9739,68 -6732,96 0,000000 0,9322 -55,15 -456,55 -311,04 0,000000 0,9856 -882,63 -9739,68 -6732,96 0,000000 0,8869 -111,43 -917,73 -625,23 0,000000 0,9576 -1783,27 -19578,24 -13534,28 0,000000 0,9940 -111,43 -917,73 -625,23 0,000000 0,9997 -1783,27 -19578,24 -13534,28 0,000000 0,8999 -111,43 -917,73 -625,23 0,000000 0,9645 -1783,27 -19578,24 -13534,28 0,000000 0,9955 -111,43 -917,73 -625,23 0,000000 0,9998 -1783,27 -19578,24 -13534,28
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0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,020 0,020 0,020 0,020 0,020 0,020 0,020 0,020 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,020 0,020 0,020 0,020 0,020 0,020 0,020 0,020
PT
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
r=c∙f v 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95
ED
p
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
d=g∙o∙e g 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
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Table 5. Numerical experiments for the learning coefficient (L).
AC
3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 4
6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8 6 8
Elasticity of L with respect to: u L u H v d 0,9998 0,9904 48,00 -0,03 0,00 -0,03 0,9998 0,9763 120,00 -0,07 -0,02 -0,07 0,9998 0,9904 48,00 -0,05 0,01 -0,04 0,9998 0,9685 160,00 -0,12 -0,03 -0,13 0,9998 0,9904 48,00 -0,03 0,00 -0,03 0,9998 0,9763 120,00 -0,07 -0,02 -0,07 0,9998 0,9904 48,00 -0,05 0,01 -0,04 0,9998 0,9685 160,00 -0,12 -0,03 -0,13 0,9998 0,9904 48,00 -0,03 0,00 -0,03 0,9998 0,9763 120,00 -0,07 -0,02 -0,07 0,9998 0,9904 48,00 -0,05 0,01 -0,04 0,9998 0,9685 160,00 -0,12 -0,03 -0,13 0,9998 0,9904 48,00 -0,03 0,00 -0,03 0,9998 0,9763 120,00 -0,07 -0,02 -0,07 0,9998 0,9904 48,00 -0,05 0,01 -0,04 0,9998 0,9685 160,00 -0,12 -0,03 -0,13 0,9998 0,9681 162,00 -0,11 -0,05 -0,10 0,9998 0,9222 405,00 -0,24 -0,18 -0,24 0,9998 0,9526 243,00 -0,24 -0,05 -0,19 0,9998 0,8504 810,00 -0,58 -0,42 -0,65 0,9998 0,9681 162,00 -0,11 -0,05 -0,10 0,9998 0,9222 405,00 -0,24 -0,18 -0,24 0,9998 0,9526 243,00 -0,24 -0,05 -0,19 0,9998 0,8504 810,00 -0,58 -0,42 -0,65 0,9998 0,9681 162,00 -0,11 -0,05 -0,10 0,9998 0,9222 405,00 -0,24 -0,18 -0,24 0,9998 0,9526 243,00 -0,24 -0,05 -0,19 0,9998 0,8504 810,00 -0,58 -0,42 -0,65 0,9998 0,9681 162,00 -0,11 -0,05 -0,10 0,9998 0,9222 405,00 -0,24 -0,18 -0,24 0,9998 0,9526 243,00 -0,24 -0,05 -0,19 0,9998 0,8504 810,00 -0,58 -0,42 -0,65
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0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,020 0,020 0,020 0,020 0,020 0,020 0,020 0,020 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,010 0,020 0,020 0,020 0,020 0,020 0,020 0,020 0,020
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r=c∙f v 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95 0,90 0,90 0,90 0,90 0,95 0,95 0,95 0,95
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
d=g∙o∙e g 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
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FIGURES
Inputs
Process
Outcomes
Context
Audit partner compensation Abnormal audit fees Non-audit fees Audit fee premium – Big N auditors and industry specialist Auditor tenure
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Figure 1. Indicators of audit quality (Knechel et al. 2012).
AC
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Judgment in the audit process Audit production Assessing risk Analytical procedures Obtaining and evaluating evidence Auditor-client negotiations Review and quality control
Adverse Outcomes o Restatements o Litigation Financial reporting Quality o Discretionary accruals o Accounting conservatism Audit reports Peer review and PCAOB inspections
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Incentives and motivation Professional skepticism Knowledge and expertise Within firm pressure
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Inputs
Process
Outcomes
Probability to detect the misstatement (detection risk) Effectiveness of the audit procedures Performance of the auditor (Probability to report a material misstatement when detected)
Context
Total number of potential audit paths (complexity of the audit case) Probability that a branch point includes a material misstatement (inherent risk of the audit case) Number of stages constrained by budget (resources of the audit project)
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Figure 2. Parameters of the model in the framework by Knechel et al. (2012).
AC
Probability to discover at least one misstatement under budget constraint (overall audit quality) o An ordinary auditor o The perfect auditor
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Fraction of paths identified (experience of the auditor) Fraction of paths able to follow (operative expertise of the auditor) Range of vision (visionary expertise of the auditor, investigative intuition)
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Appendix A. Partial derivatives and elasticities of Q, Q*, and L
Panel 1. Partial derivatives of Q
Q (1 rp) d ( d
v
1) /( d 1) ( H v ) d v
(A.1)
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v v Q r (d v ( H v) d (d v 1) /( d 1))(1 rp) d ( H v ) d ( d 1) /( d 1) 1 p
v v Q p(d v ( H v) d (d v 1) /( d 1))(1 rp) d ( H v ) d ( d 1) /( d 1) 1 r
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v v Q (1 rp) d ( H v )d ( d 1) /( d 1) ln(1 pr )(vd v /( d 1) d (d v 1) /( d 1) 2 d (d v 1) /( d 1) ( H v)vd v1 ) v v Q d v (1 rp) d ( H v )d ( d 1) /( d 1) ln(1 pr ) H
Panel 2. Partial derivatives of Q* H
1) /( g 1)
1 P *
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Q* (1 p) g ( g
(A.3)
(A.4)
(A.5) (A.6)
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v v Q (1 rp) d ( H v )d ( d 1) /( d 1) ln(1 pr )(d v ln( d )( H v) d v d v1 ln( d ) /( d 1)) v
(A.2)
(A.7) (A.8)
H Q * ln(1 p)(1 p) g ( g 1) /( g 1) ( Hg H /( g 1) g ( g H 1) /( g 1) 2 ( g H 1) /( g 1)) g
(A.9)
H Q * g H 1 ln( g ) ln(1 p)(1 p) g ( g 1) /( g 1) /( g 1) H
(A.10)
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H Q * ( g ( g H 1)(1 p) g ( g 1) /( g 1) 1 ) /( g 1) p
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Panel 3. Partial derivatives of L
L u (( H v )(1 H v ) / 2) d
v
(A.11)
v L d v ( H v)(v H 1)(1 / 2)u (( H v )(1 H v ) / 2) d 1 u
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(A.12)
v L (1 / 2)u (( H v )(1 H v ) / 2) d ln(u )( H v)(v H 1)vd v 1 d
(A.13)
v L (1 / 2)u (( H v )(1 H v ) / 2) d ln(u )d v (2 H 2v 1) H
(A.14)
= 0 when
2 ln( d ) H ln( d ) 4 ln( d ) 2 2 ln( d )
v
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Panel 4. Elasticities of Q
Q (1 rp) d ( d
(A.15)
(A.16)
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v L (1 / 2)u (( H v )(1 H v ) / 2 ) d ln(u )(d v (ln( d )( H v)(v H 1) v (2v 2 H 1))
1) /( d 1) ( H v ) d v
(A.17) (A.18)
Qr rp(d v ( H v) d (d v 1) /( d 1))(1 rp) 1 Qr
(A.19)
Qd d ln(1 pr )(vd v /( d 1) d (d v 1) /( d 1) 2 (d v 1) /( d 1) ( H v)vd v1 ) Qd
(A.20)
QH Hd v ln(1 pr ) QH
(A.21)
Qv v ln(1 pr )(d v ln( d )( H v) d v d v1 ln( d ) /( d 1)) Qv
(A.22)
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Panel 6. Elasticities of Q*
Q* (1 p) g ( g
H
1) /( g 1)
1 P *
(A.23)
Q * p p( g ( g H 1)(1 p) 1 ) /( g 1) Q * p
(A.24)
(A.25)
Q * H Hg H 1 ln( g ) ln(1 p) /( g 1) Q * H
(A.26)
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Panel 5. Elasticities of L v
Lu d v ( H v)(v H 1)(1 / 2) Lu
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Q * g g ln(1 p)( Hg H /( g 1) g ( g H 1) /( g 1)2 ( g H 1) /( g 1)) Q * g
(A.27) (A.28)
(A.29)
LH H (1 / 2) ln(u )d v (2 H 2v 1) LH
(A.30)
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Ld (1 / 2) ln(u )( H v)(v H 1)vd v Ld
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Lv v(1 / 2) ln(u )d v (ln(d )( H v)(v H 1) (2v 2 H 1)) Lv
(A.31)