By Rebecca L. Rosner, Christie L. Comunale, and Thomas R. SextonJUNE 2006 - Two difficulties that auditors face in assessing materiality are the need to make a binary decision (material versus not material) and the need to weigh certain qualitative factors in doing so. This article shows how a “fuzzy logic” expert system permits auditors to assess materiality on a continuous scale from 0 to 1, and allows for explicit consideration of important qualitative factors relevant to materiality.
In practice, an auditor must make an oversimplified, binary decision for each omission and misstatement, both individually and in the aggregate: It is either material, or it is not. A fuzzy logic model allows omissions and misstatements to possess a degree of value—that is, each omission or misstatement is material to a greater or lesser degree, measured on a scale from 0 to 1.
Auditors tend to view materiality as a quantitative concept; the larger the fluctuation, the more likely the auditor is to consider it material. This is natural, because size is easier to measure and analyze than nonquantitative factors. Both the SEC and FASB, however, recognize that although such thresholds and rules of thumb can be useful starting points, exclusive reliance on numerical thresholds has no basis in accounting literature or law. Indeed, materiality assessment requires the consideration of many qualitative factors beyond the size of the misstatement or omission. (The authors consider errors, which may or may not be material, to include misstatements, in which a numerical or textual item is reported incorrectly, and omissions, in which a required item is absent from the financial statement.) SEC Staff Accounting Bulletin (SAB) 99 lists qualitative factors that “render material a quantitatively small misstatement of a financial statement item.” These factors ask whether the misstatement does the following:
FASB has also emphasized that materiality is not strictly a quantitative concept (Statement of Financial Accounting Concepts 2). In fact, FASB rejected a formulaic approach for determining materiality in favor of one that incorporates all relevant circumstances.
Typically, qualitative factors are more difficult to assess than the size of a misstatement or omission. Qualitative factors often require subjective judgment and evaluation in light of other information that may not be readily available to the auditor during the audit.
Consequently, auditors tend to rely on quantitative evaluations and to do so using simplistic numerical thresholds and rules of thumb. The authors reviewed the materiality worksheets of three national public accounting firms and found no specific guidance regarding how to evaluate the qualitative factors in the materiality assessment process. One worksheet made no mention of qualitative factors. A second worksheet reminded the auditor to consider such factors but provided only one example, that of an illegal payment. The third worksheet listed the qualitative factors from SAB 99 but provided no methodology for incorporating such factors into the overall materiality assessment. In practice, qualitative factors, while recognized as important, are likely to be overlooked.
Fundamentals of Fuzzy Logic
Lofti A. Zadeh first introduced the concepts of fuzzy sets and fuzzy logic, which rests on the conceptual premise that an item can have partial membership in a set [“Fuzzy Sets,” Information and Control, 8: 338–353 (1965)]. In a simple example, consider the set of all “tall” men. The classical approach requires establishing a threshold height (e.g., six feet) and declaring that a man is “tall” only if he is at least six feet tall. Must we be so arbitrary? Using the fuzzy logic approach, membership in the set of all tall men could be equal to 0 for men who are five feet tall, equal to 1 for men who are seven feet tall, and equal to 0.5 for men who are six feet tall. Exhibit 1 shows one such membership function.
The classical logical functions “not,” “or,” and “and” are built on the classical theory of sets. For example, consider the set of all men who are “tall and wealthy.” Classical logic stipulates that a man must be a member of both sets, the set of tall men (at least six feet tall) and the set of wealthy men (which may be defined as having a net personal wealth of at least $1 million). In a parallel fashion, fuzzy logic computes the membership of a man in the set of all men who are both tall and wealthy using the man’s membership in the individual sets. For example, suppose that a man has 0.6 membership in the set of tall men and 0.2 membership in the set of wealthy men. Then he would have 0.2 (the minimum of 0.6 and 0.2) membership in the set of tall and wealthy men. Similarly, he would have 0.6 (the maximum of 0.6 and 0.2) membership in the set of tall or wealthy men. Finally, he would have 0.4 (1 minus 0.6) membership in the set of men who are “not tall.” All of the standard logical operations can be used with the principles of fuzzy logic.
Classical and Fuzzy Rule-Based Expert Systems
A rule-based expert system can be thought of as an advisory board consisting of many advisors. In this application, the auditor—the decision maker—would ask each advisor to review a given omission or misstatement and decide whether it is material. Imagine that each advisor focuses on only one aspect of the omission or misstatement in making this determination. For example, one advisor may consider only the size of the omission or misstatement, whereas a second advisor may consider only the extent to which it increases management compensation. Thus, each advisor represents one of the rules in the expert system. Finally, the decision maker would assimilate the various opinions of the advisors and reach a conclusion on the materiality of the omission or misstatement. In doing so, the decision maker would also incorporate her assessment of the aptitude of the advisors in making materiality judgments.
In a classical system, each advisor would be required to make a binary decision: the omission or misstatement is material or it is not material. For example, the first advisor may respond that, based on the size of the omission or misstatement, the omission is not material; the second advisor may say that, because it increases management compensation, it is material.
In a fuzzy system, each advisor would be allowed to express his materiality judgment as a number anywhere between 0 and 1, with 1 meaning the omission or misstatement is material, and 0 meaning it is not material. Their responses represent initial materiality assessments that would be modified later based upon the aptitude of the advisor.
In a classical system, every rule is assumed valid, meaning that every advisor’s viewpoint is assumed equally relevant to the materiality assessment. In a fuzzy system, not every rule is equally relevant to the materiality assessment. The decision maker must assign to each rule a validity value that corresponds to the auditor’s assessment of the aptitude of the advisor. For example, the auditor may assign a modest 0.35 validity value to the rule:
In the fuzzy system, the auditor then multiplies the initial materiality membership produced by each rule by its validity value, to produce the misstatement’s final value. Therefore, the final materiality assessment involving the medium size of the omission or misstatement would be 0.14 (0.4 x 0.35). The management compensation issue would yield a result of 0.765 (0.9 x 0.85). No such computations are required in the classical system, because the classical system simply produces a collection of binary (material or not material) assessments, one for each rule.
The last step is to assimilate the final assessments produced by the rules into an ultimate materiality assessment. The standard approach is to select the maximum final materiality value produced by any rule:
In the classical system, the misstatement would be assessed as material if one or more of the rules indicated that the misstatement was material. In the fuzzy system, the auditor would use the two formulas above to determine a value indicating the level of materiality.
Using Fuzzy Materiality Assessments
Each auditor might decide on a policy of how to convert fuzzy materiality values into audit actions. One approach is to interpret materiality assessments as follows:
In the example, the auditor would conclude that the ultimate materiality assessment is high on the materiality scale; indeed, close to very high. The firm’s policy would then dictate the appropriate actions for the auditor to follow, depending upon the materiality assessment category.
An effective fuzzy logic expert system to assess materiality must incorporate rules that capture both the quantitative and the qualitative aspects of a misstatement. The quantitative aspects should include both the size of the misstatement and the precision with which it is measured. The qualitative aspects should include the SAB 99 factors listed above. They could also be defined as fuzzy sets, as shown in Exhibit 2.
Greater Insight for Auditors
Auditors routinely encounter misstatements during the course of an audit. By assigning a fuzzy degree of materiality anywhere between 0 and 1, the auditor has more flexibility and precision in materiality assessment, and greater insight regarding subsequent testing and investigation.
By providing a formal model structure, the fuzzy system formalizes and documents the materiality assessment process. It requires an auditor to evaluate each qualitative and quantitative factor explicitly. It also requires that the system designers state each rule specifically and assign validities to each rule unambiguously. This facilitates better communication within the audit team and with the client, and enhances process consistency across auditors, engagements, and years.
To build a valid and reliable fuzzy expert system to assess materiality, auditing experts need to carefully extract the quantitative and qualitative factors that auditors should apply, the fuzzy rules they should use, and the validities of these rules. Once the initial design is complete, a formal feedback system may be incorporated to improve future performance. If materiality assessments later prove to be inaccurate, the system can modify itself to improve future assessments of similar situations.
In the era of SOX and increased auditor responsibility, where auditors are expected to do a more thorough job in assessing the risk of material misstatements, there may be value in considering the fuzzy logic approach.
Rebecca L. Rosner, PhD, CPA, CISA, and Christie L. Comunale, PhD, CPA, are both associate professors in the school of professional accountancy at Long Island University—C.W. Post Campus, Brookville, N.Y.
Thomas R. Sexton, PhD, is a professor in the college of business at Stony Brook University, Stony Brook, N.Y.