Belief-Function Formulas for Audit Risk

Abstract

This article relates belief functions to the structure of audit risk and provides formulas for audit risk under certain simplifying assumptions. These formulas give plausibilities of error in the belief-function sense.

We believe that belief-function plausibility represents auditors’ intuitive understanding of audit risk better than ordinary probability. The plausibility of a statement, within belief-function theory, measures the extent to which we lack evidence against the statement. High plausibility for error indicates only a lack of assurance, not positive evidence that there is error. Before collecting, analyzing, and aggregating the evidence, an auditor may lack any assurance that a financial statement is correct, and in this case will attribute very high plausibility to material misstatement. This high plausibility does not necessarily indicate any evidence that the statement is materially misstated, and hence, it is inappropriate to interpret it as a probability of material misstatement.

The SAS No. 47 formula for audit risk is based on a very simple structure for audit evidence. The formulas we derive in this article are based on a slightly more complex but still simplified structure, together with other simplifying assumptions. We assume a tree-type structure for the evidence, assume that all evidence is affirmative and that each variable in the tree is binary. All these assumptions can be relaxed. As they are relaxed, however, the formulas become more complex and less informative, and it then becomes more useful to think in terms of computer algorithms rather than in terms of formulas (Shafer and Shenoy 1988).

In general, the structure of audit evidence corresponds to a network of variables. We derive formulas only for the case in which each item of evidence bears either on all the audit objectives of an account or on all the accounts in the financial statement, as in Fig. 1, so that the network is a tree. Usually, however, there will be some evidence that bears on some but not all objectives for an account, on some but not all accounts, or on objectives at different levels; in this case, the network will not be a tree.

We assume that all evidence is affirmative because this is the situation treated by the SAS No. 47 formula and because belief-function formulas become significantly more complex when affirmative and negative evidence is combined. This complexity is due primarily to the renormalization involved in Dempster’s rule for combining belief functions.

The variables in the network or tree represent various audit objectives, accounts, and the financial statement as a whole. We assume these variables are binary. For example, we assume that an account either is or is not materially misstated. This assumption is clearly too restrictive for most audit practice. Often, for example, an auditor must consider immaterial errors in individual accounts that could produce a material error in the financial statement when they are aggregated.

We derive formulas for plausibility of material misstatement at three levels: the financial statement level, the account level, and the audit objective level. The formula at the audit objective level resembles the SAS No. 47 formula,^1 but the formulas at the other two levels are significantly different. Because our model does distinguish evidence gathered at the three different levels, audits based on our formulas are sometimes significantly more efficient^2 than audits based on the SAS No. 47 model or on the simpler Bayesian models.