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### Allocating Materiality and Aggregating Results

Trevor Stewart

28 mei 2008 - Symposium Statistical Auditing

Slide 1

I will focus on materiality allocation in group audits(which is the inverse of aggregation)

- This will keep the discussion concrete and practical
- Subject is topical: ISA 600 on group audits will apply starting in 2010 (audit periods starting on or after December 15, 2009)
- Group engagement partner is required to set component materiality at level lower than group
- Minimal guidance how to do so
- No published research
- Big problem

Group Audits

Characteristics of group audits

Types of groups

Group audit strategy, including the determination of component materiality, depends on how the group is organized and managed

In some groups, components are independently managed and audited

For example, a group may have a manufacturing subsidiary in Pittsburgh and a leasing subsidiary in Paris that are run and audited independently

Materiality allocation required

Some groups are run as one virtual single entity

For example, the components may simply represent legal entities that are operationally and systemically irrelevant and which do not require separate audits

Materiality allocation may not be required

Some groups are somewhere in between

Materiality allocation will also be somewhere in between the extremes

We will assume independent components

- A group audit is comprised of multiple components or locations that are reported in consolidated or group financial statements
- The group engagement partner must rely on the work of the component auditors
- The group engagement partner determines or approves:
- Group materiality, and
- Component materiality levels.
- Component materiality drives the extent of work and the resulting assurance at the component level
- Component materiality must be set such that the group auditor achieves the desired level of group overall audit assurance (assuming the audit goes as planned)

Component materiality:Wide variation in practice; little guidance; no published research; no generally accepted conceptual basis

Guidance in ISA 600, paragraph A43

- “To reduce the risk that the aggregate of detected and undetected misstatements in the group financial statements exceeds the materiality level for the group financial statements as a whole, the component materiality level is set lower than the group materiality level.
- “Different materiality levels may be established for different components.
- “The component materiality level need not be an arithmetical portion of the group materiality level and, consequently, the aggregate of the component materiality levels may exceed the group materiality level.”
- For example, component materiality should be less than $100 but need not be as small as $10

Need for research

Intense interest among regulators and practitioners in conceptually sound practical guidance in view of wide variety of working practices.

“No research that we are aware of has investigated how planning materiality (or its allocation) or evaluation materiality is handled on multilocation audits. Given the diverse nature of, and/or multinational operations of, enterprises today, research in this area is needed.” Messier, Jr., William F., NonnaMartinov-Bennie, and AasmundEilifsen. “A Review and Integration of Empirical Research on Materiality: Two Decades Later.” Auditing: A Journal of Practice and Theory 24.2 (Nov. 2005): 153-

A solution is proposed within the framework of aGeneral Unified Assurance Model (GUAM)

GUAM

Representing and aggregating assurance

- Auditor’s professional (subjective) assurance about potential misstatement in a component is represented by a probability distribution
- Known as an assurance profile in GUAM
- The assurance profile is refined as assurance about the component is accumulated and evidence is obtained
- This is consistent with but a considerable extension of the profession’s standard Audit Risk Model, AR = RMM×DR
- GUAM is used to aggregate results across components to derive a group assurance profile
- The group assurance profile defines the 95% upper limit to potential misstatement
- If the audit is properly planned and goes as expected then the evaluated upper limit should equal group materiality
- Which will allow the group auditor to conclude with 95% confidence that total misstatement does not exceed group materiality

Allocating materiality

- Group materiality is the target 95% upper misstatement limit for the group
- Based on component size and other factors, GUAM works backwards to determine target assurance profiles for each component
- Component materiality is the 95% upper misstatement limit for the component
- Component materiality is used to determine the extent of work sufficient to achieve the target assurance for the component
- If the audits go as expected and target assurance is achieved for each component, then component assurance profiles will aggregate to deliver the desired group assurance profile, and thus 95% confidence relative to group materiality

Audit assurance is typically expressed as one point, e.g., “We are 95% confident that total misstatement does not exceed $300K.”

In GUAM this is just one point on a continuum expressing assurance in relation to potential misstatement.

The continuum is the assurance profile—a probability distribution—expressing the auditor’s professional judgment about the potential for undetected misstatement.

The exponential distribution is a simple (but important) form of assurance profile

It is the source of many tables used in practice—for example, the Reliability Factors Table 6.1 of the AICPA, Audit Sampling Guide

It is a member of the family of gamma distributions.

Assurance profiles (prior probability distributions) are a key concept in GUAM63%

86%

95%

Assurance profiles in GUAM are represented by gamma probability distributions

- Intuitively appealing variety of shapes
- Closely related to (a conjugate prior of) the Poisson distribution used in audit sampling, especially MUS
- Already used in auditing (shape α = 1 is the exponential distribution)
- Widely used in fields similar to auditing

x = Total Misstatement

Distribution of x1 + x2

95%

Distribution of x2

95%

95th Percentile = 4.74β

95%

95th Percentile = 3.0β

Aggregation and allocation:Two equal components, exponential assurance profilesAggregation

Planning: Allocation

Group auditor expects to be 95% confident total misstatement does not exceed 4.74β

Therefore group materiality is M = 4.74β

Therefore component materiality should be 3.0/4.74M = 0.63M

- Component auditor may be 95% confident total misstatement does not exceed 3.0β
- Therefore component materiality is 3.0β
- Group auditor can be 95% confident total misstatement does not exceed 4.74β

α =2

Component materiality for groups of equal-sized components(95% confidence assumed)

- For example, for 3 components component materiality should be 0.48 times group materiality
- The materiality multiple is 1.43
- Component materiality is 1.43 times “standalone” materiality for the component

Notes

- Equal components is ordinarily a “worst-case” situation
- The allocation assumes complete independence of the component audits, an assumption that results in smallest component materiality levels
- In most groups a number of components also require statutory audits with a materiality level lower than the component materiality level
- Lower materiality does not translate into proportionately more work as much audit work is fixed, regardless of materiality, or otherwise does not scale proportionately

Many components

A multidimensional problem

Unequal component sizes

Causes technical problems with the convolution of component gamma distributions

Expected misstatement might not be zero for some components

Materiality may be “pre-determined” for some components

For example, where statutory audits are involved

Various optimizations may be required

Minimize amount of work

Minimize cost

Components are not necessarily all audited “independently” in the stochastic sense

Etc., etc., etc,…

Real groups are more complicatedThis works irrespective of how the components are weighted as long as the weights sum to 1.

Therefore the auditor is free to weight the components to achieve secondary goals, such as work or cost minimization.

Typically, the secondary goal is to minimize work across the group while achieving the required level of group audit assurance

Mathematically, this is a classic constrained optimization problem

If Yi is the “size” of component i, then work will be approximately minimized for weights

Group materiality and confidence are determined

Component materiality

Group M

Confidence (95%)

M1

M2

:

MN

3

1

ALGORITHM

Component materiality is computed

w1

w2

:

wN

If the audits using component materiality go as planned aggregate assurance will meet group audit objectives.

Weights are assigned to components

4

2

∑wi = 1

Component materiality is set to achieve group audit objectivesSoftware solution in Microsoft Excel

- Despite underlying complexity, software is easy to use
- User just needs to specify overall group parameters and sizes of components
- Software computes component materiality
- More complex group situations can also be dealt with by the software

This panel is typically hidden from the user

Sub-Optimal: X = Group Materiality, Y = 50% Group Materiality, Z = M × √RelativeSize

The “Materiality Horizon”Various optimizations are possibleExample: Two equal-sized components,Group M = $100 (Confidence = 95%)

Final thoughts

GUAM is a step towards a General, Unified Assurance Model

General: It works whether assurance is subjective professional judgment, statistically based, or a combination of both

Unified: It provides a common framework for the accumulation of assurance at the account level as well as the aggregation (roll-up) to the financial statement and group level

It is an extension of the standard Audit Risk Model, not totally new

It provides a conceptual and practical computational framework

- The GUAM materiality allocation algorithm explained here establishes a lower bound for component materiality
- When other factors are considered larger component materiality levels may be indicated
- Where separately-reporting entities share services and audit work is performed at the service center, GUAM may be used to determine how much of that assurance may be taken at the entity level
- GUAM can also be used to determine tolerable misstatement for individual financial statement accounts/assertions and to aggregate results

Upcoming paper, Summer 2008

Assurance and Materiality in Group and Other Multi-Component Audits

Trevor R. StewartDeloitte & Touche LLP

William R. Kinney, Jr.University of Texas at Austin

28 mei 2008 - Symposium Statistical Auditing

Finis

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