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

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Group Audits

Characteristics of group audits 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)

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

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Component materiality:Wide variation in practice; little guidance; no published research; no

generally accepted conceptual basis

Allocation of $100 group materiality to 10 equal components—various methods in use

Group materiality (ISA upper extreme), 1×$100 $100

Half of group materiality, 0.5×$100 $50

Proportionate to square root of size, √0.1×$100 $32

Arithmetic proportion (ISA lower extreme), 0.1×$100 $10

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

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Need for researchIntense 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., Nonna Martinov-Bennie, and Aasmund Eilifsen. “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)

Representing and aggregating assuranceAuditor’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 profileThe group assurance profile defines the 95% upper limit to potential misstatementIf 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 materialityGroup materiality is the target 95% upper misstatement limit for the groupBased on component size and other factors, GUAM works backwards to determine target assurance profiles for each componentComponent materiality is the 95% upper misstatement limit for the componentComponent materiality is used to determine the extent of work sufficient to achieve the target assurance for the componentIf 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

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GUAM

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

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.

- 1β 2β 3β 4β 5β

95%86%63%

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.

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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

0/β

1/β

0.0β 1.0β 2.0β 3.0β 4.0β 5.0β 6.0β 7.0β 8.0β

α =1

α =2

α =3α =4

x = Total Misstatement

1 /( / )( ; , ) , 0; , 0

( )

xx eg x x

Aggregation and allocation:Two equal components, exponential assurance profiles

Aggregation

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

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

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- 1β 2β 3β 4β 5β 6β

- 1β 2β 3β 4β 5β 6β

95%

95%

95th Percentile = 3.0β

- 1β 2β 3β 4β 5β 6β

95%

95th Percentile = 4.74β

Distribution of x1

Distribution of x2

Distribution of x1 + x2

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

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Number of Components

Component Materiality

Groupwide Multiple

1 1.00 1.002 0.63 1.263 0.48 1.434 0.39 1.555 0.33 1.646 0.29 1.717 0.25 1.778 0.23 1.829 0.21 1.87

10 0.19 1.91

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Real groups are more complicated

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,…

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Component materiality is set to achieve group audit objectives

This 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

ii

j

Yw

Y

Group M

Confidence (95%)

M1

M2

:

MN

Group materiality and confidence are determined

Weights are assigned to

components

Component materiality

w1

w2

:

wN

∑wi = 1

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

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1 3

4

ALGORITHM

Component materiality is

computed

Overall planning parametersConfidence Level 95.0%Group Materiality 100

Component materiality computed

Component Name Size (Y) Likely Misstatement Materiality √Y w α β E = αβ V = αβ2

Component 1 9,000 - 63 95 0.5000 1.0000 21.08 21.08 444.36 Component 2 4,000 - 48 63 0.3333 1.0000 15.88 15.88 252.29 Component 3 1,000 - 28 32 0.1667 1.0000 9.51 9.51 90.48 Component 4 - - - - - - - - - Component 5 - - - - - - - - -

14,000 190 1.0000 46.48 787.13

Group summary AggregateAchieved group Mat. 100 α = E2/V 2.74

β = V/E 16.94

Chart Scale 1:Mat 1:100

- 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Software 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

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This panel is typically hidden

from the user

This panel is typically hidden

from the user

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The “Materiality Horizon”Various optimizations are possible

Optimizations M1 M2

A Work minimization. M1 and M2 minimize the total work across the group.

63 63

B Super precision. Component 1 has one easily verified asset and a computable income stream, so that the auditor is virtually 100 percent assured that there is no misstatement. For group purposes it is sufficient to set M2 equal to group M.

0 100

C Forced precision. Component 1 requires a separate statutory audit with M1 = 50, an amount less than that required for group purposes. The algorithm is run backwards to compute M2 given M1 and group M.

50 75

D Cost minimization. Components are essentially the same except that the marginal cost of auditing Component 1 is 25% that of Component 2.

48 77

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

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

M1

M2

A

B

D

C

Cost Min(48,77)

Work Min(63,63)

X

Y

Z

Example: Two equal-sized components,Group M = $100 (Confidence = 95%)

Final thoughts

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

28 mei 2008 - Symposium Statistical Auditing

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 bothUnified: 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 levelIt is an extension of the standard Audit Risk Model, not totally newIt provides a conceptual and practical computational framework

Assurance and Materiality in Group and Other Multi-Component Audits

Trevor R. StewartDeloitte & Touche LLP

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

Upcoming paper, Summer 2008

Finis