AN ANALYTICAL APPROACH
MATERIALITY
Byung T. Ro*
TG
ACCOUNTING
Introduction
The concept and standard of materiality play a key role in determining
what (set of) accounting information is made public through accounting
channels. The essence of the materiality issue is that an item of accounting
information should be made public if it (or its magnitude)' is material. Alternatively, any item that (or whose magnitude) is not material may be ignored for
disclosure as a separate item.2 This implies that, as Rose et al. (1970) argue, not
all financial information need be disclosed, nor need even all errors be corrected.
Despite its importance, the materiality issue has not been fully explored
within an analytical framework. Rather, accounting researchers as well as
accounting rule-making bodies have used a piecemeal approach to the issue and
have often hastily attempted to make suggestions or draw specific conclusions?
The previous approaches to accounting materiality do not hold much prospect
for understanding as well as resolving materiality issues for at least two reasons.
First, the approaches are based on the assumption that it is feasible to derive a
fKed materiality standard for an accounting disclosure. With this assumption,
researchers, like accounting rule-making bodies, are primarily interested in
searching for the variable that should be used as a base of materiality standard
and/or the specific materiality standard that should be .followed. However, the
question of whether the assumption is valid Seems overlooked.
Second, the previous studies lack an analytical model, or a well-structured
theory, of materiality that can be used in examining basic materiality issues.
Some of the studies merely attempt to defme what materiality is or should be,
while others suggest a general direction to be followed, without examining some
basic materiality issues (e.g., Dohr (1950), Rappaport (1964), Hicks (1964),
Bernstein (1967), Woolsey (1973), Theil (1972, p.321), and O'Connor and
Collins (1974)). Also, some others simply suggest a materiality standard (for
example, five percent of net income or its border zone) derived from their
research using a sample of hypothetical (or actual) users (or producers) of
accounting information, again without examining basic issues concerning materiality (e.g., Frishkoff (1970), Rose et al. (1970), Pattillo (1973), Boatsman and
,
*The author is Assistant Professor of Accounting at the Krannert Gmduate
School of Management, Purdue Universiry, Indiana This pmject was generously
supported by the Purdue Research Foundation and the Peat, Marwick, Mitchell
Foundation The author wishes to thank his colleagues at Atnlue University,
Robert W. Ingram, Daniel W. COUins, Gemld L. Saimnon, and the other partiicipants m the accounting and ftnance workshop at the University of Iowa, as
wll (IS an anonymous referee, for their hebfil comments and suggestions on
an earlier &aft of this paper. (Paper received May 1981, revised November
1981)
Joumal of Business Finance & Accounting 9,3( 1982)
39 7
Robertson (1974), Moriarity and Barron (1976), Abdel-Khalik (1977), and
Ingram (1978)):
The purpose of this study is to explore certain conceptual issues of
accounting materiality using a theory of choice under uncertainty. The basic
issues explored are: (a) the meaning of materiality of an accounting information
item, (b) the measurement of materiality, and (c) the feasibility of establishing
a materiality standard for an accounting information item.
The next section discusses the concept of materiality, the basic issue (a)
above, in a decision context. The third section deals with the basic issue (b) in
a simplified situation where there is only one financial information user who has
given tastes and beliefs, materiality judgment consistent with the judgment by
information producers, and homogeneous decision problems. Traditional
approaches to accounting materiality have failed to distinguish the materiality
of an information item from that of its particular magnitude, although this
distinction is important. Therefore, discussions in these two sections will focus
on that distinction. The fourth section examines the materiality issue (c) in the
simplified situation described above. The fifth section discusses the issue (c) for
a more general case where those simplifying conditions are relaxed, with a group
of information users instead of a single information user. The conclusions are
presented in the fmal section,
Concept of Accounting Materiality
A Brief Review
The concept of materiality has long been discussed in the accounting literature. Traditionally, it has been argued that materiality (or’the determination
of materiality) is a matter of individual or group judgment by financial information users or its producers. Using the notion of judgment, Kohler (1970,
p.279) defines materiality as:
The characteristic attaching to a statement, fact, or item whereby its disclosure . . wodd be likely to influence the judgment of a reasonable person.
According to KoNer’s defmition of materiality, probably the most frequently
cited definition in the accounting literature, the person in question should make
a judgment as to whether an item is material. Reflecting who makes materiality
judgment, previous approaches to accounting materiality fall into three categories: the information-producer approach, the informationuser approach, and
the producer-user integrated approach. The information-producer approach
takes, at least implicitly, the position that accounting information producers,
such as management and auditors, can,make materiality judgment on behalf of
information
The informationuser approach adopts the idea that it should be financial
information users who ultimately have to make materiality judgment. Since
inconsistency in materiality judgment between producers and users is possible,
those favoring this approach argue that materiality issues should (or can) be
resolved directly from the standpoints of accounting information users?
.
398 R o
The producer-user integrated approach differs from the above two approaches
in admitting that information producers and users are both important in making
materiality judgment, but that there can exist an asymmetry of their materiality
judgment.’
ConceptsAdopted
Accounting information is produced to serve its users’ need for financial
information in making informed economic decisions (FASB,1978). If producers
make materiality judgment, they do so as agents to provide financial information
as needed by users, Therefore, whether an accounting information item is
material is a matter of a user’s judgment. For this reason, the present study
adopts a user-oriented materiality approach in a decision context, but without
assuming any particular type-average or professional-of users?
If materiality is viewed as a user-oriented concept in a decision context, it
can be said that financial information users as decision makers’ make materiality
judgment about uncertainty surrounding decision problems. Such an uncertainty
is generally resolved in terms of a probability assessment, which is a form of
(Hogarth 1975), made to help a decision maker take an optimal
action. A probability assessment depends upon information available. Logically,
then, the materiality concept should be linked to the role of information in
assessing uncertainty about economic decisions by financial statement users.
For this reason, in defming materiality, the present study focuses on the role
of information in a decision context and its effect on users’ economic
decisions.’
The role of information in decision analysis is reflected in the expected
payoff consequence of that information, which is referred to as the value of
information (e.g., Feltham (1968 and 1972), Demski (1972) and Winkler (1972,
p.85)). In the present study, therefore, materiality is viewed as a matter of
information value.’ ’The value of an information item itself, however, is not the
same as the value of the item’s particular magnitude since the former is an aggregation of the information values of all possible magnitudes (signals) that are
expected to be realized (disclosad) eventually for that item.
In addressing materiality issues, it is important to distinguish the materiality
of an information item from that of its particular magnitude as indicated above.
Thus the following two separate definitions of materiality are adopted in this
study:
Definition: An item of accounting information is material only if it has information value to a financial statement user.
If an accounting information item is considered in a decision problem before
the item is realized, materiality is an ex m t e concept. In an ex ante sense, an
accounting information item is equivalent to the set of all its possible magnitudes expected to be observed by users. although a particular magnitude of the
item (i.e., an element of the set) is actually observed at a particular point in time.
An Analytical Appraach to Accounting
399
However, er ante, it is not known exactly which particular magnitude is going to
be realtzed at a given point in time. Therefore, it is conceivable that users
consider all an item’s expected magnitudes (as a set) in the decision analysis.
Thus, if an item as a set has information value, the item’s particular magnitude
itself as an element of the set can have its information value, although the latter
may not be as large as the former.’
Since it is important to distinguish an item’s information value from the value
of its particular magnitude in addressing the question of materiality, the following definition of materiality for a particular magnitude of an accounting information item is also adopted:
Definition: Let an accounting information item be material. Then, a particular
magnitude of the item is also material only if it contributes to the materiality
of the item.
The first definition suggests that the existence of information value is
necessary for an item to be material. The second definition implies that, ex ante,
the condition for an item to be material is a necessary condition for the item’s
particular magnitude to be material: if an item is not material, it is not material
no matter how large its ex ante magnitude is. In a world of uncertainty, a large
number of an item is not necessarily considered more material than the item’s
smaller magnitude since the former is not always more informative (or more
likely t o be realized) than the latter, and vice versa. This statement is consistent
with the view that magnitude (signal or number) itself is not “a sufficient
basis for a materiality judgment” (FASB 1980); in order to be material, a particular magnitude should be attached to an item that is material.
Measurement of Materiality
In the previous section, the concept of materiality was discussed in a decision
context. One advantage of viewing materiality in a decision context is the fact
that materiality can be quantified within the framework of a decision model.
Using the expected utility model for making decisions, this section demonstrates
how to measure materiality.
Setting
A financial information user (a decision maker) has a decision problem that
can be described by means of the state set S , action set A, and probability
function p(-), all finite and continuous. An outcome xeX, the finite outcome
space, of an action aEA that the decision maker can take is determined by the
payoff function, x(s, a) where s E S is an uncertain future state. For a given s E S,
he knows what outcome will result from an action choice. However, he does
not know which state will occur. Therefore, the outcome of an action choice is
uncertain.
The decision maker chooses among rules of action. Choosing the best rule of
action is equivalent to choosing the best action available (Marschakand Radner
1972, Ch.2). A decision to choose a rule of action (i.e., a decision rule can be
made with or without considering an information item in decision analysis.
400, K O
When a particular signal (magnitude)’ y e Y, the finite signal set, representing
the item is considered in decision analysis, the choice of a decision rule depends
upon information in that signal. Thus a decision rule 4.)is viewed as a function,
a = 4y), which maps Y to A. The particular signal y is determined jointly by
the occurrence of a state and an information system q. That is, yq(s).
In a state-action-independentworld, an outcome x e X changes as either
s E S or a e A (or both) changes. Since uncertainty about future states is assessed
using information available and since a = a@), information in y e Y can affect
the determination of x e X through its effect on either s E S or a e A (or both).
As in choosing a decision rule, he can assess the probability distribution function
p(s) for uncertain future states with or without considering an information
item represented by y E Y. If he assesses p(s) without the item considered, his
assessment is based on other decision-relevant factors, and the result is Pds)
where the subscript “0” implies “without the item in question considered”.
If he assesses p(s) with the item considered, his assessment is based on information in y E Y, as well as other decision-relevant factors, and the result is
P(S/Y).’
Similarly, a decision maker can choose a rule of action with or without
considering an information item in decision analysis. If he chooses an action
rule without considering the item, he does so using other available information,
and the result is a0 = a~(.). If he chooses an action rule using information in
the item as well, then the resulting action is a = 4 y ) .
The decision maker’s preference over decision outcomes can be represented
completely by his utility function, u(x). In this sense, the decision maker is
an expected utility maximizer in choosing an optimal decision rule.
The following assumptions are made as part of the decision setting described
above :
(a) Every y e Y provides either decision-relevant information or at worst no
disinformation that reduces a decision maker’s expected utility if he considers y in his decision analysis.
(b) The probability density function p(s) is differentiable on S.
(c) The decision function 4 y ) is differentiable on Y.
(d) The payoff function x(s, a(y)) is monotonic and differentiable on both
Sandy.
(e) The utility function u(x) is a real-valued, differentiable function with
domain X and is concave, nonnegative, monotonically increasing,
bounded, and unique up to a positive linear transformation.
Besides these assumptions, for simplicity, the study deals with the gross value
of information, thereby not considering the cost of information.’ Throughout
this (and the next) section, it is also assumed that there is only one fmancial
information user whose tastes and beliefs are given, whose decision problems are
homogeneous, and whose materiality judgment is consistent with that by
fmancial information producers.
An Analytical Apprwch to Accountirg
401
Measuring the Materiality of an Item
Consider a decision problem for which a decision maker wants to maximize
an attribute (expected utility) by choosing an optimal action. Without considering an accounting information item in the decision analysis, he will choose
the optimal action that yields the maximum expected utility.
Eu(x*)
= max
u(xo(s, ao)) po(s) ds . . . . . . . . . . . . . . . . (1)
a0
where
Eu(x*)
= the maximum expected utility from the decision maker's opti-
mal action
= the payoff function not based on an accounting information item
W(')
= the probability function not based on an accounting information
PO(*)
item
S
= a future state
= an action to be taken by the decision maker without considering
a0
an accounting information item in decision analysis.
Now suppose that the decision maker considers fmancial information in his
decision analysis. Further, suppose that an accounting information system q
is given" and is identical t o the item whose possible magnitudes, y = ~ S ) ,
can be generated from rl (that is, Y and r) are equivalent, both representing the
same item). Then he ,will choose the optimal decision rule that yields the
maximum expected utility
Eu(x*/q)
= sy may ss M s , a(Y))) P(S/Y?rl) P(Y/V) ds dY . . . . . . . (2)
where
Eu(x*/lr) = The maximum expected utility from the decision maker's optimal
action conditional upon accounting information system (item)
4y)
= the decision function defined on the set Y whose element is y
= the payoff function based on accounting information system 17
x(*)
p(./.)
= the probability function conditional upon y and/or q.
Note that the payoff function, x(s, a(y)), based on information in y E Y is
allowed to differ from the payoff function, xo (s, ao), not based on such information (see note 11). Thus the value of information in the item represented
by y E Y is given by the decision maker's expected utility difference
V(q)
= Eu(x*/Q) - Eu(x*)
.....................
(3)
where V($ is the information value of V .
The present study views V as a measure of the materiality of the infomation
item n (but not of its particular magnitude)," the greater the infomation
402 Ro
value V, the more material the item. The rationale behind the view is that V
measures the extent to which information improves the expected value of the
attribute that a decision maker wants to maximize, The improvement results
from the decision maker’s ju&ments affected by using the item’s information
in his decision analysis, as opposed to judgments based on factors other than that
information.
Measuring the Materiality of Magnitude
As indicated earlier, one may characterize an accounting information item
as the set of all its possible magnitudes that can be realized. In this case, the
item’s materiality can be decomposed into relative materiality of its various
possible magnitudes. Thus, the item’s materiality can be seen as a welghted
average of the materiality of such various magnitudes.
Suppose a decision maker expects to observe a particular magnitude y E Y of
the item q with some probability. Then the materiality of this particular y is
measured by a positive real number r defined as
r = (V)(w) . . . . . . . . , . . . . . . . . . . . . . . . . . . . , (4)
.
.
where w = p(y/q)’ and 0 5 w < 1. That is, the number r is iiewed as a measure
of the materiality of y in the sense that w represents a relative importance of a
particular magnitude y E Y in yielding the item’s materiality measured by V
Where a decision maker has already received a signal y from 17 and is about
to take an action, the materiality of this particular y is measured in terms of
the following expected utility difference:
v(y/n)=max
Isu (x(s, a(y))) p (s/y, q)ds - Eu(x*) . . . . . . . . . . . (5)
a
where v(y/q) is the materiality of the particular y from q. In this case, V(q)
v(y/~). That is, the item’s materiality is identical to the materiality of its
particular magnitude y since no other magnitude of the item is actually observed
by the decision maker in this case.
Since a magnitude itself is not meaningful unless it is associated with an item
and since, ex ante, a decision maker will consider in his decision analysis all
possible magnitudes (of an item) for which p(y/$ > 0, the following proposition
holds:
Proposition 1: (a) An ex ante magnitude of an accounting information item can
be material only if the item itself is material in an ex ante sense. (b) If an item is
material, there exists at least one ex ante magnitude of the item that is material.
Prmfi See Appendix.
The proposition explains how the materiality of an accounting informatim
item is related to the materiality of its various expected magnitudes. It says that
the materiality of an item is a necessary condition for the item’s particular
magnitude to be material, as discussed earlier. It also suggests that the materiality of a particular magnitude (i.e., a component) of an item cannot exceed the
An Analytical Apprwch to Accounting
403
materiality of the item (i.e., components as a whole) since r < V as seen in
ity of a particular magnitude (i.e., a component) of an item cannot exceed the
of the item whose realization is expected with some probability (i.e., whose
w’s are positive) are considered to be material, provided that the item is material.
Degree of Materiality
Defining the materiality of an item’s particular magnitude further enables one
to address certain basic materiality issues. One such issue is whether materiality
is an all-or-nothing concept (that is, either an item is material or it is not). The
following proposition suggests that materiality is not such a dichotomous
concept. Rather, there exists the degree of materiality that is defined on the
set of an item’s various magnitudes and the set of materiality measures of these
magnitudes.
Proposition 2: Suppose that an item of accounting information is material. Let
Y and R be two nonempty, finite, continuous sets such the Y = ( y 1 is the set
of the item’s possible magnitudes and R = { r (equal to the closed interval,
[0, V]) is the, set of the materiality measures of the item’s magnitudes. Then
there exists a materiality function r(y) whose domain is Y and whose range is
R.
Proofi See Appendix.
The materiality function defined above suggests that there exists the degree
of materiality for an item’s various magnitudes that are expected to be realized
with some probability. Also, the existence of r(y) implies that there exists a rule
to measure the materiality of all possible magnitudes of an item. Thus, if we
know the properties of r(y), further analysis of various materiality issues is
possible.
Proposition 3: The materiality function r(y) for an accounting infomation item
is continuous, differentiable, bounded, and nonnegative on the set Y. In general,
however, it is not monotonic.
Proofi See Appendix.
The continuity and differentiability of r(y) suggest that the materiality of
every y E Y for an item is measurable in terms of r, i.e., according to the rule
r(y). The boundedness of r(y) implies that the set R (or the interval, [0, V]) is
finite and, hence, that the materiality of both an item and its particular magnitude is fmite. The nonnegativity of r(y) suggests that a decision maker is never
worse off by using information in decision analysis. However, the nonmonotonicity of r(y) creates complexity for establishing a unique, fmed materiality
standard for an accounting information item.
Feasibility of a Unique Materiality Standard
A materiality standard for an accounting information item can be defined
as follows:
Definition: A specific materiality standard for an item of accounting information i s a cn’terion that partitions the item’s entire magnitude set Y = (y)
404/Ro
into two mutually exclusive and exhaustive subsets, the subset Yim ofimmaterial
y’s and the subset Ym of material y’s, such that each element of Ym, if considered in decision analysis, affects the expected utility of a decision maker, while
every element of Yim does not,
Mathematically, a materiality standard for an item can be defined as a
criterion partitioning Y into Ym and Y b such that
hl
. . . . . . . . . . . . . . . . . . . .(S)
V(q) = Eu(x*/Ym) - Eu(x*) 2 k . . . . . . . . . . . . . . . . . . . ( 6 )
Eu(x*/Ym) >Eu(x*JY&)=Eu(x*)
where k is a minimum threshold (e.g., one unit of utility) such that an increase in
expected utility conditional upon q by less than k is not worthwhile for a
decision maker to consider the item in his decision analysis. Note that k cannot
be an accounting materiality standard since it is expressed in utility.
The above definition implies that, given the materiality function r(y), a
specific materiality standard for an item is a criterion that divides Y into Yim
whose elements have no contribution (I = 0) to the materiality of the item and
Ym whose elements have some such contribution (I > 0). The definition says
“a criterion.” Therefore, the standard could be stated in a measurement unit of
an attribute (for example, one unit of utility) to be optimized in a decision
problem. However, if the standard were stated as a particular magnitude (ym
e Y) of an accounting item (as traditionally attempted in accounting), then the
derivation of a unique, fured materiality standard for an item is infeasible as
discussed below? *
Proposition 4: Suppose that the materiality function r(y) for an accounting
information item is not monotonic. Then, the establishment of a unique, fwed
materiality standard for the item, if stated in terms of a magnitude of the item,
is not feasible.
Proof: See Appendix
The above proposition suggests that it is not possible to derive a unique,
fixed materiality standard for an accounting information item even for a given
decision maker even under the simplifying conditions stated earlier. This impossibility is not only due to the nonmonotonicity of r(y), specifically uncertainty surrounding a decision problem, but also partly due to the choice
of a measurement unit for materiality.2
The infeasibility of deriving a unique materiality standard of an item even for
a given user also becomes clear if one allows tastes and beliefs to change,
materiality judgment to be inconsistent between a user and a producer, and
decision problems to be heterogeneous. Furthermore, an item’s information
value for one user is a function of the information that other users have (Beaver
and Demski 1974), as well as that user’s initial endowment of wealth? What all
the above statements imply is that materiality is not unique even for a given
individual.
An Analyticul Approach to Accounting
405
A General Materiality Standard and Materiality Standards in Accounting Disclosure Rules
A central materiality issue is whether it is possible to derive a general unique,
fmed materiality standard of an accounting information item relevant toa group
ofusem For such a materiality standard to be derived, at least the following
conditions should hold. Firs, a unique, fixed materiality standard for an item
must be derived for a single user. Second, the following simplifying conditions
should hold across different users: homogeneous tastes and beliefs, the same
initial wealth, consistent materiality judgment between information users and
producers, homogeneous decision problems, and the same amount of information
held by all users. Third, the materiality of an item must be additive across
different users. A violation of any one of the above conditions is sufficient to
prove that the derivation of a general unique, fured materiality standard for an
item is conceptually impossible.
Proposition 4 suggests that the first condition cannot generally be met. Nor
would the second set of conditions generally be satisfied. The third condition
is conceptually difficult to hold since the aggregation of materiality over a group
of individuals is a case where the “impossibility theorem” (Arrow, 1963, and
Demski, 1973) applies, although utility can be additive under certain assumptions
(Keeney and Raiffa, 1976). Thus it follows that the derivation of a general
materiality standard relevant to all users is conceptually infeasible.
The infeasibility of a unique, furcd materiality standard for all users suggests
that the institutionally derived materiality standards for certain accounting
disclosures lack theoretical foundations and may be arbitrary, as far as their
relevance to users’ decision making is concerned. To t h i s extent, they may be
inconsistent not only among themselves, but also with users’ own materiality standards (if any) for those accounting disclosures. Therefore, users’ decision
relevance does not justify the need for such materiality standards; those who
propose or prefer such materiality standards should find reasons for justification
elsewhere.
Summary and Conclusions
Materiality is an important determinant of the set of accounting information
made public in financial reports. Despite its importance, the subject remains
relatively unexplored within an analytical framework. The present study
explores some basic materiality issues analytically using a decision model. The
study examines the concept of materiality and its measurement. It also discusses
whether it is feasible to set a unique, fured materiality standard of an accounting
information item both for an individual information user and for a group of
information users.
Based on the analysis, the following conclusions are drawn:
(a) The concept of materiality should be linked to an attribute (e.g., utility)
that a decision maker maximizes for a decision problem. That is, materiality should be measured in terms of information value.
-
406 Ro
(b) The materiality of an accounting information item should be distinguished
from the materiality of its particular magnitude. The materiality of an
item is, ex mte,necessary for the item’s particular magnitude to be
material.
(c) Materiality is not an all-or-nothing concept; instead, there! exists the
degree of materiality.
(d) The materiality function of an item is, in general, not monotone. Therefore, no unique, fixed materiality standard for an item can exist even for
a given user if the standard is stated as a magnitude of the item. Thus the
derivation of a unique, fxed materiality standard relevant to all users
is not feasible.
The conclusion that it is infeasible to set a unique, fured materiality standard is consistent with the statement by FASB (1980, paragraph l-31) that no
general standards of materiality can be formulated. The conclusion may have
important implications for the existing materiality standards for certain accounting disclosures. These standards may be arbitrary in relation to users’ decision
making. Therefore, accounting rule-making bodies may need to carefully scrutinize certain basic materiality questions, for example, those addressed in the
present study, before they hastily suggest any materiality rules for purposes
of accounting regulations.
APPENDIX
Roof of Proposition 1
(a)
@)
Suppose the information item is not material. Then, V = 0.This implies that r = 0
for every y E Y.Therefore, none of y’s can be material.
Suppose the information item is materid Then, V>o. This implies that there exists
at least one r E R such that r>o since w)o for at least one particular yeY. Therefore,
there exists at least one particular y of the item that is material. (Q.E.D.)
Roof of Roposition 2
In order to prove that the function r(y) exists, we have to show that for every y E Y
there exists a single value of I E R. That is, there must exist a set of ordered pairs
(y, I) for every y E Y and every r E R since a function is generally defined as a set of
such ordered pairs. Since R is nonempty (that is, the item is material) and since Y
i s continuous (by assumption), equation (4) suggests that for every y E Y there
exists a corresponding r e R. Thus, if y e Y, then there exists at least one particular
r E R such that (y, r) belongs to the function r(y). Likewise, if r e R, then there
exists at least one particular y e Y such that (y,I) belongs to the function r(y). Since
this is true for every y E Y and every r f R, all resulting paics (y, r) belong to the
function r(y). (Q.E.D.)
Roof of Proposition 3
(a)
By Proposition 2, r(y) is defined on Y which is continuous. By assumption, u(x),
x(sp(y)), and w )are all differentiable and, hence, continuous on Y. Since y = *s),
p(s/Y, Q) is continuous on Y. Therefore, r(y) is continuous on Y and is also differentiable since it is continuous at every y E Y.
An Analyticul Approach to Accounting
407
A function is bounded if it is continuous on a finite closed interval. By assumption,
Y is a fdte set (interval). As proved in (a) above, r(y) is continuous on Y.Therefore,
r(y) is bounded on Y.
By assumption or definition, u(x), p(s/y, Q),and p(y/Q) are all nonnegative. Assume
that every y f Y provides either decision-relevant information or at worst no such
information. Then, theoptimal action, a* = u*(y), based on information in every
y E Y should be at least as good as the optimal action a0 not based on the information. This implies that x(sp*(y)) _> xo (s,ao) for all s, a, y, and, hence, that
Eu(x*/q) - E U ( X * ) ~Therefore,
.
V is nonnegative. The nonnegativity of both V
and w in eq. (4)implies that r is nonnegative. Since this is true for every r f R, the
function r(y) is also nonnegative.
Whether r(y) is monotonic depends pon whether u(x), x(s,a(y)), p(s/y, q) and
p(y/q) are monotonic on Y such that
at every y f Y. By assumption, u(x)
and x(s, a&)) are monotonic. However, in general, it is not true that a larger (smaller)
magnitude of an accounting infomation item is more (less) likely to be r
than
a smaller (larger) magnitude of the item. That is, it is not typical that -0
at
every y f Y. Rather, a general case is that p ( y / ~ )is not monotonic dyn Y. Nor is
p(s/y,q) monotonic on Y since y = q(s). Also, the product (or sum) of monotonic
and nonmonotonic functions ie generally not a monotonic function. Thus, although
u(x) is assumed to be manotonic. the product or sum of u(x), p(s/y, Q)and p ( y / ~ is
)
generally not monotonic. Since r(y) is defined in terms of this product and sum, r e )
is, in general, not monotonic on Y. (Q.E.D.)
Proof of Proposition 4
%
Since r(y) is not monotonic, there exists at least one point on r(y) at whi
= 0. This implies that there exists no unique one-to-one relationship between
y f Y and every r f R; that !is at least two particular y’s of an item share the same
particular materiality measure, r, e R. (Q.E.D.)
NOTES
Magnitude is measured in a ratio scale such as dollar or the percentage of a base.
For example, see Accounting Research Bulletin No. 43 (AICPA, 1953,paragraph 9 of
Introduction, and Chapter 8), Accounting Principles Board (APB) Statement No. 4
(AICPA, 1970,paragraph 128), APB Opinion No. 15 (AICPA, 1969,footnote 2). APB
Opinion No. 20 (AICPA, 1971,paragraph 38). APB Opinion No. 30 (AICPA. 1973,
paragraph 241, Financial Accounting Standards Board (FASB) Statement No. 14 (FASB,
1976,paragraphs 7, 15, and 17), Regulation S-X(Securities and Exchange Commission,
1968, Rules 1-02 and 3-02), Accounting Series Release (ASR) No. 147 (SEC, 1973,
B. Interpretations and Comments, paragraph 71, and ASR No. 190 (SEC,1976, A.
General Statement). Further discussions about materiality for accounting disclosures
are found in the FASB Disarssion Memorandum (1975,Chapter 11, pp.23-49) and Statement of Financial Accounting Concepts N0.2 (FASB, 1980,paragraphs 123-132,and
Appendix C).
For example, see Hicks (1964), Rappaport (1964). Bernstein (1967), Frishkoff (1970).
Woolsey (1973), Boatman and Robertson (1974). O’Connor and Collins (19741,
Moriarity and Barron (1976), Abdel-khalik (I977),Ingram (1978),and note 2 above.
See Table 2 of Moriarity and -on
(1976) and Table 1 of FASB (1980) for an
excellent summary of prior studies and accounting regulatory rules about suggested
materiality base variables and specific materiality standards.
408 Ro
For example, see Frishkoff (1970), Moriarity and Barron (1976), Ward (1976), and
Newton (1977). Researchers using this approach typically address the question of how
materiality issues are (or should be) resolved by such information producers. In conclusion, they often suggestd variables (eeg. net income) to be used asa base of
materiality criterion and/or specific materiality standards (es. certain p e r c e n w or
dollar amount of net income, or its border zone) for resolving materiality issues.
However, as Abdelkhalik (1977) points out, one difficulty with the informationproducer approach i s its assumption that there exists a symmetry between producers’ materiality judgment and users’ materiality judgment.
For example see Dohr (1950), Hicks (1964), Rappaport (1964), O’Connor and Collins
(1974), Abdel-khalik (1977), Ingam (1978), and the accounting pronouncements cited
in note 2. Thus, their studies typicaUy focus on the question of how users, actual or hypothetical, resolve materiality issues. Their suggestions, like those discussed in the
previous approach, include certain materiality.base variables and/or specific materiality
standards that they think can help in resolving materiality isma.
Thus accountants using this approach usually address the question of how both producers and uaera make materiality judgments and/or whether their judgments are consistent (e.g. Boatsman and Robertson 1974).
Even if materiality is seen as a user-oriented concept, a question s t i l l remains as to who
the users should be. Beaver (1978) asserts that the persons making materiality judgments
should be professional users such as financial analysts having a high level of accounting
knowledge. Others argue that the users should be average reasonable users such as
average prudent invators who possess less accounthg knowledge (Rule 1 4 2 of
Regulation S-X,Kohbr 1970, p.279, Hicks 1964, and O’Connor and Collins 1974, to
name a few). In the present study, however, little attention is paid to the type*fuser
quastion since the approach followed here can apply to any type of user without loss of
its generality.
Hereafter, the terns, “decision makers” and “financial statement users”, wlll be used
interchangeably.
l o This statement is consistent with the view of the subjective probability school, as.
opposed to the objective probability school.
l 1 The role of information differs depending upon whether a decision problem is
completely formulated or not (Demdri 1969, and 1972, pp.1240, Demaki and Feltham
1976, Chapters 3 and 4). If a decision problem is completely formulated, the role of
information is to revise the decision maker’s probability npecifications. However, if a
decision problem is not completely formulated, the role of information may go beyond
probability reviaions to aid in spedfying dedsion variables and parameters, in
formulating objective functions and constraints, and in predicting parameters. For the
generality of its approach, the present study assumes that a decision problem is not
completely formulated.
This view considers materiality as being similar to. but not exactly the m e as, the
notion of relevance. Infomation is relevant if it can “make a difference” in the result
of a decision (FASB 1980). However, this difference should be kuge enough not to be
ignored by a decision maker if information were considered to be material. How large
this difference should be for materiality is a question to be addressed in the context of
materiality, not relevance. Thus it can be argued that not all relevant information is
material; only that which makes a “large enough” difference in a decision consequence
is material. In this sense, materiality serves as a threshold to detenniue the lower bound
of relevance for information to be considered in a decision analysis.
An Analytical Approach to AcCounting
409
l 3 ~ h view
b of m a t e w t y IS somewhat different from the view of materiality as a matter
a cknge in the attribute due to the unexpected magnitude (i.e., a difference between
.sxpected and actual magnitudes) of an item, as some studies (e& O’Connor and Collins
1974, and Ingram 1978) 6tot assume.
The notation y will be used hereafter as a pprticular magnitude (signal) of an accounting
information item when it appears with the word “every” or “particular”. otherwise,
it denotes a random information variable reprersnting an item.
~f
y is a past magnitude of an item,p(s/y) is seen as prior. If y is a currently received
magnitude before making a F d choice of action, p(s/y) is viewed as posterior ( W i i e r
1972, pp.267-268).
Alternatively, one may interpret the outcome x = x(s, a) as the ‘ket’’ outcome reflecting
the cost of information, assuming that the net payoff function has the same properties
as those of the p r r payoff function.
The choice of an optimal q is not a concern here. So, Q is assumed to be given.
This does not mean that materiality is the same concept as relevance (see note 12).
I9 Since the set Y is assumed to be conthous.. uWn1
_ - . is intermeted as the absolute value
of p(y = y-) = p-j)
- p(y4‘j-1) where yj is a particular &gnitude of an information
item (i = 2,. .; n<+
._
ml,
.
l o The concept of a unique, fied materiality standm~
for an accounting information item
. nt with the existing practices of materiality by external auditors.
may be iIlmmme
In pratice, the auditors may not apply a fixsd materklity standard even for a given item
of information, but multiple materiality stand.rds, depending upon the size and/or
industry m e m b m of finns being audited. It may be intemting to investigate whether
the existence of such multiple standards for an accountiq information is conceptually
valid. Nevertheless, occounting researchers and accounting rule-makiugbodies have often
attempted to derive a unique, fixed materiality rtandard applicable to all materiulity
problems. A focus of this study is directed at a fixed materiality standard for a given
information item viewed from the standpoint of accounting information users, not
producers, as indicated earlier.
21 If a materiality standard were stated in a measurement unit of an attribute being maxi-
mized in a decirion problem, it might be poaiile to establish a unique, f i e d materiality
standard. For example, if the studud is sat at one unit of utility such that any magnitude of an item whose information contributes to increasing utility by one or more
units is material. then this one uZ@r,y unlt itself & wJque m d fnw&b&. However, such a
materiality standard is surely incomkent with accounting mateddity that requires the
use of an item’s measurement unit in deriving ita materiaiity stand.rd.
of thc mate-y
since the utility function u(x) is not h o r and since the siope
function b not the same at every y E Y,information value (and th m a t M t y ) measured in expected utility depends partly upon a decision maker’s initialplacement on his
utility cumbefore making a dedsion.
’’
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