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DISCUSSION PAPER
SERIES IN
ECONOMICS AND
MANAGEMENT
The Disciplining Role of Mandatory Disclosure
Anna Boisits
Discussion Paper No. 12-04
GERMAN ECONOMIC ASSOCIATION OF BUSINESS
ADMINISTRATION – GEABA
The disciplining role of mandatory disclosure
Anna Boisits1
Abstract:
The model shows that voluntary disclosure (e.g. earnings forecast) can become informative if a
subsequent mandatory disclosure (e.g. earnings announcement) bears information about the
truthfulness of the preliminary voluntary disclosure. The sequential reports may differ from one
another for two reasons. Firstly, the private information underlying the two reports may change
during the respective announcements dates. Secondly, both disclosures may be biased. With
regard to the voluntary report, the model shows that bad information is withheld. In case of a
voluntary report, the better the private signal, the higher will be the forecasted value and its bias.
One result of mandatory disclosure is that managers try to smooth their reports. As a
consequence, the bias in mandatory disclosure will be high if the voluntary report which is given
in advance is good and the subsequent private accounting information is bad.
1
Center for Accounting Research, University of Graz, Universitätsstraße 15, A-8010.
1
1 Introduction
Managers might have incentives for self-serving voluntary disclosure. Though, it is unclear
whether these disclosures are credible. Healy and Palepu (2001) mention two mechanisms which
enhance credibility, namely third-party intermediaries and financial reporting. The latter makes it
possible to verify prior forecasts of earnings or revenues by using actual realizations. However,
voluntary disclosure can only be credible if there are adequate penalties for deviations in
forecasted and actual numbers. Without costs in the case of biasing private information, the
mandatory disclosure of the actual numbers will not have any influence on the truthfulness of the
forecasted numbers. The legal system and monitoring activity through the board are important for
establishing such penalties. However, there are also other costs which can arise if forecasted and
actual numbers deviate from one another, such as litigation risk, psychic costs or reputation costs.
Empirical studies have shown that the volatility of the share price on the earnings announcement
date is small. Therefore, much of the uncertainty is resolved prior to the actual earnings
announcement by management forecast, pre-announcements, or other information sources. This
phenomenon is illustrated for example in Beyer et al. (2010). In this study, mandatory disclosure
(e.g. earnings announcement and SEC filings) provides only 12 percent of the accounting-based
information, whereas voluntary disclosure (management forecast and pre-announcements)
amounts to 67 percent. The remaining 21 percent is information which is given by analysts.
However, this does not mean that voluntary disclosure makes mandatory disclosure redundant.
On one hand, it eliminates the remaining uncertainty, on the other hand, mandatory disclosure has
a disciplining role because it confirms the forecasted numbers. For the most part, voluntary
disclosure is credible because future mandatory disclosure verifies the previously disclosed
information. Voluntarily disclosed information may not fully coincide with future mandatory
information for several reasons. Firstly, the private information which underlies these disclosures
can change. Secondly, the actual private information may be biased both by the voluntary or
mandatory disclosure. What influences both biases? Which bias is higher?
The aim of this paper is to examine the interplay between the strategies of voluntary disclosure
and subsequent mandatory disclosure, in a situation in which mandatory disclosure is partially
capable of evaluating the truth content of the previously disclosed information. One advantage of
the joint examination of both disclosure types is that a situation is created in which a manager
incorporates future consequences of her voluntary disclosure strategy. Additionally, it is
2
demonstrated that reported values which are given voluntarily have an influence on subsequent
mandatory disclosure, because managers try to smooth earnings announcements.
The following model is closely related to the model by Einhorn and Ziv (2012) who examine a
similar setting in which they illustrate that the voluntary bias function is not constant. However,
they ignore the fact that a subsequent mandatory disclosure may contain incomplete information
on voluntary disclosure. Yet, this joint examination is important in order to gain insight into the
manner in which voluntary and subsequent mandatory disclosures interact with one another.
In the present model, the manager receives information concerning the terminal value on two
sequential dates, whereby the initial information is noisy information about the second signal.
The manager is obliged to disclose her information on the second date (e.g. earnings
announcement). However, she can decide whether to disclose a forecast voluntarily (e.g. earnings
forecast) on the first date or not. Voluntary disclosure is based on private, noisy information. The
private information which underlies a forecast is not verifiable and thus, opens up opportunities
for biasing. Since the information is no verifiable, the manager cannot be punished directly for
misreporting. The market only observes the forecasted value, not the private signal which
underlies it. Mandatory disclosure is also observable, but this report occurs later and thus is based
on more precise information. Nevertheless, private signals are correlated and, thus, the difference
between the forecasted and the actual mandatory disclosure can be used to assess the credibility
of the manager. However, a manager who always reports her actual beliefs can face credibility
costs, because the information which underlies these disclosures can change.
The model shows that mandatory disclosure restricts the possibility to bias voluntarily, but it does
not fully eliminate it. One result is that managers withhold bad information, whereas good private
signals are disclosed with a bias. The lowest bias is reached if the information is just good
enough to be disclosed. Furthermore, the biasing behavior increases the better the private
information is. However, the market discerns the incentive to bias and, thus, corrects for the
expected bias. The considered equilibrium is a fully separating equilibrium upon disclosure. In
this equilibrium the market can perfectly derive the true signal from the disclosed information.
However, reporting the truth or disclosing another information is too expensive, due to the beliefs
of the market and the credibility costs which arise because of the difference between mandatory
and voluntary disclosure values.
3
The section on comparative statics examines how the mandatory and voluntary disclosure
strategies depend on the characteristics of signals, the importance of market prices on disclosure
dates for the manager and the credibility and biasing costs. Some results of the model are
consistent with empirical findings. The following model, for example, shows that the expected
value of the difference between voluntary forecasted value and expected value before the
disclosure is positive. It also analyzes the characteristics of the variability of change in stock
price on the voluntary and the mandatory announcement date.
Most theoretical models of voluntary disclosure are either based on the assumption that voluntary
information disclosure must be truthful, or that the information is not verifiable and misreporting
is costless, cheap talk. If misreporting bears no costs, any kind of disclosure is ignored when the
market expects that the manager favors specific information (“babbling equilibrium”). Some
papers deal with credibility in a cheap talk setting. Gigler (1994) argues that a manager can
credibly communicate information if there are conflicts between goals concerning the disclosure
of specific information. Wagenhofer (2000) examines the disclosure decisions of a firm which
wants to sell an operation, while private information can either be verifiable or unverifiable.
The following contributes to models which assume that a manager can disclose voluntarily
distorted information, whereas misreporting is costly (Korn, 2004, Einhorn and Ziv, 2012 and
Beyer and Guttman, 2010). Due to the fact that disclosure is voluntary, the manager has
discretion about whether to disclose private information or not. If she opts for disclosure, she can
decide whether to bias the information or not.
There are also other studies which focus on the interaction between voluntary and mandatory
disclosure. Kwon et al. (2009) analyze how the quality of subsequent mandatory accounting
reports influences the bias in voluntary disclosure and examine the likelihood of such a report;
the higher the quality of mandatory disclosure, the lower the bias in voluntary disclosure. The
reason for this is that a higher quality of mandatory disclosure leads to a situation in which the
valuation of the firm rather depends on this kind of information than on voluntary information. In
this model voluntary information and mandatory information are two different sources which
provide information about a firm’s liquidating value. Investors use them in order to evaluate a
firm on the announcement date of the mandatory disclosure. In contrast, the following model
analyzes the effect of the subsequent mandatory disclosure on the prior voluntary disclosure,
whereas the latter is an information source about the former.
4
Einhorn (2005) also focuses on the relation of voluntary and mandatory disclosure. In this model,
a manager observes two signals about the firm value before disclosure. The manager has to
disclose one signal mandatorily. The author examines the influence of mandatory disclosure, and
its underlying parameters on the voluntary disclosure strategy, rather than investigating the
credibility of voluntary disclosure. Thus, it is assumed that both disclosures are credible
communicable. Bagnoli and Watts (2007) also assume that disclosures must be truthful. In this
setting, the private information complements the mandatory disclosure. Thus, the voluntary
disclosure of this private information makes mandatory disclosure more informative.
In contrast to the afore mentioned approaches, this paper argues that the relationship between
mandatory and voluntary disclosure is characterized by the assumption that mandatory disclosure
can be used in order to evaluate the accuracy of past voluntary disclosure. Gigler and Hemmer
(1998) and Lundholm (2003) also argue that mandatory disclosure has a confirmatory role.
Gigler and Hemmer (1998) consider a moral hazard model in which the manager exerts value
creating action in several periods, but this action is unobservable. Aside from the moral hazard
context, the assumption about the content of voluntary and mandatory information is also
different from the following model. It is assumed that private information is superior to the
subsequent public signal. The following model considers voluntary disclosure as more timely
information, but the private information which underlies it is noisy information about the
subsequent mandatory disclosure. They consider an optimal contract in which telling the truth is
achieved and the manager has incentives to work hard. Furthermore, they analyze the implication
of the frequency of mandated financial reports. An increased frequency leads to less voluntary
disclosure and, thus, reduces the informational efficiency. Lundholm (2003) considers the
confirmatory role of mandatory disclosure in a repeated game setting. Most of the time, voluntary
disclosure will be disclosed truthfully, because there is the possibility of punishment if voluntary
information differs from the mandatory one. He uses the strong assumption that the private
information coincides with the ex-post mandatory disclosed information.
None of these theories consider the interplay between the voluntary reporting strategy and the
subsequent mandatory disclosure strategy which is the focus of the following model. It proceeds
as follows: first of all, the model will be presented. Secondly, the characteristics of the
equilibrium will be illustrated. In addition, a description of the effects of various parameters on
5
disclosure strategies will be provided. Section 4 will analyze the relation between the model and
empirical research, before final section will end with a conclusion of the main ideas.
1. The model
The aim of this paper is to examine the influence which mandatory disclosure (e.g. earnings
announcement) has on voluntary disclosure (e.g. management forecast) which is provided in
advance. Voluntary disclosure can become informative if it is verified by a subsequent disclosure
and if any kind of deviation is punished. Figure 1 depicts the sequence of events. The firm’s
uncertain terminal value is a random variable , which is normally distributed with mean and
variance (~, ). This value is realized on date three. On date two, the accounting
system reports a noisy signal about the future value , = + ̃, where the noise term ̃ is
normally distributed with mean zero and variance . This is the manager’s private information.
However, she is obliged by a regulator to issue a report about this information. Yet, by engaging
in earnings management, the manager can disclose a value which differs from her private signal.
Thus, the market receives an information , which is a biased information about the manager’s
private information, = + , while denotes the earnings management. On the first date, the
manager receives a noisy signal concerning the subsequent accounting information, ̃ = +
̃ + , where is normally distributed with mean zero and variance . She can decide whether
to disclose a voluntary report after receiving the information or not. Because this information is
not verifiable the manager can provide any information she wants. The distribution of the two
signals and the terminal value is common knowledge. The realized signals are only known to the
manager.
Insert Fig. 1 about here
Both signals provide information about the same terminal value. However, they differ in
precision and appearance. It is noteworthy that is a sufficient statistic of ̃ . This setting captures
the fact, that the manager will have a conjecture about the terminal value prior to the date of the
6
mandatory disclosure. However, on the date of a given forecast, the private information differs
from the private information on the date of the mandatory report. Thus, there are two reasons
why voluntarily and mandatorily given reports can vary. Firstly, the private information which
underlies the two reports may have changed between the two dates, because manager’s signal is
only a noisy estimator of the later value. Secondly, management may bias actual private
information on both voluntary and mandatory disclosure.
The manager’s goal is to maximize her utility during these two periods, which is given by:
(1)
= + − − − .
The utility function is common knowledge. As can be seen in the equation (1), the manager is
interested in the magnitude of the firm’s market prices, and . ( ) is the weight on the
market price during the first (second) period. Both weights are assumed to be positive, as a higher
price is favorable for both periods. The manager may be interested in high market prices because
she wants to sell shares which she owns. Another reason may be that her wage depends on the
market price. The manager has two kinds of disutilities. The first one consist of costs which arise
from manipulating the accounting earnings for the mandatory report. These costs could be the
effort needed in order to conduct the manipulation, psychic costs, negotiation costs with the
auditor or litigation costs. The convexity covers the fact that it is increasingly hard to manipulate
accounting information. The weight of this cost equals . The second type of disutility are
credibility costs which arise ex post if the voluntary information which is given in advance differs
from the subsequent mandatory report. If, for example, the manager reports months before the
actual earnings announcement that the firm will make a tremendous profit but discloses a loss on
the mandatory announcement date, she will lose credibility. Although this difference can result
from a change in the information endowment, it is more likely that biasing behavior plays a role
if discrepancies are high. The convexity illustrates that the higher the difference the more likely it
will be that the manager manipulates her information which leads to an even stronger decrease in
credibility. A high discrepancy between the two reported values can create psychic costs, because
the manager wants to fulfill the forecasts. If the voluntary report and the subsequent mandatory
report strongly differ from one another, the manager could be considered as a fraud. This will
affect her reputation as well as the occurrence and the condition of future transactions. As a
consequence manager’s utility will be lower if the discrepancy is high. Penalties for deviations in
forecasted and actual numbers can also be enforced by the board of directors. It is noteworthy
7
that credibility costs are endogenous. In period two, the manager has already chosen the
voluntary report . Thus, by choosing , she has an influence on these specific of costs, because
= + . The assumption of credibility costs is important, because without these costs, in the
case of diverging numbers, the mandatory disclosure of the actual numbers would never have any
kind of influence on the truthfulness of the forecasted numbers. If the manager decides against a
voluntary report, credibility costs do not arise.
2 The Equilibrium
On the first date, the manager receives the signal . She is not obliged to report this information
and the information is not verifiable by other market participants. Therefore, the manager can
either disclose any value ℝ or nothing (Ø). Thus, the set of disclosure alternatives (" ∈ $) is
given by: $ = %Ø& ∪ ℝ. The disclosure strategy is a function: (: ℝ → $. The market expectation
+ : ℝ → $. The market observes the voluntary disclosure
of the disclosure strategy is depicted by: (
and evaluates the firm given the expected disclosure strategy: : $ → ℝ. If there is no disclosure,
the market price will be: Ø ∈ ℝ. If voluntary disclosure occurs the market price will depend
on the reported value and on the expected disclosure strategy: . , : $ → ℝ depicts the
expected pricing strategy. By assumption, the market is risk-neutral and rational. Thus, the firm is
+ = "0.
evaluated with the expected value conditional on the information available: = -.|(
On the second date, the accounting system provides the information and, by regulation, the
manager has to disclose her information. However, the reported value can differ from the private
signal, due to the fact that the manager can conduct earnings management. Thus, the disclosure
+: ℝ →
strategy is a function: 1: ℝ → ℝ. The market beliefs about the strategy are depicted by: 1
ℝ. The market observes the disclosure . Based on this information and on the expected
strategy, the market assesses the firm with :ℝ → ℝ. The manager has expectations about the
pricing behavior, , :ℝ → ℝ.
In equilibrium, actual strategies coincide with the expected ones, so that: = , , = , ,
+ and 1 = 1
+.
(=(
8
2.1 The optimal mandatory disclosure strategy
On the first date, the manager will choose a report which maximizes her expected utility, based
on his private signal . However, although she does not know the signal at this point, the
subsequent mandatory disclosure influences the bias of the voluntary disclosure. Thus, in order to
find the optimal voluntary disclosure strategy, the optimal mandatory disclosure strategy has to
be considered first. The equilibrium which is used for this second period is one with linear
strategies. The reason for using this equilibrium is that it is Pareto optimal.2 The price will be
linear in the report : = 3 + 4 and the bias in the report will be linear in signal : =
5 + 6. On the second date, the manager has already chosen either report or a non-disclosure.
In addition, the price is already realized as well. Thus, the manager takes these values as
given. The reporting strategy in the second period depends on whether she has disclosed a value
in the first period or not. Concerning the report , the manager chooses the bias, , to maximize
the following utility function:
max: = + − − − ,
(2)
while the expected pricing function equals: , = 3; + 4< and the mandatory report is depicted
by: = + . Thus, the optimal bias is given by: =∗ =
+
?@ A
BC DB@ + BC
B =EF
.
C DB@ The market assesses
the firm with the expected value given the expected strategy: = 5< + 6<: = -G|H = +
+ NOEP
+N
IJ@ KLEKDM
+ NKIJ@ DIQ@ N .
KDM
In equilibrium, the actual strategies should coincide with the expected ones:
5< = 5, 6< = 6, 4< = 4 and 3; = 3. Thus, the equilibrium strategies equal:
=∗ =
(3)
BC =EF
BC DB@ Q
= 3 + 4 = RKI@DI
@N −
I@
J
Q
+
?@ IJ@
B@ KIJ@ DIQ@ N
IJS ?@ BC DB@ @
KIJ@ DIQ@ N B@@
and
J C
− KI@DI
@ NB T +
I@B =
J
Q
@
IJ@ BC DB@ L
KIJ@ DIQ@ NB@
.
If there is no report (Ø), the manager aims to maximize the following utility function:
max: = + − .
(4)
Based on the linear equilibrium structure, the optimal strategies are:
(5)
2
Ø∗ = B
?@ IJ@
@
@
@ KIJ DIQ N
Q
and = 3 + 4 = RKI@ DI
@N −
I@
J
Q
IJS ?@
@
KIJ@ DIQ@ N B@
T + KI@JDI@ N.
I@L
J
Q
The proof of this statement is given in the appendix.
9
As can be seen in equation (3) and (5), the market is able to perfectly infer the signal from report
. In the case of a voluntary report, the mandatory report increases in the private information
and, thus, the equilibrium is separable. Therefore, it is not possible to mislead the market in
equilibrium. However, the manager biases her information in equilibrium, whereas the market
can infer the true signal. Therefore, on date one, the market price will actually be the expected
value of conditioned on the expected signal which corresponds to the realized signal in the
equilibrium ( = + KIJ @
I@ FEO
@
J DIQ N
, this is obtained by inserting the optimal strategy of the manager
into the optimal pricing function). However, the manager’s biasing strategies differ depending on
the reporting strategy in the first period and in the case of a voluntary report in the first period it
depends also on the received signal in period two. If there is no voluntary disclosure in the first
period, there will be no credibility costs and, thus, the optimal bias will be constant. Hence, the
bias does not depend on signal (see (5)). This result is similar to the result of Fischer and
Verrecchia (2000). However, in the case of a voluntary report in the first period, the manager also
cares about credibility costs. Thus, the bias in report will also depend on the voluntary reported
value and the received information . The bias in the situation of a report in the first period will
be higher (lower), compared to the bias in the case of no disclosure, if < ( > ). Both
optimal biasing functions share the same fix part. This comes from the influence of the bias on
the market price. In the case of a voluntary report, the fix part is adjusted by
minimize the sum of the credibility and the biasing costs.
BC =EF
BC DB@ in order to
2.2 The optimal voluntary disclosure strategy
On the first date, the manager receives the signal . Based on this information, she chooses her
optimal voluntary disclosure strategy. She can either report any kind of value W or nothing (Ø).
Her goal is to maximize the expected utility which is either expressed by the expected value of
(2) or (4), taking into account the optimal strategies of the second period, which are given by (3)
and (5). If she opts for a voluntary report, she will choose a report which maximizes her expected
utility:
(6)
max= -G= |H = , + - G |H − - G − |H − - G |H.
In the case of no disclosure, her utility would be:
(7)
-GØ |H = , Ø + -G |H − -G |H.
10
The price in the second period depends on the future mandatory disclosure and indirectly on the
private information . Given the equilibrium strategy of the market in the second period, it is
optimal to bias the information according to the bias function given in (3) or (5). In both cases,
this leads to a price which is equal to = + KIJ @
price is equal to - G |H = +
IJ@ XGF|YHEO
KIJ@ DIQ@ N
I@ FEO
@
J DIQ N
in equilibrium. The expected value of this
. This price and its expected value do not depend
on the strategy in the first period. If is known, the knowledge of is redundant, because the
latter is just noisy information about the former. For this reason, if the market receives the
mandatory report it can infer the private information and, thus, the market will not consider
the voluntary report or the information derived from the non-disclosure in evaluating the firm in
the second period.
The expected costs in the case of a report are given by:
(8) -G − |H + - G |H = B @DBC − - G|H + B
B B
while \]| =
@ @
KIJ DIQ@ N
I^
@
IJ@ DIQ@ DI^
C
@
and 4 = KIJ@
I@ BC DB@ @
J DIQ NB@
C DB@
Z
?@ A [ + B @DBC \]|,
B B
C
@
. These costs are minimal if = - G|H, which
means that exactly disclosing the expected value of the accounting information , given the
information leads to the lowest disutility from reporting. Thus, if the manager’s goal would be
to minimize the total expected costs, she would issue a forecast about the future mandatory report
which is equivalent to the expected value of the accounting information given her signal . This
disclosure behavior is reasonable, because the conditional expected value is the best estimator for
the future accounting information .
If the manager decides not to issue a report on the first date, the disutility comes only from
biasing earnings in the second period. Given the strategy depicted in (5) the expected costs are
given by:
(9)
-.Ø _0 = R
?@ IJ@
T .
B@ KIJ@ DIQ@ N
As can be seen in (8) and (9), the expected costs in the case of providing a voluntary report are
higher, independent of the reporting strategy. Even if the manager discloses her actual
expectations about future information, the disutility is higher than if nothing is disclosed in period
11
one. The difference between the costs in (8) for = - G|H and the costs in (9) can be interpreted
as the costs of voluntary disclosure3:
5 = R
(10)
?@ IJ@
@ NT +
B@ KIJ@ DIQ
B@ BC
BC DB@
\]|.
Due to the fact that the manager wants to maximize her utility, she will decide to provide
information voluntarily if the utility in (6) is higher than the utility under non-disclosure (7). She
will be indifferent if the utility is the same and she will chose no disclosure if - GØ H >
max= -G= |H.
Lemma 1: There exists a unique threshold value . If the private signal lies above (below)
this threshold value, > ( < ), then the manager will decide to voluntarily disclose
(withhold) information.4
Bad signals are too expensive to disclose. Verrecchia (1983) and Dye (1985) have obtained a
similar result. Withholding information leads to a higher utility for the manager if the signal is
bad: s < s. The reason for this is that the expected disutility from issuing a voluntary report is
higher than the expected costs under no disclosure, independent of the reporting strategy.
2.3 Truthful reporting
The following section illustrates that truthful reporting cannot be an equilibrium. A truthful
disclosure would occur if the manager discloses the actual expected future accounting
information conditional on her information: = - G|H. This reporting strategy would also
minimize the expected costs from reporting in period one. Assuming that investors believe that
+ = = - G|̂ H) if ̂ > ̂ , the pricing function would
the manager reports the actual forecast ((
be: = + KIJ@ DI@ N. The optimal strategy for the manager given these beliefs would be:
I@ =EO
J
(11)
r ∗ = argmax= R + KIJ@
e
3
4
Q
B@ BC
BC DB@
I@ =EO
T + - G |H −
@
J DIQ N
− - G|H +
B
C DB@
Z
?@ A C J C
@
[ + B @DBC \]|f = - G|H + I
@ DI @ B B .
B B
C
@
? I@ B DB J
Q
@ C
See Verrecchia (1983).
The proof is given in the Appendix.
12
In this case, the actual reporting strategy given the beliefs of the market (r ∗ ) does not correspond
+ = = - G|̂ H). Thus, reporting the actual forecast is not an
with the expected strategy ((
equilibrium.
2.4 Biased reporting
As demonstrated in the previous section, the manager has an incentive to overstate her actual
expectation ( ≥ - G|H). In equilibrium, the market will assess the firm with a value which is
lower than the value assumed in the previous section, because the market corrects for the
expected bias, < + KIJ@
I@ =EO
@
J DIQ N
. This lower value reduces the profitability of biasing the
forecast and as a result the bias will be lower in equilibrium. In equilibrium, the market is able to
correct the report exactly by the actual bias.
The voluntary disclosure equilibrium considered in this paper is one in which a full separation
upon disclosure occurs. This is the case if two signals do not lead to the same reported value,
except the signals which lead to withholding of information. As a result of these separable
strategies the market is capable of deducing the private information from the disclosure if the
manager provides information. Despite the fact that voluntary and mandatory disclosure is not
verifiable, the market is able to infer information from reports. The following proposition
summarizes the main characteristics of this equilibrium.
Proposition 1: There is an equilibrium which is characterized by a disclosure threshold
level: . All private information lower than this threshold level leads to non-disclosure,
( = h for all ≤ . All signals larger than this threshold induce a voluntary report,
( = for all ≥ . The reporting function is strictly monotonically increasing,
j Y > 0. If the manager possesses the threshold signal , she is indifferent between a non-
disclosure or a voluntary disclosure in which exactly the beliefs about future accounting
information is disclosed: KN = -.|0. The bias in excess of the actual, expected value
− - G|H increases in and converges to
?C IJ@ BC DB@ 5
.
IJ@ DIQ@ B@ BC
There is an infinite number of separating voluntary disclosure equilibria in this model setting
because there are infinitely many expectations about the bias at the threshold signal. According
5
The proofs are provided in the Appendix.
13
to the D1 criterion by Cho and Kreps (1987) all equilibria which have a positive bias at the
threshold value can be eliminated. One single separating equilibrium survives this criterion.
This equilibrium is characterized by disclosing the actual expectation about future mandatory
information if the manager possesses the threshold information.6 Thus, there will be no bias at
the threshold level which exceeds the expected value of the accounting information conditional
on that signal.
The equilibrium reporting function is an increasing function. Therefore, investors can infer the
private information which the manager possesses from this disclosure, in case of a voluntary
report. Although the manager biases her information, the market can deduce the true private
information. In equilibrium, the expected signal ̂ corresponds to the realized signal . Therefore,
on the first date, the market price will be the actual expected value of the terminal value ,
conditioned on the signal , although is not directly observable. Despite the fact that voluntary
information is not verifiable, the market can deduce the information which underlies it from the
report. Thus, voluntary disclosure leads to price reactions, although the information underlying
the voluntary disclosure is not verifiable ex ante and ex post. There exists only one optimal report
for every signal. The reason for that lies in the expected credibility costs and the beliefs of the
market. There is a trade-off, because of the credibility and the biasing costs. If a manager with a
low signal decides to disclose a high report , there is a great risk that is also low, because and correlate with one another. In order to avoid the credibility cost, the manager can bias the
future accounting information but at high manipulation cost. These ex post appearing costs lower
the incentive to extremely overstate the actual beliefs about future accounting information. In
equilibrium, the market is able to correct for the actual bias. Thus, the manager cannot mislead
investors. Nevertheless, the manager will bias her information. The reason for this is that
choosing a lower report which is closer to the expected value of the accounting information,
would lower the expected credibility cost. However, the market would believe that the voluntary
report was given by a manger with a lower signal. Therefore, the market evaluates the firm with a
lower price. The reduction in price would be higher than the positive effect of lower credibility
costs. Therefore, such a report would not be optimal.
The equilibrium voluntary reporting function is given by:
6
This refinement method is also used by Einhorn and Ziv (2012).
14
= - G|H +
(12)
while - G|H = +
?C IJ@ BC DB@ IJ@ DIQ@ B@ BC
IJ@ DIQ@ YEO
@
IJ@ DIQ@ DI^
l1 + m n−o
E
@ @
@Kp@
J qpQ N r@ rC KstsN
E
@
@
@
uC p@
J ZpJ qpQ qp^ [rC qr@ vw,
and m GH is the Lambert W function, with mGH ∈ G−1,0.
Insert Fig. 2 about here
Figure 2 depicts the voluntary reporting function in equilibrium. The manager will never issue a
forecast if the signal is low. In this case, the disutility of a voluntary report is too high, thus
reporting is not optimal. Regarding good information ( ≥ ), the manager has an incentive to
mislead the market in order to convince the market that the firm value is higher than the actual
expected firm value based on private information. It is not possible to be misled in equilibrium.
The reporting function for all ≥ lies between - G|H and - G|H +
?C IJ@ BC DB@ IJ@ DIQ@ B@ BC
. The report at
the threshold value equals the expected value of the accounting information. Proposition 1 asserts
that the reporting function is a strictly increasing function in the signal which is the case
because:
x=
xY
? I@ B DB C J C
@
= =EXGF|YHKI
@ DI@ DI @ NB
J
Q
^
@ BC
> 0 for > - G|H. In addition, for a higher signal , the
C J C
@
reporting function is closer to - G|H + I
@ DI @ B B . Thus, the deviation of the reported
? I@ B DB J
Q
@ C
value from the expected value of the accounting information is an increasing function in .
3 Comparative Statics
This section will focus on the consequences of changes in parameter on the disclosure strategies.
3.1 The mandatory disclosure strategy
The optimal bias of the mandatory disclosure strategy is given by equation (3), in case of a report
in the first period. As can be seen in this equation if the voluntary report is high whereas the
accounting information on date two is bad (low ), the earnings management will be high in the
15
second period. In order to lower the credibility costs, higher manipulation costs are accepted. The
higher the voluntary report, the higher will be the bias (
y:z∗
y=
> 0). Hence, the manager tries to
y:z∗
smooth her reports. Conversely, a better signal in the second period leads to a lower bias (
0).
yF
>
What are the effects on the optimal bias if the manager cares more about the credibility costs? A
higher weight on credibility costs means a higher . Differentiating the optimal bias, in equation
(3), with respect to shows that the optimal bias in mandatory disclosure increases if > and
decreases otherwise (
y:z∗
yBC
=
B@
BC DB@
− ). The optimal bias consists of two parts, the fix part
which depends neither on nor on ,
?@ IJ@
B@ KIJ@ DIQ@ N
, and the part
BC =EF
.
BC DB@ The first part emerges due
to the positive effect which a high report has on the market price. The second part guarantees that
the influence of the optimal bias on both credible and biasing costs is optimally apportioned in
order to minimize the manager’s disutility from these costs. As increases the fix part remains
the same. However, an increase of enlarges the importance of the credibility costs compared to
the bias costs. In order to reduce credibility costs, the manager’s willingness for a higher bias
increases if > .
Conversely, if the weight on the costs for manipulating accounting earnings increases it is
optimal to reduce these costs by decreasing the bias. Differentiating the optimal bias with respect
to shows that the optimal bias decreases (
otherwise.
y:z∗
yB@
< 0) if =∗ > −
?@ IJ@ BC
@
B@ KIJ@ DIQ@ N
, whereas it increases
y:z∗
y?@
The higher the weight on the price in the second period, the higher will be the bias (
> 0). For
a higher weight, the incentive to manipulate the second price is higher and thus there will be a
higher bias in equilibrium. The weight on the price in the first period, , does not influence the
optimal bias in the second period. During the second period, the manager cannot influence the
past price, .
If the variance of the noise term increases, the bias will be lower (yIz@ < 0). A higher variance
y:∗
Q
means that the signal is worse which leads to a lower price reaction to the mandatory report .
This lowers the incentive to bias information. On the other hand, a higher variance of the terminal
16
y:z∗
yIJ@
value increases the bias (
> 0), because a high risk of the terminal value makes the report
more important. Thus, the reaction to the report increases and hence also the incentive to bias.
3.2 The voluntary reporting strategy
As shown above a threshold level exists in which all signals higher than the specific value result
in voluntary disclosure, whereas bad signals lead to a withholding of information. Thus, the
probability of a voluntary disclosure is given by 1 − {KN, where { is the cumulative
distribution function of signal (̃ ~K, + + N). If increases the probability of a
voluntary report decreases. The threshold value depends on the parameters. For example the
higher the weight on the credibility costs, the higher will be the threshold value. Thus, voluntary
disclosure is less likely. The reason for this is that if is increases the disutility of disclosing
voluntarily will be higher and, thus, reporting is worthwhile only for managers with good signals.
An increase of has a direct and an indirect effect on the costs. On one side, an increase in directly increases the expected costs of biasing accounting information in the second period. In
case of a voluntary report the expected bias is higher than that of withholding information in the
first period. This direct effect increases the difference between these costs and makes voluntary
report less attractive. On the other side, a higher induces the manager to bias less because it is
more expensive. This indirect effect works in favor of voluntary reporting. If is small, the
direct effect dominates. Thus, the threshold value increases if gets higher.7 However if is
high, decreases, which makes a voluntary disclosure more likely.
Voluntary disclosure is also less frequent if the weight on the price in the second period gets
higher. A higher weight on the second price causes a stronger incentive to bias the accounting
information in the second period. Because the disutility of costs is higher in case of a voluntary
report, the difference between the costs of a voluntary report and the costs of withholding will be
stronger. Thus, a higher leads to a lower probability of a voluntary report. A higher weight on
the price in the first period increases the incentive to report voluntarily. The price in the first
period is compared to the cost more important for the manager and, thus, the threshold value
decreases and voluntary disclosure becomes more likely. Voluntary disclosure will be less
frequent if the signal in period one is less precise (higher variance). On the one hand, a higher
uncertainty increases the expected disutility in the case of a voluntary report compared to the
7
The threshold level for θ is given in the Appendix.
17
situation of providing no information. This leads to a higher threshold value. On the other hand, a
lower precision induces that the price reaction in period one is lower, whereas this reduction is
higher than the price reduction if there is no information. This also reduces the incentive to
disclose voluntarily. Thus, a lower precision of the signal in period one reduces the probability of
a voluntary disclosure. The more the two signals correlate (which is ceteris paribus the case if is lower) the more likely it will be that the manager discloses voluntarily. The second effect also
applies if increases. In this case, the precision of the accounting signal in period two
decreases. Because of this, the precision of the signal in period one, which is a signal of this
accounting information, decreases as well. The effect on the cost is not that unambiguous. On the
one hand, a lower precision of the accounting signal decreases the price reaction of the market
and so also the incentive to bias which would lead to lower expected costs and a lower threshold
value. However, a lower precision increases the expected uncertainty costs because it increases
the conditional variance of the signal given .
The voluntary disclosure strategy is given by equation (12). The better the signal , the higher
will be the bias in excess of the actual expected accounting information conditioned on the signal
(}∗ = − - G|H). This bias approaches
?C IJ@ BC DB@ IJ@ DIQ@ B@ BC
because the term in brackets in equation
(12) approaches 1 if the signal approaches infinity. The bias in the limit will be higher the
higher the weight on the price in the first period or the lower the precision of the terminal value.
A lower precision of the accounting information has the exact opposite effect. A higher weight on
the credibility cost (a higher ) decreases the limit value. Likewise, if the weight on the bias
costs increases the limit is lower. Due to these changes in the limit value, the effect on the bias of
high signal values is obvious. However, changes in parameter also influence the curvature of
the bias function and the threshold value. Thus, a change in the bias function is ambiguous for
signal values slightly above the threshold value.
In equilibrium, both the voluntary and the mandatory report are biased. However, which of these
two reports is biased to a larger extent? When comparing the ex-ante expected value of the bias
of the voluntary report to that of the mandatory report, it is detectible that the expected bias in the
voluntary
?@
B@
<
?C
BC
report
l1 + - nm n−o
is
higher
if
@ @
@Kp@
J qpQ N r@ rC KstsN
E
@
@
@
uC pJ ZpJ qp@
Q qp^ [rC qr@ E
the
following
inequality
holds:
v~ ≥ vw. The term -.m G∙H| ≥ 0 in brackets
18
on the left side is negative and higher than minus one (-.m G∙H| ≥ 0−1,0). Thus, both sides
are positive. It can be shown that the difference between the expected voluntary bias and the
expected bias in the mandatory disclosure decreases in .8 The manner in which other variables
influence this difference is ambiguous.
4 Empirical implications and evidence
In line with the presented model also empirical research displays that forecasts induce price
reactions. Waymire (1984), for example, analyzes price reactions in consequence of management
forecasts. The empirical study shows that forecast deviation is generally positive and that there is
a positive relationship between forecast deviation and cumulative abnormal return around the
announcement date. Forecast deviation is defined as the normalized difference between the
forecasted value from the manager and the analyst forecast prevailing prior to the disclosure of
the management. Because a disclosure induces a stock price reaction, the capital market derives
information from this voluntary disclosure. Thus, management forecast must be informative. The
forecast deviation equals − in the model, in which is the voluntarily given information and
stands for prior expectation. The expected value of the forecast deviation is positive (-. −
| ≥ 0 > 0) which is consistent with the empirical finding. However, this is not surprising.
Firstly, only managers with good information, ≥ , disclose. Secondly, managers bias their
actual expectation upwards. The model predicts in common with the empirical study a positive
relationship between forecast deviation and change in market price ( − ) on the
announcement date. Waymire (1984) computes the correlation between the two variables.
However, the relation depicted in the model is not linear. Computing the correlation would
suppress the actual relationship.
In line with McNichols (1989), the model shows that stock price reactions contain information
beyond that of the management report. The model shows that the market is able to correct for the
strategic behavior in forecasting. If a voluntary disclosure is observed the market corrects the
disclosed value for the expected bias.
Empirical studies have shown that the volatility of the share price on the earnings announcement
date is small. Thus, most of the information is resolved prior to the actual earnings announcement
8
Proofs are given in the Appendix.
19
by management forecast, for example. Ball and Shivakumar (2008) find empirical evidence that
management forecasts provide substantial information. The information is discretionary (which
means that it is provided when manager prefers to reduce information asymmetry) and forwardlooking, whereas earnings announcement is not discretionary and backward-looking. The
empirical study shows that the “surprise content” of earnings announcements is substantially
lower than that of management forecasts. Thus, earnings announcement might not have the role
of providing timely, new information. However, this corporate information is important in order
to curb management forecasts. One important role of earnings announcement is to discipline prior
information. The confirmation role of backward-looking financial outcomes on private forwardlooking information is also examined in Ball et al. (2012). They test the hypothesis that audited
financial reporting and previous given voluntary disclosures are complements. They show that
market reaction to forecasts increases in the resources committed to audit. Hence, financial
statement verification augments the credibility of management forecasts.
The present model is able to show under which circumstances price reactions which are caused
by voluntary disclosure are higher than reactions on the mandatory disclosure date. For this
comparison only firms which disclose voluntarily are considered, ≥ . The expected price
reaction for those firms is positive (-. − | ≥ 0 > 0) because only manager with good signal
report voluntarily. The volatility of this price reaction is given by: \]. − | ≥ 0 =
IJS
@ e1 +
IJ@ DIQ@ DI^
€KYNYEO
EKYN
−
@
@
N
€KYN KIJ@ DIQ@ DI^
ZEKYN[
@
f. The term in brackets lies between zero and one
because of Sampford’s law (1953).
Lemma 2: The variance \]. − | ≥ 0 decreases in and in .9
The higher the threshold value the lower will be the variance. The conditional variance is lower if
a smaller amount of signals, , leads to a report. The previous section shows that is higher the
higher , the higher , the lower and the higher . An increase in has two effects on the
conditional variance. On the one hand, it increases the threshold value, with the result that fewer
signals lead to a voluntary report. This reduces the conditional variance. On the other hand, a
higher makes the voluntary report less informative which leads to a lower market reaction (
is lower). This effect lowers the conditional variance as well.
9
The proof is provided in the Appendix.
20
If there was a disclosure in the first period, is known and already inherent in the price. The
expected price reaction is zero (- G − |H = 0) because all firms which have disclosed
voluntarily are obliged to provide a subsequent mandatory report. The volatility of the price
^
reaction on the mandatory announcement date is given by: \]G − |H = I@ DIJ@ DI@ I@ DI
@.
J
IS
Q
^
Lemma 3: The variance \]G − |H decreases in and increases in and .
I@
J
Q
The higher the poorer is as a signal for . Thus, the variability on the mandatory
announcement date will be higher for high because contains more new information. The
main effect which lead to a decreasing conditional variance, \]G − |H, in the case of an
increasing variance of the signal , , is the lower weight on the signal compared to the
Q
J
weight on the mean, , in the firm price = I@ DI
@ + I@ DI @ . The opposite is true for a higher
variance of the terminal value, .
I@
J
Q
I@
J
Q
If is a good signal about the future accounting information (which is the case if the two
signals highly correlate with one another, low ) then the price reaction on the date of voluntary
disclosure will on average vary more than on the day of the mandatory disclosure. The
intermediate value theorem proves that there is a threshold value for : 0 < < + . If
is small, < , then the variance on the voluntary disclosure date, \]. − | ≥ 0, is
higher than on the mandatory disclosure date, \]G − |H. If, for example, the date of the
forecast is close to that of the mandatory disclosure then there will be a high correlation between
the two private signals. Therefore, the price reaction will be high on the voluntary disclosure date
whereas it will be low on the mandatory announcement date.
5 Conclusion
This paper shows that voluntary disclosure can become informative if a subsequent mandatory
disclosure bears information about the truthfulness of the preliminary voluntary disclosure. The
voluntary report and the subsequent mandatory report may deviate because of two reasons.
Firstly, the private information which underlies these two reports may change during the two
announcement dates. Secondly, management may bias actual private information. Both
voluntary and mandatory disclosure can be biased. Although neither voluntary nor mandatory
disclosure is verifiable, the biasing and credibility costs create a situation in which the market can
21
deduce information from the reports. In equilibrium, the manager will provide a forecast if her
private information is above a threshold level. Thus, bad information is withheld because it is too
costly to disclose. Due to the non-verifiability of the report, the manager has an incentive to
overstate her actual expectations in order to affect the market price. However, manipulating the
voluntary forecast so that it exceeds the actual expectation has an effect on future credibility and
indirectly on biasing costs. To avoid credibility costs the manager can bias the future accounting
information but at the risk of high manipulation costs. Given this trade-off of costs and the beliefs
about the pricing behavior, there is only one optimal voluntary report for every signal. In
equilibrium the market is able to correct for the actual biasing strategy and, thus, the market price
will be the expected value of the firm given the private information.
The model shows that both the mandatory as well as the voluntary report will be biased. The
magnitude of the biases depends on the uncertainty of the terminal value, the precisions of the
signals and the realized signal values. In addition, the frequency of the occurrence of voluntary
disclosure also depends on the aforementioned variables. For example, the more the two signals
correlate (which is the case if is lower) the more likely it is that the manager discloses
voluntarily. The bias in the mandatory report also depends on voluntarily disclosed information if
the manager provides a voluntary report in the first period because, in this case, the manager is
trying to smooth the announced earnings values.
6 Appendix
Mandatory disclosure strategy
The mandatory disclosure equilibrium considered in this paper is separable.10 In such an
equilibrium managers with different information provide different reports. Therefore, a
manager with the signal should not have a higher utility if she would imitate a + Δ manager
(with Δ > 0) by reporting + Δ instead of disclosing :
(13)
, ≥ + Δ, .
The price in the first period is not dependent on the message because it is the past market
price. Likewise, the strategy of the voluntary report is already selected and, thus, is taken as
10
The approach is similar to the approach of finding the voluntary disclosure strategy which is discussed more
precisely in the next section.
22
given. In equilibrium, the price in the second period is given by KN = +
IJ@ FEO
IJ@ DIQ@
, if the
market observes , because investors believe that a message is sent by a manager with
information . If, however, a report of + Δ is observed, the market price would be
K + ΔN = +
. The costs are given by: − + if the manager
IJ@ FDƒFEO
IJ@ DIQ@
opts for a voluntary report in the first period whereas they are: if the manager decides to
withhold information in the first period. In the following, the existence of a preceding voluntary
report is assumed. The approach of finding the optimal mandatory reporting function when
information is withheld in the first period is similar. The minimal costs are realized if the
manager reports: = B
BC
C DB@
+B
B@
C DB@
. However, this is not an equilibrium strategy, because the
manager has an incentive to overstate the true accounting information. Thus, in equilibrium,
>
BC
BC DB@
+
B@
BC DB@
. By inserting both the price in the second period and the cost function into
the aforementioned inequality (13) and rearranging terms, the following inequality can be
obtained:
(14)
+
≥ B
B
B
B
? I@ ƒF
†„ − @ − C … + @ @@ J
e− „ − @ − C …
f≡Θ
−
KIJ DIQ NBC DB@ ƒF ≤ ƒF
BC DB@
BC DB@
BC DB@
BC DB@
ƒL
The second solution area ( ƒF ≤ Θ) can be neglected because for positive ∆, ∆ has to be
ƒL
positive as well. According to the rule of l’Hospital, the limit when Δ goes to zero of the right
side of the inequality in (14) equals lim∆F→Œ Θ =
?@ IJ@
KIJ@ DIQ@ NBC DB@ ZLE
rC
r
=E @ F[
rC qr@
rC qr@
. Furthermore,
in order to be a separating equilibrium, a manager with the information + Δ must have a
higher utility by disclosing + Δ than by imitating a -type and disclosing . Analog
to the approach above it can be shown that this is the case if
ƒL
ƒF
lies between two values while
one of these is negative and, thus, can be neglected. The limit of the positive value when Δ goes
to zero is equal to the limit calculated above. Thus, the following inequalities must hold:
?@ IJ@
r
r
@
@
KIJ DIQ NBC DB@ ZLE C =E @ F[
rC qr@
rC qr@
≤
equation equals:
(15)
xL
xF
=
xL
xF
≤
?@ IJ@
r
r
@
@
KIJ DIQ NBC DB@ ZLE C =E @ F[
rC qr@
rC qr@
?@ IJ@
r
r
@
@
KIJ DIQ NBC DB@ ZLE C =E @ F[
rC qr@
rC qr@
. Hence, the differential
.
23
Solving the equation leads to a mandatory reporting function of:
(16) =
BC
BC DB@
if
+B
BC
BC DB@
B@
C DB@
+
B@
BC DB@
+
?@ IJ@
KIJ@ DIQ@ NB@
+ KI@@DIJ@ NB > > B
? I@
J
Q
1 + m Ž
BC
C DB@
@
+B
EKIJ@ DIQ@ NBC DB@ B@
C DB@
?@ IJ@
o
@ @
t@Kp@
J qpQ Nr@ FD
u@ p@
J rC qr@ ‘,
, where m G∙H is the Lambert W
function11 which lies between zero and -1 since the argument is negative. And for
+
B@
BC DB@
?@ IJ@
KIJ@ DIQ@ NB@
(17) =
BC
BC DB@
< the mandatory message is given by:
+
B@
BC DB@
+
?@ IJ@
KIJ@ DIQ@ NB@
1 + m Ž
KIJ@ DIQ@ NBC DB@ ?@ IJ@
o
@ @
t@Kp@
J qpQ Nr@ FD
r
u@ p@
qr
C
@
J
BC
BC DB@
+
‘.
The third solution of the differential equation is:
= B
BC
C DB@
(18)
+B
B@
C DB@
+ KI@@DIJ@ NB .
? I@
J
Q
@
The reporting function in (16) can be ruled out, because there exists no constant of integration so
that B
BC
C DB@
+B
B@
C DB@
+ KI@@DIJ@ NB > > B
? I@
J
Q
@
BC
C DB@
+B
B@
C DB@
for all . For low the function is not defined. This is not the case for equation (17), there infinitely many equilibria exist because
there is an infinite number of beliefs about the “constant of integration”. In addition, the reporting
function in (18) is a potential equilibrium strategy. Because each of these equilibria are
separating, the market is able to deduce the actual signal underlying the report. Thus, the market
is indifferent towards the different equilibria. However, this is not true for the manager, since the
bias of all equilibrium disclosure strategies given in (17) (independent of the value of the constant
of integration) is higher than the bias given if (18) would be the mandatory equilibrium strategy.
Hence, the Pareto optimal equilibrium is the one where the disclosure strategy is given by (18),
which is a linear strategy in the signal .
Voluntary reporting function
The equilibrium is separable upon disclosure if a manager who possesses the signal does not
have a higher expected utility if she imitates a + Δ manager (with Δ > 0) by reporting
+ Δ:
The Lambert W function is the inverse function of ’ = “m = mo ” . Important characteristics of this relation
are that m’ ≥ −1 and that m’ < 0 for ’ < 0, m’ > 0 for ’ > 0 and m0 = 0.
11
24
-G=Y _H ≥ -G=YDƒY _H.
(19)
In a separating equilibrium, the market assesses a firm with = +
IJ@ YEO
@
IJ@ DIQ@ DI^
if a report is
observed. If the market observes a report of + Δ the firm is valued with: = +
IJ@ YDƒYEO
@
IJ@ DIQ@ DI^
in the first period. The expected disutility is given by equation (8). The last two terms
of equation (8) are independent of the reporting strategy only the first term differs, which equals
B@ BC
BC DB@
− - G|H if the -type discloses and
B@ BC
BC DB@
+ ∆ − - G|H if the -type
discloses + Δ. The change in the report is the difference between the report of a manager
with the signal + Δ and the report from a -type manager (Δ = + Δ − ). By
inserting the terms into inequality (19), the following is obtained through rearangment:
Δ + 2. − - G|H0Δ − @C
KI
B DB@ ?C IJ@ ƒY
@
@
J DIQ DI^ NB@ BC
(20)
≥0
This inequality holds if:
(21)
Δ
≥
+
B DB ? IJ@ ƒY
− . − - G|H0 †. − - G|H0 + KI@C @@ C@ NB
≤
−
J DIQ DI^ @ BC
J
The second solution area, Δ ≤ −. − - G|H0 − †. − - G|H0 + KI@CDI@@DIC@ NB
neglected because for positive ∆, ∆ has to be positive as well.
Dividing (21) by Δ > 0 leads to:
(22)
∆=
∆Y
≥Γ≡
@
J
Using the rule of l’Hospital, the limit equals: lim∆Y→Œ Γ =
zero.
J
Q
^
@ BC
, can be
r qr u p@ ˜s
E.=EXGF|YH0D—.=EXGF|YH0 D @C @@ C@ J
Zp qp qp [r@ rC
∆Y
B DB ? I@ ƒY
Q
^
.
?C IJ@ BC DB@ @ NB B
=EXGF|YHKIJ@ DIQ@ DI^
@ C
if ∆ goes to
Furthermore, in order to be a separating equilibrium, the utility of a + Δ-type must be larger
than the utility of imitating a -type:
(23)
-G=YDƒY _ + ΔH ≥ -G=Y _ + ΔH.
The solution of this equation is:
25
−. − - G| + ΔH0 − —. − - G| + ΔH0 +
∆
+ Δ
K + + N −. − - G| + ΔH0 + —. − - G| + ΔH0 +
∆
=Ψ≤
∆
š=
≤Ψ
∆
+ Δ
K + + N The limit of the upper boundary can be calculated by using the rule of l’Hospital:
š=
lim∆Y→Œ Ψ
?C IJ@ BC DB@ @ NB B
=EXGF|YHKIJ@ DIQ@ DI^
@ C
. Due to that:
equation equals:
x=
xY
(24)
? I@ B DB whereas: Κ = KI@CDIJ@ DIC @ NB@
J
Q
^
@ BC
›
=EXGF|YH
≤
x=
xY
≤
›
=EXGF|YH
, the differential
= =EXGF|YH.
›
. The manager has an incentive to overstate her actual forecast and
thus, ≥ - G|H, which is why equation (24) is positive.
The differential equation (24) can be solved by transforming the variables: ’ = 2K −
$ +  N, where $ +  = - G|H, $ = @ J
KI
function with respect to leads to:
(25)
KI@ DIQ@ N
@
@
J DIQ DI^ N
xž
xY
=2
x=
xY
and  = KI@
@
I^
O
@
@
J DIQ DI^ N
. Differentiating this
− 2$.
By inserting (24) into (25) the following separable differential equation is obtained:
xž
xY
(26)
If 2Κ − 2Az = 0, we know from (25) that
for leads to =
£
¤
=
x=
xY
›EŸ ž
.
= $ follows. Inserting
x=
xY
= $ into (24) and solving
+ $ + . If 2Κ − 2Az ≠ 0 then (26) is a separable differential equation
with the following solution:
(27)
¦ ln|¦ − $’| − ¦ − $’ = + ¨ −2$ ,
whereas C is the constant of integration. For ¦ − $’ > 0, which occurs if $ +  ≤ <
$ +  the reporting function is given by:
£
¤
+
26
(28)
= - G|H + I@DI@ B
?C IJ@ BC DB@ J
1 + m Ž−
@ BC
Q
©
tK@ª@ Nsq«
¬
›
‘,
where m GHis the Lambert W function, with m GH ∈ G−1,0 because < ¤ + $ + .
For ¦ − $’ < 0, which is the case if >
(29)
= - G|H +
£
¤
£
+ $ +  the reporting function is given by:
?C IJ@ BC DB@ IJ@ DIQ@ B@ BC
1 + m Ž
©
tK@ª@ Nsq«
¬
›
‘,
where m GHis the Lambert W function, with m GH > 0 because > 0 in this case.
Threshold value
The expected utility for a manager with a signal who discloses the report equals in
equilibrium:
(30)
while \]| =
- G= |H = R +
−e
B@ BC
BC DB@
IJ@ YEO
@
IJ@ DIQ@ DI^
− -G|H +
@ @
KIJ DIQ@ N
I^
@
IJ@ DIQ@ DI^
, 4 = KIJ@
I@ BC DB@ @
J DIQ NB@
T + R +
BC DB@
?@ A Z
[ +
IJ@ XGF|YHEO
KIJ@ DIQ@ N
B@ BC
BC DB@
T
\]|f,
and is given by eihter =
£
¤
+ $ + ,
equation (28) or (29). Equation (30) is obtained by inserting (8), the expected price in the second
period and the equilibrium price in the case of a report into (6). The constant of integration
¨ in (28) and (29) is determined by the expectation of the manipulation in excess of the expected
value -.|0 at the threshold value, K̂ N − -.|̂ 0. This bias in excess of the expected value is
assumed to be ­ = K̂ N − -.|̂ 0 (this value will be discussed at a later point). In equilibrium,
the expected ­ has to be the same as the actual one. If this is the case, then the reporting function
coincides with the expected reporting function. If ­ >
reporting function is given by 29 (28).
?C IJ@ BC DB@ IJ@ DIQ@ B@ BC
(­ <
?C IJ@ BC DB@ IJ@ DIQ@ B@ BC
), then the
If investors assume that all signals which are lower than ̂ lead to a non-disclosure then observing
no disclosure would induce the following price in the first period:
(31)
Ø = -.| ≤ ̂ 0 = − €KŶ N
KŶ N
,
27
whereby “∙ is the density function of the signal and { ∙ is the cumulative distribution
function. In equilibrium, the expected price coincides with the actual price: , Ø = Ø, thus
̂ = . The expected utility from non-disclosure is given by:
(32)
- GØ|H = R − T + R +
N
€KŶ N
KŶ
IJ@ XGF|YHEO
KIJ@ DIQ@ N
T − R
?@ IJ@
T .
B@ KIJ@ DIQ@ N
Equation (32) is obtained by inserting (9) and the expected price in the second period and the
price in the case of no disclosure (equation (31)) into (7).
In equilibrium, the manager voluntarily reports her information if: - G= |H ≥ - GØ|H and she
will disclose nothing if : -G= |H ≤ -GØ|H. denotes a signal for which the manager is
indifferent: -.= |0 = -.Ø|0. In order to prove that there is only one threshold value, the
intermediate value theorem is used by establishing a function ® j :
(33)
® j = - G= | j H − - GØ | j H = R +
−
B@ BC
BC DB@
­ − R
?@ IJ@
IJ@ KY¯ EON
@
IJ@ DIQ@ DI^
T −
B@ KIJ@ DIQ@ N
T − Z − B@ BC
BC DB@
\]| j ,
[
€KY¯ N
Y¯ whereby - G= | j H equals the expected utility of equation (30) at the signal j with j =
- G| j H + ­ and - GØ | j H which is given by equation (32) with ̂ = = j . The function ® j must be equal to zero if the utility of reporting is equal to the utility of withholding the
information (® j = ®KN = 0). If ® j is a strictly monotone function and there exists a
signal jj for which ® jj > 0 and a jjj for which ® jjj < 0 then the threshold value is
unique. The first derivative equals:
y°KY¯ N
(34)
whereby ± j =
Y¯ EO
yY ¯
@
†IJ@ DIQ@ DI^
=
?C IJ@
@ −
IJ@ DIQ@ DI^
?C IJ@
Z± j +
¯
€KY± ¯ N
@ Y±
IJ@ DIQ@ DI^
[,
€KY± ¯ N
Y± ¯ . The first term of equation ((34) is positive. Sampford’s (1953)
inequality implies that: 0 <
j
¯ Ẑ +
€KY± ¯ N
Y±
[=
€KY± ¯ N
Y± ¯ €}
E}
Z−² +
€}
[ < 1 follows, where
E}
± j = −². Thus ® j is a strictly monotonically increasing function (
y°KY¯ N
yY¯
> 0).
Next, it must be shown that ® j can be positive and negative. Calculating the limits (if it was
necessary, the rule of L’Hospital was used) shows that:
28
(35)limY¯ →³ ® j = ∞ − −
­ − R
B@ BC
BC DB@
(36) limY¯ →E³ { j j − = limY¯ →E³
limY¯ →E³
EKY¯ EON
t¶Ks¯ N
s¯ tµ
E
@ eEp@ qp@ qp@ f
¯
J
Q
^
Z¶Ks N[
N “ j = 0
(37) limY¯ →E³
(38)
= limY¯ →E³
@
KY¯ NKY¯ EONDKIJ@ DIQ@ DI^
N€KY¯ N
@ NY¯ KIJ@ DIQ@ DI^
KY¯ N
C
Ks¯ tµN
= limY¯ →E³
EKY¯ EON
t¶Ks¯ N
s¯ tµ
E
@ eEp@ qp@ qp@ f
¯
J Q
^
Z¶Ks N[
= limY¯ →E³ KI@
C
@
Due to (35) and (38) and the fact that
which ®KN = 0.
y°KY¯ N
yY¯
@ NT −
B@ KIJ@ DIQ
limY¯ →E³ ® j = 0 − B @DBC ­ − RB
B B
?@ IJ@
E
€KY¯ N
C
@
Ks¯ tµN
B@ BC
BC DB@
= limY¯ →E³
C
¶Ks¯ N
J
KY¯ N
Q
=
Y →E³
=limY¯ →E³
€KY¯ N
E€Y¯ Y¯ EO
T − B @DBC \]| j <0
KI@ DI@ N
?@ IJ@
@
EKY¯ EON
= ¯lim − 2K + +
@
@
¯
J DIQ DI^ N€Y @
\]| j = ∞ > 0,
B B
C
=0
@
> 0, there exists only one threshold value, for
The bias in excess of the expected value of the accounting information
As stated above, there is an infinite number of voluntary disclosure functions depending on the
beliefs about ­. Each of these equilibria is characterized by a threshold value ­ which
depends on ­ whereby all signals higher (lower) than ­ lead to a voluntary disclosure
(withholding of information).
All of these equilibria require out-of-equilibria beliefs so that no type has an incentive to deviate
from the equilibrium strategy. These out-of-equilibrium beliefs will determine the market price if
a report which is lower than -.|0 + ­ is observed. In equilibrium, a manager with the signal
­ must be indifferent between disclosing KN = -.|0 + ­ and disclosing nothing. But
what happens if the manager with signal ­ decides to disclose: -.|0 + ­ − · with ­ > · >
0? How high will the market price be in the first period? If ≥ ,Y , then disclosing -.|0 +
­ − · is just as good as disclosing -.|0 + ­ or disclosing nothing, whereby ,Y is defined by:
(39)
,Y K-.|0 + ­ − ·N − 5 − B @DBC ­ − · = R − KYNT
B B
C
@
= -.|0 − 5 −
€KYN
B@ BC
BC DB@
­ ,
29
where 5 = R
?@ IJ@
@ NT +
B@ KIJ@ DIQ
B@ BC
BC DB@
\]|. The last equality follows from the fact that a
manager with the signal ­ is indifferent between disclosing KN = -.|0 + ­ and
disclosing nothing. It is noteworthy that the expected disutility is smaller if the manager deviates
(­ − · < ­ ), thus, -.|0 > ,Y K-.|0 + ­ − ·N. For all out-of-equilibrium beliefs
which lead to a higher market price if a report of -.|0 + ­ − · is observed K-.|0 + ­ −
·N > ,Y K-.|0 + ­ − ·N, the manager with the signal ­ would deviate from her disclosure
strategy.
Assume a manager with a signal lower than ­, for example ­ − ¸ (¸ > 0) would report
-.|0 + ­ − ·=-.| − ¸0 + $¸ + ­ − ·. She will be indifferent if:
(40)
,YE¹ K-.|0 + ­ − ·N − 5 −
B@ BC
BC DB@
$¸ + ­ − · = R − €KYN
T.
KYN
Due to that $¸ + ­ − · > ­ − · the price in period one where a ­ − ¸ type is
indifferent between disclosing -.|0 + ­ − · and nothing is higher than ,Y K-.|0 + ­ − ·N,
the price where a ­ type would be indifferent. A type ­ − ¸ prefers to deviate for all
prices which are at least as high as ,YE¹ . For such a price reaction she either strictly prefers
reporting -.|0 + ­ − · or is indifferent. If ≥ ,YE¹ a type strictly prefers to deviate from
the equilibrium strategy. Thus, according to the D1 criteria of Cho and Kreps (1987), the market
should not believe that the report came from a type ­ − ¸ because a ­ type prefers the
deviation stronger. Thus, the market should believe that the report was sent from a manger with a
signal which is at least as high as ­. However, if the market believes this, the report
-.|0 + ­ − · should lead to a price in the first period which is at least -.|0, K-.|0 +
­ − ·N ≥ -.|0. If this is the case, then a manager with signal ­ would deviate from the
equilibrium strategy (KN = -.|0 + ­), because the disutility would be lower whereas the
price is at least as high as that of reporting KN = -.|0 + ­. Thus, all equilibria with positive
­ can be deleted by using this refinement method. The only surviving equilibrium is one in
which ­ = 0. If a report j ≤ -.|0 is observed, in such an equilibrium, the market will
evaluate the firm with j = j because the strongest incentives for this report has a firm of
30
type j whereas j = - G| j H. Thus, for all < it is not valuable to report voluntarily, because
the utility of withholding is higher.
If ­ = 0 in equilibrium, the constant of integration in equation (28) equals:
¨=
(41)
›º» £E
E¤@
− .
The manager with the signal discloses her actual beliefs about the future accounting
information and, thus,: m Ž−
©
tK@ª@ NKsq«N
¬
›
 = −1 as can be seen in equation (28). This occurs if
the constant of integration is given by equation (41). By inserting (41) into (28), the reporting
function (12) is obtained.
Disclosure strategies
The differential equation (24) guarantees that, if a manager decides to issue a report, a type j
will report j and a type jj will report jj , for all j , jj .,∞. Deviating from the
equilibrium reporting strategy, , would lead to a lower utility. However, is it possible that a
better type jj > j decides for a non-disclosure while a type j decides to disclose voluntarily?
Firstly, it can be shown that the expected utility is increases in the signal . The higher the signal
the higher will be the expected utility of reporting. The expected utility is given by equation (30).
Differentiating this equation with respect to leads to:
(42)
yX.¼zs _Y0
yY
J
Q
= I@ DIJ@ DI@ + I@ DIJ@ DI@ − B @DBC 2 − - G|H RxY − I@ DI
@ DI @ T
J
I@
Q
= ^
IJ@
@
@
IJ DIQ@ DI^
J
I@
Q
+ B @
B BC
C DB@
^
B B
C
@
2 − - G|H
x=
IJ@ DIQ@
@
@
IJ DIQ@ DI^
> 0.
I@ DI@
J
Q
^
Because ≥ - G|H equation (42) is positive. Even without the positive effect of the expected
price in the second period, the utility of a better type is higher,
(43)
yX.¼zs _Y0EXG½@ |YH
yY
- „=KY¯ N ¾ j … − - G | j H > -.=Y _0 − -. |0,
≥ 0. Thus,
if j > . Since a manager with the threshold information is indifferent between disclosing and
withholding information (-.=Y _0 = -.Ø |0). The term on the right side of the inequality is
31
equal to -.Ø|0 − -. |0. Based on this equality and the inequality in (43) the following
inequality can be calculated:
- „=KY¯ N ¾ j … > -.Ø |0 − -. |0 + - G | j H = - GØ | j H.
(44)
Therefore, a manager of type j is better off if she issues a report. If jj > j , then
- „=KY¯¯ N ¾ jj … − - G | jj H > - „=KY¯ N ¾ j … − - G | j H > - GØ | j H − - G | j H and, thus, also
- „=KY¯¯ N ¾ jj … > - GØ | jj H. Thus, if j discloses information, a firm with better information will
issue a report as well.
Frequency of voluntary disclosure
The threshold value is calculated by setting the function in (33) equal to zero (®KN = 0). Thus, is defined by this implicit equation. Due to of this equation, the derivative of the threshold level,
with respect to a parameter, can be determined by using the implicit function theorem. It is the
ratio of the derivative of ®∙ with respect to the parameter and the derivative of this function
with respect to multiplied with minus one. As already known from above,
y°KY¯ N
yY¯
> 0. In order
to calculate whether increases or decreases depending on a change of a parameter, the partial
derivative of ® ∙ with respect to the parameter has to be calculated:
y°
yBC
(45)
= −R
y°
yB@
(46)
y°
y?C
(48)
(49)
(50)
y°
@
yI^
=
=
?@ IJ@
y°
=
Á−± −
@ NT
B@¿
J
Q
KIJ@ DIQ
IJ@ KYEON
@ N¿⁄@
KIJ@ DIQ@ DI^
yIQ@
= R
B@@
C DB@
− B
@ @
KIJ DIQ@ N
I^
R
@
BC@
C DB@
= − I@DI@J@CB @@B
+ @
IJ@ DIQ@ DI^
?C IJ@
y°
B@ KIJ@ DIQ
y?@
(47)
@ NT − B
?@ IJ@
?C IJ@
KYN
=
+ ±
N
€KY± N
KY±
€KYN
ISB ?
@ N¿⁄@
KIJ@ DIQ@ DI^
@
@
C DB@ KY±
€KY± N
KY± N
@
IJ@ DIQ@ DI^
< 0,
T,
ê +
€ZY± [
KY±
TÂ −
N
B@ BC
KY± N
@
†IJ@ DIQ@ DI^
R± +
N
T < 0,
@ @
KIJ DIQ@ N
I^
R
IJ@
€KY± N
Á−̂ −
@
IJ@ DIQ@ DI^
€KY± N
+ ̂
€KY± N
KY±N
f > 0,
@
KIJ@ DIQ@ N
BC DB@ KIJ@ DIQ@ DI@ N@
R± +
€KY±N
KY± N
TÂ
^
< 0,
32
−
S
B@ BC I^
@ N@
BC DB@ KIJ@ DIQ@ DI^
The derivative in (46) is negative if θ < †
+
BC ?@@ IJS
¿
B@@ KIJ@ DIQ@ N
,
@
IJ@ DIQ@ ¿ I^
BC B@¿
@
S
@
@
IJ IJ DIQ DI^ BC DB@ @
. The partial derivative, with
respect to the weight on the second price, is negative. The derivative, with respect to , is
positive due to Sampford’s law.
Equation (49) is negative for all ± . This is obvious for ± > 0. Because
€KY± N
KY± N
ê +
€ZY±[
KY± N
f 0,1, the
upper bound of the third term in square brackets equals ± . Thus, the addition of the first term
and this term is always negative and all the other terms are negative as well. If ± < 0 additional
conditions are needed. The expression in the square brackets can be transformed to:
(51)
whereby à = −± > 0, W =
„Ã −
EÅ
ہ
Ä
− à Z−à + [… < 0,
Ä
Ä
, “Ã = “ −à and 1 − { à = { −Ã. The last two
equalities follow from the characteristics of the normal distribution function. Because the Mills’
ratio W lies between 2⁄KÆÃ + 4 + ÃN < W < 4⁄KÆÃ + 8 + 3ÃN 12, the term in (51) must be
negative for all à > 0.
Equation (50) is at least negative if —
¿
S B ¿ KI @ DI @ N
I^
Q
@ J
@
@ N IS B DB KIJ@ DIQ@ DI^
@
J C
> , which can be derived from the
last two terms of equation (50). If this inequality does not hold, it can be positive as well. The
derivative with respect to is not unambiguously which is why it is neglected.
Proof of the difference between the expected biases
The ex-ante expected difference between the two variables is given by:
(52)
-.=∗ | ≥ 0 − -.}∗ | ≥ 0 = K1 + -.m G∙H| ≥ 0N R
E?C IJ@
KIJ@ DIQ@ NBC
T+
?@ IJ@
KIJ@ DIQ@ NB@
.
The inequality in the text can be derived from (52). Differentiating (52) with respect to leads
to:
12
See Baricz (2008).
33
yX.:z∗ |YÊY0EX.:Ë∗ |YÊY0
y?@
(53)
=l
w l @ C @JNB w + @ J @ NB > 0.
yY ÍÏ y?
KIÍÎÍ
KIJ DIQ @
Ñ@
ÌÍÍÎÍ
ÌÍ
ÍÏC
J DIQ
yX.”G∙H|YÊY0 yY
E? I@
Ҍ
Ќ
I@
Ќ
This derivative is positive.
yX.”G∙H|YÊY0
yY
(54)
= ÓY m G∙H Ô
³
R
€Y
€KYN
EKYN EKYN
−
YEO
Is@
TÕ " < 0
Since the function in curly brackets in (54) has only one root and is positive on the left side and
negative on the right side and since the integral of this term equals zero, weighting it with a
negative and increasing function (mG∙H) must now add to a negative number.
Proof of Lemma 2
The
conditional
@
@
N
€KYN KIJ@ DIQ@ DI^
ZEKYN[
@
variance
f=
IJS
@
@
IJ DIQ@ DI^
is
given
is given by:
y}
yY
=
€}
@
†IJ@ DIQ@ DI^
y×Ø=.½C EO|YÊY0
@
yI^
.
=−
yM
yY
=
The
derivative
y×Ø=.½C EO|YÊY0
IJS
@
@
IJ DIQ@ DI^
IJS
@
@
IJ DIQ@ DI^
e1 +
€KYNYEO
EKYN
−
1 − 5. Based on Sampford (1953) we know that: 0 < 5 =
Z−² + E}[ < 1 and y} > 0 while ² =
E}
€}
\]. − | ≥ 0 =
by:
IJS
@
@
IJ DIQ@ DI^
R−
YEO
@
†IJ@ DIQ@ DI^
yM y}
y} yY
. Thus, the derivative with respect to T < 0. This is negative because
with
respect
to
is
yM
y}
> 0 and
given
by:
J
−
l1 − 5 −
w < 0.
Ù yY
y}
y} Ù yI
Ñ^@ KIJ@DIQ@ DI@ N@ ÌÍÍÍÎÍÍÍÏ
yM y} yY
Ҍ Ҍ Ҍ
IS
^
Ҍ
yM }
34
Figure 1 Timeline
35
Figure 2 Biasing function
36
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38