Regulation FD, Analysts’ Information Acquisition, and the
Public Goods Problem
Manabu Kotani

Faculty of Commerce, Kumamoto Gakuen University
Japan Accounting Association
E-mail: kotani@kumagaku.ac.jp

Abstract
Analysts collect their own information to improve the precision of theirearnings forecast. However, once the Regulation Fair Disclosure (FD) takes effectand information includes the aspect of public goods, individual analysts are
expected to begin free-riding on information acquired by other analysts. This paper
develops a model in which analysts decide whether or not to collect costly
information, and an investor predicts earnings based on the analysts’ behavior. The
findings of this paper include, under Regulation FD, as the number of analysts
grows, the motivation to collect information becomes weaker, and analysts depend
on public information only. Further, without Regulation FD, an increase in thenumber of analysts has a nonnegative effect on the investor ’s predictive ability.
Finally, under Regulation FD, the investor ’s predictive ability is at a low levelcompared to a case without the Regulation. It is noteworthy that Reg FD, which
intends to improve the information environment surrounding security market
participants, has these unintended consequences.
Keywords
Earnings forecast, Forecast accuracy, Regulation FD, Free-rider, Public
information(1) Introduction
Mitigation
asymmetrysts without simultaneously disclosing
of

among

the

information

security

the same information to the public. As amarketresult, enactment of Reg FD has banned

participants is seen as major purpose of

so-called

disclosure regulation. The Regulation

comprises that firms transmit material

Fair Disclosure (hereafter “Reg FD”),

information

which was enacted in 2000 in theparticipants. Further, a regulation with

United States, prohibits firms fromthe same effect as Reg FD will beprivately

introduced in Japan in the near future.

disclosing

important

information to select investors or analy-

The
23

selective

purposedisclosure,

only

of

to

this

which

particular

paper

is

to

investigate the effect of Reg FD on

representative

analysts’ information acquisition andfollowing. First, if the firm discloses

the investor ’s predictive ability.

information to all players (two analysts

There

are

abundant

prior

and

an

investor,

investor),

revealsantheinformation

researches that empirically inquire howcascade will arise among them so that

Reg FD affected analysts and investors.

the investor ’s predictive ability will

Concerning analysts, enactment of Reg

worsen. Second, if the information is

FD has (1) increased (or decreased) theoffered to only one analyst (in othernumber of analysts who follow the firmwords,(Irani and Karamanou 2003; Mohanram

disclosure), cascade formation may be

and Sunder 2006; Gomes et al. 2007 );restrained and the investor is better off.

(2)

Lastly, under Reg FD, the firm may

rendered

analysts’

informationacquisition

active(orinactive)(MohanramandSunderdecide

2006;

the

not

firm

to

issuesdiscloseaselective

information

intentionally in order to improve theJanakiraman et al. 2007; Mensah andinvestor ’s predictive ability.

Yang 2008; Kross and Suk 2012; Hahn

The work of Arya et al. (2005) isand Song 2013); and (3) improved (orclosely related to this paper.

worsened) the precision of analysts’

studies pay attention to the nature of

forecasts(Irani and Karamanou 2003;

the influence of Reg FD on analysts and

Heflin et al. 2003; Bailey et al. 2003;

investors. They differ in the following

Agrawal et al. 2006; Francis et al. 2006;

point: Arya et al. (2005) is concerned

Mohanram and Sunder 2006; Gomes etwith the information cascade caused byal. 2007; Srinidhi et al. 2009; Kross and

Reg FD, while this study focuses on a

Suk 2012). With respect to investors,

particular phenomenon, namely thatenactment of Reg FD has (1) changed

Reg FD renders information a public(orinformation

good. Therefore, Arya et al. (2005) see

environment (Heflin et al. 2003; Francis

analysts as rational decision-makers,

et al. 2006; Ahmed and Schneible 2007 )but they do not consider the strategicandinteractions of analysts. On the othernotchanged)

(2)

their

reducedinformation(or

expanded)
among

hand,

2004;

strategic agents who rationally expect

Chiyachantana et al. 2004; Gomes et al.

other agents’ incentive. In particular,

2007; Sidhu et al. 2008; Duarte et al.

the2008; Chen et al. 2010). These priormotivation of analysts to be free-riders

studies do not necessarily come to thein information collection. In general,same conclusion. On the contrary, somemost firms are covered by multiple

opposing results are presented.

analysts. Therefore, to gain deeper

investors

asymmetry(Eleswarapu

et

al.

this

study

study

pays

sees

Both

analysts

attention

to

as

the

Arya et al. (2005) investigate Reg

insight into analysts’ decision-making,

FD analytically. Their model, whicha study of their behavior in the presence

includestwo

analystsandaof rivals is needed. As mentioned above,24

a model is developed to examine theinfluenceofReg

informationFDonanalysts’(private) information (or signals) with

and

the

an effort from manager, and 𝑐𝑖 denotes

ability,while

the cost of effort incurred by analyst 𝑖.

taking analysts’ strategies into account.

Assume that signal 𝑥̃ is represented

investor ’s

collection

Analysts can extract additional

predictive

The paper is organized as follows.
Section2

presents

the

model

by:

and

𝑥̃ = 𝜃̃ + 𝜀̃,derives an equilibrium. Section 3 shows
the comparative statics and discusses

where 𝜀̃ represents the prediction errorand 𝜀̃~𝑁(0, 1⁄𝛽 ). One of the examples of

the case where Reg FD is not enforced.

𝑥̃ is the manager ’s own prospect, which

Section 4 concludes.

is not expressed in the management
forecast. To avoid being pursued as(2) Model

responsible for the wrong prediction,

There

are

representative
forecasting

analysts𝑛

investor

the

and

who

firm’sathis study assumes that the manager

arewill not venture to release his prospect

forthcoming

voluntarily.

However,

if

he

is

earnings. The variable 𝑛 is a positive

interviewed by analysts, he will reveal

integerThe

it in a passive manner. The cost offorthcoming earnings of the firm, which

acquiring 𝑥̃, which is dependent on theare denoted by 𝜃, are the realization ofa random variable 𝜃̃ , where 𝜃̃ ~𝑁(0, 1⁄𝜔 ).

ability to extract information from theNote

Suppose that each analyst observes his

and

that

larger

tilde

than

(“~”)

one.

represents

manager,arandom variable.
All

analysts

differs

among

analysts.

𝑐𝑖 , but cannot know another analyst’s
and

the

investor

cost.

He

only

knows

that

anotherobtain public information (or signals) 𝑦̃analyst’s cost is a random variable that

released by the firm at no charge.

isAssume that public information 𝑦̃ is

Meanwhile, the investor does not haverepresented by:access to a manager, and he cannot
obtain 𝑥̃ on his own. Assume that 𝜃̃ ，𝜂̃，𝑦̃ = 𝜃̃ + 𝜂̃,
whereuniformly

distributed

in[0,1]. 1

represents the prediction
error and 𝜂̃~𝑁(0, 1⁄𝛼 ). Managers’ initialand 𝜀̃ are mutually independent. 2

earnings forecast, which is made public

behavior is considered. According to the(so-calledso-called projection theorem, an analyst,

𝜂̃“management

To

forecast”),

begin

with,

the

analysts’

illustrates 𝑦̃.

who has observed 𝑦 and 𝑥, will expect

Because there is no reason to specify the
distribution of 𝑐𝑖 a priori, it is assumed that 𝑐𝑖
distributes uniformly on [0,1]. Alternatively,
even if the distribution function of 𝑐𝑖 isassumed to be a monotone increasing functionthat is continuous on [0,1], the main results(except for Corollary 1) remain unchanged. For
convenience, it is assumed that 𝑐𝑖 is a

uniformly distributed random variable on [0,1].
2 Due to the release of signals 𝑦 or 𝑥, a firm
may incur proprietary cost s. However, thedecision-making of disclosure by firms isbeyond the scope of this work, since proprietary
cost is not considered in this model.

125

where Π（ ≥ 0）represents a fixed rewardthat 𝜃̃ is represented by:E(𝜃̃ |𝑦, 𝑥) =
In

this

case,

prediction

𝛼𝑦 + 𝛽𝑥
𝜔+𝛼+𝛽

the

error

varianceisfor analysts, 𝑎𝑖 represents estimates(1)

submitted by the analyst, and 𝑐𝑖 is the

of

the

cost of acquiring 𝑥̃. 5 In contrast, whenrepresentedbyan analyst 𝑖 does not collect 𝑥 , the1⁄(𝜔 + 𝛼 + 𝛽) . In contrast, when an
analystdoes

notcollect

payoff is denoted as follows:
𝑈𝑖 = Π − (𝑎𝑖 − 𝜃)2additional

information, he can observe only 𝑦, andhis

conditional

expectation

As stated above, when an analyst
observes {𝑦, 𝑥} , then 𝑎𝑖 = E(𝜃̃ |𝑦, 𝑥) .

isrepresented by:E(𝜃̃ |𝑦) =𝛼𝑦𝜔+𝛼Further, when he observes {𝑦} , then𝑎𝑖 = E(𝜃̃ |𝑦). Therefore, in each case, the(2)expected

The variance of the prediction error issimultaneously,

theirforecast

but

without

𝑥 to other analysts are prohibited. 3 In
isassumedanalystreveals

his

that

as

(5)

2
E𝑈𝑖 = E [Π − {E(𝜃̃ |𝑦) − 𝜃} ]paper, it is assumed that sales of signal
itrepresented

1
=Π−
− 𝑐𝑖
𝜔+𝛼+𝛽

communication among them. In thisaddition,is2
E𝑈𝑖 = E [Π − {E(𝜃̃ |𝑦, 𝑥) − 𝜃} − 𝑐𝑖 ]after the formation of expectation, thereleasepayofffollows:

represented by 1⁄(𝜔 + 𝛼). Assume that,analysts

(4)

1
=Π−
𝜔+𝛼

each

expectation(6)

Π − 1⁄(𝜔 + 𝛼 + 𝛽) of equation (5) can betruthfully and without garbling.

interpreted as the reputation that an

Based on prior studies (Laster et al.

analyst gains in the capital market. The

1999; Lim 2001; Morgan and Stocken

remaining amount after deducting 𝑐𝑖 is

2003;

Kameda et al. 2011), the analyst’s

analyst 𝑖’s expected payoff. Meanwhile,Π − 1⁄(𝜔 + 𝛼) of equation (6) shows that,

payoff function is represented. 4 When

if the analyst does not collect signal 𝑥,

an analyst 𝑖 collects signal 𝑥 , his

his reputation declines, but the cost of

payoff is denoted as follows:
𝑈𝑖 = Π − (𝑎𝑖 − 𝜃)2 − 𝑐𝑖information gathering is saved.

FischerandStocken

2010;

(3)

Now consider the behavior of a

The assumption that sales of signal 𝑥 toother analysts is prohibited is based on thefollowing. There are various rules that require
financial institutions to build information
barriers.
For example, to prevent premature leakage ofmarket-moving information, several SEC andstock exchange rules mandate that the Chinese
Wall be set up as an imaginary barrier between
the investment banking, corporate finance, andresearch departments of a brokerage house, andthe sales and trading departments ( Downes andGoodman 2014). As long as these information

barriers exist, transactions of signal 𝑥 among
analysts seem impossible to carry out.
4 These prior researches consider forecasters’
incentive to minimize forecast error byincorporating the quadratic loss term into theirutility function.
5 Unlike the investor, analysts may have aninterview with managers. In that case, it isassumed that information acquisition cost isincurred to constant Π. However, none of theconclusions are changed on the assumption thatΠ = 0.

326

representative investor. The investor ’s
objective is to maximize the precision of

based on analysts’ forecasts. In the final
stage, the realization of 𝜃̃ (and 𝑥 if

the forecast, and the analyst’s forecast

manager offered) are disclosed. In thiscan be utilized at the prediction ofgame, once analysts decide whether orforthcoming earnings. Therefore, thenot to collect 𝑥 during the first stage,investor’s action is described as follows.
When all analysts submit 𝛼𝑦⁄(𝜔 + 𝛼) as

analysts and the investor do not face

their forecast, the investor adopts𝛼𝑦⁄(𝜔 + 𝛼) as forecast. In contrast,

make decisions strategically. Therefore,

when an estimate that is different from𝛼𝑦⁄(𝜔 + 𝛼) is submitted by at least one

making during the first stage.

analyst, the investor adopts the value,

where Reg FD is in force. 6 Under Reg

forbelow.

FD, selective disclosure by the firm isKnowledge about signal 𝑦 raises theinvestor’s expectation that E(𝜃̃ |𝑦) =prohibited. Hence, if a firm provides

𝛼𝑦⁄(𝜔 + 𝛼). Therefore, when the investor

participants,

observes that the analyst’s forecast issimultaneously release the information

equal

investor

to the public. In other words, if at least

conjectures that the analyst does not

one analyst extracts signal 𝑥 with aknow signal 𝑥. On the other hand, when

cost, other analysts can simultaneously

the analyst’s forecast is not equal toacquire 𝑥 without a cost. Therefore,𝛼𝑦⁄(𝜔 + 𝛼), the investor guesses that thethe information set is {𝑦, 𝑥} for eachanalyst does know signal 𝑥 , unlike
himself. As a result, the precision of theanalyst, and his estimated results are(𝛼𝑦 + 𝛽𝑥)⁄(𝜔 + 𝛼 + 𝛽). On the other hand,investor’s forecast is 𝜔 + 𝛼 when all ofwhen

the analysts submit 𝛼𝑦⁄(𝜔 + 𝛼), and isanalyst’s information set is {𝑦} and theestimate is 𝛼𝑦⁄(𝜔 + 𝛼). Now, under Reg

thereasontooutlined𝛼𝑦⁄(𝜔 + 𝛼) ,thethe situation afterwards where they
this study focuses on analysts’ decisionHereafter, the case is consideredinformation

𝜔 + 𝛼 + 𝛽 when at least one analyst

no

to

one

particular
the

firm

acquires𝑥 ,marketmust

every

submits the forecast different from𝛼𝑦⁄(𝜔 + 𝛼).

FD, the investor comes to know the

There are four stages in this game.

information disadvantage over analysts

the𝑖disappears. Therefore, note that theobserves 𝑐𝑖 and decides whether toanalyst’s forecast can no longer havecollect signal 𝑥 or not. In the second

information content, and it becomesstage, analysts simultaneously release

redundant for the investor under Reg

forecasts

FD.

In

first

by

stage,eachrelyinganalystonsame information as analysts, and an

publicinformation 𝑦 (and signal 𝑥 if it had

An expected payoff of analyst 𝑖 isbeen collected). In the third stage, thestudied in three different cases. First,

investor expects forthcoming earnings

when analyst 𝑖 collects signal 𝑥 by

6

The case where Reg FD is not enforced isinvestigated in Section 3.
27

which each type 𝑐𝑖 , with 𝑐𝑖 ≤ 𝑐 ∗ ,acquires 𝑥 , whereas other types 𝑐𝑖,

himself, the expected payoff is denoted
as equation (5). Second, if not onlywith 𝑐𝑖 > 𝑐 ∗ , do not acquire 𝑥 . 7 In
addition, there exists a unique 𝑐 ∗ ∈(0, 1) that satisfies the following
equation:analyst 𝑖 but also any other analysts
do not collect signal 𝑥 , the expected
payoff is represented by equation (6).
Lastly, suppose that analyst 𝑖 does notcollect signal 𝑥 but that someone else

(1 − 𝑐 ∗ )𝑛−1
− (𝜔 + 𝛼 + 𝛽)(𝜔 + 𝛼)𝛽 −1𝑐∗collects it. In other words, suppose thatanalyst 𝑖 free-rides on others. Then,

(8)

=0

the expected payoff is represented by:
E𝑈𝑖 = Π −1
𝜔+𝛼+𝛽[Proof] All proofs are described in the(7)

Appendix.

Free-riding on others makes it possible
for an analyst to improve his reputation
without

theinformationdifficult to solve for 𝑐 ∗ explicitly withgathering. Consequently, the expected𝑛 ≥ 3. However, the implicit function

payoff of equation (7) is higher thantheorem shows the behavior of 𝑐 ∗ . In

that of equation (5).

theDuring

cost

the

of

As shown in equation (8), it is

firststage,

each

next

section,the

resultofcomparative statics that focus on theanalyst 𝑖 conjectures other analysts’

threshold 𝑐 ∗ is shown.

actions and decides whether or not tocollect

signal𝑥 . In this game, if(3) Analysis

even just one analyst acquires signal 𝑥,

1.

each

Acquisition

analyst’s payoff will

However,

costonInformation

The following Proposition statesacquisition generates the incentive to

how parameters (such as 𝑛， 𝜔， 𝛼 andbe𝛽free-riderofImpactinformationatheincrease.

The

among

analysts.

Hereafter, the Bayesian Nash equilibria,)

affectanalysts’

informationacquisition.

consisting of symmetric pure strategies
𝑐𝑖 ≤ 𝑐 ∗ ,Proposition 1 The sign of the partial

acquires 𝑥 , whereas every other typederivative of 𝑐 ∗ with respect to each of
the parameters is represented by:in which each type 𝑐𝑖 , with
does

not

acquire𝑥 ,issolved.

Consequently, the following Lemma is

𝜕𝑐 ∗
< 0，
𝜕𝑛

obtained.

𝜕𝑐 ∗
< 0，
𝜕𝜔

𝜕𝑐 ∗
< 0，
𝜕𝛼

𝜕𝑐 ∗
> 0.
𝜕𝛽

Lemma The Bayesian Nash equilibrium

We can interpret the result ofin this game is the pair of strategies in

Proposition 1 as follows. First, theIn the case of 𝑐𝑖 = 𝑐 ∗ , it is indifferent in termsof expected payoff whether or not to acquiresignal 𝑥. In that case, the model assumes thatanalysts acquire signal 𝑥.

728reasonwhy

theis that an increase in 𝛽 enhances thelarger 𝑛 becomes, the more analysts

attractiveness of signal 𝑥 , and hence

rely

the

on

𝜕𝑐 ∗ ⁄𝜕𝑛 < 0

is

that

other analysts’ informationcollection.

In

other

words,

thetolerable

level

of

the

cost

of

information

gathering

motivation to be a free-rider arises

Consequently,

theamong analysts, and the tolerable level

information activates analysts’of the cost of information gatheringcollection. In that sense, the effect of 𝛽declines. Therefore, an increase of 𝑛

is in contrast to that of 𝛼.

8increases.
qualityofleads analysts to neglect to extract
information from the manager, so that2.

analysts become dependent on public

Predictive Ability

information only. 9

The

ImpactontheInvestor ’s

The discussions are now extended𝜕𝑐 ∗ ⁄𝜕𝜔 < 0Second, the reason why

to investigate the impact of parameters

is shown below. The payoff difference

on the investor ’s predictive ability. It is

between collecting and not collecting

evident that his predictive ability is

information

dependent

reducesas𝜔

becomesonanalysts’

information

large. Consequently, the incentive toacquisition behavior. If all analysts do

acquire 𝑥 in exchange for 𝑐𝑖 weakens.
In other words, analysts cease to gather

not collect 𝑥, then the investor accepts
the analyst forecast 𝛼𝑦⁄(𝜔 + 𝛼) as

additional information as the volatility

estimate, or predicts by himself, basedof earnings decreases.

on signal 𝑦. Consequently, the precision

Third, the reason why 𝜕𝑐 ∗ ⁄𝜕𝛼 < 0of the prediction results in 𝜔 + 𝛼. On

is that an increase in 𝛼 reduces thethe other hand, if at least one analyst

payoff difference between collecting andcollects 𝑥 ,then the investor accepts

not

the

analystcollecting

information,

so

that

forecast

analysts lose the motivation to collect(𝛼𝑦 + 𝛽𝑥)⁄(𝜔 + 𝛼 + 𝛽)

information. Accordingly, if sufficiently

predicts by himself, based on signals 𝑦precisebecomesand 𝑥 . Hence, the precision of theinformation

prediction increases to 𝜔 + 𝛼 + 𝛽 . The

acquisition by analysts will be crowded

probability that all analysts do notcollect 𝑥 is (1 − 𝑐 ∗ )𝑛 , and the

publicavailable,

information

additional

out. Thus, an increase in the value ofparameters (such as 𝑛 ， 𝜔 ,and 𝛼 )

asestimate,orforprobability that at least one analyst
collects 𝑥 is 1 − (1 − 𝑐 ∗ )𝑛 . Then the ex-

information gathering, and increases

ante expected precision of the forecast

their dependence on public information.

is represented by:weakens

analysts’

incentive

Finally, the reason why 𝜕𝑐 ∗ ⁄𝜕𝛽 > 0Intrinsically, 𝑛 is an integer. However, forconvenience, it is assumed here to be a real
number.
9 In contrast, Frankel and Li（ 2004）show that,as the number of analysts covering a firm grows,

confidential
information
leaks
out
and
information asymmetry between insiders andinvestors diminishes.

829

reduces 𝑔 due to the reduction of 𝑐 ∗ .

(1 − 𝑐 ∗ )𝑛 × (𝜔 + 𝛼) + {1 − (1 − 𝑐 ∗ )𝑛 }Here,× (𝜔 + 𝛼 + 𝛽)

it

is

shown

that

a

set

of

The above function can be rearranged to𝜔 + 𝛼 + 𝛽 − 𝛽(1 − 𝑐 ∗ )𝑛 . Let it be function

parameters
existsthat
satisfies𝜕𝑔⁄𝜕𝛼 < 0. For example, let 𝑛 = 2, 𝜔 =𝑔. Hereafter, this study inquires how

0.5 , 𝛼 = 1 ,and 𝛽 = 4 .

an increase in 𝑛 affects the value of 𝑔.

parameter
values,and𝜕𝑔⁄𝜕𝛼 = −0.187. If the firm controls 𝛼 toCorollary 1 The sign of 𝜕𝑔⁄𝜕𝑛 depends

increase 𝑔, then the firm must reduceon the value of threshold.
∗
Specifically, when 𝑐 > 1⁄2 , then𝜕𝑔⁄𝜕𝑛 < 0 is satisfied. When 𝑐 ∗ = 1⁄2 ,rather than increase 𝛼 in this case.

then 𝜕𝑔⁄𝜕𝑛 = 0 holds.
when 𝑐 ∗ < 1⁄2 , theninformation leads to improved decision-

For

these

𝑐 ∗ = 0.327𝑐∗This result disputes the conventional
view

In

addition,
𝜕𝑔⁄𝜕𝑛 > 0 is

that

high

qualitypublicmaking by investors. Thus, under Reg

satisfied.

FD, as the quality of public information
improves, analysts’ incentive to collect

As stated above, the effect of 𝑛 on

additional information becomes weaker,𝑐∗.

and the investor ’s predictive ability

𝑔 varies depending on the value of

When 𝑐 ∗ is high or, in other words,

may decline.
Lastly, the partial derivative of 𝑔when analysts’ motivation to collectadditional information is weak, thewith respect to 𝛽 is represented by:value of 𝑔 decreases in response to the𝜕𝑔= 1 − (1 − 𝑐 ∗ )𝑛𝜕𝛽

growth of 𝑛. The growth of 𝑛 has a twosided effect. The one is negative: the+ 𝛽𝑛(1 − 𝑐 ∗ )𝑛−1

growth of 𝑛 lowers 𝑔, because analysts’
incentive to collect information will be

𝜕𝑐 ∗
𝜕𝛽

(10)

>0

dampened. Another is positive: theTherefore, the quality of signal 𝑥 is

growth of 𝑛 increases 𝑔 , because thepositively correlated with the value ofpossibility that at least one analyst

𝑔. The improvement of 𝛽 increases 𝑔collects𝑥

increases.

thenot only directly, but also indirectly

threshold𝑐∗is high, the negative effectthrough an increase in the probabilityWhenoutweighs the positive effect. 10that signal 𝑥 is acquired.

Next, the partial derivative of 𝑔with respect to 𝛼 is represented by;𝜕𝑐 ∗𝜕𝑔
= 1 + 𝛽𝑛(1 − 𝑐 ∗ )𝑛−1𝜕𝛼𝜕𝛼The sign of𝜕𝑔⁄𝜕𝛼becauseincreasean

3. The Case where Reg FD is not
Enforced(9)

So far, the case where Reg FD is in

is not definite,inforce has been considered. Next, thedirectlycase where Reg FD is not enforced isincreases 𝑔 , while an increase in 𝛼

considered, and the difference between𝛼Now, 𝑐 ∗ = 1⁄2 represents the mean of theprobability function of 𝑐𝑖 .
1030

the cases is illustrated. In the absence

predictive ability. As described in 3. 2,of Reg FD, an analyst cannot free -ride

an increase in 𝑛 has both positive andon other’s information, because signal

negative effects under Reg FD. On the𝑥 is not shared by analysts. Hence,other

analyst 𝑖 collects 𝑥 only if the value of

enforced, an increase in 𝑛 does notequation (5) equals or exceeds that of

bring(6). This condition can be simplified as

Accordingly,

follows:

increase in the number of analysts has𝑐𝑖 ≤𝛽(𝜔 + 𝛼)(𝜔 + 𝛼 + 𝛽)threshold

of

cost

when

about

a

Reg

FD

is

negative

withoutRegnoteffect.
anFD,a nonnegative effect on the investor ’s(11)

predictive ability.
Lastly, the following Proposition

The right-hand side of equation (11)denotes thehand,evaluates Reg FD based on the investor’sthatdetermines an analyst’s action. 11 Let

perspective.

the threshold be 𝑘 ∗ . It is clear that 𝑘 ∗is independent of 𝑛. That is, when Reg

Proposition 2 With respect to the ex-

FD is not enforced, the number of rivals

ante expected precision of the investor’s
forecast, 𝑔 < ℎ is satisfied.

is irrelevant to the analyst’s decisionmaking.
Now, define an expected precision

Thus, in terms of expectation, Reg FDof the investor ’s forecast as function ℎ.

reduces the precision of the investor ’sThen, the following Corollary is derived.

forecast. The reason for this is that 𝑐 ∗ < 𝑘 ∗or, in other words, Reg FD undermines

Corollary 2 Suppose that Reg FD is notanalysts’ incentive to collect information. It𝑘∗ ≥ 1 ,enforced. Ifℎ = 𝜔 + 𝛼 + 𝛽 holds
regardless of 𝑛 . Meanwhile, if 𝑘 ∗ < 1 ,is𝜕ℎ⁄𝜕𝑛 > 0 is satisfied,𝛽 ⁄[(𝜔 + 𝛼)(𝜔 + 𝛼 + 𝛽)].

environment

where

noteworthy that

Reg

FD,

which

intends to improve the information𝑘∗ =marketsurrounding

participants,

security

has

these

unintended consequences.
Thus,

ℎ

is

a

constant

function

independent of 𝑛 , or an increasing(4) Conclusion

function of 𝑛. That is, in the absence of
Reg

FD,

an

increase

mightwhich analysts, who observed public

increase ℎ, but it will never reduce ℎ.

information, decide whether or not to

In contrast, Corollary 1 states that,collect costly information. Under Reg FD,

under Reg FD, an increase in 𝑛 may

which prohibits selective disclosure, when areducefirm

𝑔 . Hence,

the

of

This paper develops a model in𝑛presenceorprovidesabsence of Reg FD brings about aparticularsignificant difference in the investor ’s

release11

In the case of 𝑐𝑖 = 𝛽 ⁄[(𝜔 + 𝛼)(𝜔 + 𝛼 + 𝛽)], it is31

information

analyst,thethe

information

to

a

firm

must

to

other

assumed that analysts acquire signal 𝑥.

analysts at the same time. Hence,studies are required to clarify these

individual analysts have the incentive

considerations.

to free-ride on information acquired by

Acknowledgement

other analysts, rather than to collect by

The financial support of Kumamotothemselves.

GakuenThe findings of this paper include,
under

Regulation

numberofmotivationFD,

analysts
to

that

as

grows,

collect

University

isgratefully

acknowledged.

the
their

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Kross, W., and I. Suk. (2012). Does

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248.

Differentialon

Effects

ofShort-

andForecasts.

Laster, D., Bennett, P., and I. Geoum.
(1999).

Rational

Bias

Macroeconomic

inAppendix

Forecasts.

Proof of Lemma

Quarterly Journal of Economics,

It is assumed that any analyst 𝑖

114 (1), pp.293-318.
Lim,

T.

(2001).

adopts a strategy in which he collects 𝑥
andonly if 𝑐𝑖 ≤ 𝑐 ∗ . Suppose that analyst 𝑖Thedoes not acquire signal 𝑥. In addition,

Journal of Finance, 56(1), pp.369-

suppose that other analysts do not385.

acquire signal 𝑥 . Then, a value of

Analysts'

Rationality

Forecast

Bias.

Mensah, Y., and R. Yang. (2008). An

multiplying occurrence probabilities of

Empirical Evaluation of Analysts’

the event by analyst 𝑖’s expected payoff

Herding

is represented by:

Behavior

Following

Fair

Disclosure.

Regulation(1 − 𝑐 ∗ )𝑛−1 × (Π −

1
)
𝜔+𝛼

Journal of Accounting and Public
Policy, 27 (4), pp.317-338.

Next, suppose that analyst 𝑖 does not

Mohanram, P., and S. Sunder. (2006).

acquire 𝑥 , but at least one analyst

How Has Regulation FD Affectedacquires 𝑥. Then, a value of multiplying

theoccurrence probabilities of the event by

Operations

Analysts?

AccountingofFinancialContemporary
Research, 23 (2),analystexpected{1 − (1 − 𝑐 ∗ )𝑛−1 } × (Π −Morgan, J., and P. Stocken. (2003). An
of’spayoffisrepresented by:

pp.491-525.
Analysis

𝑖

1
)
𝜔+𝛼+𝛽

Stock

Lastly, suppose that analyst 𝑖 acquires

Recommendations.

RAND

𝑥 by himself. Then his expected payoff

Journal34(1),

is represented by:ofThe
Economics,

pp.183-203.

Π−Osborne, M. (2009). An Introduction to

1
− 𝑐𝑖
𝜔+𝛼+𝛽

Game Theory: International ed.

Hereafter, I 𝑐 ∗ is solved for using

N.Y.: Oxford University Press Inc.

the same procedures as Osborne (2009,p.290). For any

Sidhu, B., T. Smith, R. Whaley, and R.
Willis.

(2008).

Regulation

Fair𝑖 , if𝑐𝑖 ≥ 𝑐 ∗ , thefollowing inequality is satisfied:Disclosure and the Cost of Adverse34

(1 − 𝑐 ∗ )𝑛−1 × (Π −Let the left-hand side of equation

1
)
𝜔+𝛼

(8) be function 𝑓 . Then 𝑐 ∗ of 𝑓 is afunction implicitly represented by 𝑛，𝜔，

+ {1 − (1 − 𝑐 ∗)𝑛−1 }× (Π −≥ Π−𝛼, and 𝛽. Therefore, from the implicit

(12)

1
)
𝜔+𝛼+𝛽

functioninequality is satisfied:

=

1
)
𝜔+𝛼

+ {1 − (1 − 𝑐× (Π −
≤ Π−

(13)

1
)
𝜔+𝛼+𝛽

=−

the

𝛽 −1 (2𝜔 + 2𝛼 + 𝛽)(𝑛 − 1)(1 − 𝑐 ∗ )𝑛−2 (𝑐 ∗ )−1 + (1 − 𝑐 ∗ )𝑛−1 (𝑐 ∗ )−2

1
)
𝜔+𝛼

=−

𝛽 −1 (2𝜔 + 2𝛼 + 𝛽)(𝑛 − 1)(1 − 𝑐 ∗ )𝑛−2 (𝑐 ∗ )−1 + (1 − 𝑐 ∗ )𝑛−1 (𝑐 ∗ )−2

𝜕𝑐 ∗
𝜕𝑓⁄𝜕𝛽
=−
𝜕𝛽
𝜕𝑓⁄𝜕𝑐 ∗

(14)

1
)
𝜔+𝛼+𝛽

𝛽−2 (𝜔 + 𝛼)2=(𝑛 − 1)(1 − 𝑐 ∗ )𝑛−2 (𝑐 ∗ )−1 + (1 − 𝑐 ∗ )𝑛−1 (𝑐 ∗ )−2

1
− 𝑐𝑖
𝜔+𝛼+𝛽

>0

follows:■− (𝜔 + 𝛼 + 𝛽)(𝜔 + 𝛼)𝛽

−1 ∗

𝑐 =0

Let the left-hand side of the above
equation be 𝐺(𝑐 ∗ ) . Then 𝐺(𝑐 ∗ ) is a

Proof of Corollary 1

function that is continuous onand∗∗𝜕𝐺 ⁄𝜕 𝑐 < 0 for any 𝑐 ∈ ℝ. In addition,

respect to 𝑛 is represented by:

The partial derivative of 𝑔 with𝑐∗lim 𝐺(𝑐 ∗ ) = 1

𝑐 ∗ →0andlim 𝐺(𝑐 ∗ ) < 0𝑐 ∗ →1(17)<0Equation (14) can be simplified as(1 − 𝑐

(16)

𝜕𝑐 ∗
𝜕𝑓⁄𝜕𝛼
=−
𝜕𝛼
𝜕𝑓⁄𝜕𝑐 ∗

+ {1 − (1 − 𝑐 ∗)𝑛−1 }∗ )𝑛−1(15)<0= 𝑐∗ ,

following equation holds:= Π−(1 − 𝑐 ∗ )𝑛−1 (𝑐 ∗ )−1 log(1 − 𝑐 ∗ )(𝑛 − 1)(1 − 𝑐 ∗ )𝑛−2 (𝑐 ∗ )−1 + (1 − 𝑐 ∗ )𝑛−1 (𝑐 ∗ )−2

𝜕𝑐 ∗
𝜕𝑓⁄𝜕𝜔
=−
𝜕𝜔
𝜕𝑓⁄𝜕𝑐 ∗

1
− 𝑐𝑖
𝜔+𝛼+𝛽

× (Π −

following<0∗ )𝑛−1 }Hence it follows that, if 𝑐𝑖(1 − 𝑐 ∗ )𝑛−1 × (Π −theinequalities are derived:
𝜕𝑐 ∗
𝜕𝑓⁄𝜕𝑛
=−
𝜕𝑛
𝜕𝑓⁄𝜕𝑐 ∗

1
− 𝑐𝑖
𝜔+𝛼+𝛽

In contrast, if 𝑐𝑖 ≤ 𝑐 ∗ , the following(1 − 𝑐 ∗ )𝑛−1 × (Π −theorem,𝜕𝑔= −𝛽(1 − 𝑐 ∗ )𝑛 {log(1 − 𝑐 ∗ )
𝜕𝑛

hold.

𝑛 𝜕𝑐 ∗−}
1 − 𝑐 ∗ 𝜕𝑛

Hence, there is a unique 𝑐 ∗ ∈ (0,1) thatsatisfies 𝐺(𝑐 ∗ ) = 0.

(19)Equation (19) can be simplified asfollows, by using (15):■Proof of Proposition 1
35

(18)

Suppose that 𝑛 = 2. Compare 𝑐 ∗ ,𝜕𝑔= 𝛽(1 − 𝑐 ∗ )𝑛 log(1
𝜕𝑛
− 𝑐∗)

which is obtained by equation (8), and(20)

2 − (𝑐 ∗ )−1𝑛 − 2 + (𝑐 ∗)−1𝑘 ∗ . Then it is shown that the following
inequality is satisfied:

It is shown that, in the above equation,islog(1 − 𝑐 ∗ )negative,

and

𝑐∗ =

the

denominator 𝑛 − 2 + (𝑐 ∗ )−1 is positive.
Therefore, the sign of 𝜕𝑔⁄𝜕𝑛 depends𝛽(𝜔 + 𝛼)(𝜔 + 𝛼 + 𝛽) + 𝛽< 𝑘∗ =

on the sign of the numerator 2 − (𝑐 ∗ )−1 .

𝛽(𝜔 + 𝛼)(𝜔 + 𝛼 + 𝛽)

Suppose that 𝑛 > 2. Then 𝑐 ∗ < 𝑘 ∗ is alsosatisfied.

■

The result follows from the fact that
Proof of Corollary 2
When𝜕𝑐 ∗ ⁄𝜕𝑛 < 0, as asserted in Proposition 1,

parameterssatisfytheand that threshold 𝑘 ∗ is independent ofcondition that 1 ≤ 𝛽 ⁄[(𝜔 + 𝛼)(𝜔 + 𝛼 + 𝛽)]

the value 𝑛.

all of the

As we have seen, 𝑔 is represented

analysts are sure to acquire signal 𝑥.

by 𝜔 + 𝛼 + 𝛽 − 𝛽(1 − 𝑐 ∗ )𝑛 . On the otherHence, ℎ = 𝜔 + 𝛼 + 𝛽 holds in this case,hand, Corollary 2 shows ℎ = 𝜔 + 𝛼 + 𝛽 −regardless of the value 𝑛. In contrast,𝛽(1 − 𝑘 ∗ )𝑛 such that 0 < 𝑘 ∗ < 1, and ℎ =when parameters satisfy the condition

𝜔+𝛼+𝛽

that

Consequently,

or, in other words,

1 ≤ 𝑘∗ ,𝛽⁄[(𝜔 + 𝛼)(𝜔 + 𝛼 + 𝛽)] < 1other word,

or,

in

𝑘 ∗ < 1, only analysts whose

such

that

from

.

1 ≤ 𝑘∗𝑐∗ < 𝑘∗,thefollowing inequalities are satisfied.

cost of information gathering is lower

𝜔 + 𝛼 + 𝛽 − 𝛽(1 − 𝑐 ∗)𝑛than 𝑘 ∗ acquire signal 𝑥. If all 𝑛 of the< 𝜔 + 𝛼 + 𝛽 − 𝛽(1 − 𝑘 ∗ )𝑛analysts do not collect 𝑥, the precision< 𝜔+𝛼+𝛽That is, 𝑔 < ℎ holds for 𝑛 ≥ 2.

of the investor ’s prediction results in𝜔 + 𝛼. On the other hand, if at least one■

analyst collects 𝑥, the precision of theinvestor’s prediction rises to 𝜔 + 𝛼 + 𝛽.
The probability that the former state
occurs is (1 − 𝑘 ∗ )𝑛 , and the latter state
occurs is

1 − (1 − 𝑘 ∗ )𝑛 .

(Received: January 16, 2018)(Accepted: March 15, 2018)

Consequently, ℎis represented as follows:𝜔 + 𝛼 + 𝛽 − 𝛽(1 − 𝑘 ∗ )𝑛
The partial derivative of ℎ with respect
to 𝑛 is represented by:𝜕ℎ= −𝛽(1 − 𝑘 ∗ )𝑛 log(1 − 𝑘 ∗ ) > 0
𝜕𝑛

(21)
■

Proof of Proposition 2
36