Voluntary Disclosure and the Duty to Disclose

Voluntary Dislosure and the Duty to Dislose Ronald A. Dye February 15, 2013 Abstrat This paper ealuates firms dislosure deisions when they hae a duty to dislose the material information in their possession. We posit that that, een when mandatory dislosure requirements exist, and firms inur penalties when they are aught iolating the requirements, firms engage in ost-benefit alulations in deiding whether to omply with the requirements. In this framework, the paper makes preditions regarding the dependene of a firm s equilibrium dislosure poliy on arious parameters of the model, inluding: the frequeny the firm reeies information, the threshold defining material information, the size of damage payments, the expeted fration of the firm s shares purhased while information is withheld, and the probability the firm is eentually deteted haing withheld material information. The paper onludes by establishing a robust relationship between a firm s equilibrium dislosure poliy and its equilibrium inestment leel. 1 Introdution Many SEC regulations impose affi rmatie duties on firms to make dislosures of their material priate information. For example, SEC Rule 10b-5 states that: "It shall be unlawful for any person, diretly or indiretly, by the use of any means or instrumentality of interstate ommere, or of the mails or of any faility of any national seurities exhange, to...make any untrue statement of a material fat or to omit to state a material fat neessary in order to make the statements made, in the light of the irumstanes under whih they were made, not misleading..in onnetion with the purhase or sale of any seurity." 1 If all publily traded firms dutifully adhered to these and related mandatory dislosure requirements, there would be little alloational role or informational role for oluntary dislosures to play. Voluntary dislosures would beome an addendum, limited to instanes in whih firms deide whether to dislose the priate immaterial information in their possession. Our main thesis is that, in pratie, these mandates do not eliminate all firms disretion in hoosing what material information ("MI") they possess to dislose. 2 Just as posted maximum speed limits don t eliminate some motorists propensity to speed, mandated dislosure requirements don t result in all firms 1 As another example, onsider SEC Rule 12b-20. It stipulates that: "In addition to the information expressly required to be inluded in a statement or report, there shall be added suh further material information, if any, as may be neessary to make the required statements, in the light of the irumstanes under whih they are made not misleading." 2 Sunder [1997] emphasizes the point that the distintion between "mandatory" and "oluntary" ations is often artifiial (see pages nine and ten of Sunder [1997]). 1

obeying those requirements. The plethora of SEC enforement ations and priate iil litigation inoling firms aused of withholding MI is testament to firms exerising their disretion in deiding whether to obey suh requirements. Imperfet monitoring and enforement of dislosure requirements reate sope for disretion in adhering to these requirements, as do limited and unertain penalties for being aught iolating the requirements. While regulatory ations making the dislosure of ertain piees of information mandatory may alter the ost-benefit alulus that firms go through in deiding whether to dislose or withhold the priate information they possess, it does not eliminate the importane or usefulness of ostbenefit alulations to firms in making their dislosure deisions. This paper deelops a model that ealuates expeted alue-maximizing firms dislosure deisions when the firms hae a duty to dislose MI in their possession. withholding MI, they are liable for damages. When firms are aught iolating that duty by The damage payments we examine reflet seeral features of damage payments in 10b-5 and related litigation: (1) the damage payments are due shareholders who purhased, but not shareholders who sold, shares in a firm during the time the firm withheld MI; (2) the damage payments are the produt of: (2a) the differene between the "artifiially high" market prie of the firm that preailed during the time the firm withheld MI and the ("as if") market prie that would hae preailed during this time had the firm dislosed the MI; (2b) the number of shares inestors purhased at the "artifiially high" prie; and (2) a damages "multiplier" determining what fration of inestors oerpayments for the firm s shares are reimbursed; (3) the damage payments are paid by the firm itself, whih effetiely means these payments are paid by the firm s shareholders of reord after the withholding of MI is deteted; and (4) the firm bears additional legal osts, beyond the damages payments, when it is aught withholding MI. The primary goal of the paper is to deelop preditions about how firms dislosure deisions are affeted by mandatory requirements to dislose MI. To make suh preditions, we must onfront the thorny problem of determining how potential purhasers of a firm s shares alue the shares in light of the potential damage payments the firm may hae to pay them if it is subsequently aught withholding MI. This problem arises in part beause inestors who purhase a firm s shares while the firm withheld MI are both potential reipients of the damage payments (as parties "wronged" by the firm s withholding of MI) and also potential payers of the damage payments (as shareholders of the firm responsible for making damage payments to "wronged" parties), and also beause the damage payments both depend on the atual prie of the firm while the MI was withheld (as the damage payments are a funtion of the differene between this atual prie and the "as if" prie of the firm had the firm dislosed the MI) and also help to determine that atual prie (as the antiipation of reeiing damage payments affets inestors assessments of the alue of purhasing the firm s shares). These interdependenies are further exaerbated by the irular nature of the notion of materiality in 10b-5 matters. To desribe the soure of this irularity, we reall that withheld information is onsidered to be material only if a reasonable inestor would iew knowledge of that information to be important in assessing the firm s finanial ondition. 3 Operationally, this means that withheld information is material to 3 While there are many ariations in the way materiality is desribed erbally (e.g., as Oesterle [2011] notes, in one of its early attempts at expliitly defining materiality, the SEC in 1947 in Rule 405 defined material information as those matters 2

inestors only if the differene between the atual prie of the firm while the information is withheld and the "as if" prie of the firm were the withheld information dislosed is suffi iently large. But, sine (as we noted aboe) the prie of the firm while it withheld information depends on inestors prospet of reeiing damage payments, and the prospet of reeiing damage payments depends on whether the withheld information is onsidered material, it follows that whether the withheld information is onsidered material is defined in terms of itself. So, in the ourse of deeloping a model that allows us to make empirially erifiable preditions about firms dislosure deisions in the presene of "duties to dislose," we are fored to resole these irularities that arise in eah of: the definition of materiality, the determination of damage payments, and the speifiation of firms equilibrium pries in the eent they fail to dislose MI. The analysis inludes omparatie statis results that yield preditions about how a firm s propensity to dislose MI it reeies hanges with hanges in: the fration of the firm s shares expeted to be purhased by inestors during the "damages period" (when the MI was withheld), the size of the damages multiplier, the probability some fat finder will determine that the firm withheld MI, the size of the other litigation osts (besides damage payments) the firm inurs, and the definition of what is onsidered to be MI. While some of our findings are intuitie (e.g., firms withhold MI less often as the fration of shares they expet to be traded during the damages period rises), some of our findings are less so (e.g., sometimes, firms withhold MI more often as either the size of the damages multiplier or the probability that some fat finder detets ex post that the firms withheld MI inreases). These ounterintuitie findings arise beause of the preiously mentioned irularities that arise in 10b-5 litigation. Another goal of the analysis is to onnet these empirial preditions regarding a firm s dislosure poliy to the effi ieny of the firm s inestment deisions. Establishing a relationship between a firm s dislosure poliy and its inestment poliy is aluable for its own sake, but is also desirable beause it onnets one endogenous real ariable (the effi ieny of a firm s inestment poliy) that is diffi ult to assess with another easily obsered, and hene empirially measurable, endogenous ariable (the firm s dislosure poliy). main finding here is that any hange in an exogenous parameter of the model that leads the firm to dislose information less (resp., more) often in equilibrium seres to worsen (resp., enhane) the effi ieny of the firm s inestment deisions and hene leads to lower (resp., greater) expeted firm alue. eonomis underlying this result later in the Introdution. As far as we are aware, these results are new to the dislosure literature. Our We disuss the In the remainder of this introdution, we preiew a sample of our results - those related to hanges in the materiality threshold ("MT") that determines whether the information a firm withholds is material and hene whether the subsequent disoery of the firm s withholding of this information subjets the firm to to whih there is a substantial likelihood that a reasonable inestor would attah importane"), and the notion of materiality has hanged oer time (as desribed in the next footnote), the substane of "materiality" in all these ariations, as far as we are aware, always inoles what impat inestors knowledge of the information would hae on inestors pereptions of a firm s finanial ondition. 3

legal liability. 4 5 Our main finding here is to show that the dislosure "utoff" that determines whether the firm seeks to dislose or withhold the priate information it reeies is not monotoni in the MT. Speifially, we show that when the MT is suffi iently low, the utoff rises, so an expeted alue-maximizing firm is less likely to dislose its priate information if the MT is raised somewhat, but if the MT is suffi iently high, the utoff falls, so an expeted alue-maximizing firm is more likely to dislose its priate information if the MT is further inreased. What explains these results? When the MT is initially ery low and then is raised somewhat, the situation firms fae hanges from one in whih irtually all the information they reeie is onsidered material - and hene subjets them to potential liability for withholding that information - to a situation where they hae to opportunity to withhold some information without being penalized for the nondislosure. Firms naturally exploit this expanded opportunity to withhold some information without being subjet to penalty that, in a world where irtually all the information they reeie is onsidered material, they would dislose. But, when the MT is already high, a firm has onsiderable disretion in deiding what information to dislose or withhold without fear of faing legal reprisals for withholding information. Inestors onsequently beome onerned that, in suh an enironment, a firm s lak of dislosure is more likely to be attributable to the firm haing reeied but withheld unfaorable information rather than being attributable to the firm not haing reeied information. If the MT inreases from this already high leel, inestors beome progressiely more skeptial about the motiation underlying a firm s nondislosure, and so they disount the prie of nondislosing firms more the higher the MT rises. The result that firms dislose more when the MT rises from an already high leel is a refletion of the fat that firms respond rationally to inestors inreased disounting of nondislosing firms shares by dislosing more of the priate information they reeie. In addition to generating empirial preditions about the relationship between the MT and a firm s dislosure deisions, this finding also leads to preditions about the effets of hanging the MT on a firm s equilibrium inestment leel and its expeted market alue. We show that wheneer the MT goes up (resp., down), the firm s equilibrium inestment leel and its ex ante market alue goes down (resp., up) proided the MT starts at a suffi iently low (resp., high) leel. Alternatiely put, if a hange in the MT threshold results in the firm s equilibrium "no dislosure" prie going up, that same hange in the MT results in the firm s equilibrium inestment leel and (ex ante) equilibrium market alue going down. We show that this latter phenomenon is general and robust: for any parameter of the model, not just for the parameter desribing the MT, a (ommon knowledge) hange in the parameter that results in the firm in equilibrium dislosing its information less (resp., more) often results in the firm s equilibrium inestment and its ex ante market alue dereasing (resp., inreasing). This outome ours beause when a parameter hange results in a hange in the firm s equilibrium "no dislosure" prie, this prie hange is a result of a hange in inestors inferenes about its alue, rather than being a result of a hange in objetie eidene 4 Sine the interpretation of materiality has hanged oer time, deeloping omparatie statis inoling materiality is not just an aademi exerise. For example, Arthur Leitt s 1998 speeh at NYU, "The Numbers Game," was onsidered to hae hanged the notion of what onstitutes material information. (See, e.g., Zabel and Benjamin [2002]). Related, questions inoling the desirability of hanging materiality threshholds frequently hae arisen in poliy disussions (see, e.g., AICPA [1997] CPA Letter Supplements (Finane and Aounting setion) at http://aipa.om/pubs/paltr/no97/suppl/fin.htm or the Enironmental Law Institute s News and Analysis at http://elr.radampaign.om/sites/default/files/artiles/35.10666.pdf). 5 This is just one illustration of our results. We leae the disussion of the eonomi effets of hanging the other parameters of the model to the body of the text. 4

dislosed about its alue. Sine a alue-maximizing firm will dislose the information it reeies only when dislosure results in a higher market prie for the firm than does nondislosure, an inrease in the firm s "no dislosure" prie reates a higher hurdle the realized alue of the firm s priate information must exeed to make dislosure of that information worthwhile for the firm. Suh a hange thereby expands the set of the irumstanes under whih the firm will make no dislosure - and hene expands the set of irumstanes under whih the firm is pried based on inestors inferenes about the firm s alue (as opposed to being based on onrete information the firm disloses about its alue). At the time a firm makes its inestment deision, it antiipates this effet - that its future market alue will be determined more often by inestors inferenes about the firm s alue than based on the firm s dislosures about its alue - and so it responds to a hange in a parameter that inreases its "no dislosure" prie by inesting less. This linkage between the effet of hanging a parameter on a firm s equilibrium propensity to dislose information and the effet of hanging that same parameter on the firm s equilibrium inestment hoies might seem obious, or at least intuitie. But, this linkage stands in striking ontrast to the onlusion of many extant models in the literature on endogenous dislosure poliies that inreased (that is, either more preise, or more frequent) dislosures lead to redued equilibrium firm market alues and (weakly) redued equilibrium inestment leels. 6 In summary, the paper makes the following ontributions. First, it determines firms equilibrium dislosure poliies in the presene of a "duty to dislose." Seond, it deries a ariety of empirial preditions related to firms equilibrium dislosure poliies. Third, it determines firms equilibrium pries when firms are obligated to pay damages when they are aught withholding MI, where the damage payments are determined in a "10b-5 fashion," i.e., by the extent to whih firms pries during the period the MI was withheld are oerstated. resoled. inestment deisions. Fourth, it shows how arious reursions that result from 10b-5 damage payments an be And fifth, it shows how firms equilibrium dislosure poliies are linked to their equilibrium In terms of related literature: Heitzman, Wasley and Zimmerman [2010] is one of the few ontemporary papers that studies the empirial impliations of firms dislosure deisions in the presene of requirements to dislose material information. Apart from the fat that HWZ is an empirial study and the present paper is a theoretial study (albeit one whose goal is to make empirial preditions), HWZ and the present paper adopt ery different oneptual approahes to addressing firms dislosure deisions when mandatory dislosure requirements are present. Speifially, HWZ proeed by assuming that all firms obey all mandated requirements to dislose material information, whereas our approah, as noted aboe, is premised on firms undertaking ost-benefit alulations in deiding what information to dislose, regardless of whether the information is stipulated to be dislosed under some mandatory dislosure requirement. Other papers onerning materiality and dislosures inlude Patillo [1976] and Ward [1976] and Messier, Martino-Bennie, 6 For example, Pae [1999] notes that firm alue in his model is highest when the preision of the priate information the firm s manager an aquire and dislose is lowest (whih is akin to dislosing the least amount of information possible in the present model), beause that redues the inenties of the manager to waste resoures on aquiring and dislosing the information. In a different model, Pae [2004] obseres that the manager s equilibrium effort hoie in his model, whih is akin to the manager s equilibrium effort leel in the present model, is first-best, independent of the manager s propensity to make dislosures. As a third example, expeted firm alue in Verrehia [1990] dereases in the frequeny with whih a firm in his model makes a dislosure, as Pae and others hae obsered preiously (see, e.g., remarks by Pae [1999] at p. 474). 5

Aasmund Eilifsen [2005]. Pae [2005] identified some of the issues that would hae to addressed were the existing literature on oluntary dislosures extended so as to inlude legal penalties for failure to dislose, 7 but Pae [2005] did not himself ondut suh a study. Finally, our paper is premised on the idea that firms at least oasionally reeie priate information pertinent to their aurate priing whih, sometimes, they do not dislose in equilibrium. For this premise to be alid, a setting must be onsidered where the "unraelling" results of Grossman [1981] and Milgrom [1981] do not apply. Dye [1985] and Jung and Kwon [1988] identified a natural setting in whih firms in equilibrium seletiely dislose the information they reeie, whih subsequently has been adopted by other researhers in their examination of firms oluntary dislosure poliies (see, e.g., King and Wallin [1991], Pae [2005], Shaell [1994]), based on the simple notion that inestors are often unertain what priate alue-releant information firms hae in their possession. We make use of this natural setting here. The paper proeeds as follows. The next setion, setion 2, introdues the base model. In the base model, inestment is taken as exogenous, and damages related to the withholding of MI are assessed against the firm when a fat finder subsequently disoers the firm withheld MI from inestors. In setion 3, the equilibrium of the base model is defined. Setion 4 analyzes the equilibrium of the base model. Setion 5 extends the base model to inlude endogenous inestment. Setion 6 ontains onlusions. The Appendix ontains the proofs of most results not proen in the text or aompanying footnotes. 2 Introdution to the Base Model 8 At time 0, a firm knows that, by some later time 1, with probability p it will learn priately the realization x of some alue-releant random ariable x with umulatie distribution funtion ("df") L(x), density l(x), and support [x, x]. information by time 1. At time 0, the firm also reognizes that with probability 1 p, it will reeie no If the firm learns the realization of x, it may dislose that information or dislose nothing at time 2 (a time greater than time 1). Dislosing or withholding x are the firm s only options. It annot, for example, "partially" dislose the information it reeies, and it annot distort the information it reeies before dislosing it. 9 If the firm does not learn information, it neessarily makes no dislosure. If the firm reeies information but does not dislose it by time 2, it neer disloses it (at least within the time horizon onsidered in the model). The goal of the firm in hoosing its dislosure poliy is taken to be to maximize the (per share) prie of the firm at time 2. There is some stritly inreasing funtion (x) representing the disounted expeted present alue of the firm s ash flows per share (gross of any damage payments) as of time 2. If the firm 7 See Pae [2005], pp. 402-403. 8 We introdue the base model in the text in as streamlined a manner as possible, with a minimum of asides, leaing to the aompanying footnotes ommentary on the realism of the assumptions, alternatie ways of formulating the model, et. 9 The presumption here that the firm would not partially dislose its information need not be assumed: it an be deried as a feature of any equilibrium (beause any inomplete dislosure must be interpreted by inestors as skeptially as possible in equilibrium, so a firm that dislosed that it reeied information by time 1 would be obliged to dislose the exat alue of its information to aoid unfaorable priing by skeptial inestors; this is a manifestation of the "unraelling" result of Grossman [1981] and Milgrom [1981]). Also, the presumption that the firm annot distort the information it reeies is based on the impliit premise that antifraud statutes are suffi iently great and suffi iently well enfored so as to deter distorted dislosures. For a model that studies biased oluntary dislosures, see, e.g., Einhorn and Zi [2012]. 6

disloses x, eah share is pried at (x). 10 If the firm disloses nothing at time 2, the prie of eah share, denoted by P nd, is the expeted alue of (x) net of any expeted damage payments, as desribed below, the firm may be liable for. P nd is endogenously determined. All trading in the firm s shares takes plae at time 2. Short sales are prohibited. 11 Neither the firm nor the firm s managers trade in the firm s shares. 12 learn information about the firm only through the firm s dislosures. All traders are risk-neutral, homogeneously informed, and Traders are of two types: "liquidity" traders whose trading deisions are exogenous and random (as of time 0), and many "rational" traders who buy or sell shares at time 2 depending on whether the firm s time 2 prie is below or aboe their pereptions of the expeted alue of ( x) adjusted for any expeted damage payments. 13 The total fration of the firm s shares purhased by liquidity (resp., rational) traders at time 2 is desribed by the realization of the random (resp., deterministi) ariable f l (resp., f r ). f l (resp., f r ) is exogenous (resp., endogenous) and in partiular independent (resp., dependent) on the prie P nd of the firm in the eent the firm makes no dislosure. Gien realization f l of f l and f r, the total fration of the firm s shares purhased at time 2 is denoted by f f l +f r. If the firm withholds information at time 2, there is a fat finder - e.g., auditor, regulator, reporter, et. - who, by time 3 (a time later than time 2) learns with probability q that the firm withheld information at time 2. When the fat finder learns the firm withheld information, the fat finder also learns, and disloses to the inesting publi, the alue of the withheld information. type of error: The fat finder is assumed to make only one the fat finder may fail to detet the firm possessed undislosed information at time 2, but he does not err by laiming the firm withheld information at time 2 when it did not. The firm is assumed to be liable for damages if and only if the fat finder detets that the firm withheld MI at time 2. 14 Withheld information x is deemed material if the differene between the atual time 2 prie of a firm s share when the firm makes no dislosure, P nd, and what the time 2 prie of a share would hae hae been had x been dislosed, (x), exeeds materiality threshold ("MT") m. That is, withheld information is material if and only if the following inequality holds: P nd (x) m. In the following, we take the MT m to be fixed 15 and publi information. 10 Consistent with the preeding desription, unless something expliitly to the ontrary is stated, all firm alues should be taken to be amounts per share. 11 Either simply as an exogenous restrition on trades or beause there is no follow-up market after time 2 on whih time 2 short positions an be oered. 12 As we are not interested in fomulating or studying a model of insider trading. In any eent, suffi ient large penalties for inside trading would prohibit insiders from engaging in suh trading, absent dislosure of their priate information. 13 More preisely, there are presumed to be enough rational traders so that the equilibrium prie is determined by their pereptions of the expeted alue of buying a share in the firm. (This assumption might seem to render the introdution of liquidity traders into the model superfluous. But, it will be lear from the disussion below that the presene of liquidity traders does in fat affet the equilibrium priing of the firm s shares beause the expeted damage payments the firm will owe inestors who purhase shares while material information is withheld will ary with the ombined number of shares purhased by both rational and liquidity traders.) 14 There are interpretations of SEC rule 10b-5 under whih firms are not required to dislose all the new material information in their possession. Rather, firms are required to dislose the new material information they reeie only if their preious dislosures of related information would result in inestors haing a misleading interpretation of a firm s finanial ondition absent "orretie" dislosures based on the new information. If one subsribes to this interpretation, the model in the text still applies without hange, exept that the distribution of the random ariable x of the firm s priate information must now be understood to be that of the distribution of x onditional on whateer information related to x the firm preiously dislosed. We would like to thank Katherine Shipper for this obseration. 15 In a working paper we allow m to be desribed by a random ariable, and we obtain results similar to those we report 7

When the fat finder disoers that the firm withheld MI, the firm is obliged to pay all of the inestors who purhased the firm s shares at the prie P nd the damage payment d(p nd (x)) per share, where d > 0 is a "damages multiplier" determined by a judge, some dispute resolution mehanism, et. 16 Sine there is substantial doumentation that "wronged" shareholders are reimbursed for only a fration of their atual losses, 17 we take d < 1 throughout the following. Gien f and the firm is subsequently found to hae withheld MI x, it follows that the the damage payments redue the per share alue of the firm by d f (P nd (x)). When the firm is liable for damages, it is posited to inur legal osts aboe and beyond the damage payments themseles. If there are in total n shares of the firm outstanding, the firm s total legal osts are taken to be the sum of osts that ary with the number of shares purhased at time 2, k n f, and osts that ary with the total dollar alue of the damage payments, t n d f (P nd (x)), for some onstants 18 19 20 k, t 0. 3 Definition of Equilibrium in the Base Model A firm s dislosure poliy is desribed by a set ND of the realizations of the firm s priate information x that, when reeied, the firm elets not to dislose. Sine the firm s goal is posited to be to maximize its below (exept for those omparatie statis below onerning the effets of a hange in the (here deterministi) materiality threshhold). 16 Note that een if liability rules were hanged so that damage payments in priniple ould be awarded sellers, as well as purhasers, of the firm s shares, no sellers of the firm s shares would be awarded damage payments in this model, beause sellers who sell shares while the firm withholds information reeie more, not less, for those shares than they would had the firm dislosed the withheld information (beause P nd > (x) for all x ND). 17 See, e.g., Ryan and Simmons [2009]. 18 Obiously, these total legal osts orrespond to per share osts of kf and tdf(p nd (x)) respetiely. 19 In this footnote, we make seeral omments onerning the orrespondene between damage payments as portrayed in the model and atual damage payments in 10b-5 and related seurities litigation. First, onsistent with the "fraud on the market" theory of damages (Basi, In.. Leinson, 485 U.S. 224 (1988)), inestors purhasing shares at time 2 need not demonstrate reliane on the firm s dislosures (or, if the model were embellished to inlude them, the firm s finanial reports) to be awarded damage payments. All that is neessary for an inestor to be entitled to reeie damage payments is that the inestor demonstrate he purhased shares during the period in whih the firm withheld MI from the apital market. Seond, that the magnitude of the assessed damages is proportional to the differene between the "artifiially high" prie of the firm arising from the firm haing withheld MI and the "as if" prie of the firm were the firm to hae dislosed the MI, seems onsistent with pratie (see, e.g., Booth [2009]). Also, the model defines damages in terms of the losses of inestors who purhased shares at time 2. The profits that shareholders who sold shares while the firm s alue was oerstated are disregarded, een though arguments hae been made that, on effi ieny grounds, suh shareholders should be required to disgouge their profits (see, e.g, Posner [2007], at p. 483). Third, the representation of materiality is "output-based" in the sense that whether the withheld information is deemed material is determined by the amount by whih the prie of the firm would hae hanged had information withheld by the firm been dislosed. This representation fits with the U.S. Supreme Court s notion of materiality as reported, for example, in the ase of Basi In.. Leinson, where the Court found that "materiality depends on the signifiane the reasonable inestor would plae on the withheld or misrepresented information." That is, omitted information is onsidered to be material if its dislosure would hae had a suffi iently signifiant influene on the firm s prie. (In ontrast, an "input-based" notion of materiality would be desribed by riteria suh as: earnings were oerstated by x% or total assets were understated by y%, et., without regard to how muh the market alue of the firm hanges with those perentage hanges in the firm s earnings, assets, et.) Fourth, when a firm is found to hae withheld MI, the damages paid to the shareholders who purhased shares at the "artifiially high" prie P nd established by the apital markets are ompensated after the fat finder has disoered the material withholding. That is, in effet the shareholders who pay for these damages are not the shareholders of reord at the time the tort (of withholding information) was ommitted, but rather are the shareholders of reord as of (or shortly after) the time the tort is disoered. While it may be argued that better inenties to indue the firm to dislose its information would be ahieed were the former set of shareholders made liable, the impratiality of identifying those former shareholders and making them pay is generally thought to rule out that alternatie as infeasible. Fifth, while we do not bother to adjust the notation to reflet this, we note that many of the parameters in the model ould be replaed by random ariables the realizations of whih are not known at the time the firm must make its dislosure deision without affeting the results below. 20 A numerial example of the effets of the damange payments and legal osts is presented in footnote 23 below. 8

expeted market alue as of time 2, and sine (x) is taken to be monotone inreasing in x, it is lear that if a firm prefers to dislose information x it reeies at time 2, it also prefers to dislose any information x > x too. for some x [x, x]. 21 Thus, the "no dislosure" set ND is of the onentional "utoff" form: ND = {x x < x } (1) We simplify notation somewhat in the following by introduing the transformed random ariable ṽ = ( x) with df G() Pr(ṽ ), density g(), and support [, ] [(x), ( x)]. 22 We assume that the density g( ) of ṽ is ontinuous on its support. In this new notation, the "no dislosure" set ND is gien by ND = { < } for some [, ], where = P nd. Inestors who purhase shares of the firm at time 2 when the firm makes no dislosure take into aount both their pereptions of ṽ and the expeted damage payments they might reeie (in the eent the fat finder subsequently determines that the reason the firm did not dislose information was that it was withholding MI) when deiding how muh to pay for the firm s shares. In the eent the fat finder subsequently determines that MI x with assoiated alue = (x) was withheld, buying a share at prie P nd at time 2 entitles an inestor to the sum of (i) the damage payment per share of d(p nd ) and (ii) a share of the firm whih - after paying damages to the fration f of traders who bought shares at time 2 and also after paying the other legal osts - is worth (per share): df(p nd ) k dtf(p nd ). It follows that when the fat finder disoers that the firm withheld MI, the "ex post" alue of one share of the firm to an inestor who purhases the share at time 2 is: 23 d(p nd ) + df(p nd ) k dtf(p nd ) = + d(1 f(1 + t))(p nd ) k. (2) To identify the equilibrium relationship between ND and P nd, we first proide some additional detail regarding traders assumed trading behaior. Rational traders are assumed to know the distribution, but not the realization, of the fration f l of shares that will be purhased by liquidity traders at time 2 before they submit their own demands for the firm s shares then. Rational traders make a onjeture about the utoff the firm used in deiding whether to dislose its information, and they also make a onjeture about the aggregate fration f r of shares purhased by rational traders at time 2. In equilibrium, these onjetures are orret. The apital market is assumed to be ompetitie in that rational traders pereie their indiidual trading deisions to hae no impat on either P nd or f r. 21 Without loss of generality, we assume that if the firm is indifferent between dislosing its information and not dislosing it, the firm disloses. 22 Realling that L(x) denotes the df of x, it follows that G() = L( 1 ()). Inidentally, the results that follow also hold for a random ariable ṽ with infinite support proided that its mean E[ṽ] is finite and its density g( ) is ontinuous and bounded on its support. 23 The following numerial example may help to larify (2). Suppose k = t = 0 and there are in total 100 shares of the firm. Suppose also that the prie per share when x = x, would hae been (x) = $30 were x dislosed. Also suppose that if the firm does not dislose x, then the no dislosure prie is P nd = 50, that d =.6, and that 25 shares are purhased at time 2. If the fat finder disoers the firm withheld x, then the damages paid per share are.6 (50 30) = $12, for a total of $300 = 12 25. Aordingly, the total alue of the firm at time 3, after payment of damages, is 100 30 300 =$2700, or $27/share. Alternatiely, one an alulate the ex post alue of the firm per share diretly as in the text as: (x) df(p nd (x)) = $30.6.25 (50 30) = $27 If an inestor who intended to purhase a share of the firm at time 2 antipated that the firm was withholding information whih, if dislosed, would result in its alue per share being $30, and so ould ollet damages of $12 per share, the inestor would be willing to pay $39=27+12 per share (or, alternatiely omputed ia (2) as $30+.6.75 (50 30) = $39). 9

EV F (, f r ) denotes a rational trader s pereptions of the expeted alue of purhasing a share of the firm at time 2 gien his onjetures about f r and, and his knowledge of the distribution of f l. EV F (, f r ) is omputed by employing Bayes rule; the details are in Appendix A. There, we show that: EV F (, f r ) = pg( m )q 1 p + pg( ) E[ṽ + d (1 (f r + f l )(1 + t)) (P ND ṽ) k ṽ < m ]+ (3) (1 pq( )G( ( ) 1 p + pg( ) ) p(g( ) G( m ))q 1 p + pg( E[ṽ m ṽ < ]+ ) G( )p(1 q) 1 p + pg( ))(1 q) E[ṽ ṽ < ] + (1 G( ) )p(1 q) 1 p + pg( )(1 q) ) E[ṽ]. Holding and f r fixed, the first line of this expression is the probability at time 2 the firm will subsequently be found by the fat finder to hae withheld MI, gien that the firm makes no dislosure at time 2, times the onditional expeted alue of a share in the firm gien the firm is found haing withheld MI. The seond line is the probability the firm will be found by the fat finder to hae withheld immaterial information at time 2, gien that the firm makes no dislosure at time 2, times the onditional expeted alue of a share in the firm gien that the firm is found haing withheld immaterial information. The final line is the probability the fat finder will not find that the firm withheld any information at time 2, gien that the firm makes no dislosure at time 2, times the onditional expeted alue of a share in the firm gien that the fat finder fails to detet the firm withheld any information at time 2. below. This speifiation of EV F (, f r ) is entral to the definition of an equilibrium of the model, presented Definition 1 A ompetitie equilibrium of the base model where some rational traders hoose to own shares of the firm at time 2 onsists of: (a) a utoff ; (b) a no dislosure set ND = { < }; () the prie P nd in the eent the firm makes no dislosure; (d) the sum f r of the frations of shares purhased at time 2 by all rational traders who purhase shares at time 2; and (e) prie P (x) if the firm disloses x, suh that: (i) P (x) = (x) for all x; (ii) P nd = EV F (, f r ); and (iii) = P nd. The definition requires that: (i) P (x) = (x) wheneer the firm disloses x; (ii) the prie of a share when the firm makes no dislosure, P nd, equals the alue of the ash flows a trader expets to reeie from purhasing a share of the firm at time 2 net of all potential damage payments and related legal osts; (iii) the utoff the firm uses in deiding whether to dislose information it reeies at time 2 equals P nd. 24 4 Analysis of the Base Model In the introdution to the paper, we noted that to determine both the equilibrium prie of the firm and the size of the damage payments in 10b-5 and related litigation settings one has to take into onsideration seeral irularities and reursions that naturally arise in suh settings. Three speifi irularities that must be 24 If a firm makes no dislosure, the expeted alue a urrent shareholder gets from selling a share of the firm at time 2 is less than EV F (, f r), beause of the feature of 10b-5 and related litigation that sellers of a firm s shares, unlike purhasers of the firm s shares, do not ollet damage payments in the eent the fat finder subsequently reeals that the firm withheld material information at time 2. Notwithstanding this differene, the equilibrium prie of a share in the eent the firm makes no dislosure, P nd, must equal EV F (, f r) if any rational traders own any shares in the firm in equilibrium, sine any lower prie would result in an arbitrage opportunity for rational traders (they would buy the shares as long as the expeted alue of what they purhased exeeded the purhase prie) and any higher prie would deter all rational traders from possessing any shares in the firm. 10

taken into aount are: 1. the dependene of the size of damage payments on the firm s equilibrium prie, and the dependene of the firm s equilibrium prie on the size of damage payments; 2. the dependene of inestors pereptions of the alue of a share of the firm on their iew of the utoff the firm uses to define its dislosure poliy at the same time the utoff the firm uses to define its dislosure poliy depends on the firm s iew of inestors pereptions of the alue of a share of the firm; and 3. materiality 25. All of these irularities and reursions are embedded in the single equation the irular definition of = EV F (, f r ), (4) whih follows diretly from the definition of an equilibrium in the base model. By substituting (3) into (4) and simplifying, in the next lemma we rewrite equation (4) in a way that failitates analysis of the equilibrium of the base model. Lemma 1 Fix f r and set f ( fl + f r )(1 + t). Then, the equation E[ṽ] = p [ G()d pq 1 p 1 p ((1 f)d m k)g( m ) + (1 f)d is a rearrangement of the equation EV F (, f r ) =. The proof is in Appendix B. m G()d To see how (5) helps to identify features of an equilibrium for a partiular example before we proeed to obtain general omparatie statis, we onsider the solution to equation (5) when ṽ is uniformly distributed on [0, ] (so G() = for [0, ]). When both and m are in [0, ], then equation (5) is the same as: /2 = p [ 1 p d pq ] 1 p ((1 f)d m k) m + (1 f)d m d. (6) After performing some algebra, one an show that, when 1 > f and solution of (6) is gien by: 27 = ( 1 p p + k) + ( 1 p p ] (5) 1 p 1+ 1 p > m, 26 the unique positie + k)2 1 p + (1 qd(1 f))( p 2 qd(1 f) 2 m + 2k m ) 1 qd(1 f). (7) While arious omparatie statis an be deried from this expression for the utoff, the only one we mention presently is one inoling m. Viewed as a funtion of m, the quadrati qd(1 f) 2 m + 2k m is obiously not monotoni in m, and so it follows from (7) that itself is not monotoni in m. More preisely, (7) shows that k inreases in m as long as qd(1 f) > m, and dereases in m for m > k. The explanation for this nonmonotonity was disussed in the Introdution. We show below that qd(1 f) this nonmonotoniity is a feature of the equilibrium utoff for any distribution for ṽ, not just for ṽ that is uniformly distributed. In the next setion, we derie general preditions about the equilibrium of the base model. 28 25 See the fifth paragraph of the Introdution. 26 The latter ondion ensures that m > 0. 27 Deriing this expression simply entails soling a quadrati equation, and piking the positie root. Aordingly, the demonstration of the alulations leading to this expression are omitted. 28 Beause tehnial issues related to the existene and uniqueness of an equilibrium need not be broahed in order to either make or understand preditions deried from the model, we forego a detailed disussion of these tehnial issues here. (These issues are disussed in a working paper aailable from the authors.) We note here only that, as is often the ase in priing models where risk-neutral traders are present, that the equilibrium total fration f r of shares purhased by rational traders is not uniquely determined. Aordingly, in some of the results that follow, we ondition the results on the size of f r. 11

5 Analysis of the base model The following general preditions emerge from the base model. (In the statement of the theorem, we fix f r and set f ( f l + f r )(1 + t).) Theorem 1 (The Comparatie Statis of the Equilibrium Cutoff ) If either (a) f 1 or (b) f < 1 and m < 1 p pb, then the equilibrium no dislosure prie/equilibrium utoff is: (a) stritly dereasing in the probability p that information arries at the firm; (b) stritly dereasing in eah of f, t, and k. () stritly dereasing in the materiality threshold m when (1 f)d m k > 0 and stritly inreasing in the materiality threshold m when (1 f)d m k < 0; (d) stritly inreasing in the damages multiplier d when f < 1 and stritly dereasing in the damages multiplier when f > 1; (e) stritly inreasing in the probability q the fat finder disoers that the firm withheld MI when (1 f)d m > k and stritly dereasing in this probability when f > 1. 29 The proof of Theorem 1 is in Appendix C. Theorem 1(a) onfirms and extends a result known from the original Dye[1985] and Jung and Kwon[1988] models: if inestors know that information arries at a firm more often (i.e., p inreases), they will beome more skeptial of a firm that does not dislose information, and they redue the firm s equilibrium "no dislosure" prie aordingly. Theorem 1(a) douments that this result, whih was known preiously to be true only when all dislosures are oluntary, extends to ases where firms are subjet to penalties for failing to dislose MI. Theorem 1(b) is also lear: as f inreases, and so more inestors are antiipated to buy the firm s shares and hene beome eligible to ollet damages from a firm that withholds MI, a alue-maximizing firm beomes less inlined to withhold the information it reeies. inreases in either of the litigation ost parameters t or k. The same fores also apply with respet to Theorem 1() shows that the nonmontoniity of the equilibrium no dislosure prie/utoff in MT exhibited by the uniform distribution disussed aboe around (7) is a general phenomenon. What is perhaps surprising is that the MT leel m = k (1 f)d at whih the utoff hanges from inreasing in m to dereasing in m is exatly the same in the general ase - for an arbitrary distribution- as it was for the uniform distribution disussed aboe. Theorem 1(d) reports that hanges in the damages multiplier also has non-monotoni effets on the inenties of a firm to dislose its priate information. If f < 1, inreases in the damages multiplier auses firms to withhold information more often. On its fae, this is a surprising result: inreases in the damages multiplier inrease the penalty/liability the firm must pay in the eent it is aught withholding information, and so it might seem that firms would respond by withholding information less often. To explain the result, note that inreasing the damages multiplier d inreases the "ex post" alue of purhasing a share of the firm 29 Note that there is a "hole" in this last haraterization of omparatie statis, that is the set f > 1 and the set (1 f) β m > k, while mutually exlusie, are not exhaustie. As inspetion of the proof of this result shows, in the omitted region of parameter spae, omparatie statis appear to be ambiguous. 12

(2) in the eent the firm withholds MI when f < 1. 30 So, inreasing the damages multiplier when f < 1: (1) inreases the alue of a share of the firm in the eent the firm makes no dislosure, and (2) has no impat on the alue of a share of the firm when the firm disloses its information. Combined, these last two effets must indue the firm to dislose MI less often as d inreases. When f > 1, 31 this effet reerses itself: then, the ombination of payouts to other shareholders and payouts in other legal osts are so large that inreases in the damages multiplier under these onditions lead to a derease in the alue inestors at time 2 are willing to pay for the firm s shares in the eent the firm makes no dislosure. In turn, this last effet indues the firm to dislose its MI more often. As Theorem 1(e) reports, an inrease in the probability the fat finder detets the firm withheld MI may result in either an inrease or a derease in the firm s propensity to dislose information it reeies, depending on the speifi alues of arious the parameters of the model. This ambiguity is also, at first blush, surprising, as it might seem that improements in the fat finder s ability to detet the firm withheld MI would always result in the firm withholding information less often. The ambiguous effet arises beause inestors who buy the firm s shares at time 2 are both payers of damage payments (as shareholders of reord at time 3) and reipients of damage payments (as members of the lass of shareholders entitled to reeie damage payments in the eent the fat finder disoers the firm withheld MI). If the payouts to others beome too large (speifially, if f > 1) when the firm is aught withholding MI, the alue of a share in the firm to a trader purhasing the share drops as the fat finder s effetieness inreases, whereas if the payouts to others are suffi iently small (speifially i.e, f and k are suffi iently small so that (1 f)d m > k), the amount traders are willing to pay for a share of the firm when the firm makes no dislosure is inreasing in the fat finder s effetieness. explained in the same way it was in the disussion of Theorem 1(d): The dependene of the firm s dislosure poliy on a hange in q is now the firm beomes more (resp., less) inlined to dislose its information when q inreases when parameter alues hange so as to derease (resp., inrease) the alue inestors attah to the firm s shares when the firm makes no dislosure. Theorem 1 also impliations for both the diretion and size of the firm s prie reation to the nondislosure of information at time 2. In partiular, it implies that there will be a negatie prie reation attending the lak of a dislosure by the firm at time 2, assuming the parameters determining the litigation osts (k and t) are suffi iently small. To see this, first note that when k = t = 0, then een though there are damage payments in the model, those damage payments are between groups of shareholders in the firm, and hene the damage payments are, on net, zero sum. Therefore, the prie of the firm at time zero is just the firm s unonditional expeted alue E[ṽ]. When k and t are suffi iently small, it follows that the prie of the firm at time 0 will be nearly E[ṽ] too. Next notie that k 0 and 1 > q(1 f)d (learly, the latter inequality 30 To see this, just substitute f for f(1 + t) in (2) and reall that P nd > for all for whih the firm prefers not to dislose its information, to onlude that the "ex post" alue + d(1 f)(p nd ) k inreases in d when f < 1. 31 Reall f ( fl + f r)(1 + t), so it is entirely plausible for f > 1. 13

always holds sine q < 1, d < 1 and f 0) imply: = RHS(5) p 1 p p 1 p p 1 p ( ( G()d pq(1 f)d 1 p [ m G( m ) + m G()d m G( m ) m G()d m G( m ) ) m G()d ) G()d ] = p m 1 p (G(ψ) G( m )) > 0, (8) (here, ψ is a number (by the mean alue theorem for integrals) in the interal ( m, )). 32 LHS(5) = RHS(5) in equilibrium, and sine we just demonstrated that RHS(5) > 0, it follows that LHS(5) = E[ṽ] > 0 too. Therefore, sine E[ṽ] is the time 0 prie of the firm, and is the prie of the firm at time 2 when the firm makes no dislosure, we onlude there is a negatie prie reation to no dislosure at time 2. Furthermore, sine E[ṽ] does not hange with hanges in the parameters of the model, whereas - as Theorem 1 reports - does hange with hanges in the parameters of the model, Theorem 1 s onlusions immediately translate into omparatie statis about the size of the negatie prie reation to the firm s nondislosure at time 2. Sine For example, Theorem 1() implies that the negatie prie reation to nondislosure will be bigger as the MT m inreases if (1 f)d m k > 0 and the negatie prie reation will be smaller as the MT m inreases when (1 f)d m k < 0. The omparatie statis reported in Theorem 1(a), (d), and (e) similarly an be translated into statements about the size of the prie reation to the firm s nondislosure at time 2. 6 Expanding the model to inlude endogenous inestment In this setion, we extend the analysis of the preeding setion by onsidering some of the real effets of oluntary dislosures when firms hae a duty to dislose the material information they reeie. 33 We do this by modifying the model of the preious setion by making the distribution of ṽ depend on the firm s time 0 endogenous inestment hoie. Our primary goal in deeloping this extension is to onnet the (not easily obsered) effi ieny of firms equilibrium inestment hoies with obserable, and empirially testable, preditions deeloped in preeding setions regarding firms equilibrium dislosure poliies. Our main finding is that hanges in exogenous parameters that lead firms to engage in more (resp., less) dislosure lead to inreased (resp., redued) inestment effi ieny. Formally, we now suppose that at time 0, the firm under inestigation priately hooses an inestment 32 We know that ψ ( m, ), as opposed to simply ψ [ m, ], sine G(), as a df, is stritly inreasing oer its support. 33 The idea that finanial reporting and dislosure deisions should be ealuated in terms of their real effets, and in partiular, in terms of their effets on a firm s equilibrium inestment hoies, was initiated by Kanodia: see, e.g., Kanodia [1980], Kanodia and Lee [1998], and Kanodia and Mukherji [1996]. 14

leel I 0. 34 The ost of inestment I is $I. The df of ṽ is affeted by this inestment hoie, whih we indiate by writing G() = G( I); the assoiated density is written g( I), with support [, (I)]. 35 All other aspets of the model of this setion oinide with the model deeloped in prior setions. We assume that the family of dfs {G( I) I 0} indexing the distributions of the random ariable ṽ generated by inestment in a firm is ontinuously stritly ordered by stohasti dominane, as defined below. (In this definition, G I ( I) and g I ( I) respetiely refer to the partial deriatie of G( I) and g( I) with respet to I.) Definition 2 The family of dfs {G( I) I 0} is said to be ontinuously stritly ordered by stohasti dominane proided, for eah I > 0 and eah > : (i) G I ( I) 0; (ii) G I( I)d < 0; (iii) G I ( I) and g I ( I) both exist and are (jointly) ontinuous on (, (I)). The ruial requirement in this definition is part (i), as it ensures that inreasing the inestment parameter I stritly "shifts the distribution of ṽ to the right" in the sense of first-order stohasti dominane (i.e., if I 1 > I 2, then G( I 1 ) G( I 2 ) for all ). Condition (ii) ensures that this "right-shifting" is strit in the sense that it ours to some extent oer any nondegenerate interal (, ); and ondition (iii) ensures that this right-shifting of the distributions ours "ontinuously." One example (among many) of a family of dfs that is ontinuously stritly ordered by stohasti dominane is the family of umulatie distribution funtions orresponding to the set of uniform random ariables {ṽ I I > 0} where ṽ I is uniformly distributed on the interal [0, I]. 36 The first natural question to ask when the distribution of a firm s alue ṽ is affeted by inestment is how hanges in inestment 37 hange the firm s equilibrium utoff and hene hange the firm s "no dislosure" prie. Operationally, this question inoles replaing the df G() that appears in (5) with a df G( I) drawn from the family {G( I) I 0} and determining how the utoff = (I) that soles that equation aries with I. The following orollary answers this question. Corollary 1 (How the Equilibrium Cutoff Varies with Inestment) Assume: the family of dfs {G( I) I 0} is ontinuously stritly ordered by stohasti dominane and there exist positie onstants w and B suh that I E I[ṽ] w and G I ( I) B for all I > 0. Then, the equilibrium utoff is stritly inreasing in I. The proof of Corollary 1 is in Appendix C. Corollary 1 demonstrates that, under mild tehnial onditions, when a firm inreases the leel of its inestment I, then the firm s equilibrium no dislosure prie inreases. Intuitiely this outome ours 34 Een when firms are obliged to produe balane sheets reording their inestments and statements of ash flows that inlude an inesting setion that aptures ash expenditures on inestments, inestors annot be sure whether the inestments reorded in these aounts onstitute well-direted expenditures that inrease the alue-generating potential of the firms, rather than being friolous and/or unprodutie inestments. Aordingly, een with the publiation of balane sheets and statements of ash flows, it is appropriate to onsider the atual produtie inestments firms engage in as priate information to firms. 35 As this notation indiates, we take the lower endpoint of the supports to be ommon aross all I, and we allow the upper endpoints of the supports to ary with I. 36 To see this, just note that G( I) = I for [0, I] and G( I) = 1 for > I, so for all > 0 and all I > 0, 0 G I( I)d = min{,i} 0 I 2 d = (min{,i}) 2 2I 2 < 0. 37 This is the only part of this setion where we take I, or hanges in I, to be exogenous. Thus, this orollary is a omparatie stati inoling a fixed distribution for ṽ just as the results of Theorem 1 are also omparatie statis for a fixed distribution of ṽ. 15

beause inestors know that, as I inreases, the firm s nondislosure is not as indiatie of bad news (as ompared to when I is smaller), beause high alues of ṽ are more likely to our whether or not the firm makes a dislosure. To proeed with identifying the firm s equilibrium hoie of inestment (whih, reall, we take to be priate information to the firm 38 ), we must distinguish between the inestment hoie the firm atually undertakes from the inestment hoie inestors onjeture the firm undertakes. We (ontinue to) write I for the firm s atual inestment hoie, and we now introdue the notation Î to represent inestors onjetures about the firm s inestment hoie. (In equilibrium, these will be one and the same.) The utoff inestors think the firm will use will be determined by the utoff = (Î) that soles equation (5) with G( Î) replaing G(). When there is no ambiguity about what inestment leel Î inestors think the firm has made, we sometimes write (Î) as ˆ. It follows that, gien Î (and hene ˆ = (Î)) and I, the expeted alue of the firm gross of the ost of the atual inestment, as pereied by the firm (i.e., the firm s managers) is gien by: V (ˆ, I) (1 p + pg(ˆ I))ˆ + p(1 G(ˆ I))E[ṽ ṽ > ˆ, I] = (1 p + pg(ˆ I))ˆ + p ˆ g( I)d. (9) That is, when inestors think the firm uses utoff ˆ, the expeted alue of the firm gross of the ost of inestment is the probability the firm makes no dislosure (1 p + pg(ˆ I)) times the prie inestors assign to the firm gien no dislosure (ˆ ), plus the probability the firm makes a dislosure (p g( I)d) times ˆ its expeted alue gien that the firm makes a dislosure ( ). So, the firm s expeted alue net of the ost of the ost of the atual inestment I is gien by 39 : ˆ g( I)d ˆ g( I)d W (ˆ, I) V (ˆ, I) I. (10) It follows that the firm s preferred inestment hoie is determined by the first-order ondition I W (ˆ, I) = 0, i.e., by: pg I ( (Î) I) (Î) + p The following definition formalizes the notion of an equilibrium in this setting. (Î) g I ( I)d 1 = 0. (11) Definition 3 An equilibrium onsists of an inestment leel I and a no dislosure prie ˆ suh that, (i) gien the prie ˆ : (ii) the no dislosure prie ˆ is gien by: I arg max I 0 W (ˆ, I); (12) ˆ = (I ). In words, an equilibrium must satisfy two onditions: first, taking inestors onjeture Î = I about the firm s inestment - as well as the "no dislosure" prie ˆ = (I ) implied by that onjeture I - as 38 Notwithstanding that, in pratie, reporting of inestment leels ours on a firm s balane sheet, and reporting of CAPX ours on a firm s statement of ash flows, we beliee it is appropriate for modeling purposes to iew I as priate, sine the publily dislosed expenditures on inestment do not indiate whether those expenditures were effi ient or effetie. 39 We ignore disounting in the following (or what amounts to the same thing, we assume that is disounted to time zero). 16

gien, the firm s preferred inestment leel I maximizes the expeted alue of the firm net of the ost of making the inestment. That is, the inestment leel the firm atually adopts is the same inestment leel that inestors onjetured the firm adopted. Seond, the firm s "no dislosure" prie ˆ based on inestors onjeture I about the firm s inestment is the "orret" no dislosure prie in that it is the "no dislosure" prie that orresponds to the inestment leel the firm atually adopts. 40 We now show that all of the preditions of the model in Theorem 1 and Corollary 1 aboe where the distribution of ṽ was exogenous and fixed translate immediately into preditions about the firm s equilibrium leel of inestment and the firm s equilibrium expeted market alue in the model of this setion where the distribution of ṽ is endogenous and aries with the firm s inestment deision. Speifially, we show that a hange in any exogenous parameter that results in the firm s equilibrium "no dislosure" prie inreasing (resp., dereasing) results in a lower (resp., higher) equilibrium leel of inestment and a lower (resp., higher) equilibrium expeted alue for the firm. Theorem 2 below states this result formally. In preparation for the statement of Theorem 2, we let γ generially denote any one of the exogenous parameters of the model, i.e., γ { m, f, d, q, p, t, k}. We adopt the onention that when γ is assigned to be one of these parameters, all other parameters are held fixed. We let (γ) denote the equilibrium utoff that soles (5) in the model of the preeding setion, where the distribution G() of ṽ was taken as exogenously gien. We let I(γ) denote the firm s equilibrium inestment hoie as a funtion of exogenous parameter γ. And finally, we let P (γ) denote the equilibrium expeted alue of the firm as of the start of the model, i.e., P (γ) E[ṽ I(γ)] I(γ). Theorem 2 (The Comparatie Statis of Equilibrium Inestment) Assume the assumptions of Corollary 1, and either of onditions (a) or (b) of Theorem 1, hold. In the model of endogenous inestment, for all the exogenous parameters γ { m, f, d, q, p, t, k}, we hae: (i) sgn di(γ) = sgn d (γ) and (ii) sgn dγ dp (γ) dγ = sgn d (γ) dγ. The proof of Theorem 2 is in Appendix D. dγ Theorem 2(i) states that a hange in any parameter that inreases the firm s equilibrium no dislosure prie redues the firm s equilibrium inestment leel, and ie ersa. desribed in the Introdution. The intuition for this result was Theorem 2(ii) adds that a hange in any parameter that inreases the firm s equilibrium no dislosure prie also redues the firm s equilibrium ex ante expeted market alue. latter result obtains beause if inestors do not get to see a firm s leel atual of inestment diretly, the firm inariably inests too little relatie to first-best in equilibrium. Consequently, a hange in any parameter that leads the firm to inest more inariably leads to inreases in the firm s expeted market alue, both beause the firm s original (pre-parameter hange) inestment leel was too low (relatie to first-best) and beause, een with the hange in the parameter alue, there is no possibility that the firm will oerinest (relatie to first-best). Symmetri omments apply with respet to a hange in any parameter that leads a firm to inest less: suh a hange inariably redues the firm s equilibrium expeted market alue. 40 Just as was the ase in the preeding setion, we are muh more interested in haraterizing features on an equilibrium in what follows rather than addressing tehnial issues related to the existene of equilibrium. Disussion of suffi ient onditions for the existene of an equilibrium are aailable in a working paper aailable from the authors. The 17

7 Conlusions This paper extends the literature on dislosures by studying a firm s dislosure deisions in the presene of a regulatory-imposed duty to dislose material information. Speifially, it examines an expeted aluemaximizing firm s dislosure deisions in a model where, if the firm gets aught withholding material information, the firm itself is obliged to pay damages to inestors who purhased the firm s shares while the material information was withheld. Consistent with 10b-5 litigation, the damage payments are taken to be a (typially frational) multiple of the amount the inestors oerpaid for the firm s shares relatie to what they would hae paid for the shares had the information been dislosed. An extensie set of omparatie statis inoling the firm s equilibrium dislosure poliy is obtained. Among our results, we show that the probability a firm disloses its priate information is inreasing in eah of the following parameters: the ex ante probability the firm reeied information; the fration of shares of the firm expeted to be purhased while the material information is withheld; the size of the firm s other litigation osts, in addition to the damage payments, the firm inurs assoiated with determining and settling damage payments; the size of the materiality threshold if that threshhold is initially high, and also - when a suffi iently large fration of the firm s shares are expeted to be purhased while the firm withheld material information - both the damages multiplier and the probability some fat finder will subsequently disoer the firm withheld material information. In the ourse of deeloping these preditions, we show how arious irularities that arise naturally when a firm may be liable to make 10b-5 damage payments (suh as: the potential damage payments depend on the firm s equilibrium "no dislosure" prie at the same time the firm s equilibrium "no dislosure" prie depends on these potential damage payments) an be resoled. The paper onludes by establishing how a firm s equilibrium dislosure poliy and its equilibrium inestment leel are linked. The analysis shows that, generally, a hange in any exogenous parameter that redues (resp., inreases) a firm s propensity to withhold its priate information also results in the firm inreasing (resp., reduing) its equilibrium inestment leel. 8 Referenes Alexander, J., "Rethinking Damages in Seurities Class Ations," Stanford Law Reiew 48, No. 6 (July 1996), pp. 1487-1537. Booth, Rihard, "Diret and Deriatie Claims in Seurities Fraud Litigation," Virginia Law and Business Reiew 4, 2 Fall 2009, pp. 277-294. Dye, Ronald A., "Dislosure of Nonproprietary Information," Journal of Aounting Researh Vol. 23, No. 1, Spring 1985, pp. 123-147. Einhorn, E. and A. Zi, "Biased Voluntary Dislosures," Reiew of Aounting Studies Volume 17 June 2012 No. 2 pp. 420-442. Grossman, Sanford, "The Informational Role of Warranties and Priate Dislosure About Produt Quality," Journal of Law and Eonomis, 1981, 461 483. 18

Heitzman, Shane, Charles Wasley and Jerold Zimmerman, "The joint effets of materiality thresholds and oluntary dislosure inenties on firms dislosure deisions," Journal of Aounting and Eonomis 49 (2010), pp. 109 132. Jung, Woon-Oh and Young K. Kwon, "Dislosure When the Market is Unsure of Information Endowment of Managers," Journal of Aounting Researh Vol 26, Spring 1988. Kanodia, Chandra, "Effets of Shareholder Information on Corporate Deisions and Capital Market Equilibrium," Eonometria 48 No. 4 (May 1980), pp.923-953.. Kanodia, Chandra and D. Lee, "Inestment and Dislosure: The Disiplinary Role of Periodi Performane Reports," Journal of Aounting Researh 36 No. 1 (Spring 1998), pp.33-55. Kanodia, Chandra and A. Mukherji,"Real Effets of Separating Inestment and Operating Cash Flows," Reiew of Aounting Studies 1 No. 1 (Marh 1996), pp.51-71. King, Ronald R. and Daid E. Wallin, "Voluntary Dislosures When Seller s Leel of Information Is Unknown," Journal of Aounting Researh Vol. 29, No. 1, Spring, 1991, pp. 96-108. Messier, William F., Nonna Martino-Bennie, Aasmund Eilifsen, "A Reiew and Integration of Empirial Researh on Materiality: Two Deades Later," Auditing: A Journal of Pratie & Theory (Noember 2005) Milgrom, Paul, "Good News and Bad News: Representation Theorems and Appliations," Bell Journal of Eonomis,Vol. 12, No. 2. (Autumn, 1981), pp. 380-391. Oesterle, Dale, "The Oerused and Under-defined Notion of "Material" in Seurities Law," Uniersity of Pennsylania Journal of Business Law Volume 14:1 (Fall 2011), pp. 167-187, Pae, S. Seletie Dislosures in the Presene of Unertainty about Information Endowment. Journal of Aounting and Eonomis, Vol. 39, No. 3 (September 2005): 383-409. Pae, Suil, "Aquisition and Disretionary Dislosure of Priate Information and Its Impliations for Firms Produtie Atiities," Journal of Aounting Researh Vol. 37 No. 2 (Autumn, 1999), pp. 465-474. Pae, Suil, "Corrigendum to: Aquisition and Disretionary Dislosure of Priate Information and Its Impliations for Firms Produtie Atiities," Journal of Aounting Researh, Vol. 42, No. 1 (Mar., 2004), p. 121. Patillo, James, The Conept of Materiality in Finanial Reporting, New York Finanial Exeuties Researh Foundation 1976. Posner, R., Eonomi Analysis of Law 7th ed. Wolters Kluwer, New York, 2007. Ryan, Ellen and Laura Simmons, "Seurities Class Ation Settlements 2009," Cornerstone Researh. Shaell, Steen, "Aquisition and Dislosure of Information Prior to Sale," RAND Journal of Eonomis, Volume 25, Number 1, Spring 1994, pages 20-36. Spindler, J. C., "Viarious Liability for Bad Corporate Goerane: Are We Wrong About 10b-5?" Uniersity of Southern California Working Paper, Otober 2008. Sunder, Shyam, Theory of Aounting and Control, SouthWestern College Publishing, New York, 1997 Verrehia, R. "Information Quality and Disretionary Dislosure." Journal of Aounting and Eonomis 12 (Marh 1990): 365-80. Ward, Bart H., "An Inestigation of the Materiality Construt in Auditing," Journal of Aounting Researh Vol. 14, No. 1 (Spring, 1976), pp. 138-152. 19

Zabel, Rihard and J. Benjamin Jr., "Reiewing Materiality in Aounting Fraud," New York Law Journal Jan. 15, 2002. 9 Appendix A Deriing the expression for EV F (, f r ) Fix f r and set f (f r + f l )(1 + t). Then, the expeted alue of the firm gien the firm made no dislosure, EV F (, f r ), is the sum of the expressions in the following bullet points: the onditional probability the firm is subjet to damages times the firm s expeted alue gien that it is subjet to damages. That is: 41 pg( m )q 1 p + pg( ) E[ṽ + d (1 f) (P ND ṽ) k ṽ < m ]. (13) the probability the fat finder disoers that the firm withheld immaterial information times the firm s expeted alue gien that the firm withheld immaterial information, i.e., p(g( ) G( m ))q 1 p + pg( )) E[ṽ m ṽ < ]. the probability the fat finder does not subsequently disoer that the firm withheld any information, 1 pqg( ) 1 p + pg( ), times the onditional expeted alue of the firm gien that the firm made no dislosure and the fat finder does not disoer that the firm withheld any information: ( 1 G( )p(1 q) 1 p + pg( )(1 q) E[ṽ ṽ < ] + 10 Appendix B Proof of Lemma 1 G( )p(1 q) 1 p + pg( )(1 q) ) E[ṽ]. Fix f r and set f (f r + f l )(1 + t) and f (f r + f l )(1 + t). Then, using the expression for EF V (, f r ) appearing in (3), multiply both sides of the equation = EF V (, f r ) by 1 p + pg( ) to get: (1 p + pg( )) (14) = pqg( m ) E[ṽ k + (1 f) d (P ND ṽ) k ṽ < m ]+ pq(g( ) G( m )) E[ṽ m < ṽ < ]+ (1 p + pg( ) pqg( )) { G( )p(1 q)e[ṽ ṽ < ] G( )p(1 q) 1 p + pg( + (1 )(1 q) 1 p + pg( )(1 q) ) E[ṽ]}. 41 The denominator of the first term in (13) is the probability the firm makes no dislosure. It is the sum of: the probability the firm reeied no information 1 p and the probabily the firm reeied information and deided not to dislose the information pg( ). The numerator of the first term in (13) is the probability the fat finder disoers the firm withheld material information, and so the quotient of the first term in (13) is the probability, gien that the firm makes no dislosure, that the fat finder will subsequently disoer that the firm withheld material information and hene is subjet to damages. (Similar explanations for the other probability ratios appearing in the bullet points below are omitted.) 20