Credibility of voluntary disclosure.

I examine the credibility of a manager's disclosure of privately observed nonverifiable information to an investor in a repeated cheap-talk game setting. In the single-period game no communication occurs. In the repeated game, however, the manager almost always truthfully reveals his private

information provided the manager is sufficiently patient, the accounting report is sufficiently useful for assessing the truthfulness of the manager's voluntary disclosure, and the manager's disclosure performance is evaluated over a sufficiently long period. These factors may explain a manager's propensity to release private information to investors.

1. Introduction

* I examine the credibility of the voluntary disclosure of a corporate manager's imperfect, private information. The credibility of firms' voluntary disclosure has long been of concern to the Securities and Exchange Commission (SEC). The SEC traditionally restricted disclosure by publicly traded companies to information that was fundamentally historical in nature and subject to independent verification. This policy seems to have presumed that forecasts and other nonverifiable, forward-looking information lacked credibility and that private incentives were insufficient to eliminate management manipulation (see Pownall and Waymire, 1989). However, in 1973 it reversed its longstanding "exclusionary policy" that prohibited the voluntary inclusion of forward-looking information in SEC filings. The SEC subsequently issued several pronouncements encouraging the release of forward-looking information and in 1979 adopted a "safe-harbor" provision to shelter managers from litigation arising from unattained projections in SEC filings. Recently, the enactment of the Private Securities Reform Act of 1995 further strengthened the "safe-harbor" provisions introduced by the SEC by limiting management's liability for any forward-looking statements about a firm's prospects to those cases where the projections were not made in good faith (Pincus, 1996). This and other provisions of the Private Securities Reform Act were vigorously contested by state securities regulators, plaintiffs' counsel, the SEC, accounting firms, and other constituencies: opponents of the legislation "predicted that the United States' securities markets would become a magnet for fraud ..." (Grundfest and Perino, 1997, p. 1).

The forecasting experience of Allen Group, Inc., reported by Lees (1981), suggests that the market may response to a firm's misleading signals by ignoring its future communications and that this respond is sufficient to force the firm to alter its behavior. In this case, the firm had repeatedly failed to dispute analysts' forecasts that had been optimistic for a number of years. Analysts interpreted this acquiescence as confirmation of their forecasts and hence viewed the firm's disclosure practices with suspicion. Subsequently, when the firm believed that a number of analysts' forecasts were excessively biased, it was unable to have the forecasts lowered despite repeated verbal and written communication. In response to the perceived lack of credibility, Allen Group altered its disclosure policy to one in which it released its private information; its new policy included the guidelines "[w]e will attempt to bring a realistic attitude toward our estimates which will discount future uncertainties as best we know them and can quantify them" (p. 28) and "[w]e will recognize and hope stockholders will appreciate that forecasting is difficult and imprecise in any business and that it involves the art of judgment rather than the science of accounting technique" (p. 28).

This article proposes that a manager's concern with the credibility of his disclosure is often sufficient to ensure that he will almost always truthfully reveal his private information. I examine this issue within the context of a repeated cheap-talk game that has the following key features.(1) In each stage, the firm may pursue an opportunity with unknown earnings. The manager privately observes a noisy signal about the opportunity's earnings. He then sends a costless message to an investor who decides whether or not to provide the firm with the finance necessary to exploit the opportunity. Thereafter, if the opportunity is financed, the firm releases a mandatory accounting report that provides a noisy signal about the opportunity's earnings and about the manager's private information; unlike the manager's voluntary disclosure, the accounting report cannot be manipulated by the manager. The manager's payoff is increasing in the investor's valuation of the opportunity's contribution to firm value determined after the release of the accounting report. The investor's payoff is increasing in the firm's value. Once the payoffs are determined, the stage game is repeated.(2)

The single-state game has a unique equilibrium in which no communication is observed and the investment decision is based upon the expected value of the opportunity. In the repeated game, an equilibrium is constructed by supposing that the players divide time into a sequence of review and punishment phases. During the review phase, the investor uses the manager's voluntary disclosure when deciding to provide the firm with the requested financing, and this raises the expected contribution of the opportunity. The investor also assesses the truthfulness of the manager's voluntary disclosure in light of the accounting report. If the manager has an incentive to truthfully reveal his private information, then the investor continues believing the manager. If, however, the manager no longer has such an incentive, then depending upon the history of the game, the players may either restart the review phase or enter a punishment phase and follow the single-stage game equilibrium strategies. At the end of the punishment phase, the review phase is restarted.

The main result of this article is that the manager almost always truthfully discloses his private information provided that the manager's discount factor is sufficiently high, the accounting system that generates the mandatory accounting report is sufficiently useful for assessing the credibility of the manager's disclosure, and the review phase is sufficiently long. Thus the threat of ignoring the manager's disclosure, as the above Allen Group case suggests, may ensure that the manager almost always truthfully discloses his forward-looking information that is useful to investors for valuing and allocating capital to firms.

Whether an efficient payoff can be approximated in a repeated game is sensitive to the specification of the game.(3) Fudenberg, Levine, and Maskin (1994) analyze infinitely repeated games with imperfect monitoring. They show that any individually rational payoff in a single-stage game can be approximated in an infinitely repeated game, even if an imperfect monitor is used to evaluate the players' actions, provided the players are sufficiently patient and some conditions on the identification of the players' actions are met. Their analysis, however, does not address cheap-talk games.

In fact, it is generally unclear whether efficient payoffs can be approximated in repeated cheap-talk games.(4) Sobel (1985) examined the effect of reputation on the strategic reporting of information in a repeated cheap-talk game. His model, like most models that examine the role of reputation (see Wilson, 1985), is one of incomplete information in which the sender's type is unknown to the receiver and fixed for the duration of the supergame. The sender may be either a nonstrategic type who always truthfully reveals his private information or a strategic type who manipulates his message to maximize his expected payoff. The sender observes the value of a binary random variable without noise. Thus, if the sender issues a message that differs from the realization that the receiver ultimately perfectly observes, he reveals that he is a dishonest type. The sender's reputation is then lost for the remainder of the (finite) game and no further communication occurs.

Benabou and Laroque (1992) and Morris (1998) extended Sobel's model to a game where the sender, who again may be one of two types, observes a noisy signal of a binary random variable. Benabou and Laroque examine a manager's attempt to manipulate the firm's stock price through the release of strategically distorted messages. The sender has noisy information and can thus repeatedly manipulate the stock price without revealing whether he is a nonstrategic type who truthfully reveals his private information or a strategic type who manipulates his private information. The upshot of this is that the strategic sender's credibility does not vaporize, so he is able to transmit information indefinitely even though his message is distorted with some positive probability. Morris extends Benabou and Laroque's model by considering two types of senders who strategically manipulate their messages but differ in that one sender has the same preferences as the receiver and the other wants the receiver to take as high an action as possible. Morris finds that in an infinite-horizon game, truth telling is not an equilibrium even as the sender's discount factor approaches one.

Kim (1996) differs from Sobel, Benabou and Laroque, and Morris: while Kim also considers a repeated cheap-talk game, the sender's type, which may be high or low, is not fixed for the supergame but is rather a random variable drawn independently in each period. The receiver can verify the sender's type at some cost to both players. Kim shows that in the single-period game no communication occurs, whereas repeated interaction allows for communication. Generally, however, approximately complete information transfer does not occur; in particular, if the prior probability that the sender is a high type is sufficiently large, then there are no equilibria that feature communication in the repeated game.

In this article, the players' actions in the repeated game are coordinated by supposing that review strategies are played. Review strategies have been used to study repeated principal-agent games. Radner (1985) examined a game in which a risk- and effort-averse agent chose some unobservable level of effort to produce an output, and the principal used the publicly observed output to determine whether the agent rendered the efficient level of effort. Radner used review strategies to support equilibrium payoffs in the repeated game that were not supportable in a single-period game. The review strategies I use here differ from those employed by Radner: he, like others who have employed review strategies, used strategies that have a review period of fixed length. This results in the equilibrium strategies being inefficient for some histories of the game.(5) The principal, nevertheless, continues to comply with the equilibrium strategies in the review phase for fear that if he deviates the agent will punish him by shirking for some future period. In contrast, in the game analyzed here, the investor participates in a competitive capital market where the investor is free to refuse the firm financing. Also, any stock mispricing that arises when the investor responds to the manager's disclosure while knowing that it is dishonest will be arbitraged away by competing investors. Thus, the investor will ignore the manager if he does not disclose truthfully. Accordingly, the review strategies characterized here use a review phase that may be shortened in response to the history of the game.

The rest of the article proceeds as follows: Section 2 provides the extensive form of the game, Section 3 examines the one-stage game, Section 4 analyzes the repeated stage game, and Section 5 draws conclusions.

2. Model

* This section introduces an infinitely repeated stage game that has a firm and two long-run, risk-neutral players: a corporate manager and a representative investor (or financier).(6) The time line of a stage game is summarized in Figure 1.

[Figure 1 ILLUSTRATION OMITTED]

In each stage game, the firm has net assets in place of unknown economic value and an opportunity that requires investment if it is to be exploited. The assets in place are generated by opportunities pursued in previous stages and, except for the firm's cash holdings, are valuable while the firm is a going concern but not so in the event of the firm's liquidation.(7) The opportunity that arises during the stage can be financed through the issuance of stock, the sale of bonds, or the utilization of internal cash resources. The opportunity, however, is not atomistic, and the firm is not sufficiently liquid to finance the project out of its existing cash resources. Consequently, the manager is obliged to approach an investor for financing if the opportunity is to be pursued.

At the beginning of each stage, denoted t, nature draws the economic earnings for the opportunity from a set of possible discounted economic earnings [Omega] = {[[Omega].sub.1], [[Omega].sub.2], ..., [[Omega].sub.n]}. The economic earnings equal the sum of economic earnings accruing to stockholders, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and debtholders, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; formally, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].(8) Economic earnings have the property that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for j = e, d. The players share prior beliefs that the firm's unknown earnings are distributed with probabilities p = (p([[Omega].sub.1]), p([[Omega].sub.2]), ..., p([[Omega].sub.n])), where P([[Omega].sub.i]) [is greater than] 0 for all i and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].(9)

The investor needs to take care when investing since she is at risk of investing in opportunities that have negative earnings. Such a realization is detrimental to an equity provider (because it reduces the value of the firm) and to a debt financier (because the firm's non-cash assets have value only if the firm is a going concern), so the lender, given the firm's limited liability, suffers a loss to the extent that the firm is unable to repay the loan plus interest out of the cash flows generated by the project. The nature of the investor's decision, and the role of information in it, thus does not depend on whether the investor provides debt or equity financing. Henceforth, to simplify the exposition and without loss of generality, I assume that the opportunity is financed purely from the proceeds of an equity offering; thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i.

During the stage game, the manager privately obtains nonverifiable, forward-looking information: for instance, the manager has some "gut feeling" about the firm's performance or gathers information about a business opportunity. The forward-looking information, f, is an element of the set F = {G, B}. The conditional probability of f, given state [[Omega].sub.i], is denoted [[Pi].sub.f](f|[[Omega].sub.i]), where [[Pi].sub.f](G|[[Omega].sub.i]), [[Pi].sub.f](B|[[Omega].sub.i]) [is greater than] 0; thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is positive. The conditional posterior distribution of economic earning, given f, is described by the vector p([Omega]|f), whose ith component is p([[Omega].sub.i]|f). Also, G is more favorable than B in the sense that the conditional posterior distribution p([Omega]|G) strictly first-order stochastically dominates p([Omega]|B) (see Milgrom, 1981).

After the manager observes f, he chooses a costless message m from the set of possible messages M = {g, b} to voluntarily send to the investor.(10) Generally g and b may be any forward-looking disclosure about [Omega], but for simplicity g and b are labelled as g = G and b = B. The manager is unconstrained by f when choosing his message: his self-serving message may be false, vague, or even completely uninformative about f. The manager's strategy is a function over his private information defined by [Tau]: F [right arrow] [Delta](M), where [Delta](M) is the set of all probability distributions over M.

The investor's payoff is increasing in the firm's value, which in turn is increasing in the opportunity's payoff. The investor observes the manager's message and decides whether or not to provide the requested financing given her posterior beliefs about the opportunity's payoffs. The investor's set of actions is [Phi] = {0, 1}, where the withholding of investment is indicated by [Phi] = 0 and investment by [Phi] = 1. The investor's strategy is a function over the manager's disclosure and is defined by [Phi]: M [right arrow] [Phi]. The investor's objective in stage t when the manager discloses m is to choose [Phi] such that

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where, using Bayes' theorem, the investor's posterior beliefs are

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all m [element of] M and f [element of] F when [[Sigma].sub.F] [Tau](m|f) [multiplied by] [[Pi].sub.f](f) [is greater than] 0, and the investor believes the manager observed f = B when m [is not an element of] M.(11)

If the investor declines to finance the project, the opportunity expires and is not proposed later. If, on the other hand, the investor provides financing, the manager forthrightly applies the investor's contribution toward the project.

At the end of the stage game, the firm publicly releases a mandatory accounting report to comply with the SEC requirement that publicly traded firms periodically issue an accounting report. The accounting report discloses a value in the set A = {[a.sub.1], ..., [a.sub.j], ..., [a.sub.l]}, where [a.sub.j] [is less than] [a.sub.j+1] for all j = 1, 2, ..., l - 1, when the project is financed, and a value of zero otherwise. Further, in light of the SEC regulation requiring an independent audit of the accounting report, it is assumed that the manager is restricted from nefariously manipulating the report.(12) The accounting report is prepared under Generally Accepted Accounting Principles (GAAP), which include arbitrary revenue recognition and expense matching policies (Brealey and Myers, 1984), and therefore it reveals economic earnings with noise. The conditional probability of [a.sub.j], given state [[Omega].sub.i], is denoted [[Pi].sub.a]([a.sub.j]|[[Omega].sub.i]). For all [a.sub.j] and [[Omega].sub.i], [[Pi].sub.a]([a.sub.j]|[[Omega].sub.i]) [is greater than] 0, and thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is positive. Also, the conditional probability of [a.sub.j], given state f, is denoted [[Pi].sub.a|f])([a.sub.j]|f). As seems natural, a lower (higher) value of a is more likely than a higher (lower) value of a when f = B(G), respectively; that is, [[[Pi].sub.a|f]([a.sub.j]|B)]/[[[Pi].sub.a|f]([a.sub.j]|G)] is strictly decreasing in [a.sub.j].

When the firm proceeds with the project, the posterior probability of economic earnings [[Omega].sub.i], given f and a, is p([[Omega].sub.i]|f, a). Since GAAP require a firm to apply its accounting policies consistently across time, its accounting policies are independent of the manager's private information; hence conditional on [[Omega].sub.i], f and a are independent and p([[Omega].sub.i]|f, a) = [[Pi].sub.f](f|[[Omega].sub.i]) [multiplied by] [[Pi].sub.a](a|[[Omega].sub.i]) [multiplied by] p([[Omega].sub.i])/[Pi](f, a).

After the firm releases the accounting report, the investor values the firm, and the manager's payoff is determined. The manager's payoff for stage t, denoted [U.sub.t], is increasing in the opportunity's contribution to firm value that the investor expects given the voluntary disclosure, investment decision, and accounting report; specifically, the manager chooses m to maximize his expected payoff

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, using Bayes' rule and the observation that conditional on f, a and m are independently distributed,

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all a [element of] A, m [element of] M, and f [element of] F when [[Sigma].sub.F] [[Pi].sub.a|f](a|f) [multiplied by] [Tau](m|f) [multiplied by] [[Pi].sub.f](f) [is greater than] 0.(13)

This valuation marks the end of the stage game. The stage game is then repeated. At the beginning of the next stage, a new opportunity arises with economic earnings that nature randomly draws from the same underlying distribution p as for the previous stage. The actual economic earnings for a stage are never perfectly observed; all that is observed by the manager or investor is the imperfect accounting report. Moreover, in any stage, the accounting report is useless for inferring the economic earnings of projects undertaken in earlier stages because assets and liabilities are carried at their historical cost under GAAP, so changes in their economic values are not generally recognized in the accounting report.

I now define aspects of the repeated game. A history of length t, denoted [h.sub.t], is a sequence of manager disclosures, investor decisions, and accounting reports. The set of public histories that contains sequences of publicly observed outcomes of length t is denoted [H.sub.t]; thus, [H.sub.t] = [M.sup.t] x [[Phi].sup.t] x [A.sup.t], where t = 1, 2, ..., and [H.sub.0] = ??.

The manager's public strategy is a sequence of functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] over the set of public histories and his privately observed information set where [[Tau].sub.t]: [H.sub.t-1] x F [right arrow] [Delta](M). The set of the manager's strategy profiles is denoted as [Theta]([Tau]) and his discount factor as [Delta], where 0 [is less than] [Delta] [is less than] 1.

The investor's public strategy is defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] over the set of public histories and the manager's disclosure, where [[Phi].sub.t]: [H.sub.t-1] x M [right arrow] [Phi]. The set of the investor's strategy profiles is denoted as [Theta]([Phi]). The investor's posterior beliefs over F, given history [h.sub.t-1] and m, are defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[Eta].sub.t]: [H.sub.t-1] x M [right arrow] [0, 1]. Subsequently, after the project is financed and the accounting report released, the investor's updated beliefs over F, given history [h.sub.t], are defined as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[Eta]'.sub.t]: [H.sub.t-1] x M x A [right arrow] [0, 1]. The set of strategy profiles for the players is given by [Theta] = [Theta]([Tau]) x [Theta]([Phi]). The manager's payoff function is defined as U: [F.sup.[infinity]] x [A.sup.[infinity]] x [Theta] [right arrow] R.

All aspects of the game except the manager's privately observed signal are common knowledge. The Bayesian-Nash equilibrium concept (see Osborne and Rubinstein (1994) for a definition) is employed in the one-stage game and attention is focused on perfect public equilibria in the repeated game. A perfect public equilibrium is a profile of public strategies that from any period t, given any public history [H.sub.t-1], constitute a Nash equilibrium from that point onward (see Fudenberg, Levine, and Maskin, 1994).

Before proceeding, some discussion of the model is necessary. In each stage, the manager is presented with an opportunity and obtains some private information, the firm requires external funding to pursue the opportunity, and it releases an accounting report. More generally, however, firms have numerous opportunities, and many of them can be funded without approaching the capital markets. Also, firms voluntarily release much forward-looking information during the course of an accounting period; for instance, management may discuss the firm's strategies, industry prospects, and firm growth projections in regular meetings with analysts.

The restriction to a single voluntary disclosure during each stage is purely for simplicity. It should not be interpreted as suggesting that an investor updates her beliefs about a manager's reporting credibility in response to a single disclosure in each accounting period. Rather, an investor may use each interim quarter or annual accounting report to revise her assessment of a manager's reporting credibility in response to each of a potentially large number of voluntary disclosures in an accounting period. Thus, in practice, very few accounting periods may be necessary for an investor to develop fairly precise beliefs about a manager's reporting credibility.

Also, it is supposed that the firm approaches the capital market in every stage. Typically, however, firms do not approach the capital market every accounting period. But they do periodically seek external debt or equity funding, and at this time investors need to decide, given their beliefs, whether or not to provide financing. The model focuses on this event. The model could be changed to include periods where the firm does not approach the capital market but continues to release voluntary and mandatory disclosures, and also periods where the firm both releases private information and seeks external funding. This change would complicate the analysis without adding much insight.

The model may be extended to include a dividend distribution in each stage. If it is assumed that dividends are exogenously fixed for the supergame, which comports nicely with firms' reluctance to change their dividend payout from a long-term target payout (see Brealey and Myers, 1984), then a dividend will not signal information about a stage's economic earnings. Consequently, the dividend distribution is useless to the investor for assessing the manager's private information, and including it would not alter the results.

3. One-stage game

* This section provides sufficient conditions for the manager's private information, if truthfully disclosed, to affect the opportunity's contribution to firm value and, in turn, the manager's payoff. It then characterizes the unique equilibrium to the one-shot game when the manager is free to make misleading or completely uninformative disclosures.

But first some definitions are required. The expected contribution of the proposed opportunity to the firm's value when the investment decision is made in the absence of the manager's private information is

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The expected contribution of the opportunity if the investment decision is made after the manager reveals his private information and the investor believes the manager (i.e., [Eta](f|m = f) = [Tau](m = f|f) = 1) is

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The expected contribution of the opportunity before observing f if the investment decision is made after the manager fully reveals his private information and the investor believes the manager (i.e., [Eta](f|m = f) = [Tau](m = f|f) = 1) is

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A sufficient condition for the manager's truthful voluntary disclosure to affect the opportunity's expected contribution to firm value is that it should affect the investor's decision. In particular, there should exist a conditional posterior probability distribution p([Omega]|f) such that when B is disclosed, discounted economic earnings are expected to be negative and the opportunity is not financed, and when G is disclosed, earnings are expected to be positive and the opportunity is financed. The manager's forward-looking information then allows the investor to better avoid providing either debt or equity financing to a firm that more likely will have negative earnings. The manager's disclosure thus affects the posterior distribution of earnings and, hence, raises the opportunity's expected contribution to firm value. This observation is formalized in Lemma 1.

Lemma 1. If the investor knows that the manager truthfully reveals his private information, then E([Omega]|B) [is less than] 0 [is less than] E([Omega]|G) if and only if Q([Omega]) [is less than] Q([Omega]; F).

Proof. Follows directly from Jensen's inequality. Q.E.D.

I avoid the trivial outcome that the manager has no incentive to distort his disclosure by assuming that E([Omega]|B) [is less than] 0 [is less than] E([Omega]|G) for the remainder of the article. Thus if the manager is forthright and the investor knows this, then the investor will finance the project if m = g but not if m = b, and Q([Omega]|B) = 0 [is less than] Q([Omega]|G).

When the accounting report is sufficient for the manager's private information, the manager's disclosure does not affect the investor's firm valuation at the end of the stage. The manager therefore has no incentive to misrepresent his information. This is likewise true for the case when the accounting report reveals the firm's earnings without noise. So for the manager's message to potentially affect the valuation, the accounting report should not be sufficient for the manager's private information with regard to economic earnings. Nevertheless, it may be more informative than the manager's information, as is usually the case in practice.

The equilibrium to the one shot-game is now characterized. In an environment where the manager's privately observed signal G is more favorable than signal B and the investor believes the manager's voluntary disclosure, even partially, the manager's dominant strategy is to bias his message upward and always disclose g irrespective of f. Given these incentives, he cannot credibly disclose his information. The investor's best response is therefore to ignore the manager's disclosure and provide either debt or equity financing to the firm if the opportunity's expected contribution is positive but not otherwise. This argument is summarized in Lemma 2.

Lemma 2. The single-stage game has a unique Bayesian Nash equilibrium in which no communication occurs and the investor provides financing if and only if E([Omega]) [is greater than] 0.

Proof Follows from the above discussion. Q.E.D.

Notice that if the investor ignores the manager's voluntary disclosure and invests in the firm (i.e., Q([Omega]) = E([Omega]). [is greater than] 0), then the manager who observes f = B would want to send a message truthfully conveying his private information to discourage investment. If believed, his expected payoff increases to Q([Omega]|B) = 0 [is greater than] E([Omega]|B). The manager who observes f = G would not mimic the message sent by the manager who observes f = B. Nevertheless, the fact that the investor may use the manager's disclosure at the end of the period to value the firm creates incentives for the manager to distort his message.(14) Consequently, in equilibrium no communication occurs and the expected contribution of the proposed opportunity to firm value is Q([Omega]).

When a manager is protected by "safe-harbor" provisions from litigation that arises when a firm's performance fails to meet a manager's earlier projections, then in a one-shot game no communication occurs, which in turn diminishes the efficiency of the capital markets at allocating resources. This observation results despite the existence of the accounting report that the investor could use to assess the integrity of the manager's disclosure.

4. Repeated game

* In this section I show that there exists an equilibrium in which the manager almost always truthfully discloses his information, and the expected contribution of the opportunity to the firm's value approximates that observed in a complete-information environment.

An equilibrium in review strategies is characterized. This technique divides time into a sequence of review phases. The review phase has a maximum length of R stages. During a review phase the manager sends a message and the investor maintains a trust index. The trust index, however, is not part of the physical environment, in that it does not describe the investor's belief formation process. Rather, the investor commits to update the trust index in a particular way, and this index then serves to coordinate the players' behavior over time.

The investor performs two tasks at the end of each stage t during the review phase (1 [is less than or equal to] t [is less than or equal to] R). First, the investor assesses the credibility of the manager's disclosure. When m = g and the opportunity is financed, the manager's disclosure is assessed to be credible if a [element of] {[a.sub.l-s], [a.sub.l-s+1], ..., [a.sub.l]}, and when m = b and the opportunity is not pursued, the manager's disclosure is assessed to be credible with probability [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where s maximizes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].(15) The probability that the investor assesses the manager's truthful disclosure to be credible, S, captures the strength or usefulness of the accounting information system for assessing the credibility of the manager's disclosure. The assumptions imposed on the distribution of f and of a are sufficient to ensure S [is greater than] 0.

If the disclosure for stage t is assessed to be credible, the trust index becomes [I.sub.t] = [I.sub.t-1] + 1; otherwise the trust index remains as [I.sub.t] = [I.sub.t-1]. The value of the trust index at the beginning of the review phase is [I.sub.0] = 0.

Second, at the end of each stage during the review phase, the investor determines whether the manager has an incentive to continue to report truthfully given [h.sub.t] and the values of R, S, [Delta], Q([Omega]), and Q([Omega]; F). If the manager no longer has such an incentive, then the review phase is terminated (perhaps before stage R) and the manager either passes or fails the review depending upon the value of index [I.sub.t] relative to a credibility threshold L(t). If [I.sub.t] [is greater than or equal to] L(t), then the manager passes the review and the players restart the review phase; if, on the other hand, [I.sub.t] [is less than] L(t), then the manager fails the review and the players enter a punishment phase. During the punishment phase, the investor ignores the manager. This phase ends after K stages, where K = [Mu]R and [Mu] [is greater than] 0. Thereafter, the review phase is restarted.

The credibility threshold L(t) = [(R [multiplied by] S - J)/R] [multiplied by] t reflects the number of disclosures during the t periods of the review phase that are expected to be assessed as being credible adjusted for a measurement error parameter J that compensates for using an imperfect monitor to evaluate the manager. Set J = [Beta][R.sup.[Rho]], where [Beta] [is greater than] 0 and 1/2 [is less than] [Rho] [is less than] 1.

The manager's normalized discounted expected payoff at the beginning of the review phase is

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

An equilibrium where U([I.sub.0]) approximates Q([Omega]; F), which is observed when the manager fully reveals his private information, is characterized in Proposition 1.

Proposition 1. For all [Epsilon] [is greater than] 0, there exists [Delta] discount factor [Delta] an accounting system credibility assessment value S, and a review phase of length R such that for each [Delta] [element of] [[Delta], 1) and S [element of] [S, 1), there is a review phase of R [is greater than] R that supports a perfect public equilibrium of the repeated game in which U([I.sub.0]) [is greater than] Q([Omega]; F) - [Epsilon] and the manager almost always truthfully discloses his private information.

This equilibrium is characterized by the following strategies:

(i) The manager voluntarily discloses his private information truthfully during the review phase but does not disclose it during the punishment phase.

(ii) The investor responds to the manager's voluntary disclosure during the review phase and invests only if Q([Omega]|f) [is greater than] 0 and ignores the manager during the punishment phase and invests only if Q([Omega]) [is greater than] 0.

Proof. See the Appendix.

In an environment where a manager's preferences over the investor's beliefs are uncorrelated with the manager's private information, this proposition shows that the manager truthfully discloses his information even though he misrepresents his information in a one-period game. Consequently, in a multiperiod setting, a manager's voluntary disclosure may improve the efficiency with which an investor allocates resources to firms. The key to obtaining the approximately efficient outcome is that the investor can credibly threaten to ignore the manager's disclosure when she doubts its veracity. It is credible for the investor to ignore the manager during the punishment phase because the players revert to the single-period, unique Bayesian Nash equilibrium. As shown in Lemma 1, this response reduces the project's expected contribution and, in turn, the manager's expected payoff.

The valuation of the opportunity in the punishment phase is reminiscent of what stock traders call a "liar's discount": this is the discounting of a firm's stock price below that of its competitors when financial analysts refuse to trust a firm's management after its prior voluntary disclosure is revealed to be misleading (King, 1988).

I now provide some intuition for why an efficient outcome is attained for sufficiently large values of R, S, and [Delta]. An increase in the review phase R (for S and [Delta] sufficiently large) lengthens the punishment phase and therefore heightens the manager's incentive to disclose truthfully. In addition, an increase in R reduces the probability of inappropriately punishing a manager who has been reporting in good faith and thus mitigates the expected inefficiency of the longer punishment phase.(16) In fact, by increasing R, the manager's ex ante probability of failing the review and entering the punishment phase when he discloses truthfully can be bounded arbitrarily closely to zero. Consequently, as R increases, the increasing incentive to report honestly coupled with the decreasing measurement error results in the opportunity's expected contribution converging asymptotically to Q([Omega]; F).

A better allocation of capital is associated with an accounting report that is more useful for assessing a manager's voluntary disclosure. Increases in S improve the investor's ability to deservedly punish a deviant manager and not punish him otherwise. Hence, if S is sufficiently large, the manager will continue to report truthfully and avoid the costly punishment phase even though he may have had many unfavorable assessments and be close to being in a position that he can never pass the review. In contrast, for the same history of unfavorable assessments, if S is low, then the players would enter the punishment phase during which time the investor allocates capital less efficiently to each opportunity.

An increase in the discount factor heightens the prominence of the punishment phase in the manager's payoff function. A more patient manager is less inclined to behave myopically and opportunistically release information at the expense of the long-term gains that follow from building investor trust.

The following example provides some sense for the magnitudes of the parameters that approximate an efficient outcome. Botosan (1997) examined the relationship between a firm's disclosure level and its cost of equity capital within the machinery industry; she documents that firms with low financial analyst following (i.e., those that benefit most from additional voluntary disclosure) had a mean market value and cost of capital at the end of 1990 of $129.5 million and 22.1%, respectively, and that the most informationally forthcoming firms enjoyed a reduction in their cost of capital of 9.7% relative to the least forthcoming ones. This implies that if the average firm's value is determined by valuing earnings as a perpetuity, then earnings attributable to equity are estimated to be $28.62 million. Now assuming that a firm's cost of capital can range from 17.25% to 26.95%, its market value varies from $106 million to $166 million according to its level of disclosure; hence, suppose Q([Omega]) = 106 and Q([Omega]; F) = 166. Assume a cost of capital per quarter of 5.12% (i.e., an effective rate of 22.1% compounded annually), or equivalently, [Delta] = .95. Also set S = .9, [Beta] = .85, [Rho] = .95, [Mu] = .625, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Against this background, R = 8 quarterly accounting reports are sufficient to obtain a value for U([I.sub.0]) that is at least 95% of Q([Omega]; F) and to ensure that the manager will not deviate from disclosing truthfully if a deviation increases a firm's value by less than 106% of Q([Omega]; F). Further, the players are expected to enter the punishment phase that lasts for K = 5 quarters with a probability of no more than .02.

Thus, provided the manager's gains from iniquitously disclosing his private information are not too great, it is not necessary that the discount factor be too large, the usefulness of the accounting system for assessing the manager's credibility be too high, or that the review phase be unrealistically long for the model to yield magnitudes that are institutionally plausible.

This analysis suggests a number of factors that may affect a manager's voluntary disclosure of forward-looking information. First, a manager with a low discount factor, perhaps because of a short-term employment horizon or doubts about his firm's continued survival, is more likely to behave myopically. Thus, he is less able to credibly communicate with investors. Second, an accounting report of poor quality may be associated with a manager providing little additional voluntary disclosure. For instance, a manager may not release forward-looking information about the performance of a business segment because the firm may not provide segmental information in its accounting report that could be used to assess the credibility of the manager's earlier disclosure. Third, as a manager's information becomes more proprietary in nature, discretionary disclosure is less probable. Proprietary information may be helpful to competitors but harmful to the firm. Therefore, Q([Omega]; F) may be less than Q([Omega]), and the manager cannot be forced to disclose credibly. Finally, communication is less probable in the future for a manager with a history of unfavorable assessments because the players are more likely to enter a punishment phase, much like the experiences of Allen Group discussed in the Introduction.

5. Conclusions

* This article examines the credibility of the voluntary disclosure of a corporate manager's privately observed imperfect information. In a single-stage game, no communication occurs. In the repeated game, however, the manager is able to develop reporting credibility and communication results. More important, the manager's concern about his reporting credibility matters enough to ensure that he almost always truthfully discloses, in contrast to the extant repeated cheap-talk literature where this outcome is not generally the case. Thus the investor efficiently allocates capital to a firm in the absence of legal provisions to enforce truthful reporting, such as requiting that an independent auditor verify the manager's disclosures. This finding may ameliorate the concern expressed by opponents of the recently enacted Private Securities Reform Act that the legal safeguards afforded to firms that fail to meet earnings projections, and the like, will induce iniquitous reporting practices.

While the equilibrium strategies used in this article are ex ante approximately efficient in the sense that the information asymmetry between the players is almost eliminated, they are not ex post efficient. For instance, in equilibrium, the investor responds to a manager who fails a review by ignoring him for some punishment period even though he truthfully released his private information during the review phase. This punishment is inefficient. Whether strategies that are immune to this criticism can be identified awaits further research.

(1) A cheap-talk game is a signalling game in which the players' payoffs do not depend on the sender's costless message but merely on the receiver's action it induces and the sender's private information (Farrell, 1993). See Farrell and Rabin (1996) for an introduction to cheap-talk games.

(2) This setting may be modelled as a "persuasion game" in which the manager may withhold information, but the investor can verify any message sent (see Milgrom and Roberts, 1986; Shin, 1994). It seems, however, that a cheap-talk setting is more appropriate for the analysis of a manager's voluntary disclosure because often the manager's messages (such as his opinion of the firm's future performance) are not verifiable.

(3) For instance, Radner (1985) shows that in a repeated principal-agent game, the efficiency loss may be negligible. In a repeated partnership game, however, Radner, Myerson, and Maskin (1986) find that with imperfect monitoring any equilibrium payoff in the repeated game is bounded away from an efficient outcome. See Fudenberg, Levine, and Maskin (1994) for a reconciliation of these results.

(4) It is well known that partial or incomplete revelation typically occurs in single-period cheap-talk games; see, for instance, Crawford and Sobel (1982).

(5) For instance, the agent may have satisfied the requirements to pass the review well before the end of the review phase and thus profits by behaving opportunistically for the remainder of the review phase.

(6) While managers and investors do not have infinitely long lives, modelling the game as an infinite-period game eliminates distortions in the players' strategies caused by the end game. Equivalently, the game may be thought of as a finite-period game with a stochastic terminal stage.

(7) The noncash assets in place may include intangible assets (such as goodwill) that are not transferable, or tangible assets (such as specialized machinery designed for the firm's particular purpose) that can be sold only as scrap.

(8) Modigliani and Miller (1958) show that under certain conditions, this equality holds provided the capital structure does not affect the economic earnings. Thus the firm may propose pure debt, equity, or a mixture of debt and equity to finance the new project, as in venture capital financing, and this will not affect the economic earnings.

(9) The subscripts on random variables are suppressed when it will not lead to confusion.

(10) Information is either proprietary or nonproprietary. Nonproprietary information alters the firm's stock price but not the objective distribution of the firm's future, unknown economic earnings, and proprietary information changes the objective distribution of economic earnings (see Dye, 1986). This article deals with nonproprietary information. The results are unchanged, however, if the manager discloses proprietary information, provided its release is not too costly.

(11) This article considers the effect of the manager's voluntary disclosure on the investor's investment decision. The investor's choice has therefore been specified directly rather than deriving it from her payoff function. To eliminate redundant notation, her payoff function, which is potentially complex, has not been defined formally.

(12) Specifically, the SEC regulations, in addition to requiring an audit of the annual report, require that an independent public accountant review the quarterly financial information included with the annual financial statements (SEC, 1975).

(13) The manager's payoff, in expression (3), is set equal to the opportunity's contribution to firm value. Introducing a positive proportionality factor would not alter the results.

(14) Thus when expected economic earnings are positive, a neologism-proof equilibrium to the game does not exist. In contrast, when expected earnings are not positive, there is a unique neologism-proof equilibrium in which no communication occurs. See Farrell (1993) for further details.

(15) Recall that the largest value that a assumes is [a.sub.l].

(16) The possibility of inappropriately invoking the punishment phase is a natural consequence of using an imperfect monitor to assess the manager's truthfulness.

References

BENABOU, R. AND LAROQUE, G. "Using Privileged Information to Manipulate Markets: Insiders, Gurus, and Credibility." Quarterly Journal of Economics, Vol. 107 (1992), pp. 921-958.

BOTOSAN, C.A. "Disclosure Level and the Cost of Equity Capital." Accounting Review, Vol. 72 (1997), pp. 323-349.

BREALEY, R.A. AND MYERS, S. Principles of Corporate Finance. New York: McGraw-Hill, 1984.

CRAWFORD, V.P. AND SOBEL, J. "Strategic Information Transmission." Econometrica, Vol. 50 (1982), pp. 1431-1451.

DYE, R.A. "Proprietary and Nonproprietary Disclosures." Journal of Business, Vol. 59 (1986), pp. 331-366.

FARRELL, J. "Meaning and Credibility in Cheap-Talk Games." Games and Economic Behavior, Vol. 5 (1993), pp. 4514-4531.

-- AND RABIN, M. "Cheap Talk." Journal of Economic Perspectives, Vol. 10 (1996), pp. 103-118.

FUDENBERG, D., LEVINE, D.I., AND MASKIN, E. "The Folk Theorem with Imperfect Public Information." Econometrica, Vol. 62 (1994), pp. 997-1039.

GRUNDFEST, J.A. AND PERINO, M.A. "Securities Litigation Reform: The First Year's Experience." Working Paper no. 140, Stanford University Law School, 1997.

KIM, J.-Y. "Cheap Talk and Reputation in Repeated Pretrial Negotiation." RAND Journal of Economics, Vol. 27 (1996), pp. 787-802.

KING, R. "The Liar's Discount." Forbes, May 30, 1988, pp. 80-82.

LEES, EA. Public Disclosure of Corporate Earnings Forecasts. New York: The Conference Board, Inc., 1981.

MILGROM, P.R. "Good News and Bad News: Representation Theorems and Applications." Bell Journal of Economics, Vol. 12 (1981), pp. 380-391.

-- AND ROBERTS, J. "Relying on the Information of Interested Parties." RAND Journal of Economics, Vol. 17 (1986), pp. 18-32.

MODIGLIANI, F. AND MILLER, M.H. "The Cost of Capital, Corporate Finance and the Theory of Investment." American Economic Review, Vol. 48 (1958), pp. 261-297.

MORRIS, S. "An Instrumental Theory of Political Correctness." Mimeo, Department of Economics, University of Pennsylvania and Northwestern University, 1998.

OSBORNE, M.J. AND RUBINSTEIN, A. A Course in Game Theory. Cambridge, Mass.: MIT Press, 1994.

PINCUS, A.J. "The Reform Act: What CPAs Should Know." Journal of Accountancy, September 1996, pp. 55-58.

POWNALL, G. AND WAYMIRE, G. "Voluntary Disclosure Credibility and Securities Prices: Evidence from Management Earnings Forecasts, 1969-73." Journal of Accounting Research, Vol. 27 (1989), pp. 227-245.

RADNER, R. "Repeated Principal-Agent Games with Discounting." Econometrica, Vol. 53 (1985), pp. 1173-1198.

--, MYERSON, R., AND MASKIN, E. "An Example of a Repeated Partnership Game with Discounting and with Uniformly Inefficient Equilibria." Review of Economic Studies, Vol. 53 (1986), pp. 59-69.

SECURITIES AND EXCHANGE COMMISSION (SEC). "Notice of Adoption of Amendments to Form 10-Q, Regulation Regarding Interim Reporting." Accounting Series Release, No. 177, 1975.

SHIN, H.S. "News Management and the Value of the Firm." RAND Journal of Economics, Vol. 25 (1994), pp. 58-71.

SOBEL, J. "A Theory of Credibility." Review of Economic Studies, Vol. 52 (1985), pp. 557-573.

WILSON, R.B. "Reputations in Games and Markets." In A.E. Roth, ed., Game-Theoretic Models of Bargaining. New York: Cambridge University Press, 1985.

Appendix

Proof of Proposition 1. The proof uses the following definitions:

[Psi] denotes the probability of failing the review conditional on being at the beginning of the review phase when the players use the specified review strategies;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the conditional probability that the investor will assess disclosure of m = g to be truthful when the manager observes f = B;

U([I.sub.R-n] = i) is the manager's normalized discounted expected payoff when there are n stages left in the review phase and the trust index at stage R - n equals i; [Lambda] = [[Sigma].sub.A] ([[Sigma].sub.[Omega]] [Omega] [multiplied by] p([Omega]|g, a))[[Pi].sub.a|f](a|B) - Q([Omega]|B) is the one-stage gain that the manager expects before observing a when he observes f = B, discloses m = g, and the investor believes him; and finally,

q [equivalent] R [multiplied by] S - J.

First, it is observed that neither player has a strict incentive to deviate from the equilibrium strategies by construction.

Second, it is shown that the manager's equilibrium strategy yields almost full revelation by establishing that U([I.sub.0]) [is greater than] Q([Omega]; F) - [Epsilon]. This result is obtained by proving that, first, the manager will only fail the review when [I.sub.t] [is less than] max(0, t - (R - q)), and second, the manager will almost always truthfully disclose his information.

The manager truthfully discloses when L(t) [is greater than or equal to] [I.sub.t] [is greater than] max(0, t - (R - q)), if he discloses truthfully when [I.sub.R-n] = q - n.

Claim A1. Let [I.sub.R-n] = q - n and consider n such that 1 [is less than or equal to] n [is less than or equal to] q. There exists [Delta]' and S' such that for all [Delta] [element of] [[Delta]', 1) and S [element of] [S', 1), there exists R' such that for all R [is greater than] R' the manager discloses truthfully.

Proof of Claim A1. Consider the case where the manager observes f = B. For any R, [I.sub.R-n] = q, and any n such that 1 [is less than or equal to] n [is less than or equal to] q, the manager reports truthfully if

(A1) (S - x)/1 - [Delta] [Delta]{U([I.sub.R-(n-1)] = q - (n -1)) - [(1 - [[Delta].sup.K]Q([Omega]) + [[Delta].sup.K]U([I.sub.0])]} [is greater than or equal to] [Lambda].

After substituting in

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which holds for all n such that 1 [is less than or equal to] n [is less than or equal to] q, and taking the limit as [Delta] [right arrow] 1, the left-hand side of expression (A1) converges uniformly in R and S to

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now substituting for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and taking the limit as S [right arrow] 1 yields

(A4) (1- x){(Q([Omega]; F) - U([I.sub.0]))(n - 1) + K(U([I.sub.0]) - Q([Omega]))}.

Note that 1 - x [is greater than] 0, U([I.sub.0]) [element of] (Q([Omega]), Q([Omega]; F)), and [Lambda] is bounded. Given that K = [Mu]R, there exists R' such that for all R [is greater than] R', the manager reports truthfully that f = B. Finally, because the manager's expected gain from sending the message m = b when he observes f = G is negative and the probability of being assessed as disclosing truthfully is the same for m = g as for m = b, it is immediate that the manager will reveal that f = G. Q.E.D.

If the manager passes the review for some [h.sub.t], denote the stage that the review phase is terminated as t'. For [Delta] [element of] [[Delta]', 1), S [element of] [S', 1), and R [is greater than] R', it can be shown that the manager truthfully discloses when [I.sub.0] = 0, and therefore t' [is greater than] 0 for [H.sub.t].

Claim A2. For any [Epsilon] [is greater than] 0, there exists [Delta] [is greater than or equal to] [Delta]' and S [is greater than or equal to] S' such that for all [Delta] [element of] [[Delta], 1) and S [element of] [S, 1), there exists R [is greater than or equal to] R' such that for all R [is greater than] R, U([I.sub.0]) [is greater than] Q([Omega]; F) - [Epsilon].

Proof of Claim A2. It follows from Claim A1 that a sufficient condition for obtaining a lower bound on U([I.sub.0]) is to assume that during the review phase the manager reports in good faith but if he fails the review, then he fails at the earliest possible stage, t = R - q. Suppose that the manager passes the review at some t'. The Markovian property of this repeated game implies that

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solving for U([I.sub.0]) and then taking the limit as [Delta] [right arrow] 1 yields

(A6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that t' [is greater than] 0. Further, given J = [Beta][R.sup.[Rho]], where [Beta] [is greater than] 0 and 1/2 [is less than] [Rho] [is less than] 1, and using Lemma A1 below, it follows that [Psi], which is bounded above by [R [multiplied by] S [multiplied by] (1 - S)]/[J.sup.2], approaches zero as R increases without limit. Also, K = [Mu]R, where [Mu] [is greater than] 0, implies [lim.sub.R[right arrow][infinity]] K/(R - q) = [Mu]/(1 - S) [is greater than] 0. Hence, for any [Epsilon] [is greater than] 0, there exists [Delta] [is greater than or equal to] [Delta] and S [is greater than or equal to] S' such that for all [Delta] [element of] [[Delta], 1) and S [element of] [S, 1), there exists R [is greater than or equal to] R' such that for all R [is greater than] R, U([I.sub.0]) [is greater than] Q([Omega]; F) - [Epsilon]. Q.E.D.

Thus, the manager almost always reveals his private information. Q.E.D.

Lemma A1. For all [Delta] [element of] [[Delta]', 1), S [element of] [S', 1), and R [is greater than] R', [Psi] [is less than or equal to] [R [multiplied by] S [multiplied by] (1 - S)]/[J.sup.2].

Proof of Lemma A1. Given [Delta] [element of] [[Delta]', 1), S [element of] [S', 1), and R [is greater than] R', in equilibrium the manager discloses truthfully and may pass the review before stage R and will fail the review if for any t [is less than] R, [I.sub.t] [is less than] max(0, t - (R - q)); see Claim A1. Since once the manager passes the review, the review phase is terminated, it can be shown that [Psi] is less than or equal to the probability of failing the review if the manager can pass the review only at the end of period R but can fail the review before period R if, for any t [is less than] R, [I.sub.t], [is less than] max(0, t - (R - q)). It now can be established that the probability of failing the review under the latter specification equals the probability of failing the review when the manager is evaluated at period R only and is said to fail the review if [I.sub.R] [is less than] q and pass if [I.sub.R] [is greater than or equal to] q, denoted Pr([I.sub.R] [is less than] q).

If the manager discloses truthfully and is evaluated at period R only and is said to fail the review if [I.sub.R] [is less than] q and pass if [I.sub.R] [is greater than or equal to] q, then [I.sub.R] is distributed as a binomial random variable. It then follows from Chebyshev's theorem that Pr([I.sub.R] [is less than] q) [equivalent] Pr([I.sub.R] [is less than] R [multiplied by] S - J) [is less than or equal to] [R [multiplied by] S [multiplied by] (1 - S)]/[J.sup.2]. Q.E.D.

Phillip C. Stocken(*)

(*) University of Pennsylvania; pstocken@wharton.upenn.edu.

This article has benefited greatly from helpful discussions with Kalyan Chatterjee, John Fellingham, Vijay Krishna, and James McKeown, and from comments by the Editor, two anonymous referees, and participants in workshops at the University of California-Berkeley, Carnegie Melon University, University of Chicago, Harvard University, London Business School, University of North Carolina, University of Pennsylvania, The Pennsylvania State University, University of Rochester, University of Southern California, the Eighth Annual Conference on Financial Economics and Accounting (Buffalo, New York, 1997), and the 1997 American Accounting Association annual meeting. Financial support from the Deloitte & Touche Foundation and the G.H.R. Edmunds Memorial Trust is gratefully acknowledged.