Journal of Accounting and Economics 12 (1990) 365-380. North-Holland

INFORMATION QUALITY AND DISCRETIONARY DISCLOSURE

Robert E. VERRECCHIA*

Urwersi(~ oj Pennsylvmiu, Philudelphiu, PA I91 04, USA

Received January 1989, tlnal version received November 1989

Using the model of discretionary disclosure suggested in Verrecchia (1983), I show that an increase in the quality of private information received by a manager results in more disclosure on his part. I also discuss how this result influences a manager’s choice over levels of quality.

1. Introduction

The purpose of this paper is to discuss how the quality of information available to a manager affects his incentives to disclose or withhold that information in the presence of external parties who have rational expectations about his motivation.’ Previous work on managers’ incentives to voluntarily disclose proprietary and nonproprietary information includes Darrough and Stoughton (1990) Dontoh (1988) Dye (1986), Joh and Lee (1988) Jung and Kwon (1988) Lanen and Verrecchia (1987) and Wagenhofer (1990), as well as earlier papers by Dye (1985) and Verrecchia (1983).* Typically these papers deal with the relation between the manager’s incentives to disclose and the realization of the manager’s private signal (e.g., whether the information is ‘good news’ or ‘bad news’) and not the quality of the signal per se.3 In contrast, I am concerned with how the quality of the manager’s private signal

*I gratefully acknowledge financial assistance from Arthur Young & Company and the Wharton School of the University of Pennsylvania. I would also like to thank Gerald Feltham (the referee).

‘The idea for this paper was suggested to me by Judy Rayburn and Gordon Potter, who grapple with some of these issues in Potter and Raybum (1988). Conversations with Ron Dye helped in expanding on the original thrust of the discussion, and Rat% Indjejikian assisted in some numerical analysis. Helpful comments by Ray Ball and participants at the accounting workshops at the Universities of Alberta and Maryland are gratefully acknowledged.

‘See also the discussion of Darrough and Stoughton (1990) in Verrecchia (1990).

31t is easy to confuse notions of information ‘realizations’ with notions of information ‘quality’, as I do in the following passage: ‘He [the manager] exercises discretion by choosing the point, or degree of injormatiort quali(v, above which he discloses what he observes, and below which he withholds his information’ [Verrecchia (1983, p.197). emphasis added]. Presumably, information quality involves the distributional characteristics of an uncertain event (e.g., its variance), whereas a realization is simply the outcome of the event itself.

0165-4101/90/$3.5001990, Elsevier Science Publishers B.V. (North-Holland)

366 R. E. Verrecchia. Informurion qua& and discretionaty disclosure

influences his decision to disclose or withhold that information (holding the realizations fixed), and how the answer to this question affects the manager’s decision to choose, or precommit, to levels of quality. Before addressing these questions, I briefly review a model of discretionary disclosure that provides the basis for my discussion.

2. A model of discretionary disclosure

To expedite the discussion, I rely on the analysis offered in Verrecchia (1983). That paper analyzes a model in which a manager of a risky asset (e.g., ‘the firm’) exercises discretion in the disclosure of information in the presence of market traders who have rational expectations about his motivation to withhold unfavorable reports. The information is a signal y” that reveals the true, liquidating value of the risky asset ii perturbed by some noise E”. That is, y” = ii + E, where the tilde is used to suggest events that are unknown before their realizations. The market’s prior beliefs about the liquidating value of the risky asset, ii, are that this value has a normal distribution with mean y, and precision (i.e., the inverse of variance) of h,. The random noise in the manager’s signal, El, has a normal distribution with mean zero and precision S. I interpret s as the quality of the manager’s information about the uncertain liquidating value of the risky asset. This is consistent with the idea that as s increases (decreases) the manager’s private information about ii is more (less) precise.

The manager exercises discretion by choosing the point i above which he discloses what he observes, and below which he withholds his information. This model also assumes that when the manager reveals his private signal, the liquidating value of the risky asset is reduced by some amount c, which is interpreted as the cost associated with disseminating proprietary information about the firm to external parties.

The manager’s incentive is to maximize the market price of the risky asset, conditional on observing his signal. The discussion in this paper focuses primarily on that circumstance in which the risky asset is priced at its expected value (conditional of whatever information is available to the market at that time). That is, there is no adjustment for firm-specific risk.4 In this event, when a manager observes J = y and discloses it, the price of the risky asset

41n Verrecchia (1983), the function /3(.) captures the degree to which the market prices (firm-specific) risk in its determination of the risky asset’s value, where p(.) is a continuous, nonnegative. nondecreasing function of its argument, which is measured in units of variance. The assumption that the market prices the risky asset at its expected value is equivalent to setting fl(.) = 0. In section 3, I briefly describe how relaxing the assumption that /?(.) = 0 can change the results of the analysis.

R. E. Verrecchiu, Informution qualicv and discretionan, disclosure 367

adjusts to

w=Y) =yo+ &Y-YJ-c.

When a manager withholds the observation 9 =y, the price of the risky asset becomes

P(j=ylx) =yo- 4hb) G(x) ’

where

G(x) = 1‘ g(t)dt. --cz

Therefore, the manager discloses his signal when the market reaction to that information (adjusted by the deadweight loss of c) equals or exceeds the market reaction to its withholding (assuming that this reaction rationally anticipates a manager’s motivation to withhold unfavorable reports).

To provide some intuition about the relation between disclosure and infor- mation quality, consider the extreme cases in which 9 provides no information

about u (i.e., s + 0) and +F provides perfect information about u (i.e., s + co). If J provides no information, the price of the risky asset if y” is disclosed becomes

The price of the risky asset if _F is withheld becomes

hiI)P( j =y 2 x) =y,, (2)

since it can be shown that

l.m h&d4 .,‘-o G(x)

--f 0.

This implies that regardless of whether j is disclosed, the market’s expecta- tions about ii are that its mean and variance are equal to its prior mean of yO and prior variance of hi’. However, disclosing j requires that the firm absorb

368 R. E. Verrecchia, Information quality and discretionaT disclosure

a proprietary cost of c. Consequently, as should be clear by comparing eqs. (I) and (2) above, the market price of the risky asset is always higher if _fj is withheld. Therefore if j provides no information, a manager wants to pre- clude the possibility that the signal is disclosed since it has no effect on market expectations and, in addition, entails a cost. Thus the equilibrium threshold level of disclosure, 2, is set to be infinite.

Suppose, alternatively, that 2 provides perfect information. Here, the price of the risky asset if jj is disclosed becomes

lim P(y”=y)=y-c. (3) s - 01

The price if I; is withheld becomes

lim P(jj=ylx) =yO- lim hhb)

s-cc s-0 G(x) . (4)

This implies that if j is disclosed and provides perfect information, it reveals that the true liquidating value of the risky asset is y with no uncertainty. If, however, F is withheld, the price of the risky asset does not fall to negative infinity, thereby forcing the manager to fully disclose. That is, lim, -c ,P( jj = y I X) is a finite-valued function for finite values of x.~ This has two implica- tions. First, even in the event that j provides perfect information about U, the threshold level of disclosure 2 is finite. In equilibrium some (perfect) informa- tion is disclosed and some (perfect) information is withheld. Second, because 2 is finite-valued when information is perfect, the threshold level of disclosure falls below infinity, thereby suggesting that more disclosure results from perfect information versus no information.

Can the suggestion that more disclosure results from information of higher quality be established more generally? In general, requiring a manager to maximize the price of the risky asset on the basis of his choice of a threshold level of disclosure, and rational inferences in the event the information is withheld, is equivalent to determining an 2 as a solution to the equation c = F(2), where

F(x) = &-&b -Yd + hii’d G(x)

[see Verrecchia (1983)].6 Therefore if it can be shown that .? [determined as

‘For example, when the market prices the risky asset at its expected value and s -+ co, .C is determined as a solution to x =.vo + c-J(x), where J(X) = exp( -h,[x -4b]*/2)/ (/q,J”exp( - h,[t -.v0]*/2) dr). This implies that Z is finite-valued.

6This expression differs from the one in Verrecchia (1983) in that here it is assumed that the asset is priced at its expected value, i.e.. p(.) = 0.

R. E. Verrecchia, Information quality and discretionaty disclosure 369

a solution to the equation c = F(2)] falls as s increases, this will establish that higher-quality information implies more disclosure.

3. Information quality when assets are priced at their expected value

When assets are priced at their expected value, the relation between the quality of the manager’s information and the threshold level of disclosure is unambiguous. As the information becomes more (less) precise, the threshold level of disclosure falls (rises). This implies, for example, that in contrasting information sources of two distinct qualities, e.g., ‘high’ versus ‘low’, there exist realizations of high-quality signals that a manager would be forced to disclose even though identical realizations from low-quality sources would be withheld (and the reverse is never true).

The intuition for this result is that a manager’s incentive to disclose or withhold is motivated in part by the market’s expectations in the absence of the information. As the quality of the information increases, the market exerts more pressure on the manager to disclose the information by discounting the value of the risky asset more sharply in the event the information is withheld. This induces the manager to reduce the threshold level of disclosure below the one he adopts when the information is of lower quality.

Corollary I. Suppose that the market prices the risky asset at its expected value. Then the threshold level of disclosure decreases as the precision of a manager’s information increases.’

Proof. See the appendix.

At a very simple level, Corollary 1 is consistent with a number of stories one might tell. For example, if a manager has an astrologer’s report as to the firm’s risky prospects (say, prospects ‘look poor’), the market may be indifferent to its disclosure because an astrologer is perceived as a ‘low-quality’ information source. But the withholding of a detailed, financial analysis that reports exactly the same thing (i.e., prospects ‘look poor’) might ignite a sharp, negative market response because this report is perceived as originating from a ‘higher-quality’ information source, and thus its withholding promotes more discounting.

A result that might be interpreted as similar to Corollary 1 is found in Jung and Kwon (1988) who extend a model suggested by Dye (1985). In that paper Dye imagines that the manager observes privately, with some probability, the realization of the liquidating value of the firm (i.e., ii = u in the model discussed above) and observes nothing otherwise. Unlike Verrecchia (1983)

‘AS I show in the appendix, this relation is strict for all finite 1.

370 R. E. Verrecchiu, Information qua1it.v and discretionan, disclosure

the existence of private information is uncertain from the perspective of external parties in Dye (1985). This uncertainty provides the ‘noise’ that supports the withholding of information in the same fashion that the propri- etary cost provides ‘noise’ in the model sketched here. In their extension of Dye (1985) Jung and Kwon (1988) show that as the probability that the manager observes ii = u increases, the threshold level of disclosure falls (see Proposition 2 of their paper). If one interprets an increase in the probability that a manager is privately informed as an increase in the quality of his information (since in the absence of being informed the manager knows nothing beyond what is commonly known), then Proposition 2 of Jung and Kwon is similar to Corollary 1. It is not entirely clear, however, whether an increase in the probability that a manager is informed should be interpreted as a measure of quality.

There is a series of results concerning information quality and disclosure that are related to Corollary 1 through the behavior of the threshold level of disclosure, and 1 briefly review three of them here. The first involves demon- strating that as the precision of the prior beliefs about ii (i.e., h,) increases, the threshold level of disclosure increases. The intuition here is that the more is known about the risky asset a priori (or commonly), the less pressure is exerted by the market on a manager to reveal what he knows privately.

Corollary 2. Suppose that the market prices the risky asset at its expected value. Then the threshold level of disclosure increases as the precision of prior beliefs about ii, h,, increases.8

Corollary 2 raises the following question. How are the precisions of prior beliefs and a manager’s private information related? If, for example, assets whose liquidating values are very uncertain (i.e., low prior precision) require closer monitoring on the part of managers (i.e., high quality of private information), then Corollaries 1 and 2 above are reinforcing in that they drive threshold levels down simultaneously. Alternatively, if prior precision and the quality of a manager’s information are positively related, Corollaries 1 and 2 are countervailing, thereby confounding any testable hypotheses. I leave this as an unresolved empirical question.

Information quality influences the probability of disclosure, as well as the threshold. The probability that a manager discloses what he observes is 1 - G(Z), where G(.) is as defined in section 2 and f is the threshold level of disclosure. Although it might seem self-evident from Corollary 1 that the probability of disclosure increases as s increases because the threshold level of disclosure falls, the issue is more subtle than that. As s shifts it also changes

‘Because of the similarity of this result to Proposition 3 of Jung and Kwon (1988). I simply state it without proof. The proof is available on request.

R. E. Verrecchiu, In/ormation qua&v und discretionary disclosure 371

the distribution of J, and this can have a countervailing effect. For example, when the threshold level of disclosure is above the market’s priors, i.e., yO, a small increase in s lowers the threshold but also concentrates more signals around y,, and hence below the threshold. The lowered threshold serves to increase the probability of disclosure, but the greater mass around y0 tends to decrease it. Nonetheless, the probability of disclosure does increase as the precision in a manager’s observations rise.

Corollary 3. Suppose that the market prices the risky asset at its expected value. Then the probability of disclosure increases as the precision of a manager’s information increases.

Proof. See the appendix.

Similarly, as the precision of the prior beliefs about ii (i.e., h,) increases, the probability of disclosure falls.

Corollary 4. Suppose that the market prices the risky asset at its expected value. Then the probability of disclosure falls as the precision of prior beliefs about ii, h,, increases.g

In short, Corollaries 3 and 4 on the probability of disclosure parallel Corollaries 1 and 2 on the threshold level, but they are not equivalent mathematical statements.

Although the economic intuition underlying Corollaries 1 through 4 is compelling, there are reasons why these results might not hold more generally. Two specific concerns are that these results depend on how ‘information quality’ is characterized, and the fact that risk is not priced.

To demonstrate problems that arise with a different characterization of information quality, consider the following example. (Details to this example are offered in tables 1, 2, and 3.) Let P(.) represent the probability of the term in its argument, and imagine that the realizations of the liquidating dividend ii are u = 3, u = 2, u = 1, with probabilities P(u = 3) = 0.89, P(u = 2) = 0.1, and P( u = 1) = 0.01, respectively. Consider two cases that differ only in the quality of the private information the manager receives. In case I, suppose that the private signal observed by the manager is the realization of the liquidating dividend uncontaminated by noise. That is, when u = 3 he observes y = 3, when u = 2, y = 2, and when u = 1, y = 1. If the proprietary cost of disclosure c is equal to 0.1, then the manager always discloses his private information when he observes y = 3 (which implies u = 3) and withholds otherwise since the proprietary cost does not warrant discriminating between an observation

‘The proof is available on request.

J.A.E.- B

312 R. E. Verrecchiu, Inform&ion quality and discretionary disclosure

Table 1

An example of lower-quality information resulting in more disclosure.

Characteristics of the distribution of the liquidating dividend

Realization of the dividend u=3 u=2 u=l

Probability of realization P(u=3)=0.89 P( u = 2) = 0.1 P(u=l)=O.Ol

Expected liquidating value E[ a] = 2.88

Characteristics of case I Signal

Realizations of the signal y = 3 y = 2 ,!J=l

Probability of realization P( y = 3) = 0.89 P( .v = 2) = 0.1 P( I’ = 1) = 0.01

Probability of realization of signal conditional on P(y=3]u=3)=1 P(y=2]u=2)=1 P(y=l]u=l)=l realization of dividend

Expected liquidating value conditional on the signal E[ir]y = 3]= 3 E[ir(y=2]=2 PJir]_V=l]=l

Characteristics of case II Signal

Realizations of the signal .v=3 .v = 2 .v = 1

Probability of realization P(v=3)=0.9 P( y = 2) = 0.045 P( y = 1) = 0.055

Probability of realization of signal conditional on P(?,=3(u=3)=1 P(y=2]u=2)=0.45 P(y=l(u=2)=0.45 realization of dividend P(?,=3(u=2)=0.1 P(y=l(u=l)=l

Expected liquidating value conditional on the signal E[ ii (y = 3]= 2.988 E[ii(y = 2]= 2 E[ii]y = 1) = 1.818

of y = 2 versus y = 1 (see table 2 for further details). In particular, this implies that the threshold level of disclosures lies above y = 2.

In case II, suppose that the manager’s information system is such that when u = 3 he observes y = 3, and when u = 1 he observes y = 1, but when u = 2 his signal is contaminated with noise such that he observes y = 3, y = 2, or y = 1 with the following probabilities: P( y = 3 (u = 2) = 0.1, P( y = 2 (u = 2) = 0.45, and P( y = 1 (u = 2) = 0.45. One feature of this example is that in both case I

Table 2

Case I equilibrium demonstrating a threshold level above 2.

Discretionary disclosure equilibrium for case I when proprietary cost is c = 0.1

E[ii]~v=3]-r=2.9>E[li]y=1,2or3]==E[fi]=2.88

E[ii(,r = 2]- c = 1.9 < Qii]y = 1 or 2, but not 3]= 1.909

E[ii]~=l]-c=0.9iEjii]y=1or2,butnot3]=1.909

R. 15. Verrecchio, Information quuliy und discretionur?, disclosure 373

Table 3

Case II equilibrium demonstrating a threshold level of 2.

Discretionary disclosure equilibrium for case II when proprietary cost is c = 0.1

E[li]~, = 31 - c = 2.888 > E[ii(_r = 1, 2, or 3]= F$2i] = 2.88

E[ii]y = 2]- c = 1.9 = E[ii]y = 1 or 2, but not 3]= 1.9

E[iiJr = l] - c = 1.718 < F$ii]v = l] = 1.818

and case II a signal of y = 2 reveals that u = 2. Nonetheless, in case II the threshold level drops to exactly y = 2. The manager discloses his private information when y = 3 or y = 2, but withholds otherwise (see table 3 for further details). lo What drives this example is that when the manager with- holds information in case I, the realization of ii is interpreted as 2 versus 1 with a probability distribution proportional to the unconditional probability of 2 versus 1. However, if the manager withholds information when y = 2 or 1 in case II, the probability that the realization of 6 is 2, versus 1, is fess likely than in case I, because a realization of u = 2 also implies a signal of y = 3. Therefore, in case II, a manager is motivated to discriminate y = 2 from y = 1.

The significance of this example is that the manager’s information in case II is clearly inferior to that in case I, using any reasonable definition of informa- tion quality. Nonetheless, the threshold level of disclosure falls in going from case I to case II. This demonstrates that Corollary 1 is more subtle than it might appear initially, and might not extend to alternative notions of informa- tion quality. In particular, Corollary 1 depends on the symmetry implicit in all realizations of ii being contaminated with the same distribution of error.

In addition to problems with its extension when the notion of information quality is changed, Corollary 1 does not necessarily extend to cases in which risk is priced. ‘I The reason for this is that when the precision of a manager’s information increases and risk is priced, two countervailing effects can result. As captured by Corollary 1, a first-order effect lowers the threshold level of disclosure through the market exerting more pressure on the manager. How- ever, when the market prices risk, the price of the (risky) asset might be

“‘An artifact of the model, which shows up prominently in this example, is that even though a manager ‘withholds’ J’ = 1, it is revealed through the fact that y = 3 and _r = 2 are disclosed. Nonetheless, by ‘withholding’ .v = 1. the liquidating dividend absorbs no deadweight loss in this event. This artifact exists because the proprietary cost is c or zero depending upon whether the manager discloses or withholds. In particular, the cost is independent of how much information is revealed through the manager’s disclosure strategy. (In this context, one could interpret the proprietary cost as a fee paid by the manager to certify the veracity of his disclosure.) One refinement of the model would be to suggest a proprietary cost that depends upon the withholding interval (i.e., ( - M, a]) in the event that information was withheld. This approach invites other problems: see, for example, the discussion in Lanen and Verrecchia (1987).

“In Verrecchia (1983), the market prices risk whenever /3(.) f 0.

374 R.E. Verrecchia, Information quality und discretionuy disclosure

reduced by a risk premium. ‘* Therefore, increasing the precision of the manager’s information has a potential second-order effect of reducing uncer- tainty, which might decrease the risk premium. This is because higher-quality information reduces the discount for risk vis-8-vis lower-quality information. Consequently, increasing the quality of the information can raise the thresh- old level of disclosure, provided that the reduction in the risk premium offsets the increased pressure on the manager through expectations in the informa- tion’s absence.13

In other words, the existence of risk premia might confound the relation between information quality and threshold levels of disclosure. The relation suggested in Corollary 1 might be stronger for assets (or, perhaps, markets for assets) where uncertainty plays a smaller role. For example, larger-sized firms might attract more, better diversified groups of shareholders (e.g., large institu- tional investors), whose greater risk tolerance leads to less discounting. In this event, higher-quality information might be more positively related to greater disclosure for larger- versus smaller-sized firms, although this remains an unresolved empirical question.

Finally, note that while I focus my attention on Corollary 1 in discussing problems with extending my results more generally, these problems are en- demic to Corollaries 2, 3, and 4 as well. This is because all these results are linked through the behavior of the threshold level of disclosure. To suggest one illustration, note that in the example above concerning alternative notions of information quality, the probability of disclosure in case I is 0.89 because here only the signal y = 3 is disclosed. However, the probability of disclosure rises to 0.945 in case II because here both the signals y = 3 and y = 2 are disclosed. Therefore, the probability of disclosure increases as the quality of information falls, using any reasonable notion for information quality.

4. Endogenous information quality

To this point, I have treated the quality of the information, S, as exogenous. This is consistent with assuming that the manager is endowed with informa- tion of quality s and has no control over its choice. The final question concerns how a manager might choose s in light of the discussion above.

Note that in Verrecchia (1983) the manager’s private information has no intrinsic value. That is, the information does not assist in managing the asset better or otherwise improving its liquidating performance.14 Assuming that a

“This assumes, of course. that shareholders cannot diversify away firm-specific risk.

13The key to constructing an example that demonstrates this claim is to allow the rate of discount for risk [e.g.. the gradient of /3(_)] to fall as the precision of information increases. An example of this is available on request.

14Lanen and Verrecchia (1987) discuss how different ways of precommitting to use and disclose information affect various types of discretionary disclosure equilibria, when information has intrinsic value.

R. E. Verrecchia, Information qurrlig md discretionary disclosure 375

manager could choose s before he observed 9, and he wanted to maximize the ex ante market price of the asset, and the information itself had no intrinsic value, and risk is not priced, the manager would always precommit never to receive any information. That is, he would elect s = 0, since at this level of precision he learns nothing. The reason for this is that for any positive S, there is a possibility of disclosing what he observes, and this deadweight loss reduces the asset’s value by c. Thus, by receiving no information whatsoever, the manager ensures no deadweight loss and thereby maximizes the ex ante

market price of the asset.

Corollary 5. Suppose that the market prices the risky asset at its expected value.

If the manager seeks to maximize the ex ante price of the risky asset, then he

always precommits to receive no information, i.e., s = 0.

Proof. See the appendix.

Corollary 5 should not be interpreted too broadly. When risk is priced, a manager might be motivated to obtain costless private information even in the presence of deadweight losses associated with its disclosure. This is because the reduction in discounting when information is revealed, thereby increasing the expected market price, might offset the proprietary cost associated with its disclosure.” Furthermore, the possibility that the information has intrinsic value also encourages its receipt (whereas the possibility that the private information is costly to acquire militates against its acquisition). My point is simply to suggest that all these elements should be considered in a model in which s is endogenous.

5. Conclusion

In this paper I show how a change in the quality of information received by a manager affects the manager’s threshold level of disclosure and the probabil- ity of disclosure. Information of higher quality implies a lower threshold level of disclosure and a greater probability of disclosure, holding realizations themselves fixed. The intuition for this is that when a manager withholds information of higher quality, the market discounts the value of the asset further than it would otherwise. This forces the threshold level down. Further- more, the threshold falls far enough to increase the probability of disclosure even though the distribution of the manager’s private information shifts simultaneously. These results are applied to show that a manager might precommit to receive no information, so as to preclude the possibility of disclosing what he observes.

Note, however, that all the results depend on a representation of informa- tion quality implicit in Verrecchia (1983). Furthermore, when risk is priced,

“An example of this is available on request.

316 R. E. Verrecchiu, Information quuli<v and discretionury disclosure

the relations are not necessarily unambiguous because higher-quality informa- tion has a potential secondary effect of lowering the discount on the asset imposed by the market for uncertainty. This, in turn, might raise the threshold level. Despite these caveats, the intuition that higher-quality information is accompanied by more disclosure appears to be a robust economic notion (at least as a first-order effect), and thus might be useful for assisting future empirical investigations.

Appendix

A. 1. Proof of Corollary I

The technique for proving relations based on Mill’s Ratio was first sug- gested by Sampford (1953), and the following proof benefits from those suggestions. When the market prices the risky asset at its expected value, the threshold level of disclosure 2 is determined by

(A-1)

from eq. (9) of Verrecchia (1983, p. 188) and the fact that p(.) = 0. What I want to establish is that ax/as -C 0. Because the proof is complicated, I will prove this relation through a series of lemmas. My first lemma is to establish an expression for ax/as. To ease the notational burden, I drop the zero subscript on y, and h,, and designate the threshold level of disclosure simply by x (without the caret).

Lemma 1.

ax h -1 _=_ as 2s(h+s)

hk(x) - & H(x),

where

H(x) = h-k(x)

&(X-Y)+ G(x> +b-djhk(x)-&). Proof. Taking the derivative with respect to s on both sides of eq. yields

/h+s\/ d h-‘g(x)\dx (A.3)

64.2)

(A-1)

R. E. Verrecchiu, Informution quuli(v and discretionary disclosure 371

The following relations are simple exercises to show:

h_‘g(x) 5

(‘- G(x) = h+s i 1 - b-Y>

from eq. (A.1)

d h-‘g(x)

ds G(x) =---

- &(x 4’

d h-‘g(x)

dx G(x) =/&(x)-l,

where

k(x) =h-l--

Substituting in for these relations in eq. (A.3) and rearranging terms yields the expression for dx/as given in eq. (A.2). Q.E.D.

My objective is to show that the right-hand side of eq. (A.2) is negative for all finite x (henceforth, when I say ‘all x’, I mean all finite x). Note that the right-hand side of eq. (A.2) is negative provided that H(x) > 0 for all x, since k(x) > (h + s)-i for all x [see lemma 3 of Verrecchia (1983)]. For the same reason it should be clear that H(x) > 0 for all x 2 y, since k(x) > (h + s)-l. Consequently, if 1 can show that H(x) > 0 for all x <y, I will have estab- lished that ax/as < 0. The way I show that H(x) > 0 for x <y is to assume otherwise and then demonstrate that a contradiction results. Consider first what it means for there to exist some x < y such that H(x) I 0.

Lemma 2. Suppose there exists some finite z E ( - cc), y) such that H(z) I 0. Then there must also exist some$nite t E (- co, y) such that

(i) H(t) ~0 and (ii) &H(t) =O.

Proof. I can show that lim,, _m H(x) + 0. Therefore, H(x) --, 0 as x + - 00 and H(x) is positive when x > y. This implies (from Rolle’s Theorem) that if there exists a z such that H(z) I 0, then there exists a t E ( - 00, y) satisfying (i) and (ii) above. Q.E.D.

My next step is to prove that there exists no t E (- co, y) satisfying (i) and (ii) above, and thus there exists no z cs ( - CO, y > such that H(z) 5 0. First, I need an intermediate result that will be useful in proving my final lemma.

Lemma 3. Define the function L(x) such thut

ThenL(x)>OjorallxE(-oo,y).

Proof. Note that since 1~-*sk(x)~(h-Cs)-‘,forall xe(-GO,~)

Finally, I make use of Lemma 3 to show that there exists no t E (- CO, y) satisfying the conditions of Lemma 2 above.

Lemma 4. There exists no t E ( - 00, y) such that H( t > I 0 and d H( t )/dx = 0.

Proof. Note that

R. E. Verrecchia, Information quality and discretionary disclosure 379

substituting in first the expression for H(x), and then L(x), as given in Lemma 3 above, and rearranging terms. Suppose that H(t) I 0. Then note that when dH(.)/dx is evaluated at x = t, from eq. (A.4) dH(t)/dx must be positive (for all finite t). That is, dH(t)/dx > 0. This is because: first, h-’ > k(x) > (h + s)-l for all x [see lemma 3 of Verrecchia (1983)]; second, (y - t)hL(t) > 0 from Lemma 3 above and the fact that t < y; and, third, (t -y)h(l - hk(t))H(t) 2 0 since H(t) I 0 by assumption and t <y. There- fore there exists no t E (-cc, y) such that H(t) < 0 and dH(t)/dx = 0.

Q.E.D.

Since there exists no t E (- 00, y) such that H(t) < 0 and dH(t)/dx = 0, there can exist no z E (- cc, v) such that H(z) I 0. Consequently, H(.) is a positive function, which, in turn, implies that ax/as -=z 0 from eq. (A.2). That is, the threshold level of disclosure falls as a manager’s information becomes more precise. This proves Corollary 1.

A.2. Proof of Corollary 3

The probability of disclosure is 1 - G(f). Taking the derivative of this

expression with respect to s yields

a(1 -G(2))

a.9 = g+j-;W$g(‘)dr 1

ho

i

-1

=di.) 2(h,,+s)’

h,k(z?) - $-_ 0

/

z G(t) dt

X -m

G(1) “’ substituting in the expression for a.?/ds offered in the proof to Corollary 1, using integration by parts to establish that

I ’ h, + s

--oc G(t)dr = (x -y)G(x) + hsg(x),

0

and noting that k(x) > (h, + s)-l for all finite x. This establishes that the probability of disclosure increases as s increases. Q.E.D.

380 R. E. Verrecchia. Information qualit.v and discretionmy disclosure

A.3. Proof of Corollary 5

When risk is not priced, the asset is priced at E[ii]y” = y] - c when infor- mation is disclosed and E[ ii ]J = y 5 a] when information is withheld. Let the ex ante market price of the firm be represented by Q, where

Q=lm{Etalg=yl-c)g(y)dy+li E[fi(y”=ysZ]g(y)dy z -CC

=y,- [l- G(R)]c,

and g(.), G(.) are as defined in section 2. However, from Corollary 3,1 - G(P) increases as s increases, and thus taking the derivative of Q with respect to s yields

ap/as < 0

Since the ex ante market price of the firm declines as s increases, an optimal choice of s 1s s = 0. Q.E.D.

References

Dontoh. Alex, 1988, Voluntary disclosure, Working paper (University of British Columbia, Vancouver, BC).

Darrough. Masako N. and Neal M. Stoughton, 1990, Financial disclosure policy in an entry game, Journal of Accounting and Economics 12. 219-243.

Dye, Ronald, 1985, Disclosure of nonproprietary information, Journal of Accounting Research 23, 123-145.

Dye, Ronald. 1986, Proprietary and nonproprietary disclosures, Journal of Business 59, 331-366. Joh. Gun-Ho and Chi-Wen Jevons Lee. 1988, Strategic timing of financial disclosure, Working

paper (San Diego State University, San Diego, CA and Tulane University, New Orleans, LA). Jung. Woon-Oh and Young K. Kwon, 1988, Disclosures when the market is unsure of information

endowment of managers, Journal of Accounting Research 26, 146-153. Lanen. William N. and Robert E. Verrecchia, 1987. Operating decisions and the disclosure of

management accounting information, Journal of Accounting Research 25, 165-189. Potter, Gordon and Judy Rayburn, 1988, Interim reporting: An empirical study of factors

determining the reporting frequency of firms, Working paper (University of Minnesota, Minneapolis, MN).

Sampford, A.M., 1953, Some inequalities on Mill’s ratio and related functions, Annals of Mathematical Statistics 24, 130-132.

Verrecchia. Robert E., 1983. Discretionary disclosure, Journal of Accounting and Economics 5, 179-194.

Verrecchia. Robert E., 1990, Endogenous proprietary costs through firm interdependence, Journal of Accounting and Economics 12, 245-250.

Wagenhofer. Alfred, 1990, Voluntary disclosure with a strategic opponent, Journal of Accounting and Economics, this issue.