REPORTING BIAS AND INFORMATIVENESS IN CAPITAL MARKETS WITH NOISE TRADERS

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1 REPORTING BIAS AND INFORMATIVENESS IN CAPITAL MARKETS WITH NOISE TRADERS MARTIN HENRIK KLEINERT ABSTRACT. I discuss a disclosure model in which a manager can bias earnings reports. Informed traders acquire knowledge about the uncertain objectives of the manager and trade based on their superior information with noise traders. In the unique linear equilibrium, the market price corrects the report only for an expected bias. The price efficiency is influenced by the cost of biasing, the number of informed traders and the prior precision of the report. I calculate the liquidity of the market and the utilities of the market participants and I show that the noise traders benefit from more and better informed traders. KEY WORDS: Bias, Disclosure, Price Efficiency JEL Classification: M4, G4. Introduction Fischer and Verrecchia 2000) analyzed how a reporting bias affects the informativeness of prices when the reporting objective is uncertain for the capital market. They defined the bias as the difference between the realization of earnings and the manager s actual earnings report. Using a regression of the market price on the

2 2 earnings report, they showed, how the information content of the price is reduced by and varies with the bias in the report. My paper extends their model by introducing a market microstructure. Thereby, I can answer additional questions concerning the bias in the report and the market price: what happens if only some traders are capable to understand the incentives of the disclosing entity? 2 What is the influence of noise traders that trade for liquidity reasons or even irrationality? If a regulator aims to provide liquidity or to protect uninformed investors, what is her optimal action in this environment? In my model a manager who receives a noisy signal of the true firm value reveals a biased signal regarding the firm s value to some investors. If this signal is a public report, one can assume that only some investors actually read it or are able to detect the earnings management between the lines of the report. These informed investors obtain information about the distribution of the manager s incentives to influence the share price and her costs to in- or deflate the report; thereby, the utility of the manager is similar to Fischer and Verrecchia 2000). The price formation is modelled along the lines of Kyle 985), but I extend it to a situation with several informed traders. The risk neutral informed traders are aware of their influence on the price. They submit a market order to market makers who observe only the net order flow of informed and noise trading. Based on this order flow, they try to derive information on the report and the bias to assess the firm s true value. At the end of the game, the firm value is realized and each participant takes away her profit or loss. 2 In Framework 25 in IASB 2003), it is demanded: However, information about complex matters [...] should not be excluded merely on the grounds that it may be too difficult for certain users to understand.

3 3 In this setting, I show that a unique linear equilibrium exists and the results provided by Fischer and Verrecchia 2000) are robust. Additionally, the interdependence of the decision to bias of the manager and the order flow of informed traders can be analyzed. The informed traders correct the report for the expected bias and trade the more aggressive, the higher the variance in noise trading and the more precise their own signal is. The unexpected bias and the noise trading influence the equilibrium price. The market becomes more efficient, the more the informed investors trade. Although the marginal cost of biasing decreases the bias in the report and makes the price more efficient, it has also a negative effect on the market depth. Afterwards, I analyze the profits of the participants. The profit of the informed traders grows with the cost of biasing and the precision of the prior knowledge on the manager s objective and the variance of noise trading. Noise traders always loose on average, thereby, their loss decreases with the cost of biasing and all the precisions in the model. If the regulator wants to limit the losses of the noise traders and to increase liquidity, then it is necessary to increase the prior precision of the firm s value and the number of informed traders. The latter can be achieved, in the model, for example, by increasing the precision of the report of the manager to the informed traders. The remainder is organized as follows. Section 2 introduces the model, and section 3 shows that a unique linear equilibrium exists. Section 4 analyzes the

4 4 characteristics of the equilibrium and the price efficiency. Then, section 5 concentrates on the utilities of the participants and the number of informed traders. Finally, section 6 summarizes my results and discusses limitations of the model. 2. Description of the model The one-period disclosure game consists of four distinct events as depicted in the timeline in Figure. There are four groups of participants: A manager, market makers, N informed traders and a group of noise traders. In the following, the utilities and the assumptions about the random variables that are relevant at the different time points are explained chronologically. Figure. Description of events. Manager observes v + n and u and chooses bias b Informed traders learn i v + b and submit market orders x n i) Noise trading and price are established Realization of firm value and consumption The utility of the risk neutral manager is ) up cb2 2. Correspondingly, it depends on the share price p of the firm. Nevertheless, the direction and the magnitude are determined by the realization of the random variable ũ. Ex ante, this incentive uncertainty is normally distributed with mean µ u and precision τ u. Usually, one would expect that managers try to inflate the price. This

5 5 can be achieved in the model with a probability arbitrarily close to one by assuming a large expectation µ u and a high precision τ u. But there are also situations in which the manager tries to deflate the current share price, and this is reflected in the model. 3 The manager has a disutility for the square of a bias b multiplied with the cost parameter c > 0. The parameter can be interpreted as the severity of the punishment in case of a detection of the bias by the regulatory institution. 4 The manager receives a noisy signal about the per share value of the firm ṽ + ẽ; thereby, v and e are assumed to be normally distributed with mean zero and precisions τ v and τ e a priori. 5 He adds the bias b to this signal and discloses the report 2) r v + e + b to the N informed traders. Furthermore, there is an exogenously given net amount z of noise trading. It can be, for example, liquidity trading or irrational trading of uninformed investors. 6 In the model, it is the realization of a normally distributed random variable z that has mean zero and variance σz. 2 3 Fischer and Verrecchia 2000) give some examples: If a large long-term stockholder must repurchase shares in the near term to cover employee stock options. If the manager wants to engage in a management buyout. Or if he is about to receive a new option grant and attempts to drive down the price in order to lower the strike price. Another reason could be that a manager intends to leave the firm soon and then her objectives may be uncertain. 4 See e.g. Korn 2004) for an explicit modelling of the costs as the expected punishment in case of a detection. 5 All random variables are assumed to be independently distributed. This assumption is necessary to find the closed form solution. 6 See Black 986) for a definition of noise trading as trading on noise as if it were information.

6 6 Modelling the pricing mechanism, I stick to Kyle 985). The N informed traders submit a market order to the market makers. They choose their demand x n for n,..., N as to maximize their profits conditional on the private information i n and their influence they have on the equilibrium price p. Ex post, the profit is 3) π n v px n ))x n. The market makers observe only the sum 4) y n x n + z of the demands of the informed traders n x n and the liquidity traders z. They stand in perfect competition to each other, and, therefore, the price is set efficiently as the expected firm value given the net order amount y. 5) p E[ṽ n x n + z y]. After trading, the firm s value v and the profits are realized. 3. The linear equilibrium In this setting, a unique linear equilibrium is derived. Consequently, the players make linear conjectures about the behavior of other participants of the form 6.) b αu 6.2) x n η + βr n 6.3) p θ + γy.

7 7 The manager reacts only due to her incentive to bias the firm price 7. The informed traders order on the report. And the market price is conjectured to depend on the net order amount y in a linear way. In equilibrium, these conjectures must be self-fulfilling, i.e. the parameters are chosen as to maximize the individual utilities. To find the equilibrium assignments of the parameters, first, the utility of the manager in ), the profits of the informed traders in 3) and the price condition in 5) are evaluated. By comparing the coefficients with 6), a system of equations is derived. This system of equations for the parameters α, β, γ, η and θ has a unique solution as stated below as Proposition and proved in Appendix A. 3.. Utility maximization by the manager. The first order condition for the utility of the manager in ) with respect to the bias b is 7) u pb) cb 0. b The market price in 6.3) can be written using 4) ) p θ + γ x n + z n 6.2)2) θ + γ Nη + βṽ + ẽ + b)) + β n ẽ n + z ). The evaluation of 7) yields an equation for the parameter α in 6.) 8) α c Nβγ. The second order condition for a maximum is satisfied by definition, since c > 0. 7 The independence from her information about the firm value and the nonexistence of a constant term follow from the first order condition below.

8 Maximization of expected profits by the informed traders. The informed investors maximize their expected profits conditional on the report r with respect to their order volume x n. Inserting 6.3) into 3) yields E[ π n r] E[ṽ θ γỹ))x n r] 4) )) E[ṽ r] θ γ x k + E[ z r] k x n There are n first order conditions 9) E[ṽ r] θ γ k x k γx n k x k x n 0 for n,..., N. In a Nash equilibrium a marginal change in the demand of one informed trader does not change the demands of the others see Hirth and Neus 200)). Hence, it is 0) k x k x n x n x n. Due to symmetry the total demand equals N times the individual demand of each informed trader. The first order conditions can, therefore, be written ) E[ṽ r] θ γnx n γx n 0, which means x n E[ṽ r] θ N + )γ n. The second order conditions demand 2) γ > 0.

9 9 The expected firm value given the report is 3) E[ṽ r] τ r τ v r αµ u ) with 4) τ r + ) + α2. τ v τ e τ u Comparing coefficients with 6.2) yields 5) 6) η βαµ u β τ r. N + )γ τ v θ N + )γ 3.3. The efficient price. Finally, the market makers determine the price with 5). It is p E[ṽ ỹ y] E[ṽ Nη + βnṽ + Nαũ + n ẽ n ) + z y] τ y τ v y Nη Nβαµ u ) with 7) τ y V ar ỹ). Comparing coefficients with 6.3) yields 8) 9) θ γnη + Nβαµ u ) γ Nβ τ y τ v.

10 The solution. The system of equations 8), 6), 5), 9) and 8) is solved in Appendix A. Its solution is stated here as Proposition. A unique linear equilibrium for the disclosure game exists. The bias is b αu; thereby, α is given endogenously as a root of a polynomial and it lies in the interval 20) α 0, N c N + The equilibrium order amount x n η + βr is + τ ). v τe 2) x n and the price p θ + γy becomes τr σ 2 z N r αµ u) 22) p cα y Nτr σz 2 with 23) τ r + ) + α2. τ v τ e τ u Proof. See Appendix A. If the variance of noise trading vanishes, the model will collapse. Consequently, the market makers can infer from the net order flow y the individual order volumes x n. Hence, they loose their informational advantage and stop trading. In the borderline case in which the informed investors learn ṽ for sure τ e and τ u ) and N, the same solution as in Kyle 985) realizes, as x n τ v σzv 2 p y. 2 τv σz 2

11 Accordingly, if the marginal cost of biasing becomes large c ), then the manager does not bias anymore and the price simply updates the information regarding the private signals conveyed by the order stream. The case N is very similar to the model of Fischer and Verrecchia 2000) in that the solution for α is the root of the polynomial 24) cα) 3 + cα + ) c 2 c2 0. τ u τ v τ e 2τ v In their model, cα is the coefficient of the report in the equilibrium price. In my model, this effect is indirect via the informed trading and the noise has an additional influence on the price. For N the constant term is multiplied with 2 in contrast to their model. 4. Characterization of the equilibrium In this section, the behavior of the manager, the informed traders and the resulting price is analyzed. The calculations of the comparative statics summarized in the tables of this section can be found in Appendix B. 4.. Choosing the bias and trading volumes. The bias b αu influences the report r v+e+b the informed traders receive. It is proportional to the realization of the random variable ũ, i.e. the degree of the manager s incentive to manipulate the share price. Because α is always positive, the manager exaggerates the firm s value if a higher price is in her interest u > 0) and deflates the report if he desires a lower price u < 0). Table shows the behavior of the bias b αu for positive u. In the case u < 0, the results hold vice versa. It is immediate that a positive bias decreases with the marginal cost of biasing c. The manager increases the bias with the number of

12 2 Table. The coefficient of the bias. c N τ v τ u τ e σ 2 z α informed traders 8 because due to informed traders the bias of the report is reflected in the market price see the discussion of equation 3) below). Accordingly, the influence of the precisions can be explained. If the precision τ r of the report r v + e + αu of the informed traders increases see Table 2), the coefficient β grows and hence, they trade more on a deviation of the report from its expectation. Therefore, the manager biases more, the higher the precision of τ r. Only for the precision of the firm value τ v, there is a countervailing effect. If the knowledge about the profits is more certain ex ante, the information on v in the report is used less. Therefore, an increasing precision of the firm s value leads to less influence of the bias on the price and, subsequently, to a lower profit of the biasing for the manager. The precision of τ r grows with the three precisions τ v, τ u and τ e see Table 2). A higher coefficient α of the bias leads to more variance in the report because it makes the signal noisier. Nevertheless, the direct effect of the precisions τ v, τ u and τ e prevails. Table 2. The precision τ r. c N τ v τ u τ e σ 2 z τ r Note that α < for all N. c

13 The demand of the informed traders is ex ante x n η + β r with β τ r σ 2 z N η βαµ u. Because β > 0, the informed traders demand more if the report is better. Table 3 shows the influence of the parameters. The informed trader relies 3 and Table 3. The coefficient of the demand. c N τ v τ u τ e σ 2 z β more on the report if its precision τ r is higher. If the variance in noise trading is higher, there is more noise in the net order flow y and the report can be exploited more aggressively. An increase in the number of informed investors N has two countervailing effects. The direct effect is due to more information in the price and the indirect effect is due to a higher bias αu and, accordingly, lower precision τ r. Ex post, the realized order flow is 25) x n βv + e + αu µ u )). The informed trader subtracts the expected bias from the report Price efficiency and market depth. The equilibrium price p is given by p θ + γy with γ cα Nβ and θ 0. It increases with the net demand for the shares y because γ > 0. In a perfect capital market, the bias in the report would be backed out and not be reflected in the price. In my model, there are two sources of noise. First, noise trading prevents market makers to anticipate the private information of informed traders and due to the uncertainty about the manager s objective, the bias has an impact on the price.

14 4 Ex post, by inserting 2) into 22) the price becomes ) 26) p cα v + e + αu µ u ) + Nτ r σz 2 z. The coefficient cα can be interpreted as a measure of the information content of the net order flow y. The higher cα is, the more pronounced is the price reaction triggered by a change in y. If the price efficiency Eff is defined with the portion of the explained variance of the firm value by observing the price to the prior variance, cα will be a measure of the informational efficiency because Eff V arṽ p) V arṽ) ) τ v V arṽ) Cov2 ṽ, p) V ar p) 26) τ v τ v c 2 α 2 τ 2 v c 2 α 2 τ v + τ e + α2 τ u + Nτ r ) 23) cα) cα. Due to 20), it is 27) 0 < Eff < N N + + τ <. v τe A value of zero means that no information about v is reflected in the price and a value of one means a fully revealing price. In the model, the price is never fully revealing. Its upper bound increases with the number of informed traders N and the precision in the error term τ e. However, it decreases with the prior precision of the firm value τ v.

15 5 The equilibrium price discounts the average private information minus its expected bias and a term for the noise trading. The discounting cα < adjusts for the uncertainty in the following way. Note that 28) cα 37) N N + /τ v < τ v + τ e + α2 τ u /τ v τ v + τ e + α2 τ u τ r τ v. The market maker assumes that y contains approximately the same information about v as the report r. But she attaches an additional amount of variance to y and in this way accounts for the uncertainty of the noise trading. However, the variance in noise trading σz 2 does not influence her updating. Table 4 shows the effect of the parameters on the price efficiency cα. Consequently, Table 4. The price efficiency. c N τ v τ u τ e σ 2 z Eff a greater enforcement of disclosure regulation i.e. a higher c) increases the price efficiency. The more informed traders N participate in trading, the smaller is the influence of noise traders relative to informed trading, and, hence, the price contains more information. Higher precisions τ v, τ e and τ u increase the precision in the report and via informed trading the precision of the price in the case of τ e and τ u. More transparency of the manager s objectives, for example of incentive plans, increases the price efficiency. Only for the prior precision of the firm s value τ v, a countervailing effect occurs. If the prior of the firm s value is more certain, the market makers rely more on their prior information than on the net order flow y.

16 6 The negative effect of τ v on the efficiency can be put into perspective by noting that the precision of the firm value given the price 29) V ar ṽ p) τ v cα increases with the prior precision of the firm value τ v. Note that the price corrects only for an expected bias. The unexpected bias enters the price with the same weight as the firm value. If the manager wants to increase the price more than expected u > µ u ), she succeeds only when the effect is large enough so that the discount of the price is outweighed cα < ). More precisely, she is successful, only if 30) αu µ u ) > cα)v and the other parts influence of error terms and noise) are neglected. For 0 < u < µ u, the price will be lower than the true firm value, although the manager wants to increase it. If she wants to deflate the firm price more than expected i.e. for u < µ u < 0), it is easier for her to influence the price in the desired direction at least if v > 0). The influence of the number of informed traders N can be seen by writing 3) p v vcα ) + cαe + cα 2 u µ u ) + cα Nβ z. The influence of the error terms e n and the noise trading vanish with a growing N. The influence of the firm value v decreases since cα moves closer to ). But the influence of u µ u increases with N. However, this increasing influence is only in the manager s interest if the realized incentive to bias is higher than expectations i.e. if u > µ u and u > 0 or u < µ u and u < 0).

17 7 More punishment of the biasing has also its downside. This issue can be identified by its influence on one aspect of liquidity, the market depth d. The market is less liquid if the cost of biasing increases and this harms informed traders and liquidity traders. The market depth d can be measured by the size of the order flow required to change prizes by one dollar. This is exactly the inverse of the order flow coefficient γ. 32) d Nβ cα Nτr σz 2 cα The market depth increases with the number of informed traders, with the precision of the private information and with the variance in noise trading. It decreases with the information content of the net order flow y. Table 5 shows the influence of the parameters. A higher cost of biasing c and higher precisions τ u and τ e increase Table 5. The market depth. c N τ v τ u τ e σ 2 z d the precision τ r but this effect is outweighed by a higher information content of the price. For the precision of the firm value τ v, it is the other way around, the influence on τ r dominates that on cα. It is intuitive that market depth increases with the number of informed traders N and the variance in noise trading σz The utilities of the participants In this section, the utilities of the three participants - the manager, the informed traders, and the group of the noise traders - are analyzed.

18 8 5.. The manager. The results for the manager have the same form as in Fischer and Verrecchia 2000) and are stated here without proof. By using 26), the manager has an ex ante - profit of 33) E[ũ p 2 cα2 ũ 2 ] ) 2 cα2 µ 2 u. τ u and 34) E[ũ p 2 cα2 ũ 2 ũ u] 2 cα2 u µ u ) 2 µ 2 ) u ex post. The manager s utility is high, when the capital market assumed that he has no or little incentive to bias at all µ u 0). The expected utility is positive if there is uncertainty about the type of the manager, i.e. if he wants to increase or decrease the price. It can also be positive if the precision τ u is low and if the realization of u is far away from expectation. Table 6. The utility of the manager. c N τ v τ u τ e σ 2 z cα 2? The absolute value of the expected utility grows with cα 2 see Table 6). cα is the coefficient of the bias αu µ u ) in the price. The profit, therefore, increases with the coefficient of the bias α and the informativeness cα of the order flow y. The utility, therefore, grows with the parameters, that increase the coefficient α of the biasing. Only for the influence of c, there is a countervailing effect. An increase in the marginal costs of biasing increases the informativeness of the price but it also decreases the bias. If there is less knowledge on the incentives of the manager small

19 9 τ u ) relative to the precisions τ v and τ e, the profit of the manager even grows with a higher marginal cost of the biasing. Otherwise, it is falling with higher c. The ex ante expected utility falls if incentives of the manager are known better a priori τ u grows). This stands in contrast to the result in Table 6 and the ex post case The informed traders. By using 26) and 25), the expected profit of the informed traders is π i E[ṽ p) x n ] β cα + ) + α2 ) τ v τ v τ e τ u 37) β τ v N +. The profit is always positive. Table 7 shows the influence of the parameters. The Table 7. The profit of the informed traders c N τ v τ u τ e σ 2 z π i profit of the informed traders increases with the precision of the report τ r because their information is then more precise. It also grows with the variance in noise trading. It decreases with the number of informed traders N and the precision in the firm value τ v. By introducing a constant cost of information acquisition C, the number of informed traders can be made endogenous. Then, a zero profit condition π i C 0

20 20 yields 35) NN τr σ 2 z τ v C. The left hand side increases in N, and, therefore, the number of informed traders increases 9 if the prior precision in the firm value decreases, if the precision of the report increases, if the amount of noise trading increases and of course if the costs of information acquisition decrease The noise traders. The union of noise traders has an expected profit of π n E[ṽ p) z] σ2 z Nβ. The noise traders always have a negative utility. Table 8 shows the influence of the parameters. Noise traders loose less if the costs of biasing c increase, if the Table 8. The loss of the noise traders. c N τ v τ u τ e σ 2 z π n number of informed traders N increases and if the precisions τ v, τ u and τ e increase. It increases with the variance in noise trading σz. 2 If the objective of the regulation is to protect the noise traders, it is desirable to increase the number of informed traders without increasing the loss of the noise traders. In my model, this favors an increase in the marginal costs of biasing c, and higher precisions τ e and τ u. 9 Note that τr also increases in N via α, but is bounded in N.

21 2 This could be achieved by a higher punishment for a bias, better information systems in the firms higher τ e ) and more information about the incentives of the management, e.g. about their payment contracts. 6. Summary and Conclusions This paper examines an extension of the model of Fischer and Verrecchia 2000). Introducing a market microstructure, I show that a unique linear equilibrium exists. In equilibrium, the manager biases the disclosure r of the firm s value according to her incentive regarding the share price. The N informed traders correct the report for the expected bias and trade on this corrected information. The market makers try to derive the information on the firm s value of the informed traders from the trading flow. However, they discount the share price further because there is an additional source of uncertainty, namely the noise trading. In my model, the price is never fully revealing - but the efficiency increases with the number of informed traders and the marginal cost of biasing. The efficiency increases in the precisions of the error term in the signal of the manager and of the incentives of the manager. Nevertheless, a higher prior precision in the firm s value decreases the price efficiency because the order flow is less informative for the market makers to determine the price relative to the prior information. Informed traders always make positive profits and that of noise traders is always negative. Both can be made better off by increasing the marginal costs of biasing and the precisions of the variables in the report. The utility of the manager is positive if there is a high uncertainty about its type i.e. if she wants to over- or understate the firm s value) and, ex post, if the realization of her incentives is far away from the expectation of the market.

22 22 One assumption is risk-neutrality. The model of Kyle 989) showed, for example, how the role of σz 2 changes if participants are risk averse. One can expect that also the price is even less informative because the informed traders do not trade as aggressively. But in Kyle 989) this result reversed, if the number of informed traders is made endogenous by introducing a cost of information acquisition. A limitation of the explicit modelling of the capital market is that the manager does not participate in the market. Henceforth, the model cannot analyze stockbased compensation or insider trading effects. One extension could be a more explicit modelling of the behavior of noise traders. Assuming that they do not understand the report, maybe they know at least the headline earnings r of the report. It is appropriate to model their trading proportional to the difference between the reported firm value and the expected firm value. As an alternative, one could use the difference between the price and the expected firm value. Appendix A Proof of Proposition. First, only the equations for α, β and γ in 8), 6) and 9) are considered. 6) can be solved for βγ 36) N + )βγ τ r τ v. Using this for 8), it is cα Nβγ 37) N N + + τv τ e + α 2 τ v. τ u

23 23 and α is hence given by the root of the polynomial 38) cα 3 τ v + cα + τ ) v N τ u τ e N + 0. This polynomial has one and only one solution for α. It suffices to note, that the derivative with respect to α is always positive and the left hand side is negative for α 0 and positive for α c N N+ + τ v τ e The precision τ y in 7) is. β 2 N 2 39) τ y + σz 2 τ r ) Inserting 9) into 36) yields β 2 Nτ y τ v τ r N+ τ v 39) β 2 NN + ) τ r σ 2 z + N 2 β 2 β 2 N τ rσ 2 z. β must be positive, because α > 0, γ > 0 see 2)) and due to 8), it is 40) β cα Nγ. This equation can also be solved for 4) γ cα Nβ cα. Nτrσz 2 Now, the parameters η and θ are left. Inserting 8) into 5) yields η βαµ u N + )γ γ)nη + Nβαµ u) and this can be solved for 42) η βαµ u.

24 24 θ is given by inserting this solution into 8) as 43) θ γn βαµ u ) + Nβαµ u ) 0. Appendix B In this appendix, the comparative static results of section 4 are calculated. In Appendix A, it was shown see equation 38)), that α is the unique root of the polynomial 44) p : α3 + + ) α τ u τ v τ e N. N + cτ v Its derivative 45) p 3 α2 τ u + τ v + τ e is positive for all α. Furthermore α satisfies 46) 0 < α < N c N + + τ < v τe c. Table. α c p ) N N+ τ v c 2 < 0, α τ v α N ) p τv 2 N+ c 46) < p τv 2 c N N+ + τ v τ e c ) N N+ < 0, α τ u α 3 p τ 2 u > 0, 47) α τ e α p τ 2 e > 0. The variance in noise trading σ 2 z has no effect on α. To see the influence of N on α, write the equation for α in 44) in the form 48) cα 3 + cα + ) N τ u τ v τ e N + τ v

25 25 The right hand side increases with N. To hold the equation, α has therefore to increase with N. Table 2. The influence of the parameters behaves in the opposite way to 49) σ 2 r τ v + τ e + α2 τ u. The parameters influence σ 2 r also indirectly via α. The statics for τ v, c and N follow from Table. For τ e and τ u, it is σ 2 r τ e 2α τ u α τ e τ 2 e 50) 47) 2α p α τ u τe 2 3 α2 + + ) ) < 0. τe 2 τ u τ v τ e 5) σ 2 r τ u τ 2 u 2α α ) τ u α 2 τ u 52) 47) p τ 2 u 2ατ u α 3 τ 2 u α 2 3 α2 + + ) ) < 0. τ u τ v τ e Table 3. The results follow immediately with the results for τ r in Table 2 and from β τr σ 2 z N.

26 26 Table 4. The results for N, τ u, τ e, τ v and σ 2 z follow immediately from Table. The behavior for c is cα c α + c α c 47) α c p N N + τ v c ) 2 45) α3 α2 + + ) N ) c p τ u τ v τ e N + τ v c ) 2 53) 44) p 2 α3 τ u ) > 0. The behavior of the alternative definition of the efficiency remains the same except for τ v. Here, it is τ v τ v cα cα τ α ) v c ) cα) 2 τ v 47) cα + τ vc α N ) cα) 2 p τv 2 N + c ) 45) cα)3 α2 cα) 2 p cα) 2 p + + ) ) N τ u τ v τ e τ v N + cα) cα)3 α2 τ u + τ v N N + ) + τ e ) + ) cα > 0. N + )τ v Table 5. The behavior of d Nσ 2 z σ 2 r cα is immediate for N, σ2 z and τ v.

27 27 It remains τ u, τ e and c. It suffices to look at σ 2 rcα. σ 2 rcα c 2 σ 2 r σr 2 c cα + cα) σr 2 c ), 53) ) 2α 2 N σr 2 p τ u N + τ v c cα + 2α 3 2 2σ2 r p τ u ) 2 2α 2 N + 2σrα 2 σr 2 p τ u N + cτ v 44) 2 σ 2 r 2α 2 p τ u N > 0. N + cτ v σ 2 r σrcα 2 c τ u 2 α + c α σ σr 2 r τ 2 u τ u 44), 52) c 2 σr 2 c 2 σ 2 σr 2 p τu 2 r > 0. α α 2 α2 + + ) ) p τu 2 τ u τ v τ e α 3 α 3 + 2σr 2 p τu 2 ) σ 2 rα τ e 2 σr 2 α + α σ σr 2 r τ 2 e τ e 44), 50) 2 σ 2 r 2 σ 2 r α σ2 r p τ 2 e ασ 2 r p τ 2 e > 0. ) + 2σr 2 α p τe 2

28 28 Table 6. cα 2 behaves like α see Table ) except for c: cα 2 c α 2 + 2cα α c 47) α 2 + 2cα p N N + τ v c ) 2 45) α 2 3 α2 + + ) N ) 2cα p τ u τ v τ e N + τ v c ) 2 44) α N p N + τ 2α + ) ). vc τ v τ e At least for small τ u and high τ e and τ v, this will be positive. Table 7. The comparative statics of β N+)τ v follow immediately with Table 3. Table 8. The statics of σ2 z is clear from Table 3 except for N and τi Nβ σ2 σz z. With β 2 N and τ i decreases in N, the other results follow, too. References [Black 986)] Black, F Noise. The Journal of Finance 4: [Fischer and Verrecchia 2000)] Fischer, P. and R. Verrecchia Reporting bias. The Accounting Review 75 April): [Hirth and Neus 200] Hirth, H. and W. Neus Ad-hoc-Publizitt und Wettbewerb beim Insiderhandel. Wirtschaftswissenschaftliches Studium 30: 0-04 [IASB 2003)] International Accounting Standards Board International Financial Reporting Standards. [Korn 2004] Korn, E Voluntary disclosure of partially verifiable information. Schmalenbach Business Review April) [Kyle 985] Kyle, A Continuous Auctions and Insider Trading. Econometrica 53 November): [Kyle 989] Kyle, A Informed speculation with imperfect competition. Review of Economic Studies 56: