Why Do Large Investors Disclose Their Information?

Transcription

1 Why Do Large Investors Disclose Their Inforation? Ying Liu Noveber 7, 2017 Abstract Large investors often advertise private inforation at private talks or in the edia. To analyse the incentives for inforation disclosure, I develop a two-period Kyle (1985) type odel in which an infored short-horizon investor strategically discloses private inforation to enhance price efficiency. I show that inforation disclosure is optial when the scope of private inforation is large and when the large investor has a high reputation. Short investent horizons induce inforation disclosure aong investors and are beneficial for price efficiency. However, strategic inforation disclosure reduces trading before disclosure and hars price discovery. Keywords: Inforation Disclosure, Price Discovery, Asyetric Inforation, Market Microstructure I would like to thank y supervisor Noran Schürhoff for his helpful coents and advice. For insightful coents I thank Marco Della Seta, Jeroe Deteple, Darrell Duffie, Sabine Eliger, Thierry Foucault, Arte Neklyudov, Chaojun Wang and participants at UNIL brown bag seinar, SFI research day, Asian Finance Association conference and EFA doctoral tutorial. University of Lausanne and Swiss Finance Institute; Address: University of Lausanne, Quartier UNIL- Chaberonne, Batient Extranef 243, 1015 Lausanne, Switzerland; E-ail: ying.liu@unil.ch. 1

2 Large investors, such as hedge fund anagers, often advertise their trading ideas at private talks or in the edia. Acadeic researchers have docuented that this is a systeatic phenoenon. In their recent epirical work, Ljungqvist and Qian (2016) exaine available public reports fro soe hedge funds and individual investors about shorting 124 overpriced copanies. They found that these investors anaged to correct the ispricing by encouraging long-ter investors to sell their positions on the target copanies. Swe (2017) also discovered that hedge funds usually acquire inforation, trade and then strategically disclose their inforation to utual funds via analysts, expecting utual funds to trade on their inforation and, thus, accelerate the incorporation of inforation into the asset price. The above evidence leads to the following questions: Why do investors spread their inforation freely, considering that inforation production is very costly? Why do these investors not trade and profit fro the valuable inforation, instead of disclosing it? Furtherore, does this behavior iprove or har price efficiency and trading intensity? To answer these questions, I developed a two-period odel based on Kyle (1985) in which endogenous inforation disclosure arises due to the short horizon of the infored investor. In the odel, there is one risky asset, whose payoff is only known by the infored investor. Trading takes place in both periods and the payoff of the asset is exogenously realized at the end of the second period. The infored investor trades the asset in the first period and liquidates the position in the second period. Before liquidation, the infored investor decides to disclose the private inforation to other uninfored investors or to liquidate the position silently. The infored investor trades rationally in the first period by taking the inforation disclosure into consideration. Both silent liquidation and endogenous inforation disclosure lead to early inforation revelation and, thus, iprove price inforativeness and accelerate the speed of ispricing 2

3 correction. Silent liquidation is achieved through the aggressive trading of the infored investor hiself or herself, and endogenous inforation disclosure is achieved through the participation of new investors who receive the inforation. The inforation disclosure decision depends on the price realization in the first period and the trading intensity of the new investors. The infored investor tends to disclose the inforation when the nuber of followers is large. Inforation disclosure is ost likely to occur when the infored investor has a higher reputation. With inforation disclosure, the infored investor reduces the trading intensity. The short horizon of infored investors is natural in the odel. In contrast to long-ter investors for who the liquidation price is the known fundaental value of the asset, the short-ter investor faces uncertainty about the liquidation price. Thus, the horizon liit results in different trading behaviors. Long-ter investors trade cautiously to hide their private inforation and alleviate price ipacts. However, short-ter investors are better off hiding inforation and eliinating price ipacts when entering the arket and disclosing inforation and creating price ipacts when leaving the arket. The two opposite targets cannot be achieved by purely adjusting the trading intensity because the infored investor s trading volues are the sae at the two points of tie. Therefore, endogenous inforation disclosure arises when the infored investor liquidates his or her position. The disclosure attracts new investors and additional capital flows into the arket and pushes prices closer to their fundaental values. Silently liquidating the position could be optial in this odel because the infored investor has price ipacts. The infored trader incurs no financial constraint and is able to ake a large order that oves the price. In case the nuber of followers is sall or the financial capital brought by the followers is not large enough, the infored trader is better off taking a large position and signal the inforation to the arket aker. However, there 3

4 is always a risk that the inforation ight be revealed early. One question is under which conditions does inforation disclosure happen? I find that the investor discloses the inforation when the trading volue of the new investors is at least two ties the trading volue of the infored investor. The arket aker gets the inforation through the total nuber of orders she observes: the presence of large orders indicates that there is uch inforation. Without new investors, the total order in the second period is only the liquidation plus the noise deand. With new investors, half of the deand fro new investors is unobservable to the arket aker because it offsets the liquidation of the infored investor. The reaining half, ixed with the order fro noise traders, is observed by the arket aker. That is why the disclosure decision is related to two ties the trading volue of new investors. Another interesting question is about the ipact of inforation disclosure on price discovery and arket efficiency. It is evident that inforation disclosure iproves price efficiency after disclosure because additional infored investors participate in the arket after inforation disclosure occurs, and thus, the prices are highly inforative. Related literature, such as that of Boel (2003), Kovbasyuk and Pagano (2015), Ljungqvist and Qian (2016) and Schidt (2017) also confir this result. However, they focus on sall or copetitive investors who incur financial constraints and do not have any price ipacts. I focus on large investors and find that inforation disclosure reduces trading intensity and hars price discovery before disclosure. In the presence of inforation disclosure, an infored investor can borrow financial firepower fro his or her followers in the second period. Thus, he or she only focuses on hiding the inforation in the first period and, therefore, trades less intensively. In the odel I show that with the option to disclose inforation, the short-ter infored trader trades even less than a long-ter trader. 4

5 The paper generates iportant policy iplications in ters of inforation sharing aong investors. Previous studies proote inforation disclosure because it eliinates the liit on arbitrage and encourages the participation of short-ter investors. I copleent these results by showing that the ipact of inforation disclosure by large investors is different than that of others. I find that strategic inforation disclosure reduces the trading intensity before disclosure and hars price discovery. Inforation disclosure is beneficial for the infored investor but detriental to the noise traders because the expected loss of the noise traders increases with inforation disclosure. Related Literature Other papers on the incentives of inforation disclosure include the theoretical works of Boel (2003), Kovbasyuk and Pagano (2015), and Schidt (2017) and the epirical paper of Ljungqvist and Qian (2016). The infored investors (ruourongers or speculators) in these papers are sall and copetitive, and face capital constraints. The orders fro the investors are negligible and do not have price ipacts. In contrast, I investigate the behavior of large investors, who usually have large deand and cause price adjustents. Inforation disclosure is a strategic decision that takes price ipacts into account. Moreover, I show that an infored investor strategically reduces deand before disclosure to hide inforation and eliinate price ipacts. My paper is also related to two other strands of literature. One is about inforation sales and exchanges. Adati and Pfleiderer (1986) docuent that, for a onopolistic inforation owner, it is optial to add noise to the inforation when it is sold and that it is ore profitable to sell different signals to different traders than just one signal to all traders. Adati and Pfleiderer (1988) discussed whether an infored owner wants to sell inforation or alternatively trade strategically on the basis of the inforation. Fishan and Hagerty 5

6 (1995) explain that selling inforation is a way for infored traders to aggressively coit to trades. The authors also point out that a risk-averse trader is better off selling inforation than just trading on it. Inforation disclosure is siilar to inforation selling with zero price. However, I found that zero-price inforation does not exist in the equilibriu results of the above papers. Because the infored investor has to copensate his or her losses fro copeting with new investors using the profit fro inforation sales, the price can not be zero. Studies on inforation exchange, such as that of Stein (2008), shows that the inforation is transitted aong investors. The copleentarity of inforation structures akes it optial to exchange inforation with other investors. Because the infored investor does not necessarily expect inforation feedback fro the receivers, the inforation transission echanis can not explain the disclosure. The second part is about the liit on arbitrage. In the survey of Grob and Vayanos (2010), the authors suarize the costs incurred by arbitrageurs, including fundaental and non-fundaental risks, short-selling costs, and leverage, argin, and capital constraints. In particular, De Long et al. (1990) docuents noise trader risk. Shleifer and Vishny (1997) discuss perforance-based arbitrage, whereby investors face capital constraints. Abreu and Brunnereier (2002, 2003) introduce the synchronization risk: arbitragers uncertainty about when other arbitrageurs will start exploiting a coon arbitrage opportunity causes delayed arbitrage with holding costs. My paper copleents the literature by pointing out a way of eliinating the liit on arbitrage. The paper is also related to the andatory disclosure literature, such as that of Steven Huddart (2001) and Yang and Zhu (2017). These researcher also use the two-period Kyle (1985) trading echanis in their papers. In contrast, I endogenize the inforation disclosure decision and derive a sub-gae perfect equilibriu. Steven Huddart (2001) shows that andatory inforation disclosure is beneficial to price discovery and arket efficiency while 6

7 Yang and Zhu (2017) conclude that the presence of order flow traders hars price discovery in the first period and its ipact on arket liquidity is ixed. Yang and Zhu (2017) find that the price discovery is hared because infored investors add noise to deand, but I show that the price discovery is hared because the infored investor reduces the trading intensity. I proceed as follows. In Section 1, I describe the odel. In Section 2, I solve the benchark case in which the short-ter trader does not disclose the inforation. Then I introduce the inforation disclosure in Section 3 and derive the sub-gae perfect equilibriu. I discuss the iplications of inforation disclosure in Section 4. Section 5 concludes the paper. 1 The Model In the odel, one single risky asset is traded by four arket participants: one infored trader, one copetitive arket aker, noise traders and followers. There are three tie periods t {1, 2, 3}, the risky asset is traded in periods 1 and 2 before it pays off at the third period. The asset s payoff θ is either high (θ H ) or low (θ L ) with probability β 0 and 1 β 0. The initial price of the asset equals to the expectation of the payoff, that is p 0 = β 0 θ H +(1 β 0 )θ L. For siplicity, I assue that β 0 = 1. 2 At t = 1, the infored trader arrives in the arket with probability µ [0, 1, whether the infored trader is present or not is unknown to anyone else. The infored trader is risk neutral and has short horizon, that is, the infored trader trades in the first period and liquidates the position in the second period. In addition, the short horizon of the infored trader is coon knowledge. The infored trader knows the payoff of the risky asset and trades strategically with the inforation. I follow Grossan and Stiglitz (1982) by intro- 7

8 ducing noise traders in the arket, who do not have any inforation and trade randoly. The existence of noise traders ensures that the inforation of infored trader will not be fully reflected in price and the infored trader does not take all the surplus. The deand fro noise traders in each period are independently uniforly distributed 1, u t U( 1, 1). As in Kyle (1985), the price of the risky asset in each period is set by a copetitive arket aker. The arket aker only observes the total order flow y t and sets the price p t to clear the arket: p 1 = E[θ y 1, p 2 = E[θ y 1, y 2. The infored trader has to close the position in period 2, while the payoff is realized in period 3. The infored trader thus faces uncertainty about the liquidation price. In order to eliinate the price uncertainty, the infored trader ay disclose the private inforation to followers. The followers are uninfored at period 1 and do not participate in the arket. They only get infored and start trading if the infored trader discloses the inforation at period 2. The nuber of followers is observable to the infored trader at period 1. Credibility is essential in the inforation revelation, in this odel, I siply restrict that the infored trader only reveals true inforation. The assuption is reasonable since the objective is large investors in the financial arket, for exaple, fund anagers, who concerns a lot of the reputation. Once the large investor is confired as spreading false ruours and anipulating the arket, her reputation will be draatically ipaired and she ay lose the job or face large investent withdraw. Moreover, a recent paper by Schidt (2017) also states that short-ter ruouronger prefers to share her inforation truthfully. 1 Sae assuption in Schidt (2017) 8

9 2 Benchark: No inforation disclosure In this section, I derive the equilibriu of the trading gae without inforation disclosure. In particular, I characterize the pricing equations and copute the optial strategy of the infored trader. The equilibriu is solved using forward-backward induction. I first characterize the price function in the first period. Then I derive the price equation in the second period. Finally, I copute the optial deand of the infored trader. 2.1 Price in First Period In the first period, if the infored trader appears, she buys x 1l if she learns that θ = θ H and sells x 1s if she learns that θ = θ L. The noise traders always appear in the arket and trade u 1 in the first period. The total order flow in the first period is: x 1l + u 1 if θ = θ H and infored trader appears y 1 = u 1 if infored trader does not appear x 1s + u 1 if θ = θ L and infored trader appears The arket aker is uninfored, who tries to extrapolate inforation fro the order flow she observes. Due to the copetitiveness, the arket aker sets the price equal to the expected value of the risky asset conditional on the total order flow y 1. Since the deand fro the noise traders are uniforly distributed and bounded on [ 1, 1, the total order in the first period is distributed on [ x 1s 1, x 1l + 1. Depending on the value of x 1l and x 1s, the price equations are as follows: 9

10 (1) 0 x 1l < 1 and 0 x 1s < 1, θ H if 1 < y x 1l p M if 1 x 1s < y 1 1 p 1 = E[θ y 1 = p 0 if x 1l 1 y 1 1 x 1s (1) p M if 1 y 1 < x 1l 1 θ L if x 1s 1 y 1 < 1 (2) 1 x 1l < 2 and 1 x 1s < 2, θ H if 1 < y x 1l p M if x 1l 1 < y 1 1 p 1 = E[θ y 1 = p 0 if 1 x 1s y 1 x 1l 1 (2) p M if 1 y 1 < 1 x 1s θ L if x 1s 1 y 1 < 1 (3) x 1l 2 and x 1s 2, θ H if x 1l 1 y 1 x 1l + 1 p 1 = E[θ y 1 = p 0 if 1 y 1 1 θ L if x 1s 1 y 1 1 x 1s where p M β 1 θ H + (1 β 1 )θ L and p M (1 β 1 )θ H + β 1 θ L. β 1 = β 0 β 0 µ+1 µ = 1 2 µ, which is the updated belief of arket aker conditional on observing a total order y 1 in-between 1 x 1s and 1 in case (1) or between x 1l 1 and 1 in case (2). 10

11 Different fro Kyle (1985), the price equation is piecewise throughout the entire doain, that is due to the binoial distribution of the payoff θ. The price is closer to the true payoff when the total order size y 1 is large. For instance, when y 1 is in-between 1 and 1 + x 1l in case (1) and (2), or between x 1l 1 and x 1l + 1 in case (3), the arket aker can infer the existence of infored trader since the order fro noise trader can not be larger than 1. Fro the positive order, the arket aker can further deduce that the payoff is high and set the price as θ H. As shown by case (3), when the infored trader trades very aggressively, the private inforation is fully revealed through trading in the first period, and the infored trader gets zero profit. Therefore, very aggressive order, that is x 1l 2 and x 1s 2 are not optial and will not considered thereafter. When the total order size y 1 is oderate, the arket aker can not differentiate whether the order coes fro purely noise trader, or fro both noise trader and infored trader. But conditional on the existence of infored trader, the arket aker can deduce the true payoff fro the order direction, so she akes an adjustent on the price. For exaple, when y 1 is between 1 x 1s and 1, the arket aker deduces that the order coes fro both infored trader and noise trader with probability β 0 µ, the order coes fro purely noise trader with probability 1 µ. So the arket aker updates the belief of high payoff fro β 0 to β 1 = β 0 β 0 µ+1 µ, and sets the price as p M. Siilarly, when the arket aker observes a negative order between 1 and x 1l 1, her belief of low payoff is adjusted to and sets the price as p M. 1 β 0 1 β 0 +1 µ = β 1, At last, when the order size y 1 is very sall, which is the intersection of x 1l + u 1, u 1 and x 1s + u 1, the arket aker can not extrapolate any inforation fro the total order, so she keeps the price as p 0. Next, I characterize the prices in the second period. 11

12 2.2 Price in Second Period: no inforation disclosure In the second period, the total order includes the deand fro noise traders u 2, and, in presence of infored trader, the liquidation fro the infored trader: x 1l + u 2, if y 1 = x 1l + u 1, y 2 = u 2, if y 1 = u 1, x 1s + u 2, if y 1 = x 1s + u 1. The arket aker sets the price equal to the expected value of risky asset conditional on the current order flow y 2 and historical order y 1 : p 2 = E[θ y 1, y 2 = E[θ p 1, y 2. Since p 1 contains the sae inforation as y 1, the arket aker thus adjusts the price p 2 based on p 1 and y 2. Conditional on high payoff θ H and the appearance of infored trader, there are three possible outcoes of price p 1 : θ H, p M and p 0. Firstly, It is obvious that once the inforation is fully revealed in the first period, that is p 1 = θ H, the price p 2 = p 1 = θ H. Then I show the price equations when (1) p 1 = p M ; (2) p 1 = p 0 in below. (1) When p 1 = p M θ H, if 1 x 1l y 2 1, p 2 = p M, if 1 < y 2 1 x 1l, p 0, if 1 x 1l < y 2 1. (3) 12

13 (2) When p 1 = p 0 : θ H, if 1 x 1l y 2 < 1, p M, if 1 y 2 < x 1s 1, p 2 = p 0, if x 1s 1 y 2 1 x 1l, (4) p M, if 1 x 1l < y 2 1, θ L, if 1 < y 2 x 1s + 1. In contrast to the pricing rule in the first period, a large positive total order in the second period iplies low payoff and a large negative total order iplies high payoff That is due to the liquidation of the infored trader. Since the short-ter horizon of infored trader is coon knowledge, the arket aker realizes that in the presence of the infored trader, the order ust be negative in the first period if she observes a large positive order in the second period, and then she adjusts her belief downwards, vice versa. If the trading in the first period partially reveals the inforation and p 1 = p M, the arket aker ruled out the possibility that y 1 = x 1s + u 1. Then in the second period, the total order is either y 2 = x 1l + u 2 or y 2 = u 2, and the price p 2 is irrelevant with x 1s, as shown by equation (3). If the trading in the first period does not reveal any inforation, that is when p 1 = p 0, the price p 2 depends on both x 1l and x 1s, as shown by equation (4). 2.3 The deand of infored trader Given the price equations in each period, I can copute the expected profit of the infored trader and then derive the optial deand. Since the probability of high payoff equals to the probability of low payoff, the equilibriu is syetric. So I only study the case when the payoff is high. For coparison, I calculate the optial deands of infored trader with short-ter horizon and that with long-ter horizon. Furtherore, I ipose one tie-breaking 13

14 rule: when two orders result in the sae expected profit, the infored trader strictly prefers the saller order. At first, the expected profit of the infored trader with long-ter horizon is given by E[π1 L θ = θ H = E[x 1l (θ H p 1 ) θ = θ H [ x 1l (θh p M ) x 1s = + (θ 2 H p 0 ) 2 x 1l x 1s 2, 0 x1l < 1, 0 x 1s < 1, [ x 1l (θh p M ) 2 x 1l 2, 1 x1l < 2, 1 x 1s < 2. The first order condition gives the optial strategies in each subdoain as x L 1l = 1 β 1 β 0 2(1 β 0 ) x 1s, 0 x 1l < 1, 0 x 1s < 1, 1, 1 x 1l < 2, 1 x 1s < 2. Since β 0 = 1, the infored trader plays syetric strategy when the true payoff is high and 2 low, that is x 1l = x 1s, which yields the optial deand of infored trader with long-ter horizon, denoted by x L as x L = x L 1l = x L 1s = 2(1 β 0) = β 1 3β 0 β Next, when the infored trader has short-ter horizon, the expected profit is E[π1 S θ = θ H = E[x 1l (p 2 p 1 ) θ = θ H = [ x 1l (θh p M ) x 1s 2 [ x 1l (θh p M ) 2 x 1l 2 x 1l + (θ 2 H p 0 ) 2 x 1l x 1s x 1l + (p 2 2 M p 0 ) 2 x 1l x 1s 2 x 1s 2, 0 x1l < 1, 0 x 1s < 1, x 1l 2, 1 x1l < 2. 14

15 Solving the first order condition gives the optial strategies in each subdoain as x S 1l = 2(1 β 0 ) 2(β 1 β 0 )x 1s (β 1 β 0 )(4β 1 β 0 3)x 2 1s 2(1 β 0)(β 1 β 0 )x 1s +4(1 β 0 ) 2 3(1 β 0, 0 x ) 1l < 1, 0 x 1s < 1, 4, 1 x 3 1l < 2. Taking x 1l = x 1s and then coparing the expected profit in each subdoain by substituting the x 1l, x 1s into the profit function, gives the optial deand as x S = x S 1l = x S 1s = 4, 0 µ < β 1 +1 = 4 µ 5β µ, µ 1, where at µ = , the deand x 1 = 4 3 and x 1 = 4 µ 3+µ result in equal profits. The optial deands of infored trader with different horizons are illustrated in figure 1, the solid line represents the deand of trader with long-ter horizon, the dashed line represents the deand of trader with short-ter horizon. Firstly, infored trader with short-ter horizon trades ore aggressively than that with long-ter horizon. Copared with long-ter trader, short-ter trader incurs price uncertainty at liquidation. The liquidation price is a rando variable which depends on the trading intensity for the short-ter trader, while for long-ter trader, the liquidation price is the true payoff Since ore intensive trading induces ore inforation revelation, and drives the price closer to the true payoff The short-ter trader thus increases the deand, in order to reduce the price uncertainty in the liquidation. Secondly, both deands reduces with the probability of infored trading µ. The paraeter µ captures the sensitivity of arket aker to the total order, when µ is sall, the arket aker is less sensitive to the order flow, and the price adjustent is sall. But when µ increases, the arket aker becoes ore sensitive to the order flow and the price ipact 15

16 Figure 1: The deand of the infored trader also increases. Therefore, the infored trader trades ore cautiously. For long-ter trader, the deand is always saller than 1, which is the axiu deand fro noise traders. But for short-ter trader, the deand is larger than 1 but saller than 2 when µ < When µ is larger than , the trader reduces the deand to below 1. 3 Inforation Disclosure In this section, I derive the sub-gae perfect equilibriu of the two-period trading gae with inforation disclosure. The infored trader has the option to disclose the inforation to other followers before the liquidation in the second period, Z and 1. Each follower is rational, risk neutral, uninfored and does not trade in the first period. None of the follower has financial constraint. After receiving the private inforation, each follower trades strategically in the second period. Because the fundaental value will be revealed in the third period, each follower only trades one period. When 2, every follower trades copetitively with each other. 16

17 The followers can be regarded as business partners or fellow investors who have connections with the infored trader, so the nuber of followers is observable by the infored trader. Knowing the nuber of followers, the infored trader can derive each follower s trading strategy and the expected price in the second period. The infored trader then copare the expected price with and without inforation disclosure and decides whether to disclose the inforation. The tie line is shown in Figure 2. Figure 2: Tie line of the odel Infored trader observes the nuber of followers and decides whether to share the inforation, and subits order x 1, noise traders subit orders u 1. Market aker announces the price p Nature chooses the realizations of the fundaental value θ of risky asset. Market aker announces the price p 1 Infored trader liquidates his position x 1, followers and noise traders subit their orders x 2, u 2 Fundaental value of risky asset is revealed The equilibriu is defined as followers. Definition 1. An equilibriu is defined as a series of trading strategies X 1, X 2, prices P 1, P 2 and inforation disclosure strategy D, such that the following conditions hold: (1) For given inforation disclosure strategy D and given nuber of followers, the infored trader obtains highest expected profit with trading strategy X 1, that is, E[π 1 (X 1, D) θ E[π 1 (X 1, D) θ. 17

18 (2) Conditional on inforation disclosure (D = 1), each follower obtains highest expected profit with trading strategy X 2 : E[π 2 (X 2, P 1 ) θ, D = 1, P 1 E[π 2 (X 2, P 1 ) θ, D = 1, P 1. (3) The prices p 1 and p 2 are set as p 1 = E[θ y 1, p 2 = E[θ y 1, y 2. (4) The inforation disclosure strategy D {0, 1} satisfies D arg ax E[π 1(X 1, D) θ. D The odel is solved by backward induction. Firstly, I characterize the price equation in the second period, then I derive the follower s trading strategy. Secondly, I copute the trading strategy of the infored trader with inforation disclosure in the first period. At last, I derive the inforation disclosure decision of the infored trader. 3.1 Price in second period: with inforation disclosure After receiving the inforation, each follower buys x 2l when θ = θ H and sells x 2s when θ = θ L in the second period. The infored trader liquidates the position, which is x 1 when 18

19 θ = θ H and x 1 when θ = θ L. The noise traders have order u 2. The total order is thus x 2l x 1 + u 2 if y 1 = x 1 + u 1 y 2 = u 2 if y 1 = u 1 x 2s + x 1 + u 2 if y 1 = x 1 + u 1 The price in the second period depends on the price p 1 and the total order flow y 2. Siilar as the previous section, I only discuss the case when θ = θ H, so two possible values of p 1 are considered : (1) p 1 = p M ; (2) p 1 = p 0. (1) When p 1 = p M : (a) If x 1 x 2l x 1 < 0: θ H if x 2l x 1 1 y 2 < 1 p 2 = p M if 1 y 2 x 2l x (5) p 0 if x 2l x < y 2 1 (b) If 0 x 2l x 1 < 2: θ H if 1 < y 2 x 2l x p 2 = p M if x 2l x 1 1 < y 2 1 (6) p 0 if 1 y 2 x 2l x 1 1 (c) If x 2l x 1 2: θ H if x 2l x 1 1 < y 2 x 2l x p 2 = p 0 if 1 y 2 1 (7) 19

20 (2) When p 1 = p 0 (a) If x 1l x 2l x 1 < 0 and x 1 x 2s x 1 < 0: θ H if x 2l x 1 1 y 2 < 1 p M if 1 y 1 1 (x 2s x 1 ) p 2 = p 0 if 1 (x 2s x 1 ) < y 1 < x 2l x (8) p M if x 2l x y 1 1 θ L if 1 < y 1 1 (x 2s x 1 ) (b) If 0 x 2l x 1 < 1 and 0 x 2s x 1 < 1: θ H if 1 < y x 2l x 1 p M if 1 (x 2s x 1 ) y 1 1 p 2 = p 0 if x 2l x 1 1 < y 1 < 1 (x 2s x 1 ) (9) p M if 1 y 1 x 2l x 1 1 θ L if (x 2s x 1 ) 1 y 1 < 1 (c) If 1 x 2l x 1 < 2: θ H if 1 < y x 2l x 1 p M if x 2l x 1 1 y 1 1 p 2 = p 0 if 1 (x 2s x 1 ) < y 1 < x 2l x 1 1 (10) p M if 1 y 1 1 (x 2s x 1 ) θ L if (x 2s x 1 ) 1 y 1 < 1 20

21 (d) If x 2 x 1 2: θ H if x 2 x 1 1 y 1 x 2 x p 2 = p 0 if 1 y 1 1 θ L if (x 2 x 1 ) 1 y 1 1 (x 2 x 1 ) There are two differences in the price functions p 2 with and without inforation disclosure. The first one lies in the infored trading intensity: without inforation disclosure, the total order y 2 contains the sae inforation as the total order y 1, which is purely the order x 1. But with inforation disclosure, the infored part is x 2 x 1 in the second period, which is different fro x 1. The inforation content depends on the total orders x 2 fro the followers, if the followers trade oderately, the total order x 2 x 1 ight be saller than x 1 and inforation disclosure results in less infored trading. If the followers trade aggressively such that x 2 x 1 is larger than x 1, the inforation disclosure results in ore infored trading. The second difference lies in the arket aker s belief. Note that without inforation disclosure, the infored trader reverse her order in the second period, thus a negative order y 2 actually iplies high fundaental value and a positive order y 2 iplies low fundaental value. But with inforation disclosure, the followers trade on the inforation, and subit buy orders when the fundaental value is high and sell orders when the fundaental value is low, so a positive order y 2 iplies high fundaental value and a negative order y 2 iplies low fundaental value. 21

22 3.2 Second period: followers strategies In this section, I derive the follower s trading strategy in the second period. After receiving the inforation, the followers becoe infored and participate in the arket fro the second period. Since the fundaental value will be revealed in the third period, each follower only trades one period and tries to axiize the expected profit E[π 2 θ, p 1, x 1 = E[x 2 (θ p 2 ) θ, p 1, x 1, by taking the price function p 2 as given. Siilarly, I only consider the case when θ = θ H. As derived in section 2.1 that when the fundaental value is high and when the infored trader appears in the arket, the price p 1 takes three possible values: θ H, p M and p 0. If p 1 = θ H, the inforation is fully revealed in the first period and there is no ore trading in the second period. If p 1 = p M, the inforation is partially revealed in the first period. In the second period, each follower chooses x 2l to axiize the expected profit, by taking the other followers deand into account and by taking the p 2 forulas (5)(6)(7) as given, which yields the expected profit function [ x 2l (θ H p M ) x 2l+( 1)x 2 x 1+2, if 0 < x 2 2l x 1 ( 1)x 2, [ E[π 2 θ = θ H = x 2l (θ H p M ) 2 x 2l ( 1)x 2 +x 1, if x 1 ( 1)x 2 x 2l x ( 1)x 2, 2 0, if x 2l x 1 ( 1)x 2. 22

23 The expected profit of each follower depends on the infored trading intensity in the second period, that is, the value of x 2l x 1. Due to the copetition, each follower trades by taking the deand fro other followers, denoted by x 2, as given. In order to derive the optial strategy of each follower, I first solve the constrained axiization proble in each interval, and get the deand function x 2l, which is a function of x 2. I then take x 2l = x 2 and derive the value of x 2 in each interval. Next I substitute x 2l with the value of x 2 and copute the iplied profit in each interval. Then I copare the profits in each interval and derive the optial deand x 2, which is the one results in the highest profit. The trading strategy of the follower is as follows. Proposition 1. If the price equals to p M in the first period, conditional on inforation disclosure, each follower deands x 2 = x The proof is given in Appendix. On one hand, the optial deand of each follower decreases with the nuber of followers, that is due to the price ipact. Each follower reduces the deand so that the total deand fro all the followers is not too large, in order to hide the private inforation and eliinate the price ipact. On the other hand, the total order fro the followers x 2 = (x ) increases with the nuber of, that is due to the copetition fro the other followers, sae result as Fishan and Hagerty (1995) and Foster and Viswanathan (1996). At last, if p 1 = p 0, no inforation is revealed in the first period. Siilar as the previ- 23

24 ous case, the expected profit function of each follower is given by E[π 2 θ = θ H = x 2l [(θ H p M ) x1 x2s 2 + (θ H p 0 ) x 2l+( 1)x 2l 2x1+x2s+2 2 x 2l [(θ H p M ) x2s x1 2 + (θ H p 0 ) 2 x2s+2x1 x 2l ( 1)x 2 2, if 0 < x 2l x 1 ( 1)x 2,, if x 1 ( 1)x 2 < x 2l x ( 1)x 2, x 2l [(θ H p M ) 2+x1 x 2l ( 1)x 2l 2, if x ( 1)x 2l < x 2l x ( 1)x 2, 0, if x 2l > x 1 ( 1)x 2. Each follower axiizes the expected profit by choosing the optial deand x 2l, taking the price p 2 as given, and also taking the other 1 followers deands as given: Proposition 2. When the price equals to p 0 in the first period, conditional on inforation disclosure, an equilibria falls into the following categories: 1. Aggressive trading: each follower trades x 2 = x if x 1+1 2(1 β 1 ). 2. Moderate trading: each follower trades x 2 = x 1+ 1 β 1 if < x (1 β 2β 1 ) 1 The proof is given in Appendix. The following graph illustrates the optial deand of each follower when p 1 = p 0. 24

25 Figure 3: The follower s deand in the second period when p 1 = p 0 Figure 3 shows that the optial deand x 2 depends on the nuber of followers and the probability of infored trading µ (note that β 1 = 1 ). When x µ 2(1 β 1, each follower ) trades aggressively and deand x 2 = x 1+2. In this case, the total order in the second period +1 is also aggressive and 1 x 2 x 1 < 2. When < x 1+1 2(1 β 1, the copetition aong the ) followers becoes saller, the probability of infored trading increases and arket aker becoes ore sensitive to the order, so each follower reduces the deand to x 1+ 1 β β 1. The total order also reduces and 0 x 2 x 1 < 1. In conclusion, the trading intensity of each follower is positive correlated with the nuber of followers and negative correlated with the probability of infored trading µ. Copared with the trading strategy when p 1 = p M, the follower trades less intensively when p 1 = p 0. Note that the expected profit contains two parts: the order size x 2 and the arginal profit θ H p 2. When p 1 = p M, inforation has been partially revealed in the first period, the probability of ore inforation revelation by trading in the second period is thud higher. So the price ipact when p 1 = p M is higher and the arginal profit θ H p 2 is thus lower for each follower. Therefore, the follower takes a larger deand by copensating the 25

26 loss fro the arginal profit. When p 1 = p 0, no inforation is revealed in the first period. The price ipact is saller in the second period and the arginal profit for each follower is higher, so the followers would rather to have a saller position. But when the nuber of followers increases, the follower increases the deand due to ore fierce copetition. 3.3 Inforation disclosure strategy When deterining whether to disclose the inforation or not, the infored trader has to copare the expected price p 2 with and without inforation disclosure. Obviously, the optial disclosure decision results in higher price p 2. Given the optial deand of each follower and the pricing rule in the second period fro in section 3.1 and 3.2, I then derive the inforation disclosure strategy of the infored trader. Without loss of generality, I ipose one tie-breaking rule: whenever the infored trader is indifferent between disclosing and not disclosing the inforation, the infored trader strictly prefers disclosing the inforation. In the first case, when p 1 = p M, it is derived in section 2.2 that conditional on the appearance of infored trader and θ = θ H, the price p 2 without inforation disclosure becoes θ H if 1 u 2 x 1 1 p 2 = p M if x 1 1 u 2 1 So the expected price without inforation disclosure is E[p 2 θ H, x 1, p 1 = p M, d = 0 = x 1 2 θ H + (1 x 1 2 )p M, (11) where d = 0 denotes no inforation disclosure and d = 1 denotes inforation disclosure. If the infored trader discloses the inforation, each follower trades x 2 = x , the price p 2 26

27 becoes θ H if 1 ( (x 1+2) x +1 1 ) < u 2 1 p 2 = p M if 1 u 2 1 ( (x 1+2) x +1 1 ) and the expected price with inforation disclosure is E[p 2 θ H, x 1, p 1 = p M, d = 1 = 2 x 1 2( + 1) θ H + (1 2 x 1 2( + 1) )p M (12) Coparing equation (11) and (12), we find that the infored trader is better off disclosing the inforation when x 1 2, and she is better off not disclosing the inforation when +2 x 1 < In the second case, when p 1 = p 0, the price p 2 conditional on no inforation disclosure is given by θ H if 1 u 2 < x 1 1 p 2 = p M if x 1 1 u 2 2x 1 1 which yields the expected price p 2 as p 0 if 2x 1 1 < u 2 1 E[p 2 θ H, x 1, p 1 = p 0, d = 0 = x 1 2 θ H + x 1 2 p M + (1 x 1 )p 0. (13) If the infored trader discloses the inforation, proposition 2 shows that each follower trades x 1 = x when x 1+1 by p 2 = θ H if 1 ( (x1+ 1 β 1 ) + 1 p M if 1 2( (x1+ 1 β 1 ) β (1 β 1 ) and trades x 2 = x 1+ 2β 1 x 1 ) < u 2 1 x 1 ) u 2 1 ( (x1+ 1 β ) 1 x 2β ) 1 2β 1 p 0 if 1 u 2 < 1 2( (x1+ 1 β ) 1 x ) 2β 1 when < x 1+1. The price p 2(1 β 2β 1 ) 2 is given 1 θ H if 1 ( (x1+2) +1 x 1 ) < u 2 1 or p 2 = p M if 1 u 2 1 ( (x1+2) +1 x 1 ) 27

28 yields the expected price p 2 as E[p 2 θ H, p 1 = p 0, < x (1 β 1 ), d = 1 = 2 x 1 4β θ H + 2 x 1 4β p M + (1 2 x 1 2β )p 0, E[p 2 θ H, p 1 = p 0, x (1 β 1 ), d = 1 = 2 x 1 2( + 1) θ H + (1 2 x 1 2( + 1) )p M. (14) Coparing the expected prices in equation (13) and (14), we see that the infored trader is better off disclosing the inforation when x 1 when x 1 >. β 1 +1 β 1, and she is better off not disclosing +1 The inforation disclosure decision is contingent on the price p 1, which also depends on the value of x 1 and. I suarize and illustrate the inforation disclosure strategy in the following figure. Unconditional Disclosure Conditional Disclosure No Disclosure 0 β x 1 Figure 4: The inforation disclosure decision of infored trader Unconditional Disclosure eans that the infored trader always discloses the inforation for any price p 1. Conditional Disclosure eans the infored trader only discloses the inforation when p 1 = p M and not disclose the inforation when p 1 = p 0. No Disclosure eans the infored trader does not disclose the inforation for any price p 1. Fro Figure 4 we can see that if the infored trader initially decides to disclose the inforation in period one, she will keep the coitent and discloses the inforation in the second period whenever p 1 = p 0 or p 1 = p M, if she trades x 1 [0, β 1 ; she will deviate to +1 not disclose the inforation in the second period when p 1 = p 0 but keeps the coitent when p 1 = p M, if she trades x 1 ( x 1 > 2 +2, 2 β ; she will deviate for any price p 1 if she trades in the first period. Moreover, Figure 4 shows that the inforation disclosure is negative related to the deand of the infored trader: the inforation disclosure is ore likely when the infored trader has a sall deand in the first period. 28

29 3.4 The first period: the infored trader s strategy In the first period, the infored trader tries to axiize the expected profit by choosing an optial deand x 1. Section 3.3 shows that the inforation disclosure is a price contingent strategy, which also depends on the value of x 1. In the sub-gae perfect equilibriu, the optial deand x 1 is chosen by taking the inforation disclosure decision into account. Then I copute the expected profit function of the infored trader, which is a piecewise function. In general, there are four cases: (1) when 0 x 1 2(1 β 1 ) 1, the infored trader discloses the inforation whenever p 1 θ H and each follower trades x 2 = x in the second period; (2) when 2(1 β 1 ) 1 < x 1 β 1, the infored trader discloses +1 the inforation whenever p 1 θ H, each follower trades x 2 = x 1+ 1 β 1 if p = p 0 and trades 2β 1 x 2 = x 1+2 if p +1 1 = p M ; (3) When < x β , the infored trader only discloses the +2 inforation when p 1 = p M and each follower trades x 2 = x ; (4) when x 1 > 2 +2, the infored trader does not disclose the inforation in the second period. I characterize the expected profit function of the infored trader by the value of and µ. Since the expected profit is a piecewise function, when solving the optial deand, I first solve the constrained axiization proble and get the optial deand in each subdoain; then I substitute the optial deand in the profit function and get the iplied profit in each subdoain; in the end I copare the iplied profit in each subdoain and get the axiu profit, as well as the corresponding deand x 1. The profit function and the optial deand are given as follows. = 1 At this point, 2(1 β 1 ) 1 < 0 < β 1 +1 < 2 +1 < 1, the expected profit of infored 29

30 trader is E[π 1 θ H = x 1l [ x 1s 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x1s 2 )(θ H p M ) 2 x 1l 2(2β 1+1) +x 1l [(1 x 1l 2 x1s 2 )(p M p 0 ) 2 x1s 2(2β 1+1), if 0 x 1l β 1+1, [ x x 1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x1s 2 )[(θ H p M ) x 1l 2 + (p M p 0 ) x1s 2 if β < x 1+1 1l 2 +2, [ x x1s 1l 2 (θ H p M ) x 1l 2 + (1 x 1l 2 x1s 2 )[(θ H p M ) x 1l 2 + (p M p 0 ) x1s 2 2, if +2 < x 1l 1, x 1l (1 x 1l 2 )(θ H p M ) x 1l 2, if 1 < x 1l 2. The optial deand is solved as 4 x 3 1 =, 0 µ < (15) 2β 1+1 5β = 4 µ µ, µ 1 The optial deand reduces with the probability of infored trading µ. The infored trader trades very aggressive when the probability of infored trading µ is sall, and trades oderately when µ is large, at µ = , the expected profit of trading 4 3 equals to the expected profit of trading 4 µ 3+µ. Moreover, both 4 3 and 4 µ 3+µ are larger than the disclosure threshold 2, so the infored trader does not disclose the inforation +2 when = 1. Indeed, the optial deand is the sae as the optial deand derived in section 2.3, where there is no inforation disclosure. = 2 When 1 2 β 1 3 4, 0 < 2(1 β 1) 1 < E[π 1 θ H = < 2 β [ x x1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x 1s 2 )(θ H p M ) 2 x 1l 2(+1) +x 1l [(1 x 1l 2 x 1s 2 )(p M p 0 ) 2 x 1s 2(+1) x 1l [ x1s 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x 1s +x 1l [(1 x 1l 2 x 1s 2 )(p M p 0 ) 2 x 1s 2(2β 1 +1) 2 )(θ H p M ) 2 x 1l = 1, the expected profit is 2(2β 1 +1) [ x x1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x 1s 2 )[(θ H p M ) x 1l 2 + (p M p 0 ) x 1s 2 if 0 x 1l 2(1 β 1 ) 1, if 2(1 β 1 ) 1 < x 1l if β 1 +1 < x 1l 1, x 1l (1 x 1l 2 )(θ H p M ) x 1l 2, if 1 < x 1 2. β 1 +1, 30

31 When 3 4 < β 1 1, 2(1 β 1 ) 1 0 < E[π 1 θ H = < 2 β = 1, the expected profit is [ x x 1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x1s 2 )(θ H p M ) 2 x 1l 2(2β 1+1) +x 1l [(1 x 1l 2 x1s 2 )(p M p 0 ) 2 x1s 2(2β 1+1) [ x x 1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x1s 2 )[(θ H p M ) x 1l 2 + (p M p 0 ) x1s 2 if 0 x 1l if β 1+1, β 1+1 < x 1l 1, x 1l (1 x 1l 2 )(θ H p M ) x 1l 2, if 1 < x 1 2. The optial deand is 4 x 3 1 =, 0 µ < β 1+16β β 1+580β β β4 1 2(2+3β 1+8β1 2) µ 1, (16) We can see that when µ is sall, the infored trader still trades very aggressively, even though the nuber of followers increases to 2. When µ becoes larger, the infored trader reduces the deand and trades oderately. At µ = , the expected profit of trading aggressively equals to the expected profit of trading oderately. Note that 1+30β 1+16β β β β β4 1 2(2+3β 1 +8β 2 1 ) is saller than the disclosure threshold β 1, which iplies that the infored trader would disclose the inforation in the +1 second period when µ is large. 3 When 1 2 β , 0 2(1 β 1) 1 < profit function is E[π 1 θ H = < 1 < 2 β [ x x1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x 1s 2 )(θ H p M ) 2 x 1l 2(+1) +x 1l [(1 x 1l 2 x 1s 2 )(p M p 0 ) 2 x 1s 2(+1) [ x x1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x 1s +x 1l [(1 x 1l 2 x 1s 2 )(p M p 0 ) 2 x 1s 2(2β 1 +1) 2 )(θ H p M ) 2 x 1l 2(2β 1 +1) [ x x1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x 1s 2 )[(θ H p M ) x 1l 2 + (p M p 0 ) x 1s 2 < 2, the expected if 0 x 1l 2(1 β 1 ) 1, if 2(1 β 1 ) 1 < x 1l if β 1 +1 < x 1l 1, x 1l (1 x 1l 2 )(θ H p M ) 2 x 1l 2(+1), if 1 < x 1l 2 +2, x 1l (1 x 1l 2 )(θ H p M ) x 1l 2, if 2 +2 < x 1l 2. β 1 +1, 31

32 When < β 1 1, 2(1 β 1 ) 1 < 0 < profit is < 1 < 2 β < 2, the expected E[π 1 θ H = [ x x 1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x1s 2 )(θ H p M ) 2 x 1l 2(2β 1+1) +x 1l [(1 x 1l 2 x1s 2 )(p M p 0 ) 2 x1s 2(2β 1+1) [ x x 1s 1l 2 (θ H p M ) 2 x 1l 2(+1) + (1 x 1l 2 x1s 2 )[(θ H p M ) x 1l 2 + (p M p 0 ) x1s 2 if 0 < x 1l if β 1+1, β 1+1 < x 1l 1, x 1l (1 x 1l 2 )(θ H p M ) 2 x 1l 2(+1), if 1 < x 1l 2 +2, x 1l (1 x 1l 2 )(θ H p M ) x 1l 2, if 2 +2 < x 1l 2. The optial deand is x 1 = { (+1)(1+2β 1 ) (1+2β 1 ) 2 (+1) 2 24(1 β 1 )(+1)+48(1 β 1 ) 2, 0 µ µ 6(1 β 1 ) (1 10β 1 ) (1+2β 1 )(1+2β 1 2 )+ ((1 10β 1 ) (1+2β 1 )(1+2β 1 2 )) 2 32β 1 (+1)(1 5β 1 (1 β 1 +4β1 2)) 2(1 5β 1 (1 β 1 +4β1 2)), µ < µ 1, where µ is the solution of 2(1 β 1 ) 1 = (+1)(1+2β 1) (1+2β 1 ) 2 (+1) 2 24(1 β 1 )(+1)+48(1 β 1 ) 2 6(1 β 1 ). The optial deand of the infored trader is still a decreasing function of µ. However, the optial deand is not aggressive any ore when µ is sall. Instead, the infored trader trades oderately. Note that x 1 is saller than the disclosure threshold β 1 +1 for any value of µ, which eans that the infored trader discloses the inforation unconditionally when the nuber of follower is at least 3. Furtherore, at µ = µ, the optial deand equals to 2(1 β 1 ) 1 and is discontinuous at this point. That is because, the deand of each follower in the second period is discontinuous at this point. When µ µ, each follower has an aggressive deand x , but when µ > µ, each follower reduces to a oderate deand x 1+ 1 β β 1 if p 1 = p 0. Considering the discontinuity of follower s deand, the infored trader also alter the deand at µ = µ. (17) Taking the deand x 1 as given by equation (15), (16) and (17), I plot the inforation 32

33 disclosure strategy in Figure 5 with varying and µ. Figure 5: The inforation disclosure strategy 4 No Disclosure Disclosure µ Figure 5 shows that when there is zero or one follower, the infored trader does not disclose the inforation. That is because the follow up capitals fro the followers are not large enough if inforation is disclosed, the infored trader would rather signal the inforation to the arket aker by the liquidation. When the nuber of followers increases to two, the follow up capitals increase due to the copetition aong the followers, the infored trader discloses the inforation, but conditional on the probability of infored trading µ. The infored trader discloses the inforation when µ is larger than and does not disclose the inforation when µ < Note that the inforation disclosure is ore likely when the deand fro infored trader in the first period is sall. So when µ is large, the inforation is ore likely to be revealed through trading in the first period, the infored trader thus trades very cautiously. Therefore, she is ore likely to disclose the inforation in the second period. But when µ is sall, the infored trader can hide the inforation ore easily and thus trades ore intensively in the first period, so the inforation disclosure is less likely in the second period. When is at least 3, the infored trader plays fully 33

34 disclosure strategy. As docuented by Foster and Viswanathan (1996) that when there are ultiple infored traders, each trader trades ore intensively and induces larger order flow. So when there are at least three followers, the infored trader is better off disclosing the inforation because the followers will bring ore capital in the second period. One ore interesting question is how the infored trader adjust the deand with inforation disclosure and with different nuber of followers, I plot the optial deand x 1 for different in Figure 6. Firstly, the deand x 1 is an decreasing function of µ. Secondly, in ters of, the deand when = 1 is the sae with the benchark, since there is no inforation disclosure when = 1. When increases to 2, the infored traders reduces the deand conditional on inforation disclosure, as shown by the blue line. When is larger than 3, the infored trader always discloses the inforation. On one hand, the deand becoes less aggressively, that is, x 1 < 1. On the other hand, the deand increases with the. That is because when the nuber of followers increases, high copetition in the second period induces aggressive deand and thus ore inforation revelation to the arket aker, and the price is ore close to the fundaental value. When the price uncertainty becoes saller, the infored trader is ore brave to trade intensively in the first period. 34