Manipulation, Panic Runs, and the Short Selling Ban

Transcription

1 Manipulation, Panic Runs, and the Short Selling Ban Pingyang Gao Booth School of Business The University of Chicago Xu Jiang Fuua School of Business Duke University Jinzhi Lu Booth School of Business The University of Chicago Very Preliminary and incomplete.

2 Abstract This paper identifies conditions under which a short-selling ban improves the ex-ante firm value. Short selling improves price discovery and enables stakeholders to make better investment decisions. However, manipulative short selling can arise as a self-fulfilling euilibrium and mislead the investment decisions. The adverse effect is amplified by the firm s vulnerability to panic runs. Overall, short selling reduces the ex-ante firm value if both manipulative short selling is strong and the firm is very prone to runs. The results contribute to our understanding of the function of short selling in the capital markets and to the controversy around the regulations against short selling. JEL classification: Key words:

3 Introduction This paper presents a model to evaluate the effi ciency conseuences of banning short sales. Ever since the first regulation against short selling was enacted by Amsterdam exchange in 60, such regulations have been controversial (e.g., Bris et al. (007. A salient feature of the restrictions on short selling is that they are often imposed on financial stocks in the bad times. For example, in September 008 the SEC banned short-sales of shares of 799 companies for two weeks and the U.K. and Japan declared a ban on short selling for as long as it takes to stabilize the markets. Similarly, in August 0, France, Spain, Italy, and Belgium imposed temporary bans on short selling for some financial stocks during the European sovereign debt crisis. Proponents for short selling makes a straightforward argument. Like other selling and buying, short selling allows investors to express their negative opinions through trading and improves the informativeness of stock prices. The price discovery, in turn, leads to better decisions and more effi cient allocation of capital in the economy. In contrast, opponents of short selling have argued that short sellers may manipulate the market through bear raids. Speculators with initial short positions may employ various tactics, including spreading rumors, to drive down the share prices in order to close their initial short positions at a lower cost. Goldstein and Guembel (008 (hereafter referred to as GG presents a model in which manipulative short selling, whereby an uninformed speculator nonetheless shorts a firm s stock and earns a profit, can arise in a self-fulfilling euilibrium. In this paper, we combine both arguments to evaluate the short-selling (SS ban on the ex-ante effi ciency. We augment a coordination game with the manipulative short selling from GG. There is a speculator who receives information about the state with a known probability. After privately learning about whether he has received an informative signal or not, the speculator trades in a market in the style of Kyle (985 in which the stock price endogenously reflects some of the speculator s information. A continuum of investors then observe the stock price and make their decisions that collectively affect the firm s cash flow. Conditional on the stock price, the coordination subgame yields a uniue euilibrium using

4 the global games methodology. We then solve for the speculator s trading strategy, determine the entire euilibrium, and compare the euilibrium outcomes under two regimes of allowing and banning short selling. Our main result is that the SS ban improves the ex-ante effi ciency if the firm s vulnerability to runs is suffi ciently high and the speculator s information uality is not suffi ciently high. Financial firms are more vulnerable to runs, while crises are often associated with high degree of uncertainty in the market. As a result, our main result provide a possible justification for the SS ban on the financial firms stock during the financial crisis. To see why the high vulnerability to run is necessary, it is useful to consider a benchmark where the investment decision is made by representative investor, as studied in GG. In this case, the SS ban always reduces the ex-ante effi ciency despite the euilibrium existence of manipulative short selling. The intuition for this somewhat surprising result is compelling. The investor can always ignore the stock price in making the decision and thus cannot be worse off with the stock price. The implication is that a short selling ban that cannot discriminate between informed and uninformed short selling always reduces the ex-ante effi ciency. The existence of rational bear raids is not suffi cient to justify the ban on short selling. The medium level of the speculator s information uality is also necessary. To see this, consider two extremes. At one extreme, if the market is populated mainly by informed speculators, then manipulative SS arises but very infreuently. The informational benefit from informed short selling dominates the cost of the infreuent manipulative SS. As a result, the SS ban strictly reduces information and effi ciency. At the other extreme, if the market is mainly populated by uninformed speculators, then based on the intuition in GG, manipulative short selling does not arise in euilibrium. As a result, the SS ban removes the informed short selling and thus reduces the effi ciency. The intuition behind the main result is as follows. In a coordination game, the investors investment decisions are driven not only by information about the fundamental but also by their concern about others actions. Since investors use the stock price to update their beliefs about both the state and others decisions, the investors responsiveness to the stock price results from both motivations. The former improves while the latter reduces the effi ciency

5 of the investment decisions from the society s perspective. We show that the SS ban essentially reduces the stock price informativeness and makes investors less sensitive to the stock price. Such a reduction in the sensitivity to the stock price reduces the use of information in the investment decision but also mitigates the coordination failure. When the speculator s information is not suffi ciently good and when the coordination friction is suffi ciently high, the informational loss is dominated by the improvement of coordination, and, as a result, the effi ciency is improved. Our paper is related to the literature on short sales. Short sales are a basic component in modern finance theories of asset pricing and portfolio choice. Most theoretical studies thus have viewed short sales as an institutional constraint and focused on identifying its conseuences (e.g., Miller (977, Diamond and Verrecchia (987, Duffi e et al. (00, Abreu and Brunnermeier (003, Scheinkman and Xiong (003. In most of these studies, banning short sales have an adverse effect on effi ciency. A few papers have studied the ex-post conseuences of short selling in the presence of rigid frictions. The study that is most closely related to ours is GG. We built on the manipulative short selling in GG and extend GG to model a coordination decision-making game. This extension generates our main result that the SS ban can improve effi ciency, while in GG the SS ban cannot improve effi ciency despite the manipulative short selling. Brunnermeier and Oehmke (03 show that short selling forces firms with market-based leverage reuirement to liuidate the illiuid assets. In their model there is no informational feedback from the stock price to real decisions. The effect of stock price on the liuidation decision is assumed. Liu (04 also studies a coordination game with short selling. In his model, investors in the coordination game receive private information and observes the stock price as public information. Short selling is assumed to add noise into the stock price and makes the public information noisier. In contrast, short selling in our model allows speculator s private information to be endogenously impounded into the stock price and makes the stock price more informative. Liu (06 studies the interaction between a investors coordination game with interbank market trading and focuses on the feedback loop between interbank market rate and the coordination game. Interbank market serves as a provision of liuidity 3

6 and banks do not learn any information from the interbank market. In contrast, we focus on the interaction between managers learning from price and the coordination game. Our model makes a methodological contribution to the coordination with market manipulation. We use the global games methodology to obtain the uniue euilibrium of the coordination game to conduct welfare analysis. However, the market manipulation component from GG employs two rounds of trading and is too complicated to be combined with the coordination game. We use a one round trading setting to simplify the market manipulation component and integrate it with the coordination game. This formulation of market manipulation may be used in other settings. Finally, our paper is related to the literature on the welfare effects of public information in coordination games (Angeletos and Pavan (007. Morris and Shin (00 is probably the first to show that in settings with coordination motives, more precise public information may decrease welfare when private information is suffi ciently precise. In our setting there is no private information per se so even with coordination, more public information should increase welfare. The detrimental effect of more public information comes from the interaction of coordination with feedback effect. The rest of the paper is organized as follows. Section introduces the model set up. Section 3 highlights the multiple euilibria problem of the model. In section 4, we pin-down the uniue euilibrium using the global game techniue. In section 5, the main results, specifically the result that short selling could be detrimental to bank value, are presented. Section 6 concludes. Model setup Our model augments a coordination game with the manipulative short selling from GG. We start with a coordination game. Consider a risk-neutral economy with no discounting and four dates (t 0,,, 3, one firm with an underlying project, and a continuum of investors. The underlying state θ is either good or bad with eual probability, i.e., θ {H, L} with Pr(θ H. 4

7 At t, after observing a signal yet to be described below, each investor makes a binary investment decision, i.e., a i {0, }. If the investor does not invest, she receives a payoff normalized to 0. If she invests, then her payoff u is jointly determined by the state θ and the aggregate investing population l a i di: i [0,] u θ δl. The parameter δ 0 captures the degree of strategic complementarity among investors investment decisions. δ is often referred to as the project s vulnerability to runs. The project s aggregate value is v ( l (θ δl. So far we have a standard coordination game. The only information source for investors is the stock price endogenously determined in a Kyle setting. Specifically, there are three types of traders. The first is a speculator who learns perfectly about θ with probability α (0, ] and nothing with the complementary probability, that is, the speculator observes a signal s {H, L, φ}. We call a speculator with s H(L as a positively (negatively informed speculator and a speculator with no information (s φ as an uninformed speculator. The speculator chooses an order d(s {, 0, }. The second group of traders are liuidity traders who trade for reasons orthogonal to state θ. Their aggregate order is denoted as ũ, which is normally distributed with mean zero and variance σ n. Finally, the third group of traders is the market maker who observes the total order flow d + ũ and sets price eual to the expected firm value: P E[v ]. The stock price and the order flow have identical information content. For simplicity, we assume that investors observe the order flow (instead of the stock price. In sum, the timeline of the events is as follows. At t 0, the speculator s information endowment s is realized. At t, the trading occurs. Both the stock price and the order flow are observed. 5

8 At t, investors observe the order flow and make decisions. At t 3, the firm s terminal cash flows are realized. As a benchmark, GG s setting is special case of our model with δ 0. [There is another difference between our setting and that of GG is the assumption that noise trading is normally distributed rather than discretely distributed. As will be discussed below, this assumption allows us to solve the uniue euilibrium with only one round of trading instead of two rounds of trading as in GG, resulting in tractable analysis of welfare when we introduce the coordination friction into the feedback model.] We make a few assumptions before proceeding. A : H > 0 > L A : H + L > δ A3 : c r > H + L A states that it is socially optimal to invest in the good state and not to invest in the bad state. A guarantees that in the absence of any information, the default choice is to invest. It enhances the assumption in GG that r H +r L > 0 to accommodate the coordination game represented by δ. Finally, A3 reuires that the feedback effect is suffi ciently strong so that the investors decisions can be influenced by the stock price even in the absence of the coordination failure (see Proposition 3 of GG. c r is defined by euation (4 in the Appendix. To simplify the notation, we normalize H and L r with r (0,. r is a measure of the strength of the informational feedback effect. A higher r indicates that the investment decision is more sensitive to information. In addition, we introduce a tier-breaker. If the negatively (positively informed speculator is indifferent among d {, 0, }, he always choose d (d. This rules out a degenerate euilibrium where no trading takes place. The effi ciency is defined as the expected firm value that aggregates the payoffs to all 6

9 investors: V E[( l(r θ δl]. ( A perfect Bayesian euilibrium (PBE of our model consists of the speculator s trading strategy d(s, each investor s withdrawal strategy a i ( and beliefs about the fundamental θ such that ( both the speculator and the investors maximize their respective objective functions, given their beliefs and the strategies of others and ( each investor uses Bayes Rule, if possible, to update beliefs about θ. 3 The euilibrium In this section, we solve for the euilibrium using the backward induction. 3. The coordination subgame We first solve for the subgame after the investors have observed the order flow. We conjecture and later verify that satisfies maximum likelihood ratio property (MLRP, that is, a higher indicates that θ H is more likely. Intuitively, positively informed speculator always chooses d and negatively informed speculator always chooses d. Given the conjectures, regardless of the uninformed speculator s choice of d, a higher order flow indicates that it is more likely that the order flow comes from positively informed speculator and therefore a higher probability of θ H. Now consider the strategy of investor i when observing order flow. If she chooses to withdraw, then she gets 0 for sure. If she chooses to continue, then her expected payoff differential is ( E[θ ] δe[l ]. As is standard in coordination game, multiple euilibria arise if is common knowledge, making it diffi cult to conduct comparative statics. We apply the global games methodology to obtain the uniue euilibrium. Specifically, we assume that each investor observes the order flow with noise and focus on the euilibrium when the noise converges to 0. This 7

10 results in an uniue euilibrium of the coordination game. Lemma The investors play a common threshold strategy. The common threshold is determined by the following euation. E[r ] δ 0. ( Lemma characterizes the euilibrium common threshold in an intuitive manner. Collecting these two expectations and imposing the euilibrium condition that the marginal investor has to be indifferent between continuing and running at the threshold, i.e. ( 0, we obtain euation that uniuely determines the common threshold. A standard result in the global games methodology results in E[l ]. An investor uses the signal to forecast other investors signals and actions. At i, she conjectures that exactly half of the other investors will get a signal higher than and stay whereas the other half will get a signal lower than and withdraw. Therefore, she expects that half of the investors with stay: E[l ]. The key uantity left is thus E[θ ], the marginal investor s expectation of the fundamental. Since θ is binary, the expectation is fully characterized by the conditional probability β(j Pr(θ H j. A higher β means that the marginal investor has to be more optimistic about the state to invest. β( j δ + r + r. (3 β(j is an euilibrium variable and depends on the trading strategy of the speculator. to which we turn now. 3. The trading decision and the euilibrium Anticipating the uniue euilibrium for the subgame of coordination, the speculator decides his trading strategy. The trading strategy and the investors investment decisions are then jointly determined by solving a fixed point problem. When short selling is banned, it is straightforward to show that both the negatively informed speculator and the uninformed speculator find it optimal not to trade. As GG 8

11 has shown, the manipulation is only one-sided. It is never optimal for the speculator to buy when he is uninformed. It is also straightforward to show that the positively informed speculator buys. Given the trading strategy, both the market maker and the investors make their decisions accordingly. Proposition Suppose short sales are banned. Then the positively informed speculator buys while other speculators do not trade, i.e. d (H and d (L d (φ 0. The investment threshold is B. When the short selling is allowed, the possibility of manipulative short selling complicates the derivation of the euilibrium. Short-selling lowers the order flow and induces more investors not to invest. This has two effects on the speculator s profit. First, even in the absence of changes to the investors investment decisions, the market maker interprets the lower order flow as a bad signal about the state and lowers the price accordingly. Since the market maker cannot distinguish whether the short-selling originates from the negatively informed speculator or the uninformed speculator, the price is set as an weighted average conditional on the speculator is negatively informed or uninformed. Thus, shorting is profitable for the informed speculator but not profitable for the uninformed speculator. Second, the lower price also has an informational feedback on the investment decisions. Agents use the stock price to make inference about the state and about other investors decisions. A lower price is a bad signal about the state and thus discourages investors from investing. The reduction in the investment indeed reduces the firm s terminal cash flow and creates a self-fulfilling euilibrium. Both the informed and uninformed speculator can profit from this informational feedback effect. Therefore, while the negatively informed speculator always shorts, the uninformed speculator shorts if and only if the profit from the informational feedback effect compensates the cost from his informational disadvantage, a condition satisfied when the fraction of informed speculator is suffi ciently large. Proposition Suppose short sales are allowed. If α > α (δ where α (δ is defined in the appendix, then 9

12 . the informed speculator trades in the direction of his information: i.e., d (H and d (L ;. the uninformed speculator short sells, i.e., d (φ ; 3. the investment threshold is A. Proposition generates a suffi cient condition for manipulative short selling to arise, namely, α > α (δ. Setting δ 0, Proposition replicates the main result in GG that manipulative short selling arises when the probability that the speculator is informed is sufficiently high. We extend this result to our setting of a coordination game in which δ > 0. 4 The analysis Having characterized the two euilibria, we are ready to present our main result about the effi ciency of banning short selling. We have defined the effi ciency as the ex-ante expected firm value V in euation (. After some algebra, for a given regime j {A, B}, we can write it generally as V V F B (εh + rε L. Note that V F B r H is the effi ciency in the first-best case when θ is publicly known and the investment decision is made by a single investor. In this case, the representative investor invests in the good state and does not invest in the bad state. Relative to the first-best, the effi ciency is ultimately reduced by two types of errors in the investment decisions, underinvestment in the good state or over-investment in the bad state. Denote their respective probabilities as ε H and ε L. Underinvestment reduces unit of effi ciency by foregoing the payoff H in the good state, while overinvestment generates a loss of L r when the state is bad. The ban affects the speculator s trading strategy, the information content of the order flow, the investors investment decisions and ultimately the effi ciency. We analyze each effect in turn. 0

13 Table : Investment errors in various scenarios j A B B A B A Prob*cost ε H Ij Φ( A Φ( B Φ( B Φ( A + α ε L Ij Φ( A + Φ( B σn Φ( A + Φ( B σn + αr ε H Uj Φ( A + Φ( B σn ε L Uj Φ( A + Φ( B Φ( B σn Φ( A + ( α Φ( B σn + ( α r σn Φ( A + First, the ban affects the speculator s trading strategy and investors investment decisions. It does not affect the buy order from the positively informed speculator but replaces the sell order from both the negatively informed speculator and the uninformed speculator with no trading. Accordingly, the ban does not affect the order flow distribution when the speculator is positively informed but moves the distribution to the right by one unit in other cases. Rationally anticipating the conseuences of the SS ban on the information content of the stock price, the investors adjust their investment decisions accordingly. They increases the order flow threshold above which they will invest by an amount smaller than. Lemma The SS ban changes the investment threshold as follows: A < B < A +. Given the order flow distribution and the investment threshold, the investment errors in various scenarios are summarized in Table. We examine the effi ciency of the resulting investment decisions. We break down the effi ciency effect further by the speculator s type, whether the speculator is informed (I or uninformed (U. Note that the expected error ε θ j satisfies ε θ j αε θ Ij + ( α ε θ Uj. Consider first the informed speculator who buys in the good state and shorts in the bad state. In the good state, the order flow has a mean of and variance of σ n because the positively informed speculator buys. Thus, the probability of underinvestment is ε H Ij Φ( j for j {A, B}. This explain the second row in Table. Similarly, in the bad state, the order flow has a mean of if SS is allowed and 0 if SS is banned. The probability of overinvestment is thus ε L Ij Φ( j d Ij (L, where d Ij (L is the negatively informed

14 speculator s trading strategy in regime j {A, B}. This explains the third row in Table. Therefore, when the speculator is informed, the ban on the uality of the investment decision is captured by ε θ I ε θ IB ε θ IA. Now consider the uninformed speculator who shorts when SS is allowed and does not trade when SS is banned. The order flow has a mean of d Uj (θ in regime j and state θ, and the investment errors can be expressed as in the fourth and fifth row of TABLE Y. Therefore, when the speculator is uninformed, the effect of the ban on the investment effi ciency is captured by ε θ U ε θ UB ε θ UA. We can characterize the ban s effect on investment effi ciencies as follows. Lemma 3 The SS ban affects the accuracy of the investment decisions as follows.. when the speculator is informed, it reduces the investment accuracy, that is, ε θ I > 0 for any θ;. when the speculator is not informed, the ban increase investment accuracy in the good state and reduces investment accuracy. That is, ε H U < 0 and εl U > ε L I εl U εh U ε 0. Lemma 3 is intuitive. First, the ban suppresses the information from the negatively informed speculator and increases the overinvestment in the bad state, despite the rational adjustment by investors. The ban removes the sell order from the negatively informed speculator and thus degrades the informational value of the order flow. Hence, ε L I > 0. Second, the ban also increases the investment error when the speculator is positively informed, that is, ε H I > 0. Even though the ban does not affect the buy strategy of the positively informed speculator, it changes the strategy of other types of speculators whose order flow cannot be distinguished from the positively informed speculator. In particular, the suppression of the

15 sell orders dilutes the information content of the positively informed speculator s buy order, which adversely affects the investors use of information in the investment decisions. Collectively, these two channels explain Part of Lemma 3 and capture the conventional wisdom that the SS ban reduces the effi ciency by degrading the information value of the stock price. When the speculator engages in manipulative short selling, that is, shorting when he is uninformed, how the SS ban affects the investment accuracy depends on the state. In the good state, the ban reduces investment and leads to more underinvestment, that is, ε H U > 0. In the bad state, the ban also reduces investment, but the reduction leads to less underinvestment, that is, ε L U < 0. This explains Part of Lemma 3. Finally, Part 3 of Lemma 3 shows an articulate relationship among the investment errors. First, since the speculator is not informed, the investment error is symmetric. When the ban reduces the underinvestment in the good state, it increases the overinvestment in the bad state by the same amount. that is, ε H U εl U. Second, in the bad state, both the uninformed and informed speculators use the same trading strategy across the two regimes. Thus, the ban has the same effect on the investment accuracy, that is, ε L I εl U. In sum, when the speculator is informed, the ban reduces effi ciency ε H I > 0 and ε L I ε 0 > 0. When the speculator is uninformed, the ban reduces effi ciency in the bad state but increases effi ciency in the good state, that is, ε L U εh U ε 0 < 0. There is a trade-off. Collecting these errors and weighting them by their probabilities and associated conseuences, we can write out the effi ciency difference across the two regimes as V (V Ban V Allowed α ε H I (αr ( α( r ε 0. (4 The next proposition presents our main result. Proposition 3 Banning short-selling improves the effi ciency if and only if α (α(δ, α (δ, where α (δ is defined in the appendix. This set is empty at δ 0 and nonempty when δ is suffi ciently large. 3

16 We illustrate Proposition 3 with two special cases, one in the absence of coordination friction δ 0 and the other with extreme coordination friction δ > r. Consider first the case with δ 0, resulting in a single-person decision making setting. In this case, the argument from GG shows that the ban reduces the effi ciency. Since the single investor can always ignore a signal, she cannot be worse off from having additional information. Since the ban reduces the stock price informativeness, it reduces the effi ciency. Alternatively, we can also use the expression of V to analyze the ban s conseuences. In particular, there is an endogenous connection between δ and the possible value of α for manipulative short-selling to be optimal, which is best illustrated by focusing on the extreme values of δ. Note that the first term of V is always negative, as discussed above. Thus, the sign of V crucially depends on the sign of the second term. When δ 0, manipulative short-selling to be optimal (i.e. A >, it is necessary that α > r. Otherwise, if α r, then the stock price cannot be informative enough to change the investor s decision. In other words, short sales can come from both the negatively informed speculator and the uninformed speculator. The cost of suppressing the former dominates the benefit of suppressing the latter because α has to be suffi ciently large. α > r results in αr ( α( r being positive, i.e. conditional on the state being bad, the beneficial effect from preventing manipulative short-selling of the uninformed speculator is dominated by the cost of preventing informed short-selling. This explains why the set is empty when δ 0. Before explaining the case when δ approaches r, it is useful to present the following Lemma. Lemma 4 α(δ is decreasing in δ. Lemma 4 is intuitive. As the coordination concern becomes severe, investors are more pessimistic and they are only willing to stay if the probability of the good state is suffi ciently high. Anticipating this fragility, manipulative short selling is more likely. In fact, as δ approaches r, an uninformed investor short sells even as the fraction of informed speculators approaches 0, i.e. α(δ 0, a case which we now turn to. 4

17 Now consider another extreme case in which δ approaches r.in this case, α (δ 0.When α 0, clearly αr ( α( r < 0, i.e. conditional on the state being bad, the beneficial effect from preventing manipulative short-selling of the uninformed speculator is dominated by the cost of preventing informed short-selling. since the likelihood of informed short-selling becomes suffi ciently small. The second term of V therefore becomes positive. The first term, while still being negative, decreases in magnitude when α becomes smaller as the benefit from informed short-selling decreases regardless of whether the state is good or bad when the probability of being informed decreases. In fact, when δ r, we can calculate that V ( r α[φ( Φ( ] Note that Φ( Φ( > 0, that is, the ban reduces the sensitivity of investors decisions to the order flow. In addition, r and α represent the respective effects of coordination and information on investors investment decisions. When r > α, then the coordination effect dominates the information effect. In this case, the ban, by mitigating the investors response to the order flow, improves the effi ciency. Otherwise, the ban reduces the effi ciency. By continuity, we can prove the more general result that the short selling ban improves the effi ciency if and only if α is not suffi ciently high and δ is suffi ciently high, i.e. the set is non-empty when δ is suffi ciently large. To better understand Proposition 3, we provide a numerical example. For ease of notation we write V as V (α, δ. We assume that r H,, r L 0.5. Note that in this case δ 0.5.When δ 0, we can numerically calculate that α( In addition, SA (α(0, 0 0. (, 0 and V (α(0, < 0, i.e. banning short-selling decreases firm value. When we increase δ and keep α α(0, short-selling is clearly still optimal for the uninformed speculator. When δ 0.48, V (α(0, < V (α(0, 0, i.e. banning short-selling is even worse. However, when δ 0.48, α(0.4 decreases to around 0.03, which enlarges the range for short-selling to be optimal for the uninformed speculator. For example, when α 0. > α(0.48, V (0., > 0. In addition, note that α cannot be too large. For example, when α 0.4, V (0.4, < 0, i.e. banning 5

18 short-selling is bad when α Empirical and policy implications TBC 6 Conclusions TBC 7 Appendix: Proofs 7. Proof of Lemma : Proof. We first prove that the investors play a common threshold strategy. It is proved in two steps. In the first step, we show that all investors use the same strategy. In the second step, we show that the euilibrium strategy must be a single threshold. Note that the proof assumes that it is optimal for the positively informed speculator to choose d and the negatively informed speculator to choose d. This will be proved at the very end after step. First, suppose that investor i chooses to withdraw if and only if i S i, and investor j chooses to run if and only if j S j, where S i and S j are subsets of the real line. Suppose that S i S j. This implies that at least at least one of the sets of S i or S j must be non-empty. Without loss of generality suppose that S i is not empty. This implies that there exists a 0 such that 0 S i but 0 / S j. This implies that upon observing i 0, investor i stays but investor j withdraws, i.e. ( i 0 > 0 ( j 0, which is a contradiction as ( i 0 ( j 0 as ( only depends on for any fixed speculators strategies. Therefore all investors use the same strategy. Second, upon observing i, investor i s expected payoff of staying relative to withdrawing 6

19 is ( i Pr(θ H i (H δe[l i ] + Pr(θ L i (L δe[l i ] L + Pr(θ H i (H L δe[l i ] Assume now that positively informed speculator will choose d + and negatively informed speculator will choose d. Denote the uninformed speculator s strategy by d u [, ]. Denote the density of conditional on r as g. Then Note that Pr(θ H i g( i θ H g( i θ H + g( i θ L g( i θh g( i θl g( i θh g( i θl + g( i θ H g( i θ L i αφ( + ( αφ( i d u i + αφ( + ( αφ( i d u Thus, for any d u (,, φ( i σ α n + φ( φ( α φ( i du σ n + i + σ n + i du σ n + ( dui+d αe u αe + ( α + ( α (σ n + + ( α (+du i +d u (σ n + + ( α lim Pr(θ H i 0 and lim Pr(θ H i. Therefore i i + by continuity, both upper dominance region and lower dominance region exists. Therefore there exists finite and such that ( i < 0 if < and ( i > 0 if <. When d u, 7

20 lim Pr(θ H i α i α and lim Pr(θ H i. Therefore the upper dominance i + region exists and exists. If α α H + α ( L < 0, then the lower dominance region exists and thus exists. If α α H + α ( L 0, then the lower dominance region does not exist. When d u +, lim Pr(θ H i 0 and lim Pr(θ H i i i + α. Therefore the lower dominance region exists and thus exists. If α H + α α ( L > δ, then the upper dominance region exists and thus exists. If α H + α α ( L δ, then the upper dominance region does not exist. As a summary, in euilibrium we either have ( i < 0 if < and ( i > 0 if > or ( i < 0 if < or 3 ( i > 0 if >. We now show that under either of the three scenarios, common threshold strategy is the euilibrium. For case and, denote B sup{ i : ( i < 0}, i.e. the highest signal below which a investor prefers to withdraw. Note that it is possible for B +, which then implies that ( B 0. If all investors use threshold strategy then investor i will withdraw when i < B, for any i. Suppose in euilibrium they do not use threshold strategy, then for investor i, there exists signals smaller than B such that a investor observing i will stay. Denote A to be the largest of them, i.e. A sup{ i < B : ( i 0}. < A < B and thus is finite. This implies that investors in the range [ A, B ] and (, will withdraw for sure, while investors in the range of [, A may choose to stay or withdraw and denote their strategies by n( i [0, ]. Since investors are indifferent upon observing A, we have ( A 0 ( B On the other hand, note that E[l A ] Φ( A σε + A n( j φ( j A d j + Φ( B A σε σε σε 8

21 and E[l B ] Φ( B σε + A Φ( A B + σε n( j φ( j B σε σε n( j φ( j B σε σε where we used Φ( x Φ(x to arrive at the last euality. Thus d j + Φ( A B σε d j + Φ( B A σε E[l B ] E[l A ] Φ( B σε Φ( A σε + A n( j [φ( j B σε σε φ( j A ]d j σε Since B > A, φ( j B σε φ( j A σε < 0 for any j [, A ]. Therefore E[l B ] E[l A ] Φ( B σε Φ( A σε < 0 In addition, Pr(θ H i is increasing in i, which results in Pr(θ H B > Pr(θ H A. Correspondingly, ( B > ( A and thus, the contradiction. Therefore all investors use the same threshold strategy in euilibrium. For case and 3, denote C inf{ i : ( i > 0}, i.e. the lowest signal above which a investor will stay. Note that it is possible for C, which then implies that ( C 0. If all investors use threshold strategy then investor i will stay when i > C, for any i. Suppose in euilibrium they do not use threshold strategy, then for investor i, there exists signals larger than C such that a investor observing i will withdraw. Denote D to be the smallest of them, i.e. D sup{ i > C : ( i 0}. C < D < and thus is finite. This implies that investors in the range [ C, D ] and (, + will stay for sure, investors in the range of (, C will withdraw for sure, while investors in the range of [ D, may choose to stay or withdraw and denote their strategies by n( i [0, ]. Since investors are indifferent upon 9

22 observing D, we have ( D 0 ( C Using similar techniues as above we can show that this ineuality cannot hold. Therefore all investors use the same threshold strategy in euilibrium. We now prove the optimal strategies for the informed speculator. First, consider the strategy of the negatively informed speculator. Since the speculator knows that θ L, he knows that euity value is non-positive. Thus, when he takes action d, his profit will be π(d de[p d]. Since E[P d] 0, π( π(0 π(. Assumption A then implies that d for the negatively informed speculator. Next, consider the strategy of the positively informed speculator. Since the speculator knows that θ H, he knows that euity value is ( l(h δl. On the other hand, conditional on total order flow, stock price P ( Pr(θ H [ l(][h δl(] + Pr(θ L [ l(][l δl(] [ l(][pr(θ H (H δl( + Pr(θ L (L δl(] Therefore, when the speculator takes action d, the speculator s profit will be π(d d(e[v d] E[P d] d(e[( l(h δl d] E[( l(pr(θ H (H δl + Pr(θ L (L δl d] de[( l Pr(θ L (H L d] Since E[( l Pr(θ L ( L d] 0, π( π(0 π(. Assumption A then implies that d for the positively informed speculator. Given that each investor uses a common threshold strategy, we now solve for the common threshold. We assume that each investor will choose to run if and only if the order flow 0

23 i for some threshold where is determined by the indifferent condition. ( Pr(θ H (H δe[l ] + Pr(θ L (r L δe[l ] 0 When, E[l ] Pr( j i, which results in Pr(θ H δ r L H r L (5 As shown in the proof of Lemma, Pr(θ H is strictly increasing in. Therefore, euation (5 has at most one solution. The expression of Pr(θ H, however, depends on and therefore the uninformed speculators strategies (as the informed speculator always buy when observing θ H and sell when observing θ L, which we turn to now. Again denote the density of conditional on r as g. From Bayes rule and given the speculators strategies that the uninformed speculator short-sells, Pr(θ H A g(a θ H g(a θ H + g( A θ L αφ( A + ( αφ( A + αφ( A + ( αφ( A +

24 Insert into euation (5 results in αφ( αφ( A A + ( αφ( A + + ( αφ( A + δ L H L, which can be rearranged as φ( A σ α n +σ ε φ( A + + α δ L H δ (6 First note that euation (6 has a solution α (0, only if α < δ L, or, euivalently, H δ α > δ L H δ. When α δ L, then H δ αφ( αφ( A A + ( αφ( A + + ( αφ( A + δ r L H r L, implying that it is always optimal for the investors to stay, or, euivalently, A. Second, when α > δ L, then when H δ A 0, euation (6 becomes δ L H δ,or, euivalently, δ δ H + L r Thus, when α > δ L, one can solve for a close-form solution of H δ A to be A σ n + σ ε [ln(α ( L H δ ln α] σ δ n [ln(α ( L H δ ln α] δ when σ ε 0

25 When α δ L, H δ A When the uninformed speculator does not trade, we can similarly calculate that Pr(θ H NT g(nt θ H g(nt θ H + g( NT θ L αφ( NT + ( αφ( NT αφ( NT + αφ( NT + + ( αφ( NT Insert into euation (5 and rearranging terms results in α α φ( NT + σ n + φ( NT σ n + φ( NT σ n + φ( NT σ n + + α + α H δ δ L (7 The solution of euation (7 is uniue as the left hand side of euation (7 is decreasing in NT. This is because the numerator is decreasing in NT and the denominator is increasing in NT. When NT, the left hand side approaches + which is clearly larger than the right hand side. In addition, when NT +, the right hand side approaches 0 which is clearly smaller than the right hand side. When short-sell is banned, then both uninformed and negatively informed speculator do not trade. we can similarly calculate that 3

26 Pr(θ H B g(b θ H g(b θ H + g( B θ r L αφ( + ( αφ( B B αφ( + ( αφ( B B Insert into euation (5 and rearranging terms results in φ( B σ α n +σ ε B φ( + α δ L H δ (8 Note that the left hand side of euation (8 is increasing in B. When B +, the left hand side becomes +, which is clearly larger than the right hand side. When B, the left hand side becomes α. Thus, euation (8 has a solution if and only if α < δ L, H δ or α > δ L. Solving euation (8 results in H δ B A + (σ n + σ ε ln When α δ L, then H δ B. α [ ( δ L ] H δ + α 7. Proof of Proposition : Proof. The proposition is proved in three steps. In step, we show that it is never optimal for the uninformed speculator to buy. In step, we derive the conditions where it is optimal for the uninformed speculator to choose d under the conjecture that the uninformed speculator will choose d. In step 3, we derive the conditions where it is optimal for the uninformed speculator to deviate to d under the conjecture that the uninformed speculator will choose d 0. Combining the conditions in steps and 3 arrives in the conditions as stated in Proposition. 4

27 Step : It is never optimal for the uninformed speculator to choose d. If the uninformed speculator chooses to buy, then from his perspective N(, σ n and i N(, σ n + σ ε. Suppose the market conjectures that the uninformed speculator chooses d, from Bayes rule and given the speculators strategies, Pr(θ H g( θ H g( θ H + g( θ L φ( + αφ( + ( αφ( In addition, l( Pr( i B Φ( B where we denote the threshold when uninformed speculator chooses to buy by B and B > 0. Thus P ( Pr(θ H [ l(][h δl(] + Pr(θ L [ l(][l δl(] φ( + αφ( + ( αφ( [ Φ( B ][H δφ( B ] σ σ ε n +σ ε + αφ( + ( αφ( σ + n +σ ε + αφ( + ( αφ( [ Φ( B ][L δφ( B ] 5

28 Therefore, if the speculator chooses to buy, he expects to pay E[P (] + φ( αφ( + ( αφ( [ Φ( B ][H δφ( B ] φ( d + + φ( αφ( + ( αφ( [ Φ( B ][L δφ( B ] φ( d whereas he expects the security to pay off E[V ] [ Φ( B ][ H + L δφ( A ] φ( d Therefore, the speculator s payoff from short-selling is E[V ] E[P (] [ φ( + αφ( + ( αφ( φ( d B [ φ( + αφ( + ( αφ( φ( d + B [ φ( + αφ( + ( αφ( φ( d ][ Φ( B ](H L ][ Φ( B ](H L ][ Φ( B ](H L 6

29 When 0, E[V ] E[P (] B H L H L H L H L H L [ φ( αφ( + + ( αφ( ] (H Lφ( Q dq B B B B α( e σ n [ α( e α( e σ n [ α( e σ n ] α(e σ n [ + α(e σ n ] f(α,, d f(α,, d < 0 σ n ] φ( d φ( dq φ( + d where we used change of variables in arriving at the third ineuality. So it is suboptimal for the uninformed speculator to buy when the market s conjecture is that the uninformed speculator will buy. We now show that when the market s conjecture is that the speculator will not trade, it is still suboptimal for the uninformed speculator to buy. If the speculator chooses to buy, he expects to pay E[P (] αφ( + ( αφ( + αφ( + αφ( + ( αφ( φ( d αφ( + + ( αφ( σ + n +σ ε αφ( + αφ( + + ( αφ( φ( d [ Φ( NT ][H δφ( NT ] [ Φ( NT ][L δφ( NT ] 7

30 whereas he expects the security to pay off E[V ] [ Φ( NT ][ H + L δφ( NT ] φ( d Therefore, the speculator s payoff from short-selling is E[V ] E[P (] αφ( + ( αφ( σ [ n +σ ε αφ( + αφ( + + ( αφ( φ( d NT αφ( + ( αφ( σ [ n +σ ε αφ( + αφ( + + ( αφ( φ( d + NT αφ( + ( αφ( σ [ n +σ ε αφ( + αφ( + + ( αφ( φ( d ](H L[ Φ( NT ] ](H L[ Φ( NT ] ](H L[ Φ( NT ] When 0, E[P (] E[V ] αφ( + ( αφ( σ (H L [ n +σ ε αφ( + αφ( + + ( αφ( NT φ( d (H L NT (H L NT αφ( αφ( + αφ( + αφ( + + ( αφ( αφ( αφ( + αφ( + + ( αφ( φ( d ][ Φ( NT ] [φ( φ( + ]d 8

31 Thus sgn{e[p (] E[V ]} sgn( sgn( NT NT φ( αφ( + αφ( + + ( αφ( α + αe σ n + ( αe σ n [φ( φ( + ]d [φ( φ( + ]d < 0 To see why the sign is negative, note that using integration by parts, NT α + αe σ n NT α + αe σ n α + αe NT σ n NT + ( αe σ n + ( αe σ n [ α + αe σ n + ( αe σ n + ( αe NT σ n [ α + αe σ n + ( αe σ n [φ( φ( + ]d [Φ( Φ( + ] + σ n NT ][Φ( Φ( + ]d [Φ( NT Φ( NT + ] ][Φ( Φ( + ]d Both terms are negative. The first term is negative because Φ( NT Φ( NT + < 0. The second term is negative because [ α+αe σ n +( αe σ n ] < 0 and Φ( Φ( + < 0. Therefore given the conjecture is that the uninformed speculator will not trade, the uninformed trader will not buy as well. We can similarly show that when the conjecture is that the uninformed speculator will short-sell, the uninformed trader will not buy as well. Therefore, it is always suboptimal for the uninformed trader to buy and Step is complete. Step : It is optimal for the uninformed speculator to choose d under the conjecture that the uninformed speculator chooses d when α < α for some α. If the uninformed speculator chooses to short-sell, then from his perspective N(, σ n and i N(, σ n + σ ε. 9

32 Conditional upon observing, the market maker will set P ( E[( l(θ δl ] From Bayes rule and given the speculators strategies, Pr(θ H g( θ H g( θ H + g( θ L αφ( + ( αφ( + + αφ( + ( αφ( In addition, l( Pr( i A Φ( A Q Thus P ( αφ( + ( αφ( + αφ( + ( αφ( + [ Φ( A ][H δφ( A ] + φ( σ + n +σ ε αφ( + ( αφ( + [ Φ( A ][L δφ( A ] 30

33 Therefore, if the speculator chooses to short-sell, he expects to get E[P (] αφ( + ( αφ( + αφ( + ( αφ( φ( + d φ( + σ + n +σ ε αφ( + ( αφ( φ( + d + + [H δφ( A ][ Φ( A ] [L δφ( A ][ Φ( A ] where the last euality comes from law of iterated expectations. He expects the security to pay off E[V ] [ H + L δφ( A ][ Φ( A ] φ( + d Therefore, the speculator s payoff from short-selling is E[P (] E[V ] αφ( + ( αφ( + σ [ n +σ ε αφ( + + ( αφ( ][ Φ( A ](H L σ σ ε n +σ ε φ( + d A + αφ( + ( αφ( σ [ n +σ ε αφ( + + ( αφ( φ( + d + A + ][ Φ( A ](H L αφ( + ( αφ( σ [ n +σ ε αφ( + + ( αφ( ][ Φ( A ](H L σ σ ε n +σ ε φ( + d 3

34 When 0, E[P (] E[V ] αφ( + ( αφ( + σ [ n +σ ε αφ( + + ( αφ( ](H L φ( Q + dq σ σ n n +σ ε A When δ < δ, and α δ L, H δ A and E[P (] E[V ] αφ( + ( αφ( + σ [ n +σ ε αφ( + + ( αφ( ](H L φ( + d σ σ n n +σ ε [ αe σ n + ( α αe σ n + ( α (H L (H L ](H L α(e σ n [αe σ n f(α,, d φ( + d φ( + d σ + ( α] n where f(α,, α(e σ n [ + α(e σ n ] φ( + Thus sgn[e[p (] E[V ]] sgn[ f(α,, d] 3

35 Note that we can write f(α,, d as f(α,, d f(α,, d + f(α,, d f(α,, d [f(α,, + f(α,, ]d f(α,, d where we used change of variables to arrive at the second ineuality. Note that f(α,, > 0 > f(α,, as f(α,, σ n > 0 if and only if > 0. Note that f(α,, f(α,, + α(e σ n + α(e σ n < as e σ n < e σ n Therefore f(α,, + f(α,, < 0 and therefore f(α,, d < 0, resulting in E[P (] E[V ] < 0. Short-selling is therefore not optimal for the uninformed speculator when δ < δ, and α δ L. H δ When δ < δ, and α > δ L, H δ A < 0 and E[P (] E[V ] H L A f(α,, d Through changing variables, for any N > 0, we have 0 N A 0 f(α,, d N A f(α, N, d N N A 0 f(α, N, d 33

36 Therefore N A N A 0 N A 0 f(α,, d f(α, N, d + 0 f(α,, d [ N f(α, N, + f(α,, ]d + N A f(α,, d Recall that when σ ε 0, A σ δ n [ln(α ( L H δ ln α] Thus, when δ < δ, A is increasing in α as A α σ n δ L H δ α[α ( δ L ] H δ > 0 Therefore, the derivative of f(α,, σ A n d with respect to α is α A f(α,, d [ N f(α, A, + f(α, NA, ]N ( A α N A + α [ N f(α, N, + f(α,, ]d 0 f(α, N A, N ( A α + N A α f(α,, d f(α, A, ( A α N A + α [ N f(α, N, + f(α,, ]d + 0 Now take the limit of α N A A α f(α,, d f(α,, d when N +. The first term is positive as 34

37 f(α, A, < 0 and ( A α A α < 0. The second term is positive as lim N + α [ N f(α, N, + f(α,, ] α f(α,, (+ ( π e σ n e σ n ( > 0 when > 0 (α e σ n + The third term converges to zero as N A when N +, implying that α A +. Therefore α A f(α,, d > 0 f(α,, d > 0 as we can choose a number N 0 suffi ciently large so that we can change variables and express f(α,, σ A n d as A N0 A 0 f(α,, d [ f(α,, + f(α,, ]d + f(α,, d N 0 N 0 N 0 A A Thus f(α,, σ A n d is increasing in α. f(α,, d < 0. When α, A σ n In addition, A δ δ, δ L H δ increases in δ, resulting in A When α δ L, H δ A ln δ L H δ. Note that when δ < δ, and δ L H δ <. increasing in δ. Since when fixing α, f(α,, d increases in A when A < 0, f(α,, d increases in δ. δ L H δ and A 0 and f(α,, σ A n d > 0. When δ 0, A δ L H δ When r, implying that A is increasing in r. When r 0, then A and f(α, Q, σ A n dq < 0. When r then A 0 and f(α,, σ A n d > 0. Therefore there exists a uniue α ( δ L, which is also a function of r and δ such that short-selling is optimal form H δ the uninformed speculator if α > α and either r > c r or r c r but δ > c δ. 35

38 where c r is the uniue solution to σ n ln c r f(, Q, dq 0 (9, c δ is the uniue solution to σ n ln c δ f(, Q, σ L n dq 0 H c δ, α is the uniue solution to σ n [ln(α ( δ L H δ f(α, Q, dq 0 (0 ln α ] Step is thus proved. Step 3: It is optimal for the uninformed speculator to choose d under the conjecture that the uninformed speculator chooses d 0 when α < α for some α. When the conjecture is that uninformed speculator chooses d 0, the investors will withdraw if and only if NT. If the speculator chooses to short-sell, he expects to get E[P (] αφ( + ( αφ( + αφ( + αφ( + ( αφ( φ( + d αφ( + + ( αφ( σ + n +σ ε αφ( + αφ( + + ( αφ( φ( + d [ Φ( NT ][H δφ( NT ] [ Φ( NT ][L δφ( NT ] 36

39 whereas he expects the security to pay off E[V ] [ Φ( NT ][ H + L δφ( NT ] φ( + d Therefore, the speculator s payoff from short-selling is E[P (] E[V ] αφ( + ( αφ( σ [ n +σ ε αφ( + αφ( + + ( αφ( φ( + d NT αφ( + ( αφ( σ [ n +σ ε αφ( + αφ( + + ( αφ( φ( + d + NT αφ( + ( αφ( σ [ n +σ ε αφ( + αφ( + + ( αφ( φ( + d ](H L[ Φ( NT ] ](H L[ Φ( NT ] ](H L[ Φ( NT ] When 0, E[P (] E[V ] αφ( + ( αφ( σ (H L [ n +σ ε αφ( + αφ( + + ( αφ( NT φ( + d (H L NT (H L NT αφ( αφ( + αφ( + αφ( + + ( αφ( αφ( + αφ( + αφ( + + ( αφ( φ( + d ][ Φ( NT ] [φ( φ( + ]d 37

40 Thus sgn{e[p (] E[V ]} sgn( sgn( NT NT αφ( + αφ( + αφ( + + ( αφ( h(α,, d [φ( φ( + ]d where h(α,, αφ( + αφ( + αφ( + + ( αφ( + α(e σ n α(e σ n + + ( α(e σ n [φ( φ( + ] φ( +. f(α,, α(e σ n [ + α(e σ n ] φ( + Hence h(α,, > 0 if and only if > 0. We also know that NT < 0 and that d NT dα > 0. We can similarly derive the derivative of g(α,, σ NT n d with respect to α as g(α,, σ NT n d α g(α, NT, ( NT + + NT 0 NT 0 α N (g(α, N, + g(α,, d α g(α,, d α Taking the limit N +, we can similarly prove that when choosing N suffi ciently large, NT g(α,,d α d > 0. When α, we have NT σ n ln δ L H δ. When α 0, NT. Hence, there exists 38