November 2011 Preliminary and Incomplete

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1 Average Expectations, Liquidity, Liquidity, and Short-term and Short-term Trading* Trading Giovanni Cespa and Xavier Vives November 2011 Preliminary and Incomplete Preliminary Abstract In a market with short term investors and heterogeneous information, when liquidity trading displays persistence, asset prices reflect average expectations about both fundamentals and liquidity trading, and multiple, self-fulfilling equilibria arise. If asset prices are driven by average expectations about fundamentals, they heavily rely on public information and the market displays low liquidity; if they are driven by average expectations about liquidity trading, prices rely less on public information and the market displays high liquidity. Along the equilibrium with high liquidity, the volume of informational trading is high, and momentum arises at short horizons. Conversely, along the equilibrium with low liquidity the volume of informational trading is low and short term returns tend to revert. At long horizons reversal occurs. Keywords: Expected returns, multiple equilibria, average expectations, reliance on public information, momentum and reversal, Beauty Contest, High Frequency Trading. JEL Classification Numbers: G10, G12, G14 A previous version of this paper was circulated under the title Higher Order Expectations, Illiquidity, and Short-term Trading. We thank Bruno Biais, Marcelo Fernandes, Peter Hoffmann, Harrison Hong, Peter Kondor, Stephen Morris, Richard Payne, Ioanid Roşu, and seminar participants in the Econometrics Workshop at Queen Mary University of London, ESSEC, Paris Dauphine, the IESEG workshop on Asset Pricing (IESEG, May 2010), the 2010 European Finance Association meeting (Frankfurt), the ELSE-UCL Workshop in Financial Economics (UCL, September 2010), the 2011 ESSET (Gerzensee), the 2011 SED meeting (Gent), and the CNMV conference on Securities Markets: The Crisis and the Challenges Ahead, Madrid (11/11) for their comments. Cespa acknowledges financial support from ESRC (grant n. ES/J00250X/1). Vives acknowledges financial support from the European Research Council under the Advanced Grant project Information and Competition (no ). Cass Business School, CSEF, and CEPR. 106, Bunhill Row, London EC1Y 8TZ, UK. giovanni.cespa@gmail.com IESE Business School, Avinguda Pearson, Barcelona, Spain. xvives@iese.edu 1

2 Introduction We study the drivers of asset prices in a two-period market where short-term, informed, competitive, risk-averse agents trade on account of private information and to accommodate liquidity supply, facing a persistent demand from liquidity traders. Assuming that the demand of liquidity traders displays persistence and that informed investors have short horizons captures an important feature of High Frequency Trading (HFT). Due to persistence in liquidity trading informed investors can use their private signals to help them infer the demand of liquidity traders from the order flow to anticipate the impact this will have on the price at which they will unwind their positions. This allows high frequecny traders to move ahead of liquidity traders. We show that this has important consequences for the properties of the market. HFT represents a sizeable proportion of the trades carried out in today s exchanges. 1 HFTs use computer algorithms to detect arbitrage opportunities by typically scanning order flow data, and rapidly reacting to such opportunities to lock in profitable trades (the latency between the time at which the information is obtained by the computer and the time at which the trade is executed at the exchange is in the order of milliseconds and even microseconds). The debate on the merits and flaws of HFT is intense both at the academic and policy levels. While a number of authors find that HFT substantially boosts market quality, in particular enhancing price efficiency and improving depth (see, e.g. Hendershott, Menkveld, and Jones (2011), and Hendershott and Riordan (2011)), others argue that it can exacerbate extreme price movements, to the detriment of market stability (Kirilenko, Kyle, Mehrdad, and Tugkan (2010)). We find that with heterogeneous information and short horizons, liquidity trading persistence yields multiple equilibria. The intuition is as follows. With risk averse investors and heterogeneous information, the price accounts for adverse selection and inventory risk, reflecting the market expectations about the fundamentals and liquidity traders demand. Due to persistence, liquidity traders positions in the first and second period are positively correlated. Thus, informed investors use their private signals on the fundamentals also to infer the demand of liquidity traders from the first period price, to anticipate the impact that liquidity traders have on the price at which they unwind their positions (thus exploiting a private learning channel from the price as in Amador and Weill (2010) and Manzano and Vives (2011)). Suppose first period investors anticipate that the second period price is more correlated to the fundamentals. In this case, they speculate more aggressively on their private information, making the first period price reflect more the fundamentals, and less the demand of liquidity traders. This, in turn, leads them to put more weight on the private signal to extrapolate the demand of liquidity traders from the price, increasing the informativeness of the first and second period prices. In this case, this self-reinforcing loop leads investors to escalate their response to private information. Suppose instead that first period investors anticipate that the second period price is less correlated to the fundamentals. In this case, they speculate less aggressively 1 According to Tabb Group, in 2010 HFT accounted for roughly two thirds of all equity trades in the US and slightly more than one third in Europe. 2

3 on their private information, making the first period price reflect less the fundamentals, and more the demand of first period liquidity traders. This lowers the weight investors put on their private signals to infer the demand of liquidity traders from the price, yielding less informative first and second period prices. In this case, the self-reinforcing loop leads investors to dampen their response to private information. The above described equilibria have strikingly different features. Along the former equilibrium, prices are more informative about the fundamentals, and the market is thicker. Conversely, along the latter equilibrium, prices are less informative about the fundamentals and the market is thinner. Studying the stability of the equilibrium solutions, we show that the high liquidity-high price efficiency equilibrium is however unstable according to the best reply dynamics. Therefore, HFT may induce fragility in that it creates multiple equilibria with one unstable equilibrium. This paper is related to the growing literature on HFT. Hendershott and Riordan (2011) find that HFTs orders have a permanent price impact which is larger than that of slow human traders. This allows the price to adjust more rapidly toward the (informationally) efficient price. Based on this evidence, Biais, Foucault, and Moinas (2011) assume that HFTs are able to process information before slow traders. This imposes an adverse selection cost on the latter, creating a negative externality which is responsible for excessive investment in HFT compared to a utilitarian welfare-maximization benchmark. We provide an explanation for HFTs superior ability to impound fundamentals information. In the high liquidity equilibrium due to the the private learning channel from prices HFTs escalate their response to private information, making prices more (informationally) efficient. Our results are also related to the literature that studies the ability of the non-informational component of total imbalances to predict stock returns (see, e.g. Coval and Stafford (2007), and Hendershott and Seasholes (2009)). Hendershott and Seasholes (2009) trace such ability either to adverse selection or to inventory risk and derive and test empirical predictions to identify which of these two provide a better explanation for it. Our paper shows that when the non-informational component of order imbalances is predictable, multiple equilibria where return regularities arise, emerge. However, the informational content of prices substantially differs across the two equilibria. Therefore, while due to persistence in liquidity trading and dealers limited risk bearing capacity, returns become predictable, the arbitrage opportunities that such regularities imply can vary dramatically. The paper is also related to the literature that investigates the relationship between the impact of short-term information on prices and investors reaction to their private signals (see, e.g. Singleton (1987), Brown and Jennings (1989), Vives (1995), Cespa (2002), Albagli (2011) and Vives (2008) for a survey). Several authors have argued that when private information is related to an event which occurs beyond the date at which investors liquidate their positions, the latter act on their signals only if they expect them to be reflected in the price at which they liquidate (see, e.g., Dow and Gorton (1994) and Froot, Scharfstein, and Stein (1992)). In our context too a similar mechanism is at work. However, due to persistence, the first period demand of liquidity traders impacts both the first and second period order flows, which gives 3

4 private information two different roles: the one of predicting the fundamentals and the one of predicting liquidity traders demand. Finally, the paper is related to a growing literature that points out the relevance of Higher Order Expectations (HOEs) (that is average expectations of average expectations... of average expectations of the fundamentals) in influencing asset prices. It is a basic tenet of this literature (e.g., Allen, Morris, and Shin (2006)) that when prices are driven by HOEs about fundamentals, they underweight private information (with respect to the optimal statistical weight) and are farther away from fundamentals compared to consensus. This result holds when liquidity trading is transient, in which case a unique equilibrium arises. Instead, when liquidity trading displays persistence, we find that the price also reflects investors average expectations about liquidity trading. In this case, along the equilibrium with high liquidity the price is actually more strongly tied to fundamentals compared to consensus, and overweights average private information (compared to the optimal statistical weight). 2 Along the equilibrium with low liquidity, however, the opposite happens. Bacchetta and van Wincoop (2008) study the role of higher order beliefs in asset prices in an infinite horizon model showing that higher order expectations add an additional term to the traditional asset pricing equation, the higher order wedge, which captures the discrepancy between the price of the asset and the average expectations of the fundamentals. Kondor (2009), in a model with short-term Bayesian traders, shows that public announcements may increase disagreement, generating high trading volume in equilibrium. Nimark (2007), in the context of Singleton (1987) s model, shows that under some conditions both the variance and the impact that expectations have on the price decrease as the order of expectations increases. Other authors have analyzed the role of higher order expectations in models where traders hold different initial beliefs about the liquidation value. Biais and Bossaerts (1998) show that departures from the common prior assumption rationalise peculiar trading patterns whereby traders with low private valuations may decide to buy an asset from traders with higher private valuations in the hope to resell it later on during the trading day at an even higher price. Cao and Ou-Yang (2005) study conditions for the existence of bubbles and panics in a model where traders opinions about the liquidation value differ. 3 Banerjee, Kaniel, and Kremer (2009) show that in a model with heterogeneous priors, differences in higher order beliefs may induce price drift. Ottaviani and Sørensen (2009), in a static model of a binary prediction market where agents hold heterogeneous prior beliefs and are wealth constrained (either exogenously, by the rules of the market, or because of their attitude towards risk), show that the fully revealing REE price underweights aggregate private information. The rest of the paper is organized as follows. In the next section we analyze the static benchmark. In the following section, we study the two-period extension and present the multiplicity result, relating it to liquidity traders persistence. In the following section we relate our results to the literature on Higher Order Expectations. We then draw the implications of 2 In a related paper, we show that a similar conclusion holds in a model with long term investors (see Cespa and Vives (2011)). 3 Kandel and Pearson (1995) provide empirical evidence supporting the non-common prior assumption. 4

5 our analysis for the literature on HFT, return regularities, and volume. In the final section we summarize our results discuss their empirical implications. Most of the proofs are relegated to the appendix. 1 The static benchmark Consider a one-period stock market where a single risky asset with liquidation value v, and a riskless asset with unitary return are traded by a continuum of risk-averse, informed investors in the interval [0, 1] together with liquidity traders. We assume that v N( v, τ 1 v ). Investors have CARA preferences (denote with γ the risk-tolerance coefficient) and maximize the expected utility of their wealth: W i = (v p)x i. 4 Prior to the opening of the market every informed investor i obtains private information on v, receiving a signal s i = v + ɛ i, ɛ i N(0, τ 1 ɛ ), and submits a demand schedule (generalized limit order) to the market X(s i, p) indicating the desired position in the risky asset for each realization of the equilibrium price. 5 Assume that v and ɛ i are independent for all i, and that error terms are also independent across investors. Liquidity traders submit a random demand u (independent of all other random variables in the model), where u N(0, τ 1 u ). Finally, we make the convention that, given v, the average signal 1 s 0 idi equals v almost surely (i.e. errors cancel out in the aggregate: 1 ɛ 0 idi = 0). 6 We denote by E i [Y ], Var i [Y ] the expectation and the variance of the random variable Y formed by an investor i, conditioning on the private and public information he has: E i [Y ] = E[Y s i, p], Var i [Y ] = Var[Y s i, p]. Finally, we denote by Ē[v] = 1 0 E i[v]di investors average opinion (the consensus opinion) about v. In the above CARA-normal framework, a symmetric rational expectations equilibrium (REE) is a set of trades contingent on the information that investors have, {X(s i, p) for i [0, 1]} and a price functional P (v, u) (measurable in (v, u)), such that the following two conditions hold: 1. Investors in [0, 1] optimize X(s i, p) arg max x i E [ exp { W i /γ} s i, p]. (1) 2. The market clears: 1 0 x i di + u = 0. (2) Given the above definition, it is easy to verify that a unique, symmetric equilibrium in linear strategies exists in the class of equilibria with a price functional of the form P (v, u) (see, e.g. Admati (1985), Vives (2008)). The equilibrium strategy of an investor i is given by X(s i, p) = a α E (E i [v] p), 4 We assume, without loss of generality with CARA preferences, that the non-random endowment of rational investors is zero. 5 The unique equilibrium in linear strategies of this model is symmetric. 6 See Section 3.1 in the Technical Appendix of Vives (2008) for a justification of the convention. 5

6 where a = γτ ɛ, (3) denotes the responsiveness to private information, τ i (Var i [v]) 1, and α E = τ ɛ /τ i is the optimal statistical (Bayesian) weight to private information. equilibrium price is given by where E[u p] = a(v E[v p]) + u, and Imposing market clearing the p = Ē[v] + α E a u (4) = E[v p] + ΛE[u p], (5) Λ = Var i1[v]. (6) γ Equations (4), and (5) show that the price can be given two alternative representations. According to the first one, the price reflects the consensus opinion investors hold about the liquidation value plus the impact of the demand from liquidity traders (multiplied by the risk-tolerance weighted uncertainty over the liquidation value). Indeed, in a static market owing to CARA and normality, an investor s demand is proportional to the expected gains from trade E i1 [v] p. As the price aggregates all investors demands, it reflects the consensus opinion Ē1[v] shocked by the orders of liquidity traders. According to (5), the anticipated impact of liquidity traders demand moves the price away from the semi-strong efficient price. Indeed, as investors are risk averse, when they accommodate an expectedly positive demand of liquidity traders, they require a compensation against the possibility that the liquidation value is higher than the public expectation (if instead E[u p] < 0, investors require to pay a price lower than E[v p] to cover the risk that v < E[v p]). Such a compensation is larger, the higher is the uncertainty investors face (captured by Λ) and the wider is their expected exposure to the liquidity traders shock (their expected inventory, E[u p]). Therefore, Λ captures the inventory related component of market liquidity. According to (5), $1-increase in liquidity traders demand has an additional impact on the price, through the effect it produces on E[v p]. This latter effect comes from the signal extraction problem dealers face in this market: since a > 0, if investors on average have good news they buy the asset, and the efficient price increases, reflecting this information. However, this effect cannot be told apart from the buying pressure of liquidity traders, which also makes the efficient price increase. This adverse selection effect adds to the inventory effect, implying that the (reciprocal of the) liquidity of the market is measured by dp/du: where τ = 1/Var[v p] = τ v + a 2 τ u. λ = Λ + (1 α E ) aτ u τ, Finally, note that the private signal in this case only serves to forecast the liquidation value v. In the next section we will argue that due to persistence, liquidity traders demand impacts the order flow across different trading dates. In this case, instead, investors also use their private 6

7 signals to extrapolate the demand of liquidity traders and anticipate the impact this has on future prices. This additional use of private information will be responsible for equilibrium multiplicity. 2 A two-period market with short term investors Consider now a two-period extension of the market analyzed in the previous section. At date 1 (2), a continuum of short-term investors in the interval [0, 1] enters the market, loads a position in the risky asset which it unwinds in period 2 (3). An investor i has CARA preferences (denote with γ the common risk-tolerance coefficient) and maximizes the expected utility of his short term profit π in = (p n+1 p n )x in, n = 1, 2 (we set p 0 = v and p 3 = v). 7 An investor i who enters the market in period 1 receives a signal s i = v + ɛ i which he recalls in the second period, where ɛ i N(0, τ 1 ɛ ), v and ɛ i are independent for all i. We make the convention that, given v, the average signal 1 s 0 idi equals v almost surely (i.e., errors cancel out in the aggregate 1 ɛ 0 idi = 0). We also assume that informed investors observe equilibrium prices and submit a linear demand schedule (generalized limit order): X 1 (s i, p 1 ) = a 2 s i ϕ 1 (p 1 ), and X 2 (s i, p 1, p 2 ) = a 2 s i ϕ 2 (p 1, p 2 ), indicating the desired position in the risky asset for each realization of the equilibrium price. The constant a n denotes the private signal responsiveness, while ϕ n ( ) is a linear function of the equilibrium prices. The position of liquidity traders is assumed to follow an AR(1) process: θ 1 = u 1 θ 2 = βθ 1 + u 2, (7) where β [0, 1] and {u 1, u 2 } is an i.i.d. normally distributed random process (independent of all other random variables in the model) with u n N(0, τ 1 u ). 8 If β = 1, {θ n } 2 n=1 follows a random walk and we are in the usual case of independent liquidity trade increments: u 2 = θ 2 θ 1 is independent from u 1 (e.g., Kyle (1985), Vives (1995)). If β = 0, then liquidity trading is i.i.d. across periods (this is the case considered by Allen et al. (2006)). 9 This random process could be interpreted in the following way. Denote by x n the aggregate position of informed investors at time n: x n = 1 0 x indi. In the first period, informed investors clear the position of liquidity traders: x 1 + θ 1 = 0. Suppose 0 < β < 1, then in the second period market clearing involves (i) the reverting position of first period informed investors x 1, (ii) the position of second period informed investors x 2, (iii) a fraction 1 β of the first period liquidity traders position θ 1 (who revert), 7 We assume, without loss of generality, that the non-random endowment of investors is zero. 8 The AR(1) assumption for liquidity traders demand is not new in the literature. For instance, He and Wang (1995) and Cespa and Vives (2011) consider a model with long term investors in which liquidity trading is generated by an AR(1) process. 9 For empirical evidence on the persistence in liquidity traders demand see Chordia and Subrahmanyam (2004), Easley et al. (2008), and Hendershott and Seasholes (2009). 7

8 and (iv) the new generation of liquidity traders with demand u 2. Letting x 2 x 2 x 1, θ 2 θ 2 θ 1 = u 2 (1 β)θ 1, market clearing implies x 2 x 1 + u 2 (1 β)θ 1 = 0 x 2 + θ 2 = 0 x 2 + βθ 1 + u 2 = 0. Hence, at date 2 a fraction (1 β)θ 1 of first period liquidity trades reverts. The lower is β, the higher is the fraction of reverting first period liquidity trades. Thus, assuming persistence in liquidity trades allows to model in a parsimonious way the possibility that agents in the market have different horizons: when β = 0 each generation of informed investors and liquidity traders have the same investment horizon; as β grows, investment horizons become increasingly different (see Table 1). Please insert Table 1 here. We denote by E in [Y ] = E[Y s i, p n ], E n [Y ] = E[Y p n ], Var in [Y ] = Var[Y s i, p n ], and Var n [Y ] = Var[Y p n ], respectively the expectation and variance of the random variable Y formed by a trader conditioning on the private and public information he has at time n, and that obtained conditioning on public information only. The variables τ n and τ in denote instead the precisions of the investors forecasts of v based only on public and on public and private information: τ n = (1/Var n [v]), and τ in = (1/Var in [v]). More in detail, consider a candidate linear (symmetric) equilibrium x i1 = a 1 s i ϕ 1 (p 1 ), x i2 = a 2 s i ϕ 2 (p 1, p 2 ). Letting x n 1 0 x indi, and imposing market clearing in the first period implies (due to our convention): x 1 + θ 1 = 0 a 1 v + θ } {{ } 1 = ϕ 1 (p 1 ), (8) z 1 where the intercept of the first period aggregate demand z 1 a 1 v + θ 1 denotes the informational content of the first period order flow. In the second period the market clearing condition is x 2 + βθ 1 + u 2 = 0 x 2 βx 1 + u 2 = 0 a 2 v ϕ 2 (p 1, p 2 ) β(a 1 v ϕ 1 (p 1 )) + u 2 = 0 a 2 v + u } {{ } 2 = ϕ 2 (p 1, p 2 ) βϕ 1 (p 1 ), (9) z 2 where in the second line we use (8), while in the third we let a 2 a 2 βa 1, and denote by z 2 a 2 v + u 2 the informational content of the second period order flow. From (8) and (9) it is easy to see that the sequence {z 1, z 2 } is observationally equivalent to {p 1, p 2 }. Therefore, E n [v] = τ 1 n (τ v + τ n u t=1 a tz t ), Var n [v] = (τ v + τ n u t=1 ( a t) 2 ) 1, E in [v] = τ 1 in (τ ne n [v] + τ ɛ s i ), and Var in [v] = (τ n + τ ɛ ) 1. The consensus opinion about the fundamentals at time n is denoted by Ēn[v] 1 E 0 in[v]di. Letting α En = τ ɛ /τ in, we have E in [v] = α En s in + (1 α En )E n [v], and due to our convention Ē n [v] = α En v + (1 α En )E n [v]. 8

9 2.1 Equilibrium analysis We start by giving a general description of the equilibrium. The following proposition characterises equilibrium prices: Proposition 1. At a linear equilibrium of the market the price is given by ( p n = α Pn v + θ ) n + (1 α Pn )E n [v], (10) a n where θ n = u n + βθ n 1, and a n, α Pn, respectively denote the responsiveness to private information displayed by investors and by the price at time n. The price impact is measured by dp n /du n : λ n = α P n + (1 α Pn ) a nτ u. (11) a n τ n According to (10), at period n the equilibrium price is a weighted average of the market expectation about the fundamentals v, and the (noisy) average private information held by investors. Manipulating this expression we can separate the total impact of news from that of past information in the second period price: p 2 = α P 2 (a 2 v ± βa 1 v + θ 2 ) + (1 α P2 )E 2 [v] a 2 = ( αp2 a 2 + (1 α P2 ) a 2τ u τ 2 Similarly, in the first period we have p 1 = ( αp1 a 1 + (1 α P1 ) a 1τ u τ 1 which together with (12) justifies the expression in (11). A different rearrangement of (10) yields ) z 2 + βα P 2 a 2 z 1 + (1 α P2 ) τ 1 τ 2 E 1 [v]. (12) ) z 1 + (1 α P1 ) τ v τ 1 v, (13) p n E n [v] = α P n a n (a n (v E n [v]) + θ n ) (14) = Λ n E n [θ n ], where Λ n α Pn /a n, implying that there is a discrepancy between p n and E n [v] which, as in (5), captures a premium which is proportional to the expected stock of liquidity trading that investors accommodate at n. The assumption of short-term investment horizons, together with that of persistence in liquidity trading have important implications for the equilibrium of the market. We analyze these implications in the following results: Corollary 1. At a linear equilibrium, the price incorporates a premium above the semi-strong efficient price: where p n = E n [v] + Λ n E n [θ n ], (15) Λ 1 = Var i1[p 2 ] γ + βλ 2 (16) Λ 2 = Var i2[v]. (17) γ 9

10 Comparing (16) with (6), shows that short term trading affects the inventory component of liquidity. In a static market when investors absorb the demand of liquidity traders, they are exposed to the risk coming from the randomness of v. In a dynamic market, short term investors at date 1 face instead the risk due to the randomness of the following period price (at which they unwind). As liquidity trading displays persistence, second period informed investors absorb part of first period liquidity traders position and this contributes to first period investors uncertainty over p 2, yielding (16). Our assumptions on investors horizons also allows us to capture an important feature of High Frequency Trading. Indeed, as liquidity traders demand displays persistence, first period informed investors can infer it from the observation of the order flow using their private signals, to anticipate the impact this will have on the price at which they will unwind their positions. This allows them to trade ahead of liquidity traders. 10 Indeed, using (10) and (14): [ ( E i1 [p 2 ] = E i1 α P2 v + θ ) ] 2 + (1 α P2 )E 2 [v] a 2 ( = α P2 + (1 α P2 ) ( a ) 2) 2 τ u E i1 [v] + βλ 2 E i1 [θ 1 ] + (1 α P2 ) τ 1 E 1 [v]. (18) τ 2 τ 2 According to the above expression, when β > 0 the private signal serves two objectives: it allows to predict the impact of fundamentals on p 2, but it also creates a private learning channel from the first period price that allows investors to recover information on θ 1 from the observation of p 1 to predict the impact of θ 2 on p 2 (as in Amador and Weill (2010), and Manzano and Vives (2011)). This yields multiple equilibria: Proposition 2. Linear equilibria always exist. In equilibrium, a 2 = γτ ɛ, and a 1 is implicitly defined by the equation φ(a 1 ) a 1 (1 + γτ u a 2 ) γ 2 τ u a 2 τ ɛ = 0 If β (0, 1]: 1. There are two equilibria a 1, a 1, where a 1 < a 1 (see (A.17), and (A.18), in the appendix for explicit expressions). 2. When a 1 = a 1, a 2 βa 1 > 0, and λ 2 > 0, while when a 1 = a 1, a 2 βa 1 < 0, and λ 2 < 0. Furthermore, λ 2 < λ 2, and prices are more informative along the equilibrium with high second period liquidity. If β = 0, the equilibrium is unique: a 1 = lim β 0 a 1 = γa2 2τ u 1 + γa 2 τ u. (19) Due to persistence, liquidity traders positions in the first and second period are positively correlated. Thus, informed investors use their private signals on the fundamentals also to infer the demand of liquidity traders from the first period price, to anticipate the price at which they 10 This type of inference is a well documented feature of High Frequency Trading who use specific algorithms to detect the large orders that liquidity traders split across several trading rounds, in an attempt to minimize price impact, and profitably exploit them (see, e.g., Scott Patterson (June 30, 2010). Fast Traders Face Off with Big Investors Over Gaming, Wall Street Journal.) 10

11 unwind their positions. The more the second period price is correlated to the fundamentals, the more aggressively first period investors speculate on private information. As a consequence, the first period price reflects more the fundamentals, and less the demand of liquidity traders. This, in turn, leads first period investors to put more weight on the private signal to extrapolate the demand of liquidity traders from the price, making the first and second period price more related to fundamentals. In this case, this self-reinforcing loop leads investors to escalate their response to private information. Similarly, the less correlated is the second period price with the fundamentals, the less aggressively first period investors speculate on their private information. But then, the first period price reflects less the fundamentals, and more the demand of first period liquidity traders. This lowers the weight investors put on their private signals to infer the demand of liquidity traders from the price, and makes the first and second period price less related to fundamentals. In this case, instead, the self-reinforcing loop leads investors to dampen their response to private information (see Figure 1). Along the former equilibrium, prices are more informative about the fundamentals and the second period market is thicker. Conversely, along the latter equilibrium, prices are more informative about the demand of liquidity traders, and the second period market is thinner. When β = 0, liquidity trading is transient, and first period investors cannot use their information on θ 1 to forecast p 2. This eliminates the private learning channel from the first period price and yields a unique equilibrium. Note that along the equilibrium with high liquidity λ 2 < 0, implying that a positive realization of the news in the order flow (z 2 > 0) leads to a decrease in the second period price. The reason for this result is that a 1 > a 2 /β, and thus ( a 2 ) = a 2 βa 1 < 0, so that in the second period, informed investors reduce their exposure to the risky assed based on private information. As a consequence, if z 2 = ( a 2 ) v + u 2 > 0, it could be that v < 0 or that u 2 > 0, which leads the market to update downwards its assessment of the fundamentals, and negatively update the price. Please insert figure 1 here. The next result characterizes investors strategies: Corollary 2. At a linear equilibrium, a rational investor s strategies are given by X 1 (s i, p 1 ) = a 1 α E1 (E i1 [v] p 1 ) + α P 1 α E1 α E1 E 1 [θ 1 ] (20) X 2 (s i, p 2 ) = a 2 α E2 (E i2 [v] p 2 ). (21) When β (0, 1] and (i) a 1 = a 1, α P1 α P1 < α E1. < α E1, and when (ii) a 1 = a 1, α P1 > α E1. For β = 0, According to (21), in the second period an investor acts like in a static market. In the first period, instead, he loads his position anticipating the second period price, and scaling it down according to the uncertainty he faces on p 2 : X 1 (s i, p 1 ) = γ E i1[p 2 ] p 1. Var i1 [p 2 ] 11

12 In this case, his strategy can be expressed as the sum of two components (see (20)). The first component captures the investor s activity based on his private estimation of the difference between the fundamentals and the n-th period equilibrium price. This may be seen as akin to long-term speculative trading, aimed at taking advantage of the investor s superior information on the liquidation value of the asset, since p 2 is correlated with v. The second component captures the investor s activity based on the extraction of order flow, i.e. public, information. This trading is instead aimed at timing the market by exploiting short-run movements in the asset price determined by the evolution of the future aggregate demand. Indeed, using the expressions in Corollary 1: E 1 [p 2 p 1 ] = Var i1[p 2 ] E 1 [θ 1 ], (22) γ which implies that based on public information the investor expects reversion. However, the position he takes depends on the difference α P1 α E1 : when α P1 α E1 < 0 he engages in market making (thereby accommodating the aggregate demand), when α P1 α E1 > 0 he chases the trend (and thus follows the market). To fix ideas, suppose that E 1 [θ 1 ] > 0. Given that E 1 [θ 1 ] = a 1 (v E 1 [v]) + θ 1, an investor s reaction to this observation depends on whether he believes it to be driven by liquidity trades or fundamentals information. According to Proposition 2, when a 1 = a 1, the first period price reflects less the fundamentals (and thus more the demand of liquidity traders), which leads to infer that E 1 [θ 1 ] > 0 is due to the impact of liquidity traders orders. As a consequence α P1 < α E1, and investors speculate on reversion. When instead a 1 = a 1, the first period price reflects more the fundamentals (and thus less the demand of liquidity traders). In this case, E 1 [θ 1 ] > 0 is more likely to reflect the average private signal of informed investors, α P1 > α E1, and investors speculate on continuation. As the persistence in liquidity trading increases, in both equilibria first period informed investors speculate less aggressively on their private information. Indeed, in this case the first period price is more informative about θ 2, which leads investors to lower the weight to private information in both equilibria: Corollary 3. At equilibrium for all β (0, 1), a 1 / β < 0. Proof. The equation that yields the first period responsiveness to private information is given by: φ(a 1 ) λ 2 τ i2 a 1 γ a 2 τ u τ ɛ = 0. The result then follows immediately, since from implicit differentiation of the above with respect to β: a 1 β = γτ u a 1 (a 2 a 1 ) (1 + γτ u a 2 ) + γβτ u (a 2 a 1 ) < 0, independently of the equilibrium that arises; in the low liquidity equilibrium we have βa 1 < a 1 < a 2 γτ ɛ, and in the high liquidity equilibrium a 1 > a 2 /β > a 2 and 1 + γτ u a 2 < 0. 12

13 We conclude this section, showing that absent private information, equilibrium multiplicity disappears when β > 0. In this case prices are invertible in the demand of liquidity traders, and the model is akin to Grossman and Miller (1988): Corollary 4. When τ ɛ = 0, β [0, 1] there exists a unique equilibrium where Λ 2 = Var[v]/γ, and p n = v + Λ n θ n (23) X n (θ n ) = Λ 1 n (p n v), (24) Λ 1 = Var 1[p 2 ] γ + βλ 2. According to (24), rational investors always take the other side of the order flow, buying the asset at a discount when θ n < 0, and selling it at a premium otherwise. For a given realization of the innovation in liquidity traders demand u n, the larger is Λ n, the larger is the adjustment in the price rational investors require in order to absorb it. Therefore, Λ n proxies for the liquidity of the market. Uniqueness follows from the absence of private information. Indeed, in this case prices are invertible in the demand of liquidity traders and the feedback loop responsible for equilibrium multiplicity disappears. 2.2 Stability In this section we analyze the stability of the equilibrium solutions. We start by defining the best response function which determines the equilibrium responsiveness to private information of a first period investor, given the decision made by the other first period investors. From the expression for φ(a 1 ) in Proposition 2, an equilibrium obtains as a solution to the following fixed point equation a 1 = ψ(a 1 ) γ a 2 a 2 τ u. (25) 1 + γτ u a 2 Analizing the best response function ψ(a 1 ) it is possible to see that it is discontinuous for a 1 = (γτ u β) 1 (1 + γτ u a 2 ): lim ψ(a 1) = (26) a 1 ((γτ uβ) 1 (1+γτ ua 2 )) lim ψ(a 1) =, (27) a 1 ((γτ uβ) 1 (1+γτ ua 2 )) + and that the decisions on the weights investors put on private information are always strategic substitutes: ψ(a 1 ) βa 2 τ u = < 0. (28) a 1 (1 + γτ u a 2 ) 2 The reason for the existence of this discontinuity is that with persistence, first period investors unwind their position against the second period aggregate demand x 2 + u 2, and thus their average private information impacts both the first and second period order flows. Indeed, recall from (8) and (9) that z 1 = a 1 v + u 1, z 2 = a 2 v + u 2, 13

14 where a 2 = a 2 βa 1. This, in turn, has implications for the inference the market makes from the second period order flow. At equilibrium a large value of a 1 has two effects: first (according to the first of the above expressions), it increases the impact of fundamentals information vis-àvis that of liquidity trades in the first period price; second, if β > 0 it also changes the impact that the fundamentals information in the second period order flow has on the second period price (see the second expression above). This can potentially reduce the uncertainty faced by first period investors. Indeed, we have that Cov i1 [E 2 [v], E 2 [θ 2 ]] = a 2τ u (τ 2 a 2 a 2 τ u ) Var τ 2 i1 [z 2 ], (29) 2 and since along the equilibrium with high liquidity a 2 < 0, Cov i1 [E 2 [v], E 2 [θ 2 ]] < 0 Var i1 [p 2 ] < Var i1 [E 2 [v]] + Λ 2 2Var i1 [E 2 [θ 2 ]]. (30) Therefore, by speculating aggressively on their private information, first period investors can make the (conditional) correlation across the two components of the second period price negative, reducing the risk to which they are exposed when clearing the market. In fact, according to (30), a positive shock from liquidity traders (u 2 > 0) has two countervailing impacts on p 2 : first, a positive one, due to its effect on the expected inventory of the second period investors; second, a negative one due to the impact it has on the second period market expectation about the fundamentals. When a 1 = (γτ u β) 1 (1 + γτ u a 2 ) Var i1 [E 2 [v]] + Λ 2 2Var i1 [E 2 [θ 2 ]] = 2Λ 2 Cov i1 [E 2 [v], E 2 [θ 2 ]], so that the second period price is entirely determined by first period information, and Var i1 [p 2 ] = 0. Plotting ψ(a 1 ) together with the 45-degree line, and using (26), (27), and (28) we see that the solution space of the fixed point equation is divided in two intervals: the first one is (0, (γτ u β) 1 (1 + γτ u a 2 )), and the second one is ((γτ u β) 1 (1 + γτ u a 2 ), ). The equilibrium with low (high) liquidity falls in the first (second) region (see Figure 2). Please insert Figure 2 here. To analyze stability, consider the following argument. Assume that the market is at an equilibrium point â 1, so that â 1 = ψ(â 1 ). Suppose now that a small perturbation to â 1 occurs, which makes the first period price informationally equivalent to z 1 = â 1v+u 1. As a consequence, first period investors modify their weights to private information so that the aggregate weight becomes â 1 = ψ(â 1). If the market goes back to the original â 1 according to the best reply dynamics with the best response function ψ( ), then the equilibrium is stable. Otherwise it is unstable. Formally, we have the following definition: Definition 1. An equilibrium is stable (unstable) if and only if its corresponding value for a 1 is a stable (unstable) fixed point for the best response function ψ( ) (i.e., if and only if its corresponding value for a 1 satisfies ψ (a 1 ) < 1 ). 14

15 With this definition it is immediate to verify that ψ (a 1) < 1, ψ (a 1 ) > 1, (31) implying that the equilibrium with high responsiveness to private information and high liquidity is unstable according to the best reply dynamics. 3 Average expectations and reliance on public information In this section we show that persistence makes the first period price driven by investors HOEs about fundamentals and by their average expectations about liquidity trading. This, in turn, has implications for price reliance on public information. Starting from the second period, and imposing market clearing yields 1 Due to CARA and normality, we have 0 X 2 (s i2, z 2 )di + θ 2 = 0. (32) X 2 (s i2, z 2 ) = γ E i2[v] p 2. Var i2 [v] Replacing the above in (32) and solving for the equilibrium price we obtain p 2 = Ē2[v] + Λ 2 θ 2, where Λ 2 = Var i2 [v]/γ. Similarly, in the first period, imposing market clearing yields: 1 and solving for the equilibrium price we obtain 0 X 1 (s i1, z 1 )di + θ 1 = 0, Substituting the above obtained expression for p 2 in (33) yields p 1 = Ē1[p 2 ] + Var i1[p 2 ] θ 1. (33) γ ] Var i1 [p 2 ] p 1 = [Ē2 Ē1 [v] + Λ 2 θ 2 + θ 1 γ = [Ē2 Ē1 [v] ] + βλ 2 Ē 1 [θ 1 ] + Var i1[p 2 ] θ 1. (34) γ According to (34), there are three terms that form the second period price: investors second order average expectations over the liquidation value (Ē1[Ē2[v]]), the risk-adjusted impact of the first period stock of liquidity trades (θ 1 ), and investors average expectations over first period liquidity trades (Ē1[θ 1 ]). As liquidity trades are persistent, rational investors anticipate unwinding a fraction β of their inventory (θ 1 ) to second period investors, thereby affecting their exposure to the risky asset and thus p 2. Using their private signals, first period investors can 15

16 estimate θ 1 from the observation of the order imbalance, which makes p 1 reflect the first period market consensus over the size of liquidity traders demand. Expression (34) implies that due to persistence in liquidity trading, the weight placed by the price on investors average information is the sum of two terms: the first term captures the impact of HOEs on v, the second term reflects the impact of investors average expectations over θ 1. Computing where Ē 1 [Ē2 [v] ] = ᾱ E1 v + (1 ᾱ E1 ) E 1 [v] Ē 1 [θ 1 ] = a 1 (1 α E1 )(v E 1 [v]) + θ 1, ( ᾱ E1 = α E1 1 τ ) 1 (1 α E2 ), τ 2 which implies that the total weight the price places on average private information is given by α P1 = ᾱ E1 + βλ 2 a 1 (1 α E1 ). Note that β, ᾱ E1 < α E1. This implies that when liquidity trading is transient (β = 0) the first period price places a larger weight on public information than the optimal statistical weight. This finding is in line with Morris and Shin (2002), and Allen et al. (2006). The latter prove that with heterogeneous information, prices reflect investors HOEs about the final payoff. In this case, the law of iterated expectations does not hold, and investors forecasts overweight public information because these anticipate the average market opinion knowing that this also depends on the public information observed by other investors. The price is then systematically farther away from fundamentals compared to investors consensus. However, as shown above when β > 0, the price also reflects investors average expectations about the impact that the demand of first period liquidity traders has on the second period price. Thus, an additional term adds to ᾱ E1 a 1 > which for α E1 βλ 2 (1 α E1 ), can increase the weight placed on average private information above the optimal statistical weight. Due to Corollary 2 we then have Corollary 5. At equilibrium, 1. When β (0, 1], if { a 1, then α P1 < α E1, and Cov[p 1, v] < Cov[Ē1[v], v] a 1 = a 1, then α P1 > α E1, and Cov[p 1, v] > Cov[Ē1[v], v]. 2. When β = 0, α P1 < α E1 and Cov[p 1, v] < Cov[Ē1[v], v]. With persistence liquidity traders positions in the first and second period are positively correlated. Thus, investors use their private signals to infer the demand of liquidity traders 16

17 from the order flow, anticipating the impact this has on the next period price. As private signals are informative about the fundamentals, this increases the weight the first period price assigns to the average private information. Along the equilibrium with high liquidity, investors escalate their response to private information. In this case the extra weight that adds to ᾱ E1 is high enough to draw the price closer to fundamentals compared to consensus. In view of the results obtained in Section 2.2 this equilibrium is, however, unstable. Along the equilibrium with low liquidity the price is farther away from fundamentals compared to consensus. This equilibrium, which shares the same properties of the one found by Allen et al. (2006), is instead stable. 4 Implications In this section we discuss several implications of our results. 4.1 High Frequency Trading Our model can shed light on the impact that heterogeneous trading frequencies have on market liquidity. In particular, when β > 0, we can interpret the informed traders as HFTs that are able to turn around their positions at a higher speed compared to liquidity traders. The case β = 0 captures the extreme situation in which the technological features of HFT are available to all liquidity traders too, whereas when β = 1 the technological gap between HFT and liquidity trading is maximal. Formally, the sequence of events for the first period liquidity traders is illustrated in Figure 3. Please insert Figure 3 here. The expected profit that first period liquidity traders obtain is given by Π θ1 E[βθ 1 (v p 1 ) + (1 β)θ 1 (p 2 p 1 )] ( ( )) τ i1 τ v = βλ 1 + (1 β)λ 2 a 2 τ 1 u < 0. (35) a 1 τ i1 It is easy to see that along the high liquidity equilibrium lim β 0 Π θ1 = 0, since in this case the first period signal responsiveness diverges at β = 0 (see Corollary 3). On the contrary, along the equilibrium with low liquidity lim Π θ 1 = Π θ1 < 0. β 0 a1 = γa2 2 τu,β=0 1+γa 2 τu In general, plotting (35) along the two equilibria that arise with β > 0 one obtains Figure 4. Please insert Figure 4 here. Depending on parameters values the two plots intersect or not (we need τ v high for them to intersect), but the bottomline is that for β small, liquidity traders expected losses are always 17

18 smaller along the high liquidity equilibrium (in the second period the conclusion is immediate given the properties of the equilibrium). Several authors suggest that the introduction of algorithmic trading and high frequency trading have generated a large improvement in market liquidity accompanied by a reduction in adverse selection risk (see, e.g. Hendershott, Menkveld, and Jones (2011)). This finding would be consistent with the high liquidity equilibrium where, short of profitable speculative opportunities, informed investors in the second period lower their exposure to the risky asset (see Proposition 2), adverse selection decreases across the two periods, and the first period price under-relies on public information (and thus over-relies on private information, see Corollary 5). 11 Our model sheds light on the issue of whether HFT increases or decreases liquidity. both equilibria we see that a higher technological advantage of HFT (higher β) leads to higher expected losses of first period liquidity traders. A further implication of our model is that with differential information, the existence of a discrepancy in the speed at which different types of investors can turn around their positions can be responsible for indeterminacy. The presence of HFTs leads to equilibrium multiplicity with the outcome of either instability at a high liquidity equilibrium, or a stable equilibrium with low liquidity. 4.2 Momentum and reversal In this section we compute the autocovariance of returns at different trading horizons. A first implication of our model is that return autocorrelation depends on trading horizons, and on the equilibrium on which investors coordinate: Corollary 6. At equilibrium: 1. For all β [0, 1], Cov[p 2 p 1, p 1 v] < For β (0, 1], Cov[v p 2, p 1 v] < 0. For β = 0, Cov[v p 2, p 1 v] = For β (0, 1], along the equilibrium with high liquidity Cov[v p 2, p 2 p 1 ] > 0. Along the equilibrium with low liquidity, for τ v < ˆτ v, there exists a value ˆβ such that for all β > ˆβ, Cov[v p 2, p 2 p 1 ] > 0 (the expression for ˆτ v is given in the appendix, see equation (A.31)). If β = 0, Cov[v p 2, p 2 p 1 ] < 0. According to the above result, along the equilibrium with high liquidity, momentum occurs at short horizons (close to the end of the trading horizon), whereas at a longer horizon, returns 11 If we measure adverse selection at time n via the OLS conditional regression coefficient of the liquidation value over the time n informational addition (that is the term multiplying (1 α Pn ) in the adverse selection component of liquidity at n), we can verify that along the high liquidity equilibrium adverse selection decreases between periods 1 and 2. To see this, note that along the high liquidity equilibrium adverse selection at time 1 is a 1 τ u /τ 1, whereas at time 2 we have (a 2 βa 1 )τ u /τ 2. As in this case a 2 < 0, we look at the absolute value and check that a 1 /τ 1 > a 1 /τ 2 > a 2 /τ 2. The first inequality is trivially always satisfied, whereas the second one requires a 1 > a 2 /(1 + β), a condition that is always satisfied along the equilibrium with high liquidity. In 18

19 display reversal. 12 This is in line with the empirical findings on return anomalies that document the existence of positive return autocorrelation at short horizons (ranging from six to twelve months, see Jegadeesh and Titman (1993)), and negative autocorrelation at long horizons (from three to five years, see De Bondt and Thaler (1985)). It is interesting to relate this result with Daniel, Hirshleifer, and Subrahmanyam (1998) who assume that overconfident investors underestimate the dispersion of the error term affecting their signals and overreact to private information. This, in turn, generates long term reversal and, in the presence of confirming public information which due to biased self attribution boosts investors confidence, also lead to short term positive return autocorrelation. This pattern of overreaction, continuation, and correction is likely to affect stocks which are more difficult to value (e.g., growth stocks). In such a context, momentum is thus a symptom of mispricing and hence related to prices wandering away from fundamentals (conversely, reversal is identified with price corrections). In our model, along the equilibrium with high liquidity, investors rationally react more strongly to their private signals compared to the static benchmark, in contrast to the overreaction effect of the behavioral literature. 13 However, this heavy reaction to private information leads to stronger information impounding and to prices that track better the fundamentals (see Proposition 2). Momentum at short horizons in this case is therefore associated with a rapid convergence of the price to the full information value. To illustrate this fact, in Figure 5 we plot the mean price paths along the equilibrium with low liquidity (thick line), with high liquidity (thin line), and along the static equilibrium, that is the one that would obtain if investors reacted to information as if they were in a static market (dotted line). From the plot it is apparent that in the equilibrium with high liquidity the price displays a faster adjustment to the full information value than in the equilibrium with low liquidity (and the static equilibrium). This shows that the occurrence of momentum is not at odds with price efficiency. Please insert figure 5 here. As stated in the corollary, momentum can also occur along the equilibrium with low liquidity, provided that investors are sufficiently uncertain about the liquidation value prior to trading (that is, τ v is low) and that liquidity trading is sufficiently persistent (β high). In that equilibrium, investors respond less to private information, information impounding is staggered, and prices adjust more slowly to the full information value (see Figure 2). However, if sufficiently persistent, liquidity trading exerts a continuous price pressure which can eventually outweigh the former effect. Therefore, along this equilibrium momentum arises with slow convergence to the full information value, implying that the occurrence of a positive autocorrelation at short horizons per se does not allow to unconditionally identify the informational properties of prices. 12 The fact that Cov[p 2 p 1, p 1 v] < 0 is due to the initial effect, p 0 = v. Numerical simulations show that in a model with three periods, in the equilibrium with high liquidity, both Cov[v p 3, p 3 p 2 ] and Cov[p 3 p 2, p 2 p 1 ] are positive. 13 Indeed, the static solution calls for a 1 = γτ ɛ (see, e.g, Admati (1985), or Vives (2008)), and it is easy to verify that 0 < a 1 < γτ ɛ < a 1. In Daniel, Hirshleifer, and Subrahmanyam (1998) overconfident investors overweight private information in relation to the prior. 19

20 Finally, at long horizons, the effect of private information on the correlation of returns washes out and the only driver of the autocovariance is the persistence in liquidity trading, which generates reversal. 4.3 Expected volume and return predictability In this section we investigate the implications of our results for the expected volume of informational trading and the predictability of returns along the two equilibria. We show that the expected volume of informational trading is high (low) along the high (low) liquidity equilibrium. This implies that a high volume of informational trading predicts momentum, in line with the evidence presented by Llorente, Michaely, Saar, and Wang (2002). However, as we have argued in the previous section, also along the equilibrium with low liquidity momentum can occur, provided liquidity trading displays sufficiently strong persistence (and the ex-ante uncertainty about the liquidation value is sufficiently high). This implies that a low volume of informational trading can also predict continuation. In this case, though, momentum is a signal of slow price convergence to the liquidation value. In sum, momentum is compatible with both a high and a low volume of informational trading, but the implications that return continuation has for price informativeness are markedly different in the two situations. We start by defining the volume of informational trading as the expected traded volume in the market with heterogeneous information net of the expected volume that obtains in the market with no private information analyzed in Corollary 4. This yields: 14 and V 2 = = π V 1 = = π 1 E [ X 1 (s i1, z 1 ) ] di E [ X 1 (θ 1 ) ] di π Var [X 1(s i1, z 1 )]di 0 π Var [X 1(θ 1 )]di ( a 2 1τ 1 ɛ + τ 1 u ) τ 1 u, (36) 1 E [ X2 (s i2, z 2 ) X 1 (s i1, z 1 ) ] di E [ X 2 (θ 2 ) X 1 (θ 1 ) ] di π Var [X 2(s i2, z 2 ) X 1 (s i1, z 1 )]di 0 π Var [X 2(θ 2 ) X 1 (θ 1 )]di ( (a a 2 2)τ 1 ɛ + (1 + (β 1) 2 )τ 1 u ) (1 + (β 1) 2 )τ 1 u. (37) We measure the total volume of informational trading with V 1 + V 2, and obtain Corollary 7. At equilibrium, for all β (0, 1] the expected volume of informational trading is higher along the high liquidity equilibrium. When β = 0 only the equilibrium with a low volume of informational trading survives. 14 This is consistent with He and Wang (1995). 20