Technical Trading and Strategic. Liquidity Provision

Transcription

1 Technical Trading and Strategic Liquidity Provision Ming Guo and Chun Xia Abstract We study heterogeneous investors roles in liquidity provision and corresponding asset pricing implications over short horizons. Competitive risk-averse market makers supply explicit liquidity. Uninformed technical investors and an informed investor employ technical trading and offer implicit liquidity strategically. Informed and technical investors are contrarians. Their trades and market makers positions positively forecast future returns. With more technical trading, market quality enhances, market makers behave more myopically, autocorrelations between consecutive returns become more negative, and informed investor s unconditional expected profits can increase. The model reproduces many salient empirical patterns and provides new testable implications. Guo is from the Shanghai Advanced Institute of Finance, Shanghai Jiaotong University. Xia is from the Faculty of Business and Economics, University of Hong Kong. We are very grateful to Tarun Chordia, Pete Kyle, Avanidhar Subrahmanyam, and Sheridan Titman for their invaluable suggestions. We thank Ekkehart Boehmer, Henry Cao, Charles Chang, Chun Chang, Long Chen, Songnan Chen, Jennifer Huang, Michael Lemmon, Chengfu Ma, Jianjun Miao, Lin Peng, Mark Seasholes, Ke Tang, Jiang Wang, Tan Wang, Hong Yan, Ning Zhu, Rong Zhao, and seminar participants at the CKGSB/SAIF Capital Markets and Corporate Finance 2011 Summer Program, 2013 Asian Meeting of Econometric Society, Fudan University, Renmin University of China, Shanghai Jiaotong University, Southwestern University of Finance and Economics, and University of Hong Kong for helpful discussions. We are solely responsible for any errors and omissions. 1

2 I. Introduction Substantial progress has been made in understanding the liquidity provision in financial markets. In particular, Grossman and Miller (1988) develop an influential theory of liquidity provision under the competitive rational expectations paradigm. They demonstrate that patient risk-averse investors such as specialists, individuals or other types of investors provide liquidity to accommodate a temporary imbalance of demand and supply between impatient investors. Consequently, the price concessions offered by investors who require immediacy lead to subsequent return reversals. Recent empirical studies by Hendershott and Seasholes (2007), Kaniel, Saar, and Titman (2008), and Brogaard, Hendershott, and Riordan (2013) respectively show that the positions of specialists, and the trades of individual investors and high-frequency traders (HFTs) in the U.S. are negatively correlated with past returns and positively correlated with future returns over short horizons. 1 They interpret the findings to be consistent with Grossman and Miller (1988). We observe that Grossman and Miller (1988) neither separate liquidity provision roles of different investors nor differentiate their trades and positions. In addition, investors trading in their model is induced by exogenous liquidity shocks. As a result, the speculative motive whereby investors know that security prices deviate from their fundamental values and thus offer liquidity is not considered. To better understand the empirical evidence and address these concerns, we develop a dynamic model to delineate the liquidity provision decisions by informed investors, uninformed investors and market makers over short horizons in the strategic rational expectations framework pioneered by Kyle (1985). Instead of relying on investors trading nonsynchronization or patience difference, we link investors liquidity provision decisions to their strategic use of historical prices and emphasize the role of arbitrage capitals in moving the prices back to the fundamental values. Following Brown and Jennings (1989) and Hirshleifer, Subrahmanyam, and Titman (1994), we refer to trading based 1 High-frequency trading usually involves buying and selling stocks according to computer algorithms based on past prices and volumes, and holding stocks for only a short period of time. Chordia, Goyal, Lehmann, and Saar (2013) provide an excellent research review of high-frequency trading. 2

3 on information inferred from past prices as technical trading. The insight of our paper is as follows: While explicit liquidity is provided by risk-averse market makers, who observe current as well as historical order flows, informed investors and uninformed technical investors provide implicit liquidity to the market, who have advantages to submit orders before the market makers but cannot observe the current price. They determine whether the prices of a risky asset are overvalued or undervalued relative to their expected fundamental values using technical analysis. Although Kavajecz and Odders- White (2004) and Hendershott, Jones, and Menkveld (2011) find empirical evidence on the relationship between technical analysis and liquidity provision, it has not been examined in the existing theoretical models. We fill the gap by endogenizing this relationship and exploring the implications for different investors trading behavior and asset pricing. We demonstrate that adding this new perspective is helpful to reproduce many salient patterns documented by the empirical literature. Our strategic liquidity provision theory therefore complements and refines the theory of Grossman and Miller (1988). Market effi ciency and the incentive for investors to seek private information have been studied intensively in the existing literature. Given technical trading is widely used among institutional and individual investors, particularly HFTs, it is surprising that few papers address the effects of the competition among technical traders on market effi ciency and the incentive for investors to become informed. The answers to these questions are not as obvious as they look. Conventional wisdom may suggest that more fierce competition among technical traders would cause the asset prices to deviate less from the fundamental values and reduce the predictability of historical prices on future prices. As a result, the magnitude of return autocorrelations declines and the prices become more informative, leading to a more effi cient market. Therefore, the unconditional expected profits of the informed traders decline, implying a lower motivation for investors to acquire information. Nonetheless, the technical traders trading may affect the market makers perception of illiquidity premium adversely, because of inventory risk, and thus the autocorrelation coeffi cients of consecutive 3

4 returns. As liquidity improves, the informed traders can trade on their private information at a more favorable price so that their unconditional expected profits may increase despite of more fierce competition among technical traders. Our framework allows us to examine these questions rigorously. In our model, there are a risk-free bond and a risky stock. At date 0, the price is fairly set at the unconditional mean of the stock s payoff (cash flow) which is realized at date 3. At date 1, a risk-neutral informed trader knows the stock s future payoff. Trading takes place at dates 1 and 2. The informed trader submits orders, together with risk-averse uninformed technical traders and noise traders, to the market without observing the current price. Noise traders trade randomly and cause stock prices to deviate temporarily from the fundamental values. After observing net order flows, risk-averse market makers infer equilibrium prices and submit orders to clear the market. In other words, the market makers supply liquidity to absorb the net order flow. It is noteworthy that our model is intended to capture short-term trading and asset pricing patterns. Because the market makers are risk averse, the equilibrium stock price at each trading date comprises two components: a fair value and an illiquidity premium. The former is the expected payoff conditional on the information set of the market makers, and the latter, being proportional to the market makers stock holdings, is required to compensate them for bearing undesired inventory risk in providing immediacy service. At date 1, the technical traders choose not to trade because they know that the date- 0 price is fairly set, while the informed trader trades on his private information. At date 2, after observing a positive (negative) date-1 price change, the technical traders not only raise (reduce) their estimates of the stock payoff, but also infer that the price is overvalued (undervalued) compared to the fair value. Thus, they trade in the opposite direction to the date-1 price change, and provide implicit liquidity to the market in order to earn part of the illiquidity premium. This result holds even if the technical traders are more risk averse than the market makers. The informed trader achieves a balance between trading on remaining 4

5 information and liquidity provision at date 2. Knowing that the deviation of the date-1 price from the fair value is transitory, he trades in the opposite direction to the date-1 price change too. The technical and informed traders both trade strategically by taking into account the impact of their trades on the equilibrium prices. Consequently, the price reverts to the fair value only partially at date 2, therefore their trades at date 2 are positively correlated with the return realized at date 3. To the best of our knowledge, the findings whereby both the informed and technical traders employ contrarian trading strategies and their trades positively forecast future returns are new to the theoretical literature. 2 We can interpret the technical traders in the model as individual investors, algorithmic traders, or HFTs. 3 Our predictions not only match widely documented empirical findings that these types of traders tend to be contrarians over short horizons, 4 but also explain why the net trades of individual investors and HFTs predict future short-horizon returns, as in Kaniel, Saar, and Titman (2008), Barber, Odean, and Zhu (2009), Kelly and Tetlock (2012), and Brogaard, Hendershott, and Riordan (2013). 5 a corporate insider or an institutional investor. 6 The informed trader can be proxied by A common finding in the literature is that insiders on aggregate are contrarian investors and their trades positively forecast future returns over short horizons. 7 Interestingly, Boehmer and Wu (2008) and Chordia, Goyal, and Jegadeesh (2012) find evidence that both individual and institutional investors behave as contrarians over short horizons. 2 See Kim and Verrecchia (1991), Wang (1993), Brennan and Cao (1996), and Watanabe (2008) and discussion in subsection IV.B. 3 Individuals usually have limited resources and capabilities to conduct fundamental research and typically employ technical trading (De Bondt (1998)). Algorithmic traders buy and sell stocks according to computer algorithms based on past prices and volumes. 4 See, e.g., Choe, Kho, and Stulz (1999), Jackson (2003), Grinblatt and Keloharju (2000, 2001), Richards (2005), Goetzmann and Massa (2002), Kaniel, Saar, and Titman (2008), and Barber, Odean, and Zhu (2009), Kelly and Tetlock (2012), and Broggard (2011). 5 Barber and Odean (2013) survey the literature and point out that There is an ongoing debate in the literature regarding the origins of this short-run predictability (of individual investors) in the U.S.. Our study sheds some lights on this issue. 6 Many top traders and fund managers in investment banks, proprietary trading desks and hedge funds routinely use technical trading strategies (Schwager (2008)). 7 See, e.g., Seyhun (1986), Rozeff and Zaman (1998), Lakonishok and Lee (2001), Piotroski and Roulstone (2005), Jenter (2005), and Fidrmuc, Korczak, and Korczak (2009). 5

6 Like Grossman and Miller (1988), our model shows that the market makers inventory positions are negatively correlated with past and contemporaneous returns, but are positively correlated with subsequent returns. Intuitively, if the risk-averse market makers hold larger (smaller) positions, current prices have to be lower (higher) and future prices have to be higher (lower) to compensate them for taking undesired inventories. These results are consistent with empirical findings in Hendershott and Seasholes (2007) and Boehmer and Wu (2008). 8 Next, we turn our attention to the model s implications on asset pricing. When more technical traders compete with the informed trader to provide implicit liquidity, the market makers require an illiquidity premium of a smaller magnitude (smaller price impact due to inventory risk), and consequently, technical trading enhances the price discovery process and improves market quality. More precisely, the stock prices become more informative, price volatility declines, and the market is more liquid. Intuitively, the more fierce competition from the technical traders causes the informed trader to trade more on his private information and to provide less liquidity. Although the price impact due to information asymmetry increases, the price impact due to inventory risk drops more, so that the market becomes more liquid. These results are in line with the empirical evidence reported by Kavajecz and Odders-White (2004), Broggard (2011), Hendershott, Jones, and Menkveld (2011), and Chordia, Roll, and Subrahmanyam (2011). The risk aversion of market makers causes the equilibrium prices to deviate from the stock s fair values. Because this deviation is driven mainly by the temporary noise shock rather than information regarding the fair value, a correction is not only expected in future dates but also materialized by the contrarian trading of the informed and technical traders. As a result, returns over short horizons are negatively autocorrelated, which conforms to findings on stock return reversals that occur at horizons ranging from a week to a month 8 In addition, subsection IV.C shows that our results are in line with earlier empirical evidence documented by Hasbrouck (1988), Hasbrouck and Sofianos (1993), Madhavan and Smidt (1993), and Madhavan and Sofianos (1998). 6

7 (e.g., Jegadeesh (1990) and Lehmann (1990)). Surprisingly, we find that as the competition among technical traders becomes more fierce, although the prediction power of historical prices regarding future returns declines, the autocorrelations between consecutive returns become more negative. An increase in the number of technical traders improves market quality so that return volatilities decline. More importantly, because the informed and technical traders provide more implicit liquidity, the market makers know that a noise shock in this period will be more likely absorbed in the next period, so that the prices tend to be less mean-reverting in subsequent periods, leading to a weaker intertemporal hedging benefit. In other words, the market makers demand is determined more by the expected return in the next period and the corresponding risk, and thus they behave more myopically. Given a negative (positive) order flow shock, the stock holdings of the market makers increase (decrease), so that the current-period price or return drops (rises). As the number of technical traders increases, since the market makers become more myopic, they demand a larger (smaller) expected return for the next period in equilibrium, leading to a more negative autocovariance of consecutive returns. Because return autocorrelation equals the return autocovariance normalized by return variance, the autocorrelations between consecutive returns become more negative. Note that the market makers myopic behavior is endogenously induced by the competition among the technical traders in our model. In contrast, investors myopia is usually assumed exogenously in the existing literature (e.g., Brown and Jennings (1989) and Blume, Easley, and O Hara (1994)). 9 This result has important empirical implications. It has been widely accepted that a smaller magnitude of autocorrelations between consecutive returns represents a more effi cient market (See the discussion in chapter 2 of Campbell, Lo, and MacKinlay (1997)). Our paper shows, however, that a larger magnitude of negative return autocorrelations does not necessarily indicate less effi cient prices, because more technical trading also leads to more 9 In practice, investors myopia or short-termism may come from their liquidity needs, incentives related to the evaluation of fund managers performance, or diffi culties associated with financing long-term investment due to market imperfections. 7

8 informative and liquid prices, and a decrease in price volatility. 10 Interestingly, we also find that more intense competition among the technical traders can result in larger unconditional expected profits for the informed trader, because the increase in the profits from trading on his private information at a more favorable price (resulting from a smaller price impact) exceeds the decrease in the profits from providing less liquidity. In other words, investors can be more incentivized to seek private information when the market liquidity improves and the prices become more informative in equilibrium. 11 To our best knowledge, the above two findings are unique in our model and new to the literature, which may be useful in guiding future empirical testing. The remainder of the paper is organized as follows. Section II relates our model to the existing theoretical literature. Section III develops a dynamic strategic trading model in which a risk-neutral informed trader, risk-averse technical traders trade against competitive risk-averse market makers in presence of noise traders. Section IV examines the effects of technical trading on traders trading strategies, liquidity provision, market quality and return autocorrelations. We also present empirical evidence that is broadly consistent with the model s predictions. Section V concludes the paper and points out new testable empirical implications. The proofs are relegated to the Appendix. II. Literature Review Our paper is related to several strands of literature. We only have space to provide a short review of the vast theoretical literature on liquidity provision. Stoll (1978) is among the first to examine the determinants of the illiquidity premium component of bid-ask spreads charged by risk-averse market makers when they have to hold undesired inventory in providing immediacy to investors. Campbell, Grossman, and Wang (1993) extend Grossman and 10 We are grateful to Sheridan Titman for pointing out the importance of this finding. 11 The reason that investors benefit of being informed when facing more technical trading is different from the strategic complementarities in information acquisition (see Vives (2008, pp ) and chapters 7-8 of Veldkamp (2011)). 8

9 Miller (1988) to investigate the relationship between trading volumes and returns when market makers provide liquidity to other investors. Weill (2007) studies the liquidity provision of competitive market makers during financial crisis when facing temporary selling pressure and order execution delays. Vayanos and Wang (2012) analyze how asymmetric information and imperfect competition affect market liquidity and asset prices. Either nonsynchronized trading or varying impatience is the key ingredient in the above-cited papers. We differ from them by assuming away nonsynchronization in trades and stressing the role of uninformed but strategic technical trading in liquidity provision. Theoretical models on technical trading are built in the competitive or strategic framework. In the former, Brown and Jennings (1989) and Grundy and McNichols (1989) show that technical analysis of prices has a positive value to informed investors because while a single price cannot reveal a stock s fundamental information, a sequence of prices can under certain circumstances. Blume, Easley, and O Hara (1994) give conditions under which analysis of volumes can improve informed investors learning of fundamentals in addition to prices. Hirshleifer, Subrahmanyam, and Titman (1994) study the technical analysis and trading patterns when some investors receive information before others. In the latter, Brunnermeier (2005) demonstrates that technical trading helps an informed investor gauge the extent to which private information is reflected in public announcements. Admati and Pfleiderer (1988) explore how uninformed investors engage in strategic trading by choosing either the composition or timing of their trades. These papers either do not consider the employment of technical trading by competitive uninformed investors, or do not relate investors trading strategies to liquidity provision. There is a small but growing literature on the effects of risk-averse market makers in strategic trading models. Subrahmanyam (1991) develops a static model to investigate the effects of market makers or informed investors risk aversions on market effi ciency. In a continuous time model, Guo and Kyle (2012) suggest that the interplay of informed traders strategic trading and market makers risk aversion is important for understanding return 9

10 dynamics such as momentum, reversal, and post-earnings announcement drifts. Vayanos (2001) analyzes endowment risk-sharing of uninformed strategic traders in the presence of noise traders and competitive risk-averse market makers. Nonetheless, the research questions of these papers are different from ours. We contribute to this stream of literature by differentiating the liquidity provision roles of different investors, studying the competition among technical traders, and explaining investors trading patterns and short-term asset pricing properties. Finally, we note that even though it is diffi cult to incorporate technical trading rules based on head and shoulder patterns, resistance and support levels, and moving averages in a parsimonious theoretical model with endogenous prices, recent breakthrough has been achieved by Zhu and Zhou (2009) who show the value-added use of moving averages of exogenous prices in a partial equilibrium asset allocation model. Zhou and Zhu (2013) further extend Wang (1993) and demonstrate that the moving averages of past asset prices can forecast future prices. III. The Model We consider a dynamic strategic trading model in which a risk-free bond and a risky stock are available for trading. We assume that the interest rate for the bond is zero and its price is one. The stock s payoff (cash flow) D is realized at date 3 and is normally distributed with mean D and variance σd 2. We normalize D = 0 for expositional convenience. At date 0, traders begin with their endowments and identical prior beliefs, and trade solely for optimal risk-transfer purposes, so the stock price is given by P 0 = D. At date 1, an informed trader knows the true value of the stock payoff, but the rest of the traders only know its distribution. Trading takes place at dates 1 and 2 among four types of traders: a risk-neutral informed trader who trades on private information and historical prices, n risk-averse uninformed technical traders who observe only historical prices, noise traders who trade for exogenous 10

11 motives, and a continuum of competitive risk-averse market makers with a unit mass whose role is to supply immediacy to the market. All traders liquidate their holdings and consume wealth at date 3. Three remarks deserve attention. First, the model is more likely to capture short-term phenomena in the stock markets because the information is assumed to be shortlived. Second, a common view among market practitioners is that technical analysis is most useful in trading over short horizons. In fact, a Wall Street Journal article reports that this short-term focus has been strengthened in recent years (Rogow (2009)). Third, risk aversion is a tractable way to model market makers limited risk-bearing capacity. 12 Recent studies by Adrian, Etula, and Shin (2010) and Danielsson, Shin, and Zigrand (2011) show that when risk-neutral market makers provide immediacy services under the funding or value-at-risk (VaR) constraints, they are effectively risk averse. More specifically, following Kyle (1985), trading at date t {1, 2} occurs in two steps. First, the informed trader, each technical trader i {1,, n}, and noise traders have time priority to submit respectively their market orders x t, z it, and u t not contingent on current price. u 1, u 2 are IID normal with zero mean and variance σ 2 u, and they are independent from the stock s payoff D. Second, a representative market maker observes the net order flows ω t = x t + z t + u t, where z t = n i=1 z it, infers prices, and chooses her order y t = ω t that clears the market. All traders other than noise traders employ trading strategies to maximize their expected utilities at date 3. We depart from Kyle (1985) by assuming that each market maker has a negative exponential utility function exp ( γ m W m 3 ), where γ m is a common risk aversion parameter and W m 3 is the market maker s wealth at date 3. Similarly, each technical trader has a negative exponential utility function with a common risk aversion parameter γ z. The initial endowments of all traders are normalized to zero to facilitate the analysis. Next, we specify the equilibrium stock prices and trading strategies of the informed 12 The assumption that market makers are risk averse is standard in inventory models and competitive trading models. Traditional strategic trading models assume risk-neutral market markets because their primary purpose is to explicate the impact of asymmetric information. We, however, intend to capture the interaction between information asymmetry and inventory risk in affecting liquidity provision. 11

12 trader, the technical traders and the market makers, respectively. We follow the convention to restrict to a linear symmetric Bayesian Nash equilibrium that satisfies the following conditions simultaneously. First, the stock price at each date is a linear function of net order flows in current and previous dates. The market makers know the equilibrium prices since they observe the net order flows and the pricing rules. They maximize their expected utilities by choosing stock positions that clear the market. Second, given the market makers updated beliefs about the stock payoff, the historical stock prices, and the conjectured trades of other traders, an informed or a technical trader chooses his optimal trade which is a linear function of his state variables. Third, the trade submitted by each technical trader is symmetric, so are the positions taken by each market maker. We begin by postulating that the linear equilibrium prices are P 1 = λ 11 ω 1, (1) P 2 = λ 21 ω 1 + λ 22 ω 2, (2) where λ 11, λ 21, and λ 22 are liquidity parameters. We conjecture that the informed trader s linear equilibrium trades are x 1 = β 11 D, (3) x 2 = β 21 D + β 22 ω 1, (4) where β 11, β 21, and β 22 measure the informed trader s trading intensities (or aggressiveness). Note that at date 2 the informed trader s state variables include his private information D, public information P 1 which is informationally equivalent to the net order flow ω 1. Given his information at date 2, each technical trader i {1,, n} submits an order z i = α i ω 1 where α i measures the technical trader s trading intensity. Because their orders are symmetric, their total orders are z = αω 1, (5) 12

13 where α = nα i. The technical traders do not trade at date 1 to avoid losing money to the informed trader. Given the information set F t, we assume that the market makers beliefs updating takes the following forms: E [D F 1 ] = τ 11 ω 1, (6) E [D F 2 ] = τ 21 ω 1 + τ 22 ω 2, (7) where F 1 = {ω 1 } and F 2 = {ω 1, ω 2 }, and τ 11, τ 21, and τ 22 capture the impacts of order flows on the expected value of stock s payoff. Taking into account traders orders (3), (4) and (5) and applying the projection theorem for normally distributed random variables, we obtain where τ 11 = β 11σd 2, (8) c 0 τ 21 = c 0 c 1 τ 22 c 1 c 2 1 β 11 σd 2. (9) [β 21 + β 11 (β 22 + α)] σd 2 c 0 = V ar [ω 1 ] = β 2 11σ 2 d + σ 2 u, (10) c 1 = Cov [ω 1, ω 2 ] = β 11 [β 21 + β 11 (β 22 + α)] σ 2 d + (β 22 + α) σ 2 u, (11) c 2 = V ar [ω 2 ] = [β 21 + β 11 (β 22 + α)] 2 σ 2 d + [ (β 22 + α) ] σ 2 u. (12) In the ensuing analysis, we solve the linear equilibrium in which our conjectures are verified. A. Informed Trader s Maximization Problem The informed trader s optimization problems are derived using backward induction. After observing the date-1 price P 1, he can back out noise trade u 1. Hence, given his optimal trade 13

14 x 1, the informed trader s chooses x 2 at date 2 to maximize his final wealth: max x 2 E [x 1 (D P 1 ) + x 2 (D P 2 ) D, F 1, u 1 ], Plugging in technical traders total trades (5) and price (2) at date 2, he derives the first-order condition (FOC) with respect to x 2 and obtains x 2 = D (λ 21 + λ 22 α) ω 1 2λ 22. (13) Clearly, the second-order condition (SOC) is strictly negative when λ 22 > 0. It will be demonstrated shortly that the informed trader trades on his private information regarding the stock s payoff as well as provides liquidity to the market. At date 1, the informed trader s maximization problem can be written as: max x 1 x 1 (D E [P 1 D]) + λ 22 E [ (x 2) 2 D ]. Substituting date-1 price (1) and his date-2 trade (4), he optimally chooses The SOC is strictly negative when λ 11 > λ 22 β x 1 = 1 + 2λ 22β 21 β 22 D. (14) 2 (λ 11 λ 22 β22) 2 The risk-neutral informed trader s equilibrium trades (13) and (14) have the similar expressions described by Kyle (1985), except that in our framework, he takes into account the competition from the uninformed technical traders when submitting his trades. B. Technical Traders Maximization Problem At date 2, the technical traders observe the date-1 price P 1, which is informationally equivalent to observing F 1. The technical traders and the informed trader do not observe the net order flow at date 2, compared to the market makers. However, knowing whether the price at date 1 is overvalued or undervalued compared to the fair value, they have time priority to submit orders before the market makers. More precisely, by submitting z i 14

15 conditional on information set F 1 at date 2, each risk-averse technical trader i {1,, n} maximizes or max z i E [ exp [ γ z z i (D P 2 )] F 1 ], max z i E [D P 2 F 1 ] γ z z i 2 z2 i V ar [D P 2 F 1 ]. Given price (2), informed trader s trade (4), market makers belief updating (6), and other technical traders symmetric trades at date 2, technical trader i derives the FOC with respect to z i and obtains zi = τ 11 λ 21 λ 22 [β 21 τ 11 + β 22 + (n 1) α i ] [ ] ω (1 λ22 β 2λ 22 + γ 21 ) 2 σd 2 1. (15) z c 0 + λ 2 22 σu 2 When λ 22 > 0 is satisfied as the requirement for the informed trader s maximization problem at date 2, the SOC of each technical trader s problem is strictly negative. Each technical trader takes into account the price impact and the hedging need for his inventory risk, captured by the first and second terms in the denominator respectively. We will show shortly that the technical traders altogether trade against date-1 net order flow ω 1 and consequently provide liquidity to the market. Because the technical traders do not have information on the stock s payoff and know that P 0 is fairly set at the unconditional mean of its final payoff, they naturally do not trade at date 1. C. Market Makers Maximization Problem Since market makers are identical, we solve only for a representative market maker s optimization problem over her final wealth y 1 (D P 1 ) + y 2 (D P 2 ). At date 2, she can infer equilibrium price P 2 (2) based on her information set F 2. Given her optimal position y 1, the maximization problem can be expressed as: [ max y1 (E [D F 2 ] P 1 ) + y 2 (E [D F 2 ] P 2 ) γ ] m y 2 2 (y 1 + y 2 ) 2 V ar [D F 2 ]. 15

16 The FOC with respect to y 2 yields y 2 = E[D F 2] P 2 γ m V ar [D F 2 ] y 1. (16) The first term is the familiar demand function for the stock that increases with its expected excess return and decreases with both the risk aversion of the market maker and the fundamental risk. The second term shows that market maker s demand for the risky stock decreases with her cumulative holdings in the stock. In other words, y1 + y2 is the market maker s optimal total holding or inventory position at date 2. At date 1, the representative market maker can infer equilibrium price P 1 (1) based on her information set F 1. She chooses position y 1 to maximize her final wealth: where We solve this problem in the Appendix. max y 1 E [ exp ( γ m W m 3 ) F 1 ], W3 m = y 1 (P 2 P 1 ) + (E[D F 2] P 2 ) (D P 2 ). (17) γ m V ar [D F 2 ] LEMMA 1 At date 1, the representative market maker s optimal position is y1 = E [P 2 F 1 ] P 1 γ m V ar [P 2 F 1 ] (λ 22 τ 22 ) (λ 22 λ 21 + λ 11 ) ω 1. (18) The market maker s demand at date 1 comprises two components. The first component represents the demand of a myopic risk-averse market maker who exploits the expected price appreciation across dates 1 and 2, whereas the second component reflects her intertemporal hedging demand and its intensity is measured by (λ 22 τ 22 )(λ 22 λ 21 +λ 11 ) λ 2 22 λ 2 22 > Since the stock s payoff will be realized at date 3, the impact of a noise shock on the stock price is temporary, and as a result the market maker s intertemporal hedging demand is negatively related to the date-1 net order flow ω 1. Hence, given a positive (negative) net order flow, the magnitude of her optimal demand is smaller (larger) than when the market maker is myopic. 13 Examples of myopic demand function are examined in Brown and Jennings (1989) and Blume, Easley, and O Hara (1994), among many others. 16

17 D. Equilibrium Trades and Prices The traders trading strategies, the market makers beliefs updating, and the market clearing conditions together yield the equilibrium trades and prices, which are summarized in Theorem 1. THEOREM 1 When a linear symmetric equilibrium exists, the trades of the informed and n technical traders and the prices in the model are respectively given by x 1 = β 11 D, x 2 = β 21 D + β 22 ω 1, z i = α i ω 1, P 1 = λ 11 ω 1, P 2 = λ 21 ω 1 + λ 22 ω 2, where 1 + β 22 β 11 = 2 (λ 11 λ 22 β22), β 2 21 = 1, β 22 = λ 21 + αλ 22, 2λ 22 2λ 22 2 (τ 11 λ 21 ) c 0 α i = 2 (n + 2) λ 22 c 0 + γ z (σd 2 +, α = nα 4λ2 22c 0 ) σu 2 i, λ 11 λ 21 = λ 22 [c 1 + τ 22 γ m (c 0 c 2 c 2 1)] c 0 + γ m (c 0 c 2 c 2 1) (λ 22 τ 22 ), λ 21 τ 21 = λ 22 τ 22 = γ m [ σ 2 d ( c 0 τ c 1 τ 21 τ 22 + c 2 τ 2 22)], and τ 11, τ 21, τ 22 ; c 0, c 1 and c 2 satisfy equations (8)-(12) respectively. The second-order conditions are λ 11 > λ 22 β 2 22 and λ 22 > 0. Note that price functions (1) and (2) can be rewritten as: P 1 = E[D F 1 ] + (λ 11 τ 11 )ω 1, P 2 = E[D F 2 ] + (λ 21 τ 21 )ω 1 + (λ 22 τ 22 )ω 2 = E[D F 2 ] + (λ 21 τ 21 )(ω 1 + ω 2 ). When the market makers are risk averse, the equilibrium price at each trading date has two components: a fair value, which is the expected payoff conditional on risk-neutral market makers information set, and an illiquidity premium to compensate risk-averse market makers for holding undesired risky positions due to limited risk-bearing capacity, which is usually 17

18 termed inventory risk in the literature. 14 For example, the illiquidity premium at date 2 is given by (λ 21 τ 21 )(ω 1 + ω 2 ) = (λ 21 τ 21 )(y 1 + y 2 ), so it is positively related to the market makers date-2 stock holdings. From (6) and (7), we know that τ 11, τ 21, and τ 22 measure price impact due to the asymmetric information ( or adverse selection), and λ 11 τ 11 and λ 21 τ 21 = λ 22 τ 22 measure price impact due to the inventory risk. Before proceeding, in order to highlight the role of technical trading, we consider two special cases in which technical traders are absent. In the first, there are no technical traders, and a linear equilibrium can be characterized by Theorem 1 with n = 0 and α i = α = 0. In the second, as in Kyle (1985) the market makers are assumed to be risk-neutral. Therefore the prices satisfy the martingale property and the technical traders do not trade. Intuitively, risk-neutral market makers would have already exploited any profitable opportunity based on information inferred from past prices. In this case the endogenous parameters are determined in Theorem 1 with γ m = 0 and α i = 0. It is straightforward to show that a linear equilibrium exists as follows: COROLLARY 1 When the market makers are risk neutral, the equilibrium is characterized by: and β 11 = 2L 1 1, β 21 = 1, β 22 = λ 21, 4L 1 λ 11 2λ 22 2λ 22 2L (2L 1) σ d L σ d λ 11 = λ 21 = τ 11 = τ 21 =, λ 22 = τ 22 =, 4L 1 σ u 2 (4L 1) σ u L = λ 22 = 1 [ ( 1 ( 7 cos π arctan 27) )] λ is the unique positive solution to a cubic equation 8L 3 4L 2 4L + 1 = Although the market makers take into account the intertemporal hedging demand at date 1, they are unable to completely hedge away the inventory risk because they are uninformed and risk averse. 18

19 IV. Equilibrium Analysis and Empirical Implications We carry out the equilibrium analysis in the following subsections. First, we investigate the trading behavior of the technical and informed traders and the predictability power of their trades on future returns. Second, we assess the trading strategy of the market makers and the predictability power of their positions on future returns. Third, we examine the effects of technical trading on price quality and market liquidity. Finally we explore the model s predictions on short-term return autocorrelations. We present the corresponding empirical evidence along the analysis. The exogenous parameters in our model are cash flow volatility σ d, noise volatility σ u, and risk aversion parameters γ m and γ z. We use exogenous parameters σ d = 0.2, σ u = 1, γ m = 1, and γ z = 2 as the benchmark values throughout the comparative static analysis. Note that in our numerical exercises, the technical traders are assumed to be more risk averse than the market makers. 15 We emphasize that the main findings are robust for a wide range of parameter values, irrespective of their relative values, that guarantee the existence of linear equilibria. To preserve the space we only present the comparative statics of the cash flow volatility σ d. 16 A. Technical Traders Trading Strategy and Return Predictability We first study the technical trader s individual trade z i = α i ω 1 and their total trades z = αω 1. The returns (price changes) at dates 1, 2, and 3 are P 1 P 0, P 2 P 1, and D P 2 respectively. Proposition 1 summarizes the relationship between their trades and returns. PROPOSITION 1 Each technical trader s trade and their aggregate trades at date 2 have the following properties: Corr [z i, P 1 P 0 ] = Corr [z, P 1 P 0 ] = 1, 15 As mentioned earlier, we can interpret the technical traders as individual investors and they are often more risk averse than institutional investors. 16 All unreported numerical results are avaialbe from the authors upon request. 19

20 Corr [z i, P 2 P 1 ] = Corr [z, P 2 P 1 ] = α [(λ 21 λ 11 ) c 0 + λ 22 c 1 ] V ar [z] V ar [P2 P 1 ], Corr [z i, D P 2 ] = Corr [z, D P 2 ] = α [(τ 11 λ 21 ) c 0 λ 22 c 1 ] V ar [z] V ar [D P2 ], where V ar [z] = α 2 c 0 and the variances of price changes are given by equations (A10)-(A11) in the Appendix, respectively. When the market makers are risk averse, for a wide range of exogenous parameter values, the technical traders employ a contrarian strategy; z i and z have positive price impact and positively forecast future return. As mentioned earlier, the risk-averse market makers require an illiquidity premium as compensation for holding undesired positions. Consequently, the price P 1 deviates from the fair value by an amount proportional to the net order flow ω 1. The technical traders thus choose to trade in the direction opposite to ω 1 at date 2, as shown by Corr [z, P 1 P 0 ] < 0. In doing so, they provide implicit liquidity to the market and earn part of the illiquidity premium. We emphasize that technical traders employ the contrarian strategy even if they are significantly more risk averse than the market makers. Intuitively, the price impact of the technical traders trading and the total risk are quadratic functions of the quantities they trade, while the expected profits are a linear function of their trades. Hence, the technical traders would choose to trade if there exists an illiquidity premium, no matter how risk averse they are. 17 Because the technical traders have time priority over the market makers in providing liquidity, they can take advantage of the temporary price deviation. Understanding the positive price impact of their trades, they will not trade too aggressively to push the price completely back to the pre-trade fair value, so that they will benefit from capital gain at next date. As a result, their trades positively forecast future return and this is manifested 17 Although each technical trader s order z i and their aggregate trades z have the same correlation with the returns, numerical solutions show an important difference between individual trading intensity α i and aggregate intensity α. With more technical traders, α i reduces but α increases, therefore the competition makes each technical trader to trade less intensely, but as a whole they trade more aggressively in liquidity provision, as in a standard Cournot competition model. 20

21 by Corr [z, D P 2 ] > 0. [Insert Figure 1 Here] Panels A and B in Figure 1 show that an increase in the cash flow volatility σ d leads to larger Corr [z, P 2 P 1 ] and Corr [z, D P 2 ] respectively. Intuitively, a higher σ d generates both more information asymmetry and a larger inventory risk. Because the market makers demand higher compensation, prices deviates more from fundamentals and the illiquidity premium increases. The technical traders therefore trade more aggressively, so that their trades have a larger price impact and a larger predictability power of stock return realized at date 3. Additionally, when the number of the technical traders increases, more competition among technical traders generates a larger price impact on the price and a smaller deviation of the price from the expected fundamental value. Consequently, more technical trading reduces return predictability at date Empirical studies have shown that individuals, algorithmic traders and HFTs tend to be contrarians: They buy after prices go down and sell after they go up, especially in the short term (e.g., Choe, Kho, and Stulz (1999), Grinblatt and Keloharju (2000, 2001), Goetzmann and Massa (2002), Jackson (2003) and Richards (2005)). In particular, Kaniel, Saar, and Titman (2008) find that individual investors on the NYSE tend to buy stocks following declines in the previous month and sell them following price increases. Because individual investors on average are less informed than corporate insiders or institutional investors, they are more likely to employ technical trading. If we interpret the technical traders in our model as individual investors, these empirical findings are in line with our model s implications. Broggard (2011) shows that HFTs on the NASDAQ tend to follow a price reversal strategy driven by order imbalances. 18 As n goes to infinity, the perfect competition will make each technical trader s order size infinitesimal, but their aggregate order size is still finite. The technical traders as a whole still provide extra liquidity at date 2. In the limit, the return predictability of each technical trader s order and their aggregate trades disappears. 21

22 Our model also explains the empirical findings on the short-term return predictability of the trades of individual investors and HFTs. 19 Jackson (2003) documents that the net flows of small investors positively predict future short-horizon returns in Australia. Kaniel, Saar, and Titman (2008) and Kelly and Tetlock (2012) find that in the U.S., stocks experience statistically significant positive and negative excess returns one month after intense individual buying and selling respectively. Using signed small-trade volume as a proxy for individual trading in Taiwan, Barber, Odean, and Zhu (2009) show similar results on return patterns in the several weeks after heavy buying and selling by individuals. Brogaard, Hendershott, and Riordan (2013) report that the direction of buying and selling by HFTs predicts price changes over short horizons measured in seconds. These findings are confirmed in Panels C and D in Figure 1 respectively. Furthermore, numerical solutions in Panels C and D show that technical trading is more profitable when the cash flow volatility is higher. Kaniel, Saar, and Titman (2008) and Barber, Odean, and Zhu (2009) find that individual investors trades have more forecasting power for small stocks. Small stocks usually involve greater cash flow volatility. Hence, our theoretical predictions are consistent with their empirical evidence. These empirical studies largely attribute their findings to the competitive liquidity provision theory proposed by Grossman and Miller (1988) that focuses on the inventory risk. Our model alternatively suggests that the forecasting power of trades from individual investors and HFTs may stem from the provision of implicit liquidity when they use technical analysis to determine whether prices are overvalued or undervalued relative to the fair value. 19 Because individual investors lack an information advantage, it is not easy to understand why their trades enable short-term returns forecastability. Many studies in the behavioral finance literature build on the presumption that the trading behavior of uninformed individuals is irrational, which makes the return predictability findings even more puzzling. 22

23 B. Informed Trader s Trading Strategy and Return Predictability We assess the trading patterns of the informed trader in this subsection. The informed trader s trades can be written as x 1 = β 11 D, x 2 = β 21 (D E[D F 1 ]) + (β 22 + β 21 τ 11 )ω 1. In particular, x 2 consists of two components. The first part x I 2 = β 21 (D E[D F 1 ]) represents the informational component, which is proportional to the informed trader s remaining information. 20 The second part x NI 2 = (β 22 + β 21 τ 11 )ω 1 is the non-informational component, which is related to liquidity provision. Proposition 2 characterizes the properties of x PROPOSITION 2 The informed trader s trade at date 2 has the following properties: Corr [x 2, P 1 P 0 ] = λ 11 (β 22 + β 21 τ 11 ) c 0 V ar [x2 ] V ar [P 1 P 0 ], Corr [x 2, P 2 P 1 ] = β 21σd 2 + [τ 11 (β 22 + α) + 2α (λ 21 λ 11 )] c 0 + 2β 22 λ 22 c 1 2, V ar [x 2 ] V ar [P 2 P 1 ] Corr [x 2, D P 2 ] = β 21σd 2 + [τ 11 (β 22 α) 2λ 21 (β 22 + β 21 τ 11 )] c 0 2β 22 λ 22 c 1 2, V ar [x 2 ] V ar [D P 2 ] where V ar [x 2 ] = β 2 21σ 2 d + β 22 (β β 21 τ 11 ) c 0. When the market makers are risk averse, for a wide range of exogenous parameter values, the informed trader employs a contrarian strategy at date 2; his trade has positive price impact and can positively forecast future return. [Insert Figure 2 Here] Panels A through C in Figure 2 plot the properties of the informed trader s date-2 trade against the cash flow volatility σ d for different values of n. Remarkably, because 20 It is easy to show that when the market makers are risk-neutral, as in Kyle (1985), the informed trader s date-2 trade x 2 is independent of return P 1 P 0 and he trades only on the residual information D P 1 = D E[D F 1 ]. 21 The properties of the informed trader s date-1 trade x 1 can be analyzed in a similar way. As expected, it has positive price impact, and can positively forecast future returns. We restrict our attention to date 2 so that we can compare the informed trader s trading strategy and return predictability with those of the technical traders. 23

24 Corr [x 2, P 1 P 0 ] < 0, the informed trader employs a contrarian strategy at date 2 too. To understand this result we note that x I 2 is uncorrelated with past price change P 1 P 0, but x NI 2 is negatively correlated with P 1 P 0. Like the technical traders, the informed trader at date 2 knows that the price only deviates from the fair value at date 1 transitionally, which is expected to be corrected at future date. He therefore trades against the past price change. The informed trader s date-2 trade positively forecasts future returns (Panel C) for two reasons. First, the date-1 price is not fully-revealing regarding the stock s final payoff, so x I 2 forecasts the return at date 3. Second, x NI 2 forecasts future return because it is related to liquidity provision, which is similar to the forecastability of the technical traders trades. Further, Panels A shows that, when σ d rises, the informed trader knows that the price deviates more from the fair value at date 1, and he therefore trades less on his private information and more on liquidity provision at date 2. Panel C reveals that the expected profits at date 2 from trading on liquidity provision dominates those from trading on private information, equivalently, the return predictability of x 2 increases. The informed trader in our model can be interpreted as a corporate insider. Our findings match empirical evidence that insiders often purchase (sell) shares after periods of negative (positive) abnormal stock performance and that insiders trades can predict abnormal future stock price changes (e.g., Seyhun (1986), Rozeff and Zaman (1998), Lakonishok and Lee (2001), Piotroski and Roulstone (2005), Jenter (2005), and Fidrmuc, Korczak, and Korczak (2009)). In addition, consistent with our model s predictions, Lakonishok and Lee (2001) find that insiders are able to predict returns in smaller firms. Jenter (2005) documents that the predictability of excess returns is stronger in small firms. The trading patterns of the informed and technical traders in our model share the same properties: They employ contrarian trading strategies at date 2, their trades have positive price impact, and their trades positively forecast future returns. To our knowledge this finding is new to the literature. Rational informed and uninformed investors follow contrarian and momentum strategies respectively in competitive trading models considered by Kim and 24