Information, Imperfect Competition, and Volatility

Transcription

1 Information, Imperfect Competition, and Volatility Mahdi Nezafat and Mark Schroder May 5, 07 Abstract We analyze a model of costly private information acquisition and asset pricing under imperfect competition. We show that imperfect competition generally creates strategic complementarity in traders' information acquisition decisions. The source of strategic complementarity is the change in the liquidity of the risky asset that arises from a change in the precision of a private signal: when an uninformed trader becomes informed, the liquidity of the risky asset generally increases. The increase in liquidity encourages more private information acquisition by informed traders. We also show that imperfect competition can shut down private information acquisition, leading to signicant illiquidity and volatility. This nding implies that excess return volatility can be observed in markets with low information asymmetry. JEL Codes: G Keywords: Imperfect competition, Information production, Excess volatility, Information complementarity, Adverse selection, Exchange traded funds ETFs Mahdi Nezafat, Broad College of Business, Michigan State University, East Lansing, MI nezafat@broad.msu.edu. Mark Schroder, Broad College of Business, Michigan State University, East Lansing, MI schroder@broad.msu.edu. We thank Junya Jiang, Liyan Yang, Sergey Glebkin, and participants at the 06 SFS Finance Cavalcade, the 06 European Finance Association Annual Meeting, and the 06 Financial Management Annual Meeting for helpful comments. All remaining errors are ours.

2 Introduction A basic tenet of nance is that asset prices aggregate diverse private information dispersed among traders. In this paper, we consider the following question: how does a large trader's incentive to collect information interact with other large traders' incentives to collect private information? We show that imperfect competition generates complementarity in private information production which can severely reduce equilibrium information production resulting in excess price volatility. This nding sheds some lights on the empirical observation that volatility of rm fundamentals does not fully explain the volatilities of corporate bond and credit default swaps CDS returns see Bao and Pan 03. In the standard perfect competition setting, the information choice of one trader has no eect on asset prices nor on other traders' incentives to collect information. Even when a fraction of traders decides to collect more private information, their choice aects the incentives of other traders to collect information only through increased price informativeness, which discourages information collection. However, many markets including corporate bonds, CDSs, small-cap stocks, privately-held rms, real estate, and collectibles are imperfectly competitive. These markets are characterized by a relatively small number of traders and the demand of each trader impacts the price. In such settings, the decision of one trader to collect more private information changes not only liquidity and price informativeness, but also changes the trading strategies and information acquisition incentives of other traders. A large theoretical literature has made signicant contribution to our understanding of the eects of imperfect competition on asset prices see, e.g., Glosten and Milgrom 985; Kyle 985; O'Hara 003; and Vayanos and Wang 0, among others. Our model contrasts with the prior literature by relaxing the unrealistic assumption that investors are endowed with private information and by focusing on the second moment of asset prices. We shows that imperfect competition can generate complementarities in private information acquisition; imperfect competition generally has an adverse eect on private information acquisition for risk-averse traders, and the eect can be so severe that all traders optimally stay uninformed;

3 and 3 imperfect competition can lead to excess return volatility by adversely aecting the production of private information. Consider a market with a small number of informed and uninformed traders. As an example, the over-the-counter market for credit default swaps for which only a small number of nancial institutions act as market makers see Atkeson et al. 0. Suppose that a trader decides to collect private information about a risky asset. For example, hedge fund X decides to obtain private information about the likelihood of default of company Y and approaches bank Z to purchase a credit default swap contract. Does the decision of the informed trader hedge fund X encourage the uninformed trader bank Z to acquire private information about the creditworthiness of company Y? Presumably, the answer to this question is yes. If so, how does hedge fund X react in both information collection and price negotiation to bank Z's decision to obtain information? How does the information production by the hedge fund and the bank aect the CDS spread and other hedge funds' incentives to collect information and trade CDSs on company Y? What role does the liquidity of the CDS contract play in the production of information and the transmission of that information into the CDS spread? To answer these questions we must distinguish between the eects of private information on one's own trading strategy and on the trading strategies of other traders. Two types of incentives drive the information acquisition decision in imperfect markets. We use the term fundamental incentives to refer to the desire to trade on more precise information to facilitate better investment decisions. The term strategic incentives refers to how one's information decision can aect the trading strategies of other traders in the market. The change in these strategies may aect the informativeness of the price, the price impact of trading, the risk premium, and the variance of the asset return. In this paper, we analyze a rational expectations equilibrium model of costly private information acquisition and asset pricing under imperfect competition. We assume that the nancial market is populated by two groups of traders. Traders within each group are ex ante 3

4 homogeneous. One group, called informed, obtains private signals, and the second group, called uninformed, does not obtain private signals or equivalently, obtains zero-precision signals. The model has two stages: rst an information acquisition stage, and then a trading stage. The trading stage of the model closely follows Kyle 989. A nite number of risk-averse investors trade a risky asset after observing a private signal regarding the payo of the asset. Each trader has a self-sustaining belief that he faces an upwardly sloping price curve for the shares in the risky asset, and hence the market is imperfectly competitive. The information acquisition stage of the model follows the recent theoretical literature on endogenous information acquisition by incorporating a signal precision choice by each trader. In particular, each trader chooses the precision of his private signal, holding xed the precision choices of the others, while anticipating both the price impact at the trading stage and the information impact of his precision choice on the trading strategies of all other traders. Both fundamental and strategic incentives determine the optimal precision. In the second stage the trading stage, we assume that all precisions are known and that traders choose their optimal demand parameters, holding xed the demand parameters of all other traders. This two-stage approach has been extensively used in the literature see, e.g., Li et al. 987; Vives 988; Hwang 993; and Persico 000, among others. We show that imperfect competition generates complementarity in private information acquisition. This is an important nding because many nancial markets are imperfectly competitive. A large literature studies mechanisms that generate strategic complementarity in information acquisition. In the standard setting of Grossman and Stiglitz 980, there is never strategic complementarity in private information acquisition because an increase in the fraction of informed traders increases the price informativeness and hence reduces each trader's incentive to become informed. Manzano and Vives 0 extend the setting in Grossman and Stiglitz 980 and show that in the presence of multiple sources of private information, there can be strategic complementarity in information acquisition. However, equilibria that have strategic complementarity in information acquisition are un- 4

5 stable. Therefore, the strategic substitutability result in Grossman and Stiglitz 980 is robust. In our model, each informed trader obtains only one private signal prior to his portfolio choice both informed and uninformed traders observe the public price signal. Therefore, there is only one source of asymmetric information, and strategic substitutability complementarity in information means that an increase in one trader's precision reduces increases the optimal precisions of the other traders. We show that there can be strategic complementarity in private information acquisition in the imperfect competition model. The source of strategic complementarity is the change in the liquidity of the risky asset that arises from a change in the precision of a private signal: when an uninformed trader becomes informed, the liquidity of the risky asset generally increases. The increase in liquidity encourages more private information acquisition by informed traders. The mechanism that generates complementarity in our model is novel to the literature and demonstrates that imperfect competition alone can generate information complementarity. What are the asset pricing implications of endogenizing private information acquisition in the presence of imperfect competition? We show that imperfect competition generates excess volatility i.e., return volatility in excess of fundamental volatility. The reduction in the equilibrium information arising from negative eects of strategic incentives in private information acquisition and the increase in the illiquidity of the risky asset are the sources of excess volatility. This is an important nding because it implies that imperfect competition can endogenously generate an environment with low production of private information and excess volatility. To the best of our knowledge, our paper is the rst in the literature that studies the eects of imperfect competition on the second moment of asset prices. By focusing on the asset pricing implications of information production in an imperfect competition setting, we are able to generate a set of predictions that appear consistent with recent empirical results demonstrating that, for instance, in an imperfectly competitive setting, illiquidity and not 5

6 fundamentals can generate excess volatility in corporate bond and CDS prices see Bao and Pan 03. In addition, our model provides empirical implications regarding passive and active investments and the eects of exchange-traded-fund ownership on asset prices which we discuss in Section 3.3. The framework that we develop makes contributions to two strands of the literature. First, we contribute to the literature on the eects of imperfect competition on asset prices in the presence of exogenous information see, e.g., Glosten and Milgrom 985; Kyle 985; O'Hara 003; and Vayanos and Wang 0, among others. We show that the quality of private information, as well as the sequential nature of the trading and information acquisition problems in particular, whether precisions are common knowledge at the trading stage, plays a signicant role in how imperfect competition aects asset prices. Second, we contribute to the literature on complementarity in information production see, e.g., Admati and Peiderer 987; Hellwig and Veldkamp 009; Manzano and Vives 0; Garcia and Strobl 0; Goldstein and Yang 05, and Mele and Sangiorgi 05, among others by showing that complementarity can arise naturally when a nite number of traders participate in the market. A Strategic Model of Information Production. Setup of the Model The model is static and the nancial market is populated by n traders with identical preferences who are indexed by j,..., n. Risky Asset One risky asset with payo of x is traded. All traders have the prior belief that the risky asset payo is normally distributed, x Nµ x, σ x. The risky asset supply is denoted by z and z Nµ z, σ z. The per capita level, mean, and variance of the risky asset supply are, respectively, denoted by z = z/n, µ z = µ z /n, and σ z = σ z/n. 6

7 Private Signals Each trader j obtains a noisy private signal, x j, about the asset payo, x j = x + ɛ j, where the noise ɛ j is independent across traders and independent of x, and ɛ j N0, τ ɛ j, where τ ɛj = σ ɛ j denotes the precision of the noise in private signal j. We refer to τ ɛj as the signal precision of trader j. Private signals are the source of dispersed heterogeneous information. Preferences and Wealth Each trader has exponential utility over his end-of-period wealth. In particular, trader j's expected utility function is UW j = E [exp ρ j W j ], where his nal wealth is W j = a j x P + w j. The variable a j is trader j's risky asset demand, P is the price of the risky asset, w j is trader j's initial wealth, E[.] is the expectation operator, and ρ j is the constant coecient of absolute risk aversion. Without loss of generality, we assume that w j = 0. Imperfect Competition Following Kyle 989, we consider only linear equilibria. Each trader has a self-sustaining belief that he faces an upwardly sloping price curve for the shares in the risky asset. In particular, trader j believes that his demand, denoted by a j, is related to the price as follows: P = p 0j + λ j a j, the variable p 0j see equation 5 in the appendixincorporates all elements of the price that are not related to trader j's demand, λ j is a constant that is determined in equilibrium and represents the degree of illiquidity associated with trader j's demand. If λ j is large, the 7

8 demand by trader j has a large impact on the price, and hence the risky asset is illiquid. If λ j is small, the demand by trader j has a small impact on the price, and hence the risky asset is liquid. We call λ j the price impact parameter. We consider only linear equilibria and therefore trader j's optimal demand is ane and is related to his private signal and the price as follows: a j x j, P = α j + β j x j γ j P, 3 where parameters {α j, β j, γ j } n j= characterize traders' demand functions and are determined in equilibrium. We respectively refer to β j and γ j as the demand sensitivities to the private signal and the price. Information Acquisition We follow the theoretical literature on endogenous information acquisition in nancial markets and explicitly incorporate the information acquisition decision in traders' overall investment decisions see, e.g., Peress 004; Van Nieuwerburgh and Veldkamp 00; and Mondria 00, among others. Each trader j, at the rst stage of the model, chooses the precision of his private signal, subject to the information cost function I.. The function I. represents the cost in dollars and is assumed to be convex and increasing, with I 0 = 0 and lim τ I τ =. Sequential Nature of the Trading and Information Acquisition We assume that all traders' precisions are known in the trading stagewhich we alternatively refer to as the second stageand traders choose their optimal demand parameters, holding xed the demand parameters of all other traders. In the information acquisition stagewhich we alternatively refer to as the rst stageeach trader chooses the precision of his private signal, anticipating both the price impact at the trading stage, and the information impact of his precision choice on the trading strategies of all other traders. In our opinion, this approach in modeling information acquisition is appropriate for representing dynamic trading situations in which traders learn each other's information acquisition decisions over time. 8

9 . Denition of Equilibrium Let Θ j = α j, β j, γ j denote the vector of trader j's demand parameters, Θ = Θ,..., Θ n the collection of all traders' demand parameters, X = x,..., x n the vector of traders' private signals, and T = τ ɛ,..., τ ɛn the vector of private signal precisions. As in Kyle 989, the market-clearing condition n j= a j = z implies that the equilibrium price i.e., P is a function of traders' demand parameters, their private signals, and the risky asset supply we omit the dependence on z in the notation, that is, P Θ, X = n j= α j + β j x j z n j= γ. 4 j Trader j's realized wealth, from equation, is the product of his demand and the asset return: W j Θ j, x j,p Θ, X = [α j + β j x j γ j P Θ, X] [x P Θ, X]. We emphasize the dependence of the joint distribution of the signals X on the precisions T by introducing a superscript T in the expectation operator, and we write the ex ante utility, net of information cost, as V j = E [ T exp [ ]] ρ j Wj I j τɛj. Note that for trader j, the distribution of x j aects the distribution of wealth W j directly through his demand and indirectly through the distribution of the equilibrium price. The distributions of other traders' signals aect V j only through the price. We next provide the denition of the equilibrium in the strategic model. Denition. For any T R n +, Θ = ˆΘ T is a second-stage trading equilibrium if, for each trader j =,..., n, E T [exp ρ j W j Θ j, x j,p Θ, X] E T [ exp ρ j W j Θ j, x j,p Θ, X ], In our applications, we consider only parameters such that a unique second-stage equilibrium exists for all T R n + consistent with Theorem 5. in Kyle

10 for all Θ j, where Θ is obtained from Θ by replacing the jth element with Θ j. Further, T is a rst-stage precision equilibrium if, for each trader j =,..., n, [ E [exp ρ T j W j ˆΘj T, x j,p ˆΘ T, X ]] I j τɛj [ [ ]] E T exp ρ j W j ˆΘj T, x j,p ˆΘ T, X I j τ ɛj, for all τ ɛ j, where T is obtained from T by replacing the jth element with τ ɛ j. T, ˆΘ T is an equilibrium of the strategic model. The pair In this setting, strategic eects of information choice arise because in the rst stage, each trader anticipates the eect of his precision choice on the equilibrium demand parameters of all traders, ˆΘ T. Trader j's utility depends on other traders' demands through the equilibrium price..3 Solution of the Model We assume that the nancial market is populated by two groups of traders. Traders within each group are ex ante homogeneous. One group, called informed, obtains private signals, and the information acquisition decision of each trader in this group is the choice of the precision of his private signal. The second group, called uninformed, does not obtain private signals or equivalently, obtains zero-precision signals. Let N denote the number of informed traders, M the number of uninformed, and n = N + M the total number of traders. To ensure a unique linear trading-stage price equilibrium, with all informed traders having the same demand function and all uninformed traders having the same demand function i.e., symmetric, using Kyle's term, we assume throughout that either N 3 and M 0, or N = and M see Theorem 5. in Kyle 989. In the case N =, Kyle 989 shows that uniqueness also requires either an upper bound on the risk aversion coecient of the uninformed or a lower bound greater than on M. Therefore, our N = results typically have an additional restriction. 0

11 .3. Characterization of the Equilibrium with Exogenous Information We need the explicit solution for the equilibrium price and demand parameters in the setting with exogenous information to characterize the equilibrium precision of private signals in the endogenous information model. Trader j observes x j, P when choosing a portfolio, and his portfolio choice problem for a given signal precision can be written as max E [W j x j, P ] ρ j a j V [W j x j, P ], 5 where V[.] is the variance operator. Substituting the wealth expression and the price expression, trader j's optimal demand for the risky asset is note that P is a function of a j : 3 aj = E [x x j, p 0j] p 0j ρ j V [x x j, p 0j ] + λ j = E [x x j, P ] P ρ j V [x x j, P ] + λ j. 6 The ex ante expected utility of trader j, denoted by U j, can be written as [ U j = E exp ] φ j, 7 where φ j = E [x x j, p 0j ] p 0j V [x xj, P ] + λ j /ρ j = E [x x j, P ] P. 8 V [x xj, p 0j ] + λ j /ρ j V [x x j, P ] + λ j /ρ j Let the subscript I U denote the parameters of informed uninformed traders and use the notation τ U = V [x P ] and τ I = V [x x j, P ], where x j is informed trader j's private signal. Dene, for each trader j, the sum of the other trader's signal sensitivities, β j, and the price precision, τ Pj, which represents the contribution of the price signal to the precision 3 Note that the information to trader j in P and p 0j is the same, and therefore E [x x j, p 0j ] = E [x x j, P ] and V [x x j, p 0j ] = V [x x j, P ].

12 of the payo conditional on trader j's information β j = k j β k, and τ Pj = β j k j β k /τ. 9 ɛ k + σz The following proposition characterizes the demand parameters for an exogenous set of private signal precisions in the model. Proposition Equilibrium parameters with exogenous information. For any set of private signal precisions {τ ɛ,..., τ ɛn }, traders' demand parameters in the benchmark and strategic imperfect competition models satisfy the following system of equations: β j = τ ɛj ρ + λ j τ j + τ Pj /β j, γ j = τ j τ Pj / β j λ j, j =,..., n. 0 ρ + λ j τ j + τ Pj /β j Trader j's precision τ j = V [x P, x j ] based on both his private signal and the price is τ j = τ x + τ ɛj + τ Pj. Traders' second-order conditions are ρτ j + λ j > 0, j =,..., n. The following proposition gives the equilibrium trading parameters in the case when the exogenous signal precisions are identical. Proposition Symmetric imperfect competition equilibrium. Suppose all traders are informed i.e., n = N and M = 0 and have identical signal precision τ ɛ. The equilibrium price and demand parameters are as follows:

13 a The equilibrium price is for some constant K P = K + β γ x + n n ɛ j z. β j= b Equilibrium β is the unique positive solution to the polynomial nβ τ ɛ + βρ n β + n σ zτ ɛ = n σ zτ ɛ n. 3 n The solution β to equation 3 is strictly increasing in τ ɛ, n and σ z and strictly decreasing in ρ. Further, β/τ ɛ is decreasing in τ ɛ from ρ in τ ɛ from 0 toward n σ z n n. c Equilibrium γ and precision τ = τ I are γ = β n n toward zero, and β /τ ɛ is increasing + n β + n σ zτ ɛ τ x n n β + n σ. 4 zτ ɛ τ ɛ τ = τ x + τ ɛ + τ P = γ γ β τ x, τ P = n β n β /τ ɛ + n σ. 5 z The price precision, τ P, is increasing in τ ɛ and n and decreasing in σ z is increasing in τ ɛ. and ρ. Further, τ P /τ ɛ d The price impact parameter λ is strictly decreasing in n and σ z, and lim n λ = lim σ z λ = 0. The three components of each trader's equilibrium total precision τ are the precision of payo, τ x, the precision of the private signal, τ ɛ, and the precision of the price, τ P. The precision of the price is strictly increasing in τ ɛ as the signal quality improves and n as more signals are pooled. An increase in the supply variance σ z, when ρ > 0, strictly increases the noise in the price and reduces τ P. The demand sensitivity to a private signal, β, is increasing in τ ɛ. The demand sensitivity to the price, γ, however, is not generally monotonic, though it is strictly increasing and therefore price impact is strictly decreasing 3

14 for suciently large τ ɛ. The nonmonotonicity of γ is caused by the conicting pressures of adverse selection, which, for some parameters, can cause γ to be increasing in τ ɛ for small τ ɛ, and declining perceived risk i.e., declining τ as τ ɛ increases, which causes more aggressive and price-sensitive trading. As discussed in Section., the strategic eects of information choice are generated through the equilibrium price. In particular, the strength of the strategic eects depends on how strongly the price reacts to a trade measured by price impact, λ and how ecient the price is at aggregating signals measured by the precision of the price, τ P. In two limiting cases, these strategic eects disappear. First, as the supply variance goes to innity, σ z, the price impact vanishes, and, for ρ > 0, the price becomes completely uninformative. Second, as the number of traders goes to innity, n, the price impact again vanishes. Although the price informativeness increases in n τ P τɛ / ρ σ z as n, the limiting n price is unaected by the signal and therefore the signal precision of any one trader. In both limiting cases, the price signal and the private signal of each trader therefore become conditionally independent, with the same resulting signal sensitivity of trading, β I = τ ɛ /ρ, no price impact, and precision choice determined only by fundamental incentives..3. Equilibrium Information Production Having characterized the equilibrium price of the risky asset with an exogenous precision of private signals, we next characterizes the equilibrium precision of private signals. Trader j anticipates the eect of his precision choice on other traders' equilibrium demand functions, which, in turn, aects the price and the risk premium of the asset. In the rst stage of the model, each trader holds xed other traders' signal precisions when choosing his own optimal signal precision. Proposition 3 Equilibrium precision of private signals in the strategic model. Consider 4

15 the strategic model. The signal precision τ ɛi is a Nash equilibrium if τ ɛi arg max τ ɛd 0 + V [φd ] exp ρiτ ɛd E [φ D ], 6 + V [φ D ] where φ D is dened in 8 and E [φ D ] and V [φ D ] are given in the proof. The trader precisions satisfy and the demand parameters {β D, β I,..., β I, 0,..., 0} and {γ D, γ I,..., γ I, γ U,..., γ U } satisfy the system of equations 0 with the set of signal precision {τ ɛd, τ ɛi,..., τ ɛi, 0,..., 0}. The variables with subscript D identify the demand and precision parameters corresponding to the possible deviating precision τ D of the optimizing trader. Note that any choice of τ D aects the demand parameters β D and γ D of the optimizing trader, as well as the demand parameters β I and γ I shared by the other informed traders, who have common precision τ ɛi. The demand and precision parameters then determine the utility of the optimizing trader through the rst two moments of φ D, as well as the variance of the return of the asset. Except under restrictive assumptions, Problem 6 does not have a closedform solution. In the next section we rely on numerical solutions to provide insights on the properties of the model in general and provide closed-form solutions for particular cases. Revisiting the Sequential Nature of Trading and Information Acquisition Problems In the strategic model presented in Section., it was assumed that the precisions become common knowledge prior to trading. For comparison, we also consider a benchmark model in which each trader chooses his precision parameter and demand parameters while holding xed the choices of all other traders. Therefore strategic incentives are eliminated in the benchmark model. By contrast, in the strategic model, each trader holds xed the precisions of the other traders when choosing his precision, but he anticipates the response of all the demand functions in the trading-stage equilibrium to his precision choice. The benchmark model is appropriate in a setting in which the precision choices of other traders are not observed the demand parameters of other traders need not be observable either. This is true whether the precision and demand parameters are chosen simultaneously or in 5

16 sequence. In our opinion, this approach in modeling information acquisition is appropriate for representing static trading situations e.g., sealed-bid auctions or dynamic trading situations in which traders do not learn each other's information acquisition decisions over time. Appendix A provides the denition of the equilibrium in the benchmark model and the following proposition characterizes the equilibrium precision of private signals in the benchmark model. The key simplication in the benchmark model is that each trader j holds xed the other traders' signal precisions and demand parameters when choosing his signal precision and demand parameters {τ ɛj, α j, β j, γ j }. We emphasize that holding xed other traders' parameters does not mean that a trader knows these parameters. Proposition 4 Equilibrium precision of private signals in the benchmark model. Consider the benchmark model. A necessary and sucient condition for optimality of informed trader j's signal precision τ ɛj, given the precision choice τ ɛi by the N other informed traders, is τ j ρ + λ j τ j = I τ ɛj, j =,..., N, 7 where λ I = [N γ I + Mγ U ], trader j's precision is N βi τ j = τ x + τ ɛj + N βi /τ ɛ I + n σ, z and where β I is the positive solution to the polynomial 3, and the remaining demand parameters satisfy the system of equations 0. A Nash equilibrium always exists and is characterized by a solution to 7 with τ ɛj = τ ɛi and β I, γ I, γ U satisfying the system of equations 3. 6

17 .4 Properties of the Model The nature of the equilibrium information is illustrated through numerical examples see Appendix A for the choice of parameter values followed by analytical results for parameters for which the model is solvable in closed-form. Figure shows the equilibrium information i.e., τ ɛ as a function of the number of traders in the market i.e., n in the perfectly competitive model, benchmark model, and strategic model. 4 All traders are ex ante identical and face the same information acquisition cost function i.e., N = n, M = 0, and I j. = I. for j =,..., n. Figure shows that imperfect competition reduces traders' incentives to acquire information. As we discussed earlier, imperfect competition generates price impact, which reduces not only the marginal benet of private information but also the precision of the price signal. As n increases, the optimal precision decreases in the perfect competition model. This is because for any τ ɛ, price precision is increasing in n, which reduces the marginal benet of private information the private and price signals are substitutes. In the strategic model, we get the opposite result: equilibrium precision is increasing in n. In the benchmark model, equilibrium precision is nonmonotonic: it is initially increasing as liquidity increases, and then it is decreasing as the substitution eect increases. Exacerbating the Adverse Selection Problem Figure shows that the strategic incentives have generally a negative eect on equilibrium information. Traders acquire less information in the strategic model in comparison to the benchmark model. However, the following result suggests that this nding does not always hold true. Proposition 5 Equilibrium information with one informed trader. Assume one riskneutral informed trader and M risk-neutral uninformed traders are in the market. The informed trader's optimal signal precision τ ɛ in the strategic model is the unique solution to 4 In addition to the imperfect competition model, we also consider a perfect competition model, in which each of the n traders ignores the eect of his demands on the price of the risky asset i.e., no price impact and ignores the eect of his precision on the other traders' demand strategies i.e., no strategic eects. The strategic and the benchmark approaches are equivalent in the competitive setting. Appendix B presents an equilibrium model in which this behavior is rational. 7

18 the following equation: σ z 8 τ x + τ ɛ 3/ τɛ M M τ x M + M µ z σz = I τ ɛ. 8 In the benchmark model, the informed trader's optimal signal precision τ ɛ is the unique solution to the following equation: σ z 8 τ x + τ ɛ 3/ τɛ M M τ x M = I τ ɛ. 9 The optimal precision is always higher in the strategic model than in the benchmark model, and the dierence is increasing in µ z. With a quadratic cost function I., the ratio of the optimal strategic precision to the benchmark precision is increasing in µ z/σ z and decreasing in M. Proposition 5 highlights an important nding of the paper that strategic eects do not always have an adverse eect on traders' incentives to acquire private information. The proof of the proposition shows that the informed trader's expected utility has the following form: U I = { µ x E [p 0I ] + V [x p 0I ] }, 0 4λ I τ x + τ ɛ where the informed trader's price impact is λ I = M = Mγ U M τ ɛ. τ x + τ ɛ τ x σz The price impact λ I is increasing in the informed trader's signal precision because of adverse selection: an increase in τ ɛ makes the price more informative, which diminishes the sensitivity of uninformed trades to the price, γ U. Dening x p 0I as the external return, the equation- 0 terms in the braces are the squared external risk premium, which measures the benets to trading, even without information the proof shows µ x E [p 0I ] = 4 µ x E [P ], and 8

19 the explained variance of the external return, 5 V [x p 0I ] τ x+τ ɛ, which measures the benet of the informed trader's private signal. In the benchmark model, the informed trader holds xed λ I and the distribution of p 0I when choosing τ ɛ. This results in the rst-order condition 4λ I τ x+τ ɛ = I τ ɛ, which is equivalent to 9. In the strategic model, the trader anticipates the eect of the choice τ ɛ on both λ I and the distribution of p 0I ; this results in the rst-order condition 8. This adverse-selection eect might suggest that strategic eects would diminish information acquisition; however, the proof shows that the risk premium term µ x E [p 0I ] and the total variance term V [x p 0I ] are both increasing in the informed trader's precision at a faster rate than the price impact, which accounts for the positive contribution of strategic eects to optimal precision. Adverse selection causes the informed trader's price impact parameter to increase with precision, and the larger price impact generates larger noise in the external return, as well as a higher equilibrium risk premium to compensate for the higher price impact. Paradoxically, the informed trader therefore benets from the more severe adverse selection problem i.e., the increase in the price impact parameter associated with the higher precision. This is because the loss from the higher price impact is more than oset by the benets from the increased risk premium, as well as the increased information value i.e., the explained variance. This phenomenon arises only with small risk aversion. For high risk aversion, the risk premium depends not only on the price impact but also on the conditional variance of the payo, which is declining in the precision and negatively inuences the risk premium. Also, higher risk aversion reduces trading aggressiveness, resulting in diminished liquidity by reducing price informativeness and therefore adverse selection. Zero-Precision Equilibrium Figure also shows that when the number of traders is small, the strategic incentives may lead to no information acquisition in equilibrium. Dene 5 The explained part of the variance is the total variance minus the residual variance, which is /τ I, where τ I = τ x + τ ɛ. 9

20 M = n the number of other traders, all with zero precision, in any trader's optimal precision problem. The following proposition characterizes a necessary condition for a zero precision equilibrium. Proposition 6. Suppose all traders have zero precision and common risk aversion coecient ρ. The marginal utility of precision for each trader is negative, that is, for all j, if and only if d dτ ɛj U τ ɛj < 0 τ ɛj =0 ρ 4 σz 4 M M 3 M M + ρ τ x σz M 3 µ z +τ x M M 3 + M M µ z M M + + M 3 + M + M σz M < 0 Corollary. The marginal utility of precision for each trader is negative at zero common precision for some strictly positive interval of ρ between the two positive roots of the polynomial on the left-hand side of if and only if M + M M 4 M + M 4 M M 3 + M M 3 + M < µ z. σz If either ρ or σ z is suciently large, and therefore the price signal is suciently uninformative, then there is no zero-precision equilibrium. An uninformative price implies that increasing precision always reduces the price impact, and the private information is not shared with other traders through the price. Also, if the absolute supply µ z is small enough, then there is no zero-precision equilibrium. Note that µ z aects equilibrium utility only through E [φ j ], which is essentially a risk premium eect see equation 39. An increase in precision reduces conditional uncertainty and reduces the absolute risk premium. When µ z is small enough, this disincentive to information acquisition in the strategic problem is also small. Finally, if M is suciently large, and therefore the price impact parameter is small, then there is no zero precision equilibrium. However, if the price is informative enough that the adverse selection eect is signicant, the trading benets of information 0

21 are not too large e.g., ρ not too small, and each trader is large i.e., M small, then the marginal benet of information in a zero information setting can be negative. In summary, imperfect competition generally has an adverse eect on private information acquisition for risk-averse traders and the eect can be so severe that all traders optimally stay uninformed. Moreover, surprisingly, under the assumptions of low risk aversion and a single informed trader, strategic incentives encourage the informed trader to exacerbate the adverse selection problem faced by the uninformed traders. The utility loss from the higher price impact is dominated by the gain from the higher expected return..5 Related Models The basic setup of our model of information acquisition is dierent from that in Grossman and Stiglitz 980. Our model is an imperfect competition one in which traders choose the precision of their signals, a continuous variable, and they observe heterogeneous signals regarding the risky asset payo. The model presented in Grossman and Stiglitz 980 is the standard perfect competition one with an innite number of traders, and each trader can spend a xed cost to observe a common signal of xed precision. Therefore, there is no heterogeneity among informed traders, and traders cannot choose the precision of the common signal. Kyle 989 and Subrahmanyam 99, among others, analyze information acquisition under imperfect competition. In these papers, traders can spend a xed cost to observe independent signals of xed precision Kyle 989 or a common signal of xed precision Subrahmanyam 99. Hence, these papers are silent about how, for instance, informed traders' incentives to acquire private information change as an uninformed trader decides to become privately informed. This is because the precision of the private signals is held constant. In addition, our paper provides results on complementarity in information acquisition, the eects of strategic incentives on equilibrium precision, the existence of zero information equilibria, and the implications of information acquisition for asset prices.

22 3 Information, Imperfect Competition, and Volatility This section discusses two novel ndings of the paper regarding information complementarity and excess volatility. Unless a distinction is needed, we refer to both the benchmark and strategic models as the imperfect competition model. 3. Complementarity in Information Acquisition The literature has analyzed mechanisms that can generate complementarity in information acquisition see, e.g., Admati and Peiderer 987; Hellwig and Veldkamp 009; Manzano and Vives 0; Garcia and Strobl 0; and Goldstein and Yang 05, among others. In the standard setting of Grossman and Stiglitz 980, there is never strategic complementarity in private information acquisition. This is because an increase in the fraction of informed traders increases the price informativeness and hence reduces traders' incentives to become informed. Manzano and Vives 0 extend the setting in Grossman and Stiglitz 980 and show that in the presence of multiple sources of asymmetric information, there can be strategic complementarity in information acquisition. However, equilibria that have strategic complementarity in information acquisition are unstable. Therefore, the strategic substitutability result in Grossman and Stiglitz 980 is robust. In our model, a trader observes only one private signal although the equilibrium price of the risky asset can be interpreted as a public signal. Private signals are the source of heterogeneous information among informed traders and of information asymmetry between informed and uninformed traders. We next study if and how imperfect competition generates complementarity in information acquisition. As is common, we dene complementarity and substitutability in our model in terms of the slope of the reaction function. 6 Our notion of substitutability is similar to that used for traders' demands in other contexts see, for example, Bulow et al In particular, for any trader j, denote by τ j τ i the optimal precision of trader j given trader i's precision τ i 6 Throughout, we set aside issues of uniqueness.

23 for notational simplicity, we omit the dependence on the other n traders' precisions. Complementarity and substitutability in private information acquisition are dened as follows. Denition Information substitutability/complementarity in the strategic model. Trader i's information is locally a substitute complement for trader j's from the perspective of trader j and at the precision vector τ j, T j, if τ j τ i / τ i < > 0. The standard result is that the above denition of substitutes complements in terms of the slope of the reaction function is equivalent to a denition in terms of the eect of an increase in τ i on trader j's marginal benet of precision. To see this, let v j denote trader j's certainty equivalent utility: v j = ρ j ln U j I τ j. The marginal benet of precision to trader j, given τ i, is MB j τ j, τ i = v j τ j. Because trader j's reaction function satises the necessary rst-order condition MB j τ j τ i, τ i = 0, totally dierentiating implies τ j τ i τ i = MB τ j τ j τ i, τ i i MB τ j τ j τ i, τ i, j where τ j MB j τ j τ i, τ i < 0 is necessary for an interior maximum in trader j's information acquisition problem. That is, τ jτ i τ i and τ i MB j τ j τ i, τ i always have the same sign. We next demonstrate complementarity and substitutability in our model using two approaches. In the rst approach, we assume that there are N = informed traders and show the reaction curve of the rst informed trader i.e., his optimal information as a function of the precision of the second trader. 7 Because the two informed traders are identical, equilibrium precision is at the intersection of the reaction function and the identity line. As Panel A in Figure 3 shows, there can be complementarity in the information acquisition decisions in the strategic model. In the second approach, we show the eect of an exogenous increase in the number of informed traders while keeping the total number of traders constant on the equilibrium 7 In this setting, there are M = 0 uniformed traders and the equilibrium information is strictly positive. 3

24 information. Let τ ɛ N, M denote the equilibrium information with N potentially informed and M uninformed traders. The variable τ ɛ N, 0 τ ɛ N, captures the change in equilibrium information for a marginal increase in the number of informed traders. Panel B in Figure 3 shows τ ɛ N, 0 τ ɛ N, for N = 3,..., 00 in the strategic model. The gure shows that there is no complementarity when the number of traders is suciently small, complementarity for an intermediate range of number of traders, and substitutability when the number of traders is suciently large. For the small-n region, when the uninformed trader switches to being informed, the equilibrium information does not change. This is because the equilibrium precision is zero in this range. Two factors can contribute to a negative marginal value of information when the common precision is zero: information acquisition by a single trader, when all others remain uninformed, causes the price impact to increase as a result of adverse selection see Lemma 6, and information leakage through the price causes an increase in the total precision of the uninformed traders, which reduces the risk premium see Proposition 7. In the intermediate-n range, there is complementarity in private information acquisition. Why? An increase in the number of informed traders has two eects: it potentially increases the informativeness of the price, and it reduces the price impact parameter i.e., it increases the liquidity of the risky asset. The second eect is absent in the perfect competition model and is the main driver of complementarity in information acquisition in the imperfect competition model. When a trader switches from being uninformed to informed, the price sensitivity of his demand increases. This is because the adverse selection eect diminishes as he no longer relies completely on the price as a signal, and because trading becomes more aggressive as perceived risk diminishes. This increased price sensitivity reduces the price impact parameter faced by other informed traders, which increases the marginal benet of precision. For the large-n region, there is strategic substitutability in private information acquisition. As N becomes large, the imperfect competition model converges to the perfect com- 4

25 petition model. Then there is strategic substitutability: as the number of informed traders increases, the price becomes more informative and the increased price informativeness reduces the marginal benet of the increased private signal precision. 3. Imperfect Competition and Excess Volatility We next study the implications of imperfect competition for the return volatility. To the best of our knowledge, our paper is the rst to study the eects of imperfect competition on the second moment of asset prices. Shiller 98 and LeRoy and Porter 98 develop a variance bound inequality between the volatility of the price and the volatility of fundamentals and show systematic violations of the inequality in the U.S. nancial market. The literature has provided theories to explain the excess volatility of asset prices. For instance, Dow and Werlang 99 and Ozsoylev and Werner 0 show that the presence of ambiguous information may lead to excess volatility. We show that illiquidity results in less equilibrium information acquisition, and a less informative equilibrium price can generate large return volatility. The following proposition characterizes the return volatility in the perfectly competitive and imperfect competition models. Proposition 7. Suppose N informed traders have the identical exogenous signal precision τ ɛ. In both the imperfect-competition and perfect-competition models, the equilibrium return variance is V [x P ] = N γ I β I + Mγ U σ x + N β I /τ ɛ I + σ z [Nγ I + Mγ U ], where β i, γ i, λ i, and τ i, for i {I, U}, are equilibrium variables that are characterized in Proposition. Panels A and B in Figure 4, respectively, show the return volatility for a low level of risk aversion ρ = 3 and a high level of risk aversion ρ = 7 in the competitive, benchmark, and strategic models as a function of the number of traders i.e., n. All traders are ex 5

26 ante identical and face the same information acquisition cost function i.e., N = n, M = 0, and I j. = I. for j =,..., n. We emphasize that in this gure, the precision of private signals for each model is endogenous. The gure shows that the return volatility decreases as the number of traders increases. In the imperfect competition model, the reduction in return volatility is mainly driven by the increase in liquidity and the increase in price precision as the number of informed traders increases. A comparison between Panels A and B shows that the return volatility is an increasing function of risk aversion. Figure 4 also shows that perfect competition does not generate excess return volatility i.e., return volatility in excess of payo volatility after controlling for the return volatility arising from the variance of the supply of the risky asset, whereas imperfect competition can. More important, the adverse eects of strategic incentives in information production in the strategic model can create substantial volatility in the return of the risky asset. The following proposition establishes that for suciently large precision of private signals, the return variance is decreasing in the precision of private signals. This implies that the return volatility in the strategic model is larger than the one in the benchmark model. There is considerable slack in the proof, and therefore the condition τ ɛ > ˆτ ɛ is by no means necessary. Equation 3 gives the return variance for zero precision and shows that excess variance in this case is generated by the supply uncertainty. Proposition 8. In the symmetric model, in which all n traders have the identical exogenous signal precision τ ɛ, the return variance, V[x P ], is strictly decreasing in τ ɛ for all τ ɛ satisfying τ ɛ > ˆτ ɛ, where ˆτ ɛ is dened in 63. Further, τ ɛ = 0 = V [x P ] = + σ z n ρ, 3 τ x τx n τ ɛ = V [x P ] 0. In summary, our model implies that episodes of signicant illiquidity and volatility could be accompanied by low production of private information and hence low information asym- 6

27 metry among traders. This is an important nding given that many of the biggest movements in the stock marketmost notably the crash of 987have not been accompanied by any major news Cutler et al In a similar vein, our model suggests that in imperfect competition settings, one may observe excess volatility and the adverse eects of imperfect competition on private information production is the source of the excess volatility. Consistent with this prediction, Bao and Pan 03 document that corporate bond and CDS returns are excessively volatile and illiquidity and not fundamentals is the source of the excess volatility. Although Bao and Pan 03 suggest that time varying illiquidity is the source of the excess volatility, our model goes one step further and suggests that the distortion in private information production is the main driver of the illiquidity and hence the excess volatility. 3.3 Empirical Implications In this section, we discuss the empirical implications of our model and link the existing empirical literature to the predictions of the model. Degree of Imperfect Competition Does the withdrawal of one informed trader, for instance, a hedge fund, from trading a security in an imperfect competition setting increase or decrease price volatility? One can argue that as the informed trader withdraws from trading, resulting in a higher proportion of uninformed traders, the adverse selection problem will diminish, resulting in increased liquidity and reduced price volatility. Our model, however, suggests that in imperfect competition settings, even the withdrawal of one informed trader can substantially increase price volatility. This is because the withdrawal of the informed trader reduces the incentive to produce private information which in turn makes the market less liquid. The reduction in private information production and the increase in illiquidity of the asset, therefore, can cause substantial price volatility. Our model also predicts that the degree of market competition, measured for instance 7