A Model of Portfolio Delegation and Strategic Trading

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1 A Model of Portfolio Delegation and Strategic Trading Albert S. Kyle, Hui Ou-Yang, Bin Wei This Version: December 2008 Abstract This paper endogenizes information acquisition and portfolio delegation in a one-period strategic trading model. Linear equilibriums composed of linear prices, demands, and contracts are considered. We find that when the informed portfolio manager is relatively risk tolerant (averse), the price informativeness increases (decreases) with the hedging demand of the uninformed hedgers. Our results differ from those obtained in the traditional market microstructure literature in which the price informativeness is independent of or decreases with the amount of noise trading. Incentive contract in our model induces the manager to exert effort in information acquisition as well as influences the manager s trading intensity. A lower incentive contract may make the manager effectively less risk averse; consequently, the manager may exert more effort in acquiring information so as to trade more aggressively in the stock. Our results highlight the importance of developing an integrated model of portfolio delegation and strategic trading. Keywords: Portfolio Delegation, Information Acquisition, Strategic Trading, Price Informativeness JEL Classifications: G14, G12, G11 Albert S. Kyle is at Robert H. Smith School of Business, University of Maryland. Phone: (1) , Hui Ou-Yang is at Fixed Income Division, Nomura International Limited, Hong Kong. Phone: Bin Wei is at Zicklin School of Business, Baruch College, the City University of New York. Phone: (1) , The authors would like to thank Archishman Chakraborty, Nengjiu Ju, Mike Lemmon, Lin Peng, Bob Schwartz, and seminar participants at Baruch and HKUST for their comments. 1

2 1 Introduction Strategic trading models typically assume that private information is exogenously given and that agents trade for their own accounts. 1 A key result of this literature is that when traders are risk averse, the equilibrium asset price is less informative and more volatile when noise trading increases. On the other hand, a strategic trading equilibrium that involves private information is typically absent in the traditional principal-agent literature. 2 This literature considers incentives for effort, and a fundamental result is that a higher incentive contract induces the agent to exert a higher level of effort. Allen (2001) highlights the importance of developing an integrated model of asset pricing and agency. In this paper, we aim to develop such a model. In particular, we construct a linear equilibrium model in which asset prices, optimal contracts, and information acquisition are determined simultaneously. We then examine whether some of the key results obtained separately under strategic trading and moral hazard still hold in the integrated model. In our model, there is an uninformed risk-neutral investor, a risk-averse informed trader who serves as a portfolio manager, competitive risk-neutral market makers, and risk-averse uninformed hedgers or noise traders. There is one risky stock and one risk-free bond available for trading. The uninformed investor (the principal) entrusts her money to the informed manager (the agent), and the manager trades on behalf of the uninformed investor. 3 The manager has skill at acquiring private information about the stock s liquidation value, and bases his trades on the acquired information. The manager s trades affecttheassetpriceas market makers take into account the adverse selection effect in the determination of the asset price. At the end of one period, the asset s liquidation value is realized and trading profits are then determined. The manager is compensated according to a contract designed by the 1 See, for example, Glosten and Milgrom (1985), Kyle (1985), Easley and O Hara (1987), Admati and Pfleiderer (1988), Back (1992), Holden and Subrahmanyam (1992), Spiegel and Subrahmanyam (1992), Holmstrom and Tirole (1993), Foster and Viswanathan (1996), Back, Cao, and Willard (2000), and Vayanos (2001). 2 See, for example, Ross (1973), Mirrlees (1976), Harris and Raviv (1979), Holmstrom (1979), Grossman and Hart (1983), Holmstrom and Milgrom (1987), Schattler and Sung (1993), DeMarzo and Urosevic (2001), Prendergast (2002), Ou-Yang (2005), Cvitanic, Wan, and Zhang (2006), Sannikov (2007), He (2008), and Ju and Wan (2008). 3 We shall use principal (agent) and investor (manager or informed agent) interchangeably. 2

3 investor in the beginning of the period. Following Spiegel and Subrahmanyam (1992), we assume that the risk-averse uninformed noise traders hedge their endowment risk optimally. Moral hazard arises because acquiring information is costly to the manager and the effort spent for information by the manager is unobservable to the investor. We require the contract to be a linear function of trading profits. The investor is a Stackelberg leader in the sense that after she announces the contract, other market participants optimize their strategies accordingly. Specifically, given the investor s contract, the manager first chooses an effort level on information acquisition and then decides on the optimal portfolio allocation. The uninformed hedgers solve for their optimal demand for stock to hedge their endowment risk. As in Kyle (1985) and Spiegel and Subrahmanyam (1992), the competitive, risk-neutral, marker makers determine the equilibrium stock price, based on the total demand by the informed manager and uninformed hedgers. Consequently, as the Stackelberg leader, the investor takes the responses of other players into account when determining optimal contracts. Two effects determine the price informativeness. An increase in noise trading decreases the price informativeness, but an increase in informed trading (due to an increase in noise trading) increases it. Because most strategic trading models assume that the noise traders demand for stock is exogenously given, we first solve our integrated model for this special case. In the absence of risk aversion, portfolio delegation, and information acquisition, Kyle (1985) shows that the stock price informativeness, measured by the precision of the asset s liquidation value conditioning on the equilibrium asset price, is independent of the variance of noise trading. Subrahmanyam (1991) finds that when the informed agent is risk averse, an increase in the variance of noise trading decreases the price informativeness, because a risk-averse informed agent trades less aggressively than a risk-neutral one. Under endogenous information acquisition and portfolio delegation, we find that when the risk aversion (R a ) of the informed agent is relatively low (high), an increase in the variance of noise trading increases (decreases) the price informativeness. The intuition is as follows. When noise trading becomes more volatile, it is more difficult for market makers to distinguish between noise trading and informed trading, so private 3

4 information becomes more valuable. To induce the informed agent to trade more aggressively against noise traders, the investor designs a lower incentive contract for the agent to share a smaller percentage of the trading profits, which can make the agent effectively less risk averse. As a result, the agent is more incentivized to acquire more accurate private information about the stock payoff, so he can trade more aggressively in the stock. Because the informed agent has exponential utility function and all random variables are normally distributed, the marginal benefits of trading more aggressively and acquiring more accurate information decrease with R a. When noise trading increases, a relatively less risk averse agent first acquires more accurate information and then trades more aggressively in the stock. The second effect dominates the first one, leading to a more informative price in equilibrium. When the informed agent is very risk averse, the increases in his effort for information acquisition and trading aggressiveness are dominated by the increase in noise trading, resulting in a less informative price in equilibrium. When noise trading is endogenized, as in Spiegel and Subrahmanyam (1992), there are three factors that affect asset pricing and optimal contracting in equilibrium, that is, the number, the endowment risk, and the risk aversion of the noise traders or uninformed hedgers. Optimal contracting and informed trading are affected by the trading behavior of the uninformed hedgers. For example, when the number of uninformed hedgers increases, private information becomes more valuable. Hence, the principal designs a lower incentive contract to induce the informed agent to trade more aggressively against the hedgers. The principal, however, cannot reduce the incentive too much, because doing so would discourage the uninformed hedgers from trading, which would in turn lower the trading profits of the informed agent. As a result, if the informed agent is very risk averse (tolerant), the price informativeness declines (increases) when the demand of uninformed hedgers increases. In general, in anticipation of the uninformed hedgers responses, the principal designs a contract so that the informed manager trades less aggressively than in the case of exogenous noise trading. The optimal contract depends critically on the properties of the uninformed hedgers. When the number of hedgers increases, the aggregate hedging demand of the uninformed 4

5 hedgers increases. The principal then assigns a lower incentive contract to induce the informed agent to trade more aggressively. When the risk aversion of the hedgers increases, the hedgers are more concerned with their endowment risk, so they would like to trade more in the stock to hedge their risk. But trading more against the informed agent would increase their trading losses. When the informed agent is very risk tolerant, an increasingly risk-averse hedger is less willing to trade with a very aggressive informed agent, which would then reduce the trading profit of the informed agent. To alleviate the concern of the uninformed hedgers, the principal increases the incentive to make the informed agent effectively more risk averse, or the incentive increases with the risk aversion of the hedgers. When the informed agent is very risk averse, uninformed hedgers are not much concerned with informed trading, the principal reduces the incentive to induce the informed agent to trade more aggressively. Hence, the incentive decreases with the risk aversion of the hedgers in this case. When the endowment risk of the uninformed hedgers increases, the optimal contract exhibits a bell-shaped pattern when the informed agent is very risk tolerant and decreases when the informed agent is very risk averse. The intuition is as follows. When the informed agent is very risk tolerant, for a very small endowment risk, the uninformed hedgers are less willing to trade. To induce the hedgers to trade more, the principal increases incentive to prevent the informed agent from trading too aggressively. As the endowment risk of the hedgers increases to a certain level, the uninformed hedgers must trade anyway. Hence, the principal reduces incentive to induce the informed agent to acquire more accurate information and then trade more aggressively, resulting in a bell-shaped relationship between incentive and endowment risk. When the informed agent is very risk averse, the uninformed hedgers are not much concerned with the informed trading, so the principal reduces incentive to encourage more informed trading. Reducing incentives to induce both higher effort and more aggressive trading, as obtained in this paper, is not the usual result of the contracting literature. On one hand, given a higher share of the profit (higher incentive), an agent would like to work harder and trade more aggressively, trying to push up the profit. On the other hand, a higher share of the 5

6 profit makes the agent more risk averse, so he might be averse to actions that increase the risk of the profit. In a traditional agency model, the agent s effort increases the mean but does not affect the risk of the output. When we compute the agent s expected utility, the increase in the mean due to a higher effort dominates the increase in the risk due to a higher share of the risky output. Hence, a higher incentive always leads to a higher effort level in equilibrium. In our model, however, when the agent exerts a higher level of effort and trades more aggressively, he increases both the mean and the risk of the trading profit. A lower incentive contract makes the agent less risk averse, so the agent can increase his expected utility by exerting a higher level of effort and trading more aggressively for a higher profit. Although the agent s share of profits is lower due to a lower incentive, his total compensation can still increase because the total profit increases (in the presence of more noise trading). In addition, the agent can afford to bear more risk under a lower incentive fee. In net, the agent s expected utility may increase. 4 We now briefly review the related literature. Our paper is closely related to Kyle (1985), Subrahmanyam (1991), and Spiegel and Subrahmanyam (1992). Kyle develops a multiperiod model of strategic trading with a risk-neutral informed trader. Subrahmanyam and Spiegel and Subrahmanyam extend the one-period version of the Kyle model by introducing a risk-averse informed trader and by endogenizing noise trading, respectively. 5 Portfolio delegation is absent in these models. Dow and Gorton (1997) construct an equilibrium model with strategic trading and portfolio delegation. The portfolio manager is risk neutral, who may or may not receive a valuable signal about the asset payoff. Thereisnoeffort for the signal but there is still an agency problem. When the manager does not receive a valuable signal, no trading would be optimal for the manager and the principal, but the manager would still trade like a noise trader. In their model, inactivity is not rewarded because that would induce shirking by 4 In the absence of asset pricing and information acquisition, Guo and Ou-Yang (2006) demonstrate that if the agent s effort affects both the mean and the risk of an output process, a higher incentive does not necessarily lead to a higher level of effort from the agent. 5 Kyle (1981) considers endogenous noise trading in a different model. Mendelson and Tunca (2004) extend the endogenous noise trading models of Glosten (1989) and Spiegel and Subrahmanyam (1992) to multiple periods as well as endogenize information acquisition. 6

7 talented managers and would attract incompetent managers. Dow and Gorton illustrate that noise trading may be Pareto-improving, because it reduces the cost of hedging and results in larger hedging demands, which in turn can support a larger amount of investment by an informed fund manager and makes the manager better off. Notice that the agency problem in our model is due to a combination of the risk aversion of the manager, the cost for unobservable effort, and portfolio delegation. The manager always receives a valuable signal commensurate with his effort. Stoughton (1993) and Admati and Pfleiderer (1997) study an agency problem in a competitive partial equilibrium model in which the portfolio manager is a price-taker. 6 They obtain a striking result, that is, the manager s effort is independent of the slope of the linear contract. 7 Note that the manager s portfolio choice is undertaken after the effort is expended. When the stock price is given in a partial equilibrium model, the manager has an incentive to undo the incentive effect of the linear contract. In our strategic model, however, the market impact of the manager s trades on the asset price prevents the informed manager from undoing the incentive effect. As a result, different linear sharing rules induce different levels of effort. The rest of this paper is organized as follows. Section 2 first reviews briefly the Spiegel- Subrahmanyam (1992) model and then extends the model to incorporate portfolio delegation and information acquisition. Section 3 presents the main results on the stock price informativeness and optimal contracting. Section 4 concludes the paper. All the proofs are given in the Appendix. 6 For other competitive partial equilibrium models with information acquisition and portfolio delegation, see, for example, Ding, Gervais, and Kyle (2008) and Garcia and Vanden (2008). 7 For research on optimal contracting in delegated portfolio management, see, for example, Ross (1974), Bhattacharya and Pfleiderer (1985), Kihlstrom (1988), Allen (1990), Ou-Yang (2003), Dybvig, Fransworth, and Carpenter (2004), Cadenillas, Cvitanic, and Zapatero (2007), and Li and Tiwari (2008). 7

8 2 The Model 2.1 A Recap of Spiegel and Subrahmanyam (1992) In the Spiegel-Subrahmanyam (1992) model, there are multiple risk-neutral informed traders, a number of noise traders or uninformed risk-averse hedgers, and many risk-neutral competitive market makers. The uninformed hedgers maximize their expected utilities to hedge their endowment risk, whereas in the Kyle model, the noise traders demand for the stock is exogenously assumed to be a normal variable. buy and sell a single asset at a price ep at time 0. 8 The informed and uninformed traders At time 1, the liquidation value of the asset, eν N(v, σ 2 ), is announced, and the holders of the asset are paid. The asset price, determined by the competitive market makers who earn zero expected profit, is set to equal the expectation of the liquidation value. The informed trader obtains a noisy signal about the asset value e θ = eν + e, where e N(0,σ 2 ) is uncorrelated with eν. The informed trader thus bases his trade on the private information e θ, and his order, denoted by ex, is a function of e θ. There are m risk-averse uninformed hedgers, who have negative exponential utility with a common risk-aversion coefficient R h. Each risk-averse hedger j has an endowment ez j N(0,σ 2 z) of the asset, and his demand or order for the stock is a function of ez j, denoted by eu j. The sum of the hedgers orders is denoted by eu = P m j=1 eu j. Spiegel and Subrahmanyam construct a linear equilibrium in which the optimal strategies for the informed trader and the uninformed hedgers are linear, given by ex = β( e θ v) and eu j = γez j, respectively. The market makers observe only the total order flow ey = ex + eu and set the stock price to be ep = P (ex + eu) =v + λ(ex + eu). For convenience of comparison, we next report the main results of Spiegel and Subrahmanyam when there is only one risk-neutral informed trader. Proposition 1 If R 2 h mσ2 z σ 2 +2σ 2 2 > 4 σ 2 + σ 2, then the unique linear equilibrium is 8 Throughout this paper, a letter with the tilde symbol (e.g. p) denotes a random variable, and the letter itself (e.g. p) denotes the realization of the random variable. 8

9 given by λ = 4 p σ 2 + σ 2 R h σ 2 (2m 1) /mσ 2 +4σ 2 h R h m 1/2 σ z (σ 2 +2σ 2 ) 2 p σ 2 + σ 2 i, (1) h 2 R h m 1/2 p i σ z σ 2 +2σ 2 2 σ 2 + σ 2 β = p, (2) R h σ 2 + σ 2 [(2m 1) /mσ 2 +4σ 2 ] h 2 R h m 1/2 p i σ z σ 2 +2σ 2 2 σ 2 + σ 2 γ = R h mσz [(2m 1) /mσ 2 +4σ 2. (3) ] Moreover, the stock price informativeness Q is given by Q =[Var(ev P )] 1 = β2 σ 2 + σ 2 + mγ 2 σ 2 z σ 2 β 2 = 1 σ 2 + mγ 2 σ 2 z σ σ 2 +2σ 2. (4) Proof. See the proof of Proposition 1 in Spiegel and Subrahmanyam (1992). Note that when a hedger is more risk averse (i.e., R h is larger) or his endowment is more volatile (i.e., σ z is larger), his hedging demand is higher (i.e., γ and Var(u j ) are both larger). In an extreme case in which there is one hedger (i.e., m =1), who is infinitely risk averse (i.e., R h = ), we have γ = 2σ2 +4σ 2. This case corresponds to the Kyle (1985) σ 2 +4σ 2 model with exogenous noise trading when we let eu = P m j=1 eu j N(0,σ 2 u) with σ 2 u = γ 2 σ 2 z. From Equation (4), the price informativeness Q is independent of m, R h,andσ z,the three variables that affect the demand of the uniformed hedgers. Because the risk-averse hedgers are uninformed about the stock payoff, an increase in the uninformed hedging demand decreases Q. On the other hand, when the uninformed hedging demand increases, the informed trader will increase his demand to take advantage of the uninformed trading, which increases Q. When the informed trader is risk neutral, the two effects offset each other exactly, so that Q is independent of m, R h,andσ z. 2.2 Endogenous Information Acquisition and Portfolio Delegation In Kyle (1985) and Spiegel and Subrahmanyam (1992), the informed trader s information is exogenously assumed and he trades for his own account. We assume that the informed trader can decide on the extent to which he is informed through an endogenous information 9

10 acquisition process. In particular, upon input of an amount of effort ρ, the agent receives a noisy signal in which the variance of the noise e is inversely related to effort ρ. That is, σ 2 = σ 2 /ρ. The cost of exerting effort ρ is assumed to be C(ρ) =kρ 2 /2, wherek is a positive constant. This mechanism of information acquisition and the cost function for effort have been adopted by Verrecchia (1982), Stoughton (1993), Admati and Pfleiderer (1997), and Mendelson and Tunca (2004). Notice that this process of information acquisition is a generalization of that initiated by Grossman and Stiglitz (1980), Diamond (1985), and many others in which the agent pays a constant fee for a signal that is not related to the precision of the signal. We further assume that the informed trader sells his private information in the form of a fund in which a representative uninformed, risk-neutral, outside investor (principal) entrusts her money to the informed trader who serves as the fund manager (agent). The principal designs an optimal linear sharing rule, denoted by S( f W )=a + b f W, to induce the agent to exert effort both for information acquisition and for subsequent trading in the stock, where f W denotes the agent s trading profits. Moral hazard arises due to the inability of the principal to observe effort. We assume that the agent has a negative exponential utility function: U A ( W f ; ρ, a, b) 1 ³ exp h R a S( R f i W ) C (ρ), a where R a is the informed agent s risk aversion coefficient. As in Spiegel and Subrahmanyam (1992), all uninformed hedgers are assumed to have negative exponential utility functions with an identical risk aversion coefficient R h. For hedger j, given his endowment z j,his payoff V j as a function of u j is given by ev j (u j ; z j )=eν (u j + z j ) u j ep, and his utility is given by U H ( V e j ; z j ) 1 i exp h R hvj e, R h 10

11 where z j is the realization of his endowment ez j N(0,σ 2 z). Timeline 1. In Stage 1, the principal assigns a linear contract S = a + bw f to the agent, where fw denotes the agent s trading profits, and a and b are constants. 9 The contract is publicly announced. 2. In Stage 2, the market makers and uninformed hedgers believe that under the contract S = a + bw f, the agent would exert effort ρ m (b) that depends on b. 10 Moreover, they are committed to this belief, which turns out to be correct in equilibrium (i.e., they have rational expectations). 3. In Stage 3, under the contract and the belief ρ = ρ m (b), the informed agent exerts effort ρ = RHO (b, ρ m ) and observes a signal e θ.hererho (b, ρ m ) denotes the optimal effort, and e θ is a noisy signal about the liquidation value, i.e., e θ = eν + e, inwhich the noise component e N(0,σ 2 ) is uncorrelated with eν, and its variance is inversely related to the agent s effort, i.e., σ 2 = σ 2 /ρ. 4. In Stage 4, the informed agent chooses the optimal trading strategy and submits his order ex = X (θ; ρ, ρ m,b) to market makers. Simultaneously, uninformed hedger j chooses his optimal trading strategy and submits order eu j = U j (z j ; ρ m,b) to market makers, j =1,,m. Following Spiegel and Subrahmanyam (1992), we assume that all hedgers are identical but their initial endowments are independently distributed. Therefore, symmetric equilibrium trading strategies exist where they have identical equilibrium trading strategies: U j ( ; ρ m,b) U ( ; ρ m,b), j. 5. In Stage 5, the risk-neutral competitive market makers determine the stock price ep = P (y; ρ m,b) basedonthetotalorderflow y and their belief about the informed agent s effort ρ m. 9 If the informed agent designs the linear contract or has the bargaining power, the incentive part b of the contract would remain the same and the constant part a would be determined so that the principal s expected net profit would be zero. 10 Note that the constant payment a in the contract does not affect the agent s effort ρ because there is no wealth effect since the agent has an exponential utility function. 11

12 6. In Stage 6, the liquidation value eν is realized. The principal and the informed agent are compensated. We solve the model backward. Step 1: In stage 5, market makers set the stock price to earn zero expected profit. Given the total order flow y = x + u and the linear contract S( f W )=a + b f W, they set the price based on their beliefs about the agent s effort: h P (y; ρ m,b)=e eν y = X (θ; ρ, ρ m,b)+ P i m j=1 U (z j; ρ m,b),ρ= ρ m. Step 2: In stage 4, the informed agent and uninformed hedgers solve for their optimal trading strategies. After having exerted effort ρ and obtained signal e θ = θ, theinformed agent s expected utility is given by h i U A (x; θ, ρ, ρ m,b)=e U A (fw ; ρ, a, b) eθ = θ, where W f h ³ = x (eν P (y; ρ m,b)) = x eν P x + P i m j=1 U (ez j; ρ m,b);ρ m,b. The informed agent s optimal trading strategy maximizes his expected utility U A,thatis, X (θ; ρ, ρ m,b) = arg max x U A (x; θ, ρ, ρ m,b). Similarly, uninformed hedger j s optimal trading strategy u j = U (z j ; ρ m,b) maximizes his expected utility, which is given by h U H (u j ; z j,ρ m,b)=e U H ( V e i j ; z j,ρ m,b). Therefore, we have U (z j ; ρ m,b) = arg max u j U H (u j ; z j,ρ m,b). Step 3: To determine the agent s optimal effort, we solve the game in stage 3. The 12

13 agent s expected utility, before exerting effort ρ and obtaining signal θ, is h ³ U A (ρ; b, ρ m ) E θ U A ³X eθ; ρ, ρm,b ; e i θ, ρ, ρ m,b. Thus, the agent s optimal effort satisfies: RHO (ρ m,b) = arg max U A (ρ; ρ ρ m,b). Step 4: In stage 2, market makers and hedgers form correct beliefs about the agent s effort choice or they all form rational expectations. informed agent s optimal effort choice is That is, for a given contract, the ρ (ρ m,b)=rho (ρ m,b). Due to rational expectations, the following must hold in equilibrium ρ m (b) =ρ (ρ m,b). Step 5: In the final step, we solve for the optimal contract, which is designed by the principal in stage 1. The principal is risk neutral and her expected utility is denoted by: U P (a, b) E[ W f S( W f h )] = E (1 b) W f i a. The optimal contract then maximizes U P (a, b): (a,b )=argmax (a,b) U P (a, b), subject to various constraints to be specified next. We formally define the equilibrium as follows. Definition 1 An equilibrium consists of an optimal contract (a,b ), rational prior belief ρ = ρ m (b), an optimal effort choice ρ (ρ m,b) = RHO (ρ m,b), optimal trading strategies x (θ; ρ, ρ m,b)=x (θ; ρ, ρ m,b), u (z j ; ρ m,b)=u (z j ; ρ m,b), and an optimal pricing function: p (y; ρ m,b)=p (y; ρ m,b). The optimal contract (a,b ) maximizes the princi- 13

14 pal s expected utility: subject to the following constraints: (a,b )=argmax (a,b) U P (a, b), (5) ρ (b, ρ m )=argmaxu A (ρ; b, ρ ρ m ), (6a) x (θ; ρ, ρ m,b)=argmax A (x; θ, ρ, ρ x m,b), (6b) u (z j ; ρ m,b) = arg max U H (u j ; z j,ρ u m,b), j (6c) U A (ρ ; ρ m,b )=U, (6d) h p (y; ρ m,b)=e eν ρ m (b) =ρ (ρ m,b), (6e) y = x (θ; ρ, ρ m,b)+ P i m j=1 u (z j ; ρ m,b),ρ= ρ m. (6f) In Definition 1, Equation (5) determines the optimal contract, subject to the incentive compatibility constraints in Equations (6a c), the individual participation constraint in Equation (6d), the rational expectations constraint in Equation (6e), and the market efficiency constraint in Equation (6f). We determine the optimal linear contract by solving simultaneously the optimization problems of the principal, the agent, and the uninformed hedgers. Given the principal s contract, the agent s problem is to choose an effort level on information acquisition and then decides on an optimal portfolio allocation. The principal designs an optimal contract taking into account the responses from the informed agent, the uninformed hedgers, as well as the market makers. Proposition 2 below characterizes explicitly the equilibrium. Proposition 2 In Stage 5, given a linear contract (a, b) and the belief ρ m (b), marketmakers believe that the informed agent s strategy is X (θ; ρ = ρ m,ρ m,b)=β m (ρ m,b)(θ v) and uninformed hedger j s trading strategy is U (z j ; ρ m,b)=γ m (ρ m,b) z j. Consequently, market makers set the pricing rule as P (y; ρ m,b) v + λ m (ρ m,b) y, wherey is the total 14

15 order flow and λ m (ρ m,b)= β m (ρ m,b) [β m (ρ m,b)] 2 (1 + 1/ρ m )+m [γ m (ρ m,b)] 2 σ 2 z/σ 2. (7) Here λ m, β m,and γ m are functions of ρ m and b, andthesubscriptm indicates that these variables are determined under the belief ρ m. In Stage 4, the informed agent s optimal trading strategy is shown as X (θ; ρ, ρ m,b)= β (ρ; ρ m,b)(θ v) with the trading intensity β given by β (ρ; ρ m,b)= ρ/ (1 + ρ) 2λ m (ρ m,b)+r a b[σ 2 /(1 + ρ)+m (λ m (ρ m,b) γ m (ρ m,b)) 2 σ 2 z], (8) where ρ is the agent s effort chosen in Stage 3. U (z j ; ρ m,b)=γ (ρ m,b) z j,whereγ (ρ m,b) is given by Hedger j s optimal hedging demand is γ R h (1 λ m β (ρ m,b)= m ) σ 2 i. (9) 2λ m + R h h(1 λ m β m ) 2 σ 2 +(λ m β m ) 2 σ 2 /ρ m +(m 1) (λ m γ) 2 σ 2 z For simplicity, the arguments ρ m and b in β m and λ m are omitted in the above equation. In Stage 3, the agent s optimal effort ρ (ρ m,b)=rho (ρ m,b) satisfies a first order condition: C ρ (ρ (ρ m,b)) = bσ2 2 1 R a bσ 2 β (ρ; ρ m,b)+1 dβ (ρ; ρ m,b). (10) dρ which can be simplified to the following cubic equation: ρ (ρ +1) 2λ m + R a bmγ 2 mσ 2 zλ 2 m (ρ +1)+Ra bσ 2 = b 2k σ2. (11) In Stage 2, market makers, uninformed hedgers, and the informed agent all have rational expectations by correctly anticipating the effort choice. That is, ρ m (b) = ρ (ρ m (b),b), β m (ρ m,b)=β (ρ ; ρ m,b), (12) γ m (ρ m,b) = γ (ρ m,b), λ m (ρ m,b)=λ. (13) 15

16 Note that the optimal responses are all functions of b, hence we denote them by ρ (b), β (b), γ (b), andλ (b). In Stage 1, the optimal contract is determined through the following optimization: max (1 b) β (b) σ 2 [1 λ (b) β (b)(1+1/ρ (b))] a (b) ª, (14) b where a (b) is chosen to satisfy the participation constraint as follows: a (b) = R 1 a log UR a k [ρ (b)] 2 1 2R a log R a bβ (b)σ (15) Themainideaistosolveforoptimalb through Equation (14) in which other endogenous variables β (b),λ (b),γ (b),ρ (b) are determined as functions of b from Equations (7) (9) and (11). To achieve numerical stability, we solve for β (b),λ (b),γ (b),ρ (b) as functions of b in two steps. We first solve for β,λ,γ (as functions of ρ and b) from Equations (7) (9), keeping ρ and b fixed. We then use Equation (11) to find optimal ρ as a function of b for any given b [0, 1]. Solving the system of three nonlinear equations (7) (9) simultaneously for β,λ,γ is usually very sensitive to the choice of initialvalues. Werelyonaniterationbased algorithm for stable solutions. Equation (16), derived below from Equations (7) and (8), is especially useful in the algorithm. h i R a bσ 2 (βλ) (βλ) 2 (1 + ρ)/ρ (βλ) 1/(1 + ρ) = λ [2 (βλ) ρ/ (1 + ρ)]. (16) It contains (βλ) and λ only. Moreover, from Equation (7), βλ is bounded above by ³ ρ/ (1 + ρ). Therefore, we first randomly select an initial value of (βλ) (0) 0, and then calculate λ (0) from Equation (16), which immediately gives us β (0). Next, we plug in the values of β (0) and λ (0) into Equation (9), and calculate γ (0). Substituting λ (0) and γ (0) into Equation (8), we update the value of β (0) to β (1).Ifβ (1) is equal to β (0),thenweare done; otherwise, repeat the previous steps until β (0),β (1), converge. ρ 1+ρ 16

17 3 Main Results 3.1 The Case of Exogenous Noise Trading Becausethevastmajorityofthemarketmicrostructureliteraturetreatsnoisetradingasexogenous, we study this special case in this subsection. Differing from our general model, the uninformed hedgers aggregate demand for the stock is exogenously given by eu N(0,σ 2 u). We shall examine the impact of the trading behaviors of the informed agent and noise traders on the price informativeness and the optimal contract. Price Informativeness As in Kyle (1985), we define the price informativeness as the posterior precision of ev conditional on the equilibrium price: Q =[Var(ev P )] 1 = 1 σ σ 2 + σ 2 u/β 2 = 1 σ σ 2 /ρ + σ 2 u/β 2. (17) In the Kyle model in which the informed agent is risk neutral, the price informativeness Q is independent of the variance of noise trading σ u. When the informed agent is risk averse, Subrahmanyam (1991) finds that Q decreases with σ u because the risk-averse trader responds less aggressively to an increase in σ u. Under information acquisition and portfolio delegation, however, the relationship between Q and σ u depends on the risk aversion of the informed agent R a.whenr a is very low (high), Q generally increases (decreases) with σ u. To build intuition, we first consider the case with a risk-neutral informed trader. We can show that the optimal sharing rule is for the principal to sell the entire fund to the informed agent, that is, b =1. There is no moral hazard in this case. Hence, the problem of information acquisition and portfolio delegation collapses into that of information acquisition alone. The next proposition summarizes the results. Proposition 3 In equilibrium, the trading intensity (β), market impact measure (λ), and price informativeness (Q) all increase with effort ρ: β = σ r u ρ σ 1+ρ, λ = 1 2 r σ ρ 1+ρ, Q = 2 1+ρ σ 2 2+ρ, (18) σ u 17

18 where the optimal effort is given by ρ = q 1+4(σσ u / (2k)) 2/3 1. (19) 2 This proposition shows that with endogenous information acquisition, a higher σ u actually increases Q. Two competing effects determine Q. An increase in σ u decreases Q, whereas an increase in the trading by the informed agent increases it. In Kyle (1985), when σ u increases, the informed agent, whose information is exogenously given (independent of noise trading), increases his trading as well. It turns out that in the Kyle model, the two effects exactly offset each other, so that Q is independent of σ u. In our model, the informed trader s effort ρ increases with σ u as shown in Equation (19). A higher ρ leads to more accurate information, and subsequently more aggressive trading by the informed agent. As a result, more private information is incorporated into the stock price than in the Kyle model, so the price informativeness increases with noise trading. Proposition 3 demonstrates that to arrive at robust results in a strategic trading model, it is crucial to introduce endogenous information acquisition. We next extend the model to include a risk-averse informed trader as well as portfolio delegation. Proposition 4 Without portfolio delegation, if R a is small, Q increases monotonically with σ 2 u;ifr a is large, Q decreases monotonically with σ 2 u; for an intermediate value of R a, Q increases initially and then decreases with σ 2 u. According to the (black) dash-dotted lines in Subplots A4 A6 in Figure 1, when the agent s risk aversion coefficient R a is around 0.1, Q increases monotonically with σ 2 u,whereas when R a is around 1, it decreases monotonically. According to unreported results, the relationship exhibits a bell-shape when R a is between 0.1 and 1. In contrast, when effort is exogenously fixed at the level of 0.26, Q always decreases with σ 2 u (see the (red) dashed lines in Subplots A4 A6), which is consistent with the original result of Subrahmanyam (1991) The exogenous effort of 0.26 is the optimal effort level under information acquisition and portfolio delegation in our baseline specification of parameters where R a =1, σ 2 = k =1,andσ 2 u =2. The same 18

19 Once we introduce portfolio delegation, Q increases with σ 2 u for R a 1 and decreases with σ 2 u for large R a values, as shown by the (blue) solid lines in Subplots A1 A3 in Figure 1. To understand these results, we note that the informed agent s trading intensity β is determined by the objective function of bβe[ W f ] 1/2R a b 2 β 2 Var[ W f ], wherew f denotes the agent s trading profits. It can be seen that the marginal benefits of increasing β decrease with R a. Also note that 0 b 1, so the agent s effective risk aversion R a b can remain small even when R a =1. 12 When R a is low and when σ 2 u increases, the informed agent can afford to trade much more aggressively against noise traders, or increase β much more, because the first term can dominate the second term in the agent s objective function. As a result, the increase in Q due to more informed trading dominates the decrease in it due to more noise trading, leading to a more informative price in equilibrium. When R a is relatively large, the benefits of trading more aggressively (or increasing β) decreases. Asa result, when σ 2 u increases, a more risk-averse informed agent does not increase his trading aggressiveness as much as a less risk-averse informed agent, as confirmed by Subplots A7 A9 in Figure 2. Consequently, when R a is high, the increase in Q due to more informed trading can be dominated by the decrease in it due to more noise trading, resulting in a less informative price in equilibrium. For example, as shown in Subplot A3 in Figure 1, when R a =2, Q increases initially with σ 2 u and then decreases with it after σ 2 u reaches a certain level. 13 In addition, Figure 2 shows that the optimal incentive b decreases with noise trading, but ρ and β decrease with it. Figure 3 illustrates that when the informed agent becomes more risk averse, he trades less aggressively and the price informativeness decreases. The reason is that the marginal benefits of trading more aggressively decrease with the risk aversion of the informed agent. value is used in producing the results in Figures 2 and For example, as Subplot A2 in Figure 2 demonstrates, even when R a =1,hiseffective risk aversion R ab is less than 0.3, and it decreases with σ u. Our purposes are to study general patterns rather than attempt to calibrate the model. For example, we take the constant k in the agent s cost function for effort to be 1. If we use a larger k value, then we can obtain a much smaller b. See page 1275 of Ou-Yang (2005) for a discussion on the calibration issues. 13 We cannot prove these results analytically as in the case of without portfolio delegation, but numerous numerical calculations support them. 19

20 Optimal Contract In a traditional principal-agent model without an asset pricing equilibrium, an incentive contract is designed to induce optimal effort from the agent. The higher the incentive part b, the higher the effort ρ. In a principal-agent model with a competitive partial equilibrium setting in which the agent s trading does not affect the asset price directly, Stoughton (1993) and Admati and Pfleiderer (1997) find that ρ is independent of b. Inour strategic equilibrium trading model, we find that ρ does depend on b, but a higher b may not necessarily induce a higher ρ. Specifically, we find that the slope of the optimal linear contract, b, decreases monotonically with the variance of noise trading (see Subplot A1 A3 in Figure 2), but effort ρ increases monotonically (see Subplots A4 A6 in Figure 2). When noise trading is more volatile, it is easier for the agent to hide his private information and thus profit fromit. The optimal contract is designed in a way to encourage the informed agent to trade more aggressively, as evidenced from Subplots A7 A9 in Figure 2. More aggressive trading induced by more noise trading provides the agent with a higher incentive to acquire information. Notice that in a traditional principal-agent framework without asset trading, the agent s effort ρ increases with incentive b. The reason is that a higher ρ leadstoahighermean but does not affect the risk of the output. 14 Ahigherb does increase the risk of the agent s compensation, but the increase in the mean dominates the increase in the risk, so the agent s effort increases with incentive in equilibrium. In our model, however, the incentive contract induces the agent to first acquire information and then trade in the stock. When b is lower, the agent can effectively be less risk averse, so he would like to trade more aggressively against more noise trading even though under a lower b the agent s share of profits is lower. More accurate information in the presence of more noise trading will increase the total trading profits of the informed agent. As a result, the agent s total compensation will actually increase. In addition, under a lower incentive fee, the agent can afford to bear more risk. In net, the agent s expected utility increases. Hence, it is possible that reducing 14 In an agency model without asset pricing, Guo and Ou-Yang (2006) allow the agent s effort to affect both the mean and the risk of the output process. 20

21 incentive can increase effort in equilibrium. Figure 3 illustrates that when the informed agent s risk aversion R a increases, incentive and effort both decrease, which is consistent with the results obtained in a traditional principal-agent model. The agent must achieve his reservation utility in equilibrium. When the agent becomes more risk averse, the risk part of his expected utility increases (in the order of Ra) 2 more than the mean part. It is then more costly to provide incentives, so incentives decrease. Even with a lower b, a more risk-averse agent wants to trade less aggressively in the stock, because without an increase in noise trading, trading more aggressively will not benefit the informed agent. Consequently, the optimal effort and the price informativeness both decrease in equilibrium. 3.2 Endogenous Noise Trading When noise trading is exogenous, its impact on optimal contracting and the informed agent s trading is through the variance of the noise supply. When noise trading is endogenous, we can discuss the impact of the number (m),theriskaversion(r h ), and the endowment risk (σ z ) of the noise traders or uninformed hedgers on optimal contracting, the informed agent s effort, as well as the price informativeness. The variance of noise trading is given by σ 2 u = mγ 2 σ 2 z, we can then express the price informativeness as Q =[Var(ev P )] 1 = β2 σ 2 + σ 2 + mγ 2 σ 2 z σ 2 β 2 = 1 σ 2 + mγ 2 σ 2 z σ σ 2 + mγ 2 σ 2 z/β 2. (20) Note that a change in m or σ 2 z may result in simultaneous changes in ρ and β, whichmakes the comparative statics for Q with respect to m or σ 2 z complicated. We report comparative static results in Figures 4 6. Unless otherwise specified, we use σ 2 =1, k =1, σ 2 z =5, R h =3,andm =1, similar to those used by Spiegel and Subrahmanyam (1992). Figure 4 illustrates the relationship between the price informativeness Q and the number of uninformed hedgers, m. In Spiegel and Subrahmanyam (1992) where effort ρ is exogenously fixed and the informed agent is risk neutral, Q, as given in Equation (4), is independent of m. Subplot A1 confirms this independence result. Subplots A2 and A3 show 21

22 that when the informed agent is risk averse, Q decreases with m; the more risk averse the informed agent, the more quickly Q decreases. This result makes sense because a risk-averse informed agent does not trade as aggressively as a risk-neutral one, and the less aggressive informed trading makes the price less informative. The results for endogenous effort are reported in the second column of Figure 4. When the informed agent is risk neutral, Q increases with m, as shown in Subplot B1. On one hand, trading by more uninformed hedgers makes the price less informative. On the other hand, with more uninformed hedgers in the market, the informed agent can potentially profit more from trading against the uninformed hedgers. The informed agent thus exerts more effort in acquiring more accurate information. More aggressive trading by the informed agent with more accurate information leads to higher price informativeness. 15 The second effect dominates the first one, so the price informativeness increases with the number of uninformed hedgers. When the informed agent s risk aversion is low, say R a =0.1, Q increases with m initially when m is small. When m becomes large enough, the first effect dominates the second one, driving down Q. When we further introduce portfolio delegation and when R a =0.1, Q increases monotonically with m, as opposed to increasing initially and decreasing later in the absence of portfolio delegation. As we shall explain later, the optimal b decreases with m (to be shown in Figure 5), which makes the informed agent effectively less risk averse. In other words, with optimal contracting, the effective risk aversion of the informed agent is maintained at a low level, regardless of the number of uninformed hedgers. Hence, the second effect dominates the first one, leading to a higher price informativeness. When the informed agent is very risk averse (R a =2), Q generally decreases with m. As in the case of exogenous noise trading, the marginal benefits of trading more aggressively decrease with the informed agent s risk aversion. Hence, as m increases, the informed agent s trading intensity and effort in information acquisition do not increase as much as in thecaseofasmallr a. Consequently, the smaller increase in informed trading is dominated 15 Under different settings, Fishman and Hagerty (1992) and Leland (1992) discuss the impact of insider trading on the price informativeness. 22

23 by more uninformed trading, leading to a less informative price in equilibrium. We report the comparative statics regarding the optimal contract in Figure 5. It shows that the incentive b decreases monotonically with the number of uninformed hedgers, m. When m increases, the informed agent can profit more from trading against uninformed hedgers. As a result, the principal lowers b so as to encourage the informed agent to acquire more accurate information and trade more aggressively. From Subplots A2 and B2 of Figure 5, when R a =0.1, optimalb exhibits a bell-shaped relationship with σ 2 z, whereas when R a = 2, it decreases monotonically with σ 2 z. The intuition is as follows. When the informed agent is of low risk aversion (i.e., R a =0.1), for a very small σ 2 z, the principal increases b to prevent the informed agent from exerting too much effort and trading too aggressively. If the informed agent acquires very accurate information and trades aggressively, uninformed hedgers would trade less, which can hurt the informed agent s trading profits. For a sufficiently large σ 2 z,uninformedhedgersmust trade to hedge their endowment risk anyway. Hence, the principal reduces b to induce the informed agent to acquire accurate information and then trade aggressively, resulting in a bell-shaped relationship between b and σ z. When the informed agent is very risk averse (i.e., R a =2), he is not willing to trade aggressively. Even for a very small σ 2 z, the principal must assign a low b to encourage the informed agent to trade more aggressively, so as to take advantage of an increased σ z. Subplots A3 and B3 of Figure 5 illustrate a somewhat striking result, that is, when the informed agent is relatively risk tolerant, incentive b increases with the risk aversion of the uninformed hedgers R h, whereas when the informed agent is very risk averse, b decreases with R h. When uninformed hedgers become more risk averse, on one hand, they would like to trade more to hedge their endowment risk, but on the other hand, they are less willing to trade because they understand that the less risk-averse informed agent would trade aggressively against them. If uninformed hedgers trade less, then the informed agent would benefit less from his information. In equilibrium, the principal increases b, making the informed agent effectively more risk averse in order to induce uninformed hedgers to trade more. When the informed agent is very risk averse, however, the principal reduces b 23

24 to encourage the informed agent to trade more aggressively as the hedging demand of the uninformed hedgers increases. In Figure 6, we present the trading intensities of the informed agent and uninformed hedgers, the market impact cost, and the informed agent s effort, with respect to the risk aversion of the informed agent. In particular, the relationship between R a and the absolute value of γ (a measure of hedging demand ) is U-shaped, highlighting the inter-dependence between the trading of the informed agent and that of uninformed hedgers. We first examine the case in which ρ =1and b =1(depicted by the red dotted lines), or there is neither information acquisition nor portfolio delegation. When R a increases from zero to a small value, the informed agent trades significantly less aggressively (i.e., β decreases very rapidly). As a result, the price variability rises so much that the hedgers trade less intensely, i.e., γ decreases. When R a keeps increasing and becomes large enough, the less aggressive informed trading makes the market increasingly more liquid, which encourages more hedging demands. Hence, γ decreases first and increases later. Inthecasewithb =1and endogenous ρ (depicted by the black dash-dotted lines), γ increases more rapidly when R a is large enough. This is because the equilibrium level of ρ is much less than one, which is exogenously assumed in the previous case. Consequently, λ decreases to a much lower level at a greater speed, which induces more hedging demands. Hence, γ increases more rapidly when R a is relatively large. In the case under endogenous ρ and b, withtheoptimalb smaller than one, the informed agent s trading intensity is generally higher than that in the previous case with b =1and endogenous ρ, which has a dampening effect on the hedgers demand for stock. Hence, the amount of noise trading decreases. Notably, when R a increases to a sufficiently large level, γ increases at a smaller speed. This is because the informed agent s trading intensity and hence λ fall more slowly than in the previous two cases. 3.3 Summary of Main Results Table 1 summarizes the relationship between Q and σ 2 u in the case of exogenous noise trading or m, σ 2 z, R h in the case of endogenous noise trading. 24

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