Monitoring and Manipulation: Asset Prices When Agents are Marked-to-Market * This Draft: June 5, Abstract

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1 onitoring and aniulation: sset rices When gents are arked-to-arket Gary B. Gorton Yale University and NBER ing e Tsinghua University ixin uang Georgia tate University This Draft: June 5, 8 bstract Trading in security markets is delegated. We study the efficient markets aradigm in the context of such agency relations. rincial-investors want to monitor and comensate their agent-traders using market security rices in mark-to-market contracts. This introduces an externality because security rices are informative only insofar as the agent-traders of other rincial-investors have an incentive to roduce information and trade accordingly. If the market is dominated by such delegated traders, then these traders can attemt to maniulate the market rice by jointly shirking and buying or selling in the same direction. In this way, traders rovide market roof that they have worked hard and deserve high comensation. We show that even if agent-traders can coordinate erfectly, there is no euilibrium in which all rincial-investors ignore market rices. The extent of market efficiency, indexed by the delegated traders roensity of joint shirking, is endogenied. We thank ichael Brennan, Charlie Kahn, Chester att, seminar articiants at the University of Wisconsin at adison, Georgia tate University, and the NBER 6 ummer Institute for comments and suggestions. This is a much revised version of a aer reviously entitled sset rices When gents are arked-to-arket. chool of anagement, Yale University, New aven, CT chool of Economics and anagement, Tsinghua University, Beijing 84, China. heing@sem.tsinghua.edu.cn. J. ack Robinson College of Business, Georgia tate University, tlanta, G 333. hone: fnclhh@langate.gsu.edu.

2 . Introduction The intellectual triumh of the concet of efficient markets over the last 5 years has led to this aradigm becoming widesread, governing the comensation of rofessional security traders and money managers, and roviding foundations for accounting and risk management. arking-to-market refers to the widesread ractice of measuring and monitoring ortfolio managers and security traders erformance and risk-taking roensity based on market rices. s a risk management ractice, marking-to-market rofessional traders ositions received fresh imetus after the collase of Enron, and other derivatives scandals. oreover, accounting measurements of rofit and loss are also increasingly based on security market rices, as the best measure of fair value. The logic of efficient markets is clear: If market rices are the best measure of value, because they contain all relevant information, then these rices should be used for comensation, accounting, and risk management uroses. ence, the term market disciline. n imortant ste in the intellectual foundation of efficient markets was the solution to the roblem of how information came to be embedded into security rices if markets are efficient, somehow already containing all relevant information. Grossman and tiglit 98 addressed this issue by introducing the notion of noise a noisy suly curve for the asset, in their case. This allowed some traders to roduce costly information and trade rofitably on their findings, with the information being at least weakly imounded in the rice. ee Dow and Gorton 6 for a review. In this aer, we revisit the issue of market efficiency in the context of delegated trading, focusing on the use of security rices in rivate contracts with delegated traders. rincials hire traders to roduce and trade on rivate information. We show that when market rices are used to try to control the actions of agent-traders, agent-traders can maniulate rices resulting in less informative rices and ultimately making mark-to-market contracts undesirable. The efficiency of market rices is not exogenous to the risk management ractices adoted by rincial-investors. In fact, markets become less efficient. In the context of a security market in which some traders are trading on behalf of investing rincials, rincials must design comensation contracts to control the trading behavior of the rofessional traders or money managers they hire. In the standard Derivatives-related scandals include the failure of Orange County, California in the early 99s,, the failure of etallgesellschaft in 994, the failure of Barings C in 995, the collase of ong Term Caital anagement in 998, Enron in, the failure of China viation Oil in 5, and the most recent failure of ociété Générale in 8. Indeed, the sudden collase of Enron in late gave further imetus to the confluence of the efficient markets theory of security rices and the accounting rofession's notion of fair value. ccountants are increasingly embracing the notion that fair value for accounting uroses is best measured using security market rices. Enron was accused of mis-marking comlex derivatives using models. The issue of whether fair value is best reresented by market rices is most controversial and roblematic in derivatives markets, but is resent in all security markets. The relevant rule is largely stated in the Federal ccounting tandards Board F 57, Fair alue easurements.

3 agency aradigm, an agent s contractual comensation is only a function of the final value or rice, not any interim information. mark-to-market contract is different, as it involves interim rice information from security markets, information that arrives between the contract initiation and the final ayoff. ince interim market security rices are observable and contractible, these rices may be used to value the traders ositions as art of the otimal contract. When the interim market information is exogenously given, contracts ignoring it are subotimal. owever, when the information roduction is endogenied, contract choices that are based on the informative market rice also have an imact on market rices. We rovide conditions under which marking-to-market is art of an otimal contract. This confirms the logic of using market rices as a measure of a trader s erformance in addition to the final ayoff. 3 nother difference from the standard agency roblem is that in our setting the comensation of agent-traders deends on a ublic signal that is jointly affected by the agent-traders actions rior to the final date. In articular, agent-traders trade in a securities market and the interim market rice, which is at least somewhat informative due to the actions of other agent-traders, is used by each rincial to monitor his agenttrader s behavior. The roblem is that the trading behavior affects the security rices, which in turn, affects the mark-to-market values of the traders ositions, affecting their comensation. This externality is at the heart of the contracting roblems. In fact, traders have an incentive to coordinate to maniulate the rice, meaning that they can jointly shirk not roducing any information and trade in the same direction, leading to rices which will result in high comensation under the mark-to-market contracts. ence, the tension between monitoring agent-traders based on the security market rices and the maniulation that can occur under these contracts, and this endogenously determines the asset rice and market efficiency. The ossible maniulation of security rices by agent-traders means that the rincials choice of contract for their traders at the initial date is also intertwined. rincials can always choose the standard contract, that is, one which does not include interim market rice information, but rather is just based on final outcomes. Under a standard contract, an agent-trader can be induced to roduce information and trade in a direction consistent with the rincial s objective. But, adoting the standard contract means that the rice will be at least weakly informative, allowing other rincials to free ride on this information using mark-to-market contracts for their agent-traders. The euilibrium then must be one in which the rincials follow mixed strategies in contracts, sometimes adoting mark-to-market contracts and sometimes adoting standard 3 In the managerial contracting literature, it has been demonstrated that the market rice of the firm should be included in the otimal contract for the manager as long as it contains information not reflected by the firm s current or future rofit data; see, for examle, olmstrom and Tirole 993. Our aer goes beyond this result; we study the externality effect of these mark-to-market contracts when there are multile rincial-agent airs in the economy. In articular, we show that the agents have an incentive to coordinate to maniulate the market rice to get a higher ayoff. 3

4 contracts. This introduction of noise, interreted as randomness in rincials discretion over comensation, makes it harder for the agent-traders to maniulate their bonuses. The agency relation we study is ervasive in security and derivatives trading. rofessional traders work at hedge funds, mutual funds, money management firms, and banks. In over-the-counter markets, like those for government securities, foreign currency, cororate bonds, residential and commercial mortgage-backed securities, other asset-backed securities, and derivative securities comrising interest rate, euity, foreign currency, credit, commodity, energy and other derivatives, the entire market is based on traders working for others. These are very large and imortant markets. For examle, the notional amount of interest rate derivatives in June 7 was $346.9 trillion and the amount of credit default swas outstanding in June 7 was $4.6 trillion see Bank for International ettlements 7. 4 The cometition between rincials in choosing contracts for their cometing agents that we analye is similar to some revious literature in Industrial Organiation. klivas 987 and Fershtman and Judd 987 both study this tye of contracting roblem in a duooly setting with Cournot or Bertrand cometition. In klivas 987, the manager s ayoff is assumed to be a linear combination of the firm s revenue and rofit, while in Fershtman and Judd 987, the manager s ayoff is assumed to be linear in the firm s erformance relative to its cometitor. In our setting, we do not restrict the format of the agent s comensation; rather we endogenously determine the otimal contract. In a similar financial market setting, Gümbel 5 studies the asset allocation roblem in a multile rincial-agent environment, but the contract is restricted to be based on relative erformance, similar to Fershtman and Judd 987. Two other aers study issues to do with marking-to-market, though not in an otimal contracting context. lantin, ara, and hin 5 study the effects of an assumed mark-to-market regime when bank managers are assumed to have short horions. sset rices are assumed to be negatively related to asset sales, i.e., the rice goes down when banks sell assets. llen and Carletti 6 show that when asset rices sometimes reflect liuidity factors, rather than fundamentals, a mark-to-market regime can induce contagion. Our aer is uite different from these in several resects. First, we do not assume that marking-to-market is otimal. We determine when mark-to-market contracts between rincials and their traders are otimal. econd, we do not assume that there are effects on security rices due to suly and demand, causing rices to deviate from fundamentals. We isolate the effects of mark-to-market contracts in a standard trading environment; there are no exogenous liuidity factors that cause the rice to deviate from fundamentals and there are no other assumed frictions, e.g., short horions. 4 Even in the ublic euity market, most trading is delegated to rofessional ortfolio managers. In 4, U.. households directly held less than 4% of cororate euities, while they held about 9% in 95 and 7% in 97 see "Flow of Funds" issued by the Federal Reserve Board. lso according to the survey results released by the Investment Comany Institute ICI and the ecurities Industry ssociation I, in, 89% of the investors invested in mutual funds and 58% of the investors relied on rofessional financial advisers when making investment decisions. 4

5 Finally, our aer also belongs to the strand of the literature that examines the imact of agency roblems on asset ricing. For examle, llen and Gorton 993 show that when there is asymmetric information between investors and ortfolio managers, ortfolio managers have an incentive to churn; their trades are not motivated by changes in information about liuidity needs or risk sharing but rather by a desire to rofit at the exense of the investors that hire them. s a result, assets can trade at rices that do not reflect their fundamentals and bubbles can exist. Dow and Gorton 994 and Goldman and leak 3 examine the ricing imact of an interaction between agency roblems and time-horion and show that managers incentives can be distorted by an information externality. In our aer, while the rincial would only care about the final rice without hiring a trader, a trader s horion is endogenously shorter since the otimal contract links his ayoff to the interim rice. Dow and Gorton 997 study a model in which investors otimally contract with ortfolio managers who may have stock-icking abilities, and ortfolio managers trade otimally given the incentives rovided by this contract. Because investors cannot distinguish actively doing nothing from simly doing nothing, some managers trade with no roer reason noise trade. Noise trade causes high levels of turnover. 5 The aer roceeds as follows. In ection we set u the model and state some assumtions about the security ayoffs and the noisy signals about fundamentals. The analysis then roceeds in three stes. We first solve for the otimal standard contract in ection 3. It is the benchmark case, yet a subotimal solution. Then we solve for the otimal mark-to-market contract and comare it with the standard contract. We show that the otimal mark-to-market consists of a ste function based on the interim security rice, which is the roxy for the trader s rivate information. In order to revent the trader from shirking, the otimal contract reuires that the trader's trading decision i.e., buy or sell be consistent with the change in the security rice. The otimal mark-to-market contract dominates the standard contract because the additional information derived from the interim market rice enables the rincial to lower the cost of monitoring the agent. In ection 4 we analye the case of many rincial-agent airs. When there is more than one trader in the market, additional euilibria emerge. In these euilibria, traders either sell the security uon receiving rivate good news or buy the security uon receiving bad news--the oosite of what they would do in the absence of the externality and actions that are not in the rincial-investors interests. The interaction of the agency roblem and the externality gives rise to these seemingly irrational euilibria. We consider how the rincials comete in their choice of contract, mark-to-market or not, when they hire agent-traders. The externality caused by the feedback from traders trading activity results in a nontrivial contract choice by rincial-investors at the initial date. lthough coordinated market maniulation by traders may force some rincialinvestors to offer the standard contract at some times, there is no euilibrium in which rincial-investors only offer standard contract. 5 There is a large literature on delegated ortfolio management, which focuses on issues of the manager's comensation structure. E.g., Bhattacharya and fleiderer 985, Brennan 993, dmati and fleiderer 997, Cuoco and Kaniel, and Ou-Yang 3. For the most art, the issue in these studies concerns the choice of a benchmark to use to evaluate the manager's erformance. 5

6 We conclude in ection 5.. odel etu Our focus is on an agency roblem in which a rincial-investor hires an agent-trader to trade on his behalf in a security market. ike the standard agency roblem, the rincial must design a contract to induce the agent-trader to make an effort to imrove the value of the realiation of the roject, in this case a trading osition. 6 In our setting the rincial observes an interim rice signal which can rovide some information about the erformance of the agent-trader. ere, in addition to an initial effort choice, the trader can make another choice later, namely, he can buy or sell in the interim securities market. While the effort choice is hidden, as in standard agency roblems, the trading action/osition is observable and contractible. Thus, the strategy sace of the agent-trader is larger than in the standard roblem, introducing the issue of risk management in a way which is not resent in the usual roblem. The risk management issue concerns how the rincial can control the agent-trader s hidden action using his observable actions in combination with market conditions in the interim securities market. We first investigate this in the simlest setting in this section. We will call the agent-trader the rofessional trader or simly the trader, as he is hired by the rincial. The security market has five tyes of risk neutral articiants: a direct investor, an indirect investor, a rofessional trader, liuidity or noise traders, and a market maker. The direct investor invests for himself; he has no need to hire an agent-trader. 7 Both the rofessional trader and the direct investor have the technology to become imerfectly informed about the value of the security at a final date. The direct investor has money to invest in the security market, but the trader does not. owever, the rofessional trader can be hired by the indirect investor the rincial, who has the money to invest, but no access to the technology to become rivately informed. fter the trader is hired, he can choose not to make an effort to acuire information; not making an effort yields a shirking benefit of >. For ease of exosition, we will refer to the security as stock. There are three dates in this economy: Date : The trader and the rincial sign a contract which authories the trader to trade on behalf of the rincial and which secifies the ayoffs to the trader under each contingency observable to both arties to the contract. 6 ee alanié 5 for an exlication of the standard contracting roblem with unobservable effort choice. 7 The existence of a direct investor makes the rice informative even when the agent trader shirks i.e., does not roduce information and simly traders randomly. iuidity traders or noise traders are agents who lay a secial role in financial economics. They trade for exogenous reasons and, on average, lose money, allowing other agents to roduce costly information and still make a rofit. ee Dow and Gorton 6. 6

7 Date : Both the direct investor and the trader if he does not shirk receive a signal about the terminal value of the stock. Then each decides whether to submit a buy order or a sell order. For simlicity, investors can trade a maximum of x shares. The liuidity traders trade an amount ~ N, ². The market maker sets the rice based on the total order flow by forming the conditional exectation of the security value, given knowledge of the model, as is standard in financial economics following Kyle 985. Date : The liuidation value of the stock is realied, and the rincial ays his trader according to the contract. The liuidation value of the stock has the following distribution: and we assume v > v. 8 v v v with robability / with robability /, For simlicity, the direct investor and the trader if he does not shirk receive the same rivate signal, which can be either s or s, and the signal is correlated with the true liuidation value of the stock as follows: v with robability > / if v v v v with robability > / if v v. et s denote the unconditional robability of receiving a high signal, then we have: π s. 3 The robability distribution of v conditional on the signal s can be written as: rob v rob v s s rob v rob v s s. 4 When both the direct investor and the agent-trader trade on rivately roduced information, the distribution of the market order flow is a mixture of two Normal distributions: ~ Nx, σ N x, σ. 5 et. denote the robability density function for the distribution Nx, ²,. denote the robability density function for the distribution N-x, ², and. denote the robability density function for the distribution N, ². o: 8 The assumtion of eual robability for v and v is not crucial for our results. 7

8 x ex πσ σ x ex πσ σ ex. πσ σ 6 It is easy to show that / is increasing in and / is decreasing in. The monotonicity of the likelihood ratio / tells us that the higher the order flow, the more likely it is from the distribution Nx, ². similar argument alies for the likelihood ratio /. This roerty of the monotone likelihood ratio is imortant for characteriing the otimal contract. et denote the exected liuidation value at date 3 conditional on the signal s being received at date by the trader and the direct investor; is defined similarly conditional on the signal s being received. Then: E v s E v s v v v v. 7 The market maker will set the rice to be the exected liuidation value conditional on the order flow; see Kyle 985. Therefore, if both the direct investor and the agent-trader trade truthfully based on the roduced information, the rice can be exressed as a function of the order flow as follows:. 8 We can show d / >, lim, and lim. The market maker can only observe the total order,, but not the identity of traders or their orders. From the magnitude of the total order flow, he infers the news received by the informed agents. The larger the order sie, the more likely it is that the good news was received, and thus the market maker sets a higher rice. In the limit, when the market maker receives an extremely large buy order, he is retty sure that good news was received by the informed agents and the rice is set to, the exected value conditional on good news; when the market maker receives an extremely large sell order, he is retty sure that bad news was received by the informed agents and the rice is set to, the exected value conditional on bad news. ccording to euation 8, the interim market rice is always between and. Therefore, it is in the best interests of the direct investor to buy x shares given a good signal and sell x shares given a bad signal. In the next section, we will discuss the 8

9 euilibrium in which the agent-trader is hired with incentive comatible contracts that induce him to roduce information and trade on it, thus the interim rice defined in 8 is consistent with the market maker s rational exectation. 9 ater on, we extend the model to the case with multile rincial-agent airs in the economy, where the agent-traders might shirk in euilibrium and the interim rice would change accordingly. In ection 3 below, we will study two tyes of contracts. The first tye of contract only deends on the final value or rice, not any interim information. This is the tyical contract that is studied in the standard agency aradigm and we call it tandard Contract. ince it does not emloy all information available in the market, it is subotimal. The second tye of contract stiulates that the ayoff to the agent deends on his trading osition buy or sell,, the final liuidation value of the security, v, and in addition, the interim market rice,. We call it ark-to-arket Contract. 3. Cost aving of ark-to-arket Contract 3. tandard Contract s a benchmark case, we first solve for the otimal tandard Contract before we study contracts that are marked-to-market. ssumtion : There is an uer bound w for wage ayment to the agent-trader. The role of this assumtion is to guarantee a bounded otimal solution. This will become clear as we roceed; the interretation of w will also be discussed later. et us first characterie the otimal tandard Contract. Define W as follows: W { w λ, v w λ, v w for any λ and v}. 9 W s is the set of all feasible standard non-mark-to-market contracts. tandard Contract w, v, characteried by {w b, w b, w s, w s }, is incentive comatible if it satisfies: 9 It is easy to see that as long as the indirect investor hires a trader in euilibrium, the trader will roduce information trade truthfully, otherwise he will not be hired at the first lace. We assume agents cannot observe each other s security osition, or alternatively we can assume that this information is not verifiable or contractible. comlete contract secifies the ayoff to the trader for every ossible trading osition between -x and x. To simlify our discussion, we restrict our attention to buying or selling x shares. We can actually show that, for the many tyes of euilibrium that we study, the otimal contract ays ero for any osition other than buying or selling x shares. 9

10 and IC. IC. IC. w can be rewritten as: b w w s w wb w max wb w b s b b w s w w κ, w w b s s w w w s s s b, κ. IC. wb w ws w b s ws w wb w s b κ κ, and it is clear that IC. imlies IC.. IC. says that, when the trader receives a good signal, he will be better off submitting a buy order, and when he receives a bad signal, he will be better off submitting a sell order. IC. says that the trader will be better off by acuiring information and trading in the best interests of the rincial instead of shirking getting and randomly buying or selling the stock. The exected ayoff to the trader can be written as: wb ws ws wb, 3 and the otimal tandard Contract w λ, v solves the following rogramming roblem: min w W subject to the IC conditions in. 4 s The constraints in have to be binding; otherwise the rincial investor can always lower the trader s comensation without invalidating the IC conditions. The following roosition states that the otimal comensation contract is simle: the trader will either get a ositive ayoff or ero, deending on how his osition aligns with the final liuidation value. roosition Otimal tandard Contract ssume w κ /. Then the otimal contract w λ, v { w, w, w, w } solving 4 takes the following form: b b s s w b κ ws, and ws wb. 5

11 roof: ee endix. When all informed articiants trade truthfully, the exected ayoff derived from informed trading is: x x. 6 The exected wage ayment to the agent trader is: κ. 7 In order for a rincial investor to hire an agent-trader with a tandard Contract, two conditions have to be satisfied. First, the rincial investor needs to have enough resources. In other words, κ w. 8 econd, the rincial investor s articiation condition has to be satisfied: x κ. 9 enceforth we make the following assumtion: ssumtion : x κ and κ w. The above assumtion seaks to the two roles that a small lays: i small ensures that the exected value of an incentive comatible wage ayment is low enough such that a rincial is willing to hire an agent trader; and ii small ensures that there exists an incentive comatible contract given the value of w. The agent-trader s exected wage ayment is eual to κ /., which is greater than the cost he incurred to collect information,. The difference is κ /, which can be interreted as the information rent due to the agency roblem. ince the tandard Contract does not incororate the market information into monitoring, it is subotimal to a ark-to-arket Contract, which uses the information

12 contained in the interim rice, thus lower the monitoring cost. We will discuss this in more detail in the next subsection. 3. ark-to-arket Contract To simlify the notation, we define an agent-trader s exected ayoff as following: χ w χ, λ, v, λη where χ,, or ; b or s; and or. Basically, η χ λη is the exected ayoff to the trader when his trading strategy is buy or sell x, the informed order flow is, and the realied liuidation value is v. ark-to-arket Contract w,, v is incentive comatible if it satisfies: and IC. IC. b s b s s b b max s b b, κ, κ. s b b s s s s b s b Note that IC. can be rewritten as: IC. b κ, s b s s b s b κ 3 so it is clear that IC. imlies IC.. Intuitively, if a trader collects information at a cost ex-ante, then he will make a full use of it ex-ost. The exected ayoff to the trader can be written as: b b s, 4 s and the otimal contract w,, v solves the following rogramming roblem:

13 min w W subject to the IC conditions in 3. 5 Before we roceed to study the roerties of the otimal contract w,, v, we state the following lemma. emma In any otimal incentive comatible contract solving 5, the constraints in 3 are binding. roof: If the first ineuality is not binding, then we can lower w, b, v a bit, and both ineualities will still hold as long as the change in w, b, v is small. imilarly, if the second ineuality is not binding, then we can lower w, s, v a bit while not violating the ineualities. The result in the above lemma will be used to rove the following roosition, in which we show that the otimal comensation contract is simle; the trader will either receive a ositive fixed wage, w, or ero, deending on how his trading osition aligns with the interim rice and the final liuidation value v. roosition Otimal ark-to-arket Contract The otimal contract w,, v solving 5 is characteried by four cutoff values, denoted as {,, } b, b s s such that: i for any < bη,w, b, v, and for any bη,w, b, v w ; for any > sη,w, s, v, and for any sη, w, s, v w. ii satisfy the following conditions: σ b b s s ln, 6 x λη { Φ Φ } w w Φ b s Φ s w b w Φ s Φ s κ { Φ Φ } κ, b b 7 and x, 8 b s b s > where. is the cumulative distribution functions corresonding with the robability χ distribution functions. defined in 6. roof: ee endix. 3

14 roosition says that the otimal contract is a marked-to-market one that deends on both the interim rice and the final security market rice, as well as the interim trading osition of the trader. Because the direct investor always trades truthfully, there is information in the interim rice. This information can be used by the rincial to rovide the trader with an incentive to acuire information and trade accordingly. The otimal contract unishes the trader when both the interim market rice and the final liuidation value are against the direction of his trade at the interim date. The four euations in roosition, euations 6 7, are four euations with four unknowns,,, and, which characterie the otimal contract. s the ricing b b s s function in 8 is monotone in, it is erhas easiest to think of the cutoff values in as corresonding to rices. Then, the roosition says that the otimal contract consists of a ste function this is the result of monotone likelihood ratio roerty of the normal distribution as we discussed earlier in terms of the security rice for each air of and v. Figure ortrays roosition in terms of the -cut-offs. The cut-off order flows deend on the trader s interim trading osition buy or sell and on the final liuidation value or. ince rices and order flow cut-offs are isomorhic, one can think in terms of rice directly. o, for examle, if the trader buys the stock when the final value is and the rice at date,, was lower than a certain level, b, he will receive a ero ayoff regardless of whether the realied liuidation value is high or low. If is higher than, he will receive w regardless of the liuidation value. In between these two rice cut-offs, his ayoff will be deend on the final liuidation value, and he will receive ero if the liuidation value is v, and w, otherwise. imilarly, if the trader sells the stock and the security rice at date,, is higher than a certain level, s, he will receive a ero ayoff; if is lower than s, he will receive w for sure. In between, he will receive ero if the liuidation value is v, w otherwise. We can interret these results as a markto-market contract for the trader. In terms of the trader s exected comensation, roosition imlies that b > b and s > s, direct results of 6. o the trader receives a lower ayoff if the realied liuidation value contradicts the trader s rior trading decision, on average. In other words, he gets a lower ayoff if he bought the stock, but its final value was. lso, notice that the difference between and b or b and s b is greater when the signal is more accurate, that is, when the value of is greater. Intuitively, the trader gets a harsher unishment if the robability of receiving a wrong signal is lower. If there were no uer bound on the contract ayment, the otimal contract would be to ay an infinitely high amount to the trader when he bought the stock, but only when the rice was close to that is, when the order flow is infinitely large or when he sold and s 4

15 the rice was close to. The existence of w guarantees the boundedness of the otimal contract. One way to interret the uer bound w is to imagine that the trader is risk averse instead of risk neutral. Then aying a very high wage but only with a small robability is not otimal for the rincial since the risk remium reuired by the trader will make the exected ayment very high. With a risk-averse trader, the existence of w could be justified and endogenied. We do not ursue this here. We have already discussed the trader s incentive constraints above, and in order to demonstrate that the otimal contract characteried above can be sustained in a Nash euilibrium, we need to check two more things: i there is no other contract that dominates the contract characteried in roosition, and ii the ayoff to the rincialinvestor is higher than his best alternative ayoff without hiring an agent-trader. For i, we have shown that the otimal contract is the best one among all incentive comatible contracts, and we only need to roerly secify the trader s resonse when a non-incentive comatible contract is offered to make sure the rincial will not offer any of those. For examle, we can assume that the trader will shirk and always sell when such a contract is offered. For ii, we need to comare the ayoff to the rincial with the ayoff in the case in which no trader is hired. The exected net ayoff of the investment under informed trading is: x x. 9 The uantity is the joint exected ayoff to the rincial and the trader also the exected ayoff to the direct investor, that is,, with and denoting the ayoffs to the rincial and the trader, resectively. If the rincial does not hire a trader, but rather trades for himself, his ayoff could be one of the following three ayoffs: b, the ayoff from uninformed buying at date ; s, the ayoff from uninformed selling; or, the ayoff from doing nothing. Obviously we need to have max{ b, s, }. We have: b s x x. { } { } 3 5

16 It is easy to show b < and s <. Therefore, max{ b, s, } is euivalent to. roosition 3 Existence and Uniueness of the Otimal ark-to-arket Contract Under ssumtion, there exists a uniue otimal ark-to-arket Contract is used by the rincial-investor. roof: ee endix. The tandard Contract is a feasible choice because it also satisfies IC. in 3, but it is dominated by the ark-to-arket contract because the latter tightens monitoring through using the market rice. The gross ayoffs from the informed trading are exactly the same; however, the monitoring cost is smaller with the ark-to-arket contract. To conclude this section, we discuss how the ayoffs to the market articiants change with model arameter values. First, let us write out the exected ayoff to the liuidity trader: { x x }. N 3 With x x, some algebra leads us to N, which confirms the ero-sum game roerty of this tye of model. Recall from 9 that is the exected value of trading to an informed investor, either the direct investor or the rincial-agent air. The next lemma hels us understand how the cutoff values in associated with the otimal ark-to-arket Contract change with the cost of information roduction,, and the maximum wage, w. emma et {,,, } be the solution for an otimal ark-to-arket b b s s Contract. Then b b <, s s b b s >, >, and < κ κ κ κ w w w w roof: ee endix. s. We use the following two ineualities to show b < and s < : < <. The existence of an Otimal ark-to-arket contract reuires a weaker condition than ssumtion, which also guarantees the incentive comatibility and the rincial s rofitability of a tandard Contract. 6

17 emma can be understood with reference to Figure. Given the relationshi between and defined in 4, the lemma above says that the rincial has to ay the trader more when the cost of information roduction increases. When the cost of information roduction rises, the cut-off values move so that the trader receives w over a wider range of outcomes to cover the higher cost of information roduction. But, the comensation cannot be so high that the rincial does not want to hire a trader. The second art of the lemma relates the comensation, w, to the cut-offs. If w rises, then the cut-offs have to move to reduce the range of rices over which the trader gets comensated since when he does get comensated, the comensation is higher. The joint exected ayoff to the rincial and the trader is not affected by the arameter values of or w, which affect the contractual ayoff distribution between the rincial and his trader, but are affected by, the sie of noise trading, and, the recision of information roduction. We summarie the ayoff deendency in the roosition below. roosition 4 Comarative tatics With the otimal mark-to-market contract, i the joint exected ayoff to the rincial and the trader is increasing with and, and the exected ayoff to the liuidity traders is decreasing with and ; ii the exected net ayoff to the trader is increasing with, increasing with, decreasing with w, and decreasing with ; the exected ayoff to the rincial is decreasing with, increasing with w, and increasing with. roof: ee endix. The first result is the same as Kyle 985, where a rivately informed trader always benefits from a higher, while a liuidity trader is worse off with a higher. n informed trader can benefit from a higher i.e., a wider disersion of liuidity trade because the monitoring role of the rice is less effective since the rice is less informative; when is smaller, the rincial can offer a contract that is more sensitive to the rice and thus reduce the ayoff to his trader. owever, it is not clear how the value of affects the exected ayoff to the rincial. On the one hand, a higher leads to a higher exected wage ayment to the trader, on the other hand, the joint ayoff,, also increases with. With regard to, a higher recision in information roduction allows the information roducers to take a greater advantage of the liuidity traders, thus imroving their ayoffs. The results on and w can be interreted as follows. When the oortunity cost of information roduction,, is higher, the rincial has to ay the trader more. When we increase w, the otimal contract with the original lower w is still feasible, but we know from roosition that there is still slack with resect to the conditions for the otimal contract with the higher w, i.e., it can be imroved uon. Thus, it results in a lower ayoff to the trader with a higher w. Intuitively, when w is higher, the contract in the 7

18 form of w or nothing allows the rincial to ay more when the market rice is more extreme, thus generating a stronger incentive for the trader to roduce information. To summarie, if there is an informative, verifiable, signal namely, the interim security market rice-- in addition to the final value, then, erhas not surrisingly, it is otimal to include it as art of the otimal contract, as long as doing so is not so exensive for the rincial. This is the essential logic of the mark-to-market contract. The direct trader has no agency roblem and always roduces information and trades otimally. lthough his information is rivate, his order is going to move the market rice set by the rational market maker. The rincial makes use of this behavior of the direct trader to monitor his agent-trader. e free rides on the information in the rice that is due to the direct trader. The existence of the direct investor guarantees that the market rice is informative, at least to some extent. owever, the idea that the interim market rice is efficient does not rule out the ossibility that traders will trade in subotimal ways, and that they may alter the information reflected by the market rice. In case this haens, the rincials need to incororate it into the otimal contracts, and may even use contracts that are not marked-to-market. In the next section, we begin exloring these issues in the trading context. 4. Cometition in Contracts with ultile rincial-investors 4. Externality Effect of arking-to-arket In the revious section, the otimal ark-to-arket Contract is signed based uon three variables: the realied value or final rice, the interim security market rice, and the trader s trading osition. The interim rice is formed in resonse to the trading ositions of all the market articiants, some of whom roduce information. Because the interim rice is informative, it can be used to tighten the monitoring of the trader. Due to a ositive correlation in our case, a correlation of one of the signals received by different agents, a hard-working trader likely should have received a high signal when the interim rice is ushed u by the buying orders from other informed investors, and a low signal when the interim rice is ushed down by the selling orders from other informed investors. By examining this consistency, the rincial makes sure that the trader s good erformance is the result of hard-work, rather than the outcome of good luck. If the informativeness of the interim market rice can act as a monitoring device, then we have to address the uestion: where does the information come from? The assumtion of the existence of a direct investor rovides a trivial answer: it comes from the direct investor who trades for himself and is not subject to the moral haard roblem. What if the market is mainly dominated by delegated traders? In this case, the information has to come from other delegated traders. In other words, the information injected into the market by some traders is used to monitor other traders and vice versa. If traders 8

19 understand the mutual monitoring feature of the interim rice, then they can act strategically to jointly undo the monitoring effects of the rice. emma 3 uose there are two delegated traders hired by two associated rincials who each received the otimal ark-to-arket Contract we solved for in roosition of ection 3. Then if one trader shirks and buys sells x shares, it is the other trader s best resonse to follow. Joint shirking gives traders higher ayoffs than what they receive by not shirking. roof: ee endix. Roughly, when there are multile delegated traders, the otimal contract ays a high wage to a trader when his trading osition is consistent with the market rice, that is, the information imounded in the rice due to the trading by some traders is used to monitor the other traders. owever, when traders understand this mechanism, they have an incentive to jointly shirk and trade in the same direction, moving the rice, if they can find a way to coordinate their trading. By so doing, they can maniulate the market such that the interim rice is likely to move in line with their trading osition, enabling them to take advantage of the ark-to-arket Contract. Of course, if rincial-investors anticiate this, then the otimal contract should reflect the ossibility of this joint moral haard roblem. 3 If the robability of market maniulation is small, investors are still better off by offering the ark-to-arket Contract. Otherwise, investors may consider other contracts to disrut the traders coordinated maniulation. In articular, a tandard Contract might be aealing to an investor-rincial as it eliminates the incentive of coordinated rice maniulation. In the next two subsections, we formally study the euilibrium in which the traders coordinate to shirk and trade in the same direction due to the externality effect of a ark-to-arket Contract. 4. ure trategy aniulation Euilibrium To roceed, let us assume that there are two delegated traders in the market, 4 and we introduce an irrelevant noise signal, r, which is, with robability, observed by the traders before they make their effort to acuire information. The signal may be thought of as a news story, a rumor, a technical trading signal, a suerstitious event, and so on. Traders regularly communicate, these days by Bloomberg mail and , reviously by hone. The social structure of markets, including the culture and communication, is essentially subsumed by our assumtion of an irrelevant signal, but that background makes the idea uite lausible. ee, e.g., bolafia 996 and Cetina and Bruegger. 3 s we can show, the result in emma 3 still holds when the market rice and the otimal contract reflect the ossibility of joint shirking. 4 We take out the direct investor from the model mainly for two reasons: i With only two informed traders in the market, it is easy to comare the results in this section with those in the last section; ii This emhasies the idea that the agent-traders can coordinate to shirk and herd in trading only when they constitute a substantial fraction of market articiants and their trades could ossibly move the market. 9

20 The noise signal is indeendent of the fundamentals: the relevant signal s and the liuidation value v. It should be ignored. But, it serves as a coordination device. Further we assume that the irrelevant signal has two values: or, and they are observed with eual robability. 5 Uon observing the irrelevant signal, the traders both buy x shares if the signal is or sell x shares if the signal is, without acuiring the information. 6 In this subsection, we will discuss the euilibrium in which only mark-to-market contracts are offered. s we have discussed earlier, when mark-to-market contracts are offered to agent-traders in the economy, they have an incentive to use the noise signal as a coordination device and trade the asset in the same direction without roducing any information and, in this way, they can maniulate the rice to get a higher ayoff. We call this tye of euilibrium a aniulation Euilibrium, which we formally define below: Definition ure trategy aniulation Euilibrium ure trategy aniulation Euilibrium is a Nash euilibrium in which a trader roduces information and trades truthfully only when he does not observe the irrelevant signal. Uon observing the irrelevant signal, the traders both shirk and buy x shares if the signal is, and they both sell x shares if the signal is. In the above definition, the agent-traders coordinate on the irrelevant signal, r, to maniulate the market rice, and the robability of this noise signal,, can be interreted as the roensity of shirking. In a ure trategy aniulation Euilibrium, the ricing function is:. 3 Given the euilibrium ricing function χ, and with λη similarly defined as in 9, the otimal ark-to-arket Contract w,, v, which allows for coordinated rice maniulation by agent-traders uon observing the noise signal, must satisfy the following incentive constraints: When informed, a trader is willing to trade truthfully when the other trader is doing so: IC. b s b s s b s b This eual robability assumtion is urely for simlification, and our analysis below can be generalied to any robability secification between including ero and one. 6 Of course, the traders can buy or sell on both signal values, and that is euivalent to assuming that the robability of r or is eual to one.

21 In the absence of the irrelevant signal, a trader has an incentive to roduce information when the other trader is doing so: IC. b max b κ, κ, s b b s s s s b s b 34 which can be rewritten as: IC. b κ, s b s s b s b κ 35 Without information, a trader is willing to buy/sell when the other trader is doing so: IC.3 b. s b s s b s b 36 Uon observing the noise signal, a trader has an incentive to shirk and buy/sell instead of roducing information and trading truthfully if the other trader is doing so: IC.4 κ b κ. s b s b b b b s s s s 37 If an incentive comatible mark-to-market contract is offered to both agent-traders, the exected ayoff to a trader is: b b b b s s, 4 s s 38 and the otimal contracting roblem can be written as: min w W subject to the IC conditions in

22 We solve the otimal contracting roblem 39 by first solving a weakened roblem with only IC condition IC.. Then we show that the otimal solution for the weakened roblem satisfies IC., IC.3, and IC.4. By droing IC., IC.3, and IC.4 from the full rogramming roblem in 38, we can write an alternative simlified rogramming roblem: min w W subject to the IC conditions in This roblem is similar to the one in 5, and can be characteried in a similar way. First, as analogously to emma and roosition, we can show that the constraints in 35 are binding for the otimal solution to 4, and the solution is characteried by four cutoff values {,,, }. Define: b b s s b s µ b b s µ b s s b b s s b b s s. 4 In the following roosition, we first characterie the solution to 4, then show that it is also the solution to 39. roosition 5 Characteriation of the Otimal Contract In a ure trategy aniulation Euilibrium, i the otimal contract w,, v takes the following form: when b, there exists bη such that, for any < η,w, b, v, and for any bη w Φ and,w, b, v w ; when s, there exists Φ b sη such that, for any > sη,w, s, v, and for any sη, w, s, v w. oreover, ii satisfy the following conditions: σ µ ln ln b b x µ 4 σ µ ln ln, s s x µ w{ Φ b Φ b } w Φ s Φ s κ s s w { Φ Φ } κ, b x. 44 b s b s > b 43 λη

23 3 roof: ee endix. s before, the four euations 4 and 43 allow for the determination of the four - cutoff oints. Immediately, we can check that when, the above roosition is reduced to roosition. roosition 5 also imlies that < η η b b and < η η s s, which says that under the otimal contract, when the trader buys/sells the stock, he will be better off if the other trader is doing the same. This generates the externality effect, which is at the root of the existence of an euilibrium with maniulation. imilar to the last section, in order to demonstrate that the otimal contract characteried above can be sustained in a Nash euilibrium, we need to check that the ayoff to the rincial is higher than his best alternative ayoff if he does not hire a trader. In a ure trategy aniulation Euilibrium, the exected net ayoff of the investment under informed trading is: { }. x x 45 gain, let denote the ayoffs to the rincial. We need to have },,, max{ s b, where b is the ayoff from uninformed buying at date ; s, the ayoff from uninformed selling;, the ayoff from doing nothing, and, the offeuilibrium ayoff from offering an otimal standard contract. We have: { } { } { } { }. x x x x x x s b 46