Liquidity and the cost of market-based compensation in informationally efficient markets

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1 Liquidity and the cost of market-based compensation in informationally efficient markets Riccardo Calcagno Florian Heider February 205 Abstract We provide a model which adresses the problem of compensating a CEO when the impact of his actions extends well beyond his tenure at the firm. Weexaminetheoptimal design of market-based pay when the firm s stock price is the outcome of self-interested, asymmetrically informed trading in presence of risk-neutral market makers. We show that even if the stock price is efficient for valuation, it is not efficient for providing incentives. The reason is that the stock price does not react one-to-one to the actual effort decision of the CEO. We find that a low the sensitivity of the stock price to managerial effort enhances high market-based compensation. This property alters the comparative statics of the textbook trade-off between risk and incentives. Our results yield a number of novel empirical predictions on the impact of market conditions, and in particular stock price volatility and noise trading volume, on optimal pay structures. Keywords: Market-based compensation, liquidity, noise trade. JEL Classification Codes: G4, G34, D86. Introduction Stock markets aggregate dispersed, private information about the future value of stocks through information-based trading. Shareholders make use of this by linking the top management pay We are grateful for comments and conversations with Ron Anderson, Patrick Bolton, Arnoud Boot, Ingolf Dittmann, Uli Hege, Martin Hellwig, Bengt Holmström, Kose John, Lasse Pedersen, Jean-Charles Rochet, Jozsef Sakovics, Peter Swan, Daniel Wolfenzon and Xavier Vives. We also thank seminar participants at the London School of Economics, the ESSFM meeting in Gerzensee, the Bank of Canada, the Universitat Pompeu Fabra (Barcelona), HEC (Jouy-en-Josas), the Universidad Carlos III de Madrid, NYU, the Tinbergen Institute, ESSEC, the ECB, the University of Turin, the University of Amsterdam, the University of Edinburgh, the Ente Einaudi and the European Business School. This paper supersedes a previous paper with the title "Market based compensation, price informativeness and short-term trading". The views expressed do not necessarily reflect those of the European Central Bank or the Eurosystem. EMLYON Business School, Department of Economics, Finance and Control, Avenue Guy de Collongue 23, 6934 Ecully Cedex (France), calcagno@em-lyon.com European Central Bank, Financial Research Division, Kaiserstrasse 29, D-603 Frankfurt, florian.heider@ecb.int

2 to the stock price as a mean to align their respective interests. This in essence describes the monitoring role of financial markets (Holmstrom and Tirole (993)). But Gjesdal (98), Paul (992), Lambert (993) argue that information related to valuation purposes is not necessarily relevant for contracting purposes: the stock market does not distinguish between information useful to make inference about the stochastic element of value and information assessing the management s contribution to it. In this paper we examine the consequences on the optimal remuneration scheme of this misalignment of purposes, which in turn is due to the different interests between informed speculators who collect private information in order to earn trading profits on the stock market and shareholders who aim to provide incentives using the price. We first confirm that this "conflict of interests" crucially affects the design of executive compensation when the impact of the management s decisions extends beyond his tenure. Then, we show that even in an informationally efficient market with infinite risk tolerance the stock price is a poor signal in terms of provision of incentives since it reacts weakly to changes in managerial effort. Finally, we explicitly compute how the optimal remuneration scheme varies with characteristics of the stock market such as liquidity, the degree of adverse selection contained in the trade, the volatility of the noise trade, which all affect the sensitivity of the stock price to managerial actions. The main intuition of our results is the following. An efficient stock market aggregates private signals about the future firm value (Hellwig (980), Vives (995)) and a very liquid stock price weakly reacts to orders, which contain information about the future value of the firm (Kyle (985)). If market makers have infinite risk-tolerance, we prove that the semi-strong efficient stock price reacts to managerial shirking less than one-to-one. To counteract this low sensitivity to managerial shirking shareholders-principal needs to provide high power marketbased pay. Moreover, we find that the sensitivity of the stock price to managerial actions is proportional to the aggressiveness of market makers pricing rule. Thus market characteristics which reduce market-makers pricing aggressiveness, such as a greater noise trade, increase the pay-for-performance sensitivity. This result holds as well if we allow traders to optimally acquire more precise information. We also analyze the case in which the principal may collect other sources of information, such as for example accounting information or internal performance measures. If these signals are disclosed publicly, their value added in terms of information enters into the price if the market is informationally efficient. These additional performance signals increase the stock price informativeness, hence its value as an incentive device, but are redundant for contracting purposes when the stock market is semi-strong efficient. Only internal control measures which are not publicly disclosed are useful for contracting. Our model provides several novel empirical implications. A highly volatile stock should be associated with a higher market-based pay-for-performance sensitivity (PPS) because in asemi-strongefficient market ex-ante volatility is due to higher price informativeness. Also, contrarily to Holmstrom and Tirole (993), a high volume of noise trade should be associated with low market-based PPS because noise trading increases market liquidity, hence reducing the sensitivity of price to managerial action but at the same time increases the noise term in the price. In terms of policy interventions our findings suggest that imposing a cap on top management 2

3 compensation may prove counterproductive because the market-based PPS optimally increases when markets are very liquid. The same intervention may be effective if firms are induced to collect more internal information about performance, information that should not (necessarily) be disclosed.. Related literature Some theorists (Gjesdal (98), Paul (992) and Lambert (993)) argue that the valuation and the contracting roles of stock prices are not necessarily aligned. In particular, Paul (992) shows that stock-based pay does not strike the right balance between managerial activities in a multi-tasks setting where stock price is an unbiased assessment of total firm value. The stock price is a sufficient statistics for the whole value but not for individual contributions. Hence when the principal provides incentives through the stock price, (s)he can not provide the right incentives on each activity. Under some conditions the same result holds also in the case the firm value is created by only one managerial activity. However, the trading model used in Paul (992) is very specific since liquidity in the market is always infinite, i.e. there is no adverse selection. In other terms, there is no aggregation of information via self-interested trading. Contrarily to Paul (992) we show that the main "conflict of interest" between principal who want to provide incentives using the stock price and traders who aim to earning market profits from their superior information holds also in a setting where the firm value is generated by only one managerial activity, even when the stock market is informationally efficient. Several theoretical and empirical papers argue that greater stock liquidity is associated with more market-based compensation. Holmstrom and Tirole (993) (HT in the following) show that higher liquidity on the market encourages an informed speculator to collect more private information about the future firm value, information which is impounded in the stock price. A more informative stock price is used more for providing incentives. Garvey (997) shows that the optimal contract has a higher market-based component when stock liquidity is higher because the latter reduces the risk bared by the manager receiving power incentives. Chordia et al. (2008) find that liquidity Granger-causes informativeness of prices and a more informative price is used more for compensation purposes, as Garvey and Swan (2002), Kang and Liu (2008) verify empirically. We also predict that greater stock liquidity should increase market-based pay, but for different reasons. In our model greater liquidity is associated to a lower sensitivity of the stock price to managerial actions. To unfold this lower sensitivity the principal needs to pay high market-based PPS. Kang and Liu (200) study the effect of increased uncertainty about the future firm value on the provision of incentives. When uncertainty increases more informed stock traders come to the market and as a result the stock price becomes more informative and it is used more to provide incentives. Contrarily to our model, in Kang and Liu (200) the informed traders collect private information on the stochastic element of value, and the market on aggregate is assumed to be able to observe the actual managerial action. This in turn implies that the price reacts one-to-one to (out-of-equilibrium) shirking so that the valuation and the contracting role of market prices are fully aligned. The accounting literature is rich of contributions studying the value of accounting informa- 3

4 tion as performance measures, also relative to other sources of information. Lambert (993) and Bushman and Smith (200), among others, provide two complete reviews. Two papers are close to ours in analyzing how earnings and price are used in executive compensation contracts, i.e. Bushman and Indjejikian (993) and Kim and Suh (993). The first studies an agency model with two managerial actions and compares the case where earnings and price reflect or not the same information sets. The latter focuses on the contracting role of stock price, which enters into the incentive contract more than proportionally to its informational value. Contrarily to these two papers, public signals, such as earnings, are useless for contracting in our model because we assume the market has an infinite risk-tolerance. 2 Model set-up The model assumes a standard moral-hazard problem between the owners and the management of a publicly traded firm as in Holmstrom and Tirole (993), Kang and Liu (200). We depart from these papers by introducing active trading of the firm s shares in a large competitive market where a continuum of informed traders have heterogenous, dispersed and imperfect information about the future value of the firm. Their self-interested trading leads to an aggregation of information in the stock price that is useful for incentivizing management. Agents. A publicly held firm is run by a risk-averse manager. We normalize the amount of firm shares to one, so that the total firm value corresponds to its per-share value. The firm is entirely financed with equity and it is owned by two groups: insiders (owners) and outsiders (traders). The owners constitute a well-diversified (i.e., risk-neutral), passive, value oriented collective whose investment horizon coincides with the life of the firm. They hold a constant fraction of the firm. The remaining fraction is traded freely in the stock market by the outsiders. Outsiders buy and sell (fraction of) shares of the firm based on their private information about thefuturevalueofthefirm and based on the information publicly revealed by stock prices (informed traders). In addition to informed traders, on the market operate also uninformed (noise) traders who trade for exogenous liquidity reasons, e.g., stochastic life cycle motives, and risk-neutral market makers who stand ready to buy (sell) when the price is low (high). Manager tenure and information for contracting. There are four dates, =023. At =0the owners hire the manager and offer him an incentive contract paying an income at the end of his tenure. After signing the contract, the manager exerts an unobservable, costly effort that determines the expected long-run value of the firm, which realizes at the last period =3, when the firm is liquidated. This value is given by = +, wherethe stochastic element (0 ). The shock is a reduced form description of factors that affect firm value but are outside the control of the manager. To capture the idea that the manager s actions affect firmvaluebeyondhistenure,we assumethatthemanagerquitsthe firm at =2. The manager cannot wait to be paid until =3when the full consequences of We exclude the possibility the manager can trade stocks of the firm on the public market, since this would be considered as insider trading. The results we illustrate in the paper do not change in the case insiders can trade on the market, as long as their ownership fraction remains constant and if their information set coincides with the outsiders one. See also Kang and Liu (200) for a discussion over this hypothesis. 4

5 his actions materialize. Instead, he must be paid when he leaves the firm at the latest. As a consequence, the long-term value of the firm is not available for incentive contracting. The principal is then bound to contract upon other performance signals. We assume there are two contractible variables in the model. First, the trading of the firm s shares occurring at = results in a stock price that can be used in the incentive contract, which we describe next. The power incentives provided through define the market-based pay. In order to allow for a simple alternative to market-based pay, we assume that an unbiased signal about the future liquidation value of the firm arrives at a =2and that = +, where (0 ) is an observational error independent of all other random variables in the model. We first analyze the case in which represents a performance measure internal to the firm not publicly available. 2 According to standard practice, we limit our analysis to contracts that are linear in the contractible variables. Managerial contract. The manager s income contains a fixed wage 0,amarketbased element denoted by, and a non-market-based part contingent on the signal. The manager sincomeisthengivenby = () We follow Holmstrom and Tirole (993) and normalize the price and the managerial contract in order to separate the pricing and the incentive problem. In terms of actual implementation of such a contract, the proposed normalization assumes that the market-based incentives are paid in cash at =2when the manager leaves the firm. One could think of as a long-term incentive plan whose value depends on the stock price. Thus, the manager is paid 0 + in cash while the amount is paid in shares transferred from long-term inside owners to the manager. The amount can be considered as a performance bonus paid in the form of shares that the agent can liquidate at his departure. The fraction of shares that must be [ 0 ] since transferred to the manager to pay the accounting bonus is given by = [ 0 ] is the fair price of the firm s shares given public information at the moment the manager leaves. This normalization leaves all payoffs unchanged while the net liquidation value of the firm becomes =.Defining ˆ as ˆ = 0 +(+ ) (2) we can write the manager s income in terms of the normalized share price: 3 = b 0 + bˆˆ + b (3) 2 Thecasewhere represents a publicly observable signal, such as accounting report, is analysed in Section Note that the prices and ˆ are informationally equivalent. 5

6 where b 0 = 0 +, bˆ = + and b =. 4 The optimal incentive contract that inside owners offer to the manager at =0maximizes the expected net value of the firm, [ ], subject to the risk-averse manager acting in his own interest, =argmax 0 [( 0 )] and subject to the manager participating, [()] 0, where we have normalized his reservation utility to zero. The manager s utility is given by () = exp[ ( 2 2 )] where is the coefficient of absolute risk-aversion of the manager and 2 2 is his (monetary) cost of effort. 5 Competitive stock market. The one outside share of the firm is traded in a competitive stock market between a continuum [0 ] of risk-averse informed traders, uninformed noise traders and risk-neutral market makers. We follow Vives (995) to examine the aggregation of dispersed information through trading. At the opening of the stock market, at =,each informed trader privately receives a different, imperfect and unbiased signal about the future value of the firm, = +,where are i.i.d. random variables, (0 ). Contrarily to Kang and Liu (200) we assume that informed traders are interested in collecting signals over the whole of firm value, and not only on the stochastic element that influences it. This because the purpose of informed traders is to trade knowing more than the market maker about, and not to detect managerial action. 6 When collecting private information over the firm value the interests of speculators trading on the market and those of shareholders-principal are not perfectly aligned and we refer to this as the "conflict of interest" between trading and incentive provision. AsinVives(995),themarketasawhole knows theliquidationvalueofthefirm since 4 Substituting for = ˆ 0 + into () one obtains ˆ 0 = ˆ = so that we can identify the coefficients in the contract with normalized values as in (3). 5 More precisely, inside owners solve for the contract that maximizes the total value of the firm net of the dilution term transferred to the manager. That is, at =0the optimal contract maximizes [ 0 ] [ ] under the Incentive and the Participation Constraints. Recall that the amount [ ] is paid transferring the fraction of shares to the manager at her departure, when is known, so that [ ]=[ 0 ]. In expected terms the cost of this remuneration component for the inside owners is equal to [[ 0 ] ] =[ 0 ],. Thus, the inside owners maximize [ 0 ] [ 0 ] =( )[ 0 ] which corresponds to maximizing [ 0 ] the value of the share at the moment of contracting. 6 Given that even inside owners cannot observe the managerial action which determines firm value, it is hard to argue that external traders can disentagle the effect of this action from the noise term outside the management s control. 6

7 traders individual and possibly very large errors cancel out in the aggregate due to the law of large numbers: R 0 =0. The information about the liquidation value of the firm is, however, not directly accessible for contracting since it is dispersed among many traders. Instead, the stock price aggregates this information via decentralized, self-interested trading. 7 Few observations are in order at this point. We use the Vives (995) model since it is a very parsimonious one in order to describe the aggregation of private information in a competitive environment. This market has infinite risk-bearing capacity, a characteristics which allows to isolate the conflict of interest between trading and incentive provision in the neatest possible way. Notice that this market is extremely efficient from an informational point of view: if an additional signal over the future value is revealed publicly, then its information content is absorbed in the price. 8 An informed trader maximizes the CARA utility of the return from trading the firm s shares at price. He has a constant absolute risk aversion coefficient, and rational expectations, i.e. he uses all the information available. He submits then demand schedules that condition the amount he buys or sells on his information set. Noise traders demand is random, normally distributed, (0 ) and independent of all other random variables of the model. Thepriceoftheshareat =3is simply the liquidation value of the firm net of any cash payment that is made to the manager, 3 = = 0. A competitive, risk-neutral market making sector ensures that the stock price at =is semi-strong efficient and reflects all the information publicly available at that point of time, which we denote by.themarket maker observes the aggregate limit order book () resulting from the joint demand of informed and noise traders: Z () = ( ) + 0 Given that is a performance measure known only internally to the firm coincides with the limit order book (). The market maker sets the price efficiently = [ 0 ()] (4) Using the normalized price b allows a simpler expression for the equilibrium price: b = 0 + ( + )=[ ()] (5) given that b is informationally equivalent to and that () is linear in the price. 9 7 Notice that in our set-up there is no difference between information that is useful for trading and information that is useful for incentive contracting (see Paul 992). Speculators value information about shocks to final asset value while incentive contracting values information about the shock that garbles the impact of effort on the value of the firm. Both shocks are identical here and are given by. 8 In Appendix B we show that our results are overall robust even if we use the Kyle (985) model of financial markets. 9 To see this develop [ 0 ()] = [ ()] 0. Then using (4) obtain [ ()] = 0 +(+ ) = b given (2). Hence the result in the text, noticing that b is a linear transformation of, hence informationally equivalent. 7

8 Figure summarizes the sequence of events and the different horizons of the firm, the manager and the stock market. The red-arrows indicate the horizon of the manager relative to the horizon of the firm s owners and of traders. 3 Solution In this section we characterize the market equilibrium and the optimal contract in the class of linear contracts described above. 3. Pricing The standard CARA-normal framework admits equilibria in which the demand of informed trader is linear in prices as well as in the private signal (Vives (2009)). We conjecture that, given the pricing function (4), the demand schedule chosen by informed trader at =is equal to: ( ˆ) = + (ˆ) (6) where the variable denotes how aggressively a trader uses his private information and the function is linear in. 0 The aggregate limit order book is then equal to (ˆ) = + (ˆ)+ = ( + )+ + (ˆ) = + (ˆ) where = ( + )+ isthepartoftheorderbookwhichisinformativeaboutthelong-term value of the firm. The price setting condition (5) can be written as ˆ = [ (ˆ)] = [ ] Calculating this conditional expectation gives the following standard result (see, for example, Vives (2009)): Proposition (Pricing) The normalized stock price ˆ is given by: b = [ (ˆ)] = ( )+( + )+ (7) where is the hypothesized equilibrium effort of the manager, is the actual effort, = the inverse of the stock liquidity measure (as in the informed traders aggressiveness, = Kyle (985)), and ([ˆ ]) = + 2 the informativeness of the equilibrium price given the order flow. The stock price equals the weighted average of expected and actual firm value plus noise terms, one of which is due to uninformed noise trading and is therefore not related to the value. In order to provide incentives the principal has to distinguish between the anticipated 0 We prove this statement in the proof of Proposition. 8

9 equilibrium effort, and the actual effort in the price. In equilibrium coincides with but to derive the equilibrium they have to be kept separate. The sensitivity of the normalized price to managerial effort is proportional to the aggressiveness of the pricing function used by market makers. The equilibrium effort determines the market s prior expectation of firm value whereas an (off-equilibrium) deviation determines actual value and thus drives the signals traders use to update their information. Rewriting (7) as b = [ ()] = + ( )+ ( + ) one can identify an important characteristics of the equilibrium price. Denoting with 0 the public information available at =0,off-equilibrium we have [b 0 ]= + ( ) 6= = [ 0 ] whenever 6=, i.e. the expected (normalized) price b is a statistically biased signal of the managerial action, hence of the firm value. This is due to the fact that, in case the manager deviates from equilibrium, as long as 6=, the stock price does not react one-to-one to managerial shirking, unlike the performance signals on which the standard risk-incentive tradeoff is based (see, for example, Prendergast 999). The reason is that the information signals that traders receive contain both actual managerial effort and the realized shock : an effort lower than expected does not necessarily trigger a large amount of sales by the informed traders. In particular, it does not if managerial shirking is coupled with a low realization of the shock. The statistical quality of the expected price b as a predictor of firm value depends on market liquidity and other microstructure characteristics of the financial market. Overall a more liquid stock, i.e. one with lower, is less powerful to detect shirking since it absorbs orders without much price impact. Given that the sensitivity of the price to actual managerial effort, the market price is inherently weak for detecting shirking. Both the informativeness of the price and the liquidity of the market, measured by are endogenous and depend in particular on the aggressiveness with which traders use their information. The aggressiveness in turn depends on traders risk-aversion and on the precision of their information: 2 ahigher certainly improves the quality of ˆ as an incentive signal since it increases its sensitivity to the actual effort without increasing the noise it contains. Notice here the difference with Kang and Liu (200), where the private signals contain information only about the noise term. 2 Vives (995) shows that this same static equilibrium holds when the informed traders receive an information signal once and for all at the opening of the market and have long investment horizon, i.e. they liquidate their positions when realizes. In such a case, the informed traders enter a position after receiving their private signal and keep it until the end. There is no information trading in the subsequent periods in which the market is infinitely liquid. 9

10 3.2 Incentive contracting The optimal contract (b 0 b bˆ ) maximizes the value of the firm at =0net of the manager s income subject to his incentive compatibility and his participation constraints, subject to and max [ ] (8) 0 ˆ =argmax [] 2 [] 2 2 [] 2 [] where we have used the certainty equivalent for the CARA utility of the manager. The coefficients bˆ and b measure the total market-based and non-market based pay-performance sensitivity (PPS). The next Proposition provides the solution to problem (8). Proposition 2: The optimal weights respectively on the normalized stock price ˆ and on the performance measure are: bˆ = b = Σ + + Σ (9) Σ + + Σ (0) where Σ =( 2 + ). The optimal effort is equal to = + + Σ () In order to interpret the weights bˆ and b it is useful to view the optimal contract in Proposition 2 as the outcome of a two-step procedure (see Prendergast, 999). The first step aggregates the information contained in the stock price ˆ and in the public signal to obtain an unbiased statistical estimate of the future liquidation value of the firm. The second step incentivizes the manager to exert effort using the estimate obtained at the first step. 3 Using ˆ and, the best (linear) estimator of the long-term value, thatwedenotebyˆ is unbiased, [ˆ ] = and it minimizes the Mean Square Error [(ˆ ) 2 ]. We can write ˆ = + where is a noise term orthogonal to. This estimator is given by ˆ = Σ( ˆ + ) (2) The weights the estimator ˆ gives respectively to the stock price ˆ and to the signal are proportional to their respective precision and. 4 The manager is then given optimal 3 As for comparison, consider the hypothetical case in which the future liquidation value is available for contracting. There is then no need to estimate it and the first step is redundant. The optimal contract must only provide incentives. The manager s income would simply be = 0 +. The optimal contract would have =, which incentivizes the manager to exert an effort level = + (+ ). 4 In the Appendix (see Result 3) we show that Σ = (ˆ ) =() measures the estimation error. 0

11 incentives using the estimator ˆ, = b 0 + +( ˆ (3) + Σ) Combining (2) and (3) provides the same compensation contract as in Proposition 2. Interpreting the solution with this two-step procedure shows that the principal suffers an additional estimation error Σ when trying to infer the actual effort of the manager from (b ) than if he could contract on the long-term value : contracting on would allow to reach the second-best effort = while () represents a third-best solution. The pair (b ) then performs + worse than firm value as a contracting signal, even if (i) the stock price is informationally efficient and the stock market has an infinite risk-tolerance and (ii) the information contained in is not redundant for contracting. 5, 6 We can understand the comparative statics results of the total power incentives (bˆ + b ) contained in Proposition 3 starting from a discussion of the case where the only contracting signal is the normalized price ˆ so that the first step of information aggregation explained above does not play a role. In such a case the market-based PPS would be equal to the total incentives Σ b = 0 + ( +Σ0 ) with Σ0 =( 2 ). Let us compare this solution with the optimal contract the principal could write observing a different signal b = + where is still a noise term orthogonal to, and() =Σ 0. The incentive load on b in the optimal contract would be equal to : the lower the sensitivity of b to managerial action, the higher is the +( +Σ0 2 ) amount of power incentives needed on that signal. This explains why b as in (9) is proportional to the (inverse of the) sensitivity term Σ 0 =() =(ˆ). When providing marketbased incentives, the principal first needs to correct for the low sensitivity of the price to the managerial action, increasing the market-based loads by the factor Σ 0. Then the optimal incentive is computed taking into account the effect of risk the remuneration imposes on the agent, as in the standard procedure as illustrated for example in Prendergast (999). When the impact of the stochastic element on value is greater (low ), the moral hazard is severe. The adverse selection in the order flow is also greater because any information advantage is crucial to predict value. With high adverse selection the market makers price the stock aggressively (Kyle (985)) and the price is highly sensitive to shirking: the term Σ 0 is low. 5 Thesamesignal would be redundant however if was contractible. 6 A further interpretation of the optimal pay structure is obtained by writing ˆ = = (4) The ratio of the weights on performance measures is given by the ratio of their signal-to-noise ratios (Banker and Datar (989)). The signal-to-noise ratio of non-price information is since the sensitivity of the performance measure to effort is one. The sensitivity of the stock price to managerial effort is and its noise is [] = 2 : thus, the signal-to-noise ratio of the stock price ˆ is given by its informativeness. The ratio of market to non-market based pay is then equal to the ratio of the precisions of two exogenous shocks contained in the respective signals, i.e. the noise trading and the noise of non-price information, times an endogenous market factor.

12 A higher volatility of the shock at the same time increases the risk the manager is exposed to when she receives pay-for-performance, and this in turn reduces her incentives. Overall the first effect ("sensitivity") prevails and this explains why the contract performs low incentives b. 7 If there is a greater uncertainty on the market due to a high noise trading activity (i.e. is low), adverse selection in the order flow is low. As a consequence, market makers price the stock less aggressively and market liquidity is high. But this in turn implies that the stock price is less sensitive to managerial actions and the principal needs to pay a higher load on price to provide incentives: the term Σ 0 ishigh. Noticethatatthesametime,withlow the estimation error Σ 0 is high, which means that the risk the agent is exposed to through performance-based pay is high and this in turn reduces incentives. As for the comparative statics w.r. to the "sensitivity" effect prevails and b is high. Thus total incentives are decreasing in.sincehigh noise trading activity makes expensive to provide market-based incentives, the effort that is implemented at third-best is decreasing in as well. When the aggressiveness of informed speculators trading increases, the order flow conveys more precise private information and market liquidity goes down. This has two main effects: (i) the price is a better signal for detecting shirking and (ii) the market makers pricing rule is more aggressive. Both effects increase the sensitivity of the stock price to shirking so that the principal needs to pay less market-based incentives to provide incentives, i.e. b reduces. 8, 9 Proposition 3 collects all the comparative statics of the optimal weights bˆ and b in the full-fledged case where is contractible and informative. Proposition 3: Let bˆ and b be the optimal weights as in (9), (0) respectively. Then, if : (i) bˆ is increasing in,increasingin,increasingin and decreasing in ; (ii) b is increasing in,decreasingin,decreasingin and increasing in ; (iii) bˆ + b is increasing in,decreasingin,decreasingin and decreasing in. As in the standard principal-agent model, both (9) and (0) decrease with the noise term that hides the manager contribution to firm value, with his degree of risk-aversion and with the cost of effort provision. The comparative statics of the total performance-based pay bˆ + b have been discussed above. As intuition suggests, the principal increases the weight of a signal when its precision increases: hence bˆ increases in and, while b increases in. 7 Σ Start from ˆ = + + Σ and substitute for Σ = 2 and for = 2 + to obtain ( 2 ) ( 2 + ) +( +(2 ) ), fully explicit in the fundamental parameters (except for =, obviously increasing in ). ( 2 ) ( 2 + ) +( +(2 ) ) = ( 2 ) ( 2 + ) +( 8 +(2 ) ) ( ) = ( ) 2 9 In Appendix A we show that the optimal amount of long-term market-based incentives present the same comparative statics as bˆ with respect to the three parameters,,and. 2

13 3.3 The signal is publicly disclosed Contrarily to Bushman and Indjejikian (993) and Kim and Suh (993), in our setup the information contained in the performance measure is redundant for contracting if is public. Proposition 4: If is publicly revealed at =, the normalized stock price ˆ is given by: b = [ ()]= ( 0 )+( 0 + )( + )+ 0 + where, and are as in Proposition, while 0 = + is the inverse of the stock liquidity measure and = + The optimal weights respectively on ˆ and on are: ˆ = + + Σ Σ ( + ) = 0 where Σ =( 2 + ) as in Proposition 2. The optimal effort is equal to (). This result is a consequence of the semi-strong efficiency of prices together with the infinite risk-tolerance of the market. The market price efficiently aggregates all the information as soon as this becomes public. Proposition 4 shows that the stock price compounds as well the information about firm value (and managerial effort) included in the public signal. Intuitively, when the measure is announced publicly, the market efficiently aggregates its information content about the future firm value in the price, making redundant the first step of "aggregation of information" of the Prendergast (999) procedure illustrated above. Given that the estimation error Σ the principal makes in predicting isthesameinthetwocaseswhen is public and when it is private information, it is intuitive that the severity of the moral hazard problem is the same in the two situations. This explains why the third-best effort is the same in the two cases illustrated in Proposition 2 and 4 respectively. 4 Empirical Implications Several empirical studies have tested the effect of stock liquidity, price informativeness and volatility on pay-performance-sensitivity (PPS). If one considers market liquidity as a fundamental parameter of the model, our Proposition 3 predicts that the market-based PPS increases with liquidity (i.e. decreases with ) as in Jayaraman and Milbourn (20). Thanks to our explicit market model, we can clearly disentangle the effects that each determinant of market liquidity, i.e. the noise trade volume, the informativeness of the stock price and the aggressiveness of informed speculators trading activity 20 has on the optimal contract. Implication A more liquid stock leads to lower sensitive of price to managerial effort hence to more market based pay. 20 We refer here to the expression = as a measure of the (inverse of) stock liquidity. 3

14 The same implication can be found in Holmstrom and Tirole (993) if the increase of liquidity is due to more uninformed trades. This implication is verified empirically in Chordia, Roll and Subrahmanyam (2008), Kang and Liu (2008), Chen and Swan (20) and Jayaraman and Milbourn (20), who all find that the proportion of equity-based compensation in total compensation is higher in firms with higher stock liquidity. Notice however that the reason we predict market-based PPS is positively related to market liquidity is different from the one in the above papers. We argue that this is inherently due to the misalignment of speculators information and information relevant for contracting purposes. Implication 2 A more informative stock price leads to more market based pay. Jayaraman and Milbourn (20) show that when a firm is added to the S&P500 Index, the PPS decreases. This empirical finding is consistent with our second empirical implication since an addition in the S&P500 Index is usually associated to higher liquidity generated by more noise trade, hence it is associated to lower price informativeness. Also, Kang and Liu (2008) find that various measures of stock price informativeness are positively correlated with the stock-based PPS for CEO s pay schemes. In addition to these existing findings, our model also generates new empirical predictions. Implication 3 A more volatile stock price leads to more market-based pay. With semi-strong efficient markets the price is ex-ante more volatility if traders expect it contains more information about the fundamental. A volatile price is also more informative and the market-based PPS should then increase. Implication 4 When the volume of noise trade increases, the stock-based compensation decreases. An increase in the volatility of noise trade coeteris paribus increases stock liquidity and it then decreases the sensitivity of price to managerial effort. Notice that this last prediction is opposite to the one in Holmstrom and Tirole (993) and it is robust to the case in which speculators endogenously chose the precision of their private information signal (see Appendix B). This because in our model the market for information is competitive while Holmstrom and Tirole (993) assume that only one speculator has access to private information. 5 Conclusions In the present paper we examine the consequences on the optimal remuneration scheme of the different interests between informed speculators who collect private information in order to earn trading profits on the stock market and shareholders who aim to provide incentives using the price. We first argue that this "conflict of interests" crucially affects the design of executive compensation when the impact of the management s decisions extends beyond his tenure. We show that even if the stock price is efficient for valuation, it is not efficient for 4

15 providing incentives. The reason is that the stock price does not react one-to-one to the actual effort decision of the CEO. This property alters the comparative statics of the textbook tradeoff between risk and incentives. Our results yield a number of novel empirical predictions on the impact of market conditions, and in particular stock price volatility and noise trading volume, on optimal pay structures. In terms of policy interventions our findings suggest that imposing a cap on top management compensation may prove counterproductive because the market-based PPS optimally increases when markets are very liquid. The same intervention may be effective if firms are induced to collect more internal information about performance, information that should not (necessarily) be disclosed. 6 References Bushman, R. M. and R. J. Indjejikian (993). Accounting income, stock price, and managerial compensation, Journal of Accounting and Economics, Vol. 6, pp Bushman and Smith (200). Financial accounting information and corporate governance, Journal of Accounting and Economics, Vol. 32, pp Chen, B. and P. L. Swan (20). CEO Incentives and Institutional Trader Monitoring are Substitutes: Theory and Evidence, AFA 202 Chicago Meetings Paper; 24th Australasian Finance and Banking Conference 20 Paper. Available at SSRN: or Garvey, G. T. and P. L. Swan (2002). What Can Market Microstructure Contribute to Explaining Executive Incentive Pay?: Liquidity and the Use of Stock-Based Compensation, mimeo. Gjesdal, F. (98). Accounting for stewardship, Journal of Accounting Research, Vol. 9, pp Hellwig, M. (980). On the aggregation of information in competitive markets, Journal of Economic Theory, Vol. 22, pp Holmström, B. (979). Moral hazard and observability, Bell Journal of Economics, Vol. 0, pp Holmström, B. and J. Tirole (993). Market liquidity and performance monitoring, Journal of Political Economy, Vol. 0, pp Jayaraman, S. and T. Milbourn (20). The Role of Stock Liquidity in Executive Compensation, TheAccountingReview, Vol. 87, pp Kang, Q. and Q. Liu (2008). Stock trading, information production, and executive incentives, Journal of Corporate Finance, Vol. 4, pp Kang, Q. and Q. Liu (200). Information-Based Stock Trading, Executive Incentives, and the Principal-Agent Problem, Management Science, Vol. 56, pp Kim O., and Y. Suh (993). Incentive efficiency of compensation based on accounting and market performance, Journal of Accounting and Economics, Vol. 6, pp Kyle, A. (985). Continuous auctions and insider trading, Econometrica, Vol. 53, pp

16 Lambert, R. A. (993). The use of accounting and security price measures of performance in managerial compensation contracts, Journal of Accounting and Economics, Vol. 6, pp Paul, J. (992). On the efficiency of stock-based compensation, Review of Financial Studies, Vol. 5, pp Prendergast, C. (2002). The tenuous trade-off between risk and incentives, Journal of Political Economy, Vol. 0, pp Vives, X. (995). Short-term investment and the informational efficiency of the market, Review of Financial Studies, Vol. 8, pp Appendix We make use of the following two results (see Vives, 2009, Technical Appendix). The first result is the standard conditional distribution for normally distributed variables: Result (Conditional expectation) Let be a ( ) a multi-variate normally distributed vector with mean, i=,2, and variance-covariance matrices Σ,then 2 = ([ 2 + Σ 2 Σ ( )] [Σ 22 Σ 2 Σ Σ 2 ]) The second result gives the certainty equivalent for CARA utility when wealth is a linear combination of a normally distributed variable and its chi-square distributed square: Result 2 (Certainty equivalent) Let (0 2 ) and = then µ µ 2 [ exp( )] = q exp +2 2( +2) 2 2 ProofofProposition Applying Result, we obtain µ [ ] = = ( )+ = ( )+( + ) with = From the maximization of CARA utility, we know that = [ b ] [ ] = [ ] b [ ] = + + (+ ) 6

17 and substituting for b = [ ]= one obtains ( b) = ( b) which is indeed linear in b as conjectured in (6). Finally, we can identify =. 6. Proof of Proposition 2 The mean and variance of the manager s income are, respectively: [] = 0 + ˆ [( ) + ]+ [] = 2ˆ[() 2 while the incentive constraint is Problem (8) becomes: max ˆ (ˆ + ) 2ˆ [() ]+ 2 ( ˆ + =0 + 2 ]+ 2 ( + )+2ˆ ( ) (A.) + )+2ˆ ( ) 2 (ˆ + ) 2 The first order conditions with respect to ˆ and are: = ˆ () () = ˆ + Solving for ˆ and yields ˆ = = which rearranged and using Σ ( 2 + ) become: ˆ = = + + Σ Σ + + Σ Σ We obtain again the optimal effort from the incentive constraint (A.): = µ + + Σ Σ + = + + Σ 7

18 Result 3: Σ = (b ) =[(ˆ ) 2 ] where b is the best linear unbiased estimator of = + i.e. ˆ = Σ( ˆ + ) Proof: Recall the definition of Σ =( + ) =( 2 + ). From the definition of ˆ one obtains (ˆ) = (Σˆ + Σ ) = (Σˆ)+(Σ )+2(Σ ˆ Σ ) = (Σ) 2 (ˆ)+(Σ ) 2 ()+2Σ 2 (ˆ ) Given risk-neutral pricing, ˆ = [ ˆ] so we can decompose the variance 2 Substituting in (ˆ) for the various terms: (ˆ) = () ( ˆ) = (ˆ) = (Σ) 2 (ˆ)+(Σ ) 2 ()+2Σ 2 (ˆ ) µ µ Σ 2 2 µ Σ which after some manipulations and using = 2 = can be written as (ˆ) = Σ ( + ) Given that b is the BLUE estimator of, wehavethatb = + where the error term represents the estimation error and it is orthogonal to b (hence to ˆ and ). We can then write (ˆ) =()+() and substituting for (ˆ) and (): Σ ( + ) = + () () = Σ ( + ) = Σ 2 Notice that this ANOVA decomposition gives interesting insights: () = (ˆ)+( ˆ) = ([ ˆ]) + ( ) so the more volatile (ex-ante) is the price ˆ, the more information it contains about. Henceforth volatility is not necessarily bad for contracting. 8

19 using Σ =( + ). Notice finally that an analogous result holds if the only contractible variable is the (normalized) price ˆ, where formulas must be corrected all through eliminating =0and its precision. 6.2 Proof of Proposition 3 A) Comparative statics of bˆ : Start from bˆ as in (9) and substitute for Σ = 2 + and for = 2 + to obtain bˆ = , fully explicit in the fundamental parameters = ( ) 2 = if ( ) = if ( ) = B) Comparative statics of b : Consider b as in (0) and substitute for Σ = 2 + to get b = ( +) 2 +( +(2 +) ) + ( ) 2 0 = ( ) 2 ( +) 2 +( +(2 +) ) = ( ) 2 ( +) 2 +( +(2 +) ) = ( ) 2 ( +) 2 +( +(2 +) ) = ( ) 2 Σ(+ C) Comparative Statics results of ˆ + = ) + ( +Σ ) : Substitute for Σ = 2 + and for = 2 + to get: ˆ + = + ( 2 + ) + ( 2 + )

20 = ( ) 2 = ( ) = ( ) = 0 ( ) Proof of Proposition 4 When ispublicthemarketmakersobservetheinformationset = {} (), i.e. to {} {}. Applying Result, we obtain µ b = [ ] = = ( 0 )+ 0 + = ( 0 )+( 0 + )( + ) with 0 = and = manager s income: We compute then the mean and variance of the [] = 0 + ˆ [( 0 ) +( 0 + )]+ [] = 2ˆ[( 0 + ) 2 +( 0 ) ]+ 2 ( + )+2ˆ (( 0 + ) + ) and the incentive constraint becomes ˆ ( 0 + )+ =0 (A.2) The participation constraint will bind since the manager s utility is increasing in 0 (through expected income alone) while the objective function of problem (8) is decreasing in 0 and the incentive constraint (A.2) is independent of 0. After substituting the binding participation constraint into the objective function and using (A.2) to substitute for, problem (8) becomes: max (ˆ( 0 + )+ ) 2ˆ [( 0 + ) 2 +( 0 ) (ˆ( 0 + )+ ) 2 ˆ + 2 ]+ 2 ( 20 + )+2ˆ (( 0 + ) + )

21 The first order conditions with respect to ˆ and are: 0 + = ˆ ( 0 + ) 2 +( 0 ) ( 0 + ) 2 + (( 0 + ) + )+( 0 + ) = ˆ ( 0 + ) + +( 0 + ) Solving for ˆ and yields ˆ = = We rearrange and use Σ 2 + to obtain: ˆ = = + + Σ Σ ( + ) + + Σ Σ ( ( + )) and substituting for = + into the expression of one finds =0 The optimal effort is recovered from the incentive constraint (A.2): = = + + = + Σ Σ ( + ) + Σ Σ Σ ³ Appendix A: Endogenous information collection The idea is that, ex-ante, that is before the market opens, each trader chooses the optimal precision of the signal = + he will receive just at the moment the market opens. Collecting information is costly: the cost is linear, for simplicity (see the final solution) ( )= To compute the optimal I first compute the "ex-interim" profit thetrader makes, that is the profit he makes once he has received the signal, and given that he can observe the price: 2

22 [Π b] = ( b)([ b] b) = ( b)([ b] b) Ã = ( +! 2 + b b) + 2 b + Ã = ( +! 2 + b [ ]) + 2 b + Ã = ( +! 2 + b b) + 2 b + Ã! = ( ( b) b) = 2 ( b) = 2 ( b) 2 ( + ) The trader chooses the optimal ex-ante, when his information set is 0, that is the original distributions postulated in the paper ( = +, = +, b = [ ], = ( + )+, and all the distributions of the r.v. and their correlations). The ex-ante profit, net of information collection costs is then: and [Π 0 ]=[ 2 ( b) 2 ( + ) 0]= 2 ( + ) [( b) 2 0 ] [( b) 2 0 ] = [( + + [ ]) 2 0 ] µ µ = [ ( ) 0 ] µ = [ ] µ = [ ] µµ = [ ] µ 2 = = + 22

23 substituting into the ex-ante profit: [Π 0 ] = 2 ( + ) = = 2 + µ + where =,sofinally: [Π 0 ]= and the optimal precision solves: with f.o.c: 2 +2 = ( ) 2 ( 2 2 ) max ( ) 2 ( 2 2 ) =2 ( ( ) ) 0 2 (3 2 ( ) 3 2 ) 0 ( ) 2 ( ) 2 = = = = 0 Solution is 2 = 8 + (2 +) which is decreasing in : high precision 2 (i.e. low volume) of noise trade, low precision that is low aggressiveness Appendix B: Using the Kyle (985) model of trading As in HT and in Kyle (985), let us assume that there is only a single informed risk-neutral trader who observes his signal andtradesanamount in order to maximize profits ( )[ ˆ ]. Being the only trader with private information, the informed speculator does not need to condition his trade on the stock price. Let us then rewrite our model according to these assumptions to make a more formal comparison with HT. 22 Caveat: we need in order to insure a solution in exists in the case 2 6= 0. 23