The Value of Performance Signals Under Contracting Constraints

Transcription

1 The Value of Performance Signal Under Contracting Contraint Pierre Chaigneau Queen Univerity Alex Edman LBS, CEPR, and ECGI Daniel Gottlieb Wahington Univerity in St. Loui January 7, 207 Abtract Thi paper tudie the value of additional performance ignal in compenation and financing contract under contracting contraint, uch a limited liability, monotonicity, or upper bound to pay or incentive. We how that informative ignal may have no value for contracting, becaue the payment cannot be adjuted to reflect the ignal realization contrary to the informativene principle, which wa derived auming no contracting contraint. We derive neceary and uffi cient condition for a ignal to have value under uch contraint. Our reult have implication for pay-for-luck, option repricing, performance-baed veting, performance-enitive debt, and the condition under which a principal hould invet in cotly monitoring. pierre.chaigneau@queenu.ca, aedman@london.edu, dgottlieb@wutl.edu. We thank Jame Dow, Steve Matthew, Nicola Sahuguet, and conference participant at the Econometric Society World Congre for valuable comment. Gottlieb thank the Dorinda and Mark Winkelman Ditinguihed Scholar Award.

2 Executive contract are typically baed on multiple ignal of performance, and the ue of multi-ignal contract i increaing over time. For example, with performance-baed equity, the number of ecuritie granted typically depend on performance relative to a threhold (or et of threhold). The proportion of large U.S. firm that pay their executive with performanceveting equity ha rien from 20% in 998 to 70% in 202 (Betti et al. (206)). Moreover, 86% of uch grant employ at leat one accounting threhold, and o their value depend on factor other than the tock price the tandard output meaure for executive contract. The urvey of Murphy (203) report that companie ue a variety of financial and non-financial performance meaure when determining CEO bonue. Additional performance ignal are alo ued in financing contract. Mano, Strulovici, and Tchityi (200) document that 40% of loan have performance pricing proviion, i.e. the coupon rate depend on ignal uch a the firm credit rating, leverage, and olvency ratio. Thu, the payment to invetor depend on factor other than cah flow the tandard output meaure for financing contract. The main theoretical jutification for including additional performance meaure in a contract i Holmtrom (979) informativene principle. Thi principle tate that any ignal hould be included in a contract if it provide incremental information about the agent performance, over and above the information already conveyed in output. Indeed, Murphy (203) write that the informativene principle wa widely embraced by many academic who ued it a the theoretical jutification for... performance meaure ued in CEO contract. However, real-life contract appear to violate the principle. Even though ome contract are baed on ignal other than output, they are not baed on every potentially informative ignal. The informativene principle wa derived auming no contracting contraint. However, contracting contraint are an important feature in real life. The mot common i limited liability: limited liability of equity applie to financing contract between entrepreneur and invetor; in compenation contract, the alary paid by a firm to a manager cannot be negative. In addition, in compenation contract, regulation and notion of fairne often contrain realized pay. For example, the European Union limit banker bonue to twice the level of alary; in March 206, Irael removed tax deductibility from banker realized pay that exceed 35 time the alary of the lowet-paid worker in the intitution (or 2.5 million hekel, if thi i lower); the leader of the UK Labour Party propoed a maximum wage; in November 206, the UK government Green Paper propoed that company pay policie tipulate a cap on realized pay; and in 203, French Prime Miniter Jean-Marc Ayrault propoed extending the pay cap on executive in tate-owned firm to non-tate-owned firm. Turning to oft contraint, outrage contraint (Bebchuk and Fried (2004)) may prevent companie from paying their executive above a certain level. Contraint can apply to the level of incentive (the enitivity of realized 2

3 pay to output). The European Union Shareholder Right Directive tipulate that tock-baed compenation hould generally not exceed 50% of total variable pay, limiting the enitivity of pay to the tock price. The financial crii ha led many commentator to propoe limit to equity-baed pay to reduce rik-taking incentive, and ome hareholder propoal aim to cap equity award. Thu, to apply the informativene principle to many real-life etting, we mut firt tudy whether it hold under contracting contraint, and if neceary extend it. Thi paper derive neceary and uffi cient condition for a ignal to have value under contracting contraint, thu hedding light on the condition under which contract hould incorporate additional performance ignal. We firt conider the tandard framework of rik neutrality and limited liability on the manager, originally analyzed by Inne (990) and widely ued in a number of etting (e.g. Biai et al. (200), Clementi and Hopenhayn (2006), DeMarzo and Fihman (2007a, 2007b), DeMarzo and Sannikov (2006), and the textbook of Tirole (2006)). Similar to Inne (990), we conider up to two additional contraint. Firt, the firm alo exhibit limited liability. If the ditribution of output atifie the monotone likelihood ratio property, then high (low) output ignal high (low) effort and the optimal contract i live-or-die the manager receive zero if output fall below a threhold q, and the entire output if it exceed it. A imilar reult hold if the payment i bounded not by the level of output but intead by a maximum et by regulation or ocial contraint. Second, a monotonicity (or free dipoal ) contraint require the firm payoff to be non-decreaing in output, and o the manager cannot gain more than one-for-one with an increae in output. The optimal contract i then an option on output: the manager receive zero if output fall below a different threhold q, and the reidual q q if output exceed it. A imilar reult hold if the enitivity of pay to output i capped not at one, but a different level due to regulation or ocial contraint. Under either contraint, the only non-trivial dimenion of the contract that the firm mut decide i the threhold q (q ) under the optimal contract, the payment below the threhold i automatically zero, and the payment above the threhold i automatically the entire output or the reidual. Thu, an additional ignal of performance will only be included in the contract if it affect the threhold, i.e. it i optimal for the firm to vary the threhold according to the ignal realization the firm will not wih to ue it to change any other dimenion of the contract. Under the live-or-die contract, changing the threhold q alter the payment (from 0 to q or vice-vera) only in a local neighborhood around q. A a reult, a ignal i only ueful if it affect the likelihood ratio that output equal q, i.e. i informative about whether output Ertimur, Ferri, and Mulu (20) dicu a 2004 hareholder propoal at Motorola to cap equity grant at $ million, and a 2004 propoal at Eatman Kodak to crap equity grant. 3

4 equalling q i the outcome of high or low effort. If the ignal ugget the manager ha worked (hirked), the firm decreae (increae) the threhold. Under the option contract, changing the threhold q alter the payment for all q q. Thu, a ignal i only ueful if it affect the likelihood ratio that output exceed q i.e. i informative about whether output exceeding q i the outcome of high or low effort. In both cae, the contract depend on a ignal if and only if it i informative about effort at the threhold q (q ), i.e. at an intermediate output level. Signal that are informative about effort only above and/or below the threhold are of no value, becaue the payment i bounded by either a limited liability or monotonicity contraint. A a reult, a ignal can be informative almot everywhere (i.e. at all output level except the threhold) yet till have zero value. We then extend the model to rik averion. Now, the contract take a more general form: under an upper bound on payment, the manager till receive zero if output i below a threhold, but doe not necearily receive the entire output if it exceed it. Thu, the upper bound doe not bind for all high output level. A a reult, the firm can make ue of the ignal at output level above the threhold (not jut at the threhold), a long a the upper bound doe not bind at thee output level. Thu, the neceary and uffi cient condition for a ignal to have value are weaker under rik averion. Quite eparate from extending the informativene principle, thi model alo generate the firt et of uffi cient condition limited liability, log utility, and a linear likelihood ratio for option to be the optimal contract when the agent i rik-avere. Even though option are commonly granted, the only exiting jutification in a moral hazard model to our knowledge i Inne (990), which require the agent to be rik-neutral. Unlike in the rik-neutral model where the manager i the reidual claimant for q > q, o that the number of option i fixed at, under rik averion it need not be. The reult have a number of implication. Our main theoretical implication i that the informativene principle need to be modified under contracting contraint: a ignal ha value if and only if it i informative about effort at an output level for which contraint do not bind, rather than at any output level. A econd theoretical implication i that the value of information i non-monotonic in output. Thu, the firm hould only invet in additional ignal on manager performance at moderate output realization. If output i low, the manager i fired anyway; if output i high, he i the reidual claimant anyway. Thu, in neither cae are additional ignal valuable. Moving to applied implication, our tronger condition for a ignal to have value under contracting contraint can potentially explain why real-life contract do not depend on a 4

5 many ignal a the original informativene principle ugget they hould. 2 For example, executive contract typically do not depend on the firm recovery rate in bankruptcy, the outcome of litigation againt the firm, and citation of major patent. Relatedly, Bebchuk and Fried (2004) argue that the common practice of paying manager for luck, i.e. not filtering out indutry hock, i leading evidence that CEO pay reult from rent extraction rather than optimal contracting. However, ince the informativene principle doe not automatically apply under limited liability, thee practice are not necearily uboptimal. If a firm uffer a catatrophe, the manager i typically fired anyway, regardle of whether it wa due to bad luck (e.g. poor indutry performance) or hirking, and o cannot be punihed further. However, the model doe ugget that pay-for-luck i uboptimal at moderate output realization. The reult alo have implication for the deign of option compenation, where output i now the tock price. They ugget that option repricing (which, empirically, nearly alway involve a lowering of the trike price) can be jutified if prompted by poitive ignal of CEO effort. Thi reult implie that the practice of lowering the trike price upon poor performance i generally ineffi cient. However, it alo give condition under which repricing can be optimal, contrary to conventional widom that it necearily reult from rent extraction. In addition, our model provide condition under which the number of option granted to the manager hould depend on additional ignal, a in the cae of performance-baed veting. Depite it popularity, we are unaware of any theorie that tudy under what condition performance-baed veting i optimal, and what performance ignal hould be ued. Simple intuition may ugget that the number of option that vet hould depend on a ignal if it provide incremental information about effort over and above that contained in the tock price. Intead, we how that it hould depend on a ignal if and only if the ignal affect the rate at which the informativene of the tock price change with the level of tock price. Thi i becaue the number of option that vet i only one component of the compenation contract, and even a ignal which i incrementally informative need not affect thi component. For example, bad macroeconomic condition are not individually informative about effort if they are outide the manager control. However, if effort affect the tock price in boom more than in receion, the number of option hould be higher in the former. Thi reult alo ugget that ignal that trigger veting need not be adjuted for luck. Converely, even if a ignal (ay, revenue) i informative about effort, it hould not affect 2 Salanié (997, p28-29) write that the uffi cient tatitic theorem indicate that the optimal wage chedule hould depend on all ignal that may bring information on the action choen by the agent(...). Thi prediction doe not accord well with experience; real-life contract appear (...) to depend on a mall number of variable only. 5

6 veting if it doe not affect how the informativene of the tock price change with the tock price. Even if it doe, it may be optimal to grant the manager more option (albeit with higher trike price) upon low revenue, if the likelihood ratio i more enitive to the tock price when revenue are low the univeral practice of veting being triggered by beating a threhold i not predicted by theory. In addition to compenation, the rik-neutral model can alo be applied to a financing etting, in which cae the optimal contract i debt (Inne (990)). In theory, the promied debt repayment could depend on many ignal, but in practice it often doe not. Our reult ugget that thi practice may be optimal for example, a ignal that ugget that bankruptcy wa due to poor effort by the borrower, rather than bad luck, doe not affect the repayment ince the borrower receive zero in bankruptcy anyway. They alo give condition under which the repayment hould depend on additional ignal, a with performance-enitive debt if and only if it i informative about effort at intermediate output realization. Thi paper i related to both theoretical and applied literature. Starting with the former, Gjedal (982), Amerhi and Hughe (989), Kim (995), and Chaigneau, Edman, and Gottlieb (206b) extend the original Holmtrom (979) informativene principle, but not to etting with contracting contraint. Chaigneau, Edman, and Gottlieb (206a) tudy the effect on the optimal contract of increaing the preciion of a given ignal, but do not tudy the introduction of additional ignal and thu have implication for performance-enitive debt or performance-veting option; they alo do not allow for rik averion. Moving to the latter, Dittmann, Maug, and Zhang (20) quantify the effect on pay and firm value of variou retriction on CEO pay retriction on ex-pot payment, ex-ante expected pay, and pecific component of pay. Their calibration differ from our optimal contracting approach. Dittmann, Maug, and Spalt (203) calibrate the cot aving from incorporating peer performance in executive contract and Johnon and Tian (2000) compare the incentive provided by indexed and non-indexed option. The model of Mano, Strulovici, and Tchityi (200) offer an explanation for performance-enitive debt baed on advere election; our i baed on moral hazard. They alo provide empirical evidence for performance-enitive debt, a do Aquith, Beatty, and Weber (2005) and Adam and Streitz (206). Betti et al. (200, 206) are empirical tudie of the frequency, value, and characteritic of performance-veting equity. The Model We conider a principal (firm) and an agent (manager). The manager i protected by limited liability and ha zero reervation utility. He exert unobervable effort of e {0, }, where 6

7 e = 0 ( low effort ) cot the manager 0, and e = ( high effort ) cot C > 0. In thi ection, we aume that both the manager and firm are rik-neutral a then contracting contraint (rather than rik haring conideration) drive the contract, and o thi i a natural framework to tudy the value of a ignal under contracting contraint. Section 2 will extend the model to rik averion and a continuum of effort level. Effort affect the probability ditribution of output, which i ditributed over an interval q [0, q] where q may be +, and of an additional ignal {,..., S }. 3 Both output and the ignal are contractible. We refer to an output/ignal realization (q, ) a a tate and aume that the ditribution of (q, ) conditional on any e ha full upport. 4 Conditional on effort e and ignal, output q i ditributed according to the probability denity function ( PDF ): f (q e, ) := { π (q) if e = p (q) if e = 0. The marginal ditribution of the ignal i repreented by φ e := Pr ( = e = e ) > 0. The joint ditribution of (q, ) conditional on effort, denoted f (q, e), i determined by their product. The marginal ditribution of output i given by f (q e) = φ ef (q e, ). () Let LR (q) := φ π (q) φ 0p (q) (2) denote the likelihood ratio aociated with output q and ignal. We aume that the output ditribution atifie the trict monotone likelihood ratio property ( MLRP ): LR (q) i trictly increaing in q for all. The firm ha full bargaining power and offer the manager a vector of payment {w (q)} conditional on the tate. We aume that the incremental gain from effort E [q e = ] E [q e = 0] i uffi ciently higher than the cot of effort C that it i optimal for the firm to implement high 3 A dicrete ignal pace avoid meaurability iue but i unimportant for our reult. With a continuum of output and without limited liability on the principal, exitence of an optimal contract i typically an iue. The contract cannot involve the principal paying only in the tate with the highet likelihood ratio (a with dicrete output) ince thi i a et of meaure zero, o it mut involve her paying in a neighborhood around that tate. Without limited liability, the principal can generically improve on the contract by concentrating the payment in a maller neighborhood, in which cae an optimal contract fail to exit. 4 The reult are robut to a relaxation of thi aumption, except that the optimal contract might not be unique. There could exit other optimal contract that differ on a et of output that occur with probability zero. 7

8 effort (otherwie, the optimal contract would trivially involve a contant payment of zero). The firm thu olve the following program: min w (q).t w (q) φ π (q) dq (3) w (q) φ π (q) dq C 0 (4) w (q) [φ π (q) φ 0p (q)] dq C (5) w (q) 0 q,. (6) it minimize the expected payment (3) ubject to the manager individual rationality contraint ( IR ) (4), the incentive compatibility contraint ( IC ) (5), and the limited liability contraint ( LL ) (6). The IC (5) and LL (6) imply that the IR (4) i automatically atified, and o we ignore it in the analyi that follow.. Upper Bound on Payment In thi ubection, in addition to limited liability, we aume that there i a maximum payment to the manager, which can be output-dependent and i denoted w(q): 0 w (q) w(q). (7) We aume that w(q) i nondecreaing in q. The primary application i w(q) = q, i.e., limited liability on the firm. We conider the more general upper bound w(q) to allow the model to capture other contracting contraint; for example, a finite w(q) independent of q repreent a cap on ex-pot payment. To enure that high effort i implementable, we aume: q0 w(q) [φ π (q) φ 0p (q)] dq > C, (8) where, for each, q0 i implicitly defined by φ π (q0) = φ 0p (q0); if φ π (q) > φ 0p (q) for all q, et q0 = 0, and if φ π (q) < φ 0p (q) for all q, et q0 = q. q0 exit and i unique by MLRP. Without (8), the firm would implement low effort and the optimal contract would trivially involve a zero payment. Thi upper bound on payment i not neceary for our reult; Appendix B conider 8

9 the cae in which there are no additional contracting contraint and how that, like in thi ection, informative ignal may have zero value. However, in the abence of an upper bound, the optimal contract typically involve a very large payment in the highet likelihood ratio tate, which would vatly exceed total output and violate the firm limited liability contraint, and zero payment in all other tate. realitic contract. We thu conider an upper bound to achieve more Similar to Inne (990), the olution involve paying the minimum amount poible (zero) when the likelihood ratio i below a threhold κ, and the maximum amount poible when it exceed it. The threhold κ i choen o that the IC bind (exitence i hown in Appendix A); if more than one uch threhold exit, we chooe the larget one: κ := up { ˆκ : LR (q)>ˆκ w(q) [φ π (q) φ 0p (q)] dq = C }. (9) By MLRP, for each ignal realization, the threhold for the likelihood ratio tranlate into a threhold for output. Lemma characterize the optimal contract: Lemma The optimal contract with agent limited liability and an upper bound on payment i w (q) = { 0 if q < q (κ) w(q) if q > q (κ), (0) where and κ i determined by (9). 0 if LR (0) > κ q (κ) := q if LR ( q) < κ LR (κ) if LR (0) κ LR ( q) () Lemma yield a live or die contract: the manager receive the maximum payment w(q) if output exceed a threhold q and zero otherwie. For a given ignal realization, the threhold output level q i choen o that the likelihood ratio at thi output level equal κ. 5 In general, the threhold will depend on the ignal realization, and o the optimal contract i contingent upon both output and the ignal. Propoition how that the contract i independent of if and only if, for every, the output q aociated with a likelihood ratio of κ i the ame, i.e. q = q, and o the firm optimally et the ame threhold q. 5 For ome ignal realization, it i poible that thi threhold output level i a corner olution, in which cae the manager either alway receive the maximum or alway receive zero; if all threhold are interior, then q = LR (κ) for all. 9

10 Propoition The optimal contract with agent limited liability and an upper bound on payment i independent of the ignal if and only if LR (q ) = LR (q ) = κ,, (2) where κ i determined by (9). Propoition how that the preence of contracting contraint require u to refine the informativene principle. Intuitively, q i the threhold that would be choen in the abence of the additional ignal. A ignal ha poitive value if and only if it affect the likelihood ratio at q rather than in general a only at q doe the firm have freedom to change the contract, by making q depend on the ignal. When q = q i.e. the firm would chooe not to make the threhold depend on the ignal a ignal that only affect the likelihood ratio at q q ha zero value becaue the firm cannot make ue of it. It cannot change the contract for q < q becaue it i already paying zero, nor for q > q becaue it i already paying the maximum. A a reult, additional ignal about effort are only valuable for intermediate output level, not at tail output realization. Note the requirement for informativene in the tail doe not mean that our reult only applie to extreme ignal. Any output realization above or below the threhold i a tail realization. Intead, our reult implie that ignal that hift probability weight either only above and/or below the threhold hould not enter the optimal contract. In um, if output q i a uffi cient tatitic for effort e given (q, ), the ignal ha zero value. However, even if q i not a uffi cient tatitic, till ha zero value if it i informative about effort only in tate at which contracting contraint bind. In turn, contracting contraint bind everywhere except for at the threhold q..2 Upper Bound on Incentive In thi ubection, in addition to agent limited liability, we aume that rather than an upper bound on payment there i an upper bound on the enitivity of pay to performance: w (q + δɛ) w (q) ɛ, ɛ > 0. (3) Contraint (3) tate that, for a dollar increae in output, payment to the agent can increae by at mot δ dollar. We conider value of δ uch that an incentive compatible contract exit, by auming: δ {E [q e = ] E [q e = 0]} > C. (4) 0

11 The primary application i a monotonicity contraint, a in Inne (990), which correpond to δ =. 6 In thi cae, the manager cannot gain more than one-for-one with an increae in q. Inne (990) jutifie thi contraint on two ground. Firt, if it were violated, the manager would ecretly borrow on hi own account to increae output, ince he would gain more from hi contract than the amount he would have to repay. Second, if it were violated, the firm payoff would fall with output over ome region. Thu, it would exercie it control right to burn output, raiing it payoff. We generalize the upper bound on pay-performance enitivity to a general δ = to allow the model to capture other contraint on incentive, e.g. maximum level of equity award, or a maximum payment that increae with output (ince higher performance make higher pay more ocially acceptable.) Let LR (q) := φ q π(z)dz φ 0 q p(z)dz = Pr( q q, = e=) Pr( q q, = e=0) event ( q q, = ), which i trictly increaing by MLRP. 7 realization, contruct the threhold trike price a follow: q (κ) := denote the likelihood ratio aociated with the 0 if LR (0) > κ q LR (κ) if LR ( q) < κ if LR (0) κ LR ( q) For each fixed κ and ignal. (5) The threhold for the likelihood ratio κ i choen o that the IC bind (exitence i hown in Appendix A); if more than one uch threhold exit, we chooe the larget one: κ := up { ˆκ : LR (q)>ˆκ The optimal contract given by Lemma 2 below: δ(q q (ˆκ)) [φ π (q) φ 0p (q)] dq = C } (0, q). (6) 6 Under the primary application of δ = (the upper bound on incentive i a monotonicity contraint) and w (q) = q (the upper bound on pay arie from principal limited liability), then it doe not matter whether we retain the upper bound on pay when introducing the upper bound on incentive, ince the principal limited liability contraint never bind in the preence of monotonicity. 7 We have: d dq which i poitive if and only if π (q) p (q) < { φ q q π } (z)dz φ 0 q p (z)dz q π (z)dz q p (z)dz q = φ π (q) φ 0 π (z) q π (q) dz > p (z) q p (q) dz q p(z)dz + p (q) q π (z)dz ( ) q 2, q p (z)dz which i atified becaue, for any z > q, MLRP guarantee that π(z) p (z) > π(q) p (q). q [ π (z) π (q) p ] (z) dz > 0, p (q)

12 Lemma 2 The optimal contract with an upper bound on incentive i w (q) = δ max {q q (κ), 0}, where q (κ) and κ are determined by (5) and (6). Lemma 2 yield an option contract: if output exceed q, the manager receive a proportion δ of the reidual q q, rather than the maximum payment a in Section.. Propoition 2 give a neceary and uffi cient condition under which the trike price doe not depend on the ignal realization (i.e. q = q ). Propoition 2 The optimal contract with an upper bound on incentive i independent of the ignal if and only if where κ i determined by (6). LR (q ) = LR (q ) = κ,, (7) Propoition 2 how that the contract i independent of the ignal if and only if the likelihood ratio that q > q i alway κ, regardle of. Then, the firm optimally et the ame threhold q. Note that the likelihood ratio in Propoition and 2 concern different event. With a maximum payment in addition to limited liability, (Propoition ), the firm pay zero for output below a threhold q and the maximum payment if it exceed q, and o it can only adjut the payment by changing q. Doing o only affect the payment in a local neighborhood around q (i.e. change it from 0 to q or vice-vera). A a reult, a ignal i only ueful if it affect the likelihood ratio at a ingle point q = q i.e. provide information on whether q = q i more likely to have reulted from working or hirking. If ignal realization ugget that the manager ha worked, the firm increae the payment from 0 to q by reducing the threhold to q < q. If it ugget that he ha hirked, the firm reduce the payment from q to 0 by increaing the threhold to q > q. With an upper bound on incentive (Propoition 2), the manager i paid δ(q q ) if output exceed q. Thu, if the firm ue the ignal to change the trike price q, thi alter the payment not only at q = q (a in Propoition ) but for all q q ; it cannot change the payment at pecific output level in iolation a thi would violate the upper bound on incentive. Thu, a ignal ha value if it affect the likelihood ratio over a whole range q q i.e. provide information on whether q q i more likely to have reulted from working or hirking. Any ignal that hift probability ma from below to above the threhold (or vicevera) i valuable, a it affect the likelihood that output exceed the threhold. For example, conider q = 5 and a monotonicity contraint (δ = ). The likelihood ratio i higher for q = 7 than q = 3, and o (in the abence of a ignal), the manager receive 2 if q = 7 and 0 if q = 3. 2

13 If the event (q 5, = ) i more informative about effort than the event (q 5, = ), i.e., oberving that output i above the threhold i more informative about effort when the ignal i than when it i, then the firm will optimally increae the payment when the ignal i compared to when it i. To preerve monotonicity, thi i achieved by varying the threhold acro ignal realization, by etting a lower threhold for than for : q < q. However, any ignal that only reditribute ma below the threhold o that it tay below the threhold, or only reditribute ma above the threhold o that it tay above the threhold, ha no value. Continuing the earlier example, if the event (q 7, = ) i more informative about effort than the event (q 7, = ), but the event (q 5, = ) i not more informative about effort than the event (q 5, = ), then the firm would like to increae the payment for (q 7, = ). However, uch a change would violate monotonicity, and o the firm cannot ue the ignal. Depite the difference in the relevant likelihood ratio, Propoition and 2 both etablih imilar condition for a ignal to have value. In both cae, the firm only degree of freedom i the threhold q or q. With an upper bound on payment in addition to limited liability, changing q only ha local effect, and o condition (2) depend on the likelihood ratio aociated with q = q. With an upper bound on incentive, changing the trike price q affect payment at all higher output, and o condition (7) depend on the likelihood aociated with q q. The above reult ha a number of application for compenation contract. Firt, it identifie the etting in which board hould invet in additional ignal of manager performance, for intance through monitoring. A ignal that hift ma locally i only ueful at intermediate output level, not tail output, a only then will it affect the payment. 8 For example, in rik management, a moking gun indicate that a bad event i due to poor performance (e.g. exceive rik-taking) rather than bad luck, but the bad event will likely lead to firing anyway. For intance, invetor only noticed that Enron wa adopting mileading accounting practice when it wa already going bankrupt. Relatedly, the threhold output can be interpreted a a performance target below which the manager i fired. Signal are then only ueful if they affect thi target. If performance were very low, the manager would be fired anyway; if performance were very high, he would receive the entire output (or reidual output) anyway. Second, it implie that pay-for-luck (i.e. not obtaining ignal to verify whether an output level wa due to effort or luck) need not be uboptimal if it occur at tail output realization. 8 If the monotonicity contraint i impoed, a ignal that hift ma from (ay) 0 to q ha value, even though it doe not hift ma at intermediate output level. However, a ignal that hift ma locally only ha value at intermediate output. 3

14 In reality, intance of pay for luck typically concern very good or very bad outcome for example, Bertrand and Mullainathan (200) conider how CEO pay varie with pike and trough in the oil price, and Jenter and Kanaan (205) find that peer-group performance doe not affect CEO firing deciion but additional ignal are only valuable for moderate outcome. Third, for tock option, it provide condition under which the trike price hould depend on additional ignal, which can be implemented via option indexing or option repricing. Brenner, Sundaram, and Yermack (2000) find empirically that repricing nearly alway involve a lowering of the trike price, and follow poor tock price performance (both abolute and indutry-adjuted). Our model ugget that a reduction in the trike price hould be prompted by poitive, rather than negative, ignal of effort, uggeting that uch practice are uboptimal. 9 However, the model provide condition under which repricing i optimal under the optimal contract, uggeting that it i not univerally ineffi cient, contrary to concern (e.g. Bebchuk and Fried (2004)) that it repreent reward for failure. In thee cae, repricing hould be part of the contract negotiated at t = 0, not the outcome of a renegotiation at t =. The reult alo ha implication for debt contract. Inne (990, footnote 2) note that the model of rik neutrality and limited liability can be interpreted in two way. Firt, the firm offer a compenation contract to the manager, a in the above expoition. Second, the manager i an entrepreneur who raie financing from the firm, an invetor, which i the expoition in Inne (990). The optimal contract i debt, and o a ignal ha no value in determining the repayment chedule, which i automatically the entire firm value if performance i poor, and the entire promied repayment (the face value of debt plu interet) if performance i good. It ha value if and only if it affect the promied repayment. In theory, thi amount could depend on many ignal, but in practice it i often ignal-independent. Propoition 2 potentially rationalize thi practice even if ignal are informative about effort, they hould not enter the contract if they are only informative about effort in the tail. A ignal which ugget that bankruptcy wa due to poor effort by the borrower, rather than bad luck, doe not affect the repayment ince the borrower receive zero in bankruptcy anyway. In addition, Propoition 2 provide condition under which the repayment hould depend on additional ignal, a in performance-enitive debt, where the promied repayment i higher upon negative ignal of borrower performance. Thi i the cae if and only if the ignal affect the probability that performance exceed the threhold under high effort relative to low 9 Acharya, John, and Sundaram (2000) alo tudy the repricing of option theoretically. In their model, repricing i not undertaken to make ue of additional informative ignal, but intead to maintain effort incentive when option fall out of the money. 4

15 effort. For example, if macroeconomic condition affect the probability that output exceed the threhold for both high and low effort by the ame proportion, then the repayment hould be independent of macroeconomic condition. 2 Continuou Effort and Rik Averion Thi ection tudie the neceary and uffi cient condition for a ignal to add value under rik averion and contracting contraint. We alo generalize the model to a continuou effort deciion, but retain previou aumption unle otherwie pecified. Effort i now given by e R +. Let F (q e, ) and f(q e, ) denote the cumulative ditribution function ( CDF ) and PDF of q conditional on e and. We aume that, for each, F (, ) i twice continuouly { differentiable } (with repect to q and e). We continue to aume MLRP, which here entail d fe(q e,) > 0, dq f(q e,) where f e (q e, ) denote the firt derivative of the PDF with repect to e. We aume that the marginal ditribution of the ignal φ e i differentiable with repect to e. The manager utility of money i given by a trictly increaing, weakly concave, twice differentiable function u. The manager ha poitive outide wealth W and reervation utility u. 0 Hi cot of effort C(e) i a twice differentiable, trictly increaing, and trictly convex function. Thu, given a contract w (q) and an effort level e, hi objective function i E[u( W + w (q)) e] C(e). A in the firt tage of Groman and Hart (983), the firm induce a given effort level ê. It chooe a function w ( ), for each poible value of the ignal, to olve the following problem: ubject to min w (q) φ ê φ ê q 0 ê arg max e q 0 w (q) f(q ê, )dq (8) u( W + w (q))f(q ê, )dq C(ê) u, (9) φ e q 0 u( W + w (q))f(q e, )dq C(e), (20) w (q) [0, w(q)]. (2) 0 With rik neutrality (Section ) we aumed zero reervation utility, o that olving the incentive problem i cotly to the principal a it involve paying the agent rent (i.e. a lack IR). With rik averion, olving the incentive problem i cotly for the principal even if the agent doe not receive rent (i.e. the IR bind), ince the principal mut pay a premium for the rik the agent bear from receiving incentive compenation. 5

16 We tudy two cae. In the firt cae, only the manager i ubject to limited liability: w(q) = + for all q. In the econd cae, there i alo an upper bound w(q) on payment, a in Section.. Following Holmtrom (979), Shavell (979) and the ubequent literature on the informativene principle (e.g. Gjedal (982), Kim (995)), we aume that the firt-order approach ( FOA ) i valid; ee Chaigneau, Edman, and Gottlieb (206b) for the informativene principle without the FOA. We can thu replace the IC in (20) by the following equation: [-3em]em d de [ dφ ê de q φ ê 0 q 0 u( W + w (q))f(q ê, )dq = C (ê) (22) u( W + w (q))f(q ê, )dq + φ ê q 0 ] u( W + w (q))f e (q ê, )dq = C (ê)(23) The optimal contract i given by Lemma 3 below. Lemma 3 Suppoe an optimal contract exit and the firt-order approach i valid. Let λ and µ denote the nonnegative Lagrange multiplier aociated with the participation contraint in (9) and the incentive contraint in (20), repectively. With limited liability on the manager, the optimal contract i: { ( w (q) = max u / ( [ dφ λ + µ ē/de φ + f ])) } e(q ê, ), 0. (24) ê f(q ê, ) With a maximum payment in addition to limited liability, the optimal contract i: { { ( w (q) = max min u / ( [ dφ λ + µ ê/de φ + f ])) } } e(q ê, ), w(q), 0 ê f(q ê, ) (25) Without the ignal, the likelihood ratio at a given value of q can be written a LR(q) := f e(q ê) f(q ê). In the rik-neutral model of Section we alo conidered an upper bound on incentive, becaue it wa neceary to obtain realitic contract. Under rik averion, realitic contract can be obtained without uch a bound (ee, e.g., Propoition 4). 6

17 With the ignal, we define the likelihood ratio a LR (q) := f e(q, ê) f(q, ê) = dφ ê/de φ + f e(q ê, ) ê f(q ê, ). (26) With limited liability on the manager, for each fixed κ and ignal realization, contruct the threhold above which the payment i trictly poitive a follow: q (κ) := 0 if LR (0) > κ w(q) if LR ( q) < κ LR (κ) if LR (0) κ LR ( q) The threhold likelihood ratio κ i choen o that the IC bind for effort ê; if more than one uch threhold exit, we chooe the larget one: κ := { up ˆκ : [ u( W ) LR (q) ˆκ + u( W + w (q)) LR (q)>ˆκ [ dφ ] ê de f(q ê, ) + φ êf e (q ê, ) dq ] [ dφ ê de f(q ê, ) + φ êf e (q ê, ) ] } dq = C (ê) The contract in equation (24) i monotonic (via MLRP) and alo continuou, ince the likelihood ratio i continuou: it numerator and denominator are continuouly differentiable with repect to q. However, it hape (e.g. whether it i concave, convex, or linear above q ) depend on the hape of the utility function and likelihood ratio. With a maximum payment in addition to limited liability, for each realization of, define M a the et of value of q uch that, with the contract decribed in (25): (27) (28) ( w (q) = u / ( [ dφ λ + µ ê/de φ + f ])) e(q ê, ). (29) ê f(q ê, ) Intuitively, M i the et of output level for which neither contraint on contracting bind. The optimal contract i given by Propoition 3 below. Propoition 3 (i) With limited liability on the manager, the optimal contract i independent of the ignal if and only if LR (q) = LR (q),, q q := min {q }. (ii) With a maximum payment in addition to limited liability, the optimal contract i independent of the ignal if and only if LR (q) = LR (q),, q M. The intuition i a follow. In both the binary and continuou effort cae, a ignal ha no 7

18 value if and only if it doe not affect the likelihood ratio. Thi likelihood ratio i fe(q, ê) with f(q, ê) continuou effort and φ π(q) with binary effort. With rik neutrality, the relevant likelihood φ 0p(q) ratio i at a ingle intermediate output level, below which the manager receive zero and above which he receive the entire output. Here, the relevant likelihood ratio i at a range of output realization (q q or q M ). The intuition i a follow. A ignal i only valuable at output level where contracting contraint do not bind. With rik neutrality and a maximum payment in addition to limited liability (Section.), for a given realization of, contracting contraint bind everywhere except for at the threhold q. However, with rik averion, there are many output level (q q or q M ) where contracting contraint do not bind, and o the condition for a ignal to have value are weaker. In particular, while the manager receive zero below a threhold q, he doe not automatically receive the full output above q. Thu, with limited liability on the manager only, the firm can change the payment in repone to the ignal for any q > q ; with a maximum payment in addition to limited liability it can do o for any q M. Uing the decompoition of the likelihood ratio in (26), the contract depend on the ignal if it affect either dφ ê /de φ ê, or fe(q ê,), or both (a long a it doe not affect both in uch a way f(q ê,) that they cancel out o dφ ê /de + fe(q ê,) i independent of ). 2 Thu, a ignal can have value φ ê f(q ê,) for two reaon. Firt, it i individually informative about a local deviation in effort from the equilibrium effort level ê, i.e. output, i.e. f e(q ê,) f(q ê,) dφ ê /de φ ê depend on. Second, it affect the informativene of depend on. For example, if the ignal i a meaure of macroeconomic condition, low output during a boom (high ignal) may be more informative about a deviation in effort than low output during a receion (low ignal). In thi cae, the ignal will be ued in the contract, even if it i uninformative about effort (i.e. dφ ê /de i independent of ) becaue φ ê the manager effort doe not affect the probability of a receion. We now move to application of Propoition 3. In addition to the general implication, hared with Section, that a ignal i only valuable if it i informative about effort at output level for which contracting contraint do not bind, we can alo derive more pecific implication for how a valuable ignal hould be incorporated in the optimal contract where we can olve for it. Propoition 4 provide uffi cient condition for the optimal contract to be an option: limited liability on the manager, log utility, and a likelihood ratio that i linear in output ( fe(q ê,) = a f(q ê,) + b q o that LR (q) = dφ ê /de + a φ + b q), a with the Normaland Gamma ê 2 In Section, the likelihood ratio wa multiplicative rather than additive (equation (2)) ince effort wa binary. 8

19 ditribution. For each realization of the ignal, we define ˇq by 0 if LR (0) > 0 ˇq := q if LR ( q) < 0 LR (0) if LR (0) < 0 < LR ( q) (30) Note that for each, ˇq i unique due to MLRP. Propoition 4 (i) With limited liability on the manager, a likelihood ratio linear in q, and log utility (u(w) = ln w), the optimal contract can be written a with n 0. w (q) = n max{q q, 0}, (3) (ii) The number of option n received ex-pot by the manager i independent of the ignal f e(q ê,) if and only if d i independent of. dq f(q ê,) (iii) The trike price q i independent of the ignal if the IR i nonbinding and ˇq i independent of, or if d f e(q ê,) and dφ ê /de + fe(0 ê,) are independent of. dq f(q ê,) φ ê f(0 ê,) In Propoition 4, the manager ha n option with exercie price q. With log utility, u (/x) = x. Thu, with a linear likelihood ratio LR (q) = dφ ê /de + a φ + b q, log utility ê yield w (q) = u (/LR (q)) = dφ ê /de + a φ + b q, i.e., the contract i linear in q. To our ê knowledge, part (i) of Propoition 4 provide the firt et of uffi cient condition for option to be the optimal contract when the manager i rik-avere, in contrat to Inne (990) where he i rik-neutral. 3 While Propoition 3 tudied the condition under which a ignal affect any dimenion of the contract, part (ii) of Propoition 4 tudie the condition under which a ignal affect pecifically the number of option. Propoition 3 howed that a ignal ha value if it affect any component of the likelihood ratio: either dφ ê /de (i.e. i individually informative about effort) φ ê or a + b q (i.e. affect the informativene of output for effort). Such a ignal will, in general, alter the Lagrange multiplier µ and thu cale up or down the number of option n = µb received acro all ignal. However, the number of option actually received ex pot may 3 Jewitt, Kadan, and Swinkel (2008) how that the contract i option-like with rik averion and agent limited liability, in that incentive are zero for low output and poitive for high output, but do not identify condition under which the increaing portion of the contract i linear. Hemmer, Kim, and Verrecchia (999) identify a linear likelihood ratio and log utility a leading to the contract having a linear portion, but did not combine them with limited liability to obtain an option contract. 9

20 till not depend on the actual ignal realization. Thi will only arie if b, rather than any other component of the likelihood ratio, depend on i.e. the ignal realization affect the rate at which the informativene of the tock price change with the level of tock price. The intuition i a follow. A in any principal-agent model, pay i increaing in the likelihood ratio, and o the enitivity of pay to output (here, the number of option n = µb ) depend on the enitivity of the likelihood ratio to output, dlr(q) = b dq. If the likelihood ratio increae fater with output when i high (i.e. if > implie b > b ), the contract hould be teeper and more option hould be granted (i.e. n > n ). Thu, continuing the earlier example, even if effort doe not affect the probability of a receion ( dφ ê /de i independent of, a ignal of economic condition), it i optimal for more φ ê option to vet in bad (good) time if the likelihood ratio i more enitive to the tock price i.e. effort ha a greater effect on the tock price in bad (good) time. To our knowledge, thi i the firt theoretical jutification of the condition under which performance-baed veting i optimal. Note that it may be effi cient for more option to vet upon low ignal. Thi reult implie that the exiting practice, of veting alway being triggered by good performance, may not be optimal. However, it echoe the theoretical prediction of Edman, Gabaix, Sadzik, and Sannikov (202) who how that more equity hould be granted when firm value i low, and the empirical finding of Core and Larcker (2002) who how that more equity i granted upon poor performance (although they tudy tock rather than option). 4 (ay, revenue) i informative about effort (i.e. dφ ê /de φ ê Converely, even if a ignal depend on revenue ), it hould not affect veting if it doe not affect the enitivity of the likelihood ratio to the tock price. The likelihood ratio given could be a vertical tranlation of the likelihood ratio given, but if both are a enitive to q then there hould not be any performance-baed veting. Part (iii) identifie two et of condition under which the trike price i independent of the ignal. The firt i that the IR i lack and that the likelihood ratio equal zero at the ame q for all. If the IR i lack, the contract i determined by the IC. The number of option then depend on the enitivity of the likelihood ratio to output, and the trike price i choen uch that the payment i enitive to performance (i.e., i nonzero) if and only if the likelihood ratio i poitive. If the likelihood ratio turn poitive at the ame value of q for all, i.e., if ˇq i independent of, then the trike price i independent of. For example, improvement in tock price effi ciency or the informativene of accounting information ytem may affect the lope of the likelihood ratio, but not the level at which the likelihood ratio i zero. The 4 Note that, if a low ignal lead to more option veting, it may alo lead to their trike price increaing, o that the agent i not better off by generating a low ignal. 20

21 econd condition i that the likelihood ratio i independent of the ignal for any q. A uffi cient condition i that both the lope ( d f e(q ê,) ) and level of the likelihood ratio ( dφ ê /de + fe(0 ê,) ) dq f(q ê,) φ ê f(0 ê,) are independent of. 3 Concluion Thi paper how that the informativene principle mut be modified in the preence of contracting contraint, in turn allowing u to undertand whether and what condition real-life compenation and financing contract hould depend on performance ignal in addition to output. Specifically, a ignal i valuable if and only if it i informative about effort at output level at which contracting contraint do not bind, rather than in general. We derive neceary and uffi cient condition for a ignal to have value under variou contracting contraint. Starting with rik neutrality and bilateral limited liability (or an upper bound on payment), a ignal i valuable if and only if it i informative about effort at a ingle intermediate output. If there i alo a monotonicity contraint, or an upper bound on output-baed incentive, a ignal i valuable if and only if it provide information on whether beating the target performance level i more likely to have reulted from working or hirking. Likewie, under rik averion, a ignal i valuable if and only if it i informative about effort at output for which contraint on contracting do not bind. In addition to the theoretical contribution of new condition for a ignal to have value in the preence of contracting contraint, our reult have a number of implication for real-life contract. Starting with compenation contract, our reult offer a potential explanation a to why both pay and the firing deciion do not depend on many potentially informative ignal, why it may not be optimal to filter out luck, when option hould be repriced, and whether option hould have performance-baed veting condition. In particular, performance-baed veting i not necearily optimal even if a ignal i incrementally informative about effort; intead, it mut affect the rate at which the informativene of output change with the level of output. Moving to financing contract, they ugget whether and under what condition debt hould be performance-enitive. 2