Risking Other People's Money: Gambling, Limited Liability, and Optimal Incentives

Transcription

1 Risking Oher People's Money: Gambling, Limied Liabiliy, and Opimal Incenives Peer M. DeMarzo, Dmiry Livdan, and Alexei Tchisyi Preliminary and Incomplee Absrac We consider opimal incenive conracs when managers can, in addiion o shirking or divering funds, increase shor erm profis by puing he firm a risk of a low probabiliy "disaser." To avoid such risk-aking, invesors mus cede addiional rens o he manager. In a dynamic conex, however, because managerial rens mus be reduced following poor performance o preven shirking, poorly performing managers will ake on disaser risk even under an opimal conrac. This risk aking can be miigaed if disaser saes can be idenified ex-pos by paying he manager a large bonus if he firm survives. Bu even in his case, if performance is sufficienly weak he manager will forfei eligibiliy for a bonus, and again ake on disaser risk. When effor coss are convex, reducions in effor incenives are used o limi risk aking, wih a jump o high powered incenives in he gambling region. Our model can explain why subopimal risk aking can emerge even when invesors are fully raional and managers are compensaed opimally. Peer M. DeMarzo is from Graduae School of Business, Sanford Universiy, DeMarzo_Peer@gsb.sanford.edu. Dmiry Livdan is from Haas School of Business, Universiy of California, Berkeley, livdan@haas.berkeley.edu. Alexei Tchisyi is from Haas School of Business, Universiy of California, Berkeley, chisyi@haas.berkeley.edu. 1

2 1 Inroducion Invesors who enrus heir funds o financial insiuions such as invesmen banks, pension and hedge funds ypically have lile knowledge or undersanding of how hose insiuions operae. In paricular, i is exremely hard for invesors o observe he rue realized cash flows of financial insiuions, or o correcly evaluae heir risk exposure. This asymmeric informaion creaes an opporuniy for managers of financial insiuions o enrich hemselves a he expense of he invesors, which is aided by heir abiliy o swifly aler he risk profiles of he asses under managemen. To proec heir ineress, invesors can ry o wrie a conrac ha would align he manager's objecives wih heirs. The goal of his paper is o consider opimal incenive conracs in a seing in which a manager wih limied liabiliy privaely chooses he riskiness of he projec and can privaely diver cash flows for her own consumpion. In paricular, we consider a sylized model wih a wo-dimensional moral hazard problem. To gain some basic economic insighs we sar wih a one-period version of he model, which we hen exend o an infinie horizon using a coninuous ime conracing framework. A risk-neural agen (manager) wih limied liabiliy runs a projec wih cash flows ha depend on is riskiness. The agen can choose beween a low-risk and a high-risk projec. The high-risk projec increases he probabiliy of a high cash flow realizaion compared o he low-risk projec, bu i also resuls in high losses in a bad sae of naure, which we call disaser. The possibiliy of losses in he disaser sae is eliminaed when he low-risk projec is chosen. We assume ha he low-risk projec is he firs bes. In addiion o risk aking, he agen can also manipulae he cash flows by divering cash flows for his consumpion. Neiher he riskiness of he projec nor he cash flow realizaions are observed by he invesors eiher ex-ane or ex-pos unless he disaser sae occurs. The analysis of he one-period model generaes a number of key insighs. Firs, we find ha alhough i is possible o wrie a conrac ha provides he agen wih incenives o choose he low-risk projec and no seal he cash flows, i may be oo expensive o allow he invesors o break even. This is he case when he conrac erms depend only on cash flows repored by he agen and when he probabiliy of he disaser sae is low. The economic inuiion behind his resul is fairly ransparen. Any incenive-compaible conrac mus reward he agen for reporing he high cash flow, oherwise he agen would seal i. However, condiioning he agen's reward on he repored cash flow also creaes an incenive for he agen o ake he high- 2

3 risk projec, because he agen benefis from he high cash flow and his losses are bounded in he disaser sae due o he limied liabiliy. The conrac ha induces he agen o choose he lowrisk projec wihou sealing cash flows requires very high payoffs for he agen for he nondisaser oucomes. Our second finding is ha he conrac can be much less expensive for he invesor o implemen if he agen's payoff can be made condiional direcly on he disaser sae. In he opimal conrac, he agen receives a big bonus if he projec does no generae a big loss in he disaser sae. This is a cheaper way o provide incenives for he agen o choose he low-risk projec, because he high level of compensaion in non-disaser saes--which are much more likely o occur han he disaser sae--are no longer required. We sugges ha in pracice his conrac can be implemened by giving he manager ou-of-money pu opions on he companies ha are likely o be ruined in he disaser sae, wih a cavea ha he manager can collec he payoff from he opions only if her company remains in a good shape. In our coninuous ime seing, he agen can increase he drif of he cash flow, which is driven by Brownian moion, while creaing a possibiliy of disaser, which we model as a Poisson process. Risk aking occurs when he agen s coninuaion payoff is below a hreshold, which is inversely proporional o he inensiy of disaser arrival. Because managerial rens mus be reduced following poor performance o preven fund diversion, poorly performing agens will ake on disaser risk even under an opimal conrac. Unlike he saic model, risk aking can sill happen even when disaser saes are conracible. If performance is sufficienly weak he agen will forfei eligibiliy for a bonus, and again ake on disaser risk. When effor coss are convex, i is opimal o implemen low-powered incenives in he no gambling region, while high-powered incenives are opimal in he gambling region. Inuiively, sronger incenives no only encourage greaer effor, bu also make gambling more aracive. As a consequence, in he no-gambling region(i.e., wih high coninuaion payoff for he manager) he opimal effor is below no only he firs bes, bu also he second bes level in a seing wihou gambling. On he oher hand, in he gambling region, he opimal effor becomes subsanially higher and can be above he firs bes level. The high effor, while cosly, moves he manager s coninuaion payoff oward he no-gambling region more quickly. Compared o he one-dimensional moral hazard problem of DeMarzo and Sannikov (26), he opimal conrac has increased reliance on deferred compensaion. Anoher difference 3

4 from DeMarzo and Sannikov (26) is ha public randomizaion can someimes improve he conrac. When he agen s coninuaion payoff drops o he hreshold below which he agen sars gambling, a small reducion in performance could lead o harsh punishmens. On he oher hand, a small improvemen in performance in he gambling region could lead o a big increase in he agen s coninuaion payoff, so he agen sops gambling immediaely. Our paper bridges wo srands of lieraure: he lieraure on moral hazard and hidden acion and he lieraure on risk aking. The vas lieraure on moral hazard (see Salanie (1997) for a survey of saic models) has focused mosly on he problem of a principal who wans o induce an agen o exer opimal effor. We build on a hidden acion seing similar o he one used by DeMarzo and Sannikov (26) in coninuous ime and DeMarzo and Fishman (27a,b) in discree ime. Risk-shifing was firs inroduced by Jensen and Meckling (1976) as an agency conflic beween equiy and deb holders of a levered firm. The agency problem is ha he equiy would gain all he upside of increasing he risk of he firm's asses-in-place, whereas deb would be responsible for is downside. Recognizing he imporance of his agency problem, a large lieraure sudies he impac of non-concave porions of common execuive compensaions schemes (such as pus and calls) on he risk-shifing incenives of managers who have access o dynamically complee markes. Conribuions include Carpener (2), Ross (24), and Basak, Pavlova, and Shapiro (27). This lieraure reas he opimal conrac as being exogenous. The mos closely relaed papers o he saic version of our model are Diamond (1998), Palomino and Pra (23), Hellwig (1994), and Biais and Casamaa (1999). Like us, hey sudy a hidden-acion moral hazard problem in which he agen conrols boh effor and he disribuion of he oucome. Diamond (1998) and Palomino and Pra (23) use delegaed porfolio managemen as he economic moivaion. Diamond (1989) asks wheher, as he cos of effor shrinks relaive o he payoffs, he opimal conrac converges o he linear conrac. He shows ha if he agen has several ways o manipulae he oucome, he principal should offer he simples possible compensaion scheme, ha is, he linear conrac. While Diamond (1998) considers only hree possible oucomes, Palomino and Pra (23) allow for a coninuum of oucomes. Similar o our seing, he agen in heir model has limied liabiliy and can saboage (misrepor) he realized reurn. They show ha he opimal conrac is simply a bonus conrac-- he agen is paid a fixed sum if he porfolio reurn is above a hreshold. Also, by using an 4

5 explici parameerizaion of risk, hey are able o analyze he sign of inefficiencies in risk aking. Hellwig (1994) and Biais and Casamaa (1999) are ineresed in he opimal financing of invesmen projecs when managers mus exer unobservable effor and can also swich o less profiable, riskier venures. Boh papers find ha under some echnical condiions opimal financial conracs can be implemened by a combinaion of deb and equiy. The mos closely relaed papers o he dynamic version of our model are by Ou-Yang (23) and Cadenillas, Cvianic, and Zapaero (27). Boh papers are execued in a coninuous ime seing where he agen conrols boh he drif (effor choice) and he volailiy (projec selecion) of he underlying payoff process. However, neiher paper allows for he limied liabiliy of he agen, as in our model. In addiion, we allow for endogenous liquidaion before he erminal dae of he projec. This assumpion is criical since i faciliaes an all-or-nohing sraegy from he agen. Implemening his assumpion in he economic seing of hese wo papers is echnically infeasible since i would superimpose an opimal sopping problem over heir curren opimizaion rouine. Finally, Makarov and Planin (21) develop a dynamic model of acive porfolio managemen in which fund managers may secrely gamble in order o manipulae heir repuaion and arac more funds. They solve for he opimal conracs ha deer his behavior and show ha if invesors are shor-lived, hen he manager mus leave rens o invesors in order o credibly commi no o gamble. If invesors are long-lived, any conrac ha increases bu defers expeced bonuses afer an ousanding performance is opimal. Conrary o our paper, Makarov and Planin (21) consider only observable acions by he agen. The res of he paper is organized as follows. The one-period model is presened in Secion 2. The coninuous ime model is discussed in Secion 3. The coninuous ime model wih convex coss of effor is analyzed in Secion 4. Secion 5 concludes. 2 One-Period Model 2.1 Formulaion There are wo risk-neural players. The principal (invesor), owns he company which has value of is asses-in-place equal o A. The principal can add a new projec o he company, in which 5

6 case he has o hire he second player, he agen, o run i. The cash flows from he projec, Yq, ( ) depend on he degree of is riskiness, q {,1}, which is conrolled by he agen. The projec has hree possible cash-flow realizaions 1, wih probabiliy q Yq ( ), wih probabiliy 1 q( ). -D, wih probabiliy q The choice of q corresponds o he safe projec, in which case only cash flows and 1 are possible and he expeced value is equal o (,1). The choice of q 1 corresponds o he risky projec. In his case he probabiliy of he highes cash flow, 1, is increased by, bu a new negaive cash flow D can be realized wih probabiliy he safe projec has higher expeced cash flows han he risky projec (1). We assume ha D. (2) D can be inerpreed as he direc loss from he operaions. I could be quie large, bu no greaer han he value of asses-in-place, i.e., D<A. Under his assumpion he principal has limied liabiliy wih respec o he company, bu no he new projec. The agen can ake wo acions boh unobservable by he principal. Firs, upon privaely observing ha he realized cash flow is equal o 1, he agen can repor o he principal ha he cash flow is. By doing so he agen receives a privae benefi of [,1]. We inerpre his diversion of he firm's cash flows as sealing. We assume ha he agen can secrely ransfer money from he firm's accoun o his own accoun. However, oher hidden aciviies ha benefi he agen a he expense of he principal may fi he seing of he model as well. For insance, he agen can inefficienly use he firm's cash flows in order o receive non-pecuniary benefis. The fracion 1 represens he cos of diversion, which can be aribued o differen kinds of expenses and inefficiencies associaed wih he diversion. Such a hidden acion seing allows for one-o-one mapping of he cash flow sealing ino he hidden effor seing. In his seing he agen chooses a binary effor e {,1} a a cos of e. Here e=1 sands for high (opimal) effor, while e= sands for low (inferior) effor. The payoffs condiional on he join choice of effor and risk are given by 6

7 1, wih probabiliy e( q) Yqe (, ), wih probabiliy 1 e( q) q. (3) -D, wih probabiliy q Under he hidden effor inerpreaion he incenive compaibiliy consrain enforces he opimal effor. Privaely choosing he riskiness of he projec, q, is he second acion he agen can ake. We also assume ha he agen has limied liabiliy. We inerpre limied liabiliy as a disallowance of posiive ransfers from he agen o he principal, i.e., he agen canno be legally forced o pay back he principal. The principal does no observe eiher he realized cash flow, Y, or he riskiness, q, implemened by he agen and herefore mus rely wholly on he agen o repor he cash flow realizaions, Y. We assume ha he monioring is prohibiively cosly. Under hese assumpions neiher Y nor q is conracable. If he principal agrees o iniiae he projec, a he ime of iniiaion he principal and he agen sign a conrac ha governs heir relaionship over he life of he projec. The conrac obligaes he agen o repor realizaions of he cash flows o he principal. Wihou loss of generaliy, we assume ha he conrac requires he agen o pay he repored cash flows o he principal immediaely. Since here are only hree possible cash flows, any conrac can be described by hree possible paymens from he principal o he agen, w(), w (1), and w( D) corresponding o repored cash flows,1 and D. We assume ha he agen has limied liabiliy, so w(), w (1), and w( D) mus be non-negaive. Nex we solve for (i) an opimal conrac implemening he safe projec; (ii) an opimal conrac implemening he risky projec; and (iii) an opimal conrac when conracing on he disaser sae is possible. 2.2 Opimal conrac implemening he safe projec Here we derive an opimal incenive compaible conrac ha enforces he safe projec choice. A conrac is opimal if i maximizes he principal's expeced payoff, pwq (, ), subjec o a cerain payoff w for he agen. 7

8 The principal's problem is o choose a conrac { w(), w(1), w( D)} ha maximizes his expeced payoff: w( ) p( w,) maxe Y w( Y) q, (4) subjec o he promise-keeping (PK hereafer) consrain he incenive compaibiliy (IC hereafer) consrain and he low-risk-aking (LRT hereafer) consrain (PK) : w E w( Y) q, (5) (IC) : w(1) w(), (6) wy q w Y q (LRT) : E ( ) E ( ) 1. (7) I is opimal o apply he harshes possible punishmen for he disaser oucome, i.e., w(-d)=. Then, he LRT consrain becomes equivalen o ( w(1) w()) w() (8) The funcion pw (,) represens he highes possible payoff aainable by he principal, given any arbirary payoff w o he agen when he safe projec is implemened. The PK consrain implies ha he agen's expeced payoff is w. The IC consrain ensures ha when he cash flow of 1 is realized he agen ruhfully repors i. The LRT consrain guaranees ha he agen selecs he safe projec over he risky projec. I is opimal o apply he harshes possible punishmen for he disaser oucome, i.e., w( D). The conracing problem can be wrien as w( ) pw (,) max 1 w(1) (1 ) w(), (9) s.. (PK) : w w(1) (1 ) w(), (1) Combining (11) and (12) we obain a low bound on w(): (IC) : w(1) w(), (11) (LRT) : ( w(1) w()) w(), (12) w(). (13) 8

9 The payoff for he high cash flow w(1) mus be a leas more. Thus, he lowes expeced payoff for he agen under a conrac ha saisfies boh he IC and LRT consrains is given by s w, (14) Finally, we can subsiue he PK consrain (1) ino he objecive funcion (9) o obain s p( w,) p w, forw w. (15) We summarize our findings in Proposiion 1. PROPOSITION 1: The opimal incenive compaible conrac implemening he safe projec is s { w() w, w(1) w 1, w( D) } w w. The principal's expeced payoff is given by s p( w,) p w, forw w, where he minimum expeced payoff o he agen is equal o s w. Since can be arbirarily small (bu no necessarily he expeced losses since D can be made arbirarily large), he opimal conrac can become oo expensive for he principal o implemen. This happens since he agen has o be given high payoffs when cash flows are eiher or 1 in order o joinly saisfy IC and LRT consrains. Inuiively, in order o provide incenives o reveal cash flows ruhfully, he agen should be rewarded for reporing he high cash flow, as specified by he IC consrain. This creaes incenives for gambling, however. The LRT consrain says ha he expeced benefi of gambling for he agen ( w(1) w()) should be less han or equal o he expeced loss w(). Since he punishmen of he agen for creaing he disaser is bounded due o limied liabiliy, in order o preven gambling he agen should be given an exra ren w() for delivering zero cash flow. Since he loss from gambling w() is proporional o, he ren w() mus be increased when declines. As a resul, gambling is more cosly o preven when he probabiliy of disaser is low. 2.3 Opimal conrac implemening he risky projec 9

10 We now derive an opimal conrac ha implemens he risky projec. The conracing problem is o find a conrac { w(), w(1), w( D)} ha maximizes he principal s expeced payoff: subjec o he PK and IC consrains and Combining (16) and (17) gives w( ) p( w ;1) maxe Y w( Y) q 1, (16) (PK) : w E w( Y) q 1, (17) (IC) : w(1) w(). pw (,1) D w, (18) i.e., he principal s payoff is equal o he expeced cash flow minus he expeced payoff o he agen. Since here is no LRT consrain, boh w(-d) and w() can be as low as zero, while w(1) should be a leas. As he resul, he lowes possible expeced payoff for he agen when gambling is allowed is equal o Thus, we have Proposiion 2. g w ( ). (19) PROPOSITION 2: The opimal conrac implemening he risky projec is given by w ( ) w (1 ) w w w D w 1 1 (), (1), ( ) for ( ) principal's payoff is given by. The g p( w,1) Dw, for w w, (2) where he minimum expeced payoff o he agen is equal o g w ( ). Corollaries 1 and 2 highligh our nex se of resuls. 1

11 COROLLARY 1: The agen's lowes compensaion under an opimal conrac implemening he risky projec is smaller han he lowes compensaion under an opimal conrac implemening he safe projec, i.e., w g s w. This resul follows immediaely from he fac ha 1. Thus, he conrac allowing he risky projec is less expensive han he conrac ha implemens he safe projec in erms of he agen's compensaion. A he same ime, gambling reduces he expeced cash flow by D. Thus, he principal would choose he safe projec only if he expeced losses from gambling ouweigh he exra cos of he agen s compensaion. COROLLARY 2: When he principal can choose he lowes level of he agen's compensaions, he principal is beer off implemening he safe projec if 1 D. (21) We noe ha he righ hand side of he inequaliy (21) goes o infiniy when goes o zero. Thus, when he probabiliy of he disaser is low and D is large so ha he safe projec is sricly beer han he risky one, he principal would prefer he risky projec or would underake no projec. 2.4 Opimal conrac condiional on he disaser sae In his subsecion, we consider a seing wih an observable disaser sae, e.g., an earhquake or a major financial crisis. Thus, i is now possible o wrie a conrac condiional no only on repored cash flows, bu also on saes of naure. We modify our previous seing as follows. A ime zero naure makes a draw: i draws a disaser sae wih probabiliy, and draws a good sae wih probabiliy1. Ex ane, neiher he principal nor he agen knows he exac sae of naure, bu ex pos he rue sae of naure is public knowledge. In he disaser sae, he cash flows condiional on he risk q {,1}, are 11

12 while in he good sae he cash flows are 1, wih probabiliy, Yd ( q), wih probabiliy 1 q, - D, wih probabiliy q, 1, wih probabiliy ( q) / (1 ), Yg ( q), wih probabiliy 1 ( q) / (1 ), - D, wih probabiliy. The cash flows (22) and (23), which are condiional on he saes of naure n g, d (22) (23), are fully consisen wih our previous seup, since he uncondiional probabiliies of he cash flows are he same as in (1). We now derive an opimal conrac ha implemens he safe projec. The crucial difference from he previous seup is ha he ransfer o he agen, of he repored cash flow, ŷ, bu also depends on he sae of naure n. wyn, ( ˆ, ) is no only a funcion The principal's problem is o choose a conrac { w(, g), w(1, g), w(, d), w( D, d)} ha maximizes his expeced payoff: w( ) subjec o he promise-keeping consrain pw (,) maxe YwYn (, ) q, (PK) : w E w( Y, n) q, he incenive compaibiliy consrain (IC) : w(1, n) w(, n), and he low-risk-aking consrain w Y n q w Y, n q (LRT) : E (, ) E ( ) 1. Using (22) and (23) and he fac ha i is opimal o have w( D, d), he consrains can be wrien as (PK) : w w(1, g) (1 ) w(, g) w(, d), (IC) : w(1, g) w(, g), (LRT) : ( w(1, g) w(, g)) w(, d), The lowes possible paymens o he agen ha saisfy boh (IC) and (LRT) consrains are w(1, g), w(, g), w(, d) /. Subsiuing hem ino (PK) gives ha he lowes 12

13 g expeced paymen for he agen is equal o ( ) w, and he corresponding payoff for he g principal is w. Thus, he principal can implemen he safe projec by wriing a conrac condiional on he disaser sae while giving he agen he same minimum level of compensaion required o implemen he risky projec. This is achieved by giving a bonus b / for zero cash flow in he disaser sae. Proposiion 3 summarizes our resuls. PROPOSITION 3: An opimal conrac condiional on he disaser sae implemens he safe w() w b, w(1) w b 1, w( D) projec and is characerized by payoffs o he agen condiional on cash flows, 1, -D, and addiional bonus b / given only in he disaser sae for he zero cash flow. The principal's expeced payoff is given by g p( w,) p w, forw w, where he minimum expeced payoff o he agen a is equal o g w ( ). The inuiion behind his resul is ransparen. Depending on he sae of naure, cash flow can eiher be a bad or a good oucome -- i is bad in he good sae and good in he disaser sae. When conracing on he saes of naure is no allowed as in Proposiion 1, he agen has o be given high rewards for boh and 1 cash flows in order o implemen he safe projec, which makes he opimal conrac expensive. When conracing on he saes of naure is allowed, he conrac can differeniae beween cash flows and 1, and i is herefore no necessary o promise he agen high payoffs for eiher cash flow in he good sae of naure. Insead, i is opimal o give he agen a bonus b for cash flow in he disaser sae of naure. While his bonus becomes high when is small, he expeced bonus, b, remains small. As a resul, he cos of managerial compensaion remains small. 2.5 Implemenaion In pracice, condiioning on saes of naure is quie challenging, because saes of naure are difficul o caegorize and verify. However, our opimal conrac requires condiioning only on 13

14 he exreme saes of naure, such as a major financial crisis. During such a crisis a number of major financial insiuions eiher go bankrup or heir equiy suffers exreme losses. Therefore, one pracical way o implemen our conrac is as follows. A he ime when he whole economy is doing well, managers who have eiher he auhoriy or he abiliy o change he riskiness of heir projecs should be awarded ou-of-money pu opions on oher companies in he same line of business which are likely o be ruined in he case of disaser. This implemenaion is relaively inexpensive a he award ime since hese opions would be cheap. However hey would resul in large payoffs o heir holders in case of disaser. The necessary cavea is ha he manager would ge his payoff only if his company says afloa (payoff in he disaser sae in our model). The manager would raionally anicipae his large reward a he ouse and would implemen he safe projec. These conracs are no observed in pracice, however. One argumen agains such compensaion is ha i provides managers wih exra incenives o ake down heir compeiors. Our paper is he firs o poin ou he paricular benefis of using pu opions on oher companies as a par of managerial compensaions. In ligh of he recen financial crisis, he benefis of such compensaion can ouweigh is downsides, since he losses from excessive risk-aking i helps o preven can be quie large. A comprehensive cos-benefi analysis of such execuive compensaion should be execued in a general equilibrium seing, which is beyond he scope of his paper and is lef for fuure research. 3 Coninuous Time Model 3.1 Formulaion This secion exends he one-period model o he coninuous ime, infinie horizon seing. The agen manages a projec ha generaes a coninuous cash flow sream, Y, ha depends on he riskiness of he projec, q,1as follows dy ( q ) d dz Dq dn, (46) where Z is a sandard Brownian moion, and is he volailiy of he cash flow process. The drif of he cash flow process is in he low risk regime, q =, and is increased by > in he high risk regime, q = 1. In addiion, he high risk regime exposes he projec o he possibiliy of a disaser 14

15 oucome, in which case he projec generaes a large one-ime loss D and ges liquidaed immediaely. The disaser process N is a sandard Poisson process wih inensiy δ ha governs arrivals of he disaser sae when dn = 1. A disaser sae resuls in a disaser oucome for he projec, i.e., a loss of D, only if he projec was in he high risk regime a he momen when he disaser arrived. We consider a wo-dimensional agency problem semming from hidden acions aken by he agen. The firs dimension is ha he riskiness of projec can be privaely changed by he agen a any ime and is no observable or verifiable by he invesors, (excep when a disaser akes place). We assume ha D, i.e., swiching o he high risk regime leads o lower expeced cash flows. The second dimension is ha he agen observes he realized cash flows Y, bu he principal does no. The agen repors cash flows { Yˆ ; } o he principal, where he difference beween Y and Ŷ is deermined by he agen s hidden acions. The principal receives only he repored cash flows dŷ from he agen. The conrac specifies compensaion for he agen di, as well as a erminaion imes L and D, ha are based on he agen s repors and disaser realizaions. We begin our analysis assuming ha he conrac canno be made condiional on he occurrence of he disaser, unless he disaser oucome occurs, in which case he projec is erminaed a ime inf : dn 1& q 1. (47) D Thus, he erminaion ime L and agen s compensaion, di, depend only on he hisory of he agen s repors. In his secion we model he agency problem by allowing he agen o diver cash flows for his own privae benefi. The agen receives a fracion,1 of he cash flows he divers; if 1, here are dead-weigh coss of concealing and divering funds. The agen can also exaggerae cash flows by puing his own money back ino he projec. By alering he cash flow process in his way, he agen receives a oal flow of income of [ ˆ dy dy ] di, where ˆ [ dy ] ( ˆ) ( ˆ dy dy dy dy dy). diversion over-reporing (48) 15

16 The agen is risk neural and discouns his consumpion a rae. The agen mainains a privae savings accoun, from which he consumes and ino which he deposis his income. The principal canno observe he balance of he agen s savings accoun. The agen s balance S grows a ineres rae [ ˆ ds S d dy dy ] di dc, (49) where dc is he agen s consumpion a ime. The agen mus mainain a nonnegaive balance on his accoun, ha is, S. Once he conrac is erminaed, he agen receives payoff of zero. Therefore, he agen s oal expeced payoff from he conrac a dae zero is given by s W E e dc, s (5) where min{, }. L D The principal discouns cash flows a rae r, such ha r. Once he conrac is erminaed, she receives expeced liquidaion payoff L unless he erminaion was caused by he disaser oucome, in which case he liquidaion value of he projec is L D. The principal s oal expeced profi a dae zero is hen rs ˆ r p E e ( dy ) 1 ( )1 s dis e L L D L D DL. (51) The projec requires he sar-up exernal capial of K. The principal offers o conribue his capial in exchange for a conrac L, I ha specifies a erminaion ime L D and paymens {I ; L D} ha are based on repors a Y ˆ-measurable sopping ime. Ŷ. Formally, I is a Y ˆ-measurable coninuous process, and L is In response o a conrac L, I, he agen chooses a feasible sraegy o maximize his expeced payoff. A feasible sraegy is a riple of processes q, C, (i) q {,1}, Ŷ adaped o Y such ha (ii) Ŷ is coninuous and, if 1, Y Ŷ has bounded variaion, 1 (iii) (iv) C is non-decreasing, and he savings process, defined by (2), says nonnegaive. 16

17 The agen s sraegy q, C, Ŷ is incenive compaible if i maximizes his oal expeced payoff W given a conrac L, IAn incenive compaible conrac refers o a quinuple L, I, q, C, Ŷ} ha includes he agen s recommended sraegies. The opimal conracing problem is o find an incenive compaible conrac L, I, q, C, Ŷ} ha maximizes he principal s profi subjec o delivering he agen an iniial required payoff W. By varying W we can use his soluion o consider differen divisions of bargaining power beween he agen and he principal. For example, if he agen enjoys all he bargaining power due o compeiion beween principals, hen he agen mus receive he maximal value of W subjec o he consrain ha he principal s profi be a leas zero. 3.2 Derivaion of he Opimal Conrac We solve he problem of finding an opimal conrac in hree seps. Firs, we show ha i is sufficien o look for an opimal conrac wihin a smaller class of conracs, namely, conracs in which he agen chooses o repor cash flows ruhfully and mainain zero savings. Second, we consider a relaxed problem by ignoring he possibiliy ha he agen can save secrely. Third, we show ha he conrac is fully incenive compaible even when he agen can save secrely. We begin wih a revelaion principle ype of resul: LEMMA A: There exiss an opimal conrac in which he agen (i) chooses o ell he ruh, and (ii) mainains zero savings. The inuiion for his resul is sraighforward i is inefficien for he agen o conceal and diver cash flows ( 1) or o save hem ( r), as we could improve he conrac by having he principal save and make direc paymens o he agen. Thus, we will look for an opimal conrac in which ruh elling and zero savings are incenive compaible The Opimal Conrac wih a Non-Conracible Disaser Process 17

18 Noe ha if he agen could no save, hen he would no be able o over-repor cash flows and he would consume all income as i is received. Thus, dc di λ( dy dyˆ ). (52) We relax he problem by resricing he agen s savings so ha (52) holds and allowing he agen o seal only a a bounded rae. Afer we find an opimal conrac for he relaxed problem, we show ha i remains incenive compaible even if he agen can save secrely or seal a an unbounded rae. One challenge when working in a dynamic seing is he complexiy of he conrac space. Here, he conrac can depend on he enire pah of repored cash flows Ŷ. This makes i difficul o evaluae he agen s incenives in a racable way. Thus, our firs ask is o find a convenien represenaion of he agen s incenives. Define he agen s promised value W Ŷ,q afer a hisory of repors Ŷ s, s o be he oal expeced payoff he agen receives from ransfers and erminaion uiliy, if he ells he ruh and chooses risk process q afer ime : ˆ ( s) W( Y, q) E e dc s. (53) The following resul provides a useful represenaion of W Ŷ,q. LEMMA B: A any momen of ime, here is a sensiiviy Ŷ,q of he agen s coninuaion value owards his repor and public randomizaion process R Ŷ,q such ha ˆ ˆ ˆ ˆ dw q W d di ( Y, q)( dy ( q ) d) q W dn dr ( Y, q) E[ dr ( Y, q)]. (54) This sensiiviy Ŷ,q is deermined by he agen s pas repors Ŷ s, and he riskiness of he projec q s, s. Proof of Lemma B: See Appendix. Informally, he agen has incenives no o seal cash flows if he ges a leas of promised value for each repored dollar, ha is, if. If his condiion holds for all hen he agen s payoff will always inegrae o less han his promised value if he deviaes. If his condiion fails on a se of 18

19 posiive measures, he agen can obain a leas a lile bi more han his promised value if he underrepors cash when <. In addiion, he agen has incenives o swich o he high-risk regime o increase he projec cash flows. However, his can lead o a disaser oucome resuling in he loss of coninuaion payoff for he agen. Inuiively, he agen would be willing o ake on more risk only if his coninuaion payoff is below a cerain hreshold. Indeed, if he agen decides o swich o he high-risk regime a ime, he cash flow will be increased by d. If he agen ruhfully repors he cash flow and swiches back o he low-risk regime a ime +d, i resuls in an increase of he agen s coninuaion payoff by d. However, wih probabiliy d his could lead o a disaser oucome beween ime and +d, and he loss of coninuaion payoff W. Thus, choosing he high-risk regime is incenive compaible if and only if d Wd, or W. LEMMA C: If he agen canno save, ruh-elling is incenive compaible if and only if for all. Moreover, i is opimal for he agen o choose he low-risk regime whenever W and swich o he high-risk regime whenever W. Proof of Lemma C: See Appendix. Now we use he dynamic programming approach o deermine he mos profiable way for he principal o deliver he agen any value W. Here we presen an informal argumen, which we formalize in he proof of Proposiion 1 in he Appendix. Denoe by pw he principal s value funcion (he highes profi o he principal ha can be obained from a conrac ha provides he agen he payoff W). Because he principal has he opion o provide he agen wih W by paying a lump-sum ransfer of di and moving o he opimal conrac wih payoff W di, p( W) p( W di) di. (55) 19

20 Equaion (55) implies ha bw 1 for all W; ha is, he marginal cos of compensaing he agen can never exceed he cos of an immediae ransfer. Define W C as he lowes value such ha bw C 1. I is hen opimal o pay he agen according o di max( W W C,). (56) These ransfers, and he opion o erminae, keep he agen s promised value beween and W C. Wihin his range, Lemmas B and C imply ha he agen s promised value evolves according o dw q W d dz q W dn dr E[ dr ], (57) when he agen is elling he ruh. We need o deermine he sensiiviy of he agen s value o repored cash flows and he randomizaion process R ha maximize he principal s value. R Le d denoe he sochasic probabiliy ha he randomizaion process R resuls in a jump in R he agen s coninuaion payoff beween and +d. Le J denoe he size of his jump. Using Io s lemma, he principal s expeced cash flows and changes in conrac value are given by E[ dy dp( W )] q q Wb'( W ) p''( W ) q L D p W d 2 R R R R Ep'( W) dz p( W J ) p( W) d p'( W) J d. (58) Because a he opimum he principal should earn an insananeous oal reurn equal o he discoun rae, r, we have he following equaion for he value funcion: R R, qj,, rp ( W ) d max q q Wp '( W ) p ''( W ) q L D p W d E p'( W) dz ( p( W J ) p( W) p'( W) J ) d s.. 2 W q 1 J R d W R R R R (59) (6) Firs we noe ha opimal randomizaion assures ha funcion p is weakly concave. If p(w) is convex around W, hen i is opimal o randomize around W, since ( ( R R p W J ) p( W) p'( W) J ) is posiive 2

21 for some J R. However, he randomizaion makes p(w) affine in W, and hence p (W)=. Thus, p(w) canno be convex, and he las erm in equaion (59) is zero. Equaion (59) can be rewrien as follows: ( ) '( ) max ''( ) 1 2 ( ) '( ) rp W Wp W p W q D L p W Wp W., q (61) Given he concaviy of p, pw and L p( W) Wp'( W). In addiion, D, because he high-risk projec is dominaed by he low risk projec. As a resul, he smalles possible β and q are S S opimal. Thus,, and q=1 for W W, and q= for W W. Inuiively, since he inefficiency in his model resuls from early erminaion and gambling, choosing lowes possible β has a double benefi. Firs, reducing he risk o he agen lowers he probabiliy ha he agen s promised value falls o zero. Second, lower β reduces incenives for he agen o gamble. We require hree boundary condiions o pin down a soluion o his equaion and he boundary W C. The firs boundary condiion arises because he principal mus erminae he conrac o hold he agen s value o, so p = L. The second boundary condiion is he usual smooh pasing condiion he firs derivaives mus agree a he boundary and so pw C 1. The final boundary condiion is he super conac condiion for he opimaliy of W C, which requires ha he second derivaives mach a he boundary. The principal s value funcion herefore saisfies he following second-order ordinary differenial equaion: 1 2 2, W S W W C, (62) 2 rp( W ) Wp'( W ) p ''( W ) wih pw pw C W W C for W W C. This condiion implies ha pw C, or equivalenly, using equaion (62), C C rp( W ) W. (63) This boundary condiion has a naural inerpreaion: I is beneficial o pospone paymen o he agen by making W C larger because doing so reduces he risk of early erminaion. Posponing paymen is 21

22 sensible unil he boundary (63), when he principal and agen s required expeced reurns exhaus he available expeced cash flows. The following proposiion formalizes our findings: PROPOSITION 5: The conrac ha maximizes he principal s profi and delivers he value W, W C o he agen akes he following form: W evolves according o ˆ S S P P dw q W d di ( dy ( q ) d) q W dn J dk J dk. (64) When W, W C, di. When W W C, paymens di cause W o reflec a W C. If W W C, an immediae paymen W W C is made. The conrac is immediaely erminaed when he disaser oucome occurs. The conrac is also erminaed when W reaches. Jumps of he size J = W S W P can happen when W reaches hresholds W P or W S S P. The cumulaive inensiies and of he jumps are given by Wd ( ˆ dy ( ) d) S d 1 S, W S P W W W (65) Wd ( ˆ dy ( ) d) P d 1 P. W S P W W W (66) P S If W (W, W ), he agen s ime zero coninuaion payoff will be W P S W W wih probabiliy, S P W W and W S P W W wih probabiliy, when he projec is iniiaed. I is opimal for he agen o choose S P W W he high-risk regime whenever S W and he low-risk regime whenever S W. The principal s W expeced payoff a any poin is given by a weakly concave coninuously differeniable funcion pw ha solves W on he inerval [W, W P ], ( r ) p( W) ( ( DL)) Wp'( W) p''( W), (67) (68) 2 rp( W ) Wp '( W ) p ''( W ), on he inerval [W S, W C ], and is given by ( S P P pw ) pw ( ) P pw ( ) pw ( ) WW, S P W W (69) 22

23 on he inerval (W P, W S ). When DrL, (7) he agen is never allowed o gamble, i.e., W P =. When and DrL, ( r) p( W S ) ( ) p'( W S ) W S D L, (71) hen here are no jumps in he agen s coninuaion payoff, i.e., W P = W S. When DrL, and ( r) p( W S ) ( ) p'( W S ) W S D L, hen < W P < W S and W P solves ( r) p( W P ) ( ) p'( W P ) W P D L. (72) The addiional boundary condiions are given by p( W) 1 for W W C, (73) rpw C W C =, (74) p() = L, Proof of Proposiion 5: See Appendix Condiion (7) has a simple and inuiive inerpreaion. I says ha gambling canno be allowed under any circumsances when he expeced cash flows generaed by he projec are less han he expeced loss D due o gambling and he opporuniy cos of delaying liquidaion rl. When gambling is allowed under he opimal conrac, hen no randomizaion is needed when condiion (71) is saisfied. This condiion guaranees ha he principal s value funcion pw is concave, i.e., p( W). When condiion (71) is no saisfied, randomizaion is needed. Inuiively, he value funcion would be convex on inerval (W P, W S ) wihou jumps. This means ha i should be opimal o increase 23

24 he volailiy of W on his inerval. In he limi as he volailiy goes o infiniy, he W jumps beween W P and W S P. The size of he jump is deermined by condiion (72), which is equivalen o p( W ). One immediae implicaion of Proposiion 5 is ha he opimal conrac relies on deferred compensaion when gambling is possible more han when he agen canno increase he riskiness of he projec, i.e., = and. In boh cases he payoffs a he consumpion boundary W C and b(w C ) lie on he sraigh line (74), whose slope is negaive. Since he gambling is inefficien, he principal s expeced payoff given by he value funcion p(w) is lower. As a resul, he opimal consumpion boundary is W C is higher when gambling is possible The Opimal Conrac wih a Conracible Disaser Process In his secion, we derive he opimal conrac when he disaser process is observable and conracible. As before, he disaser oucome occurs when he disaser arrives and he risk is high, resuling in immediae liquidaion of he projec and he negaive payoff of L-D o he invesors. However, unlike he previous seing, he erms of he conrac can be condiional on he even when he disaser arrives and he risk is low. We will call coninuaion payoff adjusmens condiional on such an even bonuses. I is sraighforward o modify Lemma B for he seing wih bonuses. LEMMA D: A any momen of ime owards his repor and a bonus process B Ŷ,q such ha, here is a sensiiviy Ŷ,q of he agen s coninuaion value ˆ ˆ ˆ 1 ˆ dw W (1 q ) B ( Y, q) q W d di ( Y, q)( dy ( q ) d) (1 q ) B ( Y, q) q W dn. This sensiiviy Ŷ,q and he bonus process Ŷ,q are deermined by he agen s pas repors Ŷ s, and riskiness of he projec q s, s. (75) The main difference from Lemma B is he jump B in he agen s coninuaion payoff ha happens when he disaser occurs and he risk is low. Unlike he jump when he disaser oucome occurs, he size of bonus jump B is an endogenous variable. We could add an addiional 24

25 randomizaion erm in (3) as we did in Lemma B. However, as we argue laer, randomizaion would no improve he conrac. As before, he agen has no incenives o seal cash flows if he ges a leas of promised value for each repored dollar, ha is, if. This condiion is no affeced by he bonuses, which are condiional on he disaser process. However, he bonuses have a direc effec on he agen s incenives o gamble. In he high-risk regime, a disaser oucome resuls no only in he loss of he coninuaion payoff W for he agen, bu also in he loss of he bonus B ha he agen would ge in he low-risk regime. On he oher hand, he agen s benefis from swiching o he high-risk regime are unaffeced by he bonuses and are equal o an increase in he agen s coninuaion payoff by d over he period beween ime and +d. Thus, choosing he high-risk regime is incenive compaible if and only if d ( W B) d, or W B. The following lemma summarizes our conclusions. LEMMA E: If he agen canno save, ruh-elling is incenive compaible if and only if for all. Moreover, if i is opimal for he agen o choose he low-risk regime whenever W B and swich o he high-risk regime whenever W B. Proof of Lemma E: See Appendix. Lemma E saes ha posiive bonuses condiional on he no-disaser oucome reduce he agen s incenive o choose he inefficien high-risk projec. The minimum bonus needed o implemen he low-risk regime is equal o min{ W,}. Since he principal s expeced payoff is a concave funcion of W, i is opimal o choose he smalles possible β and B ha saisfy incenive compaibiliy and implemen a desired level of risk. Thus, in he opimal conrac, i should always be he case ha β = In addiion, he opimal bonus should beb =min{w S W,} when he low-risk regime is implemened. PROPOSITION 6: The conrac ha maximizes he principal s profi and delivers he value W, W C o he agen is characerized by hree regions: 25

26 a. High-risk regime (q =1) wihou bonuses when W, W B S B S b. Low-risk regime (q =) wih he bonus equal o ( W W ) when W [ W, W ) S C c. Low-risk regime (q =) wih no bonuses when W [ W, W ] The agen s coninuaion payoff W evolves according o ˆ B Wd ( dy ( ) d) qwdn forw [, W ) S (( ) ) ( ˆ S B S dw W W d dy d) (1 q)( W W) qwdn for W [ W, W ) ( ˆ S C Wd di dy d) qwdn forw [ W, W ] (76) When W, W C, di. When W W C, paymens di cause W o reflec a W C. If W W C, an immediae paymen W W C is made. The conrac is erminaed a ime when W reaches zero or he disaser oucome occurs. The principal s expeced payoff a any poin is given by a concave smooh funcion pw, which saisfies B S S B S S C ( r) p( W) ( ( DL)) Wp'( W) p''( W) forw [, W ) (77) ( r) p( W) p( W ) ( ) W W p'( W) p''( W) forw [ W, W ) (78) rp( W ) Wp '( W ) p''( W ) for W [ W, W ] (79) The principal s coninuaion funcion pw also saisfies he following boundary condiions p = L C and pw ( ) 1. The consumpion boundary is deermined by rpw C = W C, while he bonus S region is deermined by W and W B ha solve S B S L W p'( W ) D p( W ). (8) Equaion (8), which deermines boundary W B of he gambling region, says ha he second derivaives of he soluions of (77) and (78) mach a W B. 4 Hidden Effor Throughou our analysis so far, we have concenraed on he seing in which he cash flows are privaely observed and he agen may diver hem for his own consumpion. In his secion, we consider a principal-agen model in which he agen makes a hidden effor choice. 26

27 We now assume ha he principal observes he cash flows bu no he agen s effor. A conrac condiional on cash flow realizaions deermines he agen s compensaion and erminaion of he projec. Thus, here are wo key changes o our model. Firs, since cash flows are observed, misreporing is no an issue. Second, we assume ha a each poin in ime, he agen can choose his effor. Depending on agen s effor, he resuling cash flow process is dy ( q ) d dz Dq dn. Working is cosly for he agen, and he cos of effor is given by convex funcion c(). Specifically, we suppose ha he agen s flow of uiliy is given by dc di c( ) d. The agen s choices of effor and risk q are measurable processes wih respec o pas cash flow realizaions. As in Secion 3, we can define he agen s coninuaion uiliy afer a hisory of realized cash flows Y s, s for given measurable process and q of he agen s effor and risk choices: ( s W ) E e ( dis c( ) d). I is sraighforward o modify Lemmas B and C for he case wih hidden effor. LEMMA F: A any momen of ime, here is a sensiiviy of he agen s coninuaion value owards cash flows realizaion such ha ( ) ( )( ( ) ) dw q W d c d di Y dy q d q W dn Process is measurable wih respec o he cash flow process Y. The agen s opimal choice of effor is given by arg max { ( )} c. The agen chooses he low-risk regime whenever W and swich o he high-risk regime whenever W. (9) Proof of Lemma F: See Appendix. Given a conrac in place characerized by sensiiviy Y of he agen s payoff oward he projec s cash flows, he agen opimally chooses his effor and he riskiness of he projec. For simpliciy, we assume ha he effor cos funcion c is such ha here is an inernal soluion 27

28 for he opimal effor for any >. Thus, he opimal effor level for he agen is such ha c ( ) =. PROPOSITION 7: The conrac ha maximizes he principal s profi and delivers he value W, W C o he agen is characerized by wo regions: a. High-risk regime (q =1) when W, W G G C b. Low-risk regime (q =) when W [ W, W ]. When W, W C, di. When W W C, paymens di cause W o reflec a W C. If W W C, an immediae paymen W W C is made. The conrac is erminaed a ime when W reaches zero or he disaser oucome occurs. The principal s expeced payoff a any poin is given by a concave smooh funcion pw, which saisfies ( r) p( W) ( W) ( DL) ( ) W c( ( W)) p'( W) c'( ( W)) p''( W) forw[, W ) (91) rp( W ) ( W ) ( W c( ( W ))) p'( W ) c'( ( W )) p''( W ) for W [ W, W ] (92) G G C The agen s opimal equilibrium effor (W) is given by W G H ( W) s.. c'( ( W)) forw [ W, W ), (93) * S H C ( W) arg max c( ) p'( W) c'( ) p''( W) for W [, W ) [ W, W ], (94) where W H is such ha H * H ( W ) ( W ), (95) and W G is such ha funcion p is wice coninuously differeniable a W G. The agen s coninuaion payoff W evolves according o dw ( q ) W c( ) d di c( )( dy ( q ) d) q W dn. The principal s coninuaion funcion pw also saisfies he following boundary condiions: p = L C and pw ( ) 1. The consumpion boundary is deermined by rpw C = W C. 28

29 5 Conclusion This paper sudies opimal incenive conracs in a seing wih a wo-dimensional moral hazard problem, in which an agen wih limied liabiliy privaely chooses he riskiness of he projec and can privaely diver cash flows for consumpion. Relaive o he low risk projec, he high risk projec increases he probabiliy of a high cash flow realizaion, bu i also resuls in high losses in a bad sae of naure, named disaser. To avoid risk-aking and divering funds, invesors mus cede addiional rens o he manager. In he saic seing, we find ha he opimal conrac ha implemens he low-risk regime and ruhful reporing of cash flows may require a very high level of compensaion for he manager if he conrac erms are coningen only on he repored cash flows. The expeced level of managerial compensaion can be much lower if disaser saes can be idenified ex-pos by paying he manager a large bonus if he firm survives. I can be implemened by giving he agen ou-of-money pu opions on he companies ha are likely o be ruined in he disaser sae. In a dynamic conex, because managerial rens mus be reduced following poor performance o preven fund diversion, poorly performing managers will ake on disaser risk even under an opimal conrac. Even when disaser saes can be idenified ex-pos, if performance is sufficienly weak he manager will forfei eligibiliy for a bonus, and again ake on disaser risk. Our model can explain why subopimal risk aking can emerge even when invesors are fully raional and managers are compensaed opimally. Risk aking is likely o ake place when he disaser sae is unlikely or afer a hisory of poor performance, when he agen has lile skin lef in he game. As a resul, he opimal conrac has increased reliance on deferred compensaion. Even absen evidence of risk aking, small reducions in performance could lead o harsh punishmens. 29

30 APPENDIX A Proofs A.1 Proof of Lemma B Noe ha W Ŷ,q is also he agen s promised value if Ŷ s, s, were he rue cash flows and he agen repored ruhfully. Therefore, wihou loss of generaliy we can prove (54) for he case in which he agen ruhfully repors Ŷ = Y. In ha case, by he law of ieraed expecaions s s V e di ( Y) e W ( Y, q), (A.1) is a maringale and by he maringale represenaion heorem here is a processes and ψ such ha dv e ( Y, q)( dy ( q ) d) e Y, q dn d dr Y, q E[ dr Y, q ], (A.2) where dy ( q) d is a muliple of he sandard Brownian moion for L D. If dn =1 and q =1, hen W, since he projec liquidaed immediaely and he coninuaion payoff for he agen is zero. If dn =1 and q =, hen, since he conracual paymens o he agen canno be condiional on he disaser process realizaions ha do no lead o he disaser oucome. Thus, Yq, qwyq,. (A.3) Differeniaing (A.1) wih respec o we find e di( Y) e W( Y, q) de dw( Y, q), dv e ( Y, q)( Y ( q ) d) e q W Y, q dn d dr Y, q E[ dr Y, q ] and hus (54) holds. QED (A.4) A.2 Proof of Lemma C If he agen ruhfully repors he cash flows and chooses he equilibrium risk level q, he presen value of his fuure consumpion is given by V. Then according (5) and (A.1) and Lemma B, his expeced payoff is equal o W EV E V e, ( ), [ ]. Y q dy q d e q W Y q dn d dr E dr 3