Learning in Dynamic Incentive Contracts

Transcription

1 Leaning in Dynamic Incenive Conacs PETER M. DEMARZO AND YULIY SANNIKOV * May 28, 2008 Absac We deive he opimal dynamic conac in a coninuous-ime pincipal-agen seing, in which boh invesos and he agen lean abou he fim s pofiabiliy ove ime. Because invesos lean abou he fim s fuue pofiabiliy fom cuen oupu, which also depends upon he agen s acions, deviaions by he agen diso invesos beliefs. We chaaceize he opimal conac, and show ha he pefomance sensiiviy of he agen s payoff is se o accoun fo boh moal hazad and asymmeic infomaion. We show ha he opimal conac can be implemened by compensaing he agen wih equiy and allowing him o manage he fim s cash eseves by seing he is payou policy. Unde his conac, he fim accumulaes cash unil i eaches a age balance ha depends on he agen s peceived poduciviy. Once his age balance is eached, he fim sas paying dividends equal o is expeced fuue eanings, while any empoay shocks o eanings eihe add o o deplee he fim s cash eseves. The fim is liquidaed if i exhauss is cash eseves. We also show ha once he fim iniiaes dividends, hese dividends ae smooh elaive o eanings, and ha liquidaion is fis-bes, despie he agency poblem. * Sanfod Univesiy and UC Bekeley/NYU. This eseach is based on wok suppoed in pa by he NBER and he Naional Science Foundaion unde gan No

2 In many ypes of agency elaionships oupu is infomaive abou boh he agen s cuen acions and he fuue qualiy of he pojec iself. Company eanings depend on boh cuen manageial effo and unobsevable exogenous facos ha affec he fim s fuue pofiabiliy. Mogage paymens eflec he boowe s financial discipline and he abiliy o pay, which is influenced by such facos as income and nondisceionay expenses. The pefomance of a woke may depend on boh he woke s effo and skill. In all of hese seings, he dynamic agency poblem is complicaed by he fac ha boh he agen and he pincipal ae leaning abou he qualiy of he pojec ove ime, and ha he agen s acions can diso he pincipal s infeences. In a dynamic conex, aligning incenives in such an envionmen is complicaed by he fac ha he agen s knowledge of his own acions will povide pivae infomaion egading he pojec s qualiy. Absen appopiae long-em conacs, incenives may even beak down compleely due o he ache effec : When he pincipal has full bagaining powe bu canno commi o a long-em conac, he pincipal will measue he agen s pefomance agains a oughe benchmak if he agen eveals himself o be a moe poducive ype. As shown by Laffon and Tiole (1988), in anicipaion of he ache effec he agen will no wan eveal his poduciviy when i is high, and so will offse i by educing effo. 1 In conas, he lieaue on caee concens (e.g. Holmsom (1999)) finds ha if all bagaining powe lies in he hands of he agen, whose wage is se compeiively each peiod, hen he agen s concen ove his fuue wage will ceae incenives fo effo even if employes only offe fla wage conacs. The eason is ha wokes wih a bee oupu hisoy ge highe wage offes because hey ae peceived o be moe poducive. Though incenives o boh exe effo and eveal high abiliy exis in his seing, hey may be oo song o oo weak o achieve an opimal oucome depending on agen s abiliy o manipulae make beliefs hough effo. These incenive popeies of sho-em conacs ceainly moivae he sudy of dynamic conacs unde full commimen. Unfounaely, chaaceizing he opimal dynamic agency conac wih leaning has poven difficul because of echnical 1 See also Baon and Besanko (1987), and Feixas, Guesneie, and Tiole (1985), fo elaed esuls. 2

3 challenges. Sandad dynamic agency models assume ha he pincipal and he agen always have common knowledge abou how he disibuion of fuue oupu depends on he agen s effo. 2 This sucue faciliaes he applicaion of ecusive mehods o solve fo an opimal conac. 3 This is no he case when he pincipal is leaning abou he pojec s qualiy, as hen he pincipal and agen may have asymmeic infomaion abou fuue oupu. Asymmeic infomaion aises endogenously, as he agen s acions can diso he pincipal s infeences abou fuue pofiabiliy. Fo example, suppose he agen lowes effo, causing a empoay dop in oupu. If he pincipal anicipaes high effo by he agen, he aibues a dop in oupu o a negaive exogenous shock, and so has lowe expecaions abou fuue oupu han he agen. This effec povides an addiional channel hough which he agen s cuen acions may ale he ems of he conac going fowad, and heeby affec he agen s incenives. We consuc a model of dynamic agency wih leaning ha is boh (1) acable unde full commimen and (2) adapable o sudy a numbe of applicaions. The main ingediens of ou model ae as follows. Time is coninuous. The agen is isk neual bu has limied liabiliy. The agen s oupu is sochasic, wih an expeced flow on dae given by δ if he agen pus in full effo. The agen may also ake acions ha geneae pivae benefis, bu a he cos of educing he expeced oupu flow below δ. Impoanly, he expeced flow of benefis fom he elaionship, δ, is no diecly obsevable and evolves ove ime sochasically. The souce of he asymmeic infomaion poblem is ha he pincipal and he agen lean abou δ fom ealized oupu. Thus, he pincipal s belief abou δ becomes disoed if he agen deviaes. A conac povides he agen wih incenives by specifying paymens o he agen and condiions unde which he elaionship wih he pincipal is eminaed, boh as a funcion of he hisoy of obseved oupu. 2 See, e.g., Rogeson (1985), Thomas and Woall (1990), Phelan and Townsend (1991) o moe ecenly Albuqueque and Hopenhayn (2001), DeMazo and Fishman (2007), DeMazo and Sannikov (2006), Biais, Maioi, Planin and Roche (2007) and Biais, Maioi, Roche and Villeneuve (2007). 3 The use of ecusive mehods o solve fo dynamic conacs was pioneeed by Spea and Sivasava (1987). See also Sokey and Lucas (1989) and Ljungqvis and Sagen (2004). 3

4 Fo he sake of conceeness, ou pape sudies one applicaion in which he agen is a manage of a fim, who can pu effo in he fim and engage in ouside aciviies o poduce a pivae benefi. In his case oupu coesponds o he fim s pofis and he hidden paamee δ eflecs he fim s pofiabiliy. In geneal, howeve, he model can be applied o many ohe seings. Fo example, in he conex of egulaion and he pocuemen of public goods, oupu can be inepeed as he social benefi ne of he conaco s ealized coss, which depends on he conaco s effo, while δ epesens he conaco s efficiency. In a labo conac seing (see e.g. Lazea (1986) and Holmsom (1999)), we inepe δ as he agen s poduciviy in poducing some measued oupu. Finally, we can apply o model in he conex of mogages as in Piskoski and Tchisyi (2007), whee δ capues he boowe s expeced abiliy o pay (i.e. income ne of spending needs) and oupu coesponds o acual paymens. Ou dynamic agency model delives geneal conclusions abou (1) he sucue of incenive povision in he opimal conac and (2) inefficien eminaion ha may be equied o povide opimal incenives. Regading incenives, he key idea is o undesand how he agen s acions affec he pincipal s beliefs, which deemine he benchmak agains which he pincipal measues he agen s fuue pefomance. While he agen may be ewaded fo cuen oupu, he may be elucan o apply effo if doing so aises expecaions egading fuue oupu, and heeby makes i moe difficul fo he agen o ean fuue bonuses. To ensue he agen s incenive o exe effo, he opimal conac mus compensae he agen fo he effec of effo on cuen oupu as well as on invesos beliefs abou fuue oupu. In ha sense, he asymmeic infomaion poblem aises he opimal pefomance sensiiviy of he conac. 4 Fo ou specific applicaion, in which he agen is a manage who uns he fim, we show ha a simple way o ceae appopiae incenives is by giving he agen a facion of he fim s equiy. The idea ha equiy ceaes incenives is no new, bu he way ha equiy solves he incenive poblem in ou seing is new. The value of equiy changes when he 4 The pefomance sensiiviy we ae alluding o is wih egad o he agen s uiliy. I may no be diecly obseved in he agen s immediae compensaion o consumpion. Indeed, in he opimal conac hee ae egions fo which he agen s immediae consumpion is insensiive o he fim s cuen pefomance, bu an incease in uiliy sill follows due o an incease in expeced fuue bonuses. Thus, one mus be caeful elaing hese esuls o empiical measues of pay fo pefomance. 4

5 oupu ealizaion, which measues he agen s pefomance, diffes fom expeced oupu, which is a benchmak agains which pefomance is measued. Why does he agen no wan o ease he benchmak wih lowe effo, in ode o ceae a bee impession in he fuue? The eason is ha by loweing he expecaion abou fuue oupu, he agen also lowes he value of his equiy sake. Afe solving fo he opimal conac, we show ha i can be implemened hough a naual specificaion of he fim s payou policy and a capial sucue. When he fim is young, i makes no payous and accumulaes cash unil i eaches a age level of financial slack ha is posiively elaed o he manage s peceived poduciviy. Once his age balance is eached, he fim iniiaes dividend paymens. Fom ha poin on, he fim pays dividends a a ae equal o is expeced fuue eanings. Specifically, he fim smoohes is dividends by absobing any empoay shocks o eanings hough an incease o decease of is cash eseves (o available cedi). When he fim exhauss is available cash (o cedi), he fim is liquidaed and agency elaionship is eminaed. This payou policy capues well many of he sylized facs associaed wih obseved payou policies. Immaue fims do no pay dividends, bu insead eain hei eanings o inves, epay deb, and build cash eseves. Fo hese fims he value of inenal funds is high, as hey isk unning ou of cash and being pemauely liquidaed. Bu once he fim has sufficien financial slack, dividends ae hen paid a a level ha appeas o be a smooh esimae of he pemanen componen of eanings. 5 Because dividend changes eflec pemanen changes o pofiabiliy, hey ae pesisen and have subsanial implicaions fo fim value. The payou policy we idenify in ou model is incenive compaible fo he manage and is he unique implemenaion of he opimal mechanism ha povides he fases possible payou ae subjec o he consain ha he fim will no need o aise exenal capial in he fuue. 6 Thus, in ou model fims build up a age level of inenal funds o ensue ha hey neve need o liquidae inefficienly due o financial consains. This consain 5 See Linne (1956), Fama and Babiak (1968), as well as moe ecen sudies by Allen and Michaely (2003) and Bav e. al. (2005). 6 Thee ae many alenaive dividend policies ha ae opimal if hee is no cos o aising equiy capial in he even ha he fim uns ou of cash in he fuue. In pacice, howeve, hese coss ae subsanial. 5

6 alone canno explain dividend smoohing, as once he age is eached we would expec all excess cash flows o be paid ou as dividends. The key dive of dividend smoohing in ou model is he fac ha hee is leaning abou he fim s pofiabiliy based on he cuen level of he fim s cash flows. When cash flows ae high, he fim s peceived pofiabiliy inceases. This aises he cos of liquidaing he fim (we ae liquidaing a moe pofiable enepise), and heefoe aises he opimal level of financial slack. Thus, a poion of he fim s high cuen cash flow will opimally be used o incease is cash eseves, esuling in a smoohed dividend policy. 7 In he nex secion of he pape, we descibe he coninuous-ime pincipal-agen poblem wih leaning abou he fim s pofiabiliy. Then in Secion 2, we pesen he soluion o his poblem when moal hazad is absen, which is based puely on opion-value consideaions. This soluion is impoan o undesand he opimal long-em conac wih moal hazad, which we deive in Secion 3. In Secion 3, we also show how his opimal conac can be implemened in ems of he fim s payou policy. Secion 4 pesens a heoeical jusificaion of he opimal conac, and Secion 5 discusses seveal exensions of ou basic model. 1. Model Risk-neual ouside invesos hie a isk-neual manage o un a fim. Invesos have unlimied wealh, wheeas he agen has no iniial wealh and mus consume nonnegaively. 8 Boh he agen and invesos discoun he fuue a ae > 0. The pofiabiliy of he fim depends on he agen s manageial skill δ, which evolves sochasically wih ime. The fim povides esouces ha allow he agen o use his skills moe poducively. Tha is, inside he fim he agen can use his skills o poduce cash flows a an expeced ae of δ, wheeas ouside he fim, his poduciviy is only λδ 7 This payou policy is heefoe also consisen wih he evidence ha fims cash and leveage posiions ae songly influenced by pas pofiabiliy, even when fims ae financially unconsained (see, e.g., Fama and Fench (2002)). 8 The assumpion ha he agen has no iniial wealh is wihou loss of genealiy; equivalenly, we can assume he agen has aleady invesed any iniial wealh in he fim. The agen s limied liabiliy pevens a geneal soluion o he moal hazad poblem in which he fim is simply sold o he agen. 6

7 fo some λ (0, 1). If he agen is fied, his expeced ouside opion equals he pepeuiy value of his ouside poduciviy, and so is given by R(δ ) = λδ /. When he agen woks fo he fim, hee is moal hazad. The fim s cumulaive cash flows X ae dx = ( δ a ) d +σ dz, (1) whee σ is he volailiy of cash flows, Z is a sandad Bownian moion, and a 0 is he exen o which he agen dives his own effo and fim esouces fom he fim fo pivae benefi. I is moe efficien o use he agen s and fim s esouces inside he fim o geneae pofi ahe han fo he agen s pivae benefi. If a > 0, he agen ges a pivae benefi a ae of λa. The pofiabiliy of he fim δ, which depends on he agen s manageial skill, evolves ove ime. 9 The fim s cash flows convey infomaion abou fuue pofiabiliy. We model his by assuming ha δ sas a δ 0 > 0 and evolves accoding o dδ = ν( dx ( δ a ) d) = νσ dz, (2) as long as δ > 0. Equaion (2) can be inepeed as he seady sae of a fileing poblem in which he fim s ue pofiabiliy, δ *, is unobsevable, and he manage and invesos aemp o lean he fim s pofiabiliy based on pas cash flows. Specifically, suppose δ * 0 is iniially nomally disibued wih sandad deviaion γ 0, and is subjec o nomally disibued shocks so ha dδ * = σ dz whee Z is a Bownian moion independen of Z. Then we can apply he Kalman-Bucy file o compue he Bayesian poseio disibuion fo δ * given he obseved cash flows up o dae. In paicula, if we define δ E [ δ ] 9 Alenaively, we could inepe δ as he suiabiliy of he agen s manageial syle and skills o cuen make condiions. 7

8 hen 10 dδ = γ /σ 2 (dx (δ a )) d (3) whee γ is he sandad deviaion of he poseio disibuion and evolves accoding o dγ = ((σ ) 2 -γ 2 /σ 2 ) d. Fo simpliciy, fo mos of he pape we focus he seady sae of his poblem, in which γ 0 =γ =σσ ; in ha case, (3) educes o (2) wih ν = σ /σ. In Secion 5 we analyze an exension in which γ 0 > σσ and γ is deceasing, so hee is moe unceainy abou fim s fuue pofiabiliy when i is young. Since he agen s unobsevable effo enes he leaning equaion (2), in ou conacual envionmen he agen may have pivae infomaion no only abou his effo, bu also abou he fim s pofiabiliy. Indeed, if he pincipal expecs he agen o choose effo a, he pincipal will updae his belief δ ˆ abou fim pofiabiliy accoding o dδ ˆ = ν( dx ( δ ˆ a ) d), δ ˆ =δ. (4) 0 0 Thus, if he agen chooses a diffeen effo saegy aˆ a, he pincipal s belief δ ˆ will be incoec. Unlike in a sandad pincipal-agen seings, in ou envionmen deviaions by he agen geneae asymmeic infomaion beween him and he pincipal. The fim equies exenal capial of K 0 o be saed. The invesos conibue his capial and in exchange eceive he cash flows geneaed by he fim less any compensaion paid o he agen. The agen s compensaion is deemined by a long-em conac. This conac specifies, based on he hisoy of he fim s cash flows, nonnegaive compensaion dc fo he agen while he manages he fim, as well as a ime when he agen is fied. Fomally, a conac is a pai (C, ), whee C is a non-deceasing X-measuable pocess ha epesens he agen s cumulaive compensaion and is an X- measuable sopping ime. When he agen leaves he fim, he eceives his ouside payoff of λδ /, and he invesos eceive a payoff of 10 Noe ha in his inepeaion, he fim s cash flow pocess is given by dx = (δ a ) d + σ dz, and so (2) epesens he fim s expeced cash flows afe inegaing ove he poseio fo δ. 8

9 κδ L( δ ) = L0 +, (5) whee λ + κ < We allow he value of he fim afe he manage leaves o depend on δ o ecognize ha invesos may be able o capue some facion of he fim s pofiably, e.g. by hiing a new manage. A conac (C,) ogehe wih an X-measuable effo ecommendaion a is opimal given an expeced payoff of W 0 fo he agen if i maximizes he pincipal s pofi E e (( δ a) d dc) + e L( δ) (6) 0 subjec o W0 = E e ( λ a ) ( ) 0 d + dc + e R δ given saegy a (7) and W0 E e ( λ aˆ ) ( ) 0 d + dc + e R δ fo any ohe saegy â (8) By vaying W 0 > R(δ 0 ), we can use his soluion o conside diffeen divisions of bagaining powe beween he agen and he invesos. Fo example, if he agen enjoys all he bagaining powe due o compeiion beween invesos, hen he agen will eceive he maximal value of W 0 subjec o he consain ha he invesos payoff be a leas equal o hei iniial invesmen, K. We say ha he effo ecommendaion a is incenivecompaible wih espec o he conac (C,) if i saisfies (7) and (8) fo some W 0. REMARKS. Fo simpliciy, we specify he conac assuming ha he agen s compensaion and he eminaion ime ae deemined by he cash flow pocess, uling ou public andomizaion. This assumpion is wihou loss of genealiy, as we will lae veify ha public andomizaion would no impove he conac. 11 In pinciple, he value of he fim afe he agen leaves may depend boh on boh he agen s and invesos beliefs abou fim pofiabiliy. Howeve, because hose beliefs coincide on he equilibium pah, wihou loss of genealiy we may specify L as a funcion of a single vaiable δ. 9

10 2. The Fis-Bes Soluion. Befoe solving fo he opimal conac, we deive he fis-bes soluion as a benchmak. In he fis-bes, he pincipal can conol he agen s effo, and so we can ignoe he incenive consains (8). Then i is opimal o le he agen ake acion a = 0 unil liquidaion, since i is cheape o povide he agen wih a flow of uiliy by paying him han by leing him dive aenion o pivae aciviies. Then he oal cos of poviding he agen wih a payoff of W 0 is E e dc 0 = W0 E e R( δ ), and he pincipal s payoff is E e d e L R W δ + ( ( δ ) + ( δ)) 0. 0 Thus, wihou moal hazad he pincipal chooses a sopping ime ha solves b( δ 0) = max E e δ d+ e ( L( δ ) + R( δ)). (9) 0 This is a sandad eal-opion poblem ha can be solved by he mehods of Dixi and Pindyck (1994). Because liquidaion is ievesible, i is opimal o igge liquidaion when he expeced pofiabiliy δ eaches a ciical level of δ ha is below he level δ* such ha R(δ ) + L(δ ) = δ / See Figue 1. 10

11 b b(δ) L( δ ) + R( δ ) δ/ δ δ* δ Figue 1: Fis-bes Liquidaion Theshold and Value Funcion We have he following explici soluion fo he fis-bes liquidaion heshold and value of he fim: Poposiion 1. Unde he fis-bes conac, he fim is liquidaed if δ δ whee * νσ δ=δ.. 2 The pincipal s payoff is b ( δ 0) W 0, whee δ δ 2 b( δ ) = + L( δ)+ R( δ) exp ( δ δ) νσ if δ δ, and b ( δ ) = L(δ) + R(δ) ohewise. Poof: Noe ha b ( δ ) is he soluion on [δ, ) o he odinay diffeenial equaion b ( δ ) =δ+ ν σ b ''( δ ) (10) 2 wih bounday condiions (a) b( δ ) = L( δ ) + R( δ ), λ+κ (b) b '( δ ) = (smooh-pasing), 11

12 (c) and b( δ) δ/ 0 as δ. Le us show ha b ( δ ) gives he maximal pofi aainable by he pincipal. Fo an abiay conac (C, ), conside he pocess s = δ + δs 0 G e b( ) e ds. Le us show ha G is a submaingale. Using Io s lemma, he dif of G is e b ( δ ) + e ν σ b ''( δ ) + e δ, 2 which is equal o 0 when δ > δ and e ( L( δ ) + R( δ )) + e δ < 0 when δ < δ. Theefoe, he pincipal s expeced pofi a ime 0 is [ ] = b( δ ) W. 0 0 E e L( δ ) + e ( δd dc) = E e ( L( δ ) + R( δ )) + e δd W0 0 0 EG W G W The inequaliies above become equaliies if and only if δ δ and δ > δ befoe ime. Ou chaaceizaion of he fis-bes conac can be inepeed in ems of he fim s capaciy o susain opeaing losses. A any momen of ime, he fim mus be able o wihsand a poduciviy shock of up o dδ = (δ δ). Fom (2), his coesponds o a cash flow shock equal o dδ δ δ dx δ d =σ dz = = ν ν. (11) We can view Equaion (11) as specify he minimal level of financial slack he fim will equie in ode o avoid inefficien liquidaion. This esul will play an impoan ole in ou implemenaion of he opimal conac, which we conside nex. 12

13 3. An Implemenaion Having chaaceized he fis-bes oucome, we now conside he poblem of finding he opimal dynamic conac in ou seing wih boh moal hazad and asymmeic infomaion. The ask of finding he opimal conac is complex due o he huge space of fully coningen hisoy-dependen conacs o conside. A conac (C,) mus specify how he agen s consumpion and he liquidaion ime depend on he enie hisoy of cash flows. In classic seings wih unceainy only abou he agen s effo bu no he fim s poduciviy, hee ae sandad ecusive mehods o deal wih such complexiy. These mehods ely on dynamic pogamming using he agen s fuue expeced payoff (a.k.a. coninuaion value) as a sae vaiable. 12 Bu wih addiional unceainy and he poenial fo asymmeic infomaion abou he fim s poduciviy, hese sandad mehods do no apply diecly o ou model. Thus we will ake a diffeen appoach. We begin insead by conjecuing a simple and inuiive implemenaion fo he conac. In ou seing wih moal hazad, if he agen had deep pockes he fis-bes liquidaion policy could be aained by leing he agen own he fim. If he agen s wealh is limied, howeve, negaive cash flow shocks can lead o inefficien liquidaion. In ode o minimize his inefficiency, i is naual o expec ha in an opimal conac, he fim will build up cash eseves unil i has an opimal level of financial slack. In his secion we conside an implemenaion based on his inuiion, and hen show his implemenaion is incenive compaible. Though ou implemenaion is ahe simple, we will hen veify he opimaliy of his conac in he space of all possible conacs in he following secion. 12 Fo example, see Spea and Sivasava (1987), Abeu, Peace and Sacchei (1990) (in discee ime) and DeMazo and Sannikov (2006) and Sannikov (2007a) (in coninuous ime) fo he developmen of hese mehods, and Piskoski and Tchisyi (2006) and Philippon and Sannikov (2007) fo hei applicaions. 13

14 3.1. Cash Reseves and Payou Policy Conside an all-equiy financed fim ha uses cash eseves o povide financial slack. 13 Denoe he level of is cash eseves by M 0. Because he fim eans inees a ae on hese balances, is eanings a dae ae given by de M d + dx = ( M + δ ) d + σ dz, (12) whee we have assumed he agen s acion a = 0. If he fim uses hese eanings o pay dividends dd, hen is cash eseves will gow by dm de dd = ( M + δ ) d dd + σ dz. (13) Conside a conac in which he fim is foced o liquidae if i deplees is eseves and M = In ode o avoid inefficien liquidaion, we know fom (11) ha he fim mus have eseves M of a leas ( ) M 1 ( δ ) δ δ ν. (14) Theefoe, i is naual o suppose ha if M < M 1 (δ ), he fim will eain 100% of is eanings in ode o incease is eseves and educe he isk of inefficien liquidaion. In ha case, dividends ae equal o zeo: dd = 0 if M < M 1 (δ ). (15) Suppose he fim achieves he efficien level of eseves, so ha M = M 1 (δ ). In ode o mainain is eseves a he efficien level, using (14) and (2) we mus have 1 dδ dm = dm ( δ ) = =σdz. (16) ν 13 We noe ha ou poposed implemenaion is no unique, no is i clea ha i is opimal. The fim could also mainain financial slack hough alenaive means, such as a cedi line o loan commimen. The analysis would be simila; fo convenience we focus on he simples implemenaion in ems of cash eseves. See Biais e al (2006) fo a simila implemenaion based on cash eseves in a moal hazad seing wihou leaning. 14 Ove any finie ime peiod, he fim will expeience opeaing losses wih pobabiliy one; heefoe, absen cash o cedi, he fim mus shu down. Howeve, i is no ye clea whehe i is opimal o deny he fim funds and foce liquidaion if M = 0. We will addess opimaliy in he following secion. 14

15 Tha is, o mainain he efficien level of eseves, he fim should adjus is cash balances by he supise componen of is eanings. Then fom (13), dividends ae equal o he fim s expeced eanings: [ ] ( ) dd = E de = M +δ d if M = M 1 (δ ). (17) The following esul demonsaes ha wih his payou policy, he liquidaion policy is fis-bes: Poposiion 2. If M = M 1 (δ ) and if he fim follows he payou policy (17) afe ime, hen M = 0 if and only if δ = δ. Poof: Given he payou policy (17), he fim s cash balance evolves accoding o (16). Theefoe, M = 0 implies ( ) 1 ( ) δ =δ+ dδ =δ+ν dm =δ+ν M =δ νm δ =δ. s s Finally, because hee is no benefi fom mainaining eseves in excess of he amoun needed o avoid inefficien liquidaion, we assume he fim pays ou any excess cash immediaely. Thus, we can summaize he fim s payou policy as follows: 1 M < M δ 0 if ( ) 1 dd = ( M +δ ) d if M = M ( δ). (18) 1 1 M M ( δ ) if M > M ( δ) Unde he payou policy descibed by (18), he fim accumulaes cash as quickly as possible unil i eihe uns ou of cash and is inefficienly liquidaed, o is eseves each he efficien level. Once he efficien level of eseves is aained, he fim begins paying dividends a a ae equal o is expeced eanings. I will coninue o opeae in his fashion unless δ falls o δ, in which case M = 0 and he fim is liquidaed as in he fisbes. 15

16 Figue 2 pesens conac dynamics fo an example. Unil ime 1.5 he fim has cash balances below he efficien level, and i sands he isk of being liquidaed inefficienly. Howeve, in his example inefficien liquidaion does no happen. A ime 1.5 he fim s cash level eaches he efficien age, and he fim iniiaes dividends. Dividends coninue unil he fim s pofiabiliy falls sufficienly and i is liquidaed a dae 5. The igh panel of Figue 2 illusaes ha he oal quaely dividends ae significanly smoohe han eanings. Pofiabiliy (dela) Cum Cash Flows X 70 Efficien Reseves M1 Cash M Eanings Dividends Figue 2: Conac dynamics when = 5%, σ = 15, ν = 33%. The liquidaion heshold is δ = Compensaion and Incenive Compaibiliy The equiemen of cash eseves combined wih he payou policy descibed above deemines he liquidaion ime,, of he conac. To complee his implemenaion, we need specify he agen s compensaion, C, and hen assess whehe he conac povides appopiae incenives. Noe ha if he agen had unlimied wealh, we could povide he agen wih appopiae incenives fo effo by paying him a facion λ of he fim s cash flows. This soluion is no possible, howeve, since he fim s cash flows may be negaive and he agen has limied liabiliy. Given he implemenaion above, a naual alenaive o conside is o pay he agen he facion λ of he fim s dividends (ahe han is cash flows), which ae always non-negaive. 16

17 This compensaion can be inepeed as poviding he agen a facion λ of he fim s equiy, wih he poviso ha he agen no eceive any poceeds fom a liquidaion should i occu. This oucome could be implemened, fo example, by giving ouside invesos pefeed sock wih complee pioiy in he even of liquidaion. (Alenaively, he agen may eceive a zeo inees loan o puchase he shaes, which becomes due in he even of liquidaion.) We efe o he agen s compensaion as equiy ha is escindable in he even of liquidaion. Now we ae eady o conside he agen s incenives. To veify incenive compaibiliy, we mus deemine he agen s payoff given diffeen effo choices and payou policies. Conside he case in which he fim is aleady paying dividends; ha is, M = M 1 (δ ). Suppose he agen follows he poposed implemenaion. Then fom (17), he agen s expeced payoff W is given by ( s ) ( ) ( s ) ( ) E e λ dds + e R( δ ) = E e λ ( Ms +δ s) ds+ e R( δ) ( s ) ( ) = E e ( λ Ms + R( δ s)) ds+ e ( λ M R( )) + δ =λ M + R( δ) whee he las equaion follows fom he fac ha M s and δ s ae maingales fo s >. 15 Now conside a deviaion in which he agen cashes ou by immediaely paying ou all cash as a dividend, dd = M, and hen defauling. Unde his saegy, he agen again eceives a payoff of W = λ M + R(δ ). (19) Thus hee is no incenive fo he agen o deviae in his way. We can similaly show ha his implemenaion is obus o ohe ypes of deviaions fo he agen. Fo example, because (19) implies ha he agen s payoff inceases by λ fo each dolla of addiional cash held by he fim, hee is no incenive fo he agen o shik and engage in ouside 15 Noe ha if X s is a maingale, hen ( ) X ( s ) ( ) ( s ) ( ) X e ds e E e X s ds e = + = + X. 17

18 aciviies. The following esul esablishes he incenive compaibiliy, wih egad o boh he agen s effo choice and payou policy, of ou poposed implemenaion: Poposiion 3. Suppose he agen holds a facion λ of he fim s equiy, escindable in he even of liquidaion, and ha liquidaion occus if he fim s cash balance falls o zeo. Then fo any effo saegy and payou policy, he agen s expeced payoff is given by (19). Thus, i is opimal fo he agen o choose acions a = 0 and o adop he payou policy in (18). Poof: Conside an abiay payou policy D and effo saegy a. Define s = ( λ s +λ s) + 0 V e dd a ds e W Then using he fac ha δ is a maingale and ha [ ] [ ] ( ) E dm = M d + E dx dd = M d + δ a d dd, he dif of V is [ ] = ( λ +λ + [ ] ) = = e ( λ dd +λads ( λ M +λδ ) +λe[ dm] ) E dv e dd ads W E dw = 0 and so V is a maingale. Thus, because M = 0 so ha W = R(δ ), he agen s expeced payoff fom his abiay saegy is E e ( λ dd ) ( ) ( ) 0 +λ ad + e R δ E V e W e = + R δ = EV = V = W [ ] and heefoe he implemenaion is incenive compaible Jusificaion of he Opimal Conac Because sandad mehods do no apply diecly o ou model, in his secion we develop a new appoach o jusify he opimaliy of ou conjecued conac. While he specific 18

19 soluion is unique o ou poblem, we popose a hee-sep mehod o solve simila poblems: (1) Isolae he necessay incenive consains, which ae mos impoan in limiing he aainable expeced pofi. (2) Show ha he conjecued conac solves he pincipal s opimizaion poblem subjec o jus he necessay incenive consains. (3) Veify ha he conjecued conac is fully incenive-compaible. We conjecued a conac in he pevious secion, and veified is full incenivecompaibiliy in Poposiion 3. We need o execue seps 1 and 2 of he veificaion agumen. Befoe we poceed, we noe ha he agen mus ake acion a = 0 a all imes in he opimal conac. The eason is ha because λ < 1, i is cheape o pay he agen diecly ahe han le him ake acions fo pivae benefi. Lemma 1 (High Effo). In he opimal conac a = 0 unil ime. Poof. Conside any conac in which someimes a > 0, and le us show ha hee exiss a bee conac. Le us change i, by giving he agen an opion o ask he pincipal fo exa paymens dc. If he agen execises his opion a leas once, hen he agen s wages and eminaion ime ae deemined as if he ue pah of oupu wee Xˆ X 0 dcs ' λ =. Then he agen is indiffeen beween aising a above 0 and simply asking he pincipal fo exa money. If he asks fo money now wheneve he was loweing effo peviously, hen he agen s saegy is incenive-compaible, and he pincipal s pofi is sicly highe. Theefoe, he oiginal conac canno be opimal. Fom now on, we esic aenion o conacs wih ecommended effo a = 0. 19

20 4.1. Necessay Incenive Consains The necessay incenive-compaibiliy consains ae fomulaed using appopiaely chosen sae vaiables. 16 Fo ou poblem, we mus include as sae vaiables a leas he pincipal s cuen belief abou he agen s skill ˆ δ, which evolves accoding o d ˆ δ = ν( dx ˆ δ d), ˆ δ = δ, 0 0 and he agen s coninuaion value when he agen follows he ecommended saegy (a s ) afe ime, and he pincipal has a coec belief abou he agen s skill s ( ) ( ) W ( ) ˆ = E e dc s + e R δ δ = δ given saegy {a = 0}. The vaiables ˆ δ and W ae well-defined fo any conac (C, ), afe any hisoy of cash flows {X s, s [0, ]}. Howeve, hey do no fully summaize he agen s incenives, which depend on he agen s deviaion payoffs, he payoff ha he agen would obain if δ δ due o he agen s pas deviaions. Theefoe, we can fomulae only necessay ˆ condiions fo incenive compaibiliy using he vaiables ˆ δ and W. Lemma 2A which is sandad in coninuous-ime conacing, povides a sochasic epesenaion fo he dependence of W on he cash flows {X } in a given conac (C,). The connecion beween W and X maes fo he agen s incenives. Lemma 2A (Repesenaion). Thee exiss a pocess {β, 0} in L * such ha dw = W d dc + β ( dx ˆ δ d). (20) Poof. See Appendix A. The pocess β deemines he agen s exposue o he fim s cash flows shocks and heefoe he sengh of he agen s incenives unde he conac. I is heefoe naual 16 Fo example, in coninuous ime Sannikov (2007b) solves an agency poblem wih advese selecion using he coninuaion values of he wo ypes of agens as sae vaiables, Williams (2007) solves an example wih hidden savings using he agen s coninuaion value and his maginal uiliy as sae vaiables. 20

21 o expec ha β mus be sufficienly lage fo he conac o be incenive compaible. Fo example, conside a deviaion a ˆ = 1. The agen can gain in wo ways fom his deviaion. Fis, he agen eans he payoff λd ouside he fim. Second, fom (4), he lowe oupu of he fim educes pincipal s esimae of he fim s poduciviy. If his deviaion is he agen s fis, hen dδ ˆ = ν( dx δ d) = dδ ν d. Given hese loweed expecaions, he agen can coninue o shik and educe effo by ν fom ha poin onwad and sill geneae cash flows consisen wih he pincipal s expecaions, fo an addiional expeced pepeual gain of λνd. Because he deviaion educes he agen s conacual payoff by β d, his deviaion is pofiable if ( 1 ) β <λ+λν =λ +ν Lemma 2B below fomalizes his inuiion and esablishes a necessay condiion on β fo a conac (C, ) o be incenive compaible. Lemma 2B (Incenive Compaibiliy). Conside a conac (C,), fo which he agen s coninuaion value evolves accoding o (20). A necessay condiion fo {a = 0} o be incenive-compaible wih espec o (C,) is ha β λ(1 + ν/). Poof. See Appendix A Veificaion of Opimaliy In his secion we veify ha ou conjecued implemenaion is indeed an opimal conac. Recall fom Secion 3 ha he agen s payoff in his conac is defined by whee W = λ M + R(δ ), dm = ( M + δ ) d + σ dz and dd = 0 while M < 1 ( ) ( ) M δ = δ δ ν and dm = σ dz, I follows ha M = M 1 ( δ ) and dd = ( M + δ ) d heeafe. 21

22 dw = W d + λ(1+ν/)σ dz and dc = 0 unil W eaches W 1 (δ ), and W = W 1 (δ ) and dc = W d heeafe, (21) whee W ( δ ) = R( δ) + λ( ν + )( δ δ). This evoluion happens unil W eaches R(δ ), iggeing liquidaion. Le us show ha his conac aains he highes expeced pofi among all conacs ha delive value W 0 o an agen of skill level δ 0 and saisfy he necessay incenivecompaibiliy condiion of Lemma 2B. The se of such conacs includes all fully incenive-compaible conacs. Since he conjecued conac is incenive-compaible, as shown in Poposiion 3, i follows ha i is also opimal. Le us pesen a oadmap of ou veificaion agumen. Fis, we define a funcion b(w 0,δ 0 ), which gives he expeced pofi ha a conac of Secion 3 aains fo any pai (W 0, δ 0 ) wih W 0 R(δ) and δ 0 δ. Poposiion 4 veifies ha his definiion is indeed he expeced payoff of ouside equiy holdes in ou implemenaion. Afe ha, Poposiion 5 shows ha he pincipal s pofi in any alenaive conac ha saisfies he necessay incenive-compaibiliy condiion of Lemma 2B is a mos b(w 0, δ 0 ) fo any pai (W 0, δ 0 ) wih W 0 R(δ) and δ 0 δ. I follows ha he conjecued conac of Secion 3 is opimal. Fo W R(δ) and δ δ, define a funcion b(w,δ) as follows. (i) Fo W >W 1 (δ), le b(w,δ) = b( δ ) W. (ii) Fo W = R(δ), le b(w,δ) = L(δ). Ohewise, fo δ > δ and W (R(δ), W 1 (δ)), le b(w,δ) solve he equaion b( W, δ) = δ + Wb ( W, δ) + W λ (1 + ) σ b ( W, δ) + ν σ b ( W, δ) + λ(1 + ) νσ b ( W, δ) 1 2 ν ν 2 2 WW 2 δδ Wδ (22) wih bounday condiions given by (i) and (ii). 22

23 Fo an abiay conac (C,) wih an incenive-compaible effo ecommendaion a, in which he agen s coninuaion value evolves accoding o (21), define he pocess s = δs s + δ 0 G e ( ds dc ) e b( W, ). Noe ha on he equilibium pah we always have δ = ˆ δ. Lemma 3 helps us pove boh Poposiions 4 and 5. Lemma 3. When δ δ and C is coninuous a, hen dg = e ( νb ( W, δ ) + β b ( W, δ )) σdz e ( b ( W, δ ) + 1) dc + δ W W 1 ν ν ν ( 2 ( (1 + )) WW(, ) + ( (1 + ))( (1 + ) WW(, ) + Wδ (, )) ) σ β λ δ β λ λ δ ν δ 2 2 e b W b W b W d Poof. See Appendix A. Poposiion 4. The conjecued opimal conac of subsecion 3.1 aains pofi b(w 0,δ 0 ). Poof. Unde ha conac, he pocess G is a maingale. Indeed, fo all > 0, he coninuous pocess C inceases only when W = W 1 (δ ) (whee b W (W 1 (δ ),δ ) = -1) and ν β = λ(1 + ), so G is a maingale by Lemma 3. A ime 0, he agen consumes posiively only in ode fo W 0 o dop o W 1 (δ 0 ), and b W (W,δ 0 ) = -1 fo W W 1 (δ 0 ), so G is a maingale hee as well. Theefoe, he pincipal aains he pofi of E e b( W, δ) + e ( δd dc) = E[ G] = G0 = b( W0, δ0). 0 QED. Poposiion 5. In any alenaive incenive-compaible conac (C,) he pincipal s pofi is bounded fom above by b(w 0,δ 0 ). 23

24 Poof. Le us ague ha G is a supemaingale fo any alenaive incenive-compaible conac (C, ) while δ δ. Fis, wheneve C has an upwad jump of ΔC, G has a jump of e - (b(w +ΔC, δ ) - b(w, δ ) - ΔC ) 0, since b W (W,δ) -1 fo all pais (W,δ) (see Appendix B, which shows ha b is concave in W). Second, wheneve C is coninuous, hen β λ(1+ν/) by Lemma 2B. By Lemma 3, he dif of G is e ( b ( W, δ ) + 1) dc + W 1 ( ν ν ν 2 WW WW Wδ ) 2 2 e σ β λ b W δ β λ λ b W δ νb W δ d ( (1 + )) (, ) + ( (1 + ))( (1 + ) (, ) + (, )) < 0 since b W (W,δ) -1 and, as shown in Appendix B, (, ) 0 ν bww W δ and λ(1 + ) bww ( W, δ) + νbw δ ( W, δ) 0 (23) fo all pais (W,δ). Now, le be he ealie of he liquidaion ime o he ime when δ eaches he level δ. Then Poposiion 1 implies ha he pincipal s pofi a ime is bounded fom above by bw (, δ ). I follows ha he pincipal s oal expeced pofi is bounded fom above by E e b( W, δ) + e ( δd dc) = E G G0 = b( W0, δ0). 0 QED We conclude ha Secion 3 pesens he opimal incenive-compaible conac fo any pai (W 0, δ 0 ) such ha W 0 R(δ) and δ 0 δ. If W 0 W 1 (δ 0 ), hen his conac aains he fis-bes pofi, and liquidaion always occus a he efficien level of pofiabiliy of δ = δ. If W 0 < W 1 (δ 0 ), hen liquidaion happens inefficienly wih posiive pobabiliy. 24

25 5. Exensions of he Basic Model. In his secion we exend he basic model in a numbe of diecions of pacical inees. Specifically, we elax he assumpions ha (1) he fim s cash flows ae a sufficien saisic abou fuue pofiabiliy and ha (2) he agen s effo affecs only he cuen cash flow and no fuue pofiabiliy. The pincipal may have ohe souces o lean infomaion abou he fim s fuue pofiabiliy, e.g. pefomance of compaable fims, and he agen may also have addiional pivae infomaion abou fuue pofiabiliy, e.g. fom obseving fim s opeaions fis-hand. Moe geneally, assume ha Bownian moion Z capues he fim s idiosyncaic isk, Z M capues aggegae make (o indusy) isk obsevable by boh he pincipal and he agen, and Z O capues he infomaion ha he agen obseves pivaely. Wihou loss of genealiy, hese Bownian moions can be aken o be independen. Le he cash flows o follow M M dx = ( δ a ) d + σdz + σ dz, and expeced pofiabiliy evolve accoding o M M O O dδ = ν( dx ( δ a ) d + η dz ) + χ( δ a ) d + η dz. In his expession, paamees η M and χ (which ae ypically posiive) measue how news abou ohe companies in he indusy and he agen s cuen effo, especively, affec he fim s fuue pofiabiliy. To analyze his seing, assume a fis ha η O = 0. Then, analogously o ou basic model, he agen s coninuaion value can be epesened as dw = W d dc + β (dx - ˆ δ d) + β M dz M. A necessay condiion fo he opimaliy of effo a = 1 is β λ (1 + (ν - χ)/). (*) The logic is he following: If he agen lowes he fim s cash flow by 1 dolla, he deives an immediae pivae benefi of λ. A he same ime, he pincipal s belief abou δ goes down by ν, while he ue value of δ goes down by χ elaive o wha δ would have been had he agen no deviaed. As a esul, he deviaion fools he pincipal ino hinking ha 25

26 he value of δ is lowe by ν - χ han is ue value. The agen can cash in on his diffeence in beliefs o deive a value of a leas λ(ν - χ)/ by diveing cash fom he fim a ae ν - χ in pepeuiy. We see ha when he agen s effo adds o he fim s fuue pofiabiliy, he needs less moivaion o pu effo. The necessay condiion is in fac sufficien if i holds wih equaliy a all imes. Moeove, if η O 0 and he pincipal adjuss he agen s coninuaion value accoding o dw = W d dc + λ(1 + (ν - χ)/) (dx - ˆ δ d) + β M dz M + λ(ν - χ)/ η O d ˆ Z O, whee ˆ Z O is he agen s epo of Z O, hen he agen also has incenives o epo pivaely obseved innovaions o fim pofiabiliy uhfully. Wha abou he value of β M? One naual conjecue is ha he value of β M compleely insues he agen agains aggegae isk, ha is, β M = - λ(1+(ν-χ)/) σ M. This conjecue is incoec, which we demonsae by chaaceizing he opimal fis bes conac fo he case when η M = 0 and η O = 0. (Noe ha if η M 0 o η O 0 hen fis bes canno be aained fo any finie level of coninuaion value of he agen. Indeed, a sufficienly lage dop in Z necessaily esuls in liquidaion, which is always inefficien if hee is a compensaing ise in Z M o Z O ha keeps δ fixed). In ode o aain fis-bes when η M = 0 and η O = 0, he fim mus be able o absob a negaive cash flow shock of size (δ - δ)/ν. If fim mainains cash balances of his amoun, and if he agen holds a facion λ(1 - χ/) of fim equiy, hen he agen s coninuaion value is λδ/ + λ(1 + (-χ)/) (δ - δ)/ν, and fis-bes is aained. Noe ha his is he minimal value fo he agen equied o aain fis bes, since a dop of Z, which lowes δ o δ, pulls he agen s coninuaion value down by a leas λ(1 + (-χ)/) (δ - δ)/ν o he agen s ouside opion of λδ/. Noe ha he agen eceives he same exposue o idiosyncaic and aggegae cash flow isk in his conac. The conac gives he agen moe value when he fim becomes 26

27 moe pofiable o peven inefficien liquidaion even fo he case when he posiive shock o pofiabiliy is aggegae. 6. Appendix A. Poof of Lemma 2A. Noe ha s = s + 0 V e dc e W is a maingale when he agen follows he ecommended saegy (a s ). By he Maingale Repesenaion Theoem hee exiss a pocess {β, 0} in L * such ha since find ha s ˆ = 0 + βs s δs 0 V V e ( dx ds), dx ˆ δ ds = σdz unde he saegy (a s = 0). Diffeeniaing wih espec o, we s s s dv = e dc + e dw e W d = e β ( dx ˆ δ d) dw = W d dc + β ( dx ˆ δ d). This expession shows how W deemined by X (since ( ˆ δ ) iself is deemined by X ), and heefoe i is valid even if he agen followed an aleaive saegy in he pas. In his case W is inepeed as he coninuaion value ha he agen would have goen afe a hisoy of cash flows {X s, s } if his esimae of he fim s pofiabiliy coincided wih he pincipal s, and he planned o follow saegy (a=0) afe ime. QED. Poof of Lemma 2B. Suppose ha β < λ(1 + ν/) while ˆ δ > 0 on a se of posiive measue. Le us show ha he agen has a saegy ( a ˆ) ha aains an expeced payoff geae han W 0. Le a = δ ˆ δ when β λ(1 + ν/) and aˆ = 1+ δ ˆ δ when β < λ(1 ˆ + ν/) befoe he ime when he agen is fied. Define he pocess 27

28 ˆ λ s ˆ = + δ δ + s + λ s 0 V e W ( ) e ( dc a ds). If he agen follows he saegy descibed above, hen befoe ime ˆ, 17 and dδ d ˆ δ = ν( dx ( δ aˆ ) d ( dx ˆ δ d)) = ν( ˆ δ + aˆ δ ) = 0 o 1, dw = W d dc + β (( δ aˆ ) d + σdz ˆ δ d), dv ( ˆ λ = W δ δ) d + + e W (( ˆ ) ˆ ) ( ˆ ˆ d dc + β δ a d + σdz δd + λν δ + a δ) + λ dc ˆ ( )( ˆ ˆ + λad = β λ ν δ a δ) d + σdz. 0 o -1 The dif of V is 0 if β λ(1 + ν/), and i equals λ(1 + ν/) -β >0 when β < λ(1 + ν/). A ime he agen ges he payoff of W ( ˆ ) λ ( ˆ ) ( ˆ λ + δ δ = R δ + δ δ ) = R( δ ). Theefoe, he agen s oal payoff fom he saegy ( a ˆ) is ( ˆ λ s E e W ) e ( dc ˆ + δ δ + s + λasds) = E[ V] > V0 = W0. 0 We conclude ha β λ(1 + ν/) when ˆ δ > 0 is a necessay condiion fo he incenive compaibiliy of he agen s saegy. Poof of Lemma 3. Noe ha fo δ δ, he funcion b saisfies paial diffeenial equaion (22) even if W > W 1 (δ). Indeed, since bw (, δ) = b( δ) W and b W = -1 in ha egion, he equaion educes o b W W b ( ( δ ) ) = δ + 2ν σ ''( δ). This equaion holds by he definiion of b. 17 Noe ha dδ d ˆ δ = ν( ˆ δ + aˆ δ ) = 0 o ν implies ha δ ˆ δ fo all. 28

29 When C is coninuous a, hen using Io s lemma, ( δδ δ ) db W W d dc b W b W b W b W d (, δ) = ( ) W(, δ) + σ β WW(, δ) + ν (, δ) + βν W (, δ) + ( νb ( W, δ ) + β b ( W, δ )) σdz = b( W, δ ) d δ d b ( W, δ ) dc + δ W W ( Wδ ) σ ( β λ(1 + )) b ( W, δ ) d+ ( β λ(1 + ))( λ(1 + ) b ( W, δ ) + νb ( W, δ )) d, 2 1 ν 2 ν ν 2 WW WW whee he second equaliy follows fom (22). Fom he definiion of G, i follows ha Lemma 3 coecly specifies how G evolves. QED 7. Appendix B. We mus show ha fo all pais (δ, W), he funcion b(δ, W) saisfies (, ) 0 ν bww W δ and λ(1 + ) bww ( W, δ) + νbw δ ( W, δ) 0. I is useful o undesand he dynamics of he pai (δ, W ) unde a conjecued opimal conac fis. Fom (2) and (21), he pai (δ, W ) follows dδ = ν σ dz and dw = W d + λ(1+ν/)σ dz unil W eaches W 1 (δ ), and W = W 1 (δ ) heeafe. (24) When W eaches he level R(δ ), eminaion esuls. The lines paallel o W 1 (δ) ae he pahs of he join volailiies of (W, δ ). Due o he posiive dif of W, he pai (W, δ ) moves acoss hese lines in he diecion of inceasing W. See he figue below fo efeence. δ R(δ) δ W 1 (δ) W 29

30 The phase diagam of (W, δ ) povides wo impoan diecions: he diecion of join ν volailiies, in which dw / dδ = λ(1 + )/ ν, and he diecion of difs, in which W inceases bu δ says he same. We need o pove ha b W (δ,w) weakly deceases in boh of hese diecions. To sudy how b W (W,δ) depends on (W,δ), i is useful o know ha b W (W, δ ) is a maingale (Lemma 4) and ha b W (R(δ),δ) inceases in δ (Lemma 5). Lemma 4. When he evoluion of (W, δ ) is given by (24), hen b W (W,δ ) is a maingale. Poof. Diffeeniaing he paial diffeenial equaion fo b(w,δ) wih espec o W, we obain 2 ν ν 2 0 = WbWW ( W, δ ) + 2λ (1 + ) σ bwww ( W, δ) + 2ν σ bw δδ ( W, δ) + λ(1 + ) νσ bwwδ ( W, δ). The igh hand side of his equaion is he dif of b W (W,δ) when W < W 1 (δ ) by Io s lemma. When W = W 1 (δ ), hen b W (W, δ ) = -1 a all imes. Theefoe, b W (W, δ ) is always a maingale. QED Lemma 5. b W (R(δ), δ) weakly inceases in δ. Poof. Noe ha bw ( 0, δ0) = b( δ0) W0 E e ( b( δ) L( δ) R( δ)) δ0, W 0. (25) Tha is, he pincipal s pofi equals fis-bes minus he loss of payoff due o ealy inefficien liquidaion. Le us show ha fo all ε > 0, b(r(δ 0 )+ε,δ 0 ) - b(r(δ 0 ),δ 0 ) inceases in δ 0. Conside he pocesses (W i,δ i 1 ) (i = 1,2) ha follow (24) saing fom values δ and δ 0 = δ 0 + Δ and W i 0 = R(δ 1 0 ) + ε. Then fo any pah of Z, he pocess fo i = 1 ends up in liquidaion a a soone ime i and a a highe value of δ 1. Indeed, fom he law of 1 30

31 moion (24), i is easy o see ha while he diffeence δ 2 -δ 1 says consan a all imes, W 2 W 1 becomes lage han λ/δ afe ime 0, whee λ/ is he slope of R(δ). Since b( δ ) L( δ) R( δ) inceases in Δ, i follows ha ( ( ) ( ) ( )) 0, 0 ( ( ) ( ) ( )) 0, E e b δ L δ R W E e b L R W 1 δ 1 δ δ 1 δ 2 δ 2 δ 2 As a esul, br ( ( δ ) + ε, δ ) br ( ( δ ), δ ) = E e ( b( δ ) L( δ 1 ) R( δ 1 )) δ 1 0, W 0 + b( δ0) L( δ0) R( δ0) E e ( b( δ ) L( δ ) R( )) 2 δ 2 δ 2 0, W 0 + b( δ0) L( δ0) R( δ0) = br ( ( δ ) + ε, δ ) br ( ( δ ), δ ), whee we used (25) o deive he fis and he las inequaliy. QED We can use Lemmas 4 and 5 o each conclusions abou how b W (W,δ) changes as W ν inceases o as δ and W incease in he diecion dw / dδ = λ(1 + )/ ν. Lemma 6. b W (W,δ) weakly deceases in W. Poof. Le us show ha fo any δ 0 δ, fo any wo values W 1 0 <W 2 0, b W (W 1, δ 0 ) b W (W 2, δ 0 ). Conside he pocesses (W i,δ ) (i = 1,2) ha follow (24) saing fom values (W 1 0,δ 0 ) and (W 2 0,δ 0 ) fo δ 1 0 < δ 2 0. Then fo any pah of Z, we have W 2 W 1 0 unil ime 1 when W 1 eaches he level of R(δ ). The ime when W 2 eaches he level of R(δ ) is Since W W = R( δ ), i follows ha δ δ and W W. Using Lemmas 4 and 5, 1 2 bw( W0, δ0) = E bw( R( δ ), δ ) ( ( ), ) ( 1 E b 1 W R δ δ b 2 2 = W W0, δ0). QED Lemma 7. b W (W,δ) weakly deceases in he diecion, in which W and δ incease ν accoding o dw / dδ = λ(1 + )/ ν. 31

32 Poof. Conside saing values (W 0 i, δ 0 i ) ha saisfy δ δ 0 = Δ > 0 and W 2 1 ν 0 - W 0 = Δ λ(1 + ) / ν. Saing fom hose values, le he pocesses (W i, δ i ) (i = 1,2) follow (*). Then fo any pah of Z, a all imes δ 2 - δ 1 =Δ and W 2 W 1 ν Δ λ(1 + ) / ν (wih equaliy afe ime 0 only if W 2 = W 1 (δ 2 ) and W 1 = W 1 (δ 1 )). Theefoe, he ime 1 when W 1 eaches he level of R(δ 1 ) occus a leas as soon as he ime 2 when W 2 eaches he level of R(δ 2 ). Also, since W W +Δ λ(1 + ) / ν > R( δ ) +Δ λ/, i follows ha 2 1 ν δ δ and 2 1 W W. Using Lemmas 4 and 5, bw( W0, δ0) = E bw( R( δ ), δ ) ( ( ), ) ( 1 E b 1 W R δ δ b 2 2 = W W0, δ0). QED 8. Refeences. Abeu, D., Peace, D., and Sacchei, E. (1990) "Towad a Theoy of Discouned Repeaed Games wih Impefec Monioing." Economeica Vol. 58, pp Albuqueque, R. and Hopenhayn, H. A. (2004) Opimal Lending Conacs and Fim Dynamics, Review of Economic Sudies, Vol. 72(2), No. 247: Allen, F. and R. Michaely, 2003, Payou Policy, in G. Consaninides, M. Hais, and R. Sulz, eds., Handbooks of Economics, Noh-Holland. Akeson, Andew (1991) Inenaional Lending wih Moal Hazad and he Risk of Repudiaion, Economeica 59, Akeson, Andew, and Haold Cole (2005) A Dynamic Theoy of Opimal Capial Sucue and Execuive Compensaion, NBER woking pape No Baon, David P. and David Besanko (1984) Regulaion and infomaion in a coninuing elaionship, Infomaion Economics and Policy, 1: Benazi, Shlomo, Roni Michaely, and Richad Thale, 1997, Do changes in dividends signal he fuue o he pas? Jounal of Finance 52: Benazi, Shlomo, Gusavo Gullon, Roni Michaely, and Richad Thale, 2005, Dividend changes do no signal changes in fuue pofiabiliy, Jounal of Business 78:

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