A Learning Model of Dividend Smoothing

Transcription

1 A Leaning Model of Dividend Smoohing PETER M. DEMARZO AND YULIY SANNIKOV * Apil 24, 27 Peliminay and Incomplee 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. We show ha he opimal conac can be implemened hough he fim s payou policy. The 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 is cash eseves fall below a minimum heshold. We also show ha once he fim iniiaes dividends, his liquidaion policy 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 One of he impoan puzzles in copoae finance is he smoohness of copoae dividends elaive o eanings and cash flows. In an ealy empiical sudy, Linne (1956) developed a model of dividend policy in which he poposed ha fims adjus hei dividends slowly o mainain a age long-un payou aio. Specifically, Linne agued ha fims base hei dividend decisions on hei pecepion of he pemanen componen of eanings, and avoid adjusing dividends based on empoay o cyclical flucuaions. Numeous moe ecen sudies have confimed (e.g. Allen and Michaely (23) and Bav e. al. (25)) ha fims engage in smoohing hei dividends (and hei payous moe geneally). 1 Fuhemoe, dividends end o be paid by maue fims, and dividend changes end o esul in significan sock pice eacions in he same diecion, suggesing ha invesos view dividends as impoan indicaos of he fim s fuue cash flows. In his pape we develop dynamic conacing model of dividend smoohing. We conside a naual pincipal-agen seing in which he agen can educe effo in he fim and engage in ouside aciviies ha geneae pivae benefis. Boh he pincipal (ouside invesos) and he agen ae isk neual, bu he agen is wealh-consained. We depa fom he sandad pincipal-agen seing by assuming boh invesos and he agen lean ove ime abou he fim s expeced fuue pofiabiliy based on is cuen cash flows. A conac povides he agen wih incenives by specifying he agen s compensaion and whehe he fim will coninue o be foced o shu down as a funcion of he fim s hisoy of epoed eanings. Afe solving fo he opimal conac, we show ha i can be implemened hough he fim s payou policy and a capial sucue in which he agen holds a shae of he fim s equiy. 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 agen 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. The fim absobs any empoay shocks o eanings by inceasing o deceasing is cash 1 When discussing payou policy we will ofen, fo simpliciy, efe o dividends alone, bu ou discussion should be inepeed o include shae epuchases as well. Similaly, when discussing deb o leveage policies, we ae efeing o he fim s ne deb, which includes cash eseves.

3 eseves, and i may also boow. Howeve, when he fim s deb eaches he liquidaion value of is asses, he deb holdes liquidae he fim. This payou policy capues well he sylized facs associaed wih obseved payou policies cied above. 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. The level of dividends is based on he fim s esimae of he pemanen componen of is eanings, esuling in dividend paymens ha ae much smoohe han is eanings. Because dividend changes eflec pemanen changes o pofiabiliy, hey ae pesisen and have subsanial implicaions fo fim value. Despie he boad empiical evidence ha fims smooh hei dividends, nomaive heoeical models of dividend smoohing have poved ahe elusive. Modigliani and Mille (1961) showed ha dividend policy is ielevan if capial makes ae pefec and invesmen policy is held consan, so one could ague ha obseved dividend policy is one of many neual vaiaions ha fims could adop. Such a view is difficul o econcile wih he sock pice eacion o dividends discussed above; i is also conay o he lage body of evidence ha make impefecions ae economically significan and impoan dives of copoae financial policy moe geneally. One key difficuly wih poviding a heoeical model in which fim s opimally smooh hei payous elaive o eanings is ha, given a fixed invesmen policy, i necessaily implies ha he fim s leveage o cash posiion (is ne deb) mus be coespondingly non-smooh. This obsevaion is difficul o econcile wih he sandad ade-off heoy of capial sucue and payou policy, which pedics ha fims will mainain a age level of leveage/financial slack wih balances he ax benefis of leveage (equivalenly, he ax disadvanage of eaining cash) wih poenial coss of financial disess. In such a model, empoay cash flow shocks should be passed hough o he fim s payous as i ies o mainain is leveage age. Myes (1984) descipion of he pecking ode hypohesis does include he pedicion ha vaiaions in ne cash flow will be absobed lagely by adjusmens o he fim s deb.

4 Howeve, his conclusion is based on he assumpion ha fims dividends ae sicky in he sho un, and no heoeical jusificaion is povided fo his assumpion. 2 We ake a diffeen appoach in his pape. Fis, we use an opimal mechanism design appoach o idenify he eal vaiables of inees in ou model: he opimal iming of he liquidaion decision, and he payoffs of he agen and invesos afe any hisoy. In spii of Modigliani and Mille (1961), in ou model hee may be many opimal dividend policies ha can implemen his opimal mechanism if hee is no cos o aising equiy capial in he even ha he fim uns ou of cash in he fuue. Ye, in pacice hee ae boh insiuional and infomaional coss o aising equiy. Invesmen banks chage a 7% fee on public offeings. Moeove, new equiy could be unde-piced due o advese selecion. The pecking-ode heoy of Myes (1984) and he advese selecion agumens of Myes and Majluf (1984) imply ha fims should aise funds using secuiies ha ae leas sensiive o he fim s pivae infomaion, i.e. deb. These agumens sugges ha fims should absain fom paying dividends in ode o avoid fuue finance coss, a leas as long as he isk of unning ou of cash and iggeing inefficien liquidaion exiss. Wih his in mind, we hen idenify he unique implemenaion of he opimal mechanism he povides he fases possible payou ae subjec o he consain ha he fim will no need o aise exenal capial in he fuue. 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 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 2 Taken o is logical exeme, he advese selecion agumen in favo of deb given by Myes and Majluf (1984) would sugges ha he fim should neve pay dividends in ode o avoid fuue finance coss. If a fim wee o pay dividends (fo some ohe unknown eason), i would pesumably be balancing he maginal benefi of dividends wih he maginal value of financial slack. In his case one would expec dividends and financial slack o move ogehe.

5 will opimally be used o incease is cash eseves, esuling in a smoohed dividend policy. The fim pays ou dividends exacly a he ae of expeced eanings in ou model in ode o have jus enough cash o avoid inefficien liquidaion, given cuen pofiabiliy. Then he fim s cash balances fall afe bad news abou fuue pofiabiliy due o negaive eanings supises, and ise afe good news, due o posiive eanings supises. If he dividends wee highe han expeced eanings, he fim would un ou of cash unless is pofiabiliy unexpecedly inceases. If he fim s dividends wee lowe han expeced eanings, i would build moe cash han necessay. This would lead o inefficiency if hee ae any coss of keeping funds inside he fim. Thus, dividend smoohing in ou implemenaion of he opimal conac can be explained by wo easons (1) hee ae coss o aising ouside equiy capial and (2) hee ae sligh coss o leaving exa cash in he fim. Ou model implies ha fims smooh dividends by absobing hei cash flow vaiaions hough vaiaions in hei leveage o available financial slack. This conclusion is confimed by empiical evidence. Indeed, as documened by Fazzai, Hubbad, and Peesen (1988) and ohes, financially unconsained dividend-paying fims do appea o employ invesmen policies ha ae less sensiive o shocks o he fim s eanings o cash flows. On he ohe hand, hee is 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 (22)). In ou model, he fim gains financial sengh ove ime o miigae he inefficiencies conneced wih moal hazad. This is a common pedicion of a divese ange of dynamic conacing model, such as DeMazo and Fishman (23), Albuqueque and Hopenhayn (24), Akeson and Cole (25), Clemeni and Hopenhayn (26) and DeMazo and Sannikov (26). The key ingedien of ou seing ha dives he smooh dividend policy is he combinaion of moal hazad and leaning abou he fim s pofiabiliy. Tha is, i is impoan in ou model ha he fim s cash flows cay infomaion boh abou he agen s effo and he agen s skill.

6 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. Finally, in Secion 4, we show how his opimal conac can be implemened in ems of he fim s payou policy. 1. Model In his secion we pesen a coninuous-ime fomulaion of he fim and he pincipalagen poblem ha aises beween he manage unning he fim and ouside invesos. In ou model, isk-neual ouside invesos hie a isk-neual agen o un a fim. Boh he agen and invesos discoun he fuue a ae >. Invesos have unlimied wealh, wheeas he agen has no iniial wealh and mus consume non-negaively. 3 The fim geneaes cash flows a an expeced ae equal o a δ, whee a 1 is he agen s effo o aciviy level in he fim and δ is he expeced pofiabiliy of he fim ha depends on he agen s manageial alen. These cash flows aive wih volailiy σ >. Thus, he cumulaive cash flow pocess X fo he fim is defined by dx = a δ d + σdz (1) whee Z is a sandad Bownian moion. The fim s pofiabiliy evolves ove ime, and he ealizaion of he fim s cuen cash flow is infomaive abou hese changes in pofiabiliy. We model his by assuming δ evolves accoding o dδ = ν( dx a δ d) = νσdz. (2) 3 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.

7 We emak ha he specificaion in (1) and (2) is equivalen o he seady-sae in a model wih Bayesian updaing in which δ is he condiional expecaion of he fim s unobseved ue pofiabiliy, whee hee is an iniial common pio ha he fim s ue pofiabiliy is nomally disibued wih mean δ and is subjec o mean zeo, nomally disibued addiive shocks (which may o may no be coelaed wih he fim s cash flows). In his case he poseio esimae of he fim s pofiabiliy will evolve linealy wih he supises o he fim s cash flows, as in (2). The moal hazad poblem sems fom he fac ha he agen s aciviy wihin he fim is unobseved, and he agen can dive his own aenion (o ohe esouces) o ouside oppouniies. Given aciviy a wihin he fim, he agen can engage in aciviy wih effo 1 a ouside he fim. These ouside aciviies geneae a pivae benefi fo he agen a an expeced ae λ(1 ) δ, whee he paamee λ < 1 eflecs he fac ha a hese ouside aciviies ae less poducive han hose engaged in wihin he fim. The agen can also qui he fim a any ime. In ha even he agen devoes his full aenion o hese ouside aciviies, eaning pivae benefis a expeced ae λδ. Thus, he value of he agen s payoff in he even ha he quis he fim is epesened by λδ R( δ ) = R + (3) whee R capues any addiional pivae benefis (o losses) associaed wih no being employed by he fim. The fim equies exenal capial of K 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 fo he agen while he fim opeaes, as well as a eminaion ime when he fim is liquidaed. Fomally, a conac can be epesen by a pai (C, ), whee C is a non-deceasing X-measuable pocess ha epesen he agen s cumulaive compensaion (i.e., dc is he agen s compensaion a ime ) and is an X-

8 measuable sopping ime. In he even ha he fim is liquidaed, we assume ha invesos eceive a liquidaion value of L fo he fim s asses. The agen engages in his ouside opion and so eceives he payoff specified in (3). A conac (C, ) ogehe wih an X-measuable effo ecommendaion (a) is opimal if i maximizes he pincipal s pofi E e ((1 a) δd dc) + e L subjec o W = E e ( λ(1 a ) ) ( ) δ d + dc + e R δ given saegy (a) (4) and W E e ( λ(1 aˆ ) ) ( ) δ d + dc + e R δ fo any ohe saegy (a ). (5) By vaying W > R(δ ), 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 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 incenive-compaible wih espec o he conac (C, ) if i saisfies he consain (5). Unlike in he classic pincipal-agen seing, in ou conacual envionmen he agen may have pivae infomaion no only abou his effo, bu also fim pofiabiliy. Indeed, if he pincipal knows he agen s effo {a }, he can updae his belief ˆ δ abou fim pofiabiliy accoding o d ˆ δ = ν( dx a ˆ δ d), ˆ δ = δ. Howeve, if he agen deviaes o a diffeen effo saegy, he pincipal s belief ˆ δ becomes incoec.

9 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. 2. The Fis-Bes Soluion. Befoe we solve fo he opimal conac in Secion 3, le us deive he fis-bes soluion. Tha is, we ignoe he incenive consains (5) and imagine he case when he pincipal can conol he agen s effo, ignoing he incenive consains. Then i is opimal o le he agen ake effo a = 1 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 is E e dc = W E e R( δ ), and he goss pofi ha he fim poduces is E e δ d+ e L. Thus, wihou moal hazad he pincipal mus choose a sopping ime ha maximizes E e δ d+ e ( L+ R( δ )). This is a sandad eal-opion poblem ha can be solved by he mehods of Dixi and Pindyck (1994). I is opimal o igge liquidaion when he expeced pofiabiliy δ eaches a ciical level of δ. Poposiion 1 deives he pincipal s pofi and he opimal choice of δ fo his poblem. Poposiion 1. Le b( δ ) be he soluion o he odinay diffeenial equaion

10 1 2 2 b ( δ ) = δ + ν σ b ''( δ) 2 on [δ, ), deemined by bounday condiions b( δ ) = L+ R( δ ), b'( δ ) = R'( δ ), and b( δ) δ / as δ. On (-, δ], le b( δ ) = R( δ ). Then fo a conac ha delives value W o an agen of ype δ, b( δ) W is he pincipal s opimal pofi in a seing wihou moal hazad. I is also an uppe bound on he pincipal s pofi unde moal hazad. This uppe bound is aained if and only if he fim opeaes when δ > δ, and i is liquidaed when δ δ. b b( δ ) R(δ)+L δ/ δ δ Poof. Fo an abiay conac (C, ), conside he pocess s = δ + δs 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 νσb''( δ) + e δ, 2 which is equal o when δ > δ and e ( L + R( δ )) + e δ < when δ < δ. Theefoe, he pincipal s expeced pofi a ime is E e L+ e ( δ C) d = E e ( L+ R( δ )) + e δd W [ ] EG W G W= b( δ ) W.

11 The inequaliies above become equaliies if and only if δ δ and δ > δ befoe ime. QED 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. Inefficiency may occu when he agen has limied wealh. The lowe he agen s wealh, he geae is he inefficiency. Since i is cosless fo he isk-neual agen o pospone consumpion when he discouns fuue a he make ae, we expec ha in he opimal conac unde moal hazad he agen does no consume unil he efficien oucome can be aained. Theefoe, when he agen sas consuming, he pincipal s pofi and he liquidaion policy mus be given by Poposiion 1. We use Poposiion 1 in he nex secion, whee we deive he opimal conac unde moal hazad. 3. Deiving he Opimal Conac. The ask of finding he opimal dynamic conac in a seing like ous is complex, because hee is a 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 agen s skill, hee ae sandad ecusive mehods o deal wih such complexiy of conacs. These mehods ely on dynamic pogamming using he agen s fuue expeced payoff (a.k.a. coninuaion value) as a sae vaiable. 4 Wih unceainy abou he agen s skill, sandad mehods do no apply diecly o ou model. Solving such a poblem involves ceaiviy. While he specific soluion is unique o ou poblem, he geneal appoach o such models involves hee seps. Fis, one fomulaes necessay condiions fo incenive-compaibiliy, which include all he binding consains of he oiginal poblem. Second, one solves he elaxed opimizaion poblem 4 Fo example, see Spea and Sivasava (1987), Abeu, Peace and Sacchei (199) (in discee ime) and DeMazo and Sannikov (26) and Sannikov (27a) (in coninuous ime) fo he developmen of hese mehods, and Piskoski and Tchisyi (26) and Philippon and Sannikov (27) fo hei applicaions.

12 unde a limied se of consains idenified in he fis sep. Lasly, one veifies ha he soluion of he elaxed poblem is fully incenive-compaible. The necessay incenive-compaibiliy consains ae fomulaed using appopiaely chosen sae vaiables. 5 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 a ˆ δ d), ˆ δ = δ, and he agen s coninuaion value (when he agen follows he ecommended saegy {a s = 1} afe ime, and he pincipal has a coec belief abou he agen s skill) s ( ) ( ) W ( ) ˆ = E e dcs + e R δ. δ = δ The vaiables ˆ δ and W ae well-defined fo any conac (C, ), afe any hisoy of cash flows {X s, s [, ]}. 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. The emainde of ou deivaion poceeds as follows. Lemma 1 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 2 pesens a necessay condiion fo incenive compaibiliy of he agen s effo saegy {a =1, }. Subsecion 3.1 conjecues an opimal conac and veifies ha i is fully incenivecompaible. Subsecion 3.2 veifies he opimaliy of he conjecued conac, by showing ha i aains expeced pofi ha is a leas as high as any conac ha saisfies he necessay condiions of Lemma 2. The following epesenaion of W is sandad in coninuous-ime conacing: 5 Fo example, in coninuous ime Sannikov (27b) solves an agency poblem wih advese selecion using he coninuaion values of he wo ypes of agens as sae vaiables, Williams (27) solves an example wih hidden savings using he agen s coninuaion value and his maginal uiliy as sae vaiables.

13 Lemma 1. Thee exiss a pocess {β, } in L * such ha dw = W d dc + β ( dx ˆ δ d). (6) Poof. See Appendix. Lemma 2 povides a necessay condiion fo a conac (C, ) o be incenive compaible. Lemma 2. A conac (C, ) in which he agen s coninuaion value evolves accoding o (6) is no incenive-compaible unless β λ(1 + ν/) fo all. Poof. Suppose ha β < λ(1 + ν/) on a se of posiive measue. Le us show ha he agen has a saegy {a, } ha allows him o aain an expeced payoff geae han W. Le a pocess ˆ δ = when β λ(1 + ν/) and δ a ˆ δ 1 = when β < λ(1 + ν/). Define he δ ˆ λ s = + δ δ + s + λ s δs V e W ( ) e ( dc (1 a ) ds). If he agen follows he saegy descibed above, hen dδ d ˆ δ = ν( ˆ δ aδ )) d δ ˆ δ fo all, and dw = W d dc + β ( a δ d + σdz ˆ δ d). Theefoe, dv ( ˆ λ λ = W ) ( ( )( ˆ δ δ d Wd dc β ν aδ δ) d σdz) dc λ(1 a) δ d e λ = ( λ + ν β )( ˆ δ aδ ) d+ σdz.

14 The dif of V is eveywhee nonnegaive, and posiive on a se of posiive measue. Since ( ) ( ˆ λ R δ = W + δ δ ), he agen s payoff fom his saegy s E e R( δ) + e ( dcs + λ(1 as) δsds) = E[ V] > V = W. We conclude ha β λ(1 + ν/) fo all is a necessay condiion fo he incenive compaibiliy of he agen s saegy. QED 3.1. The Opimal Conac In his secion we conjecue he fom of he opimal conac ha delives value W R(δ ) o an agen of ype δ δ. The conac is based on wo sae vaiables: he agen s peceived skill level δ and his coninuaion value W. The evoluion of ˆ δ is no a conacual choice; i is deemined by d ˆ δ = ν( dx a ˆ δ d), ˆ δ = δ. To descibe how W evolves, define W 1 (δ) =R(δ) + (λ/ν + λ/) (δ - δ ). If W < W 1 (δ ), le W evolve accoding o dw = W d + λ(1 + ν / )( dx ˆ δ d), wih C =, unil W eaches ( ˆ 1 R δ ), esuling in eminaion, o W ( ˆ δ ). If W > W 1 (δ ), le he pincipal make an immediae paymen o he agen of C = W W 1 (δ ), o le W dop o W 1 (δ ). Once W eaches W ( δ ) a ime, le he pincipal make paymens o 1 ˆ he agen a ae dc s =W s ds fo all s, so ha W s evolves accoding o dw = λ(1 + ν / )( dx ˆ δ ds), s. s s s Then W = W ( δ ) fo all imes s, unil δ s eaches he level δ and W s eaches R(δ) = s 1 ˆ s R(δ s ) a ime. By Poposiion 1, his conac achieves fis-bes pofi W W 1 (δ ).

15 We veify ha his conac is incenive-compaible in his subsecion, and jusify is opimaliy in subsecion 3.2. We canno ely on Lemma 2 o check ha he agen s saegy {a = 1} maximizes his payoff, since Lemma 2 only gives a necessay condiion. The following poposiion veifies ha he conjecued conac is incenive-compaible fom he fis pinciples. Poposiion 2. Unde he conac oulined above, he agen of skill level δ ges he same value W independenly of his saegy. In paicula, by choosing a = 1 a all imes, he agen maximizes his payoff. Poof. Fo any saegy of he agen, he pocess ˆ λ s = + δ δ + s + λ s δs V e W ( ) e ( dc (1 a ) ds). defined in he poof of Lemma 2 has dif e λ ( λ + ν β )( ˆ δ aδ ) =. Theefoe, he agen s expeced payoff equals s E e R( δ) + e ( dcs + λ(1 as) δsds) = E[ V] = V = W independenly of he agen s saegy. QED 3.2. Jusificaion of he Opimal Conac. In his secion we veify ha he conac pesened in subsecion 3.1 is opimal. We will show ha his conac aains he highes expeced pofi among all conacs ha delive value W o an agen of skill level δ and saisfy he necessay incenive-compaibiliy condiion of Lemma 2. The se of such conacs includes all fully incenive-compaible conacs. Since he conac of subsecion 3.1 is incenive-compaible, as shown in Poposiion 2, i follows ha i is also opimal.

16 Le us pesen a oadmap of ou veificaion agumen. Fis, we define a funcion b(w,δ ), which gives he expeced pofi ha a conac of subsecion 3.1 aains fo any pai (W, δ ) wih W R(δ) and δ δ. Poposiion 3 veifies ha his is indeed ue fo ou definiion of b(w, δ ). Afe ha, Poposiion 4 shows ha he pincipal s pofi in any alenaive conac ha saisfies he necessay incenive-compaibiliy condiion of Lemma 2 is a mos b(w, δ ) fo any pai (W, δ ) wih W R(δ) and δ δ. I follows ha he conac of subsecion 3.1 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. 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 (7) wih bounday condiions given by (i) and (ii). Fo an abiay incenive-compaible conac (C, ) ha moivaes he agen o pu effo {a = 1}, in which he agen s coninuaion value evolves accoding o (6), define he pocess s = δs s + δ G e ( ds dc ) e b(, W ). Noe ha δ = ˆ δ fo any incenive-compaible conac. Lemma 3 helps us pove boh Poposiions 3 and 4. Lemma 3. When δ δ and C is coninuous a, hen ν dg = e ( νb ( W, δ ) + λ(1 + ) b ( W, δ )) σdz e ( b ( W, δ ) + 1) dc + δ W W 1 ν ν ν ( 2 ( (1 + )) WW(, ) + ( (1 + ))( (1 + ) WW(, ) + Wδ (, )) ) 2 2 e σ β λ b W δ β λ λ b W δ νb W δ d

17 Poof. See Appendix A. Poposiion 3. The conjecued opimal conac of subsecion 3.1 aains pofi b(δ,w ). Poof. Unde ha conac, he pocess G is a maingale. Indeed, fo all >, 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, he agen consumes posiively only in ode fo W o dop o W 1 (δ ), and b W (W,δ ) = -1 fo W W 1 (δ ), 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] = G = b( W, δ). QED. Poposiion 4. In any alenaive incenive-compaible conac (C,) he pincipal s pofi is bounded fom above by b(w,δ ). Poof. Le us ague ha G is a supemaingale fo any alenaive incenive-compaible conac (C, ) while δ δ. Appendix B shows ha (, ) ν bww W δ and λ(1 + ) bww ( W, δ) + νbw δ ( W, δ) (8) fo all pais (W,δ), so ha also b W (W,δ) -1 fo all (W,δ). I follows ha wheneve C has an upwad jump of C, G has a jump of e - (b(w + C, δ ) - b(w, δ ) - C ). Wheneve C is coninuous, G is a supemaingale by Lemma 3, while δ 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

18 E e b( W, δ) + e ( δd dc) = E G G = b( W, δ). QED We conclude ha subsecion 3.1 pesens he opimal incenive-compaible conac fo any pai (W, δ ) such ha W R(δ) and δ δ. If W W 1 (δ ), hen his conac aains he fis-bes pofi, and liquidaion always occus a he efficien level of pofiabiliy of δ = δ. If W < W 1 (δ ), hen liquidaion happens inefficienly wih posiive pobabiliy. In Secion 4, we sugges one capial sucue implemenaion of he opimal conac. Rahe han defining he agen s consumpion and he liquidaion ime in ems of absac vaiables δ and W, ou implemenaion gives he agen a facion of he fim s equiy, and allows him o conol he fim s payou policy. The fim may use available cash balances o boowing up o he liquidaion value of he asses L o offse opeaing losses and pay ou dividends. Defaul occus when he fim s deb eaches L. Unde he capial sucue we popose, by opimally using his disceion, he agen implemens he same oucome as he conac of subsecion A Capial Sucue Implemenaion of he Opimal Conac. In his secion we implemen he opimal conac using a emakably simple capial sucue. Le he agen hold a facion λ of he fim s equiy, and have full disceion ove he fim s dividend policy. Dividends can be paid ou of he fim s cash balances, which ae also used o absob gains and losses of cash flows. The saing cash level in he fim is deemined by M o W δ = λ, and he fim is liquidaed if he cash balances dop o a ciical level. Poposiion 5 shows ha if his ciical level is given by R /λ, hen he agen has incenives o follow a

19 dividend policy ha aains he same oucome as he opimal conac of he pevious secion. Noe ha he cash balance M inside he fim evolves accoding o dm = M d dd + dx unde his capial sucue implemenaion, whee ae he dividends paid ou by he agen, of which he agen eceives dc = λ dd. Poposiion 5. Unde he capial sucue pesened above, he agen s fuue expeced payoff a ime is given by W = λ(m + δ /). (9) The agen has incenives o choose a = 1 a all imes, o pay no dividends when W < W 1 (δ ), and o pay hem ou a ae dd = W /λ d when W =W 1 (δ ). By doing so, he agen implemens he opimal conac. The poof in Appendix C shows ha he agen is in fac indiffeen beween all effo saegies and dividend policies. If he agen chooses o educe he cash balance o he ciical level immediaely, he ges he value of W = λ(m R /λ) + R(δ ) = λ(m + δ /). If he agen chooses o un he fim wihou paying dividends, he fim poduces eanings a he expeced ae of M + δ. A facion λ of he fim s eanings compensaes he agen jus enough fo posponing he value of W, while he expeced eanings δ follow a maingale. This makes he agen jus indiffeen beween cashing ou and unning he fim. The agen may also dive aenion o pivae aciviies o ge a benefi of λ fo each dolla loss of he fim s cash flows. Since he agen can also conve one dolla of oupu ino λ unis of consumpion by paying dividends, he is indiffeen beween all effo levels.

20 If he agen efains fom paying dividends while W < W 1 (δ ) and pays dividends a he ae of expeced eanings M +δ when W = W 1 (δ ), he implemens he opimal conac, since hen W evolves in exacly he same way as in he opimal conac of subsecion 3.1, and liquidaion happens when W eaches he level λ(r /λ + δ /) = R(δ ) (he same as in Secion 3). I is emakable ha unde his implemenaion, he se of opions available o he agen depends neihe on he fim s peceived pofiabiliy ˆ δ no on he agen s coninuaion value W. The opimal ime of liquidaion is deemined by a simple vaiable, he fim s cash balance M, and he ciical heshold R /λ. 5. Appendix A. Poof of Lemma 1. Noe ha s = s + V e dc e W is a maingale when he agen follows he saegy {a s = 1}. By he Maingale Repesenaion Theoem hee exiss a pocess {β, } in L * such ha s ˆ = + βs s δs V V e ( dx ds), since dx ˆ δ ds = σdz when a s = 1. Diffeeniaing wih espec o, we find ha 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). QED. Poof of Lemma 3. Noe ha fo δ δ, he funcion b saisfies paial diffeenial equaion (7) even if W > W 1 (δ). Indeed, since bw (, δ) = b( δ) W and b W = -1 in ha egion, he equaion educes o

21 b W W b ( ( δ ) ) = δ + 2ν σ ''( δ). This equaion holds by he definiion of b. When C is coninuous a, hen using Io s lemma, ( δδ δ ) db( W, δ) = ( Wd dc) bw( W, δ) + σ β bww( W, δ) + ν b ( W, δ) + βν bw ( W, δ) d + ( νbδ ( W, δ) + βbw( W, δ)) σdz = b( W, δ) d δd bw( W, δ) dc + σ ( β λ(1 + )) b ( W, δ ) d+ ( β λ(1 + ))( λ(1 + ) b ( W, δ ) + νb ( W, δ )) d, ( Wδ ) 2 1 ν 2 ν ν 2 WW WW whee he second equaliy follows fom (7). Fom he definiion of G, i follows ha Lemma 3 coecly specifies how G evolves. QED 6. Appendix B. We mus show ha fo all pais (δ, W), he funcion b(δ, W) saisfies (, ) ν bww W δ and λ(1 + ) bww ( W, δ) + νbw δ ( W, δ). I is useful o undesand he dynamics of he pai (δ, W ) unde a conjecued opimal conac fis. The 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. (A1) 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.

22 δ R(δ) δ W 1 (δ) W 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 (A1), 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 = 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

23 Lemma 5. b W (R(δ), δ) weakly inceases in δ. Poof. Noe ha bw (, δ) = b( δ) W + E e ( L b( δ) + R( δ)) δ, W, whee b( δ ) L is he loss of value elaive o fis-bes due o pemaue liquidaion. Le us show ha fo all ε >, b(r(δ )+ε,δ ) - b(r(δ ),δ ) inceases in δ. Noe ha br ( ( δ ) + εδ, ) br ( ( δ), δ) = b( δ) R( δ) ε + E e ( L b( δ) + R( δ)) δ, R( δ) + ε L= E e ( b( δ) R( δ) b( δ) + R( δ)) + (1 e )( b( δ) R( δ) L) δ, R( δ) + ε ε Conside he pocesses (W i,δ i ) (i = 1,2) ha follow (A1) saing fom values (W i, δ i ) = (R(δ i ), δ i ) such ha δ 2 - δ 1 = >. Then fo any pah of Z, he diffeence δ 2 -δ 1 says equal o and W 2 W 1 becomes lage han λ/ fo all >. Theefoe, (i) a he ime 1 when W 1 eaches level R(δ 1 ), we sill have W 2 > R(δ 2 ), and Thus, (ii) a ime 2 > 1 when W 2 eaches R(δ 2 ), we have δ < δ since br ( ( δ ) + ε, δ ) br ( ( δ ), δ ) = ( ( δ) ( δ) ( δ ) + ( δ )) + (1 )( ( δ) ( δ) ) δ, ( δ) + ε ε E e b R b R e b R L R 2 2 E e b δ R δ b δ R δ e e b δ R δ L [ ( ( ) ( ) ( ) + ( )) ( )( ( ) ( ) ) δ 1 2 (1 e )( b( ) R( [ ( ( δ) ( δ) ( δ ) + ( δ )) ( )( ( 1 + δ 1 ) ( δ) ( δ ) + ( δ )) (1 e )( b( ) R( ) L), R( ) ] br ( ( δ ) + ε, δ ) br ( ( δ ), δ ), δ ) L) δ, R( δ ) + ε] ε E e b R b R e e b R b R δ δ δ δ + ε ε = b( δ ) R( δ ) b( δ ) + R( δ ) b( δ ) R( δ ) b( δ + ) + R( δ + ) = b( δ + ) R( δ + ) b( δ + ) + R( δ + ) b( δ ) R( δ ) b( δ ) + R( δ ), b( δ ) R( δ ) L b( δ ) R( δ ) b( δ ) + R( δ ) and

24 b( δ ) R( δ ) L b( δ ) R( δ ) L 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 δ, fo any wo values W 1 <W 2, b W (W 1, δ ) b W (W 2, δ ). Conside he pocesses (W i,δ ) (i = 1,2) ha follow (A1) saing fom values (W 1,δ ) and (W 2,δ ) fo δ 1 < δ 2. Then fo any pah of Z, we have W 2 W 1 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, b ( W, δ ) = E b ( R( δ ), δ ) E b ( R( δ ), δ ) = b ( W, δ ). 1 2 W W W W 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. Poof. Conside saing values (W i, δ i ) ha saisfy δ δ = > and W 2 1 ν - W = 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 = and W 2 W 1 ν λ(1 + ) / ν (wih equaliy afe ime 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 + (1 + ) / > ( ) + /, i follows ha 2 1 W W ν λ ν R δ λ δ δ and 2 1 W W. Using Lemmas 4 and 5, b ( W, δ ) = E b ( R( δ ), δ ) E b ( R( δ ), δ ) = b ( W, δ ) W W W W

25 QED 7. Appendix C. Poof of Poposiion 5. The vaiable W defined by (9) evolves accoding o dw = λ(m d dd + dx ) + λ/ ν (dx a δ d) = (W - λδ /) d dc + λ dx + λ ν/ (dx a δ d) = W d dc + λ (1 + ν/) (dx a δ d) - λ(1 a ) δ d. Le us show ha W is he agen s fuue expeced payoff fo any saegy fo which he ansvesaliy condiion Ee [ W ] as holds (i does fo he saegy pescibed in he poposiion). Then he pocess s = + λ s + s δs V e Wd e ( dd (1 a ) ds) is a maingale since ν dv = e W d + e dw + e λ( dd + (1 a ) δ d) = λ(1 + )( dx a δ d). Theefoe, he agen s expeced payoff is s E e λ( dds + (1 as) δsds) E[ V ] = V = W. wih equaliy if he ansvesaliy condiion holds. Thus, W eflecs he agen s fuue expeced payoff, and i is opimal fo he agen o follow he pescibed saegy. If he agen follows he pescibed saegy, hen W follows dw = W d dc + λ (1 + ν/) (dx δ d) wih dc = W d when W =W 1 (δ ), and dc = when W < W 1 (δ ), as in he opimal conac. Moeove, liquidaion occus when he fim s cash balance dops o he level M = R / λ, and he agen s value dops o R(δ ), as in he opimal conac. We conclude ha he agen s ecommended saegy implemens he opimal conac. QED

26 8. Refeences. Abeu, D., Peace, D., and Sacchei, E. (199) "Towad a Theoy of Discouned Repeaed Games wih Impefec Monioing." Economeica Vol. 58, pp Albuqueque, Rui and Hopenhayn, Hugo. A. (24) Opimal Lending Conacs and Fim Dynamics, The Review of Economic Sudies 71(2) Allen, F. and R. Michaely, 23, 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 (25) A Dynamic Theoy of Opimal Capial Sucue and Execuive Compensaion, NBER woking pape No 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, 25, Dividend changes do no signal changes in fuue pofiabiliy, Jounal of Business 78: Biais, B., T. Maioi, G. Planin, and J.-C. Roche (27) Opimal Design and Dynamic Picing of Secuiies, Review of Economic Sudies 74(2), Bav, A., Gaham, J., Havey, C. and R. Michaely, 25, Payou Policy in he 21s Cenuy, Jounal of Financial Economics Clemeni, Gian Luca, and Hugo A. Hopenhayn (26) A Theoy of Financing Consains and Fim Dynamics, Quaely Jounal of Economics 121(1). DeMazo, Pee, and Michael Fishman (23) Opimal long-em financial conacing wih pivaely obseved cash flows, Woking pape, Sanfod Univesiy. DeMazo, Pee, and Michael Fishman, (26) Agency and opimal invesmen dynamics, fohcoming in Review of Financial Sudies.. DeMazo, Pee and Yuliy Sannikov (26) Opimal Secuiy Design and Dynamic Capial Sucue in a Coninuous-Time Agency Model, Jounal of Finance 61: Dixi, A. K., and R.S. Pindyck (1994), Invesmen unde Unceainy. Pinceon Univesiy Pess.

27 Fama, Eugene and Havey Babiak, 1968, Dividend policy: an empiical analysis, Jounal of he Ameican Saisical Associaion 63: Gullon, Gusavo and Roni Michaely, 22, Dividends, shae epuchases and he subsiuion hypohesis, Jounal of Finance, 57, Gullon, Gusavo, Roni Michaely and Bhaskaan Swaminahan, 22, Ae dividend changes a sign of fim mauiy? The Jounal of Business, 75, Healy, Paul M. and Kishna G. Palepu, 1988, Eanings infomaion conveyed by dividend iniiaions and omissions, Jounal of Financial Economics 21: Linne, John., 1956, Disibuion of Incomes of Copoaions Among Dividends, Reained Eanings, and Taxes, Ameican Economic Review, 46(2), Mille, Meon H. and Fanco Modigliani (1961), Dividend Policy, Gowh, and he Valuaion of Shaes, The Jounal of Business 34(4), Philippon, T. and Y. Sannikov (27) Financial Developmen and he Timing of IPOs, woking pape, New Yok Univesiy. Piskoski, T., and A. Tchisyi (26): Opimal Mogage Design, woking pape, New Yok Univesiy. Paveen Kuma, Bong-Soo Lee: Discee dividend policy wih pemanen eanings Financial Managemen, Auumn, 21 Robes, Michael R. and Michaely, Roni, "Copoae Dividend Policies: Lessons fom Pivae Fims" (Febuay 22, 27). Available a SSRN: hp://ssn.com/absac=92782 Sannikov, Yuliy (27a) A Coninuous-Time Vesion of he Pincipal-Agen Poblem, woking pape, Univesiy of Califonia a Bekeley. Sannikov, Yuliy (27b) Agency Poblems, Sceening and Inceasing Cedi Lines, woking pape, Univesiy of Califonia a Bekeley. Spea, S. and Sivasava, S. (1987) On Repeaed Moal Hazad wih Discouning, Review of Economic Sudies, Vol. 54, Williams, Noah (27) On Dynamic Pincipal-Agen Poblems in Coninuous Time, woking pape, Pinceon Univesiy.