A Coninuous-Time Agency Model of Opimal Conracing and Capial Srucure Peer M. DeMarzo * Graduae School of Business Sanford Universiy Yuliy Sannikov * UC Berkeley This Revision: April 5, 5 ABSTRACT. We explore opimal financing in a seing when he agen can conceal and diver cash flows from a projec, and invesors only means of forcing repaymen is he hrea of erminaion. DeMarzo and Fishman (3) show ha ha an opimal conrac in his seing is a combinaion of credi line, deb and equiy. The credi line gives he agen flexibiliy o run he projec when i emporarily generaes losses, and an equiy sake gives he agen incenives no o diver cash. We look a opimal securiies in deail using he ransparency of coninuous-ime characerizaions. We explore how he opimal credi limi depends on a specific projec, and how marke values of securiies change for he duraion of he projec. We consider an exension in which he mean of cash flows depends on he agen s effor choice. The model provides a simple dynamic heory of securiy design and opimal capial srucure.. Inroducion In his paper, we consider a dynamic conracing environmen in which a risk-neural agen or enrepreneur wih limied resources manages an invesmen aciviy. While he invesmen is profiable, i is also risky, and in he shor-run can generae large losses. The agen will need ouside financial suppor o cover hese losses and coninue he projec. The difficuly is ha while he disribuion of he cash flows is publicly known, he agen may disor hese cash flows by aking a hidden acion ha leads o a privae We would like o hank Mike Fishman for many helpful commens. We are also graeful o Edgardo Barandiaran, Zhiguo He, Han Lee, Gusavo Manso, Nelli Oser, Ricardo Reis, Alexei Tchisyi, Jun Yan, Baozhong Yang as well as seminar paricipans a he Universia Auomaa de Barcelona, UC Berkeley, Universiy of Chicago, Norhwesern Universiy and Washingon Universiy. * Sanford, CA 9435. Phone: 65-736-8, e-mail: pdemarzo@sanford.edu. * 593 Evans Hall, Berkeley, CA 947-388. Phone: 65-33-749, e-mail: sannikov@econ.berkeley.edu.
benefi. Specifically, he agen may (i) conceal and diver cash flows for his own consumpion, and/or (ii) sop providing cosly effor, which reduces he mean of he cash flows. Therefore, from he perspecive of he principal or invesors funding he projec, here is he concern ha a low cash flow realizaion may be a resul of he agen s acions, raher han he projec fundamenals. To provide he agen wih appropriae incenives, invesors conrol he agen s wage, and may wihdraw heir financial suppor for he projec and force is early erminaion. We seek o characerize an opimal conrac in his framework and relae i o he firm s choice of capial srucure. DeMarzo and Fishman (3), hereafer denoed DF, consider a discree-ime model of his sor. Using a dynamic-programming approach DF show ha an opimal conrac is a combinaion of a credi line, deb and equiy. Dividends are paid when cash flows exceed deb paymens and he credi line is paid off. If deb service paymens are no made or he credi line is overdrawn, he projec is erminaed wih a probabiliy ha depends on he size of he cash shorfall. The defining feaure of his conrac is a credi limi, which can be found by compuing a fixed poin of a Bellman equaion in a discree-ime model. The coninuous-ime model of his paper has an alernaive convenien way o compue an opimal conrac using an ordinary differenial equaion. Using a coninuous-ime characerizaion, we explore he opimal conrac in more deail. We are able o show how he opimal credi limi depends on he disribuion of he projec s cash flows and he consequences of liquidaion. We describe he dynamics of securiy prices. In addiion, we also derive resuls abou how opimal projec selecion depends on he credi line balance. In all cases our differenial equaion characerizaion proves very useful for he analysis. In our coninuous-ime seing he cumulaive cash flows generaed by he projec follow a Brownian moion wih a posiive drif. We derive he opimal conrac using coninuous-ime echniques inroduced by Sannikov (3). Two differences emerge beween he opimal conracs in discree and coninuous ime. Firs, erminaion is no longer sochasic in coninuous ime, bu occurs he momen he credi line is overdrawn or here is a defaul on he long-erm deb. Second, because he projec can generae large shor-erm losses, projecs ha are very risky will no use long-erm deb bu insead require a compensaing balance wih he credi line. (A compensaing balance is a cash deposi ha he firm mus hold wih he lender o mainain he credi line.) The compensaing balance serves wo roles: i allows for a larger credi line, which is valuable given he risk of he projec; and i provides an inflow of ineres paymens o he projec ha can be used o somewha offse operaing losses. The model herefore provides an explanaion for why firms migh hold subsanial cash balances a low ineres raes while simulaneously borrowing a higher raes. For he bulk of our analysis, we focus on he case in which he agen can conceal and diver cash flows. We show in Secion 4 ha he characerizaion of he opimal conrac is unchanged if he agen makes a hidden binary effor choice. We also consider he possibiliy of conrac renegoiaion in Secion 5, and characerize he opimal renegoiaion-proof conrac... Relaed Lieraure. Our paper is par of a growing lieraure on dynamic opimal conracing models using recursive echniques ha began wih Green (987), Spear and Srivasava (987), Phelan
and Townsend (99), and Akeson (99) among ohers. (See, for example, he ex by Ljungqvis and Sargen () for a descripion of many of hese models.) As previously menioned, his paper builds direcly on he model of DeMarzo and Fishman (3). Oher recen work developing opimal dynamic agency models of he firm includes Albuquerque and Hopenhayn (), Clemeni and Hopenhayn (), DeMarzo and Fishman (3b), and Quadrini (). Wih he excepion of DeMarzo and Fishman (3), hese papers do no share our focus on an opimal capial srucure. In addiion, none of hese models are formulaed in coninuous ime. While discree ime models are adequae concepually, in many cases a coninuous-ime seing may prove o be much simpler and more convenien analyically. An imporan example of his is he principal-agen model of Holmsrom and Milgrom (987), hereafer HM, in which he opimal coninuous-ime conrac is shown o be linear. Schaler and Sung (993) develop a more general mahemaical framework for analyzing agency problems of his sor in coninuous ime, and Sung (995) allows he agen o conrol volailiy as well. Hellwig and Schmid () look a he condiions for a discree-ime principal-agen model o converge o he HM soluion. See also Bolon and Harris (), Ou-yang (3), Deemple, Govindaraj and Loewensein (), Cadenillas, Cviannic and Zapaero (3) for furher generalizaion and analysis of he HM seing. Several feaures disinguish our model from he HM problem: he invesor's abiliy o erminae he projec, he agen's consumpion while he projec is running, and he naure of he agency problem. In HM, he agen runs he projec unil dae T, and hen receives compensaion. In our model, he agen receives compensaion many imes while he projec is running, unil he conrac calls for he agen s erminaion. Also, HM analyze a seing in which he agen akes hidden acions. In our main seing he agen observes privae payoff-relevan informaion; we also consider he possibiliy of a binary hidden acion choice. Unlike HM, he erminaion decision is a key feaure of he opimal conrac in our seing. Here, as in DF, we demonsrae how his decision can be implemened hrough bankrupcy. Sannikov (3) and Williams (4) analyze principal-agen models, in which he principal and he agen inerac dynamically. Their ineracion is characerized by evolving sae variables. In heir models, he agen coninuously chooses acions (e.g. hidden effor) ha are no direcly observable o he principal, and he principal akes acions (e.g. paymens o he agen) ha affec he agen's payoff. Besides having a dynamic naure in he spiri of Sannikov (3) and Williams (4), our paper develops a new mehod o deal wih he problem of privae observaions in coninuous ime. Also, unlike in Sannikov (3) and Williams (4), hidden savings do no pose any addiional difficulies in our model. We derive an opimal conrac in a seing wihou hidden savings, and verify ha i remains incenive compaible even when he agen can save secrely. In conemporaneous work, Biais e al. (4) consider a dynamic principal-agen problem in which he agen s effor choice is binary (work or shirk). While hey do no formulae he problem in coninuous ime, hey do exam he coninuous limi of he Spear and Wang () also analyze he decision of when o fire an agen in a discree-ime model. They do no consider he implemenaion of he decision hrough sandard securiies.
discree-ime model and focus on he implicaions for he firm s balance shee. As we show in Secion 4, heir seing is a special case of our model and our characerizaion of he opimal conrac applies. This paper is organized as follows. Secion presens a coninuous-ime model. Afer ha, i derives an opimal conrac and is implemenaion wih sandard securiies: credi line, deb and equiy. In Secion 3 we analyze he properies of an opimal conrac, providing characerizaions ha canno be obained in a discree-ime seing. Secions 4 shows he opimaliy of our conrac wih hidden binary effor. Secion 5 analyzes renegoiaion-proof conracs and he issue of robusness. Secion 6 concludes he paper.. The Seing and he Opimal Conrac In his secion we describe a coninuous-ime formulaion of he conracing problem ha arises when he agen privaely observes he cash flows of he projec ha he manages on invesors behalf. We hen solve he model and derive an opimal conrac. We use a dynamic programming approach. The ulimae form of an opimal conrac is analogous o ha in discree ime, bu he echniques o derive i are somewha differen. The derivaion employs a HJB equaion, which is analogous o he Bellman equaion in discree ime, subjec o incenive compaibiliy and promise keeping condiions. The main conribuion of his secion is mehodological: o formulae he model in coninuous ime and derive an opimal conrac. We hen show how o implemen he opimal conrac hrough a choice of capial srucure, where we allow he agen o conrol he firm s payou policy. While he form of an opimal conrac and is implemenaion urns ou o be similar o ha in he discree-ime model of DeMarzo and Fishman (3), we will see ha an opimal conrac has a sharper characerizaion in coninuous ime, which can be exploied o derive comparaive saics resuls and analyze exensions in he following secions... The Dynamic Agency Model The agen manages a projec ha generaes sochasic cash flows wih mean µ and variance σ dy = µ d + σ dz, where Z is a sandard Brownian moion. The agen observes he acual cash flows Y, bu he principal does no. The agen makes a repor { Yˆ ; } of he realized cash flows o he principal. The principal does no know wheher he agen is lying or elling he ruh. The principal receives he repored cash flows dŷ from he agen and gives him back ransfers of di ha are based on he agen s repors. Formally, he agen s income process I is non-decreasing and Y ˆ-measurable. If he agen underrepors realized cash flows, he seals he difference. Sealing may be cosly: he agen is able o enjoy only a fracion λ 3
(,] of wha he seals. Also, he agen can over-repor and pu his own money back ino he projec. As a resul, he agen receives a flow of income of [ ˆ λ dy dy] + di, where ˆ λ [ dy ] ( ˆ) ( ˆ dy λ dy dy dy dy) sealing + over-reporing To make sure ha he agen does no receive income of minus infiniy, we assume ha process Y ˆ Y has o have bounded variaion. The agen is risk-neural and discouns his consumpion a rae γ. This coninues unil a erminaion ime ha is conracually specified by he principal. 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 ρ < γ: [ ˆ λ ds =ρ S d + dy dy ] + di dc, () where dc is he agen s consumpion a ime, which mus be nonnegaive. The agen mus mainain a nonnegaive balance on his accoun, i.e. S. Once he conrac is erminaed, he agen receives payoff R from an ouside opion. Therefore, he agen s oal expeced payoff from he conrac a dae is given by 3 γs γ W = E e dc s + e R. (3) The principal discouns cash flows a rae r, such ha γ > r ρ. 4 Once he conrac is erminaed, she receives expeced liquidaion payoff L. The principal s oal expeced profi a dae is rs ˆ r b = E e ( dy ) s dis + e L. The projec requires exernal capial of K o be sared. The principal offers o conribue his capial in exchange for a conrac (, I) which specifies a erminaion ime and paymens {I ; } ha are based on repors Y ˆ. Formally, I is a Y ˆ-measurable coninuous process, and is a Y ˆ-measurable sopping ime. () Noe ha () implies ha he agen pays a proporional cos ( λ) o diver funds, bu does no recover he cos if he funds are pu back ino he firm. We could also allow he agen o conceal and save funds wihin he firm, avoiding he cos ( λ) if he funds are ulimaely used o boos fuure repored cash flows (i.e., he cos is only paid if he funds are divered o he agen s personal consumpion). This change would no aler he resuls in any way. 3 We can ignore consumpion beyond dae because γ r implies i is opimal for he agen o consume all savings a erminaion (i.e., S = ). 4 Typically for a borrowing-consrained agen he ineremporal marginal rae of subsiuion is greaer han he marke ineres rae r. To represen he idea ha he agen is borrowing-consrained in a risk-neural seing, we assume ha γ > r. (The case γ = r requires eiher a finie horizon seing or inroducing a bound on he magniude of he projec s per period operaing losses; oherwise i is opimal o pospone he agen s consumpion indefiniely.) 4
In response o a conrac (, I), he agen chooses a sraegy. A feasible sraegy is a pair of processes (C, Y ˆ ) adaped o Y, such ha (i) process Y Y ˆ has bounded variaion, (ii) process C is nondecreasing, and (iii) he savings process, defined by (), says nonnegaive. The agen chooses a feasible sraegy o maximize his expeced payoff. Therefore, he agen s sraegy (C, Y ˆ ) is incenive compaible if i maximizes his oal expeced payoff W given a conrac (, I). An incenive compaible conrac refers o a quadruple (, I, C, Ŷ) ha includes he agen s recommended sraegies. We have no explicily modeled he agen s opion o qui and receive he ouside opion R a any ime. We could incorporae his by including an individual raionaliy consrain requiring ha he agen s fuure payoff from coninuing a dae, W, is no worse han his ouside opion R for all. However, in our seing his is no necessary as he individual raionaliy consrain will never bind. The agen can always under-repor and seal a rae γ R unil erminaion o obain a payoff of R or greaer. Thus any incenive compaible sraegy yields he agen a leas R. The opimal conracing problem is o find an incenive-compaible conrac (, I, C, Ŷ) ha maximizes he principal s profi subjec o delivering o 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 invesors. Remark. For simpliciy, we have specified he conrac assuming he agen's income I and he erminaion ime are deermined uniquely by he agen's repor. While his assumpion rules ou public randomizaion, because he principal's value funcion urns ou o be concave (Proposiion ), public randomizaion would no improve he conrac and his assumpion is wihou loss of generaliy. In secion 5, however, we inroduce public randomizaion when considering renegoiaion-proof conracs... Derivaion of he Opimal Conrac We solve he problem of finding an opimal conrac in several 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. Thus we consider a relaxed problem by ignoring he possibiliy ha he agen can save secrely. Our derivaion of an opimal conrac for he relaxed problem follows he framework of he discree-ime opimal conracing lieraure. Along he way we explain he echniques from sochasic calculus ha we need in coninuous ime. Finally, 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 chooses o ell he ruh, and mainains zero savings. 5
PROOF: See Appendix. The inuiion for his resul is sraighforward i is inefficien for he agen o conceal and diver cash flows (λ ) or o save hem (ρ r). We can improve he conrac by having he invesors save and make direc paymens o he agen. Thus, we can look for an opimal conrac in which ruh elling and zero savings is incenive compaible. The Opimal Conrac wihou Saving Noe ha if he agen could no save, hen he would no be able o over-repor cash flows and would consume all income as i is received. Thus, dc = di +λ( dy dyˆ ). (4) We can relax he problem by resricing he agen s savings so ha (4) holds. Afer we find an opimal conrac for he relaxed problem, we show ha i remains incenivecompaible even if he agen can save secrely. One difficuly wih working in a dynamic seing is he complexiy of he conrac space. The conrac can depend on he enire pah of repored cash flows Y ˆ, making i difficul o evaluae he agen s incenives in a racable way. Our firs ask is o find a convenien represenaion for he agen s incenives. To do so, define he agen s promised value W (Ŷ) 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 afer ime : ˆ γ( s ) γ( ) W( Y) = E e dis + e R The following resul provides a useful represenaion for W (Ŷ). LEMMA B. A any momen of ime here is a sensiiviy β (Ŷ) of he agen s coninuaion value owards his repor such ha dw ( ˆ)( ˆ = γwd di +β Y dy µ d) (5) This sensiiviy β (Ŷ) is deermined by he agen s pas repors Ŷ s, s. PROOF: Noe ha W (Ŷ) 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 (5) for he case when he agen ruhfully repors Ŷ = Y. In ha case, γ s γ = s + V e di ( Y) e W ( Y) (6) is a maringale and by he maringale represenaion heorem here is a process β such ha dv = e γ β (Y) (dy µ d), where dy µ d is a muliple of he sandard Brownian moion. Differeniaing (6) wih respec o we find and hus (5) holds. γ γ γ γ dv = e β ( Y )( Y µ d) = e di ( Y ) γ e W ( Y ) d + e dw ( Y ) 6
Informally, he agen has incenives no o seal cash flows if he ges a leas λ of promised value for each repored dollar, i.e. if β λ. 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 posiive measure, he agen can obain a leas a lile bi more han his promised value if he underrepors cash when β < λ. We summarize our conclusions in he following proposiion. LEMMA C. If he agen canno save, ruh-elling is incenive compaible if and only if β λ for all. PROOF: If he agen seals dy dyˆ a ime, he gains immediae income of λ( dy ˆ dy) bu loses β ( ˆ dy dy) in coninuaion payoff. Therefore, he payoff from reporing sraegy Ŷ gives he agen he payoff of ˆ ˆ γ γ W + E e λ( dy dy) + e β( dy dy), (7) where W denoes he agen s payoff under ruh-elling. We see ha if β λ for all hen (7) is maximized when he agen chooses dyˆ = dy, since he agen canno over-repor cash flows. If β < λ on a se of posiive measure, hen he agen is beer off underreporing on his se han always elling he ruh. 5 Now we use he dynamic programming approach o deermine he mos profiable way for he principal o deliver o he agen any value W. We presen an informal argumen, which is formalized in he proof of Proposiion. Denoe by b(w) he principal s value funcion (he highes profi o he principal ha can be obained from a conrac ha provides he agen wih payoff W). To faciliae our derivaion of b, we assume b is concave. In fac, we could always ensure ha b is concave by allowing public randomizaion, bu a he end of our inuiive argumen we will see ha public randomizaion is no needed in an opimal conrac. 6 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, bw ( ) bw ( di) di. (8) Equaion (8) implies ha b (W) for all W; ha is, he marginal cos of compensaing he agen can never exceed he cos of an immediae ransfer. Define W as he lowes value such ha b (W ) =. Then i is opimal o pay he agen according o = max(,) (9) di W W 5 The agen s repor affecs β. How do we know ha lying does no aler β o be always greaer han or equal o λ, whereas we had β <λ on a se of posiive measure under ruhelling? One way he agen can lie is by reporing dŷ = dy d when β <λ and elling he ruh when β λ. Then he probabiliy measure over he agen s repors has he same posiive probabiliy evens as he measure over he rue cash flows, so β(ŷ) λ on se of posiive measure even afer a deviaion. 6 Given he lineariy of he incenive compaibiliy condiion, public randomizaion would only be useful for allowing sochasic erminaion of he conrac. 7
These ransfers, and he opion o erminae, keep he agen s promised value beween R and W. Wihin his range, equaion (5) implies ha he agen s promised value evolves according o dw =γ Wd +βσ dz when he agen is elling he ruh. We need o deermine he sensiiviy β of he agen s value o repored cash flows. Using Io s lemma, he principal s expeced cash flows and changes in conrac value are given by ( ) EdY [ + dbw ( )] = µ+γ Wb'( W) + β σ b''( W) d Because a he opimum he principal should earn an insananeous oal reurn equal o he discoun rae, r, we have he following Bellman equaion for he value funcion: rb( W ) = max µ+γ Wb '( W ) + β σ b''( W ) () β λ Given he concaviy of b, b (W) and so β = λ is opimal. 7 Inuiively, because he inefficiency in his model resuls from early erminaion, reducing he risk o he agen lowers he probabiliy ha he agen s promised value falls o R. The principal s value funcion herefore saisfies he following second-order ordinary differenial equaion: rb( W ) =µ+γ Wb '( W ) + λ σ b''( W ), R W W, () wih b(w) = b(w ) (W W ) for W > W. We need hree boundary condiions o pin down a soluion o his equaion and he boundary W. The firs boundary condiion arises because he principal mus erminae he conrac o hold he agen s value o R, so b(r) = L. The second boundary condiion is he usual smooh pasing condiion he firs derivaives mus agree a he boundary, and so b (W ) =. 8 The final boundary condiion is he super conac condiion for he opimaliy of W, which requires ha he second derivaives mach a he boundary. This condiion implies ha b (W ) =, or equivalenly, using equaion (), rb( W ) +γ W =µ. () This boundary condiion has a naural inerpreaion. I is beneficial o pospone paymen o he agen by making W larger because i reduces he risk of early erminaion. Posponing paymen is sensible unil he boundary (), when he principal and agen s required expeced reurns exhaus he available expeced cash flows. 9 An example of he value funcion is shown in Figure. 7 In fac, we show in he proof ha b(w) is sricly concave for W W (see also foonoe 9), so ha β = λ is he unique opimum. 8 Roughly speaking, if here were a kink a W, b (W ) = and () could no be saisfied. 9 A similar argumen can be used o show ha public randomizaion is no useful. If i were, hen b would be linear (b = ) in he region in which i is used. A a boundary w of his region, he super conac condiion would require b (w) = and so rb(w) = µ + γwb (w). Bu b (w) implies rb(w) + γw µ, and hus w > W. Thus, b is sricly concave on [R, W ] and here is no role for public randomizaion. 8
µ r b, invesor s payoff Firs bes (b = µ /r W) rb = µ γw rb=µ- W b +½λ σ b L Slope b = R W W, agen s payoff Figure : The Principal s Value Funcion b(w) The following proposiion formalizes our findings: PROPOSITION. The conrac ha maximizes he principal s profi and delivers o he agen value W [R, W ] akes he following form: W evolves as dw =γwd di +λ( dyˆ µ d). (3) When W [R, W ), di =. When W = W, paymens di cause W o reflec a W. If W > W, an immediae paymen W W is made. The conrac is erminaed a ime when W his R. The principal s expeced payoff a any poin is given by a concave funcion b(w ), which saisfies rb( W ) =µ+γ Wb '( W ) + λ σ b''( W ) (4) on he inerval [R, W ] and b '( W ) = for W W, wih boundary condiions b(r) = L and rb(w ) = µ γw. PROOF: See Appendix. Hidden Savings Thus far, we have resriced he agen from saving. We now show ha he conrac of Proposiion remains incenive compaible even when we relax his resricion. The inuiion for he resul is ha because he marginal benefi o he agen of reporing or consuming cash is consan over ime, and since privae savings grow a rae ρ < γ, here is no incenive o delay reporing or consumpion. In fac, in he proof we show ha his resul holds even if he agen can save wihin he firm wihou paying he diversion cos. PROPOSITION. Suppose he process W is bounded above and solves dw =γwd di d +λ( dyˆ µ d). (5) 9
unil sopping ime = min{ W = R}. Then he agen earns payoff of a mos W from any feasible sraegy in response o a conrac (, I). Furhermore, payoff W is aained if he agen repors ruhfully and mainains zero savings. PROOF: See Appendix. This resul confirms ha conracs from a broad class, including he opimal conrac of Proposiion, remain incenive-compaible even if he agen has access o hidden savings. Proposiion will help us characerize incenive-compaible capial srucures in he nex subsecion..3. Capial Srucure Implemenaion The opimal conrac in our seing depends upon he hisory of repored cash flows. This hisory dependence is capured hrough he promised payoff W o he agen. In his secion, we show ha he opimal conrac can be implemened using sandard securiies: equiy, long-erm deb, and a credi line. We begin by describing hese securiies. Equiy. Equiy holders receive dividend paymens made by he firm. Dividends are paid from he firm s available cash or credi, and are a he discreion of he agen. Long-erm Deb. Long-erm deb is a consol bond ha pays coninuous coupons a rae x. Wihou loss of generaliy, we le he coupon rae be r, so ha he face value of he deb is D = x/r. If he firm defauls on a coupon paymen, deb holders force erminaion of he projec. Credi Line. A revolving credi line provides he firm wih available credi up o a limi C L. Balances on he credi line are charged a fixed ineres rae r c. The firm borrows and repays funds on he credi line a he discreion of he agen. If he balance on he credi line exceeds C L, he firm defauls and he projec is erminaed. We now show ha he opimal conrac can be implemened using a capial srucure based on hese hree securiies. While he implemenaion is no unique (e.g., one could always use he single conrac derived in Secion., or srip he long-erm deb ino zero-coupon bonds), i provides a naural inerpreaion. I also demonsraes how he conrac can be decenralized ino limied liabiliy securiies (equiy and deb) ha can be widely held by invesors. Finally, i shows ha he opimal conrac is consisen wih a capial srucure in which, in addiion o he abiliy o seal he cash flows, he agen has wide discreion regarding he firm s leverage and payou policy he agen can choose when o draw on or repay he credi line, he amoun of dividends, and wheher o accumulae cash balances (earning ineres r) wihin he firm. Before saing our main resul, we noe ha while i will be imporan for he pricing of he securiies, for purposes of implemenaion i is no necessary o specify he prioriizaion of he securiies over he liquidaion payoff L in he even of erminaion. We will, however, compensae he agen wih equiy, and i is imporan ha he agen does no receive par of he liquidaion payoff. Thus, we define inside equiy as idenical o equiy, bu wih he provision ha i is worhless in he even of erminaion. (Wih Inside equiy could correspond o a sock gran o he agen combined wih a zero ineres loan due upon erminaion ha equals or exceeds he liquidaion value of he equiy.
absolue prioriy, his disincion will ofen be unnecessary, as deb holders claims will ypically exhaus L.) PROPOSITION 3. Consider a capial srucure in which he agen holds inside equiy for fracion λ of he firm, he credi line has ineres rae r c = γ, and deb saisfies L rd = µ γr/ λ γ C. (6) Then i is incenive compaible for he agen o refrain from sealing, and o use he projec cash flows o pay he deb coupons and credi line before issuing dividends. Once he credi line is fully repaid, all excess cash flows are issued as dividends. Wih his capial srucure, he agen s expeced fuure payoff W is deermined by he curren draw M on he credi line: L ( ) W = R+λ C M. (7) This capial srucure implemens he opimal conrac if, in addiion, he credi limi saisfies C L = λ (W R). (8) PROOF: See Appendix. The inuiion for he incenive compaibiliy of his capial srucure is as follows. Firs, providing he agen wih he fracion λ of he equiy eliminaes his incenive o seal cash flows because he can do as well by paying dividends. Bu how can we ensure ha he agen does no pay dividends premaurely by, for example, drawing down he credi line immediaely and paying a large dividend? Given balance M on he credi line, he agen can pay a dividend of C L M and hen defaul. Bu if (7) holds, he payoff from deviaing in his way is equal o he payoff W ha he agen receives from paying off he credi line before paying dividends, and so here is no incenive o deviae. Finally, because he agen earns ineres a his discoun rae γ paying off he credi line, bu earns ineres a rae r < γ on accumulaed cash, he agen has he incenive o pay dividends once he credi line is repaid. The role of he long-erm deb, defined by (6), is o adjus he profi rae of he firm so ha he agen s payoff does indeed saisfy equaion (7). If he deb were oo high, he agen s payoff would be below he amoun in (7), and he agen would draw down he credi line immediaely. If he deb is oo low and he firm s profi rae oo high, he agen would build up cash reserves afer he credi line was paid off in order o reduce he risk of erminaion. Thus, as long as (6) holds, we say he capial srucure is incenive compaible he agen will no seal and will pay dividends if and only if he credi line is fully repaid. Under wha condiions does his capial srucure implemen he opimal conrac of secion.? Noe ha he hisory dependence of he opimal conrac is implemened One can rewrie (6) as λ (µ rd γc L ) = γr, which saes ha he agen s share of he firm s profi rae (afer ineres paymens) maches he agen s ouside opion when he credi line is exhaused.
hrough he credi line, wih he balance on he credi line acing as he memory device o rack he agen s payoff W. In he opimal conrac, he agen is paid in order o keep he promised payoff from exceeding W. Here, dividends are paid when he balance on he credi line M =. For he capial srucure o implemen he opimal conrac, hese condiions mus coincide. Solving equaion (7) for C L leads o he opimaliy condiion C L = λ (W R). There is no guaranee ha in his capial srucure he deb required by equaion (6) is posiive. If D <, we inerpre he deb as a compensaing balance. A compensaing balance is a cash deposi required by he bank issuing he credi line. The firm earns ineres on his balance a rae r, and he ineres supplemens he firm s cash flows. The firm canno wihdraw his cash, and i is seized by crediors in he even of defaul. We will examine he seings in which a compensaing balance arises in he nex secion. The implemenaion here is very similar o he implemenaion shown in he discree-ime model of DeMarzo and Fishman (3). There are hree imporan disincions. Firs, because cash flows arrive in discree porions, he erminaion decision is sochasic in he discree-ime seing (i.e. he principal randomizes when he agen defauls). Second, because cash flows may be arbirarily negaive in a coninuous-ime seing, he conrac may involve a compensaing balance requiremen as opposed o deb. Lasly, he discree-ime framework does no allow for a characerizaion of he incenive compaibiliy condiion for he capial srucure in erms of he primiives of he model, as we do here. 3. Opimal Capial Srucure and Securiy Prices The capial srucure implemenaion of he opimal conrac inspires many ineresing quesions. Wha facors deermine he amoun ha he agen borrows? When will he agen borrow for iniial consumpion? When is here a compensaing balance? Wha is he opimal lengh of he credi line? How do marke values of securiies involved in he conrac depend on he firm s remaining credi? In his secion, we exploi he coninuous-ime machinery o answer hese quesions and provide new insighs. 3.. The Deb Choice A key feaure of he opimal capial srucure is is use of boh fixed long-erm deb and a revolving credi line. In his secion we develop furher inuiion for how he amoun of long-erm deb, he size of he credi line, and he iniial draw on he credi line are deermined. To simplify he analysis, we focus on he case λ = in which here is no cos o divering cash flows. In his case, he agen holds he equiy of he firm, and finances he firm solely hrough deb. While his case migh appear resricive, he following resul shows ha he opimal deb srucure wih lower levels of λ can be deermined by considering an appropriae change o he erminaion payoffs. An alernaive implemenaion is given in Shim (4) and Biais e al. (4) for a specialized seing. Raher han a credi line, hey suppose he firm reains a cash reserve and ha he coupon paymen on he deb varies conracually wih he level of he cash reserves.
PROPOSITION 4. The opimal deb and credi line wih agency parameer and erminaion payoffs (λ, R, L) are he same as wih parameers (, R λ, L λ ) where R λ λ = λ R and L = λ L+ ( λ) r. µ PROOF: See Appendix. When λ =, he opimal credi limi is C L = W R. The opimal level of deb is hen deermined by (6), which in his case can be wrien rd = µ γr γc L = µ γw Recall also ha in he opimal conrac, W is deermined by he boundary condiion (): rb(w ) + γw = µ Combining hese wo resuls implies ha he opimal face value of deb is D = b(w ). Figure shows an example, illusraing he size of he credi line and he deb face value when he cash flow volailiy is low. From he figure, D > L, so he deb is risky. Invesors Payoff b 9 8 7 6 r b + γw = µ (W, D) 5 4 3 (R, L) Deb b (W) = (W, K) Credi Line 3 4 5 6 7 8 9 W W Agen s Payoff W Figure : The Opimal Conrac wih Low Volailiy (L = 5, R =, µ =, σ = 5, r = %, γ = 5%, λ =, K = 3) Noe ha he opimal capial srucure for he firm does no depend on he exernal capial K ha is required. However, he iniial payoffs of he agen and he invesors depend upon K as well as he paries relaive bargaining power. For example, if invesors are compeiive, he agen s iniial payoff is he maximal payoff W such ha b(w ) = K as 3
illusraed in Figure. In his example, W > W. This payoff is achieved by giving he agen an iniial cash paymen of W W, and saring he firm wih zero balance on he credi line (providing he agen wih coninuaion payoff W ). In oher words, he firm iniiaes he credi line and issues he long-erm deb. The capial raised is used o fund he projec and pay an iniial dividend of W W. The credi line is hen used as needed o cover operaing losses. Thus, he iniial capial ha is raised from invesors is b(w ), which is equal o he face value of he deb D. However, he deb is risky (D > L) and so, given coupon rae r, rades a a discoun o is face value. How does he firm raise he addiional capial o make up for his discoun? Given he high ineres rae γ on he credi line, he lender earns an expeced profi from he credi line, and so will pay his o he firm upfron. This paymen exacly offses he iniial discoun on he long-erm deb due o credi risk. Recall ha he opimal credi line resuls from he following rade-off: a large credi line delays he agen s consumpion, bu also gives more flexibiliy o delay erminaion. Paymens on deb are chosen o give he agen incenives o repor ruhfully: if paymens on deb were oo burdensome, he agen would draw down he credi line immediaely and qui he firm; if hey were oo small, he agen would delay erminaion by saving excess cash flows when he credi line is paid off. In Figure 3, we illusrae how hese inuiive consideraions affec he opimal conrac for differen levels of volailiy. Wih an increase in volailiy, he principal s profi funcion drops. Riskier cash flows require more financial flexibiliy, so he credi line becomes longer. Given he higher ineres burden of he longer credi line, he opimal level of deb shrinks. Wih medium volailiy (as shown in he lef panel of Figure 3), he face value of he deb is below he liquidaion value of he firm (D < L). Thus, if he long-erm deb has prioriy in defaul, i is now riskless. The firm will herefore raise D hrough he long-erm deb issue. However, in his case D < K. The addiional capial needed o iniiae he projec is raised hrough an iniial draw on he credi line of W W. Because b > on (W, W ), he draw on he credi line exceeds K D. The difference can be inerpreed as an iniial fee charged by he lender o open he credi line wih his iniial balance; his fee compensaes he lender for he negaive NPV of he credi line due o he firm s greaer credi risk. 4
Invesors Payoff b 9 8 7 r b + γw = µ Invesors Payoff b 9 8 7 r b + γw = µ 6 6 5 5 4 4 3 (R, L) (W, K) (W, D) 3 (R, L) (W, K) Deb Credi Line 3 4 5 6 7 8 9 W W Agen s Payoff W Credi Line W 3 4 5 6 7 Compensaing 8 9 W Balance (W, D) Figure 3: The Opimal Conrac wih Medium and High Volailiy (σ =.5, σ = 9.7) Wih high volailiy (as shown in he righ panel of Figure 3), he principal s profi falls furher. This very risky projec requires a very long credi line. Noe ha in his case D = b(w ) <. Thus, he credi line has a compensaing balance requiremen he firm mus hold cash in he bank equal o D as a condiion of he credi line. Boh he required capial K and he compensaing balance D are funded hrough a large iniial draw of W W on he credi line. Given his large iniial draw, subsanial profis mus be earned before dividends will be paid. The compensaing balance provides addiional operaing income of rd o he firm. This income increases he araciveness of he projec o he agen, prevening he agen from leaving he firm when he balance on he credi line is high. By funding he compensaing balance upfron, invesors are commied o providing he firm wih income rd even when he credi line is paid off. This commimen is necessary since invesors coninuaion payoff a W is negaive, which would violae heir limied liabiliy. The compensaing balance herefore serves o ie he agen and he invesors o he firm in an opimal way. Finally, noe ha if we increase volailiy furher in his example, he maximal profi for he principal falls below K. Thus, while such a projec is posiive NPV, i canno be financed due o he incenive consrains. Remark. While we have derived he agen s iniial payoff assuming invesors are compeiive, oher possibiliies are sraighforward o consider. For example, if he principal were a monopolis hiring he agen o run he firm, he conrac would be iniiaed a he value W ha maximizes he principal s payoff b(w ). This would no change he opimal deb and credi limi, bu in his case he firm would always sar wih a draw on he credi line. Ineresingly, as can be seen in by comparing Figure and 5
Figure 3, while higher volailiy decreases b(w ), he effec on he agen s payoff W is no monoonic. Thus he agen migh prefer o manage a higher risk projec. 3.. Comparaive Saics How do he credi line, deb, and he agen s and invesors iniial payoff depend on he parameers of he model? In he discree-ime seing, many of hese comparaive saics are analyically inracable, and mus be compued for a specific example. A key advanage of he coninuous ime framework is ha we can use he differenial equaion ha characerizes he opimal conrac o compue hese comparaive saics analyically. Here we ouline a new mehodology for explicily calculaing comparaives saics. Deails are in he Appendix. Firs, we derive he effec of parameers on he principal s profi. We sar wih he HJB equaion for he principal s profi for a fixed credi line, which is represened by he inerval [R, W ]: rb( W ) =µ+γ Wb '( W ) + λ σ b''( W ) The effec of any parameer θ on he principal s profi can be found by differeniaing he HJB equaion and is boundary condiions wih respec o θ. During differeniaion we keep W fixed, which is jusified by he envelope heorem. As a resul, we ge an ordinary differenial equaion for bw ( ) / θwih appropriae boundary condiions. We apply a generalizaion of he Feynman-Kac formula o wrie he soluion as an expecaion bw ( ) r µ γ ( λ σ ) r L = E e + Wb '( W) + b''( W) d+ e W = W θ (9) θ θ θ θ where dw =γwd di +λ dz as before. Inuiively, equaion (9) couns how much profi is gained or los on he pah of W due o he modificaion of parameers. For example, bw ( ) r = E e W = W, L which is expeced discouned value of a dollar a liquidaion ime. Once we know he effec of parameers on he principal s profi, we deduce heir effec on he deb and credi line by differeniaing he boundary condiion rb(w ) + γw = µ, and on he agen s saring value by differeniaing b(w ) = K (or b (W ) = when he principal is a monopolis). For example, he effec of L is found as follows: bw ( ) W W W r r r + b'( W ) +γ = = E e W = W <. L L L L γ r As L increases, inefficiency of liquidaion declines, so a shorer credi line opimally provides less financial flexibiliy for he projec. By similar mehods, we can quanify he impac of he model parameers on he main feaures of an opimal conrac. The derivaions are carried ou in he appendix. 6
dc L / dd/ dw / dw / db(w )/ dl + + + dr 3 + dγ ± dµ + + + + + dσ + ± dλ ± 4 + ± Table : Comparaive Saics for he Opimal Conrac While he analyic derivaion of hese resuls is echnically involved, he inuiion behind hem is clear. Consider he effec of parameers on he credi line and deb. We already know ha he credi line decreases as L increases, because i makes liquidaion less inefficien. This reduces he agen s empaion o draw he enire credi line and defaul, so he principal can exrac greaer coupon paymens on deb. If he agen s ouside opion R increases, he agen becomes more emped o draw down he credi line. The lengh of he credi line decreases o reduce his empaion, and paymens on deb decrease o make i more aracive for he agen o run he projec, as opposed o aking he ouside opion. If he mean of cash flows µ increases, he credi line increases o delay erminaion and deb increases because he principal can exrac more cash flows from he agen. If he agen s discoun rae γ increases, hen he credi line decreases because i becomes coslier o delay he agen s consumpion. On he oher hand, he amoun of deb could move eiher way due o wo effecs. For small γ deb increases in γ because he agen is able o borrow more hrough deb when he credi line is smaller. When γ becomes large, he projec becomes less profiable due o he agen s impaience, so he agen is able o borrow less hrough deb. We already saw in Secion 3. why he credi limi increases and he deb decreases wih volailiy σ riskier projecs require longer credi line and herefore he agen is able o borrow less hrough deb. In Table, b(w * ) is he maximum of he profiabiliy of he projec, i.e. he maximal amoun of capial ha he projec can raise. When L or µ increase he projec becomes more profiable, so i can poenially raise more capial. When risk of he projec σ or he agen s impaience γ increases, he projec becomes less profiable. Finally, higher ouside opion R makes i more difficul for invesors o punish he agen, so overall profiabiliy of he projec decreases. We conclude by compuing he quaniaive effec of he parameers on he deb choice of he firm for a specific example in Figure 4. Noe for example ha a compensaing balance is required if σ is high (o miigae risk), if R is high or µ is low (o increase he profi rae of he firm o mainain he agen s incenive o say), or if λ is very low (when 3 These are found for he case when he projec is profiable even if he agen does no have any iniial cash, which implies ha b (R) >. 4 dc L /dλ is negaive if R =. 7
he agency problem is small, a smaller hrea of erminaion is needed, and hus he credi line expands and deb shrinks). (Though no visible in he figure, i is also rue as γ r.) 5 5 5 C L D 5 5 L 6 4-5 5 5 µ 5...4.6.8. γ 5-4 6-5 5 5 R σ λ Figure 4: Comparaive Saics (base case: L =, R =, µ =, σ =, r = %, γ = 5%, λ = ) 3.3. Securiy Marke Values -5.5 We now consider he marke values of he credi line, long-erm deb and equiy ha implemen he opimal conrac. For his we need o make an assumpion regarding he prioriizaion of he deb in defaul. We assume ha he long-erm deb is senior o he credi line; similar calculaions could be performed for differen assumpions regarding senioriy. 5 Wih his assumpion, he long-erm debholders ge L D = min(l, D) upon erminaion. The marke value of long-erm deb is herefore r r VD( M) = E e xd+ e L D M Noe ha we compue he expeced discouned payoff for he deb condiional on he curren draw M on he credi line, which measures he firm s disance o defaul in our implemenaion. Unil erminaion, he equiy holders ge oal dividends of ddiv = di /λ, wih he agen receiving fracion λ. A erminaion, he ouside equiy holders receive he remaining par of liquidaion value, L E = max(, L D C L ) /( λ) per share, afer he deb and credi line have been paid off. 6 The value of equiy (per share) o ouside equiy holders is hen 5 Recall ha only he aggregae paymens o invesors maer for incenives; he division of he paymens beween he securiies is only relevan for pricing. 6 Lemma E in he Appendix shows ha L < D + C L when λ = and here are no ouside equiy holders, so in ha case we can se L E = o compue he shadow price of ouside equiy. 8
r r VE( M) = E e ddiv + e LE M Finally, he marke value of he credi line is r r VC( M) = E e ( dy ) xd ddiv + e LC M where L C = min(c L, L L D ). For he opimal capial srucure, he aggregae value of he ouside securiies equals he principal s coninuaion payoff. Tha is, from (7), b(r + λ(c L M)) = V D (M) + V C (M) + ( λ) V E (M). We show in he appendix how o represen hese marke values in erms of an ordinary differenial equaion, so ha hey may be compued easily. See Figure 5 for an example. In his example, L < D so ha he long-erm deb is risky. Noe ha he marke value of deb is decreasing owards L as he balance on he credi line increases owards he credi limi. Similarly he value of equiy declines o a he poin of defaul. The figure also shows ha he iniial value of he credi line is posiive he lender earns a profi by charging ineres rae γ > r. However, as he disance o defaul diminishes, addiional draws on he credi line resul in losses for he lender (for each dollar drawn, he value of he credi line goes up by less han one dollar, and evenually declines). 8 7 6 V E 5 4 3 V C V D D L 3 4 5 Draw on Credi Line M C L Figure 5: Marke Values of Securiies for µ =, σ =, λ = 5%, r = %, γ = 5%, L =, R = Figure 5 also illusraes several oher ineresing properies of he securiy values. Noe, for example, ha he leverage raio of he firm is no consan over ime. When cash 9
flows are high, he firm will pay off he credi line and is leverage raio will decline. On he oher hand, during imes of low profiabiliy, he firm increases is leverage. This paern is broadly consisen wih he empirical behavior of leverage. One surprising observaion from Figure 5: he value of equiy is concave in he credi line balance, which implies ha he value of equiy would decline if he cash flow volailiy were o increase. In fac, we can show: PROPOSITION 5. When deb is risky (L < D), for he opimal capial srucure he value of equiy decreases if cash flow volailiy increases. Thus, equiy holders would prefer o reduce volailiy. PROOF: See appendix. This is couner o he usual presumpion ha risky deb implies ha equiy holders benefi from an increase in volailiy due o heir opion o defaul. Tha is, in our seing, here is no asse subsiuion problem relaed o leverage. Noe also ha he agen s payoff is linear in he credi line balance, so ha he agen is indifferen regarding changes o volailiy. 4. Hidden Effor Throughou our analysis 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 discuss he relaionship beween his model and a sandard principal-agen model in which he agen makes a hidden binary effor choice. This model is also sudied by Biais e al. (4) in conemporaneous work. Our main resul is ha, subjec o naural parameer resricions, he soluions are idenical for boh models. Thus, all of our resuls apply o boh seings. In he sandard principal-agen model wih hidden effor, he principal observes he cash flows. Based on he cash flows, he principal decides how o compensae he agen, and wheher o coninue he projec. Thus, here are only wo key changes o our model. Firs, since cash flows are observed, here is no issue of he agen saving and using he savings o over-repor fuure cash flows. Second, we assume ha a each poin in ime, he agen can choose o shirk or work. Depending on his decision, he resuling cash flow process is if he agen works dyˆ = dy + a d, where a = A if he agen shirks We assume ha working is cosly for he agen, or equivalenly ha shirking resuls in a privae benefi. 7 Specifically, we suppose he agen receives an addiional flow of uiliy equal o λa d if he shirks. The agen canno misrepor he cash flows, since r < γ he agen will consume all paymens immediaely. Thus, if he agen shirks, dc = di +λ Ad. 7 The difference beween he wo inerpreaions amouns o shifing he agen s uiliy by a consan.
Again, λ parameerizes he cos of effor and herefore he degree of he moral hazard problem. We assume λ so ha working is efficien. Our firs resul esablishes he equivalence beween his seing and our prior model: PROPOSITION 6. The opimal Principal-Agen conrac implemening high effor is he opimal conrac of Secion. PROOF: The incenive compaibiliy condiion in Lemma C is unchanged: o implemen high effor a all imes, we mus have β λ σ. Bu hen Proposiion shows ha our conrac is he opimal conrac subjec o his consrain. I is no surprising ha our original conrac is incenive compaible in his seing, since shirking is equivalen sealing cash flows a a fixed rae. Wha is perhaps more surprising is ha he addiional flexibiliy he agen has in he cash flow diversion model does no require a sricer conrac. Of course, Proposiion 6 does assume ha implemening high effor a all imes is opimal. Under wha circumsances is his assumpion correc? If a conrac were o call for he agen o shirk afer some hisory, he projec cash flows would be diminished, bu i would no be necessary o provide he agen wih incenives. 8 Therefore, in hese saes he agen s coninuaion payoff would no longer need o be sensiive o he realized cash flows, and he agen s promised payoff would evolve as γ Wd ( ˆ di +λ dy µ d) if a= dw = γwd ( di +λ Ad) if a= A Because he principal s coninuaion funcion is concave, his reducion in he volailiy of W could be beneficial. For ha no o be he case, and for high effor o remain opimal, i mus be ha for all W, rb( W ) ( µ A) + ( γw λ A) b'( W ) () Inuiively, his equaion saes ha he principal s payoff rae from having he agen shirk would be less han under our exising conrac. 9 Define w s = λa/γ and b s = (µ A)/r = (µ γw s /λ)/r, he agen and principal s payoff if he agen shirks forever and receives no oher paymen. Then we have he following necessary and sufficien condiion, as well as a simple sufficien condiion, for high effor o remain opimal a all imes: PROPOSITION 7. Implemening high effor a all imes is opimal in he Principal-Agen seing if and only if s s γ b f ( w ) where f ( z) min w b( w) + r ( z w) b'( w). A simpler sufficien condiion is 8 Specifically, in Lemma C we can se β = in saes where he conrac called for he agen o shirk. 9 Formally, condiion () is required in he proof of Proposiion for G o remain a supermaringale for eiher effor choice.