Opimal Securiy Design and Dynamic Capial Srucure in a Coninuous-Time Agency Model PETER M. DEMARZO AND YULIY SANNIKOV * Absrac We derive he opimal dynamic conrac in a coninuous-ime principal-agen seing, and implemen i wih a capial srucure (credi line, long-erm deb, and equiy) over which he agen conrols he payou policy. While he projec s volailiy and liquidaion cos have lile impac on he firm s oal deb capaciy, hey increase he use of credi versus deb. Leverage is nonsaionary, and declines wih pas profiabiliy. The firm may hold a compensaing cash balance while borrowing (a a higher rae) hrough he credi line. Surprisingly, he usual conflics beween deb and equiy (asse subsiuion, sraegic defaul) need no arise. * Sanford Universiy and U.C. Berkeley. The auhors hank Mike Fishman for many helpful commens, as well as Edgardo Barandiaran, Zhiguo He, Han Lee, Gusavo Manso, Rober Meron, Nelli Oser, Ricardo Reis, Raghu Sundaram, Alexei Tchisyi, Jun Yan, Baozhong Yang as well as seminar paricipans a he Universia Auomaa de Barcelona, U.C. Berkeley, Chicago, LBS, LSE, Michigan, Norhwesern, NYU, Oxford, Sanford, Washingon Universiy, and Wharon. This research is based on work suppored in par by he NBER and he Naional Science Foundaion under gran No. 45686.
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 i can generae arbirarily large operaing losses. The agen will need ouside financial suppor o cover such losses and coninue he projec. The difficuly is ha while he probabiliy 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 benefi. Specifically, he agen may (i) conceal and diver cash flows for his own consumpion, and/or (ii) sop providing cosly effor, which would reduce he mean of he cash flows. Therefore, from he perspecive of he principal or invesors ha fund 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 s fundamenals. To provide he agen wih appropriae incenives, invesors conrol he agen s wage, and may also 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. We develop a mehod o solve for he opimal conrac, given he incenive consrains, in a coninuous-ime seing and sudy he properies of he credi line, deb, and equiy ha implemen he conrac as in he discree-ime model of DeMarzo and Fishman (3a). The coninuous-ime seing offers several advanages. Firs, i provides a much cleaner characerizaion of he opimal conrac hrough an ordinary differenial equaion. Second, i yields a simple deerminaion of he mix of deb and credi. Finally, he coninuous-ime seing allows us o compue comparaive saics and securiy prices, o analyze conflics of ineres beween securiy holders, and o generalize he model o broader seings. In he opimal conrac, he agen is compensaed by holding a fracion of he firm s equiy. The remaining equiy, deb, and credi line are held by ouside invesors. The firm draws on he credi line o cover losses, and pays off he credi line when i realizes a profi. Thus, in our model leverage is negaively relaed wih pas profiabiliy. 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 firm defauls and he projec is erminaed. In rare insances in which he firm pays a liquidaing dividend o equiy holders, only he ouside equiy is paid. Thus, paymens o inside and ouside equiy differ only a liquidaion.
The credi line is a key feaure of our implemenaion of he opimal conrac. Empirically, credi lines are an imporan (and undersudied) componen of firm financing: Beween 1995 and 4, credi lines accouned for 63% (by dollar volume) of all corporae deb. 1 Our resuls may shed ligh boh on he choice beween credi lines and oher forms of borrowing, and he characerisics of he credi line conracs ha are used. In our model, i is his access o credi ha provides he firm he financial slack needed o operae given he risk of operaing losses. The balance on he credi line, and herefore he amoun of financial slack, flucuaes wih he pas performance of he firm. Thus, our model generaes a dynamic model of capial srucure in which leverage falls wih he profiabiliy of he firm. In our coninuous-ime seing he projec generaes cumulaive cash flows ha follow a Brownian moion wih posiive drif. Using echniques inroduced by Sannikov (5), we develop a maringale approach o formulae he agen s incenive compaibiliy consrain. We hen characerize he opimal conrac hrough an ordinary differenial equaion. This characerizaion, unlike ha using he discree-ime Bellman equaion, allows for an analyic derivaion of he impac of he model parameers on he opimal conrac. The mehodology we develop is quie powerful, and can be naurally exended o include more complicaed moral hazard environmens, as well as invesmen and projec selecion. In addiion o his mehodological conribuion, by formulaing he model in coninuous-ime we obain a number of imporan new resuls. Firs, in he discree-ime seing, public randomizaion over he decision o erminae he projec is someimes required. We show ha his randomizaion, which is somewha unnaural, is no required in he coninuous-ime seing. Indeed, in our model he erminaion decision is based only on he firm s pas performance. A second feaure of our seing is ha, because cash flows are normally disribued, arbirarily large operaing losses are possible. In he discree-ime seing, such a projec would be unable o obain financing. We show no only how o finance such a projec, bu also how, when he risk of loss is severe, he opimal conrac may require ha he firm hold a compensaing balance (a cash deposi ha he firm mus hold wih he lender o mainain he credi line) as a requiremen of he credi line. The compensaing balance commis ouside invesors o provide he firm funds, hrough ineres paymens, ha he firm migh no be able o raise ex pos. Thus, he compensaing balance 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 he empirical observaion ha many firms hold subsanial cash balances a low ineres raes while simulaneously borrowing a higher raes. Third, in our capial srucure implemenaion, he agen conrols no only he cash flows bu also he payou policy of he firm. We show ha he agen will opimally choose o pay off he credi line before paying dividends, and, once he credi line has been paid off, o pay dividends raher han hoard cash o generae addiional financial slack. In he coninuous-ime seing, he incenive compaibiliy of he firm s payou policy reduces o a simple and inuiive consrain on he maximal ineres expense ha he firm can bear, based on he expeced cash flows of he projec and he agen s ouside opporuniy. This consrain implies ha he firm s oal deb capaciy is relaively insensiive o he risk of he projec and is liquidaion cos. However, hese facors are primary deerminans of he mix of long-erm deb and credi ha he firm will use. No surprisingly, firms wih higher risk and liquidaion coss gain financial flexibiliy by subsiuing credi for long-erm deb. Noe ha while his resul does no come ou of sandard heories, i is broadly consisen wih he empirical findings of Benmelech (4) (for 19 h cenury railroads). In addiion o enabling us o compue hese and oher comparaive saics resuls, our coninuousime framework also allows us o explicily characerize he marke values of he firm s securiies. We show how he marke value of he firm s equiy and deb vary wih is credi qualiy, which is deermined by is remaining credi. Moreover, we are able o explore no only he agen s incenives bu also hose of equiy holders. One surprising feaure of our model of opimal capial srucure is ha, despie he firm s use of leverage, equiy holders (as well as he agen) have no incenive o increase risk, ha is, under our conrac, here is no asse subsiuion problem. In addiion, for a wide range of parameers, here is no sraegic defaul problem, ha is, equiy holders have no incenive o increase dividends and precipiae defaul, or o conribue new capial and pospone defaul. For he bulk of our analysis, we focus on he case in which he agen can conceal and diver cash flows. In Secion III, we show ha he characerizaion of he opimal conrac is unchanged if he agen makes a hidden effor choice, as in a sandard principal-agen model. In Secion IV, we endogenize he erminaion liquidaion payoffs by allowing invesors o fire and replace he agen and
by allowing he agen o qui o sar a new projec. We also consider renegoiaion and solve for he opimal renegoiaion-proof conrac. Our paper is par of a growing lieraure on dynamic opimal conracing models using recursive echniques ha began wih Green (1987), Spear and Srivasava (1987), Phelan and Townsend (1991), and Akeson (1991) among ohers (see Ljungqvis and Sargen () for a descripion of many of hese models). As we menion above, his paper builds direcly on he model of DeMarzo and Fishman (3a). Oher recen work ha develops opimal dynamic agency models of he firm includes Albuquerque and Hopenhayn (1), Clemeni and Hopenhayn (), DeMarzo and Fishman (3b), and Quadrini (1). However, wih he excepion of DeMarzo and Fishman (3a), hese papers do no share our focus on an opimal capial srucure. In addiion, none of hese models are formulaed in coninuous ime. 3 While discree-ime models are adequae concepually, a coninuous-ime seing may prove o be simpler and more convenien analyically. An imporan example is he principal-agen model of Holmsrom and Milgrom (1987), in which he opimal coninuous-ime conrac is shown o be a linear equiy conrac. 4 Several feaures disinguish our model from heirs, namely, he invesor's abiliy o erminae he projec, he agen's consumpion while he projec is running, he limied wealh of he agen, and he naure of he agency problem. The erminaion decision is a key feaure of our opimal conrac, and we demonsrae how his decision can be implemened hrough bankrupcy. 5 In conemporaneous work, Biais e al. (4) consider a dynamic principal-agen problem in which he agen s effor choice is binary. These auhors do no formulae he problem in coninuous ime: raher, hey exam he coninuous limi of he discree-ime model and focus on he implicaions for he firm s balance shee. We show in Secion III ha heir seing is a special case of our model. Tchisyi (5) develops a coninuous-ime model ha is similar o our seing excep ha he cash flows follow a binary Markov swiching process, ha is, cash flows arrive a eiher a high or low rae, wih he swiches beween saes observed only by he agen. The agen s privae knowledge of he sae inroduces a dynamic asymmeric informaion problem, which Tchisyi shows can be solved by making he ineres rae on he credi line increase wih he balance. Of course, here is a large lieraure on saic models of securiy design. We do no aemp o survey his lieraure here. 6 Tha said, our model is loosely relaed o he coninuous-ime capial
srucure models developed by Leland and Tof (1996), Leland (1998), and ohers. These papers ake he form of he securiies as given and derive he effec of capial srucure on he incenives of he manager, deb holders, and shareholders, aking ino accoun issues such as he ax benefis of deb, sraegic defaul, and asse subsiuion. Here, we derive he opimal securiy design and show ha he sandard agency problems beween deb and equiy holders may no arise. I. The Seing and he Opimal Conrac In his secion we presen a coninuous-ime formulaion of he conracing problem and develop a mehodology ha can be used o characerize he opimal conrac as a soluion of a differenial equaion. We hen implemen he conrac wih a capial srucure ha includes ouside equiy, longerm deb, and a line of credi. This implemenaion decenralizes he soluion of a sandard principalagen model ino separae securiies ha can be held by dispersed invesors, giving he agen a high degree of discreion over he firm s payou policy. A. The Dynamic Agency Model The agen manages a projec ha generaes poenial cash flows wih mean µ and volailiy σ dy = µ d + σ dz, where Z is a sandard Brownian moion. For now we assume ha he agen is essenial o run he projec; in Secion IV.A we allow he principal o fire he agen and hire a replacemen. The agen observes he poenial cash flows Y, bu he principal does no. The agen repors cash flows { Yˆ ; } o he principal, where he difference beween Y and Ŷ is deermined by he agen s hidden acions, which are he source of he agency problem. The principal receives only he repored cash flows dŷ from he agen. The conrac hen specifies compensaion for he agen di, as well as a erminaion ime, ha are based on he agen s repors. In his secion we model he agency problem by allowing he agen o diver cash flows for his own privae benefi; in Secion III we show how o adap he model o he case of hidden effor. The agen receives a fracion λ (,1] of he cash flows he divers; if λ < 1, here are dead-weigh coss of concealing and divering funds. The agen can also exaggerae cash flows by puing his own money
back ino he projec. By alering he cash flow process in his way, he agen receives a oal flow of income of 7 [ ˆ λ dy dy] + di, where ˆ λ [ dy ] ( ˆ) ( ˆ dy λ dy dy dy dy). diversion + over-reporing (1) The agen is risk neural and discouns his consumpion a rae γ. 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 = ρs d + dy dy ] + di dc, () where dc is he agen s consumpion a ime. The agen mus mainain a nonnegaive balance on his accoun, ha is, S. This resource consrain prevens a soluion in which he agen simply owns he projec and runs i forever. 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 8 W = E e dc + e R γs γ s. (3) The principal discouns cash flows a rae r, such ha γ > r ρ. 9 Once he conrac is erminaed, she receives expeced liquidaion payoff L. (In Secion IV, we consider how he erminaion payoffs R and L arise, for example, from he principal s abiliy o fire and replace he agen, or he agen s abiliy o renegoiae he conrac or sar a new projec). The principal s oal expeced profi a dae is hen rs ˆ r b = E e ( dy ) s dis + e L. (4) The projec requires exernal capial of K o be sared. The principal offers o conribue his capial in exchange for a conrac (, I) ha 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. In response o a conrac (, I), he agen chooses a feasible sraegy o maximize his expeced payoff. A feasible sraegy is a pair of processes (C, Ŷ ) adaped o Y such ha (i) Ŷ is coninuous and, if λ < 1, Y Ŷ has bounded variaion, 1 (ii) (iii) C is nondecreasing, and he savings process, defined by (), says nonnegaive.
The agen s sraegy (C, ˆ Y ) 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. Noe ha we have no explicily modeled he agen s opion o qui and receive he ouside opion R a any ime. Because he agen can always underrepor and seal a rae γr unil erminaion, any incenive compaible sraegy yields he agen a leas R. In conras, his consrain may bind in a discree-ime seing because of a limi o he amoun he agen can seal per period. The opimal conracing problem is o find an incenive compaible conrac (, I, C, Ŷ) ha maximizes he principal s profi subjec o delivering he agen an iniial required payoff W. By varying W we can use his soluion o consider differen divisions of bargaining power beween he agen and he principal. For example, if he agen enjoys all he bargaining power due o compeiion beween principals, hen he agen mus receive he maximal value of W subjec o he consrain ha he principal s profi be a leas zero. REMARK. For simpliciy, we specify he conrac assuming ha he agen's income I and he erminaion ime are deermined by he agen's repor, ruling ou public randomizaion. This assumpion is wihou loss of generaliy: Because he principal's value funcion urns ou o be concave (Proposiion 1), we will show ha public randomizaion would no improve he conrac. B. Derivaion of he Opimal Conrac We solve he problem of finding an opimal conrac in hree seps. Firs, we show ha i is sufficien o look for an opimal conrac wihin a smaller class of conracs, namely, conracs in which he agen chooses o repor cash flows ruhfully and mainain zero savings. Second, we consider a relaxed problem by ignoring he possibiliy ha he agen can save secrely. Third, we show ha he conrac is fully incenive compaible even when he agen can save secrely. We begin wih a revelaion principle ype of resul: 11 LEMMA A: There exiss an opimal conrac in which he agen i) chooses o ell he ruh, and ii) mainains zero savings. The inuiion for his resul is sraighforward i is inefficien for he agen o conceal and diver cash flows (λ 1) or o save hem (ρ r), as we could improve he conrac by having he principal
save and make direc paymens o he agen. Thus, we will look for an opimal conrac in which ruh elling and zero savings are incenive compaible. B.1. The Opimal Conrac wihou Savings Noe ha if he agen could no save, hen he would no be able o overrepor cash flows and he would consume all income as i is received. Thus, dc = di + λ( dy dyˆ ). (5) We relax he problem by resricing he agen s savings so ha (5) holds and allowing he agen o seal only a a bounded rae. 1 Afer we find an opimal conrac for he relaxed problem, we show ha i remains incenive compaible even if he agen can save secrely or seal a an unbounded rae. One challenge when working in a dynamic seing is he complexiy of he conrac space. Here, he conrac can depend on he enire pah of repored cash flows ˆ Y. This makes i difficul o evaluae he agen s incenives in a racable way. Thus, our firs ask is o find a convenien represenaion of he agen s incenives. Define he agen s promised value W (Ŷ) 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 of 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 = γw d di + β ( Yˆ)( dyˆ µ d). (6) This sensiiviy β (Ŷ) is deermined by he agen s pas repors Ŷ s, s. Proof of Lemma B: 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 (6) for he case in which he agen ruhfully repors Ŷ = Y. 13 In ha case, γs γ = s + V e di ( Y) e W ( Y) (7)
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 (7) wih respec o we find γ γ γ γ dv = e β ( Y )( Y µ d) = e di ( Y ) γe W ( Y ) d + e dw ( Y ), and hus (6) holds. Informally, he agen has incenives no o seal cash flows if he ges a leas λ of promised value for each repored dollar, ha is, 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 lemma. LEMMA C: If he agen canno save, ruh-elling is incenive compaible if and only if β λ for all. Proof of Lemma C: If he agen seals dy dyˆ a ime, he gains immediae income of λ( dy dyˆ ) bu loses β ( dy dyˆ ) in coninuaion payoff. Therefore, he payoff from reporing sraegy Ŷ gives he agen he payoff of γ ˆ γ W ˆ + E e λ( dy dy) e β( dy dy), (8) where W denoes he agen s payoff under ruh-elling. We see ha if β λ for all hen (8) is maximized when he agen chooses dyˆ = dy, since he agen canno overrepor 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. 14 Now we use he dynamic programming approach o deermine he mos profiable way for he principal o deliver he agen any value W. Here we presen an informal argumen, which we formalize in he proof of Proposiion 1 in he Appendix. Denoe by b(w) he principal s value funcion (he highes profi o he principal ha can be obained from a conrac ha provides he agen he payoff W). 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. 15
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. (9) Equaion (9) implies ha b (W) 1 for all W; ha is, he marginal cos of compensaing he agen can never exceed he cos of an immediae ransfer. Define W 1 as he lowes value such ha b (W 1 ) = 1. Then i is opimal o pay he agen according o 1 di = max( W W,). (1) These ransfers, and he opion o erminae, keep he agen s promised value beween R and W 1. Wihin his range, equaion (6) implies ha he agen s promised value evolves according o dw = γw d + βσdz 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 1 E[ dy + db( W )] = µ + γwb'( 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 Hamilon-Jacobi-Bellman (HJB) equaion for he value funcion: 1 rb( W ) = max µ + γwb'( W ) + β σ b''( W ). (11) β λ Given he concaviy of b, b (W) and hus β = λ is opimal. 16 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: 1 rb( W ) = µ + γwb'( W ) + λ σ b''( W ), R W W 1, (1) wih b(w) = b(w 1 ) (W W 1 ) for W > W 1. We require hree boundary condiions o pin down a soluion o his equaion and he boundary W 1. 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 1 ) = 1. 17
The final boundary condiion is he super conac condiion for he opimaliy of W 1, which requires ha he second derivaives mach a he boundary. This condiion implies ha b (W 1 ) =, or equivalenly, using equaion (1), 1 1 rb( W ) + γ W = µ. (13) This boundary condiion has a naural inerpreaion: I is beneficial o pospone paymen o he agen by making W 1 larger because doing so reduces he risk of early erminaion. Posponing paymen is sensible unil he boundary (13), when he principal and agen s required expeced reurns exhaus he available expeced cash flows. 18 Figure 1 shows an example of he value funcion. The following proposiion formalizes our findings: PROPOSITION 1: The conrac ha maximizes he principal s profi and delivers he value W [R, W 1 ] o he agen akes he following form: W evolves according o dw = γw d di + λ ( dyˆ µ d). (14) Inser Fig. 1 here When W [R, W 1 ), di =. When W = W 1, paymens di cause W o reflec a W 1. If W > W 1, an immediae paymen W W 1 is made. The conrac is erminaed a ime, when W reaches R. The principal s expeced payoff a any poin is given by a concave funcion b(w ), which saisfies 1 rb( W ) = µ + γwb ( W ) + λ σ b ( W ) (15) on he inerval [R, W 1 ], b ( W) = 1 for W W 1, and boundary condiions b(r) = L and rb(w 1 ) = µ γw 1. B.. Hidden Savings Thus far, we resric he agen from saving and over-reporing sraegies. 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 furher, because 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 R is bounded from above and solves dw = γw d di d + λ ( dyˆ µ d) (16)
unil sopping ime = min{ W = R}. Then he agen earns a payoff of a mos W from any feasible sraegy in response o a conrac (, I). Furhermore, he payoff W is aained if he agen repors ruhfully and mainains zero savings. This resul confirms ha conracs from a broad class, including he opimal conrac of Proposiion 1, remain incenive compaible even if he agen has access o hidden savings. In he nex subsecion Proposiion will help us characerize incenive-compaible capial srucures. C. Capial Srucure Implemenaion So far, our resuls characerize he opimal principal-agen conrac. In his secion, we show how his conrac can be implemened using sandard securiies ha are held by widely dispersed invesors or inermediaries. The firm raises iniial capial K and possibly addiional cash (o fund an iniial dividend or cash reserve for he firm) by issuing he securiies a ime. Because he opimal conrac is condiional on he agen s promised payoff W, he implemenaion we describe will hold regardless of wheher he agen designs he securiies o maximize his own payoff, or he invesors design he securiies o maximize he value of he firm. (We discuss alernaive disribuions of bargaining power beween he agen and invesors in Secion II.A.) We begin by describing he securiies used in he implemenaion: 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 securiies. While he implemenaion is no unique (e.g., one could always use he single conrac, or srip he 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, how much o pay in dividends, and wheher o accumulae cash balances wihin he firm. While i is imporan for pricing he securiies, for he implemenaion i is no necessary o specify he prioriizaion of he securiies over he liquidaion payoff L. However, because we compensae he agen wih equiy, 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. 19 (Wih absolue prioriy his disincion will ofen be unnecessary, as deb claims 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 γc. (17) 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. Under 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. (18) This capial srucure implemens he opimal conrac if, in addiion, he credi limi saisfies C L = λ 1 (W 1 R). (19) The inuiion for he incenive compaibiliy of his capial srucure is as follows. Firs, providing he agen fracion λ of he equiy eliminaes his incenive o diver cash because he can do as well by paying dividends. 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 (18) implies ha he payoff from his deviaion would be equal o W, he payoff ha he agen receives from waiing unil he credi line balance is zero before paying dividends. 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 (17), is o adjus he profi rae of he firm so ha he agen s payoff saisfies equaion (18). If he deb were oo high, he agen s payoff would be below he amoun in (18) and he agen would draw down he credi line immediaely. If he deb were 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, if (17) holds, we say ha 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 B? The hisory dependence of he opimal conrac is implemened hrough he credi line, wih he balance on he credi line acing as he memory device ha racks he agen s payoff W. In he opimal conrac, he agen is paid in order o keep he promised payoff from exceeding W 1. Here, dividends are paid when he balance on he credi line is M =. To implemen he opimal conrac, hese condiions mus coincide. Solving equaion (18) for C L leads o he opimaliy condiion C L = λ 1 (W 1 R). There is no guaranee ha in his capial srucure he deb required by equaion (17) is posiive. If D <, we inerpre he deb as a compensaing balance, ha 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 examine he circumsances in which a compensaing balance arises in he nex secion. The implemenaion here is very similar o ha given for he discree-ime model of DeMarzo and Fishman (3a). 1 There are hree imporan disincions, however. 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 simple characerizaion of he incenive compaibiliy condiion for he capial srucure in erms of he primiives of he model, as we do here in equaion (17). In paricular, when γ is close o r, his condiion implies ha he oal deb capaciy of he firm,
L D+ C = µ / γ R/ λ + (1 r/ γ ) D µ / γ R/ λ, is relaively insensiive o he volailiy σ and liquidaion value L of he projec. The mix of deb and credi will depend on hese parameers, however, as we explore nex. II. Opimal Capial Srucure and Securiy Prices The capial srucure implemenaion of he opimal conrac raises many ineresing quesions. For insance, wha facors deermine he amoun ha he agen borrows? Under wha condiions does he agen borrow for iniial consumpion? When does a compensaing balance arise? Wha is he opimal lengh of he credi line? How do he marke values of he 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. A. 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 = 1 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. PROPOSITION 4: The opimal deb and credi line wih agency parameer and erminaion payoffs (λ, R, L) are he same as wih parameers (1, R λ, L λ ), where R λ 1 1 = R and L = L + (1 ). r 1 λ When λ = 1, he opimal credi limi is C L = W 1 R. The opimal level of deb is hen deermined by (17), which in his case can be wrien as rd = µ γr γc L = µ γw 1. Recall also ha in he opimal conrac, W 1 is deermined by he boundary condiion (13): rb(w 1 ) + γw 1 = µ. λ λ λ µ
Combining hese wo resuls implies ha he opimal face value of deb is D = b(w 1 ). Figure provides an example, illusraing he size of he credi line and he face value of deb when he cash flow volailiy is low. From he figure, D > L, so he deb is risky. Noe ha he opimal capial srucure for he firm does no depend on he amoun of exernal capial K ha is required. However, he iniial payoffs of he agen and he invesors depend upon K as Inser Fig. here well as he paries relaive bargaining power. If invesors are compeiive, he agen s iniial payoff is he maximal payoff W such ha b(w ) = K as Figure illusraes. In his example, W > W 1. This payoff is achieved by giving he agen an iniial cash paymen of W W 1, and saring he firm wih zero balance on he credi line (providing he agen wih fuure payoff W 1 ). In oher words, he firm issues long-erm deb o fund he projec and pays an iniial dividend of W W 1. The credi line is hen used as needed o cover operaing losses. Thus, he firm raises b(w 1 ) from invesors, which is equal o he face value of deb D. However, because he deb is risky (D > L), given coupon rae r i mus rade a a discoun. 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 pays he firm an amoun upfron ha 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 he projec more flexibiliy by delaying 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 invesors payoff 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 (he lef panel of Figure 3), he face value of deb is below he liquidaion value of he firm (D < L). Thus, if long-erm deb has prioriy in defaul, i is now riskless, in which case he firm will raise D hrough a long-erm deb issue. However, in his case we also have D < K, so he firm mus raise he addiional capial needed o iniiae he projec hrough an iniial draw on he
credi line of W 1 W. Because b > 1 on (W, W 1 ), 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 ne presen value of he credi line due o he firm s greaer credi risk. Wih high volailiy (he righ panel of Figure 3), he invesors payoff falls furher. This very risky projec requires a very long credi line. Noe ha in his case D = b(w 1 ) <. We can inerpre D < as a compensaing balance requiremen he firm mus hold cash in he bank wih a balance 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 on he credi line of W 1 W. Given his large iniial draw, subsanial profis mus be earned before dividends are paid. The compensaing balance provides he firm addiional operaing income of rd. This income increases he profiabiliy of he firm, making i incenive compaible for he agen o run he firm raher han consume he credi line and immediaely defaul. Also, 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 1 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. Inser Fig. 3 here 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 has posiive ne presen value, i canno be financed due o he incenive consrains. REMARK. While here we assume ha he agen owns he firm and he invesors are compeiive, oher possibiliies are sraighforward. For example, if he curren owners choose he capial srucure o maximize he firm s value, and he agen is hired from a compeiive labor marke, he conrac would be iniiaed a he value W ha maximizes he principal s payoff b(w ). The opimal capial srucure would be unchanged, bu he firm would always sar wih a draw on he credi line. Indeed, he iniial leverage of he firm increases wih invesors bargaining power. Comparing Figure and Figure 3, while higher volailiy decreases b(w ), he effec on he agen s payoff W is no monoonic. Thus, a hired agen migh prefer o manage a higher risk projec.
B. Comparaive Saics How do he credi line, deb, and he agen s and invesors iniial payoffs depend on he parameers of he model? In he discree-ime seing, many of hese comparaive saics are analyically inracable and can only be compued for a specific example. A key advanage of he coninuous-ime framework, on he oher hand, 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. Firs, we derive he effec of parameers on he principal s profi. We sar wih he Hamilon-Jacobi-Bellman equaion for he principal s profi for a fixed credi line, which is represened by he inerval [R, W 1 ]: 1 rb( W ) = µ + γwb'( 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 1 fixed, which is jusified by he envelope heorem. As a resul, we ge an ordinary differenial equaion for bw ( ) / θ wih appropriae boundary condiions. We hen apply a generalizaion of he Feynman-Kac formula o wrie he soluion as an expecaion, ha is, bw ( ) r µ γ 1 ( λ σ ) r L = E e + Wb '( W) + b''( W) d+ e W = W, θ () θ θ θ θ where dw =γwd di + λdz as before. Inuiively, equaion () couns how much profi is gained or los on he pah of W due o he modificaion of parameers. For example, bw ( ) E e r = W = W, L which is he expeced discouned value of a dollar a he ime of liquidaion. Once we know he effec of parameers on he principal s profi, we deduce heir effec on deb and he credi line by differeniaing he boundary condiion rb(w 1 ) + γw 1 = 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: 1 1 1 1 bw ( ) 1 W W W r r 1 r + b'( W ) + γ = = E e W W L L L L γ r = <. 1 As L increases, he 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 Inser Table I here
parameers on he main feaures of an opimal conrac. Table I repors he resuls. The derivaions are carried ou in he Appendix. The inuiion for he resuls in Table I is clear. For example, consider he mix of deb and credi. We have already shown ha credi decreases as L increases, since liquidaion is less inefficien and financial slack is less valuable. If he agen s ouside opion R increases, he agen becomes more emped o draw down he credi line and defaul. 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: For small γ, deb increases in γ because he agen is able o borrow more hrough deb when he credi line is smaller, whereas for large γ, he projec becomes less profiable due o he agen s impaience, in which case he agen is able o borrow less hrough deb. As seen in Secion II.A, he credi limi increases and he deb decreases wih volailiy σ riskier projecs require longer lines of credi and hus he agen is able o borrow less hrough deb. Finally, he effec of λ is complex. Consider he special case of R =. For his case he credi line is decreasing in λ: The cos of delaying dividends becomes larger when he impaien agen owns a larger fracion of equiy. A he same ime, however, deb increases o offse he decreased credi line. The effec of he parameers on W and b(w * ) is he same since hey boh reflec he profiabiliy of he projec. When L or µ increase, he projec becomes more profiable. The projec becomes less profiable wih an increase in he risk of he projec σ, he agen s impaience γ, he magniude of he agency problem λ, or he agen s ouside opion R. Finally, he effec of he parameers on he agen s saring value W * when invesors have all he bargaining power is deermined by he following radeoff: Larger W * delays erminaion a a greaer cos of paying he agen. In Figure 4 we conclude by compuing he quaniaive effec of he parameers on he deb choice of he firm for a specific example Noe ha in his example, 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 and mainain he agen s incenive o say), or if λ is very low (when he agency problem is small, a smaller hrea of Inser Fig. 4 here
erminaion is needed, and hus he credi line expands and deb shrinks). Though no visible in he figure, he compensaing balance arises also as γ r. C. Securiy Marke Values We now consider he marke values of he credi line, long-erm deb, and equiy ha implemen he opimal conrac. To do so, we need o make an assumpion regarding he prioriy of deb in he case of defaul. Here we assume ha long-erm deb is senior o he credi line; similar calculaions could be performed for differen senioriy assumpions. Wih his assumpion, he long-erm deb holders 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 receive oal dividends of ddiv = di /λ, wih he agen receiving fracion λ. A erminaion, he ouside equiy holders receive he remaining par of he liquidaion value, L E = max(, L D C L ) /(1 λ) per share, afer he deb and he credi line have been paid off. 3 The per share value of equiy o ouside equiy holders is hen 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 (18), b(r + λ(c L M)) = V D (M) + V C (M) + (1 λ) V E (M). In he Appendix we show how o represen hese marke values in erms of an ordinary differenial equaion, so ha hey may be compued easily. Figure 5 provides an example. In his example, L < D, hus 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 Inser Fig. 5 here equiy declines o zero 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). 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 flows are high, he firm will pay off he credi line and is leverage raio will decline. During imes of low profiabiliy, on he oher hand, he firm will increase is leverage. Finally, cash flow shocks lead o persisen changes in leverage. These resuls are broadly consisen wih he empirical behavior of leverage. D. Asse Subsiuion and Equiy Issuance One surprising observaion from Figure 5 is ha 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 + C L ), for he opimal capial srucure he value of equiy decreases if cash flow volailiy increases. Thus, equiy holders would prefer o reduce volailiy. This resul 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 wih respec o leverage. Noe also ha he agen s payoff is linear in he credi line balance, so ha he agen is indifferen o changes o volailiy. 4 In Secion I.C we demonsrae ha he opimal capial srucure implies ha he firm s payou policy is incenive compaible for he agen; ha is, he agen finds i opimal o pay dividends if and only if he credi line is fully repaid. Wha abou he incenives of equiy holders? Would hey prefer an alernaive payou policy? Moreover, could he firm raise new equiy capial o delay defaul? Tha is, could equiy holders benefi from a sraegic defaul policy? If he firm increases is payous by paying addiional dividends, for each dollar paid ouside equiy holders receive (1 λ). On he oher hand, he increased draw on he credi line changes he value of ouside equiy by (1 λ) V E (M). Thus, equiy holders prefer ha he firm no pay dividends as long as V E (M) 1. (1)
Alernaively, he firm could pay down he credi line by raising new capial hrough an equiy issue. Each dollar raised increases he value of ouside equiy by (1 λ) V E (M). Thus, he firm canno raise addiional equiy capial as long as V E (M) 1/(1 λ). () The wedge beween equaions (1) and () resuls from he fac ha he agen receives dividend paymens, bu does no conribue new equiy capial o he firm. We herefore have he following resul: PROPOSITION 6: When deb is risky (L < D + C L ), equaion (1) is saisfied and holds wih equaliy a M =. Thus, equiy holders would no wish o aler he firm s payou policy. In addiion, he firm canno raise new equiy capial if () holds for M = C L. Thus, equiy holders have no incenive o aler he firm s dividend policy. To verify ha ha equiy issues will no occur, i is only necessary o check () a he defaul boundary. Numerically, () appears o hold as long as λ is no oo small (e.g., i holds for he example in Figure 5). In Secion IV.B we consider renegoiaion-proof conracs, for which we show equaion () is guaraneed o hold. III. Hidden Effor Throughou our analysis so far we concenrae on he seing in which he cash flows are privaely observed and he agen may diver hem for his own consumpion. In his secion we consider a 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 a sandard hidden effor model, 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, misreporing is no an issue. Second, we assume ha a each poin in ime, he agen can choose o eiher 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.
Working is cosly for he agen, or equivalenly, shirking resuls in a privae benefi. Specifically, we suppose ha he agen receives an addiional flow of uiliy equal o λa d if he shirks. 5 Wih r < γ he agen consumes all paymens immediaely, so ha dc = di + λ a d. Again, λ parameerizes he cos of effor and in urn he degree of he moral hazard problem. We assume λ 1 so ha working is efficien. Our firs resul esablishes he equivalence beween his seing and our prior model: PROPOSITION 7: The opimal principal-agen conrac ha implemens high effor is he opimal conrac of Secion I. Proof of Proposiion 7: The incenive compaibiliy condiion in Lemma C is unchanged: To implemen high effor a all imes, we mus have β λ σ. Proposiion 1 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 o sealing cash flows a a fixed rae. Wha is surprising is ha he addiional flexibiliy ha he agen has in he cash flow diversion model does no require a sricer conrac. Proposiion 7 assumes ha implemening high effor a all imes is opimal. Because he reducion in cash flows due o shirking is bounded unlike he case of diversion i may be opimal o sop providing incenives and o allow he agen o shirk afer some hisories. Specifically, when he agen shirks his payoff would no need o depend on cash flows, so he agen s promised payoff would evolve according o dw γwd ( ˆ di + λ dy µ d) if a= = γwd di λad 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, he principal s payoff rae from having he agen shirk would be less han ha under our exising conrac: 6 rb( W ) ( µ A) + ( γw λa) b'( W ). (3) The agen and principal s payoff if he agen shirks forever are given by w s = λa/γ and b s = (µ A)/r = (µ γw s /λ)/r.