Security Design with Correlated Hidden Cash Flows: The Optimality of Performance Pricing

Transcription

1 Security Design with Correlate Hien Cash Flows: The Optimality of Performance Pricing Alexei Tchistyi Haas School of Business, UC Berkeley July 1, 2013 Abstract This paper stuies optimal security esign in a ynamic setting with an agency problem that arises when an agent in charge of a project can ivert cash flows for his own consumption at the expense of an outsie investor. Cash flows are unobservable an unverifiable by the investor who has the right to liquiate the project. Unlike previous analyses, we allow cash flows to be correlate over time. We solve for the optimal contract an show that it can be implemente using a creit line with an interest rate that increases with the balance on the creit line. This fining is consistent with the fact that the majority of commercial loans are lines of creit with performance pricing. In aition, we evelop a new recursive metho to eal with a correlate privately observe variable in ynamic agency settings. It allows us to reuce the imensionality of the problem an obtain a close-form solution for the optimal contract. I am extremely grateful to Peter DeMarzo for avice throughout the evelopment of the paper. I also thank Fernano Alvarez, Manuel A. Amaor, Darrell Duffi e, Gustavo Manso, Yuliy Sannikov, Ilya Segal, Ilya Strebulaev, Anrzej Skrzypacz, Bruno Strulovici an Robert Wilson for helpful comments an iscussions. I thank the Stanfor Institute for Economic Policy Research for financial support through the Lyne an Harry Braley Founation Dissertation Fellowship. 1

2 1 Introuction More than 87% of all commercial an inustrial loans mae by large omestic banks are loans uner commitment 1, otherwise known as lines of creit. A line of creit is a contract between a firm an a bank that lets the firm borrow from the bank uring the life of the contract on terms specifie in avance. Two main characteristics of a creit line are the creit limit, which stipulates the maximum amount of creit allowe, an the interest rate charge on the balance. Very often, instea of a fixe rate of interest, performance pricing schemes are use that connect the interest rate to some measure of the borrower s performance, such as the borrower s interest coverage ratio, ebt-to-ebitda ratio, leverage ratio or current creit rating. Asquith, Beatty an Weber 2005 report that more than 50% of lening agreements have performance pricing features. Moreover, most of the lening agreements require the borrower to pay a higher interest rate when the borrower performs poorly. Despite their wiesprea use, the economic rationale behin creit lines with performance pricing is not completely clear. In the friction-free worl of Moigliani an Miller 1958, the market value of the firm is inepenent of its capital structure. Manso, Strulovici an Tchistyi 2010 show that ebt obligations with performance pricing precipitate efault an increase bankruptcy cost, since they impose a higher ebt buren when the firm experiences financial strain. This paper emonstrates that performance pricing can be use to solve a moral hazar problem. We consier a moel, in which a risk-neutral agent with limite liability nees external financing for a profitable business project. If fune, the project generates stochastic cash flows. An outsie investor is unable to observe the cash flows, while the agent has the ability to ivert the cash flows for his own consumption at the expense of the investor. Before initiating the project, the agent an the investor or a group of investors sign a contract that will govern their relationship after the project is initiate. In particular, the contract obligates the agent to report the cash flows to the investor, although the investor cannot verify the agent s reports. In aition, the contract specifies payments between the agent an the investor conitional on the history of the agent s reports an specifies the circumstances uner which the control of the project s assets is transferre from the agent to the investor. The transfer of control leas to ineffi ciencies, either ue to a ea-weight cost associate with it or because the investor is less capable than the agent of running the project an cannot fin an equivalent replacement for the agent s managerial talent. We assume that the cash flows are correlate over time an follow a two-state Markov process. The correlation is an important assumption, not only because it is a more realistic assumption than inepenent cash flows, but also because it creates an aitional egree of informational asymmetry between the agent an the investor. With correlate cash flows, the agent has a superior knowlege, not only about the current cash flow realization that he observes irectly, but also about the future cash flows, since their istribution is etermine by the current cash flow. We characterize the optimal contract in this setting an its implementation using stanar securities. We fin that the optimal contract can be implemente using a combination of equity, an a creit line with an interest rate that increases with the outstaning balance on the creit line. Accoring to this implementation, the agent owns a fraction of the firm s equity, while being obligate to make interest payments on the creit line balance to the investor, who owns the rest of the firm s equity. The initial raw on the creit line is etermine by the amount of funs provie by the investor, as well as the bargaining power of the parties. The agent uses the cash flows generate by the project to repay the balance on the creit line. When the cash flow is low, the agent is allowe to raw on the creit line to make the interest payments. The agent is in efault if he is unable to pay the interest without exceeing the creit line limit. In this case, the investor takes control over the firm s assets an fires the agent. In this combination of securities, the equity s role is to rewar the agent for repaying the ebt. The role of the creit line with an escalating interest rate is more complicate. The balance on the creit line can be consiere as a memory evice that summarizes all the relevant information regaring the agent s performance. The interest rate, along with the creit limit, etermines the ynamics of the 1 E.2 Survey of Terms of Business Lening, August 2-6, 2004 Feeral Reserve Statistical Release. 2

3 creit line balance an the timing of the efault. The spee at which the balance grows is etermine by the interest rate, an this spee is greater when the balance is higher. The threat of losing control over the project inuces the agent to pay the ebt. To see why the interest rate on the creit line shoul increase with the balance, consier what will happen when the agent keeps stealing the cash flows until the creit line is exhauste an efault occurs. Because the cash flows are positively correlate, the agent has a stronger incentive to steal. As a result, the interest rate has to increase with the balance in orer to precipitate efault an iscourage the agent from stealing cash flows. The optimal interest rate structure reflects an average rate at which the agent can ivert the cash flows. An important aspect of this paper is the methoology, which has inepenent theoretical value. We evelop a new ynamic programming approach for solving for an optimal contract in a setting with a correlate privately observe variable, where the stanar ynamic programming technique oes not work. The main avantage of our approach is that it allows us to reuce the imensionality of the problem an obtain a close-form solution for the optimal contract in our setting. We also believe that this approach is not only applicable to our setting, but can also be use in other ynamic principal-agent moels with correlate hien states. A number of papers stuy optimal contracting in a setting in which an agent has the ability to ivert cash flows. In a simple one-perio moel, Diamon 1984 emonstrates that the optimal contract is ebt, where the agent s incentives to make payments to leners are given in terms of nonpecuniary bankruptcy penalties. Bolton an Scharfstein 1990 consier a similar two-perio moel, in which the investor can threaten to cut off funing in the secon perio if the firm efaults in the first. This threat inuces the firm to share the first-perio profit with the investor. In a ynamic setting with asymmetric information, Clementi an Hopenhayn 2006 emonstrate that borrowing constraints emerge as a feature of the optimal lening agreements. The two stuies that are most closely relate to ours are DeMarzo an Fishman 2007a, an DeMarzo an Sannikov Both of these papers stuy long-term financial contracting in a setting with privately observe inepenent cash flows. Unlike previous analyses, we allow cash flows to be correlate over time. It turns out that the correlation significantly changes the optimal contract between the agent an the investor. While in DeMarzo an Fishman 2007a an DeMarzo an Sannikov 2006 the optimal contract can be implemente using a creit line with a constant interest rate, we fin that the implementation of the optimal contract in our setting requires a creit line with performance pricing. Empirical stuies support the theory that performance pricing an other ebt covenants are use to mitigate agency costs. Dichev an Skinner 2002 relate the existence of ebt covenants to informational asymmetries between leners an borrowers, which may cause agency problems. Braley an Roberts 2003 claim that ebt covenants are use to reuce the agency cost of ebt. Asquith, Beatty an Weber 2005 fin that ebt contracts are more likely to inclue performance pricing when re-contracting, averse selection an moral hazar costs are higher. They also estimate that, by ollar volume, more than 50% of ebt contracts have performance pricing. Manso, Strulovici an Tchistyi 2010 show that performance pricing can be use to signal a borrower s type. This paper emonstrates that performance pricing can also be use to resolve a moral hazar problem. On the technical sie, this paper evelops a recursive metho to solve for an optimal incentivecompatible contract in a setting in which a privately observe state variable is correlate over time. The vast majority of the literature on optimal contracting assumes, however, that privately observe economic variables are inepenent over time. In this literature, an optimal contract typically epens on a history of publicly observe outcomes, which is a multi-imensional object. However, it is often possible to rewrite the problem recursively, summarizing all the relevant information in the history by a one-imensional object a continuation value. Green 1987, Abreu, Pearce an Stacchetti 1990, Phelan an Townsen 1991, Korcherlakota 1996, DeMarzo an Fishman 2007a, DeMarzo an Fishman 2007b, Biais, Mariotti, Plantin an Rochet 2007, among many others, utilize this approach. Surprisingly, there are only a few papers that allow for correlation of privately observe variables. 3

4 Fernanes an Phelan 2000 consier a ynamic moel with a risk-averse agent whose enowment follows a first-orer Markov process. Quarini 2004 uses the methoology evelope by Fernanes an Phelan 2000 to solve a ynamic principal-agent moel with privately observe persistent shocks. Doepke an Townsen 2001 evelop several new recursive methos to solve for optimal contracts in ynamic principal-agent moels with hien income an hien actions. These papers resort to numerical simulations to characterize optimal contracts. Athey an Bagwell 2008 stuy collusive equilibria in a ynamic game with persistent private information. Williams 2011 stuies persistent private information in continuos time. Battaglini 2005 an Battaglini an Coate 2003 consier moels with unobservable Markov processes in a setting in which the agent has unlimite liability. The paper is organize as follows. Section 2 introuces the ynamic contracting moel with correlate privately observe cash flows. Section 3 contains a preliminary analysis of the contracting problem. Section 4 provies the erivation of the optimal contract. Section 5 emonstrates that the optimal contract can be implemente by a combination of equity, an a creit line with an escalating interest rate. Section 6 extens the moel to the continuous-time setting. Section 7 conclues. 2 The Moel A risk-neutral agent evaluates consumption sequences {C t } accoring to t βt E[C t ], where β is the intertemporal iscount factor. The agent has a special human capital to run a project that generates stochastic cash flows {Y t }. The agent s initial wealth W is not suffi cient to initiate the project, which requires a fixe initial investment I > W. To raise the lacking capital, the agent will have to enter into a contractual relationship with an investor or a group of investors who is also risk-neutral an has suffi cient financial resources. The iscount factor for the investor is also β, which correspons to the risk-free interest rate r = 1 β 1. The project generates cash flows Y t {y L, y H } for T perios. We will refer to y L an y H as the low an the high cash flows, respectively. Without loss of generality, we normalize y L = 0, an y H = 1. The cash flows follow a two-state Markov chain with Qy enoting the probability that Y t = y H, given previous-perio cash flow y: Qy = Pr Y t = y H Y t 1 = y. The cash flows are assume to be positively correlate, which in terms of transition probabilities means that Q y H > Q y L. One can verify that this implies that the expectation of a future cash flow is always higher conitional on the high cash flow: E [Y t+k Y t = y H ] > E [Y t+k Y t = y L ], 1 where k = 1, 2,... For more on the properties of the cash flow process, see Lemma 5 in Appenix. The agent privately observes realizations of the cash flows, while the investor must rely on the agent to report the cash flow realizations. We assume that the low cash flow y L is observable an collectible by the investor, but the agent can secretly ivert the excess cash flow y H y L for his own consumption. The agent is able to enjoy only a fraction λ [0, 1] of the iverte amount. The fraction 1 λ represents ineffi ciencies associate with stealing. The agent can consume iverte cash flows immeiately or save them at the interest rate r r in his private bank account. We interpret iversion of the firm s cash flows as stealing, i.e., transferring firm s money to the agent s account. However, other activities that benefit the agent may fit the setting of the moel as well. For example, the agent can receive non-monetary benefits from spening firm s cash flows on various projects, such as corporate jets, that benefit him at the expense of investors. 2 The agent has limite liability an can quit at any time. For ease of presentation, we normalize the agent s reservation payoff to zero: R t = 0, which means that the agent will never quit voluntarily. 2 The fact that the agent cannot save non-monetary benefits is not essential here, since, as we will see later, savings cannot improve the agent s payoff. 4

5 Perio t Agent privately observes cash flow Y t Agent pays ŷ t to investor Investor pays t to agent Investor can liquiate project Figure 1: Sequence of events In exchange for the funing, the investor gains the right to take control over the project, which we refer to as the liquiation of the project. If the project is liquiate at ate t, the investor sells the project s assets an collects the liquiation value L t, which is given by L t y t = α T k=t+1 β k t E [Y k Y t = y t ]. 2 We assume that α < 1, i.e., the liquiation is ineffi cient, in the sense that the liquiation value is strictly below the expecte present value of the future cash flows. If the investor agrees to fun the project, at ate 0 the agent an the investor sign a contract that will govern their relationship until the final ate T. Accoring to the contract, the agent must report realizations of the cash flows to the investor. Of course, the reporte cash flow ŷ t can be ifferent from the true realization y t. Without loss of generality, we assume that the contract requires that the agent pays the reporte cash flows to the investor immeiately. The contract also specifies transfer payment t ŷ t 0 to the agent an the probability of liquiation p t ŷ t after any history of the agent s reports ŷ t = {ŷ 1,..., ŷ t }. The sequence of the events is illustrate in Figure 1. A contract is optimal if it maximizes the investor s continuation payoff, subject to a certain payoff for the agent. We will stuy contracts with full commitment. No renegotiation of the terms of the contract is allowe. 3 Preliminary Analysis of the Dynamic Contracting Problem We start our analysis of the contracting problem by showing that the optimal contract can be implemente using a irect revelation mechanism without savings. We then show that the optimal contract is incentive compatible if an only if there is no profitable one-perio eviation from the truth-telling strategy. 3.1 Optimality of a Truth-Telling Contract without Savings We start our analysis of the moel by showing that we can restrict our attention to the set of contracts in which the agent always tells the truth an oes not save. For any contract σ =, p, the agent chooses an optimal strategy ϕ = ŷ, C, S that specifies reporte cash flow ŷ t, consumption C t 0 an saving S t 0 as functions of y t. The agent s income i t = t + ζy t, ŷ t consists of two components: the transfer t from the investor an the ifference ζy t, ŷ t between the reporte cash flow ŷ t an the actual cash flow realization y t : 5

6 ζy t, ŷ t = λ y t ŷ t + y t ŷ t. One can use a revelation-principle type of logic to show that, for any contract σ =, p, there exists a contract σ =, p that results in the same payoff for the agent an equal or greater payoff for the investor, an for which truth-telling is the optimal strategy for the agent. Inee, suppose the reporting strategy ỹ is optimal uner the contract σ, then efine t ŷ t = t ỹ t ŷ t + ζŷ t, ỹ t ŷ t an p t ŷ t = p t ỹ t ŷ t for every history ŷ t of the agent s reports. One can see that if the agent tells the truth uner σ, then his income in each perio is equal to the income he receives uner σ when he employs strategy ỹ; an the investor s payoff uner σ is no less than uner the ol contract 3. In aition, the truth-telling is optimal uner σ, since ỹ is the optimal strategy uner σ. Savings are not necessary for the agent because the agent is risk-neutral. Since the agent fins it optimal to tell the truth uner the contract σ, he never uses his savings to misrepresent cash flows. Thus, savings translate into elaye consumption. Since the agent is risk-neutral, he receives no benefits from consumption smoothing. This leas to the following result: Proposition 1 There exists an optimal contract that inuces the agent to report cash flows truthfully an maintain zero savings. 3.2 Temporary Incentive Compatibility Constraints Given the result of Proposition 1, we will focus on irect-revelation contracts with no savings for the agent. To facilitate our analysis, we assume that the agent is not allowe to save, which implies that the agent cannot report y H when y L is realize. After fining an optimal contract without savings, we will verify that the contract remains incentive compatible when the agent is allowe to save. Let P t ŷt 1 = t 1 k=1 1 pt ŷk enote the probability that the project is active at the beginning of perio t after the history ŷ t 1 of reports, uner the contract σ =, p. If the agent reports ŷ t y t, his net income in perio t is equal to y t ŷ t + t ŷ t. Hence, the reporting strategy ŷ uner the contract σ results in the following expecte payoff for the agent: [ T a 0 y 0, ŷ, σ = E β t ŷt 1 P ŷt t λy t ŷ t + ] t Y 0 = y 0. 3 t=1 Note that the agent s payoff also epens on the initial state y 0. We say that a contract σ =, p is incentive compatible if it inuces the agent never to misreport the cash flows, i.e., [ T E β t P t y t 1 t y t ] [ T Y 0 = y 0 E β t ŷt 1 P ŷt t λy t ŷ t + ] t Y 0 = y 0, 4 t=1 t=1 for any feasible reporting strategies ŷ. Uner an incentive compatible contract, the agent s best strategy is to truthfully reveal the cash flows in every perio. We now consier a one-time eviation from the truth-telling strategy. Suppose Y t = y H. Given a history y t of the cash flows realizations, such that the project can be active in perio t, i.e., P t y t 1 > 0, the agent s continuation payoff uner the truth-telling strategy is given by 3 Since fraction 1 λ of the iverte cash flows is waste, the investor s income uner the contract σ can actually be higher than that uner the contract σ, since the agent never iverts cash flows uner the contract σ. 6

7 [ a t y t 1 T, y H = E β k t P k y t 1, y H, y k t 1 ] P t y t 1 t y t, y k t Y t = y H. k=t If the agent truthfully reveals the cash flows in each perio other than t, but cheats in perio t, the agent s continuation payoff woul be [ â t y t 1 T, y H = E β k t P k y t 1, y L, y k t 1 ] P t y t 1 t y t 1, y L, y k t Y t = y H. k=t We will call â t y t 1, y H the eviation continuation payoff after history y t 1, y H. The next theorem says that the contract is incentive compatible if an only if there is no one-time profitable eviation for the agent. Theorem 1 The contract σ =, p is incentive compatible if an only if for all t T an all histories y t 1, such that P k y t 1 > 0, a t y t 1, y H ât y t 1, y H + λyh y L. 5 Proof. See Appenix. Theorem 1 establishes that temporary incentive compatibility constraints 5 are a necessary an suffi cient conition for a contract to be incentive compatible. 3.3 Contracting Problem Definition 1 We say that an incentive compatible contract σ that implements payoffs a 0 y 0 σ an b 0 y 0 σ for the agent an the investor, respectively, is optimal at state y 0 if there is no other incentive compatible contract σ with the same payoff for the agent, but with a higher payoff for the investor: a 0 y 0 σ = a 0 y 0 σ an b 0 y 0 σ > b 0 y 0 σ. The investor s income in each perio is given by the ifference between the reporte cash flow ŷ t an the payment to the agent t, an the procees L t from the asset liquiation. Given the initial state y 0 an the continuation payoff a 0 for the agent, the investor s problem is to choose a contract σ =, p that maximizes the investor s payoff: [ b 0 y 0, a 0 = max E T 0 β t P t y t 1 Y t t y t + p t y t ] L t Y0 = y 0, 6,p t=1 subject to the incentive compatibility constraint 4 an the promise-keeping constraint [ T a 0 = E 0 β t P t y t 1 t y t ] Y 0, 7 t=1 where y t = {y 1,...y t } enotes the history of the cash flow realizations. The function b 0 y 0, a 0 represents the highest possible payoff attainable by the investor, given the payoff a 0 for the agent an the initial state y 0. Solving for the contracting problem 6, 4 an 7 means fining the optimal transfers t y t an the optimal probabilities of liquiation p t y t for each possible history y t at each time t. In orer to eal with the high imensionality of the problem, we have to rewrite the problem recursively. 3.4 Recursive Formulation of the Contracting Problem The setting of our moel is similar to DeMarzo an Fishman 2007a. However, unlike previous analyses, we allow the cash flows to be positively correlate over time. This is important not only because it is a more realistic assumption than inepenent cash flows, but also because the correlation 7

8 creates an aitional egree of informational asymmetry between the agent an the investor. With the correlate cash flows, the agent has a superior knowlege not only about the current cash flow realization that he observes irectly, but also about the future cash flows, since their istribution is etermine by the current cash flow. By misrepresenting the current cash flow, the agent also misrepresents the true quality of the project. The correlation assumption makes the contracting problem harer to solve. The vast majority of the literature on ynamic contracting assumes that privately observe variables are inepenent over time. This results in common knowlege of continuation payoffs at any point of time for a given contracts. With correlate cash flows, preferences over continuation contracts are no longer common knowlege. Therefore, stanar techniques of ynamic programming o not apply Recursive Formulation of the Contracting Problem with Inepenent Cash Flows Before approaching the contracting problem with the correlate cash flows, it may be useful to see how stanar techniques of ynamic programming can be applie in the case with inepenent cash flows, an why these techniques o not work with correlate cash flows. When the cash flows are inepenent over time, continuation payoffs for the agent an the investor associate with a contract o not epen on the current cash flow realization. This allows to rewrite the ynamic contracting problem recursively using the continuation payoff for the agent as a state variable. Let a t y L an a t y H enote payoffs for the agent uner the optimal continuation contract after the agent reports the low an the high cash flow, respectively, in perio t. The agent truthfully reveals the cash flow realization if a t y H a t y L + λy H. 8 Given incentive compatibility constraint 8, one can compute the optimal contract by backwar inuction, using the continuation payoff a t as the only state variable Recursive Formulation of the Contracting Problem with Markov Cash Flows The above approach is not applicable in the setting with correlate cash flows, since the incentive compatibility constraint 8 is no longer true. Figure 2 emonstrates how the incentive compatibility conitions are ifferent with the Markov cash flows. Suppose the expecte payoff for the agent at the beginning of perio t is a y t. In the i.i.. case, in orer to inuce the agent to reveal the cash flow realization truthfully, the agent s continuation payoff a t y H after he reports the high cash flow shoul be higher than a y t, an the agent s continuation payoff a t y L after he reports the low cash flow shoul be lower than a y t, so that IC constraint 8 hols. However, when the cash flows are positively correlate over time, a continuation contract that results in the continuation payoff a t y L for the agent in the low state will result in the agent s payoff â t y L, if the agent reports the low cash flow when the high cash flow is actually realize. Since the future cash flows are more likely to be high when the current state is high, â t y L > a t y L. Thus, the incentive compatibility constraint with the Markov cash flows shoul be written as a t y H â t y L + λy H. 9 While a t y L represents the agent s continuation payoff in equilibrium, â t y L is the agent s continuation payoff in eviation. Fernanes an Phelan 2000 use both the continuation payoff in equilibrium an the continuation payoff in eviation as state variables to formulate their contracting problem recursively. However, this metho requires computation of the continuation functions that epen on the two-state variables. Although their metho can be use in our setting for numerical simulations, the higher imensionality of their metho makes it virtually intractable analytically. Our approach to solving the contracting problem is to formulate it recursively using the agent s continuation payoff in equilibrium as a single-state variable. We will show that the incentive compati- 8

9 i.i.. cash flows a t y H Markov cash flows a t y H y a t λy H y L y a t λy H y L a t y L aˆ y t L a t y L Figure 2: Incentive compatibility with inepenent an Markov cash flows. bility constraints bin for the optimal contract. Therefore, it is possible to fin a relationship between the agent s continuation payoff in the low state a t y L an the agent s continuation payoff in eviation â t y L. Both a t y L an â t y L are base on the same continuation contract. The only ifference between these two payoffs is that the first payoff is obtaine using the istribution of the future cash flows conitional on the low state, while the secon payoff is obtaine using the istribution of the future cash flows conitional on the high state. Using backwar inuction, we can fin function ψ t that relates these two payoffs: â t y L = ψ t a t y L. We can write the incentive compatibility constraint with the Markov cash flows as follows: a t y H ψ t a t y L + λy H. 10 Equation 10 is easier to eal with than 9, since 10 epens only on the equilibrium payoff for the agent, while 9 epens on both equilibrium an eviation payoffs. 4 The Optimal Contract In this section, we erive the optimal contract by solving a sequence of optimization problems using ynamic programming techniques. In the next section, we verify using martingale techniques that the contract is inee the best possible contract. Our approach uses the two-state variables: the last cash flow an the agent s continuation payoff for a given contract. In each perio t < T, the following sequence of events takes place. First, the agent privately observes the realization of the cash flow Y t. If Y t = y H, the agent ecies whether to pay the cash flow to the investor or ivert it for his own consumption an report y L. If Y t = y L, the agent has no other choice than to report the cash flow y L. Given the agent s payment, the investor, in accorance with the contract, makes a ecision regaring the termination of the project. In the case of termination, the agent s payoff is zero an the investor s payoff is L t. If the project is not terminate, it continues 9

10 , y 1 bt yt, at ct at yl b y a y t t t b e y, a e t t t Perio t Agent privately observes cash flow Y t Agent pays ŷ t to Investor Investor pays yˆ t t to Agent Investor liquiate with probability p yˆ t t Figure 3: Continuation functions to operate in the next perio. For each perio, we consier the start-of-perio, intra-perio, an en-of-perio continuation functions b y t y t 1, a y t, b t y t, a t an b e t y t, a e t, i.e., the highest possible payoffs attainable by the investor in perio t given the agent s continuation payoffs a y t, a t an a e t at the start, mile an en of perio /t, as shown in Figure 3. We compute the continuation functions recursively. In particular, given continuation function b y t+1, we can obtain continuation function b e t, taking into account iscounting between perios. In orer to calculate continuation function b t, we calculate the optimal liquiation policy an the optimal transfer in perio t. Along the way, we calculate the agent s eviation payoff ψ t a t y L as a function of the agent equilibrium payoff a t y L in the low state. Function b y t is obtaine from b t by optimizing the investor s payoff at the beginning of perio t, subject to the incentive-compatibility constraint an the promise-keeping constraint. We start computing the continuation functions from the final perio T. At the en of the final perio, the project is liquiate an no cash flows will be generate in the future. However, a transfer from the investor to the agent is allowe. Due to the limite liability of the agent, this transfer must be non-negative. Hence, the continuation function at the en of the last perio is given by b T yt, a T = a T for a T 0, where the agent s continuation payoff a T is equal to the final payment from the investor, i.e., T = a T. 4.1 Parametric Representation of Continuation Payoffs For future analysis, it is convenient to measure the agent s continuation payoffs in terms of the values of the cash flows that the agent can steal uring a certain time interval. For continuous time τ R, let n τ be the biggest integer, such that n τ τ, an let l τ = τ n τ. We will use τ to enote a time interval of n τ perios an the fraction l τ of the next perio. The value of the cash flows generate by the project uring time τ is given by [ nτ ] V τ y E β k Y t+k + l τ β nτ +1 Y t+nτ +1 Y t = y. 11 k=1 10

11 Here, we assume that if the project operates a fraction l τ of a perio, then only the fraction l τ of the cash flow in that perio is counte. Alternatively, we can say that l τ represents the probability that the project operates in the perio n τ + 1. Function V τ has a number of goo properties. Lemma 1 Function V τ Y t is continuous, strictly increasing in τ an piecewise linear, with the righthan-sie erivative V τ y = β nτ +1 E [Y t+nτ +1 Y t = y]. 12 τ Moreover, V τ+1 Y t 1 = E [β Y t + V τ Y t Y t 1 ]. 13 Proof. See Appenix. Equation 13 is self-evient. Its left-han sie is the expecte present value of the cash flows generate uring time τ + 1, while the right-han sie represents the same value as a sum of the next-perio cash flow Y t an the value of the cash flows generate uring time τ. The erivative 12 is obtaine from 11, using the fact that l τ = τ n τ. We will use V τ to represent continuation payoffs for the agent. For example, if Y t = y L an the agent s continuation payoff at the en of perio t is a > 0, then there exists a unique τ > 0 such that a = λv τ y L. In other wors, given state y L, the agent s continuation payoff of a is equivalent to the expecte value of the cash flows that can be stolen uring time τ. Two values of the agent s continuation payoffs have special meaning: V 1 y t is the expecte present value of the next perio cash flow, while V T t y t is the expecte present value of the cash flows that the project generates if it operates until the final ate T without liquiation. The agent s continuation payoffs that are less than λv 1 y t cannot be implemente without liquiating the project with positive probability at the en of the current perio, since the agent can steal the next perio cash flow an guarantee himself expecte payoff of λv 1 y t. On the other han, the first best is achievable when the agent s continuation payoff is greater than or equal to λv T t y t. The investor can pay to the agent the ifference between the agent s continuation payoff an λv T t y t immeiately an let the agent consume the fraction λ of all subsequent cash flows. When the agent continuation payoff is below λv T t y t, the threat of liquiation must be real. Otherwise, the agent can steal all the subsequent cash flows an get the payoff λv T t y t. In this case, the continuation function at the en of perio t is given by b e t y t, a e t = T s=t+1 β s t E [Y s Y t = y t ] a e t for a e t λv T t y t Nonoptimality of Early Transfers to the Agent It is easy to see that the transfers from the investor to the agent shoul be zero until the first-best is reache, i.e., when the agent s continuation payoff a t reaches λv T t y t. Since both the agent an the investor use the same iscount factor, it is always possible to efer the agent s compensation at no cost for the investor. Inee, paying t in perio t to the agent is equivalent to paying t /β ζ t at the time ζ T when the project is liquiate. As long as the probabilities of liquiation are unchange, this will not affect the agent s an the investor s payoffs. But, since the liquiation is ineffi cient, it may be possible to improve the original contract by operating the project after time ζ using continuation value of k t /β ζ t to provie incentives for the agent. On the other han, there is no reason to elay payments to the agent when a t excees λv T t y t. Thus, we have the following result. Proposition 2 Optimal transfers from the investor to the agent are given by t = max { 0, a t λv T t Y t }. 11

12 4.3 Derivation of the Optimal Contract We now escribe the algorithm that we use to erive the optimal contract σ. This algorithm consists of three steps for each perio Step One: Liquiation Problem Consier the problem the investor faces after the cash flow announcement but before the liquiation ecision. If the project is terminate in perio t, the investor s payoff is L t, while the agent gets nothing. Given the continuation function b e t y t, an the agent s continuation payoff a t, the optimal probability of liquiation p t solves b t yt, a t = max 1 p t b e t y t, a e t + p t L t, 15 p t, a e t s.t. a t = 1 p t a e t, 16 a e t λv 1 y t, 17 p t [0, 1], 18 where equation 16 ensures that the agent s continuation payoff a t before the liquiation ecision is consistent with the continuation payoff a e t after the liquiation ecision. Constraint 17 reflects the fact that, once the project is allowe to continue into the next perio, the agent s en-of-perio payoff must be at least as high as the expecte present value of the cash flow a L y t λv 1 y t that the agent is capable of stealing in the next perio. This also implies that if the agent s intra-perio continuation payoff a t is below a L t y t, the investor must liquiate the project in perio t with positive probability. The next proposition states that the project is terminate with positive probability if an only if a t Y t rops below a L Y t. Proposition 3 The transfer to the agent an the probability of the termination in perio t are given by t = max { 0, a t λv T t Y t }, 19 { } p t = max 0, al Y t a t a L Y t. 20 The intra-perio continuation function b t Y t, a t is obtaine from the en-of-perio continuation function b e t Y t, as follows: b t Y t, a t = 1 p t b e t Y t, a e t + p t L t t for a t 0, 21 where the agent s continuation payoff a e t at the en of perio t, provie the project is not terminate, evolves accoring to: a e t = a t t /1 p t. The proof of Propositions 3-6 is in the Appenix. We combine the proof of Proposition 3 with the proofs of the other propositions in this section since by using a backwar inuction argument. The intuition behin this result is simple. The termination of the project is ineffi cient. Therefore, the investor fins it optimal to refrain from early liquiation, unless she has exhauste all other means of proviing proper incentives for the agent. If the investor has to resort to liquiation, the probability of liquiation has to be the lowest possible neee to implement the agent s continuation payoff. This probability is proportional to how much the agent s continuation payoff a t is below the minimal implementable payoff a L Y t. If a t < a L Y t an the project was not liquiate in perio t, the agent s continuation payoff at the en of perio t is increase to a L Y t. Figure 4 shows the function b e t soli line, an the function b t, which is obtaine from b e t by extening it over the liquiation region [ 0, a L] ashe line. 12

13 Investor s Payoff b b t L t e b t 0 L a Agent s Payoff a Figure 4: Continuation functions b t an b e t Step Two: Intra-Perio Agency Problem After observing the realization of Y t, the agent must report the cash flow to the investor. Let a t y t enote the continuation payoff for the agent uner the optimal contract if the agent truthfully announces the realization of y t. An important question is what woul the agent s continuation payoff be if he reporte y L instea of y H. The contract specifies future payments to the agent as a function of the cash flow reports, not actual cash flow realizations. However, Y t = y H etermines the istribution of future cash flows. As a result, the continuation payoff, as we will see, woul be higher than a t y L. For the contract σ, let ψ t a t y L enote the continuation payoff for the agent if he reports y L instea of y H in perio t. It can be expresse as a function of a t y L, since a t y L fully etermines the future terms of the optimal contract after y L was reporte. We will refer to ψ t as the eviation payoff function. The start-of-perio continuation function b y t is the solution of the following problem: b y t Y t 1, a y [ t = max E t Yt + b a t t Yt, a t Y t ] Y t 1, 22 s.t. IC a t y H ψ t a t y L + λy H, 23 PK a y t = E t [ a t Y t Y t 1 ], 24 IR a t The intra-perio continuation payoffs for the agent a t y L an a t y H must satisfy the incentive compatibility constraint IC, the promise keeping-constraint PK an the iniviual rationality constraint IR. The first constraint insures that the agent has no incentive to steal the high cash flow. The secon one says that the payoff a y t promise to the agent at the start of perio t is equal to the expecte intra-perio continuation payoff. Since the agent has limite liability, his continuation payoff cannot be negative. Proposition 4 In the optimal contract, the incentive compatibility constraint bins: a t y H = ψ t a t y L + λy H

14 Moreover, if the agent s continuation payoff a t y L in state y L is equal to λv τ y L, then the agent s continuation payoff in eviation is given by ψ t λv τ y L = λv τ y H for 0 τ T t, 27 ψ t a t y L = a t y L + λ V T t y H V T t y L for a > λv T t y L. 28 Proof. See Appenix. 4 To see why IC shoul bin, consier how the liquiation probability woul be ifferent when IC oes not bin. The liquiation is likely to occur sooner after the low cash flow realization than after the high cash flow realization, since the payoff a t y L is closer to the liquiation region than a t y H. Relaxing IC makes a t y L lower an a t y H higher. Thus, relaxing IC leas to higher chances of liquiation when they are alreay high, an lower chances of liquiation when they are low. As a result, the expecte loss associate with the liquiation increases. The secon part of Proposition 4 says that if the agent s continuation payoff in the low state is equal to the expecte present value of the cash flows that the agent can steal uring next τ perios, then the agent s continuation payoff in eviation is equal to the expecte present value of the cash flows that the agent can steal uring next τ perios conitional on the high state in the current perio. Because IC bins, the agent is inifferent between stealing cash flows an reporting them truthfully. This allows us to calculate function ψ t by representing the agent s continuation payoffs in the form of λv τ, i.e., for any a t λv T t y t, there is a unique parameter τ T t such that a t = λv τ y t. The proof is base on the observation that if the cash flow is y L, then the agent s continuation payoff in perio t is given by a t y L = E [ β 1 p t a t+1 Y t+1 Y t = y L ], 29 where a t+1 Y t+1 is the continuation payoff in perio t + 1 conitional on the cash flow Y t+1. On the other han, if the agent reports y L instea of y H, then the continuation payoff in the eviation is equal to ψ t a t y L = E [ β 1 p t a t+1 Y t+1 Y t = y H ]. 30 Given an the fact that IC bins, one can calculate function ψ t using backwar inuction. Proposition 4 allows us to calculate the evolution of the agent s continuation payoffs. Since the continuation payoffs a t y L an a t y H satisfy PK an IC with equality, we can solve equations PK an IC for a t y L an a t y H as functions of a y t. This yiels the following: Proposition 5, If the agent s start-of-perio continuation payoff is a y t = 1 β λv τ+1 Y t 1, s.t. 0 τ T t, uner the contract σ, then the agent s intra-perio continuation payoff is a t Y t = λv τ Y t + λy t. 31 The start-of-perio continuation function is given by Y t 1, 1 β λv [ τ+1 Y t 1 = E t Yt + b ] t Y t, λv τ Y t + λy t Y t 1. b y t Step Three: Discounting between Perios To complete our recursive characterization of the optimal contract, we erive the continuation function b e t 1 from b y t, taking into account iscounting between perios. If the contract results in the payoff of a e t 1 at the en of perio t 1, then the agent s payoff at the beginning of perio t must be a e t 1/β. This yiels the following: 4 The proof is interconnecte with the proofs of the other propositions in this section. 14

15 p t >0, t =0 p t =0, t =0 p t =0, t >0 Liquiation Region Continuation Region Divien Region 0 λv1 yt t a = λv Y τ t λv λ a = λv Y + Y y t+ 1 τ 1 t+ 1 t+ 1 L T t y t Agent s Payoff a Figure 5: Three regions of the agent s continuation payoffs Proposition 6 Given the start-of-perio continuation function b y t, the continuation function at the en of perio t 1 is given by b e t 1 yt 1, a e t 1 = βb y t y t 1, ae t 1. β Summary of the Algorithm Propositions 3-6 yiel the algorithm for computing the functions b e t, b t, b y t an ψ t starting from the en of the last perio T. While there is no close form solution for the functions b e t, b t an b y t, the evolution of the agent s continuation payoff can be escribe as follows: a e t 1 = λv τ Y t 1 32 a y t = λv τ Y t 1 /β 33 a t Y t = λv τ 1 Y t + λy t 34 a e t = min λv T t Y t, max a t Y t, a L Y t. 35 The agent s continuation payoff is ajuste by the iscount factor β between the en of perio t 1 to the beginning of perio t. Comparing to a y t, the agent s continuation payoff a t Y t is going to increase if Y t = y H an ecrease if Y t = y L. Transition from a t to a e t is characterize by the liquiation probability p t when a t < a L Y t an the payment a t λv T t Y t to the agent when a t > λv T t Y t. The evolution of agent s continuation payoffs can be characterize by three regions, as shown in Figure 5. In the liquiation region [0, a L y t, the transfer from the investor to the agent is always zero an the probability of liquiation is proportional to how much the agent s continuation payoff is below the liquiation bounary: p t a t y t a L y t a t y t = max a L, y t In the continuation region, there is no liquiation, but the transfer is also zero. If the high low cash flow is reporte, the continuation payoff increases ecreases. Cash flow Y t affects the agent s continuation payoff in two ways. First, the current cash flow Y t irectly affects the continuation payoff 15

16 through the term λy t. Secon, the cash flow Y t affects the agent s continuation payoff through its impact on the istribution of the future cash flows, which is reflecte in the term λv τ 1 Y t. The first-best is implemente in the ivien region. When a t y t excees λv T t y t, the investor pays t = a t y t λv T t y t to the agent. The payments to the agent in the subsequent perios are given by s = λy s, for s > t. The parameter τ that we use to express the agent s continuation payoffs can be interprete as the earliest time to efault. Accoring to 32-35, the agent s continuation payoff goes own from V τ Y t 1 to V τ 1 Y t each time the agent reports the low cash flow. If the agent reports the low cash flow n τ +1 times in a row, the liquiation will either occur in n τ perios with probability 1 l τ, or in n τ +1 perios for sure. The expecte time to efault in this case is 1 l τ n τ +l τ n τ + 1 = n τ +l τ = τ. 4.4 Justification of the Optimal Contract We have erive the optimal contract by solving a sequence of optimization problems working backwar from the last perio. This approach assumes that the optimal contract maximizes the investor s continuation payoff after every possible history of cash flows. This is the case when the cash flows are inepenent, as in DeMarzo an Fishman 2007a. Inee, the investor s expecte payoff at time zero can be increase by replacing a suboptimal continuation contract with an optimal one with the same continuation payoff for the agent. The situation is more complicate when cash flows are correlate, because replacing one continuation contract with another woul affect the incentive compatibility constraints in earlier perios. Nonetheless, using properties of the continuation functions b, we can verify the optimality of the erive contract. First, we show that ynamics of the agent continuation payoffs for any contract can be characterize by their sensitivity to the cash flow reports. Lemma 2 For any incentive compatible contract, p, there exists sensitivity µ t of the agent s continuation payoff towars his report, measurable with respect to the cash flow history y t, such that a t+1 = a t t β1 p t + µ ty t+1 QY t, for p t < Proof. After a history of cash flows such that P t > 0, for any contract the agent s continuation value, by efinition, is [ a t = 1 T ] E β k t P k k Y t. P t Taking into account the fact that P t+1 = 1 p t P t, it can be can be rewritten as follows k=t a t = t + β1 p t 1 QY t a t+1y L + QY t a t+1y H, where a t+1y L an a t+1y H represent the continuation values for the same contract given the low an high cash flow in perio t + 1. Define µ t = a t+1y H a t+1y L. Then, which implies a t = t + β1 p t a t+1y L + µ t QY t, β1 p t a t+1y L = a t t β1 p t µ t QY t, 38 β1 p t a t+1y H = a t t + β1 p t µ t 1 QY t. 39 Since y L = 0, an y H = 1, equations 38 an39 are equivalent to 37. Sensitivity µ t has a very intuitive interpretation. Since y H = 1 an y L = 0, µ t is equal to the ifference between the agent s continuation payoffs conitional on the high an low cash flow reports. 16

17 Since a t = t + 1 p t a e t, we can rewrite 37 as follows. a t+1 = ae t β + µ ty t+1 QY t. One can see that the first term represents the promise-keeping conition, while the secon term is necessary to provie incentives for the agent. Lemma 2 allows us to verify that no alternative incentive compatible contract can attain payoff for the investor greater than b 0 y0, a 0, i.e., Proposition 7 σ is the optimal contract. Proof. Define process G t = t β k Y k k + β t b e t Y t, a e t, k=0 where k an p k are transfers to the agent an the probability of liquiations specifie by some contract σ, an b e t Y t, a e t is the continuation function for contract σ. An increment of this process is given by G t+1 G t = β t β Y t+1 t+1 + b e t+1y t+1, a e t+1 b e t Y t, a e t, while the agent s continuation value uner contract σ is a e t+1 = a t+1 t+1 /1 p t+1, an the ynamics of a t, accoring to Lemma 2, is given by a t+1 = ae t β + µ ty t+1 QY t. The necessary conition for a contract to be incentive compatible is given by µ t λ 1 + V τ 1 y H V τ 1 y L, for a t = V τ y t. 40 The conitional expecte value of G t+1 G t can be written as follows E t [G t+1 G t ] = β t E t [ β Yt+1 t p t+1 b e t+1y t+1, a e t+1 + p t+1 L t+1 b e t Y t, a e t ]. Since b e t is obtaine by solving problems an 22-25, E t [G t+1 G t ] = 0, i.e. G t is a martingale, if t, p t are the same as in Proposition 3 an equation 40 hols with equality as in Proposition 4. On the other han, if t, p t, an µ t o not solve an 22-25, then E t [G t+1 G t ] 0, i.e. G t is a supermartingale. Let τ T be the time of project liquiation. For any contract, when the project is liquiate, a e τ = 0 an the investor s payoff is equal to L τ = be τ y τ, ae τ. Thus, the expecte investor s payoff uner σ is equal to E ] β k Y k k + β τ L τ = E [G τ ] G 0 = b e 0Y 0, a e 0. [ τ k=0 This proves that contract σ elivers the highest possible payoff for the investor. Below, we iscuss intuition behin Proposition 7. A creible threat of liquiation is necessary to provie incentives for the agent to share cash flows with the investor. However, liquiation is ineffi cient. Thus, the optimal contract shoul minimize the expecte cost of liquiation, while remaining incentive compatible. The contract σ oes it because of the following two properties. First, the project is liquiate only when the agent s continuation payoff rops below λv 1 y L, the lowest payoff value that can be implemente without liquiation. Secon, because the incentive compatibility constraints always bin uner the contract σ, it minimizes the volatility of the agent s continuation value a, an as a result chances that a becomes less than λv 1 y L. 17

18 4.5 Initiating the Contract The contract is initiate at time zero. The initial state Y 0 of the Markov cash flows an the continuation payoff a 0 Y 0 for the agent uniquely etermine the optimal contract between the agent an the investor. Thus, when Y 0 is commonly known, initiating the contract means choosing the payoffs a 0 Y 0 an b e 0 y L, a 0 y L for the agent an for the investor, respectively. The situation is not much ifferent when the initial state Y 0 is unknown but its istribution is common knowlege since the contract σ remains optimal in both states. If σ implements payoffs a 0 y L, b e 0 y L, a 0 y L in the low sate an a 0 y H = ψ 0 a 0 y L, b e 0 y H, a 0 y H in the high state an PrY 0 = y H = q 0 H, then σ implements payoffs a 0 = q 0 H a 0 y H + 1 q 0 H a0 y L, b e 0 a 0 = q 0 H be 0 y H, a e 0 y H + 1 q 0 H b e 0 y L, a e 0 y L. The contract sets initial continuation payoffs for the agent an the investor. Which pair of payoffs is chosen epens on the competitive environment. If the agent with initial wealth W is a monopolist facing a competitive row of investors, he chooses the initial continuation payoff a A 0, so that the investor breaks even: a A 0 = sup {a : b 0 a I W }. A higher continuation payoff for the agent leas to a lower probability of liquiation in the future. Thus, investing all the wealth results in the highest surplus of the project, which benefits the monopolistic agent. Proposition 8 In the setting with a monopolistic agent an competitive investors, it is optimal for the agent to invest all his wealth in the project at ate 0. Proof. See Appenix. If the investor has all the market power, she chooses to maximize her payoff: a I 0 = arg max b 0 a. a In general, the initial payoff a 0 for the agent can be anything between a I 0 an a A 0, epening on the market power of the agent an the investor, with a 0 increasing with the agent s market power an ecreasing with the investor s market power. 5 The Implementation of the Optimal Contract So far we have characterize the optimal contract in terms of the transfers between the agent an the investors an the probabilities of liquiation of the project. In this section, we show that the optimal contract can be implemente using the combination of a creit line with performance pricing an equity. We efine these securities as follows: Creit Line with Performance Pricing. A creit line is characterize by a creit limit Ct L an an interest rate rt C M t 1 charge on the balance M t 1. The interest rate can be a function of the creit line balance. The interest payment on the outstaning balance is ue at the en of each perio. The creit limit Ct L etermines the maximum amount of creit available for the agent in perio t. Inability to make the current interest payment without exceeing the creit line limit leas to efault. In aition, the creit line can have an initiation fee an restrictions on the ivien policy, which we will iscuss later. Equity. The agent is allowe to use cash flows to pay iviens to the equity holers in proportion to their share of ownership. Default occurs when the agent is unable to fulfill his financial obligations. Default leas to a probabilistic liquiation. Given an unmae payment z t, the firm is liquiate with probability p t z t, while with probability 1 p t z t the investor forgives the unmae payment z t an lets the agent 18

19 operate the firm in the next perio. 5 In the even of liquiation, the investor sells the firm s assets an pockets the liquiation value L t, while the agent gets nothing. Theorem 2 The optimal contract can be implemente by a combination of equity an a creit line with an escalating interest rate. The agent hols fraction λ of the equity, while the rest is hel by the investor. The creit line has the creit limit an the interest rate charge on the balance C L t = V T t y H V 1 y H for t < T, 41 rt C M t 1 τ = V T t y H V τ 1 y H 1, 42 V T t+1 y H V τ y H M t 1 τ = V T t y H V τ y H, 43 where τ is the corresponing earliest efault time. The agent can raw on the creit line to make interest payments. However, he is not allowe to borrow from the creit line to pay iviens. In the event of efault, the unmae payment z t results in the probability of liquiation p t z t = z t V 1 y H. 44 With probability 1 p t z t, the project is not liquiate an the unmae payment z t is forgiven. Proof. See Appenix. The epenence of the interest rate rt C on the balance M t 1 is expresse using the earliest efault time τ. Equations 43 an 42 shoul be rea as follows: First, for a given balance M t 1, we fin τ that solves equation 43. Then, we substitute τ into equation 42, which gives us the interest rate rt C M t 1 charge on the balance. In the propose implementation, the role of equity is straightforwar. Diviens pai to the equity holers represent the rewar to the agent for repaying the creit line ebt. Given his stake λ in the firm s equity, the agent is inifferent between stealing the firm s cash flows an paying iviens. The role of the creit line with the escalating interest rate is more sophisticate. The balance on the creit line can be consiere as a memory evice that summarizes all the relevant information regaring the past cash flow realizations. The interest rate along with the creit limit etermines the ynamics of the creit line balance an the timing of efault. The threat of losing control over the project inuces the agent to pay the creit line ebt. To prove Theorem 2, we show that the evolution of the balance on the creit line reflects the evolution of the agent s continuation payoffs inuce by the contract σ. Specifically, the parameters of the creit line are chosen so that the available creit is proportional to the agent s continuation payoff in the high state uner the optimal contract σ, minus the liquiation bounary λv 1 y H : a t y H λv 1 y H = λ C L t M t. 45 As Figure 6 illustrates, zero balance on the creit line correspons to the ivien threshol λv T t y H, while the balance equal to the creit limit correspons to the liquiation threshol λv 1 y H. An increase in the balance leas to a lower continuation payoff. Default occurs when the balance excees the creit limit. The agent uses cash flows Y t to pay the creit line. Given the outstaning balance M t 1 in perio t 1, the new balance becomes M t = 1 + r C t M t 1 M t 1 Y t. 5 Although our efinition of efault is non-stanar, it is consistent with the fact that creitors are often willing to write off a part of the ebt instea of forcing bankruptcy. We can interpret the probabilistic liquiation as an uncertainty associate with the efault proceure, which we o not moel here. 19

20 Liquiation Region Continuation Region Divien Region 0 V y 1 H V T t y H Agent s Cont. Payoff Default Region Positive Balance Zero Balance Draw on Creit Line Creit Limit L C 0 Figure 6: Agent s continuation payoff an creit line balance When a cash flow is low, the agent has to raw on the creit line to make the interest payment. On the other han, the agent uses the high cash flow to pay own the balance. The interest rate rt C M t 1 is chosen so that the evolution of the balance M t is consistent with the evolution of the agent s continuation payoffs in the high state uner the contract σ. Since the balance on the creit line tracks the agent s continuation payoff in the high state, the agent has no incentive to ivert cash flows. In the low state, there is no cash flow to ivert. However, the creit limit in the low state is too generous compare to the agent s continuation payoff. If allowe, the agent woul raw the creit line up to the limit, use all the borrowe cash to issue iviens an eclare bankruptcy afterwars. To avoi this scenario, the creit line has the covenant that prohibits paying iviens until the creit line is pai off completely. An important property of the propose implementation of the optimal contract is that it oes not epen on the way the contract is initiate, i.e., the creit limit, the interest rate an the equity share given to the agent o not epen on the amount of money the agent borrows at time zero. The time-zero borrowing is reflecte only in the initial raw on the creit line. 5.1 The Optimal Interest Rate Structure It is more convenient to examine the interest rate structure in the stationary setting with T. Let [ ] V y E β k Y k Y 0 = y. k=1 We can omit the time inex an simplify equations 43 an 42: M τ = V y H V τ y H, 46 r C M τ = V τ y H V τ 1 y H M τ. 47 We start our analysis of the optimal interest rate structure with two benchmark cases: inepenent cash flows an perfectly correlate cash flows. 20