Optimal resolution of financial distress- a dynamic contracting approach

Optimal Resolution of Financial Distress: A Dynamic Contracting Approach VO Thi Quynh Anh October 2009 Abstract This paper provides a formal analysis of the choice between liquidation and recapitalization when …rms get into …nancial di¢ culties We introduce the possibility of a costly recapitalization into a model of dynamic …nancial contracting under moral hazard We …nd that the …rm is never recapitalized nor liquidated in a period in which the high cash ‡ow is realized These two options are resorted to after a poor performance when the entrepreneur’s stake in the future cash ‡ows is low enough There exists two possible procedures to cope with …nancial di¢ culties When recapitalization cost is relatively high, the …rm should be liquidated and no recapitalization is employed By contrast, when recapitalization cost is low, in …nancial distress situation, the …rm would be recapitalized up to the extent that liquidation risk is eliminated Key words: Dynamic Financial Contracting, Moral Hazard, Recapitalization, Liquidation JEL Codes: D82, G32, G33 Introduction In general, a corporation in …nancial distress may take two routes The …rst one cor- responds to the liquidation in which the assets of the …rm are sold either piecemeal or as a going concern The proceeds from this sale are then divided among all claimants in accordance with their priority rights Alternatively, the …rm may embark on a reorganization process whose purpose is to …nd a method to overcome the trouble Typically, this process involves a negotiation between debtors and creditors with a view to establishing a new mechanism for the settlement of claims: writing oÔ some of the claims, exchanging bonds and other debts with new notes, bonds, swapping new equities for old ones, injection of new capital This paper aims at providing a formal analysis on the choice between liquidation and recapitalization - one of possible methods involved in the reorganization process - when …rms get into …nancial di¢ culties Norges Bank (Central Bank of Norway), Bankplassen 2, 0107 Oslo, Norway Email: thi-quynhanh.vo@norges-bank.no We take a dynamic …nancial contracting approach to this issue The dynamic agency model is …rst introduced by Green (1987) and Spear and Srivastava (1987) Built on the recursive method developed in these works, recently, a number of papers study the optimal long-term …nancial contract in a setting in which a risk neutral entrepreneur seeks funding from risk neutral investors to …nance a project that pays stochastic cash‡ows over many periods Their contracting relationship is subject to a moral hazard problem that comes either from the unobservability of cashows or from hidden eÔorts All of those papers assume that the entrepreneur is liable for payments to the investors only to the extent of current revenues Consequently, when facing a business failure, the …rm possesses in hand only one option, that is liquidation In this paper, to be able to simultaneously analyze the liquidation and the recapitalization decisions, we introduce into a dynamic …nancial contracting model the possibility of costly recapitalization Speci…cally, we consider a scenario where a risk neutral entrepreneur contracts with risk neutral investors to …nance a multi-period investment project This project, once funded, generates at each period of its life an observable binary cash‡ows whose distribution depends on the unobservable eÔort of the entrepreneur For simplicity, we assume that the set of feasible eÔort levels contains two elements, high or low eÔort The distribution of cashows under high eÔort dominates the distribution under low eÔort in the sense of …rst order stochastic dominance Nevertheless, exerting high eÔort is costly since it deprives the entrepreneur of a private bene…t B The novelty of our paper lines in relaxing the limited liability constraint usually imposed by previous analysis in the literature We assume, in this paper, that payments to the entrepreneur at each period can be negative Negative transfers mean a new capital injection by the entrepreneur into the …rm and so, are interpreted as recapitalization In our setup, the recapitalization is costly and voluntary The entrepreneur bears a positive cost for each additional unit of capital injected Moreover, the maximum amount the entrepreneur is willing to inject is determined by the expected discounted utility the entrepreneur can capture if the project continues in operation Hence, in our model, in front of …nancial di¢ culties, the …rm can choose either to be recapitalized or to be liquidated The properties of an optimal …nancial distress procedure are highlighted through the characterization of the optimal contract between the entrepreneur and the investors In line with numerous contributions of the literature on repeated moral hazard, to …nd the optimal contract, we rely on the dynamic programming technique We use the expected discounted utility of the entrepreneur at the start of each period as the single state variable We are able to fully characterize the optimal contract in an in…nite horizon setting We …nd that the …rm is never recapitalized nor liquidated following a good performance These two options are resorted to after a poor performance when the value of the entrepreneur’s claim to future cash‡ows is low enough In our model, there exists two possible procedures to cope with …nancial di¢ culties When recapitalization cost is relatively high, the …rm should be liquidated and no recapitalization is employed By contrast, when recapitalization cost is low, in …nancial distress situation, the …rm would be recapitalized up to the extent that liquidation risk is eliminated We also show that our optimal contract is robust to the renegotiation possibility This paper links to the literature on dynamic …nancial contracting under moral hazard whose recent contributions include DeMarzo and Fishman (2007a; 2007b); Biais, Mariotti, Plantin and Rochet (2004; 2006) (hereafter BMPR (2004, 2006)); Clementi and Hopenhayn (2006) These papers are interested in examining how …nancial contracts can be designed to solve the asymmetric information problem between contracting parties which repeatedly interact DeMarzo and Fishman (2007a) look at the optimal long-term …nancial contract between a risk neutral agent and risk neutral investors in the presence of unobservable cash‡ows problem They …rst characterize the optimal contract and then, show that the optimal mechanism can be implemented by a combination of commonly observed securities such as equity, long-term debt and a line of credit In their implementation, the long-term debt is described by a …xed charge paid at each period whereas the credit line is characterized by an interest rate and a credit limit If a required debt or credit line payment is not made, the …rm is in default, in which case it is liquidated with some probability As for BMPR (2006), their optimal contract analysis can be seen as a stationary version of DeMarzo and Fishman (2007a) but their implementation of this contract is realized via debt, equity and cash reserves When …rm can not serve its debt, probabilistic liquidation occurs Moreover, by considering the continuous time limit of discrete time model, BMPR (2006) obtain a rich set of asset pricing implications Also based on a model of multi-period borrowing/lending relationship with asymetric information, Clementi and Hopenhayn (2006) and DeMarzo and Fishman (2007b) aim at explaining some of the facts regarding …rm’s investment decisions, growth and survival rates In their model, at each period, the …rm’s size can be altered by some investment or disinvestment into the project They treat investments as observable Clementi and Hopenhayn (2006)’s paper is restricted to the moral hazard problem of unobservable cash‡ows In agreement with the empirical evidence, they …nd that investment is sensitive to innovations in the cash‡ow process; this sensitivity is decreasing with size; survival rates increase with …rm size and …rm age and size are positively correlated DeMarzo and Fishman (2007b) provide a more general model of moral hazard and so, can obtain the conclusion that the patterns, concerning …rm’s growth and investments, observed in the data arise from the genenral nature of optimal contractual arrangements in the presence of agency problems, not from the particular structure of these problems Our model is built on BMPR (2004) However, diÔerently with all of the above papers, we take into account the recapitalization as an alternative of liquidation for the …rm in case of …nancial di¢ culties Due to that, we are able to have an insightful analysis about the design of the optimal …nancial distress procedure The paper is organized as follows In the next section, we present the structure of the model and the dynamic programming formulation Section is devoted to the characterization of the optimal contract in in…nite horizon setting Section discusses the robustness of our optimal contract with respect to the renegotiation prossibility Finally, section concludes All proofs will be provided in the appendix Model A Environment Our setup closely follows the model presented in BMPR (2004) There are two types of agents: an entrepreneur and investors All are risk neutral and discount the future at the same rate r The entrepreneur is endowed with a multi-period investment project that requires a start-up capital of I The project, if undertaken, generates at each period t = 1; 2; ::::T an observable cash‡ow Rt which can take two values: high cash‡ow: Rt = R+ or low cash‡ow: Rt = R The distribution of cashows depends on the eÔort et the entrepreneur exerts to run her project The choice of eÔort is unobservable to the investors For simplicity, we assume that the entrepreneur can choose between two eÔort levels: et = (high eÔort) or et = (low eÔort) If she chooses high eÔort, the probability of getting a high cash‡ow equals p > p p which is the probability of high cashow in case of low eÔort However, if the entrepreneur decides not to exert high eÔort, she gets some private bene…ts B Cash‡ows are assumed to be independently and identically distributed across periods Since her initial wealth A is less than I, the entrepreneur must borrow to …nance her project The …nancing contract between her and the investors speci…es several decisions to be made at each period t: First, the continuation or not of the project Let xt be the probability with which the project is terminated by the investors We assume that in the event of termination, the project’s assets are liquidated at a zero price The project can also be ended if the entrepreneur decides to quit to take her outside options We normalize the reservation utility of the entrepreneur to zero Second, the payment ct to the entrepreneur conditional on the project being continued We here deviate from BMPR (2004) as well as all other papers in the literature by not imposing the condition that ct must be non negative Instead, we assume that at each period t prior to the …nal period T (i.e t < T ), after the cash‡ow is realized, the investors can require the entrepreneur to inject some capital into the …rm In that case, ct (t < T ) will take negative values This capital injection by the entrepreneur constitutes, in a dynamic setting, another punishment device available to the investors aside from the liquidation threat We interpret it as recapitalization This recapitalization is costly to the entrepreneur We assume that the cost she has to bear for each unit of capital injected equals 01 Hence, the utility function of the entrepreneur takes the following form U (ct ) = > < ct if > : (1 + ) c if t ct (1) ct < In addition, there exists a limit for the amount the entrepreneur is willing to inject It is determined by the expected discounted utility the entrepreneur can capture if the project In fact, the recapitalization costs contain some elements that are …xed and others which are proportional The …xed elements can be viewed as …xed transaction fees The proportional costs may come from proportional fees but also from dilution eÔects as in Myers and Majluf (1984) In this paper, for simplicity, we only take into consideration the proportional costs continues in operation Given a contract fxt ; ct gTt=1 , let wt denote the expected discounted utility the entrepreneur derives from all payments paid to her from period t on if she exerts high eÔort at every period Thus 6X T wt = E 6 s=t s Y xu u=t ! U (cs ) (1 + r)s t 7 Ft 7 (2) The lower bound for ct (t < T ) is then determined endogenously by the following condition: U (ct ) + wt+1 1+r for all t < T (3) This condition can also be seen as limited liability constraint It is obviously weaker than the limited liability constraint ct 0; wt+1 usually made in the literature The contract is designed and agreed upon at period 0, after which the …rm operates and the contract is fully enforced We denote: R = pR+ + (1 R = (p p)R+ + (1 p)R p+ p)R Assumption R>0>R+B i.e the project is pro…table only if the entrepreneur carefully monitors it2 B Dynamic Programming Formulation We de…ne a …nancial contract fxt ; ct gTt=1 in the above environment to be incentive compatible if it induces high eÔort and no quitting by the entrepreneur at every period3 Our interest is to …nd, among this class, the optimal contract As is standard in repeated moral hazard model (e.g Spear and Srivastava (1987)), we rely on the recursive technique to solve for the optimal contract That is, instead of writing the contract as a function of the entire history of cash‡ow realizations, we use the expected discounted utility of the entrepreneur wt at the beginning of each period t as the state variable The behavior of the optimal contract is then best characterized through it Therefore, at each period t; the investors must choose, as a function of wt , a continuation probability xt and, conditional on the project not being liquidated, payments ct ; ct as well as continuation utilities wt+1 ; wt+1 for the entrepreneur contingent on the period t-cash‡ow realizations R+ or R such that their continuation payoÔ is maximized provided that the entrepreneur always exerts high eÔort and never quits This assumption implies that R+ > and R < Negative cash‡ows mean operating losses In fact, to ensure that the optimal level of eÔort is always the high eÔort, additional restrictions are necessary For further discussion about this point, see BMPR (2004) Figure 1: Timeline of the model We de…ne a new variable wtc by wtc = wt for all t xt (4) Hence, while wt represents the expected discounted utility of the entrepreneur at the beginning of each period t prior to the liquidation decision, wtc stands for the entrepreneur’s expected discounted utility conditional on the project being continued at that period (see the …gure above) Denote by Ft (wt ) the highest possible continuation utility for the investors, given a continuation utility wt promised to the entrepreneur at the start of period t The function Ft thus satis…es the following Bellman equation: Ft (wt ) = for all wt M ax xt ;ct ;ct ;wt+1 ;wt+1 xt R pct (1 p)ct + pFt+1 (wt+1 ) + (1 p)Ft+1 (wt+1 ) 1+r (5) 0, subject to the following constraints: First, the promise keeping constraint states that what the entrepreneur was expecting to receive at the beginning of period t must be equal to the sum of the utility she derives from the payment paid to her during this period and the expected present value of her continuation utility: xt pU (ct ) + (1 p)U (ct ) + pwt+1 + (1 p)wt+1 = wt 1+r (6) Second, the entrepreneur must be given incentive to exert high eÔort at every period Since in our model, the cashows are i.i.d distributed, the following temporary incentive compatibility constraint (ICC) is su¢ cient to make the contract fxt ; ct gTt=1 in- centive compatible: xt pU (ct ) + (1 xt (p pwt+1 + (1 p)wt+1 1+r (p p) wt+1 + (1 p + p + p)U (ct ) + 1+r p)U (ct ) + p) U (ct ) + (1 i.e U (ct ) U (ct ) + wt+1 wt+1 1+r p)wt+1 B p +B (7) Hence, the provision of incentives relies on the spread in both current payments and continuation utilities This constraint illustrates the bene…t of an enduring relation which allows the principal to smooth the cost of incentive compatibility over time Third, the investors can require the entrepreneur to recapitalize the …rm but the entrepreneur is willing to that if the following condition is ful…lled for all t < T U (ct ) + wt+1 1+r for all couple (ct ; wt+1 ) Because of (7), the above constraint can be reduced to U (ct ) + wt+1 1+r (8) Fourth, the entrepreneur will never quit if at each period, the contract provides her with a continuation payoÔ at least equal to her reservation utility: (wt ; wt ) R+ (9) Finally, the feasibility condition for the continuation probability: xt [0; 1] (10) Following BMPR (2004), we introduce an auxiliary function de…ned by Vt (wt ) = Ft (wt ) + wt Vt (wt ) represents the expected social surplus generated by the project from period t on if the entrepreneur is provided with a utility wt at the beginning of this period Vt (wt ) will satisfy the Bellman equation as follows: Vt (wt ) = M ax xt R + p (U (ct ) for all wt ct ) + (1 p) (U (ct ) ct ) + pVt+1 (wt+1 ) + (1 p)Vt+1 (wt+1 ) 1+r subject to constraints (6) - (10) Before going to the characterization of the optimal contract, two important remarks are worth making here The …rst remark relates to the way we model the recapitalization as compared to the way investment is modeled in some papers of the literature Our recapitalization corresponds to the fact that some additional capital is injected into the …rm by the entrepreneur This capital serves as supplementary source of funds beside the cash‡ows to repay the creditors and whereby avoid default It does not lead to any change in the …rm’s size By contrast, in the model of Clementi and Hopenhayn (2006) as well as of DeMarzo and Fishman (2007), although at each period, there can also be some capital injected into the …rm, there are two main diÔerences with respect to our formulation First, in these papers, capital injection is realized by the investors More importantly, that capital contribution results in an expansion of the …rm’s size These papers interprete it as investment4 The second remark concerns the interpretation of x As we have seen, the primitive interpretation of x is the probability of continuation of the …rm However, it is clear that the formulation of the dynamic programming problem remains unchanged under the interpretation of x as the size of the …rm as long as the project is constant returns to scale technology Optimal Contract in In…nite Horizon Setting In this section, we consider the case where T = Since cash ‡ows are i.i.d distributed, in an in…nite horizon setting, the optimal contract is stationary So, from now on, we will skip all the time subscripts The …xed-point problem we have to solve is associated with the following dynamic programming problem: V (w) = M ax x R + p (U (c) c) + (1 x;c;c;w;w for all w p) (U (c) c) + pV (w) + (1 p)V (w) 1+r (11) 0, s.t pw + (1 p)w =w 1+r w w B U (c) U (c) + 1+r p w U (c) + 1+r x pU (c) + (1 p)U (c) + (x; w; w) [0; 1] R+ (12) (13) (14) (15) First, we establish the existence and some properties of the value function V: Proposition There exists an unique continuous and bounded solution V to the above program: (i) V is non decreasing and concave pB 1+r pB (ii) V is strictly increasing over [0; 1+r r p ) and constant over [ r p ; +1) (iii) V (w) = 1+r r R pB for all w [ 1+r r p ; +1) In reality, when a …rm announces a plan for collecting funds, there may be two purposes behind it: either new funds are used to …nance new investment opportunities and so, the …rm is expanded or they are employed to reinforce the …rm’s balance sheet to cope with commitments towards investors Our formulation is consistent with the second purpose while the literature cited above models the …rst one Proof See appendix A All the characteristics of function V that are stated in the above proposition are intuitive The main idea we should keep in mind here is that the higher the entrepreneur’s stake in the …rm’s cash‡ows, the less severe the moral hazard problem When the utility level w promised to the entrepreneur decreases, the moral hazard problem becomes more important, which leads to higher risk of liquidation or recapitalization Note that both liquidation and recapitalization are socially costly For liquidation, the reason is that even at the time of liquidation, the project remains potentially pro…table For recapitalization, it is because of the cost borne by the entrepreneur Therefore, the …rm’s value measured by the function V is non decreasing with respect to the entrepreneur’s rent w Concerning the concavity of V , this property implies that the marginal contribution in enhancing …rm’s value of a small increase in the entrepreneur’s rent becomes smaller when this rent is at a higher level What is the intuition for that? Notice that when w is small, liquidation and recapitalization risks are very high Consequently, a small augmentation of w will have signi…cant impact on …rm’s value by reducing these risks By contrast, when w is already high, these risks are small and so, reducing these risks will not result in important eÔects on rms value pB Next, the result with regard to the value of the function V over the interval [ 1+r r p ; +1) suggests that when the entrepreneur’s rent reaches 1+r pB r p, asymmetric information prob- lem does not matter anymore Accordingly, the expected social surplus generated by the project equals the present value of a perpetual annuity of R Now, we turn to the characterization of the optimal contract From the proposition 1, we see that the expected discounted continuation value of the project (i.e the sum pV (w)+(1 p)V (w) 1+r in (11)) is maximum when the entrepreneur is promised a contin- uation utilities at least equal to ous that arg max fp [U (c) 1+r pB r p in both states of nature Moreover, it is obvi- p) [U (c) c]g = [0; +1)2 Thus, if one can …nd a h 1+r pB [0; +1)2 and satisfying all constraints r p ; +1 c] + (1 c;c (x; c; c; w; w) belonging to f1g (12) - (15), then it constitues a solution It is immediately to check that the investors can structure theh entrepreneurs payoÔs in such a way if and only if the value of w belongs to pB the interval 1+r r p ; +1 The following lemma establishes …rst properties of the optimal contract when the value h pB of w falls into the interval 0; 1+r r p It addresses the choice between current payments and continuation utilities as instruments to remunerate the entrepreneur Lemma At any period, if the utility w promised to the entrepreneur belongs to the pB interval [0; 1+r r p ), then the optimal payments c; c for her conditional on the project being continued are non positive Proof See appendix B5 The idea of this proof is that as long as c > 0, by decreasing c to increase continuation utilities, we can get a higher value for the objective function Hence, given that both parties are risk neutral and discount future cash‡ows at the same rate, it is optimal to postpone current compensations for the entrepreneur, in favor of her continuation utilities, until moral hazard problem becomes totally irrelevant pB Due to the lemma 1, for w [0; 1+r r p ), we can rewrite the promised keeping constraint (12) as follows pc + (1 p)c = w x 1+ pw + (1 p)w 1+r (16) pB Replacing (16) into the objective function (11), we …nd that over the interval [0; 1+r r p ), the optimization problem (11) - (15) may be rewritten as : V (w) = M ax w+x R+ 1+ p 1+r V (w) 1+ w + p 1+r V (w) 1+ w (17) for all w [0; 1+r pB r p ), subject to (12) - (15) and the following additional constraint c; c (18) Lemma At any period, the project is optimally terminated with a positive probability if and only if the entrepreneur’s promised utility w is strictly less than pB p In that case, the optimal continuation probability equals x= w pB p Proof See appendix C Therefore, lemma states that given an expected utility w promised to the entrepreneur at the beginning of any period, the optimal probability that the project continues to be operated at that period will be 1; as follows c w = > < w > : w pB p It allows us to compute wc de…ned in (4) if w pB p (19) pB p if w< pB p In what follows, we will replace w by wc as state variable The relationship between them is determined in (19) Two previous lemmas state some optimal features of the termination probability as well as of current payments Now, we examine the continuation utilities Since the ICC pB is binding over the interval [0; 1+r r p ), we can compute (c; c) as follows c= 1+ c= 1+ (1 wc + wc p)B w p 1+r pB w p 1+r (20) (21) Combining three constraints (14), (15) and (18) with (20) - (21), we get two new constraints Note that p (U (c) c) + (1 p) (U (c) c) = (pc + (1 10 p)c) when c; c for (w; w) w (1 p)B wc + 1+r p w pB wc 1+r p (22) (23) Looking at (17), we see that optimal choices for the continuation utilities depend on the monotony property of the function V (w) may arise: either V (0) < 1+ or V (0) > 1+ 1+ w We distinguish two situations which Keeping in mind that 1+ is increasing with , these two cases correspond respectively to whether the recapitalization cost is high or low A Optimal contract when recapitalization cost is high Since the function V is concave, the condition V (0) < 1+ for all w In other words, the function V (w) Consequently, for w [0; 1+r pB r p ), 1+ 1+ implies that V (w) < w is decreasing on R+ at the optimum, we have w = (1 + r) wc + w = (1 + r) wc (1 p)B p pB p (24) (25) Replacing (24) - (25) into (20) and (21), we obtain c = c = We observe that in this case, the current payments for the entrepreneur will never be negative Moreover, note pB p that w in (24) is always larger than pB p 1+ 1+r while w is greater than pB p if and only if wc We de…ne three thresholds + r pB r p pB = 1+ p 1+r pB w = p w = w (26) (27) (28) The following proposition summarizes properties of the optimal contract when the recapitalization cost is high: Proposition When the recapitalization cost is high (i.e V (0) < 1+ ), the optimal contract is characterized as follows (i) If wc [w ; +1): the entrepreneur gets positive payments at the current period: c = wc w + B ; c = wc p w and her continuation utilities are given by w = w = w The project will be operated in the next period with probability whatever the current cash‡ow realizations 11 (ii) If wc [w ; w ): no payment is made to the entrepreneur: c = c = at the current period; the entrepreneur is entitled to continuation utilities w = (1 + r) wc + (1 p)B p ; w = (1 + r) wc pB p the project will also be in operation in the next period with probability whatever the current cash‡ow realizations (iii) if wc [w ; w ): the current payments to the entrepreneur are zero: c = c = - Following a high cash‡ow realization, the project continues in operation in the next period with probability and the entrepreneur’s continuation utility equals w = (1 + r) wc + (1 p)B p - Following a low cash‡ow realization, the project is liquidated in the next period with a positive probability equal to pB p (1+r) wc pB p We will postpone any comments about this contract until resolving the optimal contract in case of low recapitalization cost because we believe that a comparison between two cases will make its features clearer An immediate observation here is that the optimal contract described in proposition is the same as the one derived in BMPR (2004)’s setup where no recapitalization option is taken into account Corollary When recapitalization is su¢ ciently costly, whether this option is taken into consideration or not, the form of the optimal long-term …nancial contract does not change B Optimal contract when recapitalization cost is low If V (0) > 1+ , then there exists a w ~ > such that V (w) w ~ and decreasing above it In other words, V (w) following lemma shows that w ~= Lemma If V (0) > point w = 1+ 1+ 1+ w is increasing below w attains its maximum at w ~ The pB p , then the function V (w) 1+ w reaches its maximum at the pB p Proof See appendix D Since wc pB p, the right hand side of (22) is strictly greater than indicates that w satisfying (22) must be higher than w = (1 + r) wc + pB p (1 pB 1+r p , which As a consequence, at the optimum p)B p and so, c = Relatively to w, with similar reasoning, we get, using thresholds de…ned in (26) - (28), the following result: w= c > < (1 + r) w > : pB p pB p if if 12 w w wc < w wc < w The optimal choice for c is derived from the equality (21) We …nd that if wc [w ; w ), c takes negative value Summing up, we have the following proposition: Proposition When the recapitalization cost is low (i.e V (0) > 1+ ), the optimal contract is characterized as follows (i) If wc [w ; +1): the entrepreneur gets positive payments at the current period: c = wc w + B ; c = wc p w and her continuation utilities are given by w = w = w The project will be operated in the next period with probability after any realization of the current cash‡ow (ii) If wc [w ; w ): no current payment is paid to the entrepreneur: c = c = 0, she is provided with continuation utilities w = (1 + r) wc + (1 p)B p ; w = (1 + r) wc pB p the project will be in operation in the next period with probability after any realization of the current cash‡ow (iii) if wc [w ; w ): - Following a high cash‡ow realization, the entrepreneur gets zero payment: c = The project is operated with certainty in the next period and the entrepreneur’s continuation utility w = (1 + r) wc + (1 p)B p - Following a low cash‡ow realization, the entrepreneur has to recapitalize the …rm by an amount c= 1+ (w wc ) and whereby can avoid liquidation The project is operated in the next period with probability and the entrepreneur’s continuation utility is w= pB =w p Considering two optimal contracts described in two propositions and 3, we see that they share some common properties In general, the total utility of the entrepreneur increases with the cash‡ow realization: it raises following a high cash‡ow and decreases following a low cash‡ow (see region (ii) in each proposition) It stops being sensitive to the …rm’s cash‡ow only when the …rm’s accumulated performance attains some threshold (see region (i)) At that moment, the entrepreneur will be remunerated with cash payment The sensitivity of the entrepreneurs payoÔ to the rms performance serves for incentive purpose Moreover, the degree of this sensitivity depends on the magnitude of moral 13 hazard problem Indeed, de…ne k by k= B p (R+ R ) 1+ , as lemma shows, the …rst derivative of the function pB p V is lower than for all w Therefore, the investors’ continuation utility F (:) is h decreasing with the entrepreneurs payoÔ over the interval pBp ; +1 Considering the optimal contract described in proposition 3, we see that it never oÔers the entrepreneur a continuation utility less than pB p, which indicates that at any period, no other continuation contract is able to make all contracting parties better oÔ As a consequence, this contract is also renegotiation proof Proposition Both optimal contracts characterized in the previous section are renegotiation - proof 4.2 Special Cases We here consider two limiting cases concerning recapitalization cost ! +1 When recapitalization is costless (i.e on R+ , it must be that V (0) 1+ : = and = 0), since the function V is non decreasing = Thus, the optimal contract in this case is the 15 one characterized in proposition 3: Once the …rm falls into …nancial distress, it will be recapitalized and then, continues in operation with certainty When recapitalization is very costly (i.e property of the function V (w) 1+ ! +1), 1+ converges to The monotony w then depends on whether V (0) > or V (0) < If V (0) < 1, the optimal contract is the same as the one de…ned in proposition In case V (0) > 1, looking at the region (iii) of proposition 3, we see that c= 1+ (wc w ) ! when ! +1 which means that there is no recapitalization at all Hence, when ! +1 and V (0) > 1, the optimal contract involves neither recapitalization nor liquidation The reason for no liquidation is the following: hwhen V (0) > 1, the investors payoÔ is increasing with the entrepreneur’s utility over 0; pBp So, a contract that oÔers the entrepreneur a continuation utility less than pB p is Pareto inferior Conclusion In this paper, relying on a dynamic model of …nancing relationship in the presence of moral hazard problem, we address the optimal design of …nancial distress procedure According to our …ndings, a …rm falls into …nancial di¢ culties after a poor performance when the expected discounted value of future cash‡ows accruing to the entrepreneur is below some threshold In such a circumstances, there exists two alternatives as optimal mechanisms, one corresponds to the recapitalization process, the other to the liquidation Hence, no strict priority between these two options exists Based on the cost comparison, the …rm determines which option should be applied A Appendix: Proof of proposition A.1 There exists an unique, continuous and bounded solution V ? Let Cb (R+ ) be bounded continuous function space and de…ne an Bellman operator T as follows T : v(w) ! T v(w) where T v(w) = M ax x R + p (U (c) x;c;c;w;w c) + (1 16 p) (U (c) c) + pv(w) + (1 p)v(w) 1+r for all w s.t pw + (1 p)w =w 1+r w w B U (c) U (c) + 1+r p w U (c) + 1+r x pU (c) + (1 (29) p)U (c) + (30) (31) R+ (x; w; w) [0; 1] (32) To prove that there exists an unique, continuous and bounded function V , we need to show that T maps Cb (R+ ) to itself and that T is contraction Indeed, because U (c) c and v(w) are all bounded, T v(w) is also bounded Let w ^h is the ismallest point at which v(w) reaches its maximum7 For any value w ^ of w 0; pBp + 1+r , there is no loss of generality to restrict (x; c; c; w; w) in [0; 1] h i i h w ^ B w ^ ; ; [0; w] ^ Since the objective function is continuous p (1+ )(1+r) h (1+ )(1+r) i h i w ^ B w ^ and the set [0; 1] [0; w] ^ is compact, from the ; ; p (1+ )(1+r) (1+ )(1+r) h i w ^ When Maximum theorem, the function T v(w) is continuous on the interval 0; pBp + 1+r pB w ^ w ^ p + 1+r , it is clear that (x; c; c; w; w) = 1; p w 1+r w) ^ pB w ^ w ^ ; T v(w) = R + v( for w > pBp + 1+r 1+r = T v p + 1+r w> is, ; 0; w; ^ w ^ is a solution That So, T v(w) is continuous on [0; +1) T v(w) is thus both bounded and continuous This means that the operator T maps Cb (R+ ) to itself We can easily verify that the operator T satis…es two Blackwell’s su¢ cient conditions for a contraction (monotonicity and discounting) It implies that T is contraction Hence, by the Contraction Mapping Theorem, T has an unique …xed point V Cb (R+ ) A.2 V is non decreasing? Take w and w0 such that w0 > w Let (x; c; c; w; w) be solution in the program that de…nes T v(w) Construct (x; c0 ; c; w; w) such that p (U (c0 ) ing function, we have c0 w0 w x > Since U (c) is increas- > c Moreover, x pU (c ) + (1 U (c0 ) U (c)) = pw + (1 p)w p)U (c) + 1+r U (c) + w w > U (c) 1+r =x U (c) + w w x w w 1+r w + x ! =w B p Thus, (x; c0 ; c; w; w) is feasible choice in the program that de…nes T v(w0 ) Since U (c) Since v(w) is bounded function, such a w ^ exists 17 c is non decreasing function, we have x R + p U (c0 T v(w0 ) x R + p (U (c) c0 ) + (1 pv(w) + (1 p)v(w) 1+r pv(w) + (1 p)v(w) = T v(w) c) + 1+r p) (U (c) c) + (1 c) + p) (U (c) i.e T v(w) is non decreasing function A.3 V is concave? Consider T c v(w) = M ax R + p (U (c) c;c;w;w for all w pB p c) + (1 p) (U (c) c) + pv(w) + (1 p)v(w) 1+r s.t pw + (1 p)w =w 1+r w w B U (c) U (c) + 1+r p w U (c) + 1+r pU (c) + (1 p)U (c) + (w; w) R+ So, T v(w) = M ax xT c v(wc ) c x;w for all w s.t xwc = w x [0; 1] pB ; +1 p wc First, we will prove that if v(w) is concave, then T c v(w) is concave Indeed, let 0 (c; c; w; w) be solution for T c v(w) and c ; c ; w ; w )w De…ne c ; c ; w ; w w Hence, U (c ) + 1+r 0 for T c v(w ): De…ne w = w + (1 such that w = w + (1 )w w = w + (1 )w 0 U (c ) = U (c) + (1 )U (c ) U (c ) = U (c) + (1 )U (c ) and U (c ) U (c ) + w 18 w 1+r B p It implies that c ; c ; w ; w is feasible choice for T c v(w ): T c v(w ) R + p (U (c ) c ) + (1 p) U (c ) c + pv(w ) + (1 p)v(w ) 1+r The concavity of the function v(w) implies v(w ) v(w) + (1 )v(w ) v(w ) v(w) + (1 )v(w ) Since U is concave and increasing function, from de…nition of ci , we have c c + (1 h 0 [U (c) c] + (1 ) U (c ) c Similarly, U (c ) c )c and thus, U (c ) c h i 0 [U (c) c] + (1 ) U (c ) c Finally, we get T c v(w ) T c v(w) + (1 )T c v(w ) i.e T c v(w) is concave Now, we establish the concavity of T v(w) we have T v(w) = M ax w c w for all w s.t wc De…ne w ^ by w ^ = w ^ 1+r + max(w; T c v(wc ) wc pB ) p pB p Similarly to the previous parts, one can verify that T c v(w) is continuous on its domain c c ) andh constant ^ ; +1) Therefore, the mapping wc ! T v(w reaches its maximum wc i over [w c c T v(w ) pB pB c c ^ :Let arg max wc = [w ; w ], possibly reduced to a point ) wc in p; w p: w if w wc > < c c ) arg max T v(w = [w; wc ] if wc > w wc wc > pB : wc max w; p [wc ; wc ] if wc > w ( T c v(w) if w wc so, T v(w) = where A does not depend on w w A if w < wc =) T v(w) is concave A.4 V is constant over h 1+r pB ; +1 r p From previous parts, we know V (w) = R + h pB and strictly increasing over 0; 1+r r p V (w) ^ if w 1+r pB w ^ + p 1+r where w ^ is smallest point at which V (w) reaches its maximum Take w1 < w2 < V pB p + w ^ 1+r V (w2 ) = V pB p + w ^ 1+r Because V (w) is non decreasing, V (w1 ) V (w2 ) If V (w1 ) = V (w2 ), then with the concavity of V (w), we get V (w1 ) = pB p + w ^ 1+r It implies that (x; c; c; w; w) = 1; 19 B ^ w ^ p ; 0; w; is solution to the w ^ program that de…nes V (w1 ) and so, w1 = pBp + 1+r (contradiction) Hence, V (w) is strictly h h w ^ w ^ increasing over 0; pBp + 1+r Moreover, it is constant over pBp + 1+r ; +1 , that means pB p B w ^ 1+r + is the smallest point of arg max V (w) ) pB p + w ^ 1+r =w ^)w ^= 1+r pB r p: Appendix : Proof of lemma We begin by proving that at the optimum c is non positive Indeed, suppose that there exists a solution (x; c; c; w; w) such that c > Since c is strictly greater than 0, there exists a " > such that c " 0 De…ne c = c is easy to check that x; c; c ; w; w 0 " and w = w + (1 + r) " > w It satis…es all constraints (12) - (15): Because V (w) is weakly increasing function, we have8 : x R + p (U (c) " x R + p (U (c) c) + (1 c) + (1 p) (U (c) pV (w) + (1 p)V (w) 1+r c) + 0 p) U (c ) c pV (w) + (1 p)V (w ) + 1+r # which means that without loss of generality, we can restrict ourself to the solutions where c Similarly, at the optimum, c C Appendix : Proof of lemma First, from constraints (12) - (14), we have w x pB p Then, the constraints for x become (i) Now, we show that as long as w x inf pB p, 1; w pB p ! it is optimal to let x = Indeed, assuming by contradiction that for w pB p, there exists a solution x1 ; c1 ; c1 ; w1 ; w1 to the problem (17) such that x1 is strictly less than Since the quintuple x1 ; c1 ; c1 ; w1 ; w1 is solution, it will satisfy all constraints (12) - (15) and (18) De…ne three variables ( ; ; ) Note that U (c) c = for all c 0: 20 as follows = = U (c1 ) w x1 w>0 w1 w1 B 1+r p w1 = U (c1 ) + 1+r U (c1 ) + (12) implies that p + pB + p pB + p =w+ pB p where the inequality is due to the condition w Therefore p + (33) We construct from x1 ; c1 ; c1 ; w1 ; w1 another quintuple x2 ; c2 ; c2 ; w2 ; w2 such that x2 = (34) U (c2 ) = U (c1 ) " (35) U (c2 ) = U (c1 ) ! (36) w2 = w1 (37) w2 = w1 (38) where ("; !) are solutions to the following system p" + (1 p)! = (39) (40) "; ! max(0; p ) ! ; (41) p Note that due to (33), each element of the function max in (41) is less than or equal to each element of the function Moreover, because ! (1 p)! ;1 p p, we have These remarks show that there exists solution to the system (39) - (41), which implies that there exists a quintuple x2 ; c2 ; c2 ; w2 ; w2 satisfying (34) - (38) It is easily to check that the quintuple x2 ; c2 ; c2 ; w2 ; w2 constructed as above ful…ll all constraints (12) - (15) and (18) Therefore x2 ; c2 ; c2 ; w2 ; w2 are feasible choice for the problem (17) Furthermore p V (w1 ) w1 1+ 1+ 1+r 1+ p p w+R+ V (w1 ) w1 + V (w1 ) w1 1+ 1+r 1+ 1+r 1+ p p w + x2 R + w2 + w2 V (w2 ) V (w2 ) 1+ 1+r 1+ 1+r 1+ w + x1 R + < = (contradiction) So, when w p 1+r pB p, V (w1 ) w1 + it is optimal to let x = 21 h (ii) Next, we prove that for w 0; pBp , at the optimum x = w pB p The proof is similar to the previous part Assuming by contradiction that for w< pB p, there exists a solution x3 ; c3 ; c3 ; w3 ; w3 to the problem (17) such that x3 < w pB p We see that x3 ; c3 ; c3 ; w3 ; w3 satis…es all constraints (12) - (15) and (18) De…ne three variables = U (c3 ) w3 w3 B 1+r p w3 = U (c3 ) + 1+r pB w >0 = x3 p U (c3 ) + So, =p + >0 From x3 ; c3 ; c3 ; w3 ; w3 , we construct x4 ; c4 ; c4 ; w4 ; w4 satisfying x4 = w pB p > x3 (42) U (c4 ) = U (c3 ) U (c4 ) = U (c3 ) (43) (44) w4 = w3 (45) w4 = w3 (46) It is immediate to check that x4 ; c4 ; c4 ; w4 ; w4 respects all constraints (12) - (15) and (18) and thus, x4 ; c4 ; c4 ; w4 ; w4 are feasible choice for the problem (17) Moreover 1+ < 1+ p 1+r p R+ 1+r w + x3 R + V (w3 ) w + x4 V (w4 ) 1+ 1+ w3 w4 p 1+r p + 1+r + V (w3 ) V (w4 ) (contradiction) D Appendix: Proof of lemma De…ne a function H(w) by H(w) = V (w) 1+ w for w 0; + r pB r p Therefore, H(w) is determined by the following program: H(w) = M ax x R + pH(w) + (1 p)H(w) 1+r 22 1+ 1+ w3 w4 h pB for all w 0; 1+r r p s.t w w (1 p)B + 1+r x p w w pB 1+r x p ! w x = 1; pB (47) (48) (49) p Due to the properties of the function V , we see that H(0) = and H(w) is concave function Moreover, if V (0) > 1+ , then H (0) > pB p? (i) H(w) is a non increasing function when w Take wa and wb such that wa pB p wb Let (1; wa ; wa ) be solution in the program that de…nes H(wa ) Hence, wa 1+r wa + wa 1+r (1 wa p)B p pB p wb + wb (1 p)B p (50) pB p (51) which means (1; wa ; wa ) is feasible choice in the program that de…nes H(wb ) Then, we have H(wb ) R+ pH(wa ) + (1 p)H(wa ) = H(wa ) 1+r i.e the function H(w) is non increasing (ii) H(w) is a non decreasing function when Note that when w < pB p, we have x = pB p w< pB p? Thus, the program de…ning the function H can be rewritten as follows H(w) = M ax h for all w 0; pBp w R+ pB p pH(w) + (1 p)H(w) 1+r s.t w 1+r w 1+r First, we prove that H(w) B p 0 for all w h 0; pBp if H (0) > Indeed, since H (0) > 0, in the neighbourhood (0; "], H(w) is increasing function, which implies that H(wc ) for any wc (0; "] Let (wc ; wc ) be solution in the program that de…nes H(wc ): H(wc ) = wc pB p R+ pH(wc ) + (1 p)H(wc ) 1+r 23 Hence R+ pH(wc ) + (1 p)H(wc ) 1+r Obviously, (wc ; wc ) is feasible choice in the program that de…nes H(wd ) for any wd "; pBp We have H(wd ) wd pB p R+ wc pH(wc ) + (1 p)H(wc ) 1+r pB p R+ pH(wc ) + (1 p)H(wc ) = H(wc ) 1+r h for all w 0; pBp i.e H(w) Next, we show that H(w) is a non decreasing function when wf such that wf we < pB p w< pB p Take we and Let wf ; wf be solution in the program that de…nes H(wf ) wf ; wf will be feasible choice for we We get H(we ) we pB p R+ pH(wf ) + (1 p)H(wf ) 1+r wf pB p R+ pH(wf ) + (1 p)H(wf ) 1+r = H(wf ) (52) The second inequality in (52) is due to the non negative sign of the term in the parenthesis References [1] Albuquerque, R and H.A Hopenhayn (2004): "Optimal Lending Contracts and Firm Dynamics", Review of Economic Studies, 71, 285 - 315 [2] Biais, B., T Mariotti, G.Plantin and J.C Rochet (2004): “Dynamic Security Design”, CEPR Discussion Paper No 4753 [3] Biais, B., T Mariotti, G.Plantin and J.C Rochet (2006): "Dynamic Security Design: Convergence to Continuous Time and Asset Pricing Implications", Review of Economic Studies, 74, 345 - 390 [4] Clementi, G.L and H Hopenhayn (2006): "A Theory of Financing Constraints and Firm Dynamics", Quarterly Journal of Economics, 121, 229 - 265 [5] DeMarzo, P and M.Fishman (2007a): "Optimal Long-Term Financial Contracting", Review of Financial Studies, 20, 2079 - 2128 [6] DeMarzo, P and M.Fishman (2007b): "Agency and Optimal Investment Dynamics", Review of Financial Studies, 20, 151-188 [7] DeMarzo, P and Y Sannikov (2006): "Optimal Security Design and Dynamic Capital Structure in a Continuous - Time Agency Model", Journal of Finance, 6, 2681 - 2723 [8] Denis, D.K and D.K Shome (2005): "An Empirical Investigation of Corporate Asset Downsizing", Journal of Corporate Finance, 11, 427 - 448 [9] Gertler, M (1992): "Financial Capacity and Output Fluctuations in an Economy with Multi-Period Financial Relationships", Review of Economic Studies, 59, 455 472 24 [10] Hart, O (2000), "DiÔerent Approaches to Bankruptcy", NBER Working Paper 7921 [11] Holmström, B and J Tirole, (1997): “Financial Intermediation, Loanable Funds and the Real Sector” Quarterly Journal of Economics, 112, 663 - 691 [12] Shim, I., (2006): “Dynamic Prudential Regulation: Is Prompt Corrective Action Optimal”, BIS Working paper No.206, Bank for International Settlements [13] Spear, S and S Srivastava (1987): "On Repeated Moral Hazard with Discounting", Review of Economic Studies, 54, 599 - 607 [14] Spear, S and C.Wang (2005): "When to Fire a CEO: Optimal Termination in Dynamic Contracts", Journal of Economic Theory, 120, 239 - 256 [15] Tchistyi, A (2006): "Security Design with Correlated Hidden Cash Flows: The Optimality of Performance Pricing", Mimeo, New York University 25 ... diÔerently with all of the above papers, we take into account the recapitalization as an alternative of liquidation for the …rm in case of …nancial di¢ culties Due to that, we are able to have an insightful... DeMarzo, P and M.Fishman (200 7a) : "Optimal Long-Term Financial Contracting" , Review of Financial Studies, 20, 2079 - 2128 [6] DeMarzo, P and M.Fishman (2007b): "Agency and Optimal Investment Dynamics",... the optimal contract in case of low recapitalization cost because we believe that a comparison between two cases will make its features clearer An immediate observation here is that the optimal