Optimal resolution of financial distress- a dynamic contracting approach

Built on therecursive method developed in these works, recently, a number of papers study the optimallong-term …nancial contract in a setting in which a risk neutral entrepreneur seeks f

Optimal Resolution of Financial Distress: A

Dynamic Contracting Approach

VO Thi Quynh AnhOctober 2009

Abstract This paper provides a formal analysis of the choice between liquidation and recap- italization when …rms get into …nancial di¢ culties We introduce the possibility of a costly recapitalization into a model of dynamic …nancial contracting under moral haz- ard 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.

a new mechanism for the settlement of claims: writing o¤ some of the claims, exchangingbonds and other debts with new notes, bonds, swapping new equities for old ones, injec-tion of new capital This paper aims at providing a formal analysis on the choice betweenliquidation and recapitalization - one of possible methods involved in the reorganizationprocess - when …rms get into …nancial di¢ culties

Norges Bank (Central Bank of Norway), Bankplassen 2, 0107 Oslo, Norway Email: anh.vo@norges-bank.no.

thi-quynh-We take a dynamic …nancial contracting approach to this issue The dynamic agencymodel is …rst introduced by Green (1987) and Spear and Srivastava (1987) Built on therecursive method developed in these works, recently, a number of papers study the optimallong-term …nancial contract in a setting in which a risk neutral entrepreneur seeks fundingfrom risk neutral investors to …nance a project that pays stochastic cash‡ows over manyperiods Their contracting relationship is subject to a moral hazard problem that comeseither from the unobservability of cash‡ows or from hidden e¤orts All of those papersassume that the entrepreneur is liable for payments to the investors only to the extent ofcurrent revenues Consequently, when facing a business failure, the …rm possesses in handonly one option, that is liquidation In this paper, to be able to simultaneously analyzethe liquidation and the recapitalization decisions, we introduce into a dynamic …nancialcontracting model the possibility of costly recapitalization.

Speci…cally, we consider a scenario where a risk neutral entrepreneur contracts withrisk neutral investors to …nance a multi-period investment project This project, oncefunded, generates at each period of its life an observable binary cash‡ows whose distrib-ution depends on the unobservable e¤ort of the entrepreneur For simplicity, we assumethat the set of feasible e¤ort levels contains two elements, high or low e¤ort The distribu-tion of cash‡ows under high e¤ort dominates the distribution under low e¤ort in the sense

of …rst order stochastic dominance Nevertheless, exerting high e¤ort is costly since it prives the entrepreneur of a private bene…t B The novelty of our paper lines in relaxingthe limited liability constraint usually imposed by previous analysis in the literature Weassume, 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 volun-tary 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 bythe expected discounted utility the entrepreneur can capture if the project continues inoperation Hence, in our model, in front of …nancial di¢ culties, the …rm can choose either

de-to be recapitalized or de-to be liquidated The properties of an optimal …nancial distressprocedure are highlighted through the characterization of the optimal contract betweenthe 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 theexpected discounted utility of the entrepreneur at the start of each period as the singlestate variable We are able to fully characterize the optimal contract in an in…nite horizonsetting We …nd that the …rm is never recapitalized nor liquidated following a goodperformance 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 existstwo possible procedures to cope with …nancial di¢ culties When recapitalization cost isrelatively high, the …rm should be liquidated and no recapitalization is employed Bycontrast, 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 thatour optimal contract is robust to the renegotiation possibility

This paper links to the literature on dynamic …nancial contracting under moral hazardwhose 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 repeatedlyinteract DeMarzo and Fishman (2007a) look at the optimal long-term …nancial contractbetween a risk neutral agent and risk neutral investors in the presence of unobservablecash‡ows problem They …rst characterize the optimal contract and then, show thatthe optimal mechanism can be implemented by a combination of commonly observedsecurities such as equity, long-term debt and a line of credit In their implementation, thelong-term debt is described by a …xed charge paid at each period whereas the credit line ischaracterized 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 Asfor BMPR (2006), their optimal contract analysis can be seen as a stationary version ofDeMarzo and Fishman (2007a) but their implementation of this contract is realized viadebt, equity and cash reserves When …rm can not serve its debt, probabilistic liquidationoccurs 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 asymetricinformation, Clementi and Hopenhayn (2006) and DeMarzo and Fishman (2007b) aim atexplaining some of the facts regarding …rm’s investment decisions, growth and survivalrates 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 andHopenhayn (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; survivalrates increase with …rm size and …rm age and size are positively correlated DeMarzo andFishman (2007b) provide a more general model of moral hazard and so, can obtain theconclusion that the patterns, concerning …rm’s growth and investments, observed in thedata 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, di¤erently with all of the above papers,

we take into account the recapitalization as an alternative of liquidation for the …rm incase of …nancial di¢ culties Due to that, we are able to have an insightful analysis aboutthe design of the optimal …nancial distress procedure

The paper is organized as follows In the next section, we present the structure ofthe model and the dynamic programming formulation Section 3 is devoted to the char-acterization of the optimal contract in in…nite horizon setting Section 4 discusses therobustness of our optimal contract with respect to the renegotiation prossibility Finally,section 5 concludes All proofs will be provided in the appendix

2 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 atthe same rate r

The entrepreneur is endowed with a multi-period investment project that requires astart-up capital of I The project, if undertaken, generates at each period t = 1; 2; ::::T anobservable cash‡ow Rtwhich can take two values: high cash‡ow: Rt= R+or low cash‡ow:

Rt = R The distribution of cash‡ows depends on the e¤ort et the entrepreneur exerts

to run her project The choice of e¤ort is unobservable to the investors For simplicity,

we assume that the entrepreneur can choose between two e¤ort levels: et= 1 (high e¤ort)

or et= 0 (low e¤ort) If she chooses high e¤ort, the probability of getting a high cash‡owequals p > p p which is the probability of high cash‡ow in case of low e¤ort However,

if the entrepreneur decides not to exert high e¤ort, 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 herproject 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 1 xt be the probability withwhich the project is terminated by the investors We assume that in the event of termina-tion, the project’s assets are liquidated at a zero price The project can also be ended ifthe entrepreneur decides to quit to take her outside options We normalize the reservationutility of the entrepreneur to zero

Second, the payment ctto the entrepreneur conditional on the project being tinued We here deviate from BMPR (2004) as well as all other papers in the literature

con-by not imposing the condition that ct must be non negative Instead, we assume that ateach period t prior to the …nal period T (i.e t < T ), after the cash‡ow is realized, theinvestors 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 theliquidation threat We interpret it as recapitalization This recapitalization is costly tothe entrepreneur We assume that the cost she has to bear for each unit of capital injectedequals 01 Hence, the utility function of the entrepreneur takes the following form

propor-we only take into consideration the proportional costs.

continues in operation Given a contract fxt; ctgTt=1, let wtdenote the expected discountedutility the entrepreneur derives from all payments paid to her from period t on if she exertshigh e¤ort at every period Thus

wt= E

266664

37777

The lower bound for ct(t < T ) is then determined endogenously by the following condition:

U (ct) + wt+1

1 + r 0 for all t < T (3)This condition can also be seen as limited liability constraint It is obviously weaker thanthe limited liability constraint ct 0; wt+1 0 usually made in the literature

The contract is designed and agreed upon at period 0, after which the …rm operatesand the contract is fully enforced We denote:

R = pR++ (1 p)R

R = (p p)R++ (1 p + p)RAssumption 1

R > 0 > R + Bi.e the project is pro…table only if the entrepreneur carefully monitors it2

B Dynamic Programming Formulation

We de…ne a …nancial contract fxt; ctgTt=1 in the above environment to be incentivecompatible if it induces high e¤ort 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 ofwriting the contract as a function of the entire history of cash‡ow realizations, we usethe 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 characterizedthrough it Therefore, at each period t; the investors must choose, as a function of wt, acontinuation probability xtand, conditional on the project not being liquidated, payments

ct; ct as well as continuation utilities wt+1; wt+1 for the entrepreneur contingent on theperiod t-cash‡ow realizations R+ or R such that their continuation payo¤ is maximizedprovided that the entrepreneur always exerts high e¤ort and never quits

2

This assumption implies that R + > 0 and R < 0 Negative cash‡ows mean operating losses.

3 In fact, to ensure that the optimal level of e¤ort is always the high e¤ort, 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 wc

Denote by Ft(wt) the highest possible continuation utility for the investors, given acontinuation utility wtpromised to the entrepreneur at the start of period t The function

Ftthus satis…es the following Bellman equation:

Ft(wt) = M ax

x t ;c t ;ct;w t+1 ;wt+1xt R pct (1 p)ct+pFt+1(wt+1) + (1 p)Ft+1(wt+1)

for all wt 0, subject to the following constraints:

First, the promise keeping constraint states that what the entrepreneur was pecting to receive at the beginning of period t must be equal to the sum of the utility shederives from the payment paid to her during this period and the expected present value

ex-of her continuation utility:

xt pU (ct) + (1 p)U (ct) + pwt+1+ (1 p)wt+1

1 + r = wt (6)Second, the entrepreneur must be given incentive to exert high e¤ort at everyperiod Since in our model, the cash‡ows are i.i.d distributed, the following temporary

incentive compatibility constraint (ICC) is su¢ cient to make the contract fxt; ctgTt=1 centive compatible:

Third, the investors can require the entrepreneur to recapitalize the …rm but theentrepreneur is willing to do that if the following condition is ful…lled for all t < T

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 fromperiod t on if the entrepreneur is provided with a utility wtat the beginning of this period

Vt(wt) will satisfy the Bellman equation as follows:

Vt(wt) = M ax xt R + p (U (ct) ct) + (1 p) (U (ct) ct) +pVt+1(wt+1) + (1 p)Vt+1(wt+1)

1 + rfor all wt 0 subject to constraints (6) - (10)

Before going to the characterization of the optimal contract, two important remarksare 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 recapitalizationcorresponds to the fact that some additional capital is injected into the …rm by the en-trepreneur 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’ssize By contrast, in the model of Clementi and Hopenhayn (2006) as well as of DeMarzoand Fishman (2007), although at each period, there can also be some capital injectedinto the …rm, there are two main di¤erences with respect to our formulation First, inthese papers, capital injection is realized by the investors More importantly, that capi-tal contribution results in an expansion of the …rm’s size These papers interprete it asinvestment4.

The second remark concerns the interpretation of x As we have seen, the primitiveinterpretation of x is the probability of continuation of the …rm However, it is clearthat the formulation of the dynamic programming problem remains unchanged under theinterpretation of x as the size of the …rm as long as the project is constant returns to scaletechnology

3 Optimal Contract in In…nite Horizon Setting

In this section, we consider the case where T = 1 Since cash ‡ows are i.i.d distributed,

in an in…nite horizon setting, the optimal contract is stationary So, from now on, we willskip all the time subscripts The …xed-point problem we have to solve is associated withthe following dynamic programming problem:

V (w) = M ax

x;c;c;w;wx R + p (U (c) c) + (1 p) (U (c) c) +pV (w) + (1 p)V (w)

1 + r (11)for all w 0, s.t

Proposition 1 There exists an unique continuous and bounded solution V to the aboveprogram:

(i) V is non decreasing and concave

(ii) V is strictly increasing over [0;1+rr pBp) and constant over [1+rr pBp; +1)

(iii) V (w) = 1+rr R for all w 2 [1+rr pBp; +1)

4

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 tuitive The main idea we should keep in mind here is that the higher the entrepreneur’sstake in the …rm’s cash‡ows, the less severe the moral hazard problem When the utilitylevel w promised to the entrepreneur decreases, the moral hazard problem becomes moreimportant, which leads to higher risk of liquidation or recapitalization Note that bothliquidation and recapitalization are socially costly For liquidation, the reason is that even

in-at the time of liquidin-ation, the project remains potentially pro…table For recapitalizin-ation,

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 inenhancing …rm’s value of a small increase in the entrepreneur’s rent becomes smaller whenthis 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 e¤ects on …rm’s value

Next, the result with regard to the value of the function V over the interval [1+rr pBp; +1)suggests that when the entrepreneur’s rent reaches 1+rr pBp, asymmetric information prob-lem does not matter anymore Accordingly, the expected social surplus generated by theproject equals the present value of a perpetual annuity of R

Now, we turn to the characterization of the optimal contract From the tion 1, we see that the expected discounted continuation value of the project (i.e thesum 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 1+rr pBp in both states of nature Moreover, it is obvi-ous that arg max

proposi-c;c fp [U(c) c] + (1 p) [U (c) c]g = [0; +1)2 Thus, if one can …nd a(x; c; c; w; w) belonging to f1g [0; +1)2 h

1+r r

pB

p; +1 2 and satisfying all constraints(12) - (15), then it constitues a solution It is immediately to check that the investors canstructure the entrepreneur’s payo¤s in such a way if and only if the value of w belongs tothe intervalh

1+r

r

pB

p; +1 The following lemma establishes …rst properties of the optimal contract when the value

of w falls into the intervalh

0;1+rr pBp It addresses the choice between current paymentsand continuation utilities as instruments to remunerate the entrepreneur

Lemma 1 At any period, if the utility w promised to the entrepreneur belongs to theinterval [0;1+rr pBp), then the optimal payments c; c for her conditional on the project beingcontinued are non positive

Proof See appendix B5

5

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 thesame rate, it is optimal to postpone current compensations for the entrepreneur, in favor

of her continuation utilities, until moral hazard problem becomes totally irrelevant.Due to the lemma 1, for w 2 [0;1+rr pBp), we can rewrite the promised keeping constraint(12) as follows

pc+ (1 p)c = 1

1 +

wx

Lemma 2 At any period, the project is optimally terminated with a positive probability ifand only if the entrepreneur’s promised utility w is strictly less than pBp In that case, theoptimal continuation probability equals

x = pBw

p

Proof See appendix C

Therefore, lemma 2 states that given an expected utility w promised to the neur at the beginning of any period, the optimal probability that the project continues to

entrepre-be operated at that period will entrepre-be min 1; pBw

or low.

A Optimal contract when recapitalization cost is high

Since the function V is concave, the condition V0(0) < 1+ implies that V0(w) <

1+ for all w 0 In other words, the function V (w) 1+ w is decreasing on R+.Consequently, for w 2 [0;1+rr pBp), at the optimum, we have

and her continuation utilities are given by w = w = w The project will be operated

in the next period with probability 1 whatever the current cash‡ow realizations

(ii) If wc2 [w ; w ): no payment is made to the entrepreneur: c = c = 0 at the currentperiod; the entrepreneur is entitled to continuation utilities

w = (1 + r) wc+ (1 p)B

p ; w = (1 + r) w

pthe project will also be in operation in the next period with probability 1 whatever thecurrent cash‡ow realizations

(iii) if wc2 [w ; w ): the current payments to the entrepreneur are zero: c = c = 0

- Following a high cash‡ow realization, the project continues in operation in the nextperiod with probability 1 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 1 (1+r) w

c pB p 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 caseswill make its features clearer An immediate observation here is that the optimal contractdescribed in proposition 2 is the same as the one derived in BMPR (2004)’s setup where

no recapitalization option is taken into account

Corollary 1 When recapitalization is su¢ ciently costly, whether this option is taken intoconsideration or not, the form of the optimal long-term …nancial contract does not change

B Optimal contract when recapitalization cost is low

If V0(0) > 1+ , then there exists a ~w > 0 such that V (w) 1+ w is increasing below

Proof See appendix D

Since wc pBp, the right hand side of (22) is strictly greater than 1+r1 pBp, whichindicates that w satisfying (22) must be higher than pBp As a consequence, at the optimum

w = (1 + r) wc+ (1 p)B

pand so, c = 0 Relatively to w, with similar reasoning, we get, using thresholds de…ned in(26) - (28), the following result: