Research in International Business and Finance

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1 Reseah in Intenational usiness and Finane 4 (00) 3 37 Contents lists available at ienediet Reseah in Intenational usiness and Finane jounal homepage: elief updating, debt piing and finanial deisions unde asymmeti infomation Ruxing Xu a,b,, henghong i a a Depatment of Mathematis, Zhejiang Univesity, 0 Yugu Road, Hangzhou, Zhejiang 3007, China b Depatment of Mathematis, China Jiliang Univesity, Xueyuan teet, Xiasha Highe Eduation Pak, Hang Zhou, Zhejiang 3008, China atile info abstat Atile histoy: Reeived 7 Novembe 008 Aepted 9 Otobe 009 Available online 4 Novembe 009 JE lassifiation: C70 G3 G3Asymmeti infomation elief updating Capital stutue Debt valuation Finanial deisions This pape studies debt holdes belief updating, valuation of opoate debt, and equity ownes finaning deisions duing finanial distess unde asymmeti infomation. This is done within a ontinuous-time famewok, whee the elevant state vaiable is assumed to follow an Aithmeti ownian motion (AM). AM an take negative values and has vey ealisti featue ompaed with Geometi ownian motion (GM). Using Chapte of U.. ankupty Code as a ostly seening devie, we an haateize whih fim will hoose pivate wokouts (in the fom of stategi debt sevie) and whih will hoose to file fo the Chapte ankupty poedue (in the fom of debt equity swap) when the fim is in finanial distess. Using aguments simila to equilibium efinements, we give a lea pitue of how debt holdes beliefs about the fim s types ae updated aoding to the state vaiable and the fim s default behavio, and desibe optimal stategies of both paties unde those beliefs. We also povide an appoximate solution to the debt piing poblem unde asymmeti infomation. 009 Published by Elsevie.V.. Intodution In this study, we show how debt holdes beliefs about fim s types ae updated unde asymmeti infomation that equity ownes have bette infomation on thei fim o investment pojet than do Coesponding autho. Tel.: addess: xxing@jlu.edu.n (R. Xu) /$ see font matte 009 Published by Elsevie.V. doi:0.06/j.ibaf

2 4 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) 3 37 debt holdes, and examine the impat of asymmeti infomation on debt valuation and equity ownes finanial deisions duing finanial distess. The valuation of isky debt is ental to theoetial and empiial wok in opoate finane. A fuitful appoah is known as ontingent laims analysis pioneeed by lak and holes (973), Meton (974). They assume that the fim value follows a GM. The fim has a finite matuity zeo oupon bond that it will epay if its value exeeds the notional amount of the debt at debt matuity. Othewise the fim defaults on its debt. ine then, the liteatue on the valuation of opoate seuities and finaning deisions has substantially developed. lak and Cox (976) extends this setup by allowing bankupty befoe debt matuity when the fim value touhes a bankupty baie fo the fist time. Futhe extension of the basi setup intodues apital stutue hanges (e.g., in Fishe et al., 989a,b). In both papes they model the apital stutue of the fim endogenously in a ontinuous-time setting unde the assumption that equity ownes optimize the value of thei laim, and explain empiially obseved leveage atios and all pemia of allable opoate bond. The idea of equity ownes maximizing the value of thei laim when leveing the fim is futhe developed by eland (994), who fouses on the valuation of opoate debt and the sensitivity of debt value to etain model paametes, and deives a fim value level at whih equity ownes endogenously optimally tigge bankupty. His analysis is extended among othes by eland and Toft (996) to elax the assumption of infinite life debt, and eland (998) to aount fo asset substitution. eause substantial empiial evidene has doumented that bankupties ae ostly and bankupty poedues give onsideable sope fo oppotunisti behavio by vaious paties involved, the taditional stutual models ae late developed by Andeson and undaesan (996), Mella-aal and Peaudin (997), Fan (000), Fanois and Moelle (004), who take into onsideation stategi issues and enegotiation in the ontext of debt valuation and teat bankupty as a bagaining game. Almost all the papes above assume that the elevant state vaiable is given by the (unleveed) fim value, whih may pemit abitage oppotunities o othe diffiulties. Goldstein et al. (00) fist dispenses with the assumption by onsideing the Eanings efoe Inteest and Taxes (EIT) as the state vaiable, upon whih all ontingent laims an be valued in a onsistent manne. EIT is assumed to follow a GM as befoe. Within this setting, Goldstein et al. (00) onsides the impat of the option to inease futue debt levels on tax advantages, and find that both the optimal leveage atio ange and pedited edit speads ae moe in line with what is obseved in patie. Models that make use of the EIT assumption, inlude Dangl and Zehne (004), Ammann and Gense (005), Hakbath et al. (007). Commonly, EIT is assumed to follow a GM by eason of its mathematial tatability. ut eonomially the hoie of GM is debateable beause the potential negative ash flow haateisti is an impotant featue of most investment pojets (e.g., Klumpes and Tippett, 004; Maathe and Ryan, 005). AM has moe ealisti featue, fo ash flow an beome negative, whih is by definition not possible with GM. This is onfimed by Capozza and i (994) fo eal-estate net ash flows. Absent fom the papes above is the onept of infomation asymmety and its impat on debt valuation and finaning hoies. Pioneeing wok in this aea is due to eland and Pyle (977), Ross (977), Myes (977), Myes and Majluf (984). One notable ontibution in this dietion was eently made by Duffie and ando (00), who essentially extends eland and Toft (996) to inopoate a noisy vesion of the asset value, i.e., the asset value plus a noise omponent, on the pat of investos. With impefet infomation about the fim s value, they show that thee exists a default-aival intensity poess as in edued-fom models 3 and edit speads emain bounded away fom zeo as matuity goes to zeo, whih emedies an impotant itique on stutual edit isk models. Gieseke and Webe (006) extends Duffie and ando (00) to onside not only inomplete infomation about the asset value but also impefet infomation about the default baie. In ontast to Duffie and Fo example, see Asquith et al. (994), Fanks and Toous (994), Weiss (990). ee Toft and Puyk (997), Goldstein et al. (00) fo moe details. 3 As opposed to the stutual models desibed above, edued-fom models model fatos influening the default event but typially (not neessaily) leave aside the question of what exatly tigges the default event. ee, fo example, Atzne and Delbaen (995), Jaow and Tunbull (995), Jaow et al. (997), Duffie and ingleton (999).

3 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) ando (00), Cetin et al. (004) povides a diffeent appoah to inomplete infomation in that the maket infomation set is not a noisy signal, but a edued set of the infomation that is available to manages. Related liteatue also inludes Kusuoka (999), Guo et al. (007), Moelle and hühoff (007), hmidt and Novikov (008). In this pape, we pesent the model that is gounded in the fat that equity ownes know moe about thei fim o investment pojet that affets the entie ash flow than do debt holdes (e.g., the human apital of the manage, the quality o the size of the pojet). Unlike Duffie and ando (00), we assume that fim s type is equity ownes pivate infomation, whih is epesented by a positive onstant. Equity ownes know the onstant, but debt holdes do not, and they have the pio distibution of the onstant at the beginning. Howeve, if the equity ownes fail to fulfil thei debt obligations, the debt holdes an foe the fim into liquidation at some osts and find out the tue value of the onstant. ine liquidation of assets is ineffiient and ostly, two paties will play a bagaining game as in Fan (000). Using Chapte as a ostly etifie, we give a lea pitue of how debt holdes beliefs about the onstant (the fim s type) ae updated aoding to the fim s default behavio, and desibe the optimal stategies of both paties in a dynami Nash equilibium. We suppose that the elevant state vaiable follows an AM instead of a GM, sine AM has moe ealisti featue than GM aoding to Capozza and i (994) and othes. Finally we povide an appoximate solution to the debt piing poblem unde asymmeti infomation. The emainde of the pape is oganized as follows: etion intodues the infomation asymmety on the ash flow and bankupty poliy. etion 3 pesents debt valuation and equity ownes finanial deisions unde symmeti infomation, whih seves as the efeene ase. etion 4 pesents the updating ule about the debt holdes beliefs on the fim s types and optimal stategies of both paties unde those beliefs. The debt value unde asymmeti infomation is also disussed. etion 5 onludes.. Assumptions and setup.. Assumptions Thoughout the pape, the model is based on the following assumptions. Capital makets ae pefet with no tansation osts. Ageny poblems between the manage of the fim and the equity ownes ae assumed away. 4 The default-fee tem stutue is flat with a onstant ate, at whih investos may lend and boow feely. The equity ownes do not have enough ash and need to aise a etain amount of ash I to undetake a pojet at time t 0. We also assume that this pojet is the only investment oppotunity available to the fim besides isk fee saving at the ate. The fim finanes the pojet by issuing a onsol oupon bond. It equies the fim to pay a onstant oupon at a fixed ate. The pojet will podue a ontinuous ash flow whih is positively elated to a state vaiable X t and yield an instantaneous opeating pofit given by X t. The onstant an be intepeted as fim s types, e.g., the quality o the size of the investment pojet. The model setup is simila to Goldstein et al. (00) in that X t an be taken as EIT. To emphasize the infomational ole of the debt, we assume that thee ae no taxes to be paid. The state vaiable X t has a uent value of X t0 and is modeled as an AM defined on a pobability spae (, F, Q ): dx t = dt dw t, () whee and denote the dift and diffusion paametes whih ae assumed to be positive onstants. W t is a standad Wiene poess unde a isk-neutal pobability measue Q, whih ensues the absene of abitage. We suppose that the state vaiable X t is pefetly obsevable by all agents, and that the fim s type is the equity ownes pivate infomation. Equity ownes know the onstant, but debt holdes do 4 Fo a desiption of assoiated poblems and thei impat on debt value and fim s investment deisions (see Paino and Weisbah, 999; Moelle, 004).

4 6 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) 3 37 not know. They believe that U[, H ] 5 at the beginning whee > 0 and < H. The assumption about the debt holdes beliefs is ommon knowledge. We will speify late the updating ule about debt holdes beliefs whih depends on the uent and past level of X t and the equity ownes default behavio. As in Duffie and ando (00), we also suppose that equity is not taded on the publi maket, and that the fim annot tade on its own debt... ankupty poliy and bagaining fomulations Now we onside the fim s bankupty poliy when the equity ownes finane the pojet as desibed in etion.. The fim s value V t is intepeted as the value of a laim on the entie ash flow X t,sov t is defined athe natually as the expetation of the disounted value of the entie ash flow onditional on the uent level X t [ ] ( V t = E Q t X s e (s t) Xt ds = t. () Aoding to (), the ash flow to equity ownes an possibly beome negative, so the equity ownes may have to infuse money into the fim to suppot the uent obligations if they don t want the fim goes bankupt. Howeve, in ode to maximize the equity value, the equity ownes may efuse to do so. We denote the optimal shutdown level that equity ownes hoose not to pay the oupon by X (), and define the bankupty time () as () = inf{s t : X s = X ()}. (3) If at bankupty time (), the fim is liquidated, the debt holdes an find out the tue value of, and the esidual value denoted by V(X ()) is [ ] ( ) V(X ()) = E Q X () s e (s ()) X () ds = (). (4) As in eland (994), a fation 0 (known to all agents) of esidual value will be lost to liquidation osts, leaving the debt holdes with value ( )((X ()/) (/ )) and equity ownes with nothing. ine liquidation is ostly and ineffiiently, debt holdes may not want to liquidate the fim when equity holdes theaten to default, and instead enegotiate with equity holdes the tems of the debt ontat. Intuitively, thee ae gains to be ealized between equity holdes and debt holdes by enegotiation, when the fim, as a going onen, is woth moe than its liquidation value. We take the bagaining model of Fan (000), whih pesented two fomulations of eoganization: debt equity swap and stategi debt sevie. 6 If the eoganization takes the fom of debt equity swap, howeve, due to infomation asymmety about, it is not always possible that two paties an eah an ageement though pivate wokouts. In that ase, equity holdes will opt to file fo Chapte of the U.. ankupty Code fo potetion. We assume that the osts assoiated with the Chapte poedue ae smalle than liquidation osts: it takes only ˇ((X t /) (/ )) in Chapte to find out tue value of, whee 0 <ˇ< is known to all paties. The osts an be thought of as the expenses involved. o equity holdes and debt holdes will bagain ove the value: ( ˇ)((X t /) (/ )). As in Fan (000), equity holdes have a bagaining powe of (0, ), whih esults in deviations fom absolute pioity. 7 We solve fo Nash solution and the final outome afte the fim omes out of the bankupty is speified: equity holdes get ( ˇ)((X t /) (/ )) and debt holdes get ( )((X t /) (/ )) ( ) ( ˇ)((X t /) (/ )). When the eoganization takes the fom of stategi debt sevie at an endogenously detemined tigge point, the debt value should be at least the expeted outome unde the debt holdes beliefs about if the debt holdes go though Chapte 5 U[, H] means that is unifomly distibuted on [, H]. The assumption on the distibution ould be easily elaxed without essential hange to the esults. 6 ee Fan (000) fo moe details. 7 Empiial studies onduted by Fanks and Toous (989), Ebehat et al. (990) show that APR is violated in about 50% of the ases.

5 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) , and equity ownes will offe a debt sevie that is less than as an equilibium outome of the bagaining poess. 3. Valuation and apital stutue deisions unde symmeti infomation In this setion, the deivation of the ontingent laims analysis is given unde symmeti infomation, i.e., all paties know the tue value of, whih will seve as the efeene ase. Now we onside the equity value E(X) and debt value D(X) unde two fomulations of eoganization pesented in Fan (000). If equity holdes adopt fist fomulation: debt equity swap, then the elevant ash flow that goes to equity holdes is X when X>X 0, whee X0 denotes the optimal shutdown level that equity ownes hoose not to pay the oupon. It an be shown that the equity value satisfies the diffeential equation: E X E XX (X ) = E, X > X 0. (5) whee E X and E XX denote the fist and seond deivatives with espet to X, espetively. The patiula solution to the Eq. (5) is [ ] ( E p = E Q t (X s )e (s t) Xt ds = t, (6) so equity value E(X) will be of the fom E(X) = A eˇ X A eˇ X, X>X 0. (7) whee A,A ae onstants to be detemined by appopiate bounday onditions, and ˇ,ˇ ae oots of the equation ˇ (/) ˇ = 0 whih ae given by ˇ = > 0, ˇ = < 0. (8) Fom the fat that ˇ > 0, we immediately dedue that A = 0 sine othewise the solution diveges. On the othe hand, at the shutdown level X 0, the fim ownes an get a popotion of the liquidation osts fom the assumption on the elative bagaining powe, i.e. ( ) X 0 E(X 0 ) = V(X0 ) =. (9) Aoding to value mathing ondition ( ) ( ) X 0 X 0 A eˇ X0 A eˇ X0 = (0) and A = 0, we get { A = ( ) ( ) } X 0 e ˇX0. () The optimal bankupty level X 0 is obtained by invoking the smooth pasting ondition E =. X X=X 0 () olving the optimization poblem above yields the following poposition.

6 8 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) 3 37 Poposition. If equity holdes adopt debt equity swap when they ae in finanial distess, we have () The optimal bankupty level fo the debt equity swap is X 0 = ˇ () The equity value E(X) is given by E(X) = ( ). (3) ) {( ) ( ) } X 0 eˇ (X X0 ), X>X 0. (4) (3) The debt value D(X) is given by D(X) = E(X), X>X 0. (5) The seond fomulation of eoganization is stategi debt sevie: two paties negotiate at an endogenously detemined tigge point to aept a edued level of debt sevie (whih is tempoay until the fotunes impove) but ontinue to opeate the fim. That is, equity holdes stat paying some stategi debt sevie s 0 (X) whih is less than the ontatual oupon, when X falls below some tigge level X 0, and esume the oupon payment when X>X0. Theefoe the debt value D(X) satisfies the following equations: D X D XX = D, X > X 0, D X (6) D XX s 0 (X) = D, X X 0. The patiula solution to the fist equation in (6) is [ ] D p = E Q t e (s t) ds =, (7) t so debt value D(X) will be of the fom D(X) = eˇ X eˇ X, X>X 0, (8) whee, ae onstants to be detemined by appopiate bounday onditions. Fo the simila aguments above, we have = 0. If X X 0, debt holdes equie the value of the debt to be at least equal to the liquidation value plus the popotional suplus fom the bagaining fo liquidation osts: D(X) = ( )V(X) ( ) V(X) = ( ). (9) Aoding to value mathing ondition ( ) X 0 eˇ X0 eˇ X0 = ( ) (0) and = 0, we get { ( )} X 0 = ( ) e ˇX0. () Plugging (9) into the seond equation in (6), we an find out the stategi debt s 0 (X). The tigge point fo stategi debt sevie X 0 is obtained by invoking the smooth-pasting ondition D ( ) =. () X X=X 0 Using simila aguments above, we easily establish the following esults:

7 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) Poposition. If equity holdes adopt stategi debt sevie when they ae in finanial distess, we have () The tigge point fo stategi debt sevie is X 0 = ˇ ( ). (3) () The stategi debt sevie amount when X X 0 is given by s 0 (X) = ( )X. (4) (3) The debt value and equity value unde negotiated debt sevie edutions ae { ( )} X 0 D(X) = ( ) eˇ (X X0 ), X>X 0, (a) ( X ( ), X X 0. (b) (5) E(X) = D(X). (6) One of impotant fatos influening the optimal apital stutue deision is the tax benefits. o unde no taxes assumption, the two tigge levels X 0 and X0 ae the same, and the equity and debt pies ae unhanged unde the two eoganization fomulations as well. If we assume that thee is a tax advantage fo issuing debt, all these will hange sine two paties will bagain ove futue tax benefit of debt unde the stategi debt fomulation. ee Fan (000) fo moe details. 4. eliefs updating and debt valuation unde asymmeti infomation In this setion, we assume that the state vaiable X is pefetly obsevable by all agents and that only the equity ownes know the tue value of. The debt holdes believe that U[, H ]atthe beginning. We late use aguments simila to equilibium efinements to explain how the beliefs ae updated and speify optimal stategies fo the equity ownes and the debt holdes unde those beliefs. We also disuss the debt value D(X) unde asymmeti infomation. 4.. Valuation and optimal stategy of the lowest type fim We fist desibe the optimal stategy of the lowest type fim, i.e. =, unde debt holdes fixed belief that U[, H ]. In this situation, the fomulations of eoganization fo the -fim ae a bit diffeent fom above. If -fim eoganizes in the fom of debt equity swap, it will file fo Chapte potetion (unde bankupty out supevision) when X hits a etain level X = X ( ) fo the fist time fom above, beause othewise it would pay moe than that it will pay unde bankupty out supevision. To find out the tue value of, the osts assoiated with the Chapte ae ˇ ((X /) (/ )). Fom the disussion above, the equity value denoted by E (X) satisfies the diffeential equation: E X E XX ) = E, X>X. (7) o equity value E (X) will be of the fom E (X) = ) C eˇ X C eˇ X, X>X, (8) whee C,C ae onstants to be detemined by appopiate bounday onditions.

8 30 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) 3 37 Fo the same eason above, C = 0. On the othe hand, at the shutdown level X, fom the assumption on the elative bagaining powe and the osts assoiated with Chapte, the equity ownes an get ( ) X E (X ) = ( ˇ)V (X ) = ( ˇ). (9) Aoding to value mathing ondition ( ) ( ) X X C eˇ X C eˇ X = ( ˇ) (30) and C = 0, we an find { ( ) } X C = ( ˇ) e ˇX. (3) The optimal bankupty level X is obtained by invoking the smooth-pasting ondition E = ( ˇ). X X=X (3) To simplify notation, we make the following definition ˇ. (33) Now the following esults an be easily established: Poposition 3. If equity holdes adopt debt equity swap when they ae in finanial distess, we have () The optimal bankupty level fo the debt equity swap is X = ˇ. (34) () The equity value is given by E (X) = ( ) } X { eˇ (X X ), X>X. (35) (3) The debt value is given by D(X) = { ( ˇ) )} eˇ (X X ), X>X. (36) If the fim eoganizes in the fom of stategi debt sevie, the equity ownes an pay s(x) less than, whih makes debt holdes indiffeent between aepting the offe and ejeting it. If the debt holdes ejet the offe and theaten to liquidate the fim, the equity ownes will espond by filing fo Chapte. Then two paties will bagain in Chapte unde out supevision and the tue is evealed (at some osts). If the debt holdes aept the offe, we assume that the equity ownes will stat seving s(x) when X falls below a etain level X = X ( ), whee X and s(x) ae to be detemined late. Reall that we have assumed the debt holdes believe U[, H ], so the debt holdes equie that the stategi debt equals to the one offeed by an aveage type fim ˆ = ( H )/. Theefoe the debt value D(X) satisfies the following equations: D X D XX = D, X > X, D X (37) D XX s(x) = D, X X.

9 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) Then debt value D(X) will be of the fom D(X) = D eˇ X D eˇ X, X>X, (38) whee D,D ae onstants to be detemined by appopiate bounday onditions. Fo the simila aguments above, D = 0. If X X, the debt holdes equie the value of the debt to be at least equal to an aveage type fim s liquidation value plus the popotional suplus fom the bagaining fo ( ˇ)ˆ((X t /) (/ )) (liquidation osts minus Chapte osts) if the debt holdes go though Chapte : D(X) = ( )ˆ ) ( ˇ)( ) ˆ ( = ˆ ˇ). (39) Aoding to value mathing ondition ( D eˇ X D eˇ X = ˆ ˇ) (40) and D = 0, we get { ( D = ˆ ˇ) } e ˇX. (4) Plugging (39) into the seond equation in (37), we an find out the stategi debt s(x). The tigge point fo stategi debt sevie X is obtained by invoking the smooth-pasting ondition D ˆ( ˇ) =. (4) X X=X As aguments above, We an give the following esults: Poposition 4. If equity holdes adopt stategi debt sevie when they ae in finanial distess, we have () The tigge point fo stategi debt sevie is X = ˇ ˆ( ˇ). (43) () The stategi debt sevie amount when X<X is given by s(x) = ˆ( ˇ)X. (3) Unde negotiated debt sevie edution, the debt value is { ( D(X) = X } eˇ (X X ), X>X, (a) ( X, X X. (b) The equity value of the lowest type fim equals the fim s value minus the debt value, i.e. ( X E(X ) = { X } eˇ (X X ), X>X, (a) ( X ( ), X X. (b) whee (44) (45) (46) X = X, = ˆ( ˇ). (47) If the tue value of is at the lowe end of the belief: =, to avoid the Chapte osts, the equity ownes have to pay highe stategi debt sevie than they would have paid unde omplete infomation. o the equity ownes of -fim have to weigh the tade-off between the Chapte osts and the osts to avoid it. We laify this tade-off as follows:

10 3 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) 3 37 emma 5. If we assume that the debt holde s belief is U[, H ] and fixed thoughout, then the equity ownes of -fim should () hoose stategi debt sevie if /ˆ >( ˇ)/; () hoose debt equity swap if /ˆ <( ˇ)/; (3) be indiffeent between two stategies if /ˆ = ( ˇ)/. Poof:. see Appendix A. In the exteme ase whee = H, namely unde full infomation, we ae bak to pevious situations. 4.. elief updating and debt piing y now we have desibed the optimal stategy of the equity ownes of -fim unde the debt holdes fixed belief that U[, H ]. Next we desibe the optimal stategy of the debt holdes, speify the updating ule about thei beliefs, and desibe the optimal stategy of the equity ownes of a fim with any type [, H ], whee the state vaiable X is pefetly obsevable by all agents and only the equity ownes know the tue value of. The main esult is given as follows: Poposition 6. Unde the assumptions in etion and that the debt holdes an use only deteministi stategies, we have the following dynami Nash equilibium: () If /ˆ >( ˇ)/, we an get: The debt holdes adopt the following stategy: equie the equity ownes to pay the ontatual oupon ifx>x, and minimum aeptable stategi debt flow s(x) = ˆ( ˇ)X if X X, whee X = ˇ ˆ( ˇ). (48) If the equity ownes deviate, foe the fim into Chapte bankupty. The debt holdes always believe that U[, H ]. The equity ownes of a fim with any type [, H ] unde the above desibed belief adopt the following stategy: pay when X>X and s(x) when X X. () If /ˆ ( ˇ)/, we define suh that /ˆ = ( ˇ)/, whee ˆ = (/)( H ), and we an get: The debt holdes adopt the following stategy: equie the fim to pay the ontatual oupon if X>X, and minimum aeptable stategi debt flow s (X) = ˆ ( ˇ)X if X X, whee X = ˇ ˆ ( ˇ) = ˇ. (49) If the fim deviates, foe the fim into Chapte bankupty. The debt holdes update thei beliefs on as follows: at any time t, if the equity ownes have been willing to pay the oupon when X>X and s (X) when X X, the updated beliefs on ae U[, H ], X t >X, U[ t, H ], X X t >X, (50) U[, H ], X t X. whee X t = inf X s, t = 0 s t (X t ˇ ). (5)

11 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) The equity ownes of a fim with type [, ) adopt the following stategy: pay oupon if X> X () and file fo Chapte one X hits X (), whee X () = ˇ. (5) The equity ownes of a fim with type [, H ] adopt the following stategy: pay oupon if X t >X and s (X) if X t X. Poof:. see Appendix. Now we an deive an appoximate solution to the isky debt piing poblem unde debt holdes beliefs as follows: Poposition 7. esult: Unde the dynami Nash equilibium in Poposition 6, we have the following debt piing () If /ˆ >( ˇ)/, the pie of the debt unde the beliefs that U[, H ] is given by { ( D(X) = ˆ ˇ) } eˇ (X X ), X>X, (a) ( X ˆ( ˇ), X X. (b) whee X has been given in (48). () If /ˆ ( ˇ)/, the debt holdes update thei beliefs about the aoding to (50) in Poposition 6, and when the equity ownes of the fim do not file fo Chapte at X, the pie of the debt is given by If U[, H ], then D(X) (X ˇeˇ ˇ ) m( ) ( H ) { { ( (X ˇ)eˇ ˇ ) n( ) H ˇ( H ) H ˆ ( ˇ) If U[ t, H ], then D(X) (X ˇeˇ ˇ ) m( t ) ( H t ) { { ( (X ˇ)eˇ ˇ ) n( t ) H ˇ( H t ) H t ˆ ( ˇ) If U[, H ], then whee D(X) = { ˆ ( ˇ) m(y) = y ˇ ln( /y) n(y) = y ˇ( y) (53) )} eˇ (X X ) }. (54) )} eˇ (X X ) }. (55) )} eˇ (X X ). (56) ˇ ( y) y ˇ ln( /y) 3ˇ3 y ), (57) 3 y 3ˇ3 (y ) 6 3, (58) y

12 34 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) 3 37 X = ˇ ˆ ( ˇ). (59) Poof:. see Appendix C. 5. Conlusion This pape deals with the impat of asymmeti infomation on debt valuation and equity ownes finanial deisions duing finanial distess, and show how the debt holdes beliefs ae updated aoding to the state vaiable and the fim s default behavio. This is done within a famewok simila to those poposed by Goldstein et al. (00) o Moelle (004). In these papes, the ash flow dynamis ae modeled as a Geometi ownian motion, whih implies that ash flows neve beome negative. In ontast, we assume the ash flow to follow an Aithmeti ownian motion, whih allows fo negative values and has moe ealisti featue. In the study, we speify the updating ule about the debt holdes beliefs, and optimal stategies of the equity ownes and the debt holdes unde those beliefs. Using Chapte as a ostly etifie, we ae able to diffeentiate low-type fims fom high-type fims who ae moe likely to sueed in pivate wokouts duing finanial distess. In addition, an appoximate solution to the debt piing poblem unde asymmeti infomation is povided. Ou esults show that infomation asymmety heavily impat debt valuation and finanial deision of both paties. Ou esults uially depend on some key assumptions: fixed investment oppotunity, non-tading of equity. At the same time, we didn t onside the possibility of asset substitution as in eland (998) and ageny poblems as in Moelle (004). The apital stutue deisions onsideed hee is also a stati one, i.e., we haven t onsideed the impat of the possibility to hange the leveage in the futue. It would be vey inteesting to onside these questions unde asymmeti infomation. Aknowledgment The authos would like to thank the Key Gant Pojet of Chinese Ministy of Eduation (No ) fo suppot. Appendix A. Poof of emma 5 To hoose the optimal stategy between debt equity swap and stategi debt sevie, we need only to ompae the equity values in (36) and (46). o we have the following esult: The neessay and suffiient ondition fo that stategi debt sevie stitly dominates debt equity swap is { ( ) } X { eˇ (X X ) < X } eˇ (X X ). (A.) Applying Eqs. (35), (43) and (47), (A.)an be simplified e (ˇ )/() > ˆ( ˇ) e (ˇ )/(ˆ( ˇ)). ˇ ˇ (A.) eause y = xe (/x) is a monotonous ineasing funtion when x>0, (A.) is tue if and only if ˇ > ˆ( ˇ) ˇ ˆ > ˇ, (A.3) i.e., stategi debt sevie stitly dominates debt equity swap if and only if /ˆ >( ˇ)/.Onthe othe hand, debt equity swap stitly dominates stategi debt sevie if and only if /ˆ <( ˇ)/. This finishes the poof.

13 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) Appendix. Poof of Poposition 6. If /ˆ >( ˇ)/, whee ˆ = ( H )/, fo any [, H ], we have (/)( H ) ˆ > ˇ. (.) y emma 5, the optimal stategy fo the equity ownes of a -fim is stategi debt sevie, even when the debt holdes belief is that the type of the fim is unifomly distibuted on [, H ]. o all equity ownes will adopt stategi debt sevie unde the oiginal beliefs that U[, H ]. Namely, pay when X>X and s(x) = ˆ( ˇ)X when X X, with X and s(x) defined in (43) and (44). eause equity ownes of a fim with any type adopt the same tategy, the debt holdes always believe that U[, H ].. If /ˆ ( ˇ)/, Fo a low-type fim, [, ), we have (/)( H ) < ˇ. (.) Aoding to emma 5, the equity ownes of a fim of type [, ) pefe debt equity swap, i.e., filing fo Chapte one X hits X () even unde the debt holdes belief that the type of the fim is unifomly distibuted on [, H ], whee X () = ˇ. (.3) The poof of high type fims [, H ] is the same as in the above ase. How the debt holdes update thei beliefs on an be given as follows: as disussed above, it is always moe ostly fo the equity ownes of a -fim ( < ) to pay stategi debt than to file fo Chapte even unde the debt holdes belief that the type of the fim is unifomly distibuted on [, H ], so if X hits X () fom above and the equity ownes don t file fo Chapte, the debt holdes an safely move up the low end of suppot of the beliefs fo a given fim of type [, ). If X hits X fom above, the equity ownes optimal stategy is to seve stategi debt. The debt holdes belief will be fixed at [, H ]. This ompletes the poof. Appendix C. Poof of Poposition 7. If /ˆ >( ˇ)/, aoding to Poposition 6, the debt holdes always believe that U[, H ] and the optimal stategy fo the equity ownes of a -fim is stategi debt sevie. y Poposition 4, we an easily get (53).. If /ˆ ( ˇ)/, the debt holdes update thei beliefs aoding to (50) in Poposition 6. Now we show the piing fomula (54) fist. If U[, H ], then [, )o [, H ]. Next we onside two situations above: (a) [, ). When [, ), /((/)( H )) < ( ˇ)/,byPoposition 6, the optimal stategy fo the equity ownes of a -fim is debt equity swap, and the debt pie an be omputed as the expetation of debt pie D (X) on[, ), whee D (X) is the pie of a onsol issued by a fim with type [, ) who optimally files fo Chapte when X t hits X (). Aoding to (37) in Poposition 3, we an get { ( )} D (X) = ( ˇ) X () eˇ (X X()), X X (). (C.) o the expetation of D (X) on[, ) an be simplified D (X) d = ( ) H ˇ ) ( H ) ˇeˇ(X ( H ) e (ˇ)/() d

14 36 R. Xu,. i / Reseah in Intenational usiness and Finane 4 (00) 3 37 ( (X ˇ)eˇ ˇ ) e (ˇ)/() d. (C.) ˇ( H ) To ou knowledge, thee is no expliit fomula fo the definite integals in (C.), so we onside Taylo expansions of e (ˇ )/(), i.e. e (ˇ )/() ˇ ˇ 3ˇ (C.3) We an plug (C.3) into the definite integals in (C.), and get e (ˇ)/() d ˇ ln ˇ ( ) e (ˇ)/() d ˇ( ) To simplify notation, we make the following definitions m(y) = y ˇ ln y n(y) = y ˇ( y) Put (C.3) (C.7) in (C.), we an get D (X) d ( ) H ˇ ( y) y ˇ ln y ( H ) ˇeˇ(X ( H ) ˇ ln 3ˇ3 ( ) 3, (C.4) 3ˇ3 ). (C.5) 6 3 3ˇ3 y ), (C.6) 3 y 3ˇ3 ( y) 6 3. (C.7) y ˇ ) m( ) ( (X ˇ)eˇ ˇ ) n( ). ˇ( H ) (b) [, H ]. When [, H ], /((/)( H )) ( ˇ)/, bypoposition 6, the optimal stategy fo the equity ownes of a -fim is stategi debt sevie, and the debt pie an be omputed as the expetation of debt pie D (X) on[, H ], whee D (X) is the pie of a onsol issued by a fim with type [, H ]. Aoding to (45), We an get D (X) by eplaing with, i.e. { ( )} D (X) = X ˆ ( ˇ) eˇ (X X ),X>X. (C.9) (C.8) whee X = ˇ ˆ ( ˇ). (C.0) o the expetation of debt pie D (X) on[, H ] when X>X is H { { D (X) d = H H H ˆ ( ˇ) Finally, put (C.8) and (C.) togethe, we get (54). )} eˇ (X X ) }. (C.) If U[ t, H ], the optimal stategy fo the equity ownes of a -fim is debt equity swap when U[ t, ), and the optimal stategy is stategi debt sevie when U[, H ]. As disussed above, we an easily get (55) by eplaing with t in (54). If U[, H ], the optimal stategy fo a -fim is stategi debt sevie, and the debt pie an be omputed as the expetation of debt pie D (X) on[, H ] whee D (X) is given in (76), i.e. H The poof is omplete. D (X) H d = { ˆ ( ˇ) )} eˇ (X X ). (C.)

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