Contingent capital in the form of debt that converts to equity when a bank faces financial distress has been

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1 Published online hed of prin April 7, 1 MANAGEMENT SCIENCE Aricles in Advnce, pp ISSN prin ISSN online hp://dx.doi.org/1.187/mnsc INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. Coningen Cpil wih Cpil-Rio Trigger Pul Glssermn Grdue School of Business, Columbi Universiy, New York, New York 17, pg@columbi.edu Behzd Nouri Indusril Engineering nd Operions Reserch Deprmen, Columbi Universiy, New York, New York 17, bn164@columbi.edu Coningen cpil in he form of deb h convers o equiy when bnk fces finncil disress hs been proposed s mechnism o enhnce finncil sbiliy nd void cosly governmen rescues. Specific proposls vry in heir choice of conversion rigger nd conversion mechnism. We nlyze he cse of coningen cpil wih cpil-rio rigger nd pril nd ongoing conversion. The cpil rio we use is bsed on ccouning or book vlues o pproxime he regulory rios h deermine cpil requiremens for bnks. The conversion process is pril nd ongoing in he sense h ech ime bnk s cpil rio reches he minimum hreshold, jus enough deb is convered o equiy o mee he cpil requiremen, so long s he coningen cpil hs no been depleed. We derive closed-form expressions for he mrke vlue of such securiies when he firm s sse vlue is modeled s geomeric Brownin moion, nd from hese we ge formuls for he fir yield spred on he converible deb. A key sep in he nlysis is n explici expression for he frcion of equiy held by he originl shreholders nd he frcion held by convered invesors in he coningen cpil. Key words: probbiliy; diffusion; sochsic model pplicions; finnce; sse pricing Hisory: Received Augus 31, 1; cceped December 5, 11, by Gérrd P. Cchon, sochsic models nd simulion. Published online in Aricles in Advnce. 1. Inroducion Severl proposls for enhncing he sbiliy of he finncil sysem include requiremens h bnks hold some form of coningen cpil, mening equiy h becomes vilble o bnk in he even of crisis or finncil disress. Vrins of his ide differ in he choice of rigger for he civion of coningen cpil nd in how he cpil is held before riggering even. The Dodd Frnk Ac clls for regulors o sudy he poenil effeciveness of coningen cpil, nd specific definiions for riggering evens were pu forwrd in recen consulive documen issued by he Bsel Commiee on Bnking Supervision 1. Flnnery 5 proposed reverse converible debenures form of deb h convers o equiy if bnk s cpil rio flls below hreshold. His proposl uses cpil rio bsed on he mrke vlue of he bnk s equiy nd he book vlue of is deb. Flnnery 9 upded he proposl nd renmed he securiies coningen cpil cerifices. Kshyp e l. 8 proposed lock box o hold bnk funds h would be relesed in he even of crisis; in his proposl, he rigger is sysemic even, nd no risk of bnkrupcy n individul insiuion. McDonld 11 nd he Squm Lke Working Group 9 proposed coningen cpil wih rigger h depends on he helh of boh n individul bnk nd he bnking sysem s whole. The converible securiies designed by he U.S. Tresury for is Cpil Assisnce Progrm my be viewed s ype of coningen cpil in which bnks hold he opion o conver preferred shres o common equiy nd find i dvngeous o do so if heir shre price drops sufficienly low; his conrc ws sudied by Glssermn nd Wng 11. Alernive proposls for he design of coningen cpil hve led o work on vluion. McDonld 11 priced coningen cpil wih dul rigger hrough join simulion of bnk s sock price nd mrke index. Penncchi 1 compred severl cses by simulion in jump-diffusion model of bnk s sses. Albul e l. 1 obined closedform pricing expressions under he ssumpion h ll deb hs infinie muriy nd h he conversion rigger is defined by hreshold level of sses. Rviv 4 lso used n sse-level rigger nd obins closed-form expressions wih finie-muriy deb. Von Fursenberg 11 buil binomil ree for he evoluion of bnk s cpil rio. Sundresn nd Wng 1 showed h seing he conversion rigger level of he sock price my resul in muliple soluions or no soluion for he mrke price of he sock nd converible deb, rising quesions bou he vibiliy of conrcs designed wih mrke-bsed 1

2 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. riggers. Koziol nd Lwrenz 1 noed h coningen cpil cn increse incenives for risk king by mking bnkrupcy more remoe. Among recen lernives o he mechnisms considered in hese ppers, Duffie 1 proposed mndory righs offerings by bnks fcing finncil disress, McAndrews 1 proposed combinion of righs offering nd converible deb, nd Penncchi e l. 1 suggesed bundling coningen cpil wih buybck opions for equiy holders. Brennn nd de Longeville 1 esimed he overll poenil size of he coningen cpil mrke one rillion dollrs nd discussed invesor perspecives on some lernive feures. We develop model o sudy coningen cpil in he form of deb h convers o equiy bsed on cpil-rio rigger. The bnk is required o hold minimum rio of equiy o ol sses equivlenly, i fces n upper bound on leverge; if is sse vlue drops oo low, pr of is deb convers o equiy o minin he required cpil rio. Our seing is hus similr o Flnnery s 5, 9, hough he compred he mrke vlue of equiy o he book vlue of deb. Exising regulory cpil requiremens for bnks re bsed primrily on book vlues. Under Bsel rules, bnks mus minin regulory cpil equl o les 8% of heir risk-weighed sses. U.S. bnks lso fce n overll cpil-o-sses consrin wih minimum of 3% nd hreshold of 5% o qulify s well cpilized. All of hese rios re bsed on regulory ccouning mesures of deb nd cpil rher hn he mrke price of bnk s sock. Exising issunces o de he coningen core cpil CoCo bonds issued by Lloyd s Bnking Group in November 9, morgge lender Yorkshire Building Sociey in December 9, nd Credi Suisse in Februry 11, nd he principl wrie-down bonds issued by Rbobnk in Mrch 1 ll use riggers bsed on regulory cpil rios nd no mrke prices. Flnnery 5, 9 nd Penncchi e l. 1 dvoce he use of mrke d becuse i is coninuously upded, forwrd looking, nd less vulnerble o ccouning mnipulion, while noing concerns h mrke vlues could poenilly be mnipuled o rigger conversion. The resuls of Sundresn nd Wng 1 show h defining n inernlly consisen mrke-bsed rigger cn be problemic. Becuse here re good rgumens for boh mrke-vlue nd book-vlue riggers, boh ypes of securiies meri invesigion; becuse he wo require somewh differen nlysis, here we limi ourselves o book-vlue cpil rios. A disinguishing feure of our nlysis is h we model pril nd ongoing conversion of coningen cpil s bnk s cpil rio declines, consisen wih Flnnery s 5 originl proposl. Achry e l. 1, p. 166 cll his progressive conversion. Previous models hve relied on he ssumpion h converible deb is convered in is enirey s soon s hreshold is hi. Insed, we ssume jus enough conversion kes plce o minin he minimum cpil rio required, leding o process of coninuous conversion. This pril conversion process lends iself o somewh lrger rnche of converible deb hn ll--once conversion would, nd i mkes he full rnche ruly coningen, wih ech lyer convered only s needed. Wih ll--once conversion, mos of he deb is convered oo erly or oo le. Pril conversion hs imporn implicions for invesors: s coningen cpil convers o equiy, bond holders become shreholders nd hus shre in ny coss or benefis o shreholders of subsequen conversion. We will show h incresing he minimum cpil requiremen hs he effec of slowing conversion nd hus shifs more of he diluion cos from conversion o invesors who becme shreholders hrough erlier conversion of deb. A higher cpil rio cn herefore benefi he originl shreholders if he loss in sse vlue is sufficienly lrge; he vlue of he converible deb need no be monoone in he required cpil rio. We underke our vluion in srucurl model, sring from he firm s sses. The firm s cpil srucure is comprised of senior unconverible deb, coningen cpil, nd equiy. Mrke vlues of deb nd equiy re deermined, s usul, by viewing hese s clims on he sses; bu he book vlue of deb is clculed by discouning fuure coupon nd principl pymens he yield which he bond ws issued, consisen wih ccouning rules. We use he resuling book vlues in our cpil rio. The mrke nd book vlues of deb mus gree issunce nd muriy, nd we incorpore his consrin in our nlysis o fix he coupon res. In our frmework, invesors in coningen cpil hold clims on four ypes of pymens: coupons on unconvered deb, he remining principl on converible deb, dividends erned hrough deb convered o equiy, nd he vlue of his equiy he muriy of he deb. We vlue he coningen cpil s he sum of he vlues of hese pymens. Once he coningen cpil is exhused, we ssume h filure o mee he minimum cpil requiremen resuls in seizure nd liquidion by regulors. Liquidion occurs prior o bnkrupcy in he sense h bnk hs posiive equiy when i firs breches is cpil rio. We incorpore poenil liquidion coss for shreholders nd lso for bond holders in our vluion. Indeed, hese coss hve significn impc on our vluions, s does sse voliliy. Asse voliliy ffecs boh he likelihood of

3 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS 3 Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. conversion of deb o equiy nd he upside poenil of equiy following conversion. The res of his pper is orgnized s follows. Secion presens our model of he firm nd he conversion of deb o equiy, nd 3 exmines how equiy is lloced beween convered shreholders nd he originl shreholders s he vlue of he firm s sses evolve. Secion 4 inroduces dividends. Secion 5 deils he csh flows pid o invesors in he firm s senior deb, coningen cpil, nd equiy, nd 6 presens explici expressions for he vlues of hese csh flows. Secion 7 closes he model by solving for he coupons on he wo ypes of deb o eque mrke nd book vlues issunce; from hese we ge he yield spred on coningen cpil. Secion 8 exends he model o disinguish beween mrke nd book vlue of sses. Secion 9 illusres our resuls hrough numericl exmples. Deiled clculions leding o our vluion formuls re deferred o ppendices.. Model of he Firm Our model of he firm or bnk builds on long line of reserch on cpil srucure h includes Meron 1974, Blck nd Cox 1976, Lelnd 1994, nd numerous subsequen ppers. This pproch srs by modeling he dynmics of firm s sses nd hen prices deb nd equiy s clims on hose sses. In Meron 1974, he firm defuls he muriy of he deb if is sse vlue is less hn he fce vlue of he deb. In Blck nd Cox 1976, bnkrupcy occurs when sse vlue drops o n exogenous reorgnizion boundry, nd in Lelnd 1994, he ime of deful is chosen sregiclly by shreholders. In our seing, we will need o provide corresponding prescripion for he conversion of coningen cpil o equiy, s well s specifying rigger for liquidion of he firm. We inerpre he liquidion even s resuling from seizure by regulors when he firm is unble o susin is cpil requiremen, which, by design, occurs prior o rdiionl bnkrupcy even. Our sring poin is sochsic process V h models he book vlue of he firm s sses; his process drives he required level of cpil in our model, jus s ccouning-bsed mesures of sse vlue drive cpil requiremens in prcice. For rcbiliy, we ke V o be geomeric Brownin moion, dv V = r d + dw 1 where W is sndrd Brownin moion, nd is consn pyou re o he firm s securiy holders. In.1, we clcule book vlues for senior nd converible deb; subrcing he book vlue of deb from he book vlue of sses leves Q, he book vlue of shreholder s equiy, which is our mesure of cpil. In prcice, regulory cpil lso includes cerin deb insrumens no cpured in our model. Our minimum cpil requiremen is expressed s lower bound on Q /V. We use hese book vlues o model cpil requiremens nd he conversion of deb o equiy. Bu for vluion, we need o clcule mrke vlues: we ke he mrke vlue of securiy o be he expeced discouned vlue of csh flows received by invesors, irrespecive of book vlues. In he bsic version of our model, we ssume h he mrke vlue of he firm s sses equls he book vlue V in oher words, we ssume he bnk uses mrk-o-mrke ccouning for is sses. 1 In he more generl version of our model inroduced in 8, we represen mrke nd book vlues of sses hrough correled geomeric Brownin moions, hus llowing n imperfec relionship beween he wo nd creing some unceriny bou how much mrke vlue will be relized when liquidion is riggered by book-vlue-bsed cpil rio. In eiher version of he model, we clcule mrke vlues for senior nd converible deb s coningen clims on he mrke vlue of sses. We pin down he mrke vlues of hese coningen clims wih he consrin h mrke nd book vlues of deb mus coincide issunce nd muriy: when deb is issued, is book vlue is recorded is selling price mrke vlue, nd when i mures, is book vlue nd mrke vlue equl he finl pymen of principl nd ineres. In shor, we use he book vlue of sses o drive he conversion of coningen cpil, nd we use he mrke vlue of sses o drive he vluion of coningen cpil. Keeping rck of hese wo noions of vlue is essenil o pricing securiies h depend on n ccouning-bsed rigger. Our model enils severl idelizions nd simplificions. We ssume h cpil rios cn be observed coninuously; in prcice, regulory cpil is clculed qurerly, bu lrge bnks rouinely clcule inernl economic cpil on dily bsis, so he necessry informion could in principle be moniored for regulory purposes o rigger conversion. A limiion of our model is h i does no llow for jumps in sse vlue lrge jump could poenilly wipe ou ll he coningen cpil nd leve he firm bnkrup. This ype of even is beyond he scope of our model. 1 This would be he cse under Finncil Accouning Sndrd 157. Even prior o his proposed rule, using d from 1 5, Clomiris nd Nissim 7 repored h for mny bnk sses in conrs o hose of nonfinncil firms, book vlue is indeed close o fir vlue.

4 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger 4 Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org..1. Deb The firm issues ordinry senior deb s well s junior converible deb. Boh ypes of deb re issued ime zero nd mure ime T >. The senior deb hs fce or pr vlue of D due ime T nd coninuous coupon re of c, mening h i pys c D per uni of ime. The deb is issued price of D. From n ccouning perspecive, he effecive ineres re for he deb is he discoun re d h eques he csh rised D o he presen vlue of fuure pymens promised on he deb; i.e., he vlue of d h solves T [ D = De dt + c De ds ds = D e d T 1 c + c ] d d The book vlue of he deb ny inermedie de, < < T, is hen [ D = D e d T 1 c + c ] d d if he firm hs no ye filed. In oher words, hroughou he life of he deb, book vlue is clculed by discouning remining pymens he effecive ineres re which he deb ws originlly issued. In he bsence of ny oher ype of deb, we would model deful s occurring he firs ime he vlue of he firm s sses fll below he boundry defined by D, T. This is n insnce of he mechnism used by Blck nd Cox 1976, hough hey used n exponenil boundry, which corresponds o seing c =. The boundry in Blck nd Cox 1976 is ofen inerpreed s proecive deb covenn, nd h inerpreion could be pplied here. In he cse of reguled bnk, which is our focus, he boundry will serve o define minimum cpil requiremen he bnk mus minin, rher hn prively negoied covenn. The cpil requiremen will se he liquidion boundry higher by he moun of he required cpil buffer hn he deful boundry. The bnk is seized by regulors before bnkrupcy if he cpil requiremen is no minined. Nex we inroduce converible deb wih fce vlue of B, coninuous coupon re c 1, nd muriy T, issued ime zero price of B. The ssumpion h ll of he deb hs he sme muriy T is simplifying idelizion. The effecive ineres re d 1 eques B o he presen vlue of he promised pymens of coupon nd principl, T [ B = Be d1t + c 1 Be d1s ds = B e d 1T 1 c 1 + c ] 1 d 1 d 1 As pr of he originl coningen cpil issunce convers o equiy, he remining principl decreses, bu we pply he sme effecive ineres re d 1 o clcule he book vlue of he deb ousnding. If he remining principl ime is B, hen he book vlue ime is [ B = B e d 1T 1 c 1 + c ] 1 3 d 1 d 1 We ke up he conversion mechnism h deermines B in he nex subsecion. Equions nd 3 ke he coupon res c 1 nd c s given. As pr of our nlysis, we will solve for he vlues of c 1 nd c h mke he vlues of he wo ypes of deb consisen wih he overll vlue of he firm. In priculr, we will choose c 1 nd c o ensure h he iniil vlues B nd D re consisen wih mrke vlues of deb given he fce mouns B nd D nd he dynmics of he firm s sse vlue... Conversion from Deb o Equiy We denoe by V he book vlue of he firm s sses ime. Subrcing he firm s deb from is sses ime leves Q = V B D 4 we refer o Q s cpil, shreholder s equiy, or simply equiy, bu i should be inerpreed s book vlue or regulory mesure nd no s he mrke vlue of equiy, becuse nd 3 re ccouning bsed mesures of deb. Indeed, he gol of our nlysis is o clcule mrke vlues bsed on he conrcul erms of he coningen cpil. The firm is required o minin cpil rio of les, < < 1, which imposes he consrin Q V or 1 V B + D For exmple, o model bnk h is required o hold cpil equl o 5% of sses, we would se = 5. As V flucues, bnk could be in dnger of violing his requiremen; he coningen cpil convers from deb o equiy decresing B nd incresing Q o minin he consrin s long s possible. Flnnery 5 inroduced his mechnism using he mrke vlue of equiy, rher hn regulory cpil, o drive conversion. Before formlizing he conversion mechnism in our model, we consider he exmple in Figure 1. Pr of he figure shows n iniil blnce shee wih 1 in sses, 6 in senior deb nd 3 in converible deb, leving 1 in shreholder s equiy. For We cn model cpil requiremen ied o risk-weighed sses, rher hn ol sses, by djusing he vlue of. The verge rio of risk-weighed sses o ol sses over ll FDIC bnks ws 7% 75% during 3 1, so cpil requiremen of 8% of risk-weighed sses could be pproximed by requiremen of 5% 6% of ol sses. For he lrges bnk holding compnies, he sse rio is 4% 6%, corresponding o lower vlue of. The djusmen in could be ilored o specific insiuion bsed on is mix of sses.

5 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS 5 Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. Figure 1 Iniil Blnce Shee wih 1% Cpil Rio Sisfied; b Afer Drop in Asse Vlue; c Afer Conversion of Deb o Equiy Resoring he 1% Cpil Rio Asses Libiliies Asses Libiliies Asses Libiliies V = 1 D = 6 V = 95 D = 6 V = 95 B = 3 B = 3 Q = 1 Q = 5 simpliciy, we consider minimum cpil requiremen of 1%, which is jus me in. In b, he firm s sses drop o vlue of 95; he loss of 5 is bsorbed by equiy. To mee he cpil requiremen, he firm convers 4.5 of converible deb o equiy o rrive he blnce shee in c, which gin jus mees he cpil requiremen. In our model, V evolves coninuously in ime wih coninuous phs, nd we will derive he process of miniml conversion under which conversion kes plce precisely hose imes which Q = V ; i.e., imes which 1 V = B + D. We will ssume hroughou h he bnk is iniilly well cpilized in he sense h Q > V. In erms of he moun B of principl remining no convered ime, he cpil consrin is [ 1 V B e d 1T 1 c 1 d 1 [ + D e d T 1 c d + c 1 d 1 ] + c d ] 5 Once he coningen cpil is exhused, he consrin becomes 1 V D. Le b denoe he firs ime 1 V = D, which poin he firm is seized by regulors. Define L by seing 1 L { =mx s B+ De d T s 1 c /d +c /d 1 V s e d 1T s 1 c 1 /d 1 +c 1 /d 1 + } Then we show below h 1 L is he cumulive moun of principl convered up o ime. More precisely, we clim h if we se B = B 1 L, hen 5 is sisfied for ll b, nd 1 L is he les moun of conversion h mees his condiion. Equion 6 simplifies when boh kinds of deb hve consn book vlue. This holds when he deb is issued pr i.e., B = B nd D = D so he coupon res coincide wih he effecive ineres res, mening h c 1 = d 1 nd c = d. In his cse, Equion 6 simplifies o + 1 L = B + D 1 min V s 7 s 6 b c D = 6 B = 5.5 Q = 9.5 The conversion process in his cse becomes esier o visulize if we inroduce wo hresholds, = B + D 1 b = D 8 1 Under our snding ssumpion h he cpil consrin is sisfied ime zero, V >. Conversion srs when V firs his. Subsequenly, ech insn which V his level lower hn ny previously reched, ddiionl coningen cpil is convered o sisfy he consrin. Once V his b which hppens b, he coningen cpil hs been fully convered see Figure. The process L is given by { + } L = min min V s b s for ll T. 9 The widh b is 1 imes he fce vlue B of coningen cpil. A similrly rcble cse holds when he wo ypes of deb py no coupon nd hve he sme effecive ineres re h is, when c 1 = c = nd d 1 = d = d. We formlize he conversion mechnism in he following resul, in which we view 6 s mpping from ph of V o ph of L: Proposiion.1. Le D, B, c 1, c, d 1, nd d be given. The funcion L b, defined by pplying 6 o Figure V b Illusrion of he Conversion Process Noes. Conversion begins when V reches he upper boundry. The ol moun convered o ime is 1 L, where L is he disnce from he running minimum of V o, cpped b. V L

6 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger 6 Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. V b, is he only funcion wih he following properies: i L is incresing nd coninuous wih L = ; ii V B 1 L e d 1T 1 c 1 /d 1 + c 1 /d 1 D V for ll b ; iii L increses only when equliy holds in ii. Any funcion sisfying i nd ii is greer hn or equl o L on b. Condiion i is nurl for he process of cumulive conversion. Condiion ii ses h conversion occurs o preserve he required cpil rio unil b, when he coningen cpil is exhused. Condiion iii ses h conversion occurs only s needed when he firm is is minimum cpil requiremen. The resul follows from he sndrd reflecion mpping s in Hrrison 1985, p. 1 pplied o he funcion V 1 1 B [ e d 1T 1 c 1 d 1 + c ] 1 + D d 1 The proposiion deermines L only up o he ime b, when he coningen cpil hs been fully convered. Using 6 or he specil cse in 9, we cn convenienly exend he definiion of L o he inervl T, even if b < T. 3. Equiy Allocion We will vlue he coningen cpil bond by clculing he expeced presen vlue of he pymens o he holder of he securiy. The pymens include coupons pid coninuously in proporion o he unconvered deb, ny remining principl muriy, frcion of he firm s equiy erned hrough conversion, nd dividends pid on frcion of equiy. From he nlysis in he previous secion, we cn deermine how much of he coningen cpil remins unconvered ech poin in ime. To vlue he equiy componen s he bond convers, we need o nlyze wh frcion of he firm s equiy is held by invesors who were convered from coningen cpil holders o equiy holders. We limi ourselves o he cse c 1 = d 1 nd c = d, which, s explined in he previous secion, eques book vlue o remining fce vlue for boh kinds of deb. To moive he nlysis h follows, consider gin he exmple of Figure 1. Suppose, for simpliciy, h he firm srs wih 1 shres ousnding. By wriing down 4.5 in converible deb in c, he firm uomiclly dds 4.5 o equiy, bu how he ol equiy is pporioned o he prior nd new shreholders depends on how mny new shres re issued in exchnge for he convered deb. We inroduce conversion rio q >, which is he book vlue of equiy received by he coningen cpil invesors for ech dollr of fce vlue of deb convered. If q = 1, hen in c, he convered invesors need o ge 4.5 in book vlue of equiy. This is ccomplished by issuing hem 9 shres, becuse hey hen own frcion 9/1 + 9 of he firm, nd 9/19hs of he ol equiy of 9.5 is indeed 4.5. If q =, hen hey should ge 18 shres: his gives hem frcion 18/ of he ol equiy of 9.5 for book vlue of 9, which is indeed wice he book vlue of he deb hey gve up. The diluion leves he originl shreholders wih.5 in book vlue, or 1/18h of he ol equiy. The conversion rio q hs no effec on he ol moun of equiy, bu i deermines how he equiy is divided beween he originl nd convered shreholders. We need o keep rck of his llocion o deermine he mrke vlue of he converible deb. Book vlue of equiy is no, by iself, direc mesure of mrke vlue; bu he proporions of book vlue of equiy held by he wo ypes of invesors deermine how csh flows re lloced, nd he mrke vlue of he coningen cpil is he expeced discouned vlue of ll csh flows received by he invesors in hese securiies. We will derive n expression for he moun of equiy held ny ime by he originl equiy invesors. As led-in o he coninuous-ime seing, we consider discree-ime formulion wih discree rnsiion over smll inervl nd wrie V + = V + V. Suppose s in Figure 1 h he firm is jus he cpil rio boundry ime, nd i suffers n sse loss V <. From 9 nd Proposiion.1, we know h L increses when V reches new minimum nd L = V. The resuling moun of equiy following conversion is given by Q + = Q + V + 1 L = Q + V he miniml moun of ddiionl equiy required o preserve he cpil rio s in Figure 1c. Le Q o denoe he moun book vlue of equiy held by he originl shreholders, nd le = Q o/q denoe he frcion of equiy hey own. Suppose he conversion ime is he firs o occur, so h he equiy is fully held by he originl shreholders jus before conversion nd Q o = Q. Then Q o + = Qo + V q 11 L In oher words, he originl shreholders bsorb he full loss V in sse vlue, nd hey lose n moun q 11 L o he new shreholders s resul of he conversion. More generlly, if he originl shreholders own frcion of he equiy ime, hen hey bsorb frcion of he losses, nd we hve Q o + = Qo + V q 11 L 1 To formule precise resul, we work direcly in coninuous ime. We defined Q in 4. Under our

7 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS 7 Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. consn book-vlue condiion, c 1 = d 1, c = d, 4 becomes nd he expression Q = V B 1 L D 11 dq = dv + 1 dl 1 is well defined becuse V is geomeric Brownin moion nd L hs incresing phs. We inroduce he process Q o by seing dq o = Qo Q dv q 11 dl b 13 wih iniil condiion Q o = Q. We inerpre Q o s he equiy held by he originl shreholders: Equion 13 sys h he chnge in heir equiy is heir shre of he chnge in sse vlue plus heir shre of he rnsfer o new shreholders upon conversion. Using 1 o wrie his equion s dq o Q o = dq Q q1 dl Q 14 offers he following inerpreion: he percenge chnge in he book vlue of equiy held by he originl shreholders dq o/qo equls he overll percenge chnge dq /Q so long s no conversion occurs; n insn of conversion, he percenge chnge in book vlue held by he originl shreholders is reduced by he frcion of equiy rnsferred o he new shreholders. In Figure 1, 14 describes he rnsiion from o c wih q = 1, Q o = Q = 1, dq = 5, dq o = 5, nd he convered moun 1 dl = 45. Becuse dl = for > b, bsed on 14 we exend Q o beyond b if b < T by seing Q o = Q /Q b Q o b b T 15 so he frcion Q o /Q does no chnge in b T. The following resul confirms h hese definiions re meningful nd h hey led o n explici soluion. Theorem 3.1. Suppose B B nd D D > for T. Then 14 nd 15 hve excly one soluion, nd i is given by q1 / Q o = Q L T 16 Consequenly, he frcion of equiy held by he originl shreholders ime is given by q1 / L = = 1 1 L q1 / B + D { = min 1 1 min } s V q1 / s 17 B + D for T. A remrkble feure of 17 is h he frcion of equiy held by he originl shreholders ny ime depends only on he minimum sse vlue reched up o ime. Differen phs of V my produce very differen phs for he conversion process nd my resul in differen erminl vlues for equiy; nd ye, if hey rech he sme minimum sse vlue, hey leve he originl shreholders owning he sme frcion of he firm. The ol moun of coningen cpil convered o ime is 1 L, nd i is ineresing h he dependence of on his moun is nonliner ye explici. We noe some properies of 17. If L = i.e., if V never reches he cpil-rio rigger = B + D/ 1 in, hen = 1, reflecing he fc h no conversion hs occurred. If L = b i.e., if V reches he lower boundry b = D/1 which he required cpil rio cn no longer be susined, he coningen cpil is fully exhused, bu he originl shreholders re no wiped ou; hey own frcion b q1 / D q1 / = 18 B + D of he remining equiy V D = D/1. The following resul records he dependence of on he minimum rio : Corollry 3.. The proporion of equiy owned by he originl shreholders is n incresing funcion of wih min s V s held fixed if min V s < exp B + D 19 s 1 i is decresing in if he opposie inequliy holds. This resul is esily esblished by differeniing he hird expression for given in 17. We inerpre he corollry s sing, perhps surprisingly, h higher required cpil rio ulimely proecs he originl shreholders: if he loss in sse vlue is sufficienly lrge, he originl shreholders keep higher frcion of he firm under higher nd hus more sringen cpil rio. Moreover, he ol moun of shreholder equiy Q is iself n incresing funcion of ; his follows from 9 nd 11. To inerpre he condiion in he corollry, recll h conversion of deb o equiy begins when sse vlue reches = B + D/1. For smll, exp 1, so he hreshold in 19 is nerly he sme s he rigger for conversion. Thus, higher, conversion is riggered sooner resuling in lower, bu if sse vlue coninues o decline, higher resuls in higher frcion of equiy held by he originl shreholders. This phenomenon is illusred in Figure 3 for firm wih D = 5, B = 3, nd iniil sse vlue

8 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger 8 Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. Figure 3 Frcion owned by originl shreholders Comprison of he Frcion Held by he Originl Shreholders s Funcion of he Mximum Loss in Asse Vlue Up o Time, for Two Vlues of he Cpil Rio 1% lph 5% lph Mximum drop in sse vlue V = 1. The figure plos gins he mximum loss in sse vlue, V min s V s for wo differen vlues of. Conversion begins when he loss in vlue reches V B + D/1, which is pproximely 15.8 wih = 5 nd 19. wih = 1. The higher cpil rio riggers conversion sooner; however, once conversion begins he smller vlue of, he wo curves quickly cross. Indeed, from he corollry we know h once he loss exceeds V exp 1B + D/1 1, ny cpil rio greer hn 1% keeps higher frcion of equiy wih he originl shreholders. 4. Dividends nd Deb Service Pymens As is sndrd in much of he cpil srucure lierure e.g., Lelnd nd Tof 1996, we will ssume h he firm s sses genere csh re proporionl o heir vlue in our seing, book vlue, nd hese csh flows re used o service he firm s deb nd o py dividends o shreholders. If he firm pys ou consn frcion 1 of is sse vlue, hen from ime o + d, he csh flow vilble will be V d. Wih coupon re of c nd fce vlue of D, he senior deb requires pymens re c D prior o muriy. Ineres on deb is x deducible, nd we model his s in, e.g., Lelnd 1994 nd Lelnd nd Tof 1996: if he firm s mrginl x re is 1, i incurs n fer-x cos re of 1 c D in servicing he senior deb. We could pply differen mrginl x res 1, o he wo ypes of deb 3 o ge fer-x coupon res 1 i c i, i = 1 ; for simpliciy, we use common vlue. The ousnding converible deb 3 I is uncler if coupons on coningen cpil would be x deducible under he curren x code in he Unied Ses becuse he conversion feure my mke he deb oo equiy-like. This possibiliy could be modeled by king 1 =. Bu x rules could lso be chnged if regulors sough o cree incenives for bnks o hold more of heir deb in he form of coningen cpil. ime is B 1 L, requiring n fer-x pymen re c 1 1 B 1 L. The difference V 1 c 1 B 1 L + c D beween he re which csh is genered, nd he re which i is pid o deb holders is he re which dividends re pid o shreholders, whenever his difference is posiive. When he difference is negive, he firm is genering insufficien csh o service is deb. As is cusomry, we inerpre negive dividend s he issunce of smll moun of new equiy, which brings csh ino he firm. This csh is immediely pid ou o he deb holders, so he issunce hs no impc on he ol moun of cpil in he firm. We will ssume, in fc, h he new equiy is issued o exising shreholders s in righs offering nd h he originl nd convered shreholders pricipe in equl proporions. Thus, he proporion of he firm owned by he originl shreholders is unchnged. The new shreholders hen receive ne csh flow re 1 V 1 c 1 B 1 L + c D regrdless of wheher his is posiive in which cse i is dividend or negive in which cse i is he cos of rising equiy. We will need o incorpore his srem of pymens ino our overll vluion of he coningen cpil. Two prmeer rnges for he coupon nd pyou res meri specil menion. We know h s long s he firm hs no exhused is converible deb, i cn minin he minimum cpil rio by convering deb ino equiy; h is, i cn minin he bound 1 V B 1 L + D wih equliy holding he insns of conversion. I follows h if 1 > 1 mxc 1 c he firm lwys generes enough csh o service is deb, nd shreholders lwys ern dividend. In conrs, if 1 < 1 minc 1 c hen he firm will sop pying dividend nd will sr issuing smll mouns of equiy in dvnce of ny deb convering o equiy. 5. Decomposiion of Pymens on Converible nd Senior Deb In his secion, we decompose he pymens o holders of he converible deb ino principl pymen, coupon pymens, dividends on convered equiy,

9 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS 9 Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. nd erminl equiy pymen. We decompose pymens on he senior deb coningen on he firm s biliy o minin he required cpil rio. These decomposiions prepre he wy for he vluions in he nex secion. The horizon for he vluion is he smller of he deb muriy T nd he ime b which V firs his b = D/1. A b, he firm hs exhused is coningen cpil nd cn no longer susin he required cpil rio; s before, we ssume he firm is hen seized by regulors nd liquided. 4 The firm sill hs equiy his poin, bu no enough o mee he cpil requiremen. To cpure he possible loss in vlue from seizure, we ssume h shreholders recover rndom frcion X 1 1 of he equiy vlue b, he remining frcion 1 X 1 represening dedweigh cos. An lernive loss mechnism is he delyed recpilizion used by Peur nd Keppo 6. Similrly, we pply rndom recovery frcion of X 1 o senior deb. We ssume h X 1 nd X re independen of V bu no of ech oher. Indeed, o enforce bsolue prioriy of deb over equiy, we need PX = 1 X 1 > = 1. Independence beween X 1 X nd V will imply h only he expeced recovery res R i = EX i, i = 1, ener ino our vluions. These cn sisfy R 1 > nd R < 1 wihou violing bsolue prioriy. As jus one illusrion, ny R 1 R 1 cn be relized s expeced recovery res while sisfying bsolue prioriy by ssigning o X 1 X he oucomes 1 1, 1, nd wih probbiliies R 1, R R 1, nd 1 R, respecively Converible Deb We use r > o denoe fixed risk-free ineres re which o discoun ll pyoffs for vluion. The discouned pyoffs of he componens of he converible deb re s follows: principl pymen muriy, erned coupon, T e rt B 1 L T 1 e rs c 1 B 1 L s ds equiy erned hrough conversion, e rt 1 T V T B 1 L T + D 1 b >T + e r b 1 b X 1 V b 1 b T 3 ne dividends, mint b e r 1 V 1 c 1 B 1 L + c D d 4 4 An lernive inerpreion is h he firm undergoes disressed sle, so he full vlue of he sses is no recovered, bu he equiy holders need no be wiped ou. In 1, 1 L T is he ol moun of deb convered o equiy, so B 1 L T is he remining principl muriy. Similrly, in, B 1 L s is he remining principl ime s, nd muliplying his expression by c 1 yields he re which he holders of he bond ern coupons. Equion 3 breks down he clim on equiy ino wo prs, depending on wheher liquidion occurs before he muriy of he deb. In he firs erm, b > T, so he firm survives hroughou he inervl T. The mrke vlue of he firm s ol equiy T is he difference V T [ B 1 L T + D ] 5 beween he vlue of he firm s sses nd he principl pymens on he wo kinds of deb. Here we invoke our ssumpion relxed in 8 h sse vlue is mrked o mrke so h 5 is he csh pid o equiy holders fer reiring ll deb if he sses re sold T. A frcion 1 T of his residul vlue goes o he new shreholders hose who cquired n equiy ske hrough conversion of he coningen cpil. In he second cse in 3, he firm is seized nd liquided ime b when he coningen cpil is exhused. A his insn, he firm jus mees is cpil requiremen, so he residul mrke vlue is V b. A frcion X 1 of his is recovered by shreholders upon liquidion, nd frcion 1 b of he recovered vlue goes o he new shreholders. Finlly, he inegrnd in 4 is he discouned vlue of he ne dividend re in pid o he convered shreholders ime. To vlue he coningen cpil, we will need o clcule he expecions of Senior Deb The pymens on he senior deb cn be decomposed similrly bu more simply ino principl nd coupon pymens. We gin disinguish he cses b T nd b > T, he firs cse corresponding o seizure nd liquidion of he firm. The discouned pyoffs o senior debholders re s follows: erned coupon, min b T c De rs ds 6 principl, e rt 1 b >T + X e r b 1 b <T D 7 In Equion 6, coupons re pid unil eiher he muriy of he deb ime T or he liquidion b. In 7, he principl pymen is reduced from he originl fce vlue of D o X D in he cse of liquidion, reflecing rndom recovery frcion of X for he senior deb nd he possibiliy of dedweigh cos of seizure nd liquidion. If X 1, he senior deb would be enirely riskless.

10 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger 1 Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. 6. Vluion To clcule expecions of 1 7, we posi h he dynmics of he book vlue of he firm s sses re given by 1. Equivlenly, we hve, wih = r /, V = V exp + W 8 We re ssuming h he firm s sses re mrked o mrke, so h V lso represens he mrke vlue of he firm s sses; we drop his ssumpion in 8. In wriing he drif in 1 s r, we re implicily specifying he dynmics of V under risk-neurl pricing mesure h we will use o ke expecions in 1 6. Mhemiclly, his is by no mens necessry we could use ny consn drif, including one h incorpores risk premium, nd modify our vluion formuls ccordingly A Pril Trnsform Inspecion of he discouned pyoffs in 1 6 nd he proporion in 17 indices h he key remining sep for vluion is king expecions involving powers of V nd is running minimum, wih he running minimum resriced o n inervl. We herefore underke preliminry clculion of generl such expression, which we will hen use o vlue he vrious pymens. Se W = logv /V nd m = min s W s 9 hen W is Brownin moion wih drif nd diffusion coefficien. Le H v k y = H v k y = E [ expv W + k m 1 m y ] k v y 3 The funcion H depends on he prmeers nd hrough he processes W nd m; becuse hese prmeers remin fixed, we suppress his dependence nd wrie simply H v k y in referring o he funcion. The funcion is given explicily in he following resul. wih Proposiion 6.1. The funcion H in 3 evlues o H v k y = expv + v /h k y 31 h k y = + k eky+y/ y + e k+k y + k / + + k + k 3 where = + v, nd is he sndrd norml disribuion funcion. Wih y =, 3 defines he join Lplce rnsform of W nd m, nd in his sense he generl cse in 3 defines pril rnsform. In our pplicion of he formul, y will lwys ke he vlue log/v or logb/v, corresponding o he sse levels which conversion of coningen cpil srs nd ends. In severl cses, we need o ke he difference of vlues of H hese wo vlues of y wih oher rgumens held fixed, so i will be convenien o define H v k = H v k log/v H v k logb/v Principl nd Coupon Pymens The discouned expeced vlue of he principl pymen on he converible deb is he expeced vlue of Equion 1 nd is given by e rt B 1 EL T 34 Thus, o vlue he principl pymen i suffices o find he expecion of L T. Proposiion 6.. The expeced presen vlue of he coningen cpil s principl pymen is 34, where EL = H log/v bh logb/v V H 1 This expression evlues o EL = 1 b b1 + V + + where e + / b + + / / +1 V b bv +b1 35 ± 1 = ± + log/v ± b1 = ± + logb/v ± = + ± log/v ± b = + ± logb/v Figure 4 plos he expeced moun of coningen cpil convered by ime, nmely, 1 EL, over wo-yer horizon for vrious levels of nd. Recll h EL depends on hrough he boundries nd b of he conversion bnd. The figure uses V = 1 wih D = 6, B = 3, r = 5%, nd = 3%. The lef pnel fixes 5%, nd he righ pnel fixes 5%. The curves show quliively differen behvior ner ime zero: when he iniil sse level is fr from he conversion rigger eiher becuse

11 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS 11 Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. Figure 4 Expeced moun convered Comprison of 1 EL, he Expeced Amoun of Coningen Cpil Convered by Time, for Differen Vlues of he Cpil Rio Lef nd he Asse Voliliy Righ = 1% = 5% = 1% Time in yers is smll or becuse is smll, he expeced moun convered is nerly fl for smll ; he curves re seeper when he conversion rigger is closer. The expeced presen vlue of he coningen cpil coupon pymens is given by B c T 1 r 1 e rt c 1 1 e r EL d 36 We do no hve simple expression for he inegrl in 36; however, becuse EL is smooh nd monoone, he inegrl cn be ccurely pproximed by replcing i wih sum Equiy Erned Through Conversion We urn now o 3, which gives he discouned erminl vlue of he equiy cquired by he coningen cpil invesors hrough he process of conversion. We vlue seprely he wo erms in 3, he firs corresponding o he firm surviving unil T, he second corresponding o seizure nd liquidion before T. Proposiion 6.3. The vlue of he convered equiy ske in he even of survivl he firs erm in 3 is given by exp rt imes q1 / V V HT 1 V H T 1 q1 / V 1 HT 1 + V 1 q1 / V H T 1 + q1 / 37 In he even of seizure nd liquidion he second erm in 3, he vlue of he convered equiy ske is, wih Expeced moun convered σ = 3% Time in yers σ = 1% R 1 = EX 1 nd 1 = + r, b q1 / b R 1 b 1 V 1 / e rt H T 1 / logb/v Ne Dividends As discussed in 4, he difference beween he ol pyou re V nd deb service pymens crees dividend srem for equiy holders, frcion 1 of which flows o invesors who originlly held converible deb, s in 4. Tking he expeced vlue of his expression, we ge [ mint b E e r 1 = ] V 1 c 1 B 1 L +c Dd T e r E [ 1 V 1 c 1 B 1 L +c D 1 b > ] d 39 The expecion inside he inegrl cn be evlued in closed form: Proposiion 6.4. The expeced ne re which he coningen cpil invesors ern dividends i.e., he expecion on he righ side of 39 is given by V H c c 1 b H 1 1 c 1 V H 1 q1 / V [V H 1 q1 / 1 1 c c 1 b H q1 / ] 1 1 c 1 V H 1 + q1 / 4

12 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger 1 Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. The presen vlue of he cumulive dividends is he ime inegrl of his expression, which is esily nd ccurely pproximed by sum over discree se of des. I is lso eviden from his expression h he effec of he mrginl x re is simply o replce ech originl coupon re c i wih 1 c i. The formul remins vlid if we replce 1 c i wih 1 i c i o llow differen levels of x deducibiliy of he wo ypes of coupons Senior Deb The expeced vlue of he coupon pymens 6 is given by [ min b ] T E c De rs ds = D c r = D c r 1 E [ exp r minb T ] 1 e rt b > T E [ ] e r b 1 b T 41 Similrly, he discouned expeced vlue of he principl pymen 7 is given by DE [ ] e rt 1 b >T + X e r b 1 b <T = D e rt b > T + R Ee r b 1 b T 4 The probbiliy b > T coincides wih m T > logb/v, which cn be evlued direcly using Equion B1 in Appendix B; he expecion Eexp r b 1 b T is evlued explicily in Equion C3 in Appendix C. Wih hese subsiuions, he ol discouned expeced vlue of he senior deb becomes D c r + D 1 c r e rt [ + D R c [ b r b V b + V T logb/v T / ] T + logb/v V T 1 / logb/v 1 T T +1 / ] logb/v + 1 T T where, s before, 1 is he squre roo of r +. The following resul vlues he senior deb using he funcion H: Proposiion 6.5. The vlue of he senior deb, including boh coupon pymens 6 nd principl 7, is given, wih 1 = r +, by D c r + D 1 c r e rt 1 HT logb/v + D R c b 1 / r V e rt HT 1 / logb/v 7. Closing he Model: Mrke Yields In our clculions, we hve ssumed h boh he senior deb nd he converible deb re sold pr ime zero; his leds o consn book vlues for he unconvered principl, 9, nd he resuling rcbiliy. In 6, we hve clculed mrke prices for senior nd converible deb, wih coupon res ssumed given. For our model o be inernlly consisen, we need he mrke prices we clcule ime zero o coincide wih our ssumpion h he bonds sell pr. We now show h his is indeed possible nd h i deermines he coupon res for boh ypes of deb. For he senior deb, equing he expeced discouned vlue of he coupon nd principl clculed in 6.5 o he fce vlue D yields he coupon re c = r R Ee r b 1b T 1 e rt b > T R Ee r b 1b T The probbiliy nd expecion in his expression re evlued in Appendix C.4, hus llowing direc evluion of c. If R = 1, he coupon re c reduces o r: under our ssumpion h he firm is seized nd liquided when i violes is cpil requiremen before insolvency he senior deb is riskless if here is no loss of vlue liquidion. Similrly, for he converible deb, equing our vluion he sum of he expecions of 1 4 wih he fce vlue B yields he coupon re c 1 = B A 1 A 3 A 4 A + A 5 where A 1 is he expeced principl in 34, A = B T r 1 e rt 1 e r EL d from 36, A 3 is he expeced erminl equiy vlue he sum of 38 nd exp rt imes 37, nd [ mint b ] A 4 = E e r 1 V 1 c D d nd [ mint b ] A 5 = E e r 1 1 B 1 L d come from he ne dividends in 39. The resuls in 6 yield explici expressions for A 1 A 5 nd hus for he coupon re c 1.

13 Glssermn nd Nouri: Coningen Cpil wih Cpil-Rio Trigger Mngemen Science, Aricles in Advnce, pp. 1 18, 1 INFORMS 13 Copyrigh: INFORMS holds copyrigh o his Aricles in Advnce version, which is mde vilble o subscribers. The file my no be posed on ny oher websie, including he uhor s sie. Plese send ny quesions regrding his policy o permissions@informs.org. We view hese expressions s he key prcicl conribuion of our nlysis. Given he chrcerisics of he firm is sse voliliy nd he fce vlue of is senior nd converible deb hese equions give he coupon res required by he mrke. For deb issued pr, he coupon re equls he yield; so, more generlly, we inerpre hese res s he yields required by he mrke for he wo ypes of deb. These equions re herefore useful in guging he yield required by invesors in coningen cpil s compension for bering he risk h he deb hey hold convers o equiy. 8. Disinguishing Mrke nd Book Vlues of Asses To his poin, we hve ssumed h he bnk s sses re mrked o mrke so h V represens he mrke vlue of sses s well s heir book vlue. We now exend he model o cpure sochsic relion beween he wo. We use A o denoe he mrke vlue of sses. Our key ssumpion is h lhough he mrke nd book vlues of sses my differ, hey re sufficienly ligned o gree on wheher bnk is solven. If he bnk were liquided ime, deb holders would be due B + D, so he bnk is solven if is sses hve les his vlue. Our condiion, hen, is h A > B + D whenever V > B + D. To model his relionship, we inroduce second geomeric Brownin moion U, U = U exp u + u W wih W nd W he originl Brownin moion driving V hving insnneous correlion. We model A s sisfying A B D = U V B D 43 The process U cn be roughly inerpreed s mrkeo-book rio, bu wheres V B D is he book vlue of equiy, A B D is he difference beween he mrke vlue of sses nd he book vlue of deb. A nurl choice in his seing would be o ke u = u /, so h EU is consn, bu we need no limi ourselves o his cse. In his exension of our bsic model, conversion from deb o equiy is sill governed by book vlue V, jus s before; bu he vlue received by equiy holders eiher he muriy de T or seizure b now depends on he mrke vlue A. Accordingly, we modify 3 by replcing V T wih A T nd V b wih A b. This cse remins rcble under he prmeer resricion, where = u + u / 1 u 1 + r In Proposiion 6.3, 37 becomes H T 1 + u q1 / V H T 1 + u q1 / 1 H T u 1 q1 / V + 1 H T u 1 + q1 / wih = V U expr T, nd 38 becomes R 1 U b 1 [ b V b q1 / u /+ / logb/v T V T b u /++ / logb/v + + T T ] These expressions re derived hrough minor modificion of he proof of Proposiion 6.3 fer mking he subsiuion in 43. Wih he condiion h u = u /, his exension inroduces wo new prmeers, he book-o-mrke voliliy u nd correlion, s well s he iniil vlue A. Though no direcly observble, hese prmeers could be clibred using mrke vlues of firm s deb nd equiy nd book vlues from finncil semens. Becuse our model lredy hs severl prmeers, in he numericl exmples of he nex secion, we limi ourselves o he bsic model in which A = V. 9. Exmple In his secion, we use numericl exmples o invesige how he yields derived from our model chnge wih prmeer inpus nd how he inroducion of converible deb influences he spred on senior deb. Tble 1 shows he prmeer vlues we use. The firs Tble 1 Prmeers for Bse Cse I nd Modified Scenrio II Deb over sses rio D/V 9% Cpil dequcy rio 4% Risk free re r 5% 5% Voliliy of sse reurns 8% 16% Deb muriy T 1.5 Frcionl pyou of sses 3% 15% Tx re 3% Recovery re for equiy R 1 3% Recovery re for senior deb R 95% I II