he Picing of Finie Mauiy Copoae Coupon Bonds wih Raing-Based Covenans Ségio Silva Poucalense Univesiy, Pougal e-mail: segios@up.p coesponding auho) José Azevedo Peeia ISEG - echnical Univesiy of Lisbon, Pougal e-mail: jpeeia@iseg.ul.p July, 006 Absac his pape models he pice of finie mauiy copoae coupon bonds wih a aingbased covenan. Bhano 003) aleady addessed his issue, bu he coesponding model embeds wo feaues ha in some way could be consideed as significan limiaions. In he fis place, i does no ake ino consideaion any kind of paymen o he bondholdes when he aing-based covenan is iggeed. In he second place, and moe impoanly, Bhano 003) incopoaes an inconsisency egading he paymen he bondholdes ae eniled o a mauiy: he model consides his paymen when he fim has no peviously eneed in bankupcy) o coespond always o he pincipal of he deb ousanding and his can only be ue if he value of he debo s asses is geae han o equal o he pincipal, which obviously is no always he case. In his way Bhano s model ovepices he bond. Following Bhano and Mello 006), we popose o ovecome he fome weekness consideing wo alenaive pocedues: i) an incease in he coupon ae o ii) a paial amoizaion of he pincipal. he lae limiaion is addessed by explicily modeling he possibiliy of a paial paymen of he pincipal if a mauiy he make value of he asses in place is no enough o allow fo full epaymen. he values of he equiy, ax benefis, bankupcy coss and he leveaged fim ae also obained. JEL classificaion: G1, G13 1 Inoducion Since Meon s 1974)7 seminal pape which pioneeed he so called sucual appoach o picing copoae bonds, we have seen an inceasing gowh of he lieaue in his field. A an ealy sage, an impoan conibuion was povided by Black and Cox 1976)5 who, in defining defaul as a igge even ha may happen a any momen of a bond s life insead of occuing only a he mauiy, elaxed one of he simplifying assumpions pesen in Meon s model, and esablished a feaue common o almos all sucual models published heeafe. In his ype of modeling execise, defaul is iggeed when he value of he fim s asses 1 eaches some specified value, he baie level. he way in which his baie level is se, eihe endogenously o exogenously, has been a disinguishing faco beween diffeen models. One way is o conside ha he baie level is deemined by he shaeholdes, in ode o maximize he equiy value e.g. Black and Cox 1976)5, Leland 1994)0, Leland and of 1996), Goldsein, Ju and Leland 001)14, Eicsson and Reneby 003)10). he alenaive akes ino consideaion a baie level se exogenously, eflecing he pesence of some kind of covenan in he bond indenue e.g 1 O ohe sae vaiable elaed o he fim such as cash-flow o deb aio. 1
Black and Cox 1976)5, Kim, Ramaswamy and Sundaesan 1993)19, Longsaff and Schwaz 1995)3, Biys and de Vaenne 1997)6, Eicsson and Reneby 1998)9, Schobel 1999)9, Hsu, Sa-Requejo and Sana-Claa 003)15, Hui, Lo and sang 003)17, auén 1999)30, Collin-Dufesne and Goldsein 001)7, Ju and Ou-Yang 004)18, Huang e al. 003)16). Besides diffeen exensions concening he inees ae pocess, ecovey values, deb sucue, bond chaaceisics, definiion of baies and dynamic capial sucue, o name a few, he eseach in his field has also focused on aspecs elaed o he asse subsiuion poblem Mello and Pasons 199)6, Leland 1998)1, Eicsson 000)8, Bhano and Mello 006)4), he equiy/bond holdes saegic behavio Andeson and Sundaesan 1, Andeson, Sundaesan and ychon, Mella-Baal and Peaudin 1997)5, Mella-Baal 1999)4, Fan and Sundaesan 000)11), and bankupcy codes Fançois and Moelec 004)1, Moaux 00)8, Galai, Raviv and Wiene 003)13 and Yu 003)31). he sucual model poposed hee aims o pice copoae bonds whose indenue incopoaes a aing igge based covenan, which links he pay-offs o bondholdes wih he cedi aing of he fim. As pu by Bhano and Mello 006)4: a aing igge clause in a copoae bond indenue equies a fim o pepay is deb o o change he coupon ae on is deb if he fim s cedi aing eaches a specified level. Alhough he Bhano and Mello 006) famewok does ake ino consideaion his kind of bond, and consides wo ypes of aing igge covenans paial amoizaion of he deb s pincipal 3 o an accued coupon ae), hei model only applies o pepeual deb. By conas, he pesen pape poposes a famewok capable of dealing wih finie mauiy bonds. he finie mauiy case was addessed by Bhano 003)3, whose model assumes he exisence of wo possible cedi evens 4 : namely, a downgade in he cedi aing of he fim and bankupcy, which implies he liquidaion of he fim. hese wo evens wee modeled hough he specificaion of wo baies levels, V B1 and V B especively wih V B1 > V B ), esablished exogenously. Howeve, hee ae wo limiaions in Bhano 003) ha we will y o ovecome in he cuen pape. In he fis place, albei he pupose of Bhano 003) is o pice bonds wih aing based covenans, he model does no explicily assume any kind of change in bondholde pay-offs, when he covenan is iggeed. In ohe wods, when he value of he fim s asses eaches he fis baie V B1 ), he aing change only affecs some paamee values associaed wih he diffusion pocess govening he value dynamics of he fim s asses 5. Addiionally, even if he pevious emak is no aken ino consideaion, he pice fomula developed by Bhano 003) assumes ha he paymen o bondholdes a mauiy admiing ha, in he mean ime, he fim has no eneed in bankupcy and so has no been liquidaed, which is equivalen o admiing ha he second baie V B as no been eached) always coesponds he bond pincipal. his final cash-flow only makes sense in a scenaio whee he value of he fim s asses is enough o cove i. Ohewise, if he value of hese asses is insufficien o cove he face value of he bond, a mos he bondholdes will only eceive he value of he asses, since he equiy holdes will no be willing o pay he diffeence. In his sense, we may say ha Bhano 003) ovevalues he bondholdes expeced cash-flows, esuling in an ovepicing of he bond. Using he same base sucue of Bhano s model, which defines boh cedi evens aing change and bankupcy) hough he baie levels, V B1 and V B, we popose o obain a bond picing fomula ha akes ino accoun hose wo emaks. Besides obaining he bond value befoe he aing change akes place, we also deive value expessions a he momen of he aing change and immediaely afe ha. he same is done fo equiy, bankupcy coss, ax benefis and leveaged fim value. As a final noe, concening he paial pepaymen of he bond s pincipal aing igge, Bhano he pimay focus of Bhano and Mello 006) is he asse subsiuion poblem. Specifically, he auhos analyze he effecs of such covenans on he asse subsiuion poblem. 3 o be exac, Bhano and Mello 006) assume ha deb holdes eceive a facion of he iniial make value of deb when he aing of he fim changes. his amoun coesponds o a facion of he pincipal, since he analysis is esiced o deb sold a pa. 4 hese ae also assumed in Bhano and Mello 006). 5 Recall ha we have assumed a change in cedi aing causes wo chaaceisics o change: he volailiy of fim asses, and he ouflow o shaeholdes and an inceased inees expense), Bhano 003), page 6.
and Mello 006) analyzed his case aking ino accoun wo diffeen financing souces - eihe by cash infusion o by selling asses. he same will be done hee, bu insead of consideing hem sepaaely, hey will be joinly modelled, which will allow he simulaneous use of boh souces. he model developed in he cuen pape fills he gap beween Bhano 003) and Bhano and Mello 006). he pape is oganized as follows: secion esablishes he valuaion famewok, in secion 3 he bond picing fomulas ae deived, in secion 4 he value of equiy, bankupcy coss, ax benefis and he leveaged fim ae obained, in secion 5 a compaaive analysis beween ou model and Bhano s model esuls is pefomed, in secion 6 we compae he wo ypes of aing igge covenan and, finally, secion 7 concludes he pape. he Valuaion Famewok I is assumed ha he value of he fim s asses, V, is descibed by he following coninuous diffusion pocess, unde he isk neual pobabiliy: dv /V = α i )d + σ i dw Whee is he consan) isk fee inees ae, α i is he cash payou ae, σ i he consan) asses eun volailiy and dw an incemen of a sandad Bownian moion. As in Bhano 003), we allow he paamees of he diffusion pocess o ale afe a aing change so he subscip i akes he value 1 befoe) o afe) 6. Noice ha his change, once occued is assumed o be ievesible and pemanen. he deb of he fim is chaaceized by a single coupon bond wih pincipal F, coupon ae c and finie mauiy. hus, he bondholdes eceive a coninuous paymen flow cf d a leas unil he aing change. We conside he exisence of wo cedi evens, namely a fim s aing downgade and he occuence of bankupcy. Each of hese will be modeled hough an exogenous specificaion of wo hesholds, which is wo baie levels, designaed by V B1 and V B especively. In addiion, we also assume ha he bankupcy even is always peceded by he aing downgade so: V B1 > V B. he aing change will occu when he value of he asses inesecs fo he fis ime he fis igge level V B1 ). As a esul, we define he ime of he aing change as: τ 1 = inf{ 0 : V V B1 } Once V B1 is eached, wo possible oucomes may esul, depending on he fomulaion of he aing-based covenan of he bond, namely: 1. An incease in he coupon ae keeping he pincipal a he iniial level. In his case, he coninuous coupon flow o bondholdes changes o: cf d, wih > 1 c, coesponds o he new coupon ae);. A paial efund of he pincipal, keeping he coupon ae a he iniial level. his will lead o a educion in inees paymens: c F d, wih < 1, 1 ) is he facion of he nominal deb value ha is edeemed). In boh cases, he new coupon paymens will occu unil he bond maues o unil he fim goes bankup, which will happen when he value of he asses eaches he second consan) baie V B, V B < F ). hus, he bankupcy ime is defined as: τ = inf{ τ 1 : V V B } A τ, he fim is liquidaed, and he bondholdes eceive he value of he fim s asses ne of bankupcy coss: ρ 1 V τ = ρ 1 V B, 0 < ρ 1 1). Consequenly, we assume ha bankupcy coss 6 Specifically, Bhano 003) assumes ha α > α 1 and σ > σ 1. 3
ae a consan facion 1 ρ 1 ) of he value of he asses. Noice ha his implies he veificaion of he absolue pioiy ule in bankupcy, since V B < F, and ha in case of bankupcy, he shaeholdes ge nohing. hus, we have: Fo < τ 1, he asses value diffusion pocess unde he isk-neual pobabiliy measue) is given by: dv /V = α 1 )d + σ 1 dw and he coupon flow: cf d. Fo τ 1, he asses value diffusion pocess changes o: dv /V = α )d + σ dw, and he coupon flow o cf d. o value he bond, Bhano 003) akes ino accoun he paymens o he bondholdes in hee disinc siuaions: 1. he value of he asses emains above V B1 unil he mauiy of he bond τ 1 > );. he value of he asses cosses V B1 bu emains above V B unil he mauiy of he bond τ > ); 3. he value of he asses eaches V B befoe he mauiy of he bond τ < ). he disinguishing feaue of ou model elaive o Bhano s besides he fomulaion of he coupon paymens afe V B1 has been eached) deives fom he fac ha i akes ino accoun a possible defaul a he mauiy of he bond. his is absen in Bhano 003), which only consides he possibiliy of bankupcy when V B is cossed. Conside he following numeical example: V B1 = 100, V B = 50 and F = 100 values aken fom Bhano 003)). In Bhano s fomulaion, if he value of he fim s asses neve cosses he bankupcy baie, he bondholdes pay-off a mauiy is always he pincipal of he bond. In he example, 100, which is possible only if he value of he fim s asses a mauiy, V, is enough o cove he coesponding paymen V > 100). If no, he fim defauls, since i is impossible o honou he paymen. he siuaion whee Bhano s famewok is valid coesponds o he esiced case whee V B F. A geneal fomulaion would lead o disinguishing five siuaions: 1. he value of he asses emains above V B1 unil he mauiy of he bond τ 1 > ): a) and he value of he asses a mauiy is geae han o equal o he pincipal V F ); b) o he value of he asses is insufficien o epay he pincipal V < F );. he value of he asses cosses V B1 bu emains above V B unil he mauiy of he bond τ > ): a) and he value of he asses a mauiy is geae han o equal o he pincipal V F ); b) o he value of he asses is insufficien o epay he pincipal V < F ); 3. he value of he asses eaches V B befoe he mauiy of he bond τ < ). Noice ha, he scenaio 1b) is only possible if he face value of he bond, F, is geae han V B1. Consideing V B1 > F, figue 1 shows fou sample pahs, I, II, III and IV fo he asses values associaed wih he fou possible scenaios, 1, a), b) and 3 especively. I will be assumed ha, in he even of defaul a mauiy τ 1 >, V < F and τ >, V < F ), he pay-off o bondholdes will be a facion of he make value of he asses: ρ V, and 0 < ρ 1). 4
Figue 1: Fou sample pah fo he asses values associaed o he fou possible scenaio, consideing V B1 > F. Asses value emains above V B1 unil he mauiy of he bond - I; asses value cosses V B1 bu emains above V B, and a mauiy he value of asses is sufficien insufficien) o epay he bond face value - II III) and finally he asses value eaches V B befoe he mauiy of he bond - IV. 3 he Bond Value 3.1 Raing-based covenan: incease in he coupon ae In he fis place we will conside he case of an incease in he coupon ae, when he asse value his V B1. Remembe ha he new coupon, afe he aing igge, is given by cf d, wih > 1. he value, a ime, of he bond will be given by he expeced value, unde he isk neual pobabiliy Q P ), of all paymens discouned a he isk fee ae: BV,,, c, F, ) = E Q P cf e s ) 1 s<τ1 ds F e ) F 1 τ1>,v F + ρ V 1 τ1>,v <F ) F + { e τ1 ) 1 τ1< E Q P τ 1 cf e s τ1) 1 s<τ ds F τ1 { } e τ1 ) 1 τ1< E Q P e τ1) F 1 τ>,v F + ρ V 1 τ>,v <F ) F τ1 F + F } + ) e τ1 ) 1 τ1< E Q P e τ τ1) ρ 1 V τ 1 τ< F τ1 F 1) Whee 1 A is he indicao funcion, which assumes he value of one if he even A is ue and zeo ohewise. he fis line epesens he discouned expeced value of he coupon flow unil he fis baie V B1 ) is eached, he second line efes o he paymen a mauiy if he asses value emains 5
above V B1 ) pio o he mauiy dae. he hid and fouh line consides he coupon flow afe he aing change and he paymen a mauiy, especively, when he asse value cosses he fis igge level V B1 ) bu emains above he second baie V B ) pio o he mauiy dae. Finally he las line efes o he ecovey value ha accues o bondholdes when bankupcy is iggeed pio o he mauiy dae. Noice ha he sum of he hee expeced values associaed o he filaion F τ1 in he las hee lines epesens he value of he bond a τ 1, ha is o say a he ime of he aing change: BV τ1, τ 1,, c, F, ) = E Q P cf e s τ1) 1 s<τ ds τ 1 F τ 1 + e τ1) F 1 τ>,v F + ρ V 1 τ>,v <F ) F τ1 + e τ τ1) ρ 1 V τ 1 τ< F τ1 ) hus, given he coninuiy of he asses value pocess,v τi = V Bi ), we can ewie expession 1) as: BV,,, c, F, ) = E Q P cf e s ) 1 s<τ1 ds F e ) F 1 τ1>,v F + ρ V 1 τ1>,v <F ) F e τ1 ) 1 τ1< BV B1, τ 1,, c, F, ) F 3) Whose value is given by deivaion in he appendix A.1): B V,,, c, F, ) = cf 1 e ) 1 Q P τ 1 F ) V V B1 ) µm 1 µp 1 σ 1 Q m τ 1 F ) + + F e ) Q P τ 1 >, V F F ) + ρ V e α1 ) Q V τ 1 >, V < F F ) + + e τ1 ) BV B1, τ 1,, c, F, ) g P τ 1, V, V B1 ) dτ 1 4) and B V B1, τ 1,, c, F, ) = cf 1 e τ1) 1 Q P τ F τ1 ) + + ρ 1 V B cf ) VB1 V B ) µm µp σ Q m τ F τ1 ) + + F e τ1) Q P τ >, V F F τ1 ) + ρ V B1 e α τ1) Q V τ >, V < F F τ1 ) 5) 6
Whee: Q X τ 1 F ) - sands fo he pobabiliy, unde measue X, of having a aing change unil he mauiy of he bond peiod ); Q X τ F τ1 ) - he pobabiliy, unde measue X, of fim eneing ino bankupcy, fom he momen of he aing change unil he mauiy of he bond peiod τ 1 ); Q X τ 1 >, V F F ) Q X τ 1 >, V < F F )) - he pobabiliy, unde measue X, of he fim neve having a aing change, and he value of he asses a mauiy being geae han o equal o lesse han) he face value of he bond; Q X τ >, V F F τ1 ) Q X τ >, V < F F τ1 ))- he pobabiliy unde measue X, fom he momen of he aing change, of he fim neve eneing ino bankupcy, and he value of he asses a mauiy being geae han o equal o lesse han) he face value of he bond; Noe ha Q X τ i >, V < F ) = 1 Q X τ i ) Q X τ i >, V F ). Q X τ 1 F ) = N a 1 b 1 µ X 1, ) ) µx 1 V σ 1 + V B1 N a 1 + b 1 µ X 1, ) Q X τ 1 >, V F F ) = N a 1 c 1 ) + b 1 µ X 1, ) V V B1 ) µx 1 σ 1 N a 1 c 1 ) + b 1 µ X 1, ) Q X τ F τ1 ) = N ) ) µx d b µ X VB1 σ, τ 1 + V B N d + b µ X, τ 1 ) Q X τ >, V F F τ1 ) = N ) ) µx d c τ 1 ) + b µ X VB1 σ, τ 1 V B N d c τ 1 ) + b µ X, τ 1 ) a i = lnv /V Bi ) σ i ; b i µ X i, s ) = µx i s ; c i s) = lnf/v Bi) ; d = lnv B1/V B ) σ i σ i s σ τ1 Whee µ X i is he dif of he logaihm of he asses value diffusion pocess, unde pobabiliy measue Q X : µ µ P i = α i σi /; µ V i = µ P i + σi = α i + σi /; µ m ) i = P i + σ i whee he subscip i idenifies he sae of he fim i =1 befoe he aing change, i = afe he aing change), and N - cumulaive sandad nomal densiy funcion. If V B1 > F, subsiue Q X τ 1 >, V F F ) by Q X τ 1 > F ) and Q X τ 1 >, V < F F ) by 0. Afe a aing change, fo > τ 1, he bond value educes o 7 : B V,,, c, F, ) = cf 1 e ) 1 Q P τ F ) + + ρ 1 V B cf ) ) µm µp V σ V B Q m τ F ) + 7 We will use he noaion B ) as he value bond afe he aing downgade o diffeeniaes fom B ) which is he value bond befoe he aing downgade. 7
+ F e ) Q P τ >, V F F ) + ρ V e α ) Q V τ >, V < F F ) 6) Whee: Q X τ F ) = N a b µ X, ) + V V B ) µx σ N a + b µ X, ) Q X τ >, V F F ) = N a c ) + b µ X, ) ) µx V σ V B N a c ) + b µ X, ) 3. Raing-based covenan: epaymen of a facion of he pincipal Conside now a case whee he aing igge covenan, insead of geneaing an incease in coupon ae, leads o a paial epaymen. Le 1 ) be he facion of he pincipal ha is epaid < 1). his implies ha, a he aing change dae τ 1 ), he bondholdes eceive an addiional cash-flow given by: 1 )F, being F he new face value of he bond which will geneae a new coupon flow of cf d. As in Bhano and Mello 006), we will conside wo souces of financing fo his epaymen: eihe hough new equiy o hough he sale of asses. Bu insead of eaing hem sepaaely as done by hose auhos), we will inegae boh appoaches, giving ise o a moe geneal famewok whee he efund can be funded by a combinaion of he wo souces. Specifically, if we define θ as he facion of he paymen made hough he sale of asses, 0 θ 1), he epaymen of 1 )F will be funded by 1 θ)1 )F geneaed hough a new equiy issue and θ1 )F geneaed hough he sale of asses. he diffeen souce of funding influences he bond value only hough he pobabiliy of eaching he liquidaion level V B ), when he value of he asses eaches he aing igge level V B1 ). Specifically, a τ 1, when he aing of he fim is changed, if he sale of asses is used o finance he efund fo θ > 0), he asses value jumps immediaely fom V B1 o V B1 θ1 )F see figue ), aising he pobabiliy of liquidaion and educing in his manne he value of he bond. So, he geae he θ, he lowe he bond value will be. Noice also ha, if he value of he asses afe he jump is equal o lowe han V B, V B1 θ1 )F V B ), hen, he fim is immediaely liquidaed, τ 1 = τ ) 8 he bond value expessions fo his case, ae simila o he pevious one, wih he excepion of he definiion of B V τ1, τ 1,, c, F, ), whee i consides now he paial edempion 1 )F, he new face value F, and he fac ha a τ 1 he asses value is given by V B1 θ1 )F insead of V B1, hus expession ) is eplaced by: BV τ1, τ 1,, c, F,, θ) = 1 ) F cf e s τ1) 1 s<τ ds τ 1 F τ 1 + e τ1) F 1 τ>,v F + ρ V 1 τ>,v < F ) F τ1 + e τ τ1) ρ 1 V τ 1 τ< F τ1 8 his scenaio will no be aken ino accoun in he deivaion of he bond value fomula since we will assume ha V B1 θ1 )F > V B. 8
Figue : Sample pahs of he asses value consideing diffeen souces of funding. Black line - fully financed by new equiy θ = 0); Gay line - when he selling of asses is used θ > 0). Whose value is: B V τ1, τ 1,, c, F,, θ) = 1 ) F + cf 1 e τ1) 1 Q P τ F τ1 ) + + ρ 1 V B cf ) ) VB1 θ 1 ) F µm µp σ V B Q m τ F τ1 ) + + F e τ1) Q P τ >, V F F τ1 ) + + ρ V B1 θ 1 ) F e α τ1) Q V τ >, V < F F τ1 ) 7) Whee Q X τ F τ1 )is inepeed as he pobabiliy unde measue Q X ha he asses value eaches V B fom V B1 θ 1 ) F by he mauiy and is defined as Q X τ F τ1 ) eplacing V B1 by V B1 θ 1 ) F. he same is ue fo Q X τ >, V F F τ1 ) in elaion o Q X τ >, V F F τ1 ) whee addiionally F is eplaced by F. Noice ha he pevious expession assumes ha he fim is no liquidaed a he aing change, ha is: V B1 1 ) F > V B and addiionally ha F > V B. Afe he aing change, fo > τ 1, he bond value is given by expession 6), afe adjusing fo he new face value, hus he las line uns o: F e ) Q P τ >, V F F ) + ρ V e α ) Q V τ >, V < F F ) I is wohwhile poining ou ha boh pevious models wok on he pemise ha he elaionship beween he value of he fim s asses and liabiliies is he single dive of is cedi sance. 9
his is equivalen o assuming ha any excepional siuaion of insananeous insolvency, poenially leading o he inabiliy of he fim o honou a coupon paymen, would be solved wih a empoay inflow of funds, povided by he shaeholdes. 4 he Leveaged Fim Value In he pevious secion we deived he valuaion fomulae fo deb. In his secion, we obain he valuaion fomulae fo equiy, ax benefis and bankupcy coss. 4.1 Equiy Value he equiy value a < τ 1, will be given by he expeced pesen value of all cash flows aising fom he diffeen fuue scenaios he fim is facing. Applying he same easoning used in he valuaion of deb claims, we may use he following expession simila o expession 3) fo he deb): EV,,, c, F, ) = E Q P α 1 V s cf 1 ι)) e s ) 1 s<τ1 ds F e ) V F ) 1 τ1>,v F F e τ1 ) 1 τ1< EV τ1, τ 1,, c, F, ) F 8) Whee EV,,, c, F, ) and EV τ1, τ 1,, c, F, ) denoes he equiy value a < τ 1 ) and τ 1 especively. So, condiioning in he fim no suffeing a aing downgade unil mauiy of he deb fis wo lines), he value of equiy akes ino accoun he seam of dividends, defined as he cash payou α 1 V s ) minus he coupon paymens adjused fo he ax benefi of deb cf 1 ι)), whee ι is he copoae ax ae fis line), and he esidual value of he fim a mauiy afe he epaymen of he pincipal of he deb second line). Noice ha if τ 1 > and V < F, he shaeholdes eceive nohing since we ae assuming he veificaion of he absolue pioiy ule. Ohewise, if he value of he asses eaches he fis baie befoe he bond maues hid line), he value of he equiy a τ 1 is EV τ1, τ 1,, c, F, ), which in un is defined as follows: - Fo he accued coupon ae aing igge covenan > 1): EV τ1, τ 1,, c, F, ) = E Q P α V s c F 1 ι)) e s τ1) 1 s<τ ds F τ 1 V F ) e τ1) 1 τ>,v F F τ1 9) - Fo he paial epaymen of he pincipal aing igge covenan < 1): EV τ1, τ 1,, c, F, ) = E Q P α V s c F 1 ι)) e s τ1) 1 s<τ ds F τ 1 V F ) e τ1) 1 τ>,v F F τ1 1 θ)1 )F 10) 10
So, a τ 1, he value of equiy is given by he expeced pesen value of: he seam of dividends fom τ 1 o o τ whicheve comes fis fis line of he expessions). hese ae defined by he new cash payou α V s ) deduced fom he new coupon paymen adjused fo he fiscal benefi c F 1 ι)). Remembe ha fo he accued coupon ae aing igge case, > 1, whee 1) is he elaive incease in he coupon ae and fo expession 10), < 1, whee 1 ) coesponds o he facion of he pincipal ha is edeemed. he esidual value of he fim a mauiy, if he fim hasn been liquidaed in he meanime; which is defined by V F ), expession 9)) since he deb pincipal emains unchanged, and by V F ), in he second case expession 10) since a facion 1 ) of he pincipal has been paid a he momen of he aing change. Fo he second ype of aing igge covenan, we sill have o ake ino accoun he facion of he pincipal amoized hough a new cash infusion fom shaeholdes las line in expession 10)) he final expessions fo equiy value, fo < τ 1 ae deivaion in he appendix A.): E V,,, c, F, ) = V 1 e α1 ) Q V τ 1 >, V < F F ) V B1 cf 1 ι) ) ) cf 1 ι) µm 1 µp 1 V σ 1 V B1 Q m τ 1 F ) 1 e ) 1 Q P τ 1 F )) F e ) Q P τ 1 >, V F F ) + + e τ1 ) E V B1, τ 1,, c, F, ) g P τ 1, V, V B1 ) dτ 1 11) As fo he deb value, if V B1 > F, subsiue Q X τ 1 >, V F F ) by Q X τ 1 > F ) and Q X τ 1 >, V < F F ) by 0. In expession 11), EV B1, τ 1,, c, F, ) is defined as deivaion in he appendix A.): Fo he accued coupon ae aing igge covenan case soluion of 9) ): E V B1, τ 1,, c, F, ) = V B1 1 e α τ1) Q V τ >, V < F F τ1 ) V B c F 1 ι) c F 1 ι) ) VB1 V B ) µm µp σ Q m τ F τ1 ) 1 e τ1) 1 Q P τ F τ1 )) F e τ1) Q P τ >, V F F τ1 ) 1) 11
And, fo he paial epaymen of he pincipal aing igge covenan soluion of 10)): E V B1, τ 1,, c, F, ) = V B1 θ 1 ) F 1 e α τ1) Q V τ >, V < F F τ1 ) V B ) c F 1 ι) VB1 θ1 )F c F 1 ι) V B ) µm µp σ 1 e τ1) 1 Q P τ F τ1 )) Q mτ F τ1 ) F e τ1) Q P τ >, V F F τ1 ) 1 θ)1 )F 13) Whee he pobabiliies Q X ) and Q X ) ae defined as above in secion 3. Recall ha expession 13) is only valid fo V B1 θ1 )F > V B. Afe he aing change, fo > τ 1, he value of equiy is obained hough expessions 1) and 13) afe eplacing τ 1 fo, V B1 and V B1 θ1 )F ) fo V in 1) and 13) especively. In 13), Q X ae subsiued by Q X ) and he las em in he fouh line also disappeas. I is woh noing ha, fo high face value bonds face values highe han he baie level: F > V B1 ) wih a paial efund aing igge covenan, when an equiy issue is used o fund he paial epaymen θ < 1), he expession 13) may un negaive fom a ceain value of τ 1 onwads. his eflecs he fac ha, fom he equiy holdes poin of view, in his kind of siuaion, an addiional cash infusion migh have a negaive expeced ne pesen value, consequenly, in his case, he shaeholdes aional decision making pocess will lead hem o shun he coesponding paymen. If ha happens, he equiy value mus be zeo and he fim will ene ino bankupcy. Fo example, assume he following paamees value 9 : V 0 = 150, V B1 = 100, V B = 50, α 1 = 0, 07, α = 0, 10, σ 1 = 0, 30, σ = 0, 45, ι = 0, 35. Consideing a bond wih F = 130, c = 13%, a ime o mauiy of 10 yeas and a paial efund of 0% = 0, 8) paially financed by equiy θ = 0, 5), he fim immediaely enes ino bankupcy if he baie is hi afe 8 yeas if τ 1 8). his level of τ 1 8 yeas, in he example) will end o be lowe he lowe θ and and he highe F and c. hus, fo face values geae han V B1, in he inegal of expession 11), E V τ1, τ 1,, c, F, ) mus be eplaced by max E V τ1, τ 1,, c, F, ) ; 0 10,11. 4. ax Benefis and Bankupcy Cos Values he final expessions fo he value of ax benefis B) and bankupcy coss BC), a < τ 1 ae as follows deivaion in he appendix A.3): BV,,, c, F,, ι) = cf ι 1 e ) 1 Q P τ 1 F ) cf ι V V B1 ) µm 1 µp 1 σ 1 Q m τ 1 F ) + e τ1 ) BV τ1, τ 1,, c, F,, ι) g P τ 1, V, V B1 )dτ 1 14) 9 Fom Bhano 003) 10 If his adjusmen is no aken ino accoun, alhough expession 11) wih 13) will undevalue he equiy, he diffeence will be insignifican fo high values of V 0 and ime o mauiy. Fo he example of he ex, he diffeence in value is less han 0,3%. 11 Noe fuhe ha, in his siuaion we also have o adjus he bond picing fomula; hus, if we assume ha in he even of bankupcy a τ 1 when shaeholdes ae no willing o ealize he cash infusion), he same ecovey ae o bond holdes as in τ, expession 7) is eplaced by ρ 1 V B1, which is he pay-off o bond holdes condiioned on bankupcy a τ 1 ha is condiioned on max E V τ1, τ 1,, c, F, ) ; 0 = 0) 1
Whee, fo he accued coupon ae case > 1): BV τ1, τ 1,, c, F,, ι) = c F ι 1 e τ1) 1 Q P τ F τ1 ) c F ι ) µm µp VB1 σ V B Q m τ F τ1 ) 15) and fo he paial amoizaion of pincipal < 1): BV τ1, τ 1,, c, F,, ι) = c F ι 1 e τ1) 1 Q P τ F τ1 ) c F ι VB1 θ1 )F V B ) µm µp σ Q mτ F τ1 ) 16) uning o he bankupcy coss: BCV,,, c, F, ) = 1 ρ )V e α1 ) Q V τ 1 >, V < F F ) + + e τ1 ) BCV τ1, τ 1,, c, F, ) g P τ 1, V, V B1 )dτ 1 17) Whee, fo he accued coupon ae case > 1): ) µm µp VB1 σ BCV τ1, τ 1,, c, F, ) = 1 ρ 1 )V B Q m τ F τ1 ) + V B + 1 ρ )V B1 e α τ1) Q V τ >, V < F F τ1 ) 18) and fo he paial amoizaion of pincipal < 1): ) VB1 θ 1 ) F µm µp σ BCV τ1, τ 1,, c, F, ) = 1 ρ 1 )V B Q V mτ F τ1 ) + B + 1 ρ ) V B1 θ 1 ) F e α τ1) Q V τ >, V < F F τ1 ) 19) Once again, if V B1 > F, subsiue Q X τ 1 >, V < F F τ1 ) by 0, and in 16) and 19), i is assumed ha V B1 θ1 )F > V B. Afe he aing change, fo > τ 1, he values of ax benefis and bankupcy coss ae obained hough expessions 15) and 18) afe eplacing τ 1 fo and V B1 fo V in he accued coupon ae case. Similaly, fo he paial amoizaion of pincipal, he coesponding values ae obained 13
using expessions 16) and 19) afe subsiuing τ 1 and V B1 θ1 )F ) by and V especively and Q X ) by Q X ) 1. 4.3 Leveaged fim value he value of he leveaged fim, defined as v.) is now obained eihe by he sum of bond and equiy values o by he sum of he values of fim asses and ax benefis less he value of bankupcy coss: v ) = B ) + E ) = V + B ) BC ) Noice ha fo he paial epaymen of pincipal aing igge covenan case, he value of some vaiables will jump wih he change of he fims aing when he baie level V B1 is eached - a τ 1 ). Specifically: i) he bond value, iespecive of he funding souce used o ealize he paial edempion of he pincipal, declines immediaely afe τ 1, by an amoun equal o he value amoized; ii) he equiy value, when he paymen o bondholdes is fully funded hough a cash infusion, ises by he same amoun, immediaely afe τ 1. In conas, when he sale of asses is used, he equiy value emains unchanged immediaely afe τ 1 ; iii) he leveaged fim value, immediaely afe τ 1 emains he same wih he issue of equiy, and dops wih he sale of asses, by he same amoun of he asse sale which is 1 )F. he simulaneous use of boh souces of funding will, naually, lead o inemediae jump obseved fo equiy and fim value. 5 Compaison wih Bhano s Model As peviously noed, he main shocoming of Bhano s model is he absence of defaul a mauiy o, puing i diffeenly, he implici assumpion ha, a mauiy, bondholdes always eceive he full amoun of he pincipal. Such an assumpion leads o an ovepice of he bond. Following we compae Bhano s 003) esuls wih ous, using he same paamee values o highligh he diffeences, namely: V 0 = 150; V B1 = 100; V B = 50; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = 0, 5 13 ; ι = 0, 35; F = 100. Fo he coupon ae we will assume c = 9%. Since Bhano s model does no explicily assume any kind of change in bondholdes pay-offs, eihe a o afe he aing change of he fim, and o isolae only he mauiy defaul effec, we shall assume ha in ou model = 1. We sa by defining he excess pice EP ) of Bhano s model in elaion o ous as: EP = B B /B 1)100 Whee, B B and B sand fo he pice of he bond in Bhano s model and in ou model, especively. Figue 3, shows he excess pice as a funcion of ime o mauiy. Figue 3a, consides diffeen values fo he ecovey facion of he asse values a mauiy. As we can see, even in he case whee hee is no defaul cos a mauiy ρ = 1, he bondholdes eceive V if V < F ), he pice diffeence can each 3%. As expeced, a decease in he ecovey facion lowes he ue pice of he bond and so he excess pice ises. Figue 3b, highlighs he effecs of he asse value on pice diffeences consideing ρ = 0, 9). he lowe he value of he asses, he geae he pobabiliy of he fim eneing in defaul a mauiy, esuling in highe excess pices. 1 I is woh noing ha if he face value is geae han he baie level, an adjusmen, simila o he one saed in he foonoe 10 fo he bond value, would have o be made in expessions 14) and 17). 13 Alhough on he legend of exhibi in Bhano 003), page 61, i is wien ha he bankupcy coss epesen 50%, leading o a ecovey facion of 50% ρ 1 = 0, 5), he gaph assumes ha ρ 1 = 1) no bankupcy coss). ha same value ρ 1 = 1) is assumed heeafe on he ohe exhibis in Bhano. By defaul, in he cuen secion, i will be assumed ha ρ 1 = 0, 5). Even so, in ode o allow a compaison wih Bhano 003), some numeical esuls wee calculaed assuming ρ 1 = 1). 14
Figue 3: Excess pice as a funcion of ime o mauiy, wih: V B1 = 100; V B = 50; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = 0, 5; F = 100 and c = 0, 09. Figue 3b consides hee diffeen values fo ρ wih V = 150, and figue 3b consides hee diffeen values fo V wih ρ = 0, 9. In fac, as figue 4 illusaes, ovepicing ises subsanially when he value of he fim s asses eaches V B1 figue 4a), and hen appoaches he liquidaion level, V B figue 4b). Remembe ha since we ae assuming ha = 1, he pice diffeence is only aibuable o a possible defaul a mauiy which pesumes ha he face value of he bond is geae han he second baie level F > V B ). Ohewise, if V B is geae han F, hen he value of he asses a mauiy in condiioning of neve having eached V B in he meanime) will always be sufficien o epay he pincipal and, in ha case, defaul a mauiy will neve happen. So, given a ecovey facion a mauiy ρ ), he smalle he gap beween F and V B, he smalle he excess in pice will be. If V B F, ou model assuming = 1) conveges o Bhano s model. hese emaks ae illusaed in figue 5. In figue 5a, hee diffeen values fo V B ae consideed wih ρ = 0, 9 and F = 100), and in figue 5b, hee diffeen face values of deb ae also aken ino consideaion keeping he coupon ae a 9%,ρ = 0, 9 and V B = 50). In boh figues, i is assumed ha = τ 1, ha is, he value of he asses equals V B1. Fixing he ime o mauiy, we can have an infinie numbe of combinaions of coupon aes and face values o which a single value fo he bond issue coesponds. If we conside only bonds issued a pa face value equal o emission value), we ae able o compae, fo he diffeen models, he pa coupon ae c pa known as pa yield), defined as follows: c pa : BV,,, c pa, F, = 1) = F 0) Since Bhanos model ovepices he bond, fo a given value of he issue, ceeis paibus, i is expeced ha he pa coupon ae inheen in Bhano s model will be lowe han ha in ous. Figue 6 shows exacly hese esuls. I gaphs he pa coupon ae in pecenage), as a funcion of he face value of he bond equal o he emission value) consideing a 5-yea figue 6a) and a 10-yea figue 6b) ime peiod o mauiy. In each of hese figues, wo diffeen values ae consideed fo he ecovey facion upon liquidaion of he fim ρ 1 = 0, 5 and 1). Relaing o ou model, we sill coninue o assume ha = 1, so afe he aing change of he fim, he coupon and pincipal of he bond do no undego any changes, and we also conside ρ = 1 no 15
Figue 4: Excess pice as a funcion of ime o mauiy, wih: V B1 = 100; V B = 50;α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = 0, 5; F = 100 and c = 0, 09. Figue 4a consides hee diffeen values fo ρ wih V = 100 a τ 1 ), and figue 4b consides hee diffeen values fo V appoaching he liquidaion heshold) wih ρ = 0, 9. Figue 5: Excess pice as a funcion of ime o mauiy, wih: V = V B1 = 100; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = 0, 5; ρ = 0, 9 and c = 0, 09. Figue 5a consides hee diffeen values fo V B wih F = 100, and figue 5b consides hee diffeen values fo F and V B = 50. 16
Figue 6: Compaison of pa coupon aes in pecenage), as a funcion of face value, in Bhanos model and in ou model wih: V = V B1 = 100; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ = 1 fo diffeen values of ρ 1 0,5 and 1) consideing a five yeas ime lapse o bond mauiy figue 6a) and a en yeas ime o bond mauiy figue 6b). bankupcy coss when defaul occus a mauiy). As we can see, he geae he pincipal meaning a geae gap beween F and V B and consequenly a geae excess pice), he geae he diffeence will be beween pa coupon aes deived fom ou model and hose calculaed on he basis of Bhanos model. Fuhemoe, his diffeence ends o shink wih ime o mauiy, since ou model wih = 1) only diffes fom Bhano s model in espec o cash-flow a mauiy: he geae he mauiy he lowe he elaive impoance of he cash-flow specifically, he expeced pesen value) will be, in ems of pice. If, insead of vaying he face value, we vay he ime o mauiy ) in 0) fo a given face value fo he bond, we obain he pa yield cuve. Bhano 003) calculaes he pa yields fo six mauiies second column of exhibi 6, page 63 in Bhano 003)) assuming F = 100 and ρ 1 = 1. We conduc a simila analysis, using he same paamee values, in compaing ou model wih Bhano s model, bu insead of elying on he pa yield cuves, we analyze he esuling cedi speads. Noice ha he analysis is simila since he famewok elies on a consan isk-fee inees ae wih a fla isk-fee yield cuve. In figue 7a below we gaph he cedi spead cuves in basis poins) esuling fom Bhano s model and ou model. he hump-shape is pesen in boh models, alhough moe ponounced in ou model. As we can see, and as expeced, even consideing he absence of bankupcy coss, in he case of defaul a mauiy ρ = 1 in ou model), he diffeences ae expessive especially in sho mauiies. o highligh hese diffeences we have ploed hem in a sepaae gaph figue 7b). I is woh poining ou ha hese cedi spead diffeences ae independen of he ecovey value ρ 1. In fac, he influence of ρ 1 is he same in boh models. I only influences he bondholdes payoff when he fim is liquidaed befoe mauiy). So, alhough diffeen ecovey values lead o diffeen bond pices, hese changes in pice will be he same in boh models, which in un will 17
Figue 7: Compaison of cedi speads in basis poin), as a funcion of ime o mauiy, in Bhanos model and in ou model figue 7a); Cedi spead diffeences beween he models in basis poin), as a funcion of ime o mauiy, figue 7b).Wih V 0 = 150; V = V B1 = 100; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = ρ = 1 and consideing a face value of 100 issued a pa. lead o an equal change in he pa yields leaving he diffeence on cedi speads unchanged. 6 he Influence of he Raing igge Covenan In he pevious secion, since he focus was o highligh he ovepicing inheen in Bhano s model, we have assumed, in ou model, fo he pupose of compaison, ha = 1. In he cuen secion, his assumpion will be elaxed fo wo main easons: o analyze in moe deail he effecs on bond pices and cedi speads of a aing igge covenan specified as an incease in coupon ae o a paial edempion of he pincipal, and o allow compaisons wih bonds issued wihou his kind of covenan. his compaison is also pesen in Bhano 003) bu ou analysis is disinc in seveal aspecs. In he fis place, as saed peviously, he bond pice fomula pesened by his auho does no ake ino accoun any kind of change on bondholdes cash flow afe he aing change. In essence, he only influence of he aing based covenan on Bhano s bond pice is he consideaion of a geae payou aio and asse value volailiy afe he downgade of he fim. Addiionally, in picing bonds wihou covenans, hose changes ae no aken ino accoun 14 ; insead Bhano 003) uses an implied volailiy value 15. On he conay, elying on he fac ha iespecive of he deb ype wih o wihou aing igge covenans), he fim is equally subjec o aing noaion and so o aing changes), we always assume he exisence of he wo baies he downgade level and he liquidaion level). hus, fo he case of bonds wihou covenans, when he aing of he fim is changed, nowihsanding he fac ha he bondholdes cash flow emains unchanged, he fim may ale is cash payou ae and isk in he same way as is assumed in he case of bonds wih aing igge covenans. In sho, he pices of he diffeen bonds ae obained as follow: - using expession 4) and 5), wih = 1, fo bonds wihou aing igge covenans; - using expessions 4) and 5), wih > 1, fo bonds wih aing igge covenan of he ype accued coupon ae, whee 1) is he elaive change in he coupon ae; 14 If hey wee, hee would be no disincion beween he wo ypes of bonds. 15 I compue he implied volailiy of he asse pocess ha makes he model bond pice wihou covenans equal he ue pice. Page 61 in Bhano 003) 18
- using expessions 4) and 7), wih < 1, fo bonds wih aing igge covenan of he ype paial pepaymen of he pincipal, whee 1 ) is he facion of he facial value ha is edeemed; Remembe fom secion 3, whee in his las case, he paial amoizaion of he deb can be financed eihe hough selling asses, cash infusions fom shaeholdes o a combinaion of boh. In wha follows, we will assume fo he paamee values: V 0 = 150; V B1 = 100; V B = 50; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = 0, 8; ρ = 1. o compae he hee ypes of bonds, we esic ou analysis o pa issued deb. 6.1 Pa Yields Since he exisence of a covenan in he bond indenue aims o poec bondholdes inees, we would expec ha, fo a given face value of deb, bondholdes would equie a lowe coupon ae in his kind of bond as compaed o ha equied fom unpoeced deb. Such a esul is povided in ou model, as illusaed in figue 8a. he figue plos he pa) coupon ae as a funcion of bond s face value emission value), assuming a ime o mauiy of 10 yeas, fo hee ypes of deb: a bond wihou covenan, a bond wih accued coupon ae aing igge covenan wih = 1,, so a 0% incease in he coupon), and a bond wih paial efund aing igge covenan wih = 0, 8, so a 0% educion in he face value). Fo his las ype, a disincion is also made egading he financing souce fully financed by new equiy: θ = 0; o fully financed hough he sale of asses: θ = 1). Compaing boh covenans, accued coupon ae and paial amoizaion of pincipal when financed hough cash infusion θ = 0), fo he same pa value, and a same pecenage change posiive in he fome, negaive in he lae case), he equied pa coupon ae fo he fis case is always geae han ha fo he second case. In ohe wods, fo he wo ypes of bonds o have he same equied pa coupon ae hus being equivalen a he issue dae), he pecenage incease in he coupon ae mus be geae han he face value pecenage decease. Fo example, consideing a ime o mauiy of 10 yeas and a pa emission value of 100, he pa coupon ae of a bond wih a 0% educion of pincipal aing igge covenan is 10,994%, while fo he accued coupon ae aing igge covenan case, he pecenage incease in he coupon ha euns he same pa coupon ae is 37,53%. Noe howeve, ha afe he deb is in place, he especive pa yields evolve diffeenly ways as ime passes. Noice ha wha diffeeniaes hese wo ypes of bonds is he coupons eceived by bondholdes and he way he pincipal is edeemed, afe he aing downgade. Specifically, in one case, bondholdes eceive highe coupons bu ae exposed o a geae loss if bankupcy occus hey eceive ρ 1 V B insead of F ). In he ohe case, alhough bondholdes eceive a lowe coupon, he loss incued in he bankupcy scenaio is also lowe ρ 1 V B insead of F 16 ), since pa of he pincipal was aleady eceived when he fim s aing changed. Noe also ha, in eihe case, he pobabiliies of he fim eneing ino bankupcy V B be cossed) ae he same 17. On he ohe hand, when he paial efund is funded hough he sale of asses θ = 1), he downgade in he fim s cedi aing leads o a downwad jump in he value of he asses by he amoun of he efund) which in un will cause an incease in he pobabiliy of bankupcy. hus, his kind of bond is iskie o less poeced, when compaed wih he equiy issue case, and ha fac is efleced in he highe pa coupon ae equied by bondholdes a he issue dae) as illusaed in figue 8a. he geae he face value, he geae he downwad jump fo a fixed ) 16 I is impoan o emembe ha we ae assuming ha F he new face value afe he paial edempion of pincipal) is geae han V B. 17 Ye, he pobabiliy of defaul a mauiy is diffeen since F < F. 19
Figue 8: Pa coupon ae in pecenage) as a funcion of he face value consideing a 10 yeas mauiy bond figue 8a). Pa yields in pecenage) as a funcion o ime o mauiy, consideing F = 100 figue 8b)..Wih V 0 = 150; V B1 = 100; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = 0.8 and ρ 1 = ρ = 1. would be, and also he geae he diffeence would be beween he equied coupon ae on he wo bonds. In effec, fo high values of pincipal, he paial efund aing igge covenan bond, when financed by he selling of asses, can be iskie han he accued coupon ae aing igge covenan bond. he pa yield cuves fo he fou bonds ae also ploed figue 8b), assuming a 100 face value. As we can see, in confomiy wih wha was saed in he pevious paagaphs, he bond wihou he covenan yields he highes cedi speads eaching o almos 500 basis poins fo a ime lapse o mauiy of 7,5 yeas given he values fo he paamees in he example) while he lowes cedi speads ae geneaed by he paial efund aing igge covenan when financed by cash infusion whee he cedi speads do no exceed 50 basis poins). I mus be emphasised ha hese numeical esuls ely upon he specific paamee values used. Specifically i was assumed ha afe he aing downgade, he payou aio of he fim changed in he same manne iespecive of he bond ype. If insead, depending on wha happens o he coupon value afe he aing change, we had consideed diffeen changes in he payou aio, he esuls would also be diffeen. Fo example, if we had assumed no change fo he no covenan case since he coupon emains unchanged) and a decease fo he paial amoizaion case since he educion in he pincipal educes he coupon), he pa yields of hese bonds would have been lowe 18. In paicula, he pa yields of he bond wih he paial efund covenan when fully financed by he selling of asses, could even be lowe han hose fom he accued coupon ae case. 6. Equiy and Leveaged Fim Value I is ineesing o compae he values of equiy and he whole leveaged fim associaed wih he vaious kinds of deb. able 1 epos hose values consideing a 100 face value bond issued a pa wih a mauiy of 10 yeas. Fo bonds wihou he aing igge covenan and accued coupon ae aing igge covenan ype, he values ae obained fo he issue dae = 0), and fo wo hypoheical downgade daes, τ 1 = 6 and τ 1 = 8 he baie V B1 is hi when he ime o mauiy of he bond is 4 and yeas especively). Fo bonds wih a paial efund aing igge covenan, 18 Noneheless, he bond wih no covenans would sill have he highes pa yields. 0
able 1: Equiy, bond and leveage fim values a he issue dae =0) and a he aing downgade dae τ 1 = 6 and 8 fo he no covenan and accued coupon ae cases -fis panel; and τ 1 = 6 fo he paial amoizaion case - second panel). he bonds have a face value of 100 issued a pa wih a mauiy of 10 yeas, and V 0 = 150; V B1 = 100; α 1 = 0, 07; α = 0, 10; σ 1 = 0, 30; σ = 0, 45; = 0, 075; ρ 1 = 0.8 and ρ 1 = ρ = 1. besides he issued dae, values ae obained fo τ 1 = 6, and immediaely afe τ + 1 ). In addiion o some of he pevious findings, he able shows ha he equiy value, and he leveaged fim value ae insensiive in elaion o he pecenage incease on he coupon ae. he same esul does no hold fo he paial efund aing igge covenan case. Indeed, in his lae case, he geae he amoizaion of pincipal, he lowe he equiy and he coesponding leveaged fim value. 7 Conclusion Using a famewok simila o ha used by Bhano 003), we developed a model o pice finie mauiy coupon bonds wih aing igge based covenans, which esolved an inconsisency inheen in ha model. namely, he absence of defaul a mauiy. We showed ha, his limiaion in Bhano s model could lead o a significan bond ovevaluaion. In addiion, compaisons wee made consideing bonds wih diffeen ypes of aing igge covenans. Alhough, he bond wih a paial efund aing igge covenan, when fully financed by equiy infusion, offes he geaes bondholde poecion and hus he lowes cedi spead, he coesponding equiy and leveaged fim values ae lowe when compaed wih hose coesponding o he accued coupon aing igge covenan. 1
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Appendix Peliminaies he dynamics of he logaihm of he asses value, Y = lnv ae: dy = µ X i d + σ i dw X whee W X is a Wiene pocess unde he pobabiliy measue Q X and µ X i is he coesponden dif X = P, V o m). Q P - is he pobabiliy measue when he asses value pocess is nomalized by he saving accoun; Q V - is he pobabiliy measue when he asses value is used as numeaie; Q m - is he pobabiliy measue when asses paying one uni a defaul ae used as numeaie. he coesponding dif ems ae defined as follows: µ P i = α i σi / ) µ V i = µ P i + σi = α i + σi / ) µ m i = µ P i ) + σi E Q X 1 A F = Q X A F ) = pobabiliy, unde measue Q X, of even A occuing. he fis passage ime densiy a, > ), of he asses value fom V o a baie level V B, V > V B ), unde pobabiliy measue Q X is given by: ) V «ln V B g X, V, V B ) = e 1 lnv /V B )+µ X ) σ πσ ) 3 and he coesponding cumulaive disibuion funcion, is: G X, V, V B )= ) = N ln V V B µ X ) σ + V V B g X u, V, V B )du = ) µ X σ whee N sands fo he cumulaive sandad nomal densiy funcion. ) N ln V V B + µ X ) σ 1) Consideing he definiions of τ 1 and τ in he body of he ex: Q X τ 1 F ) = G X, V, V B1 ) - using µ X 1 and σ 1 in expession 1). Q X τ F τ1 ) = G X, V τ1, V B ) and Q X τ F ) = G X, V, V B ) - using µ X σ in expession 1). and E Q P e τi ) 1 τi F = e u ) g P u, V, V Bi )du = = V V Bi ) µm i µp i σ i g m u, V, V Bi ) du = V V Bi ) µm i µp i σ i Q m τ i F ), whee i = 1,. 4
E Q P e τ τ1) 1 τ F τ1 = E Q V e αiτi ) 1 τi F = = V V Bi ) µ m i µp i σ i 1«VB1 V B ) µm µp σ Q m τ F τ1 ) e αiu ) g V u, V, V Bi )du = g m u, V, V Bi ) du = V V Bi ) µ m i µp i σ i «1 Q m τ i F ), A.1 whee i = 1,. BV,, ) = E Q P cf e s ) 1 s<τ1 ds F +E Q P e ) ρ V 1 τ1> ;V <F F e ) F 1 τ1> ;V F F e τ1 ) 1 τ1< BV τ1, τ 1, ) F Fis em: E Q P cf e s ) 1 s<τ1 ds F = E Q P τ1 cf e s ) ds F = = cf EQ P e ) 1 τ1> + e τ1 ) 1 τ1 1 F = = cf = cf { } 1 e ) E Q P 1 τ1> F E Q P e τ1 ) 1 τ1 F = 1 e ) Q P τ 1 > F ) ) µ m 1 µp 1 V V B1 σ 1 Q m τ 1 F ) Second em: E Q P e ) F 1 τ1>,v F F = F e ) Q P τ 1 >, V F F ) hid em: E Q P e ) ρ V 1 τ1>,v <F F = = ρ V e α1 ) E Q P e α 1 ) ) V V 1 τ1>,v <F F = = ρ V e α1 ) E Q P e α 1 ) ) V V F E Q V 1 τ1>,v <F F = = ρ V e α1 ) Q V τ 1 >, V < F F ) noe ha e α1) ) V is a maingale unde pobabiliy measue Q P ) 5
Fouh em: E Q P e τ 1 ) 1 τ1< BV τ1, τ 1, ) F = e τ1 ) BV B1, τ 1, ) g P τ 1, V, V B1 ) dτ 1 Collecing ems and noing ha Q X τ i > ) = 1 Q X τ i ), yields he expession 4) in he body of he ex. BV τ1, τ 1, ) = E Q P cf e s τ1) 1 s<τ ds F τ 1 + e τ1) F 1 τ> ;V F F τ1 e τ1) ρ V 1 τ> ;V <F F τ1 + e τ τ1) ρ 1 V τ 1 τ< F τ1 Noice ha he fis hee ems in he above expession ae simila o hose elaing BV,, ; c, F, ), he diffeence being he filaion consideed τ 1 insead of ) and he coupon cf insead of cf ). hus, applying he same deivaion, he soluion will be he same as ha obained peviously afe eplacing V by V τ1 = V B1, by τ 1 and cf by cf. he fouh em: E Q P e τ τ 1) ρ 1 V τ 1 τ< F τ1 = ρ1 V B E Q P e τ τ 1) 1 τ< F τ1 = ) µ m µp = ρ 1 V VB1 σ B V B Q m τ F τ1 ) Collecing ems and noing ha Q X τ i > ) = 1 Q X τ i ), yields he expession 5) in he body of he ex. Afe he downgade, he bond value is defined as BV τ1, τ 1,, c, F, ), bu since > τ 1 he elevan filaion is F, hus: BV,, ) = E Q P cf e s ) 1 s<τ ds F e ) F 1 τ>,v F F e ) ρ V 1 τ>,v <F F e τ ) ρ 1 V τ 1 τ< F Whose soluion will be he same of Bτ 1, τ 1,, c, F, ) expession 5) afe eplacing V B1 by V and τ 1 by yielding expession 6). A. Equiy value a < τ 1 : 6
EV,, ) = E Q P α 1 V s cf 1 ι)) e s ) 1 s<τ1 ds F e ) V F ) 1 τ1>,v F F e τ1 ) 1 τ1< EV τ1, τ 1, ) F Fis em: E Q P α 1V s by Fubini s heoem) cf 1 ι)) e s ) 1 s<τ1 ds F = = α 1 E Q P Vs e s ) 1 s<τ1 F ds cf 1 ι) E Q P e s ) 1 s<τ1 F ds = = α 1 V e α1s ) E Q V 1 s<τ1 F ds cf 1 ι) E Q P τ1 e s ) F ds = = V E Q V τ1 α 1 e α1s ) ds F cf1 ι) E Q P 1 e ) 1 τ1> e τ1 ) 1 τ1 F = = V E Q V 1 e α1 ) 1 τ1> e α1τ1 ) 1 τ1 F cf1 ι) { 1 e ) E Q P 1 τ1> F E Q P e τ1 ) 1 τ1 } = V {1 e α1 ) Q V τ 1 > F ) E Q V e α1τ1 ) 1 τ1 F cf1 ι) 1 e ) Q P τ 1 > F ) = V 1 e α1 ) Q V τ 1 > F ) cf1 ι) ) V V B1 ) µ V V B1 «µ m 1 µp 1 σ 1 1 1 e ) Q P τ 1 > F ) ) µ V V B1 m 1 µp 1 } F = σ 1 Q m τ 1 F ) Q m τ 1 F ) m 1 µp 1 σ 1 Q m τ 1 F ) = Second em: E Q P e ) V F ) 1 τ1>,v F F = = V e α1 ) E Q V 1 τ1>,v F F F e ) E Q P 1 τ1>,v F F = = V e α1 ) Q V τ 1 >, V F F ) F e ) Q P τ 1 >, V F F hid em: 7
E Q P e τ1 ) 1 τ1< EV τ1, τ 1, ) F = e τ1 ) EV B1, τ 1, ) g P τ 1, V, V B1 ) dτ 1 Collecing ems and noing ha in he second em Q V τ 1 >, V F ) is equal o Q V τ 1 > ) Q V τ 1 >, V < F ), yields he expession 11) in he body of he ex. Equiy value a = τ 1, accued coupon ae case: EV τ1,, ) = E Q P α V s c F 1 ι)) e s τ1) 1 s<τ ds τ 1 F τ 1 e τ1) V F ) 1 τ>,v F F τ1 + Applying he same easoning as fo he wo fis ems in he pevious case bu consideing now he filaion a τ 1 yields he expession 1) in he body of he ex. A.3 ax benefis value a < τ 1 : BV,, ) = E Q P ιcf e s ) 1 s<τ1 ds F Fis em: = ιcf e τ1 ) 1 τ1< B V τ1, τ 1, ) F E Q P ιcf e s ) 1 s<τ1 ds F = E Q P τ1 ιcf e s ) ds F = { 1 e ) E Q P 1 τ1> F E Q P = ιcf Second em: 1 e ) Q P τ 1 > F ) ) µ V V B1 } e τ1 ) 1 τ1 F = m 1 µp 1 σ 1 Q m τ 1 F ) E Q P e τ 1 ) 1 τ1< B V τ1, τ 1, ) F = e τ1 ) B V B1, τ 1, ) g P τ 1, V, V B1 ) dτ 1 Collecing ems yields he expession 14) in he body of he ex. ax benefis value a = τ 1, accued coupon ae case: BV τ1, τ 1, ) = E Q P ιc F e s τ1) 1 s<τ ds τ 1 F τ 1 Applying he same easoning as fo he fis em in he pevious case bu consideing now he filaion a τ 1 yields he expession 15) in he body of he ex. 8
A.4 Bankupcy coss value a < τ 1 : BCV,, ) = E Q P 1 ρ ) V e ) 1 τ1>,v <F F + e τ1 ) 1 τ1< BC V τ1, τ 1, ) F Fis em: E Q P 1 ρ ) V e ) 1 τ1>,v <F F = 1 ρ ) V e α1 ) E Q V 1 τ1>,v <F F = = 1 ρ ) V e α1 ) Q V τ 1 >, V < F F ) Second em: E Q P e τ 1 ) 1 τ1< BC V τ1, τ 1, ) F = e τ1 ) BC V B1, τ 1, ) g P τ 1, V, V B1 ) dτ 1 Collecing ems yields he expession 17) in he body of he ex. Bankupcy cos value a = τ 1, accued coupon ae case: BCV τ1, τ 1, ) = E Q P 1 ρ ) V e τ1) 1 τ>,v <F F τ1 1 ρ 1 ) V τ e τ τ1) 1 τ< F τ1 Fis em: E Q P 1 ρ ) V e τ1) 1 τ>,v <F F τ1 = 1 ρ ) V τ1 e α τ1) E Q V 1 τ>,v <F F τ1 = = 1 ρ ) V B1 e α τ1) Q V τ >, V < F F τ1 ) Second em: E Q P 1 ρ1 ) V τ e τ τ1) 1 τ< Fτ1 = 1 ρ1 ) V B E Q P e τ τ 1) 1 τ< Fτ1 = ) µ m µp = 1 ρ 1 ) V VB1 σ B V B Q m τ < F τ1 ) Collecing ems yields he expession 18) in he body of he ex. 9