A valuation model of credit-rating linked coupon bond based on a structural model

Transcription

1 Compuaonal Fnance and s Applcaons II 247 A valuaon model of cred-rang lnked coupon bond based on a srucural model K. Yahag & K. Myazak The Unversy of Elecro-Communcaons, Japan Absrac A cred-lnked coupon bond pays a coupon assocaed wh s cred rang a he me of he coupon paymen dae, raher han an amoun equal o he nally fxed coupon. The only exsng corporae bond valuaon model for credrang-rggered producs was formulaed by Jarrow e al. However, hs model does no ncorporae he fac ha ncreases n he coupon paymen resulng from downgrades may cause a furher deeroraon of cred rangs and of he lkelhood ha he company wll be able o make fuure coupon paymens. In hs paper, we presen a cred-lnked coupon bond valuaon model ha consders hs ssue. Usng a srucural approach, we exend he classcal model of Meron by nroducng a hreshold value correspondng o each cred rang, and a volaly of he company value process ha depends on s cred rang. Gven hese exensons, our model s more flexble han he JLT model, and we are clearly able o capure he above effec va numercal smulaons. Furhermore, from he perspecve of praccal mplcaons, he JLT model ends o value cred-lnked coupon bonds more cheaply han does our model when he nal cred rang s hgh, whle he reverse s rue for a low nal cred rang. Keywords: rsk managemen, dervave prcng, cred rsk. 1 Inroducon The formulaon and use of corporae bond valuaon models daes from he work of Meron [5]. In he Meron model, he defaul of a bond s defned as a sae n whch he corporae value falls below he face amoun of he bond, and n whch he corporae value process follows a geomerc Brownan moon. As a resul of hese assumpons, he Meron model may easly be used n conjuncon wh he Black-Scholes formula o value corporae bonds. Usng valuaon frameworks of hs knd s ypcally characersed as followng a srucural approach, and WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne) do: /cf060241

2 248 Compuaonal Fnance and s Applcaons II many exensons of he Meron model have been derved. Anoher avenue for corporae bond valuaon s relavely new and s known as he reduced form approach. The laer approach assumes ha he me o defaul may be modelled as a hazard rae. Famous and represenave reduced form models nclude hose of Jarrow and Turnbull [4] (he JT model), Jarrow e al. [3] (he JLT model), and Duffe and Sngleon [2]. Among hese srucural and reduced form models, only he JLT model explcly uses a rang ranson marx n modellng he me o defaul. Gven such precedng research on he valuaon of he corporae bond, he JLT model a frs glance appears he mos suable for he valuaon of credrang-rggered bonds, such as he cred-rang-lnked coupon bond. However, n order o ncorporae he dea ha he ncreased coupon paymen due o downgradng deeroraes he poenal for fuure coupon and noonal paymens, he mpac of ncreased coupon paymens on he balance shee of he company mus be consdered, n addon o he cred-rang ranson self. In hs paper, for he purpose of valung cred-rang-lnked coupon bonds, we furher develop he deas presened by Bhano [1] by consderng an analogue of he JLT model n a srucural conex. The remander of he paper s organsed as follows. The nex secon brefly revews he Meron and JLT models, and presens he movaon for our research. Secon 3 proposes our valuaon model and s means of calbraon. Secon 4 examnes varous feaures of he model usng numercal examples. The fnal secon summarses and concludes. 2 Pror research and he movaon for our model 2.1 Meron model The Meron model assumes ha he value of he company follows a nex geomerc Brownan moon: dv = µ d +σdw, (1) V where µ, σ, and W are, respecvely, he drf and volaly of he corporae value process and a sandard Brownan moon under he usual sascal measure. In order o value a corporae bond, he Meron model frs ransforms process (1) no one under a rsk-neural probably measure, such as process (2) below: dv = rd + σdw ~, (2) V where r, σ, and W are, respecvely, he rsk-free shor rae, he volaly of he corporae value process, and a sandard Brownan moon under he usual rskneural measure. The model hen compues he rsk-neural expecaon of he payoff expressng he corporae bond value mn (V r, B), where B denoes he face amoun of he bond. Fnally, he model dscouns hs expecaon back o s WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)

3 Compuaonal Fnance and s Applcaons II 249 presen value. Therefore, he model makes convenen use of he Black-Scholes formula. 2.2 The JT and JLT models The JT model Under an approprae probably space and he assumpon ha he rsk-free neres rae process and he defaul me process are ndependen, he JT model provdes he value (F(,T)) of he T-maury dscoun corporae bond a me as gven by equaon (3): ~ * F(, T ) = p(, T )( δ + (1 δ ) Q ( τ > T ), (3) where δ s he recovery rae, p(,t) s he prce of he T-maury rsk-free dscoun bond a me, and ~ Q ( τ * > T ) s he probably under he rsk-neural probably measure ha he defaul happens afer he maury of he bond The JLT model The JLT model frs descrbes he cred rang of a company usng he sae space S = {1,,k}. The frs sae ndcaes he hghes cred rang (AAA), whle he second sae corresponds o he second-hghes cred rang (AA), and so on. The fnal sae k ndcaes defaul. The model nally adops marx (4) as he cred-rang ranson probably marx for a gven pon n me. In parcular, he emprcal cred-rang ranson probably marx s gven by q1,1 q1,2 q1, k q2,1 q2,2 q2, k. (4) Q = qk 1,1 qk 1,2 qk 1. k where q,j s he probably ha he cred rang of he company changes from k o j, and where, for all, j, 0 q = q, 1. Moreover, q and ( ) ( ), j,, 1 j, j + n. 0, n he n-perod ranson probably marx s hen compued as Q = Q Under he usual assumpons ha he marke s complee and ha he arbrage-free condon s sasfed, he JLT model hen nroduces he ranson probably marx from me o me + 1 under a rsk-neural measure: ~ Q [ ~, + 1 = q, j (, + 1) ]. (5) To rean s Markov characer, he JLT model resrcs he rsk-neural probably q~, j (, + 1) o q ~, j (, + 1) = π ( ) q (6), j for all, j, j, where π () s he rsk premum. The marx form of equaon (6) may be wren as ~ I = Π Q I, (7) Q, +1 ( )[ ] WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)

4 , j 250 Compuaonal Fnance and s Applcaons II ( ) where I s a k k un marx Π ( ) = dag π1( ),, πk 1( ),1, for all, j, π () > 0. Furhermore, q~, j ( 0, n) s defned as he probably ha he cred rang of he company jumps from cred rang o cred rang j over n perods, and hs probably s expressed as he(, j) -h enry on he lef sde of equaon (8). ~ ~ ~ ~ Q = Q Q. (8) 0, n 0,1 1,2 Q n 1, n Under he rsk-neural probably measure, he JLT model provdes he ~ probably Q ( T ) * τ > ha he a company wh he -h cred rang a me does no defaul unl he maury T of he bond as * Q τ > T = q, T = 1 q, T, (9) * where τ nf { s η k} ( ), ( ), ( ) j k j K = : s =. Usng equaon (10), he JLT model hen evaluaes he T-maury, -h cred F, T, smply by subsung * ~ ( T ) Q τ * > T n valuaon formula (3) of he JT model. ~ * F, T = p, T δ + (1 δ ) Q τ > T. (10) rang dscoun corporae bond a me, ( ) ~ τ n place of ( ) ( ) ( )( ( ) Q > 2.3 Characersc feaures of he Meron and JLT models, and he movaon for our model The Meron model Srengh: Snce negraes a defaul based on he srucure of he balance shee of he company, he model easly ncorporaes he fnancal mpac of cred-rang changes on he balance shee. Weaknesses: 1. The model does no explcly descrbe cred rangs and, herefore, s no suable for valung cred-rang-rggered producs. 2. Wh he excepons of he rsk-free neres rae r and he maury T of he bond, he model has only hree fundamenal parameers, namely he volaly of he corporae value process σ, he nal corporae value V 0, and he face amoun of he corporae bond B. Therefore, he model has oo few parameers o f he marke cred spreads of all maures flexbly. 3. In hs regard, he volaly σ of he company value process does no depend on s cred rang and s consan across all cred saes. 4. In he course of valung a coupon bond, he model mus deermne wheher he bond was n defaul a any coupon paymen dae, and hs procedure s very meconsumng. 5. The model canno ncorporae he erm srucure of rsk-free neres raes The JLT model Srenghs: 1. The model s based on cred rangs and s herefore suable for valung cred-rang-rggered producs. WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)

5 2. The model ncorporaes a cred rsk premum ( ) π ha depends boh on he me and he cred rang provded n he rsk-neural cred-rang ranson probably marx Q ~. Therefore, he model s flexble enough o f marke cred spreads for all maures. 3. In hs regard, no only he rsk premum π ( ), bu also he emprcal credrang ranson probably q,j n he marx Q, depend by defnon on he cred rang. 4. The model easly values coupon bonds. 5. The model s able o ncorporae he erm srucure of rsk-free neres raes. Weakness: Snce models a defaul usng a cred-rang ranson probably marx, he model does no ncorporae he srucure of he balance shee of he company. For hs reason, does no consder he fnancal mpac of he cred rang on he balance shee. In lgh of hese characerscs, we propose a valuaon model for he credrang-lnked coupon bond ha ncorporaes he mpac of ncreased coupon paymens on he poenal of he frm o pay fuure coupons and o make face value paymens. Our modellng approach s srucural, alhough we recognse ha srucural models are n several respecs weak n comparson o he JLT model. In shor, we aemp o ncorporae he benefs of he JLT model no an analogous srucural model. 3 Our model and s calbraon 3.1 Our model Before nroducng our model, we descrbe he correcon of several weaknesses of he Meron model: Weakness 1 As an analogue of he cred-rang sae space S = (1,...,k) n he JLT model, ( ) we nroduced k 1 hreshold values, V *, = 1,, k 1. The k 1-h hreshold *( 1) value V k ( k 1) s smply he coupon value c of he bond a he coupon paymen dae and he face amoun B + c ( k 1) of he bond a Maury. Weaknesses 2 and 3 Insead of he common volaly of he corporae value process σ, we *( ) nroduced he cred-rang-dependen volales σ, for = 1, k 1. In he case of = k, no volaly exss, because he company defauls n ha sae. The *() volaly σ essenally corresponds o he emprcal cred-rang ranson probably marx Q n he JLT model. We also nroduced a cred-rangdependen nal corporae value V 0, for = 1, k 1, o ncrease he flexbly of he model. Weakness 4 Snce we adoped a Mone Carlo smulaon mehod for he purpose of valuaon, he analyss requred very lle me. WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne) Compuaonal Fnance and s Applcaons II 251

6 252 Compuaonal Fnance and s Applcaons II Our model: Based on hese revsons, he rsk-neural company value process n our model may be descrbed as n equaons (11) and (12) below. A any me excep ha of he coupon paymen, *(1) *(1) dv = rv d + σ V dw, : V > V *( j) *( j 1) *( j) dv = rv d + σ V dw. : V > V > V (11) In addon, a he coupon paymen me l, ( j ) *( j 1) *( j) V = V c. : l l V > V > V (12) l where, V l s he jus-before- l value of he corporae bond wh nal cred rang, and where c ( j ) s he coupon of a bond wh he j-h cred rang a he dae of ssue. Valuaon procedure based on a Mone Carlo smulaon: Sep1 : Smulae he sample pah of he corporae value process gven by equaons (11) and (12), sarng wh he nal corporae value. Sep2 : Compue he cash flow (coupon + face amoun) for each sample pah. Sep3 : Inves he cash flow calculaed n Sep 2 n he rsk-free asse for he maury T of he corporae bond. Take he rsk-neural expecaon of he nvesed cash flow a me T, and dscoun backwards o s presen value. 3.2 Calbraon of our model Parameers n our model Exogenous parameers: The exogenous parameers nclude he cred-rang-dependen company value *() volales σ, for = 1, k 1, as well as he coupon and face amouns of he ( j ) bond, c and B. As menoned above, hese values correspond o he emprcal cred-rang ransonal probably marx Q n he JLT model. Parameers o be esmaed: The parameers o be esmaed ncluded he cred-rang-dependen nal corporae values V 0, for = 1, k 1, and k 2 hreshold values, such as sae () V * *( 1), for = 1, k 2, excep he defaul sae V k and he oal number of parameers was 2k 3. To faclae he calbraon of he model, we resrced he k 1 hreshold values V *( ) *, for = 1, k 2, by ( ) + 1 V = ( V0 + V0 ) 2, for *( 1) ( 1) = 1,, k 2, by V k = c k a he coupon paymen dae, and by *( k 1) ( k 1) V = B + c a maury. Therefore, he oal number of parameers o be esmaed was smply k 1. The k 1 nal company values V 0, for = 1, k 1, n our model correspond o he rsk premum π ( ) n he JLT model. We allowed he nal company values V 0, for = 1, k 1, o depend on he maury T of he WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)

7 corporae bond. Under hs allowance, he number of parameers ( ) π n he JLT model (dscree verson) maches ha of he parameers V 0 n our model Calbraon Three remarks regardng he model calbraon are n order. Frs, we allowed he nal company values V 0 o depend on he maury T of he corporae bond. Therefore, he esmaed values of V 0 could dffer by maury. Second, for each maury T, we red o esmae he k 1 nal company values by fng he k 1 model cred spreads o he marke cred spreads by numercally solvng k 1 equaons. Fnally, we assumed ha he coupon bonds observed n he marke were par bonds, and ha her coupons were he same as her yelds. 4 Numercal expermens Compuaonal Fnance and s Applcaons II 253 Specfcaon of he cred-rang-lnked coupon bond, and valuaon mehods n numercal expermens: Each cred-rang lnked coupon bond was assumed o behave as follows. If he bond bore he same cred rang ha had on ssuance, hen pad a each coupon dae he amoun of he correspondng coupon nally specfed. If he bond was n defaul a he coupon paymen dae, he corporae value a ha me was pad a he maury T of he bond. In several numercal expermens, we compared he varous bond values derved from he hree dfferen valuaon models: (1) he JLT model, n whch, a he coupon paymen dae, he coupon correspondng o he cred rang was pad, as menoned above; (2) (our model); and (3), whch was essenally he same as our model, excep ha he fall n company value resulng from coupon paymens remaned a he nal coupon amoun, alhough he company pad he coupon correspondng s cred rang a he coupon paymen dae. In oher words, we adoped a model ha was economcally ncorrec as a reference pon from whch o evaluae he oher models. Daa and he seng of exernal parameers: We adoped sx possble cred rangs: AAA, AA, A, BBB, BB, and D. Therefore, k = 6. The bond maury was fve years, and he erm srucure of he rsk-free neres rae was fla. The face amoun of each bond was yen, and he coupon of he bond wh each cred rang was he same as s yeld. Table 1: The cred spreads. Table 2: The volales. Rang Seep 5% 10% 20% 25% 35% Fla 20% 20% 20% 20% 20% Rang Seep 0.18% 0.44% 0.92% 1.85% 4.% Fla 0.16% 0.26% 0.46% 1.12% 2.05% We adoped he average emprcal cred-rang ranson probably marx Q n he JLT model ha was announced by R&I (a Japanese rang agency) beween 1994 and In hs dervaon, we lumped ogeher all of he ranson probables for cred rangs below BB, wh he excepon of he defaul sae; hese were gven he correspondng cred-rang label BB. WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)

8 254 Compuaonal Fnance and s Applcaons II Moreover, n esmang he rsk premum Π ( ) echnque adoped by JLT (1997)., we used he esmaon Table 3: The cases of numercal expermens. The Cases of Cred Spread(Fla): Volales(Fla) Volales(Seep) Rsk-free neres rae(1.21%) Case1 Case2 Rsk-free neres rae(3.21%) Case3 Case4 The Cases of Cred Spread(Seep): Volales(Fla) Volales(Seep) Rsk-free neres rae(1.21%) Case5 Case6 Rsk-free neres rae(3.21%) Case7 Case8 For boh he volaly of he company value process and he cred spread of he bond correspondng o each cred rang, we allowed wo dfferen sengs, and hese are lsed n Tables 1 and 2, respecvely. In addon, we se he rskfree neres rae alernavely a 1.21% and 3.21%. Therefore, n oal, we performed egh numercal expermens (Cases 1 hrough 8), he resuls of whch are summarsed n Table 3. The resuls of he numercal expermens, and her mplcaons: The egh valuaons, correspondng o Cases 1 hrough 8, of he cred-ranglnked coupon bond for each of he hree valuaon models are provded n Fgures 1 hrough 8, respecvely. Sragh Bond Rang Sragh Bond Rang Fgure 1: The resuls of Case 1. Fgure 2: The resuls of Case 2. Sragh Bond Rang Sragh Bond Rang Fgure 3: The resuls of Case 3. Fgure 4: The resuls of Case 4. WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)

9 Compuaonal Fnance and s Applcaons II 255 Sragh Bond Rang Sragh Bond Rang Fgure 5: The resuls of Case 5. Fgure 6: The resuls of Case 6. Sragh Bond Rang Sragh Bond Rang Fgure 7: The resuls of Case 7. Fgure 8: The resuls of Case 8. (1) Overvew of he resuls (a) All hree models valued he cred-rang-lnked coupon bond above he sragh bond when he cred rang of he bond was relavely hgh (AAA, AA, A), whle he oppose was rue when he cred rang of he bond was relavely low (BBB, BB). (b) The value of he cred-rang-lnked coupon bond derved from he JLT model ended o be lower han hose derved from and under a relavely hgh nal cred rang (AAA, AA, A); he reverse was rue under a relavely low nal cred rang. The frs resul was obaned because, under a hgher nal cred rang, he effec of he coupon ncrease resulng from a downgrade swamped he resulng decrease n he poenal of he company o make fuure coupon paymens. Under a low nal cred rang, he suaon was reversed. The second resul was obaned because he coupon paymen amoun dd no affec he cred-rang ranson probably n he JLT model, whle he ncreasng coupon amoun ncreased he defaul probably, and he magnude of hs effec was larger under a low cred rang han under a hgh cred rang. (2) The nfluence of he cred spread (comparson of Case 1 & Case 4 and Case 5 & Case 8). The frs resul (1) appeared more salen for a large, seep cred-spread curve han for one ha was small and fla. The reason underlyng he frs resul n (1) also explans hs observaon. WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)

10 256 Compuaonal Fnance and s Applcaons II (3) The nfluence of he volaly of he company value process (comparson of Cases 5 and 6) (a) In boh Models A and B, and for all cred rangs, he value of he credrang-lnked coupon bond ended o be hgher under a fla volaly srucure (20% n all cases) han under a seep volaly srucure (5, 10, 20, 25, and 35%, respecvely, from he hghes cred rang o he lowes). (b) The valuaon derved usng devaed from ha of o a greaer exen under he fla volaly srucure han under he seep one. The frs resul may be explaned as semmng from reason (1) above. The devaon of he value derved from from ha derved from resuled from boh he cred-rang probably and he dfference beween he nally se consan coupon and he cred-rang-lnked coupon. For Cases 5 and 6, he laer mpac was he same, bu he former was larger under fla volaly han under seep volaly. (4) The nfluence of he rsk-free neres rae For all of he nal cred rangs, he value of he cred-rang-lnked coupon bond was hgher when he rsk-free neres rae was low. The dfference beween he nally se consan coupon and he cred-rang-lnked coupon derved no from he rsk-free neres rae self, bu raher from he cred spread. The rskfree neres rae only affeced he value of he cred-rang-lnked coupon bond hrough s mpac on he dscoun rae of s cash flow. 5 Summary and concludng remarks In hs paper, we presened a srucural valuaon model for cred-rang-lnked coupon bonds ha ncorporaes he fac ha an ncreased coupon paymen resulng from a downgrade may deerorae he poenal of he ssung company o make fuure coupon and noonal paymens. Through numercal expermens, we demonsraed ha our model reasonably capures hs effec. A praccal mplcaon of our model s ha he valuaon of a cred-rang-lnked coupon bond based on he JLT model ends o underesmae he value of he bond when s nal cred rang s hgh. However, he reverse s rue when he nal cred rang s low. References [1] Bhano K., Prcng Corporae Bonds wh Rang-Based Covenans. The Journal of Fxed Income, March, pp. 57-, [2] Duffe, D. & Sngleon, K., Modelng Term Srucures of Defaulable Bonds. Revew of Fnancal Sudes, 12, pp. 7-0, [3] Jarrow, R.A. Davd L. & Turnbull, S.M., A Markov Chan Model for he Term Srucure of Cred Rsk Spreads. Revew of Fnancal Sudes, 10(2), pp , [4] Jarrow, R. & Turnbull S.M., Prcng Dervaves on Fnancal Secures Subjec o Cred Rsk. Journal of Fnance, 50, pp , [5] Meron, R.C., On he Prcng of Corporae Deb: The Rsk Srucure of Ineres Raes. Journal of Fnance, 29, pp , 19. WIT Transacons on Modellng and Smulaon, Vol 43, ISSN X (on-lne)