New recipes for estimating default intensities

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1 SFB 649 Dcuon Pape 9-4 New ecpe o emang deaul nene Alexande Baanov* Caen von Lee* Andé Wlch* * WeLB AG, Düeldo, Gemany SFB E C O N O M I C R I S K B E R L I N h eeach wa uppoed by he Deuche Fochunggemencha hough he SFB 649 "Economc R". hp://b649.ww.hu-beln.de ISSN SFB 649, Humbold-Unveä zu Beln Spandaue Saße, D-78 Beln

2 New ecpe o emang deaul nene Alexande Baanov, Caen von Lee and Andé Wlch all WeLB AG Abac: h pape peen a new appoach o devng deaul nene om CDS o bond pead ha yeld mooh neny cuve equed e.g. o pcng o managemen pupoe. Aumng connuou pemum o coupon paymen, he deaul neny can be obaned by olvng an negal equaon Volea equaon o nd nd. h negal equaon hown o be equvalen o an odnay lnea deenal equaon o nd ode wh me dependen coecen, whch numecally much eae o handle. Fo he pecal cae o Nelon Segel CDS em ucue model, he poblem pem a ully analycal oluon. A vey good and a he ame me mple appoxmaon o h analycal oluon deved, whch eve a a ecpe o eay mplemenaon. Fnally, hown how he new appoach can be employed o emae ochac em ucue model le he CIR model. Keywod: CDS pead, bond pead, deaul neny, ced devave pcng, pead modellng, ced modellng, loan boo valuaon, CIR model JEL clacaon: C3, C and C Dclame: he dea peened below elec he peonal vew o he auho and ae no necealy dencal o he ocal mehodology ued a WeLB AG Acnowledgemen: he paal nancal uppo om he Deuche Fochunggemencha va SFB 649 Öonomche Ro gaeully acnowledged. Inoducon CDS and bond pead cuve a well a he mpled deaul nene deved om hee pead cuve ae ey npu o many applcaon, o nance ced devave pcng o pead and ced model o managemen pupoe. ypcally, he oupu o uch pcng o model que enve o he way uch pead and neny cuve ae emaed om obevable mae quoe. I common pacce o mae cean mplyng aumpon n he emaon pocedue; a pomnen example he aumpon o pecewe conan deaul nene ha ae deved by booappng he CDS quoe obeved o deen maue. Howeve, uch pocedue ae oen no able wh epec o oule e.g. due o daa qualy ue, and geneally poduce cuve conanng dconnue and ump. h pape popoe a able emaon pocedue ha avod he hocomng oulned above and a he ame me numecally eay o mplemen. he mehod ele on a andad model cla commonly ued o he obeved em-ucue o quoe o CDS o bond pead, namely Nelon Segel ype model. he deaul neny epeened a he oluon o an negal equaon a Volea equaon o nd nd ha deved om he andad pcng appoach o CDS o deaulable deb numen by mang he aumpon o connuou pemum o coupon paymen. h negal equaon may be anomed o an odnay lnea deenal equaon o nd ode wh me dependen coecen, he numecal eamen o whch aghowad. he mechanc and he peomance o he new ng pocedue demonaed ung he example o Nelon Segel ype uncon ed o CDS pead obeved on an abaly choen ade day Nelon Segel uncon have he advanage o pemng a cloed om analycal Alexande_Baanov@WeLB.de, Caen_von_Lee@WeLB.de, Ande_Wlch@WeLB.de h model cla may be genealzed o a much boade cla wh ucen degee o eedom o accommodae almo any em-ucue hape eul wll be peened n a epaae pape.

3 oluon o he nd ode deenal equaon menoned above. Apa om h eaue, hee no necey o c o he Nelon Segel uncon o he ng pocedue o wo. In ac, we have developed a vey geneal cla o exponenal-polynomal uncon wh ucen degee o eedom o accommodae almo any em-ucue hape and conanng he Nelon Segel- o Svenonmodel a pecal cae. Deal wll be peened n a epaae acle. Fnally, he emaon o ochac deaul neny model baed on he new pocedue o deaul nene demonaed o he CIR model. Deaul neny a oluon o a Volea equaon o nd nd In he equel, he cae o CDS pead cuve condeed n ode o deve mooh deaul nene; howeve, he dea may be aneed n a aghowad way alo o bond pead. Followng he andad appoach o CDS pcng [Hull 3], he expeced value o deaul leg and pemum leg hould be equal. he expecaon o he peen value o he pemum paymen gven by E n PVP emum D E Iτ > whee denoe he quoed CDS pead o mauy, < < < n ae he pemum paymen dae, -, uually.5 yea, and τ he ochac mng o deaul. In he conex o a deaul neny model baed on a deemnc deaul neny and ho ae, h expeon ge he ollowng om: n n P E PV [ ] emum exp u u du he uncon dened a exp u u du [ ] he uual denon o dcoun aco and uvval pobably have been ued: D exp u du E I > τ > P τ exp u du he aumpon o a deemnc deaul neny and ho ae wll be elaxed below, whee a CIR model aumed o. Aumng connuou pemum paymen, equaon may be appoxmaed by E PVP d emum he peen value o he deaul leg gven by:

4 PV Deaul D τ Iτ Aumng a xed non-ochac lo gven deaul, he expecaon o he peen value o deaul paymen : E PV E D τ I Deaul whch may be ewen a τ E PV d Deaul D exp u du Summng up he pevou omulae, he pcng equaon E PV E PV become 3 d d h an negal equaon o he uncon dened n equaon. d P emúm Deaul Fo he me beng uppoe ha he CDS-pead cuve a well a he ee ho ae ae gven. Deenang he denng equaon o yeld he ollowng expeon: ' 4 Ineng equaon 4 no equaon 3, one oban he cenal eul: 5 d h a Volea equaon o nd nd o he uncon. Equvalen epeenaon a nd ode deenal equaon In pncple, he Volea equaon 5 may be olved numecally n ode o oban he deaul neny. Howeve, numecal pocedue o olvng negal equaon decly ae complex and omeme even unable. heeoe anohe appoach o olvng equaon 3 popoed ha ele on olvng an odnay lnea deenal equaon and heeoe numecally much eae o handle. By deenang equaon 5 once, one ge he ollowng expeon: 6 d Reaangng em lead o

5 7 d Fuhe deenaon o 6 yeld 8 d By neng equaon 7 n equaon 8, and eaangng em, one nally oban he ucue o a deenal equaon o : 9 h g wh g and h '. Equaon 9 a lnea deenal equaon o nd ode wh me dependen coecen and can ealy be olved numecally, ung he ollowng nal condon: ' Compaon o common pace: he booappng appoach eved Aumng he deaul neny o be conan ove he me neval [, ], he negal equaon 3 mple o [ ], Now uppoe ha obevaon o he pead ae gven o deen maue < < m e.g. m 5 and Y, 3Y, 3 5Y, 4 7Y, 5 Y and aume a pecewe conan ucue o o he om ] m I, wh. By neng h epeenaon no equaon 3, one oban he ollowng oluon: J J exp exp whee d D J exp,,,, m. h e o equaon may be olved ecuvely w...,,, m : he equaon o lead o he oluon / whch hen need no he econd equaon o. h n un poduce a nonlnea equaon o whch can be olved e.g. by he Newon mehod. Connung n a mla manne, one oban he deaul nene 3,, m o all he emanng mauy buce.

6 he pecal cae o Nelon Segel: Exac analycal oluon and numecal ecpe In ode o demonae how he emaon pocedue popoed n h pape wo n pacce, and how peom n compaon o he andad booappng appoach, appled ung a paamec cla o cuve popoed by C. Nelon and A. Segel [Nelon 987]. Howeve, he emaon pocedue doe no ely on any pecc model cla and wo wh ohe ype o cuve a well ee oonoe. In a ep, Nelon Segel cuve ae ed o he obeved CDS quoe on an abaly choen ade day 5.3.8, ouce: MaI. Alo, a Nelon Segel model o he ho ae ed o he quoed he EURIBOR em ucue. he eul ae ploed n gue. Eubo and aocaed ho ae CDS pead ae EURIBOR ho ae CDS pead [bp] AA A BBB BB B eno [monh] eno [monh] Fgue : Sho ae and CDS pead cuve om ng a Nelon-Segel model o quoe a o Subung he ed cuve and exp-.67 o he CDS pead and ho ae a well a a xed lo gven deaul.6 and he nal condon no he econd ode deenal equaon 9, a numecal negaon o he ODE lead o a oluon o. By ue o 4 one oban he deaul neny om and ubequenly he pobably o deaul PD accodng o he elaon PD exp u du he eul o and PD ae hown n gue. Deaul neny CDS mpled deaul ae on AA A BBB BB B Pobably o deaul PD em ucue AA A BBB BB B 9.% 6.% 8.% 5.% 7.% 6.% 4.% a e [% ] 5.% 4.% P D [% ] 3.% 3.%.%.%.%.%.% me [monh].% me [monh]

7 Fgue : Deaul neny and cumulave pobably o deaul deved om Nelon-Segel ng a o Compaon o booappng appoach Fo he ae o compaon, alo he andad booappng mehod appled o he daa e decbed above. Applyng ecuon omula o he aveaged quoe a o eeng o Euopean A-aed copoae, one oban he pece-we conan deaul neny epeened n gue 3. he coepondng mooh deaul neny cuve mpled by he popoed ng pocedue alo ploed. One clealy ee ha epecally o maue below 5 yea, he pecewe conan uncon dplay an economcally unnuve behavou lage ump ze and poenally vulneable o anomale n he daa zg-zag behavou. hee dadvanage ae avoded n he Nelon Segel ng appoach. deaul ae yea Fgue 3: Deaul neny o Euopean A-aed copoae a o a o Cloed om oluon o Nelon-Segel ng poblem A uhe advanage o he Nelon-Segel ng pocedue woh menonng: he deenal equaon 9 even pem a cloed om analycal oluon he po ae aumed o be conan. Fo example, he deaul pobably mpled om he Euopean A-aed CDS conac wh Nelon-Segel ng uncon.3.94 exp-γ on he adng day calculae a γ.5ν ν PD e ν e F,.85,.5ν ν whee γ.369, ν exp-γ,.434 and F a,b,z he Kumme conluen hype geomec uncon. he paamee γ goven he exponenal decay ae n a Nelon-Segel o he CDS pead. Clealy, he numecal oluon numecal o equaon 9 baed on he ed connuou po ae cuve de om he analycal oluon analycal baed on a conan aveage po ae. Bu un ou ha he PD-emae deved om numecal and analycal ae numecally dencal. Smple appoxmaon omula o deaul pobable When analyng he behavou o he ng pocedue, un ou ha ome o he paamee ae amazngly able ove me. h obevaon a he ba o an appoxmaon omula o he exac PD em ucue ha decly ee o he CDS pead cuve. he ocal pon ha he ao o CDS pead cuve and PD em ucue µ R,x CDS R,x / PD R,x,x / PD R,x able ove me o a lage ange o ang clae he ubcp R denoe ang clae, whle he agumen and x ee o em ucue and adng dae. Fo he ang clae om AA o BBB, he elaon µ R,x µ hold o he analyed peod om 7..8 unl.4.8. Numecally, one nd he ollowng elaon:

8 PD.39., x e.835e CDS, x, R x, R AA A BBB,.5 [ yea] he analyzed peod conan que an eac behavou o mpled cumulave deaul pobable, a hown n gue 4 o A-aed copoae. R deaul pobabl Fgue 4: Impled em-ucue o cumulave deaul pobable o A-aed copoae wh, 3, 5, 7, and yea. he undelyng obevaon peod cove quoe om 7..8 unl.4.8. h gve condence n he valdy o he appoxmaon omula alo o uue peod. Fgue 5 demonae he peomance o he appoxmaon by compang µ o he nonlnea µ R,x. day m-ao mauy@monhd Fgue 5: Compaon o µ doed cuve o he nonlnea µ R,x connuou cuve. Fng o ochac model o deaul neny: CIR Model he above appoach alo povde a good ang pon o ng ochac model o deaul nene and ho ae. In he equel, he cae o a ochac deaul neny obeyng he CIR Cox Ingeoll Ro model condeed a an example whch could be ealy exended o a moe geneal eng. In he amewo o he CIR model [Cox 985], he ochac deaul neny gven by θ d σ dw d α wh a Wene poce W. he equaon o genealze a ollow: 3 D E exp u du D exp A C CIR

9 wh an ane em ucue exp-a C dened hough by he coecen ee e.g. [McNel 5] o a devaon A C β β α β e αθ ln σ β α / β e β β α e β β α σ h uncon CIR olve he deenal equaon 9. he oluon can be obaned by he numecal mehod decbed above. Gven h oluon o CIR and he dcoun ae D, equaon 3 allow o emae he hee paamee α, σ and θ o he CIR poce o. Sandad nonlnea egeon echnque may be employed o olve he mnmzaon poblem: [ CIR D exp A C ] mn { α, θ, σ } he em n he bace epeen he uvval pobably mpled by he ng pocedue decbed above whle he econd em epeen he uvval pobably mpled by he CIR model dependng on α, σ and θ. Fgue 6 demonae he eul o uch a non-lnea egeon baed on he mae daa obeved on he ollowng value wee obaned o he CIR paamee: α.74, σ and θ uvval pobably yea Fgue 6: Suvval pobable mpled by he Nelon Segel ng pocedue blue cuve and CIR model wh ed paamee α, σ and θ ed cuve o Ouloo In h acle a new pocedue o emang deaul nene baed on obeved CDS pead o bond pead ha been peened. he pocedue ha wo man advanage:. he deaul neny naually become a connuou uncon o and no economcally unnuve dconnue ae.. he pocedue able w... oule and noy daa e.g. due o eoneou CDS-quoe becaue ele on a pecedng moohng pocedue. he new emaon pocedue alo eve a a able ba o ng ochac deaul neny model le.

10 Fuue eeach wll be conduced o analye he eec o moe geneal model clae n he conex o he new emaon pocedue o deaul nene. We expec uch genealzed model clae o povde neeng economc ngh no he me evoluon o CDS o bond pead and he aocaed deaul nene. Accodngly, he acal popee o uch model wll alo be ubec o uhe nvegaon. Reeence [Hull 3]: Hull, J.C.,and Whe, A., 3. he valuaon o Ced Deaul Swap Opon, Jounal o Devave, 3, 4-5 [Cox 985]: Cox, J.C., Ingeoll, J.E., and Ro, S.A., 985. A heoy o he em Sucue o Inee Rae, Economeca, 53, [Nelon 987]: C. Nelon and A. Segel, 987. Pamonou modellng o yeld cuve, J. o Bune 6 987, [McNel 5]: A. McNel, R. Fey and P. Embech. Quanave R Managemen, Pnceon Unvey Pe 5

11 SFB 649 Dcuon Pape See 9 Fo a complee l o Dcuon Pape publhed by he SFB 649, pleae v hp://b649.ww.hu-beln.de. "Impled Mae Pce o Weahe R" by Wolgang Hädle and Benda López Cabea, Januay 9. "On he Syemc Naue o Weahe R" by Guenhe Flle, Man Odenng, Oap Ohn and We Xu, Januay 9. 3 "Localzed Realzed Volaly Modellng" by Yng Chen, Wolgang Kal Hädle and Ua Pgoch, Januay 9. 4 "New ecpe o emang deaul nene" by Alexande Baanov, Caen von Lee and Andé Wlch, Januay 9. SFB 649, Spandaue Saße, D-78 Beln hp://b649.ww.hu-beln.de h eeach wa uppoed by he Deuche Fochunggemencha hough he SFB 649 "Economc R".