RELIABILITY AND CREDIT RISK MODELS

Transcription

1 Chaper 8 RELIABILITY AND CREDIT RIK MODEL I hs chaper, he reader wll frs fd a shor summary of he basc oos of relably ad he he sem-markov exesos. Afer ha, he classcal problem of cred rsk s also preseed ogeher wh a aalogy wh relably ad s show how sem-markov models are useful for hs mpora opc of face coeco wh he ew rules of he Basel Commee. CLAICAL RELIABILITY THEORY Relably heory s maly cocered wh he secury of maeral fgs. A frs dsco mus be made bewee smple ad complex srucures. For a smple srucure, s possble o defe wha s called he lfeme of he cosdered sysem, defed as he r.v. T represeg he me erval bewee me 0 ad he me of he frs falure, falure meag ha he sysem s ou. A complex sysem s composed of several smple compoes, from whch falures have a mpac, more or less mpora, o he way he sysem s workg.. Basc Coceps Le us cosder a smple srucure called he relably sysem havg r.v. T as ΩI,,P. lfeme, T beg defed o he probably space ( ) Defo. The relably fuco of s gve by he fuco U defed as U () = PT ( > ), 0,. (.) [ ) U() represes he probably ha o falure happes before. If F represes he dsrbuo fuco of T, s clear ha for all o-egave : U()= F(). (.) If he desy fuco f of T exss, we oba: U () = fudu ( ). (.3) From ow o, we always assume ha he f or he dervave of U exss. Defo. The fuco r, defed as

2 336 Chaper 8 f() U'() r () = =, > 0, F ( ) U ( ) s called he falure rae of he compoe. (.4) Is meag s smple: le us cosder a me such ha he eve {T>} occurs. From basc defos of codoal probably (relao (6.4) of Chaper ) ad from relao (.), we ca successvely wre: P ( < T + d) PT ( + dt> ) = PT ( > ) f() d = U () (.5) U'( ) = d U () = rd (). Cosequely, r()d smply represes he codoal probably of havg a falure he fesmal me erval (,+d) gve ha he compoe has o falure before or a me. o, he value of he falure rae a me s a rsk measure o have suddely a falure jus afer me. By egrao, relao (.4) gves: r( u) du 0 U () = e (.6) provded ha we suppose ha U(0)=. From he las relao, s clear ha ay o-egave fuco ca be a falure rae f he followg wo codos are sasfed: - he fuco r s egrable o he posve half real le, - rudu ( ) =. (.7) 0 The mea lfeme T s jus he mea for he d.f. F. By egrao by pars, s possble o show ha T = U() d (.8) 0 ad smlarly f he varace σ exss: σ = U ( ) d (.9) 0. Classfcao Of Falure Raes The frs classfcao of falure rae ypes was gve by Barlow ad Proscha (965) wh he followg defo.

3 Relably ad cred rsk models 337 Defo.3 There are hree ypes of falure raes: () creasg falure rae ype ( shor IFR) ff, : < r( ) < r( ), (.0) () decreasg falure rae ype ( shor DFR) ff, : < r( ) > r( ), (.) () cosa falure rae ype ( shor CFR) ff, : < r( ) = r( ). (.) I he las case, le us wre r ( ) = λ ; he relao (.6) gves: λ U () = e, 0 (.3) so ha he d.f. of T s he egave expoeal dsrbuo of parameer λ (see Chaper, seco 5.5). Laer Barlow ad Procha (965) refe hs classfcao wh he followg defo. Defo.4 () A falure rae s of creasg falure rae average ( shor IFRA) ype (respecvely of decreasg falure rae average ( shor DFRA) ype) ff he fuco x x r() d, x [ 0, ) x s creasg (respecvely decreasg). 0 () A falure rae s of ew beer ha used ( shor NBU) ype (respecvely of old beer ha used ( shor OBU) ype) ff U( x+ y) ( ) U( x) U( y), x, y > 0. (.4) The meag of hese las wo ypes s smple; for he OBU ype, for example, we ca wre equaly (.3) he form: U( x+ y) U( x) (.5) U( y) or PT ( > x+ y) PT ( > x) (.6) PT ( > y) ad fally PT ( > x) PT ( > x+ yt> y), (.7) hs las relao meag ha, gve he eve {T>y}, he codoal probably of he eve {T>x+y} s always smaller ha he ucodoal probably of he same eve for y=0. I oher erms, he fac of workg up o me x always mples a wear pheomeo called agg. Le us meo ha s possble o show (Barlow ad Proscha (965)) ha he followg clusos are rue: IFR IFRA NBU, DFR DFRA OBN.

4 338 Chaper 8 Moreover, hese clusos are src. The geeral shape of a falure rae s he bahub wh hree perods: he begg, s of ype DFR, he here s a me erval whch s of expoeal ype ad fally a hrd ad laer perod, of IFR ype: Fgure.: Bahub shape of a falure rae.3 Ma Dsrbuos Used I Relably Referrg o seco 5 of Chaper, we wll gve he prcpal dsrbuos used relably heory, ogeher wh he value of he falure rae ad s ype, f ay. () Posso dsrbuo of parameer λ ; () Gamma dsrbuo r λ e γ( λ, r), r( ) = ; (.8) r λu u e du () Webull dsrbuo of parameers λ, β, r () λβ ( λ) β = ( β > : IFR, β < : DFR, β = : EXP ); (.9) (v) log-ormal dsrbuo of parameers μ, σ, (l μ) / σ () e r = ; (.0) (l u μ) / σ e du u (v) he rucaed ormal law of parameer ( μ, σ ) for whch he desy s defed as

5 Relably ad cred rsk models 339 ( μ ) σ μ f() = e = Φ', kσ π kσ σ ( μ ) 0 μ σ k = e d =Φ, σ π σ Φ beg defed Chaper as he dsrbuo fuco of a N(0,) r.v. The falure rae has he value ( μ ) μ ' σ e Φ σ r () = =. ( μ ) μ σ e du σφ σ (.) (.).4 Basc Idcaors Of Relably Geerally speakg, le us suppose ow ha he cosdered compoe s reparable, ha s o say ha f here s a falure a me, we ca repar he compoe usg a radom me Y ad ha afer reparao, he compoe wll sar aga wh he same falure rae as before he reparao. Ths s he cocep of mmal reparao. The dsrbuo fuco G of he repar me Y s called he maaably fuco ad he equvale of he falure rae he repar rae fuco oed s, so ha: G'( ) s () =, G ( ) (.3) s( u) du 0 G () = e. The effec of cosderg possble reparaos mples ha we ca ow roduce a wo-sae sysem wh as sae space { 0, }, hese wo saes represeg respecvely workg ad repar saes. Now he relably sysem ca have raso from oe sae o he oher oe: a me 0, he sysem s he operag sae or sae or up sae; a he me of he frs faluret, goes o sae 0 or dow sae for a me Y ad so o. The me evoluo of he sysem s hus heorecally gve by he sequece { T, Y,..., T, Y,... } ad from ow o, le Z() represe he sae of he cosdered sysem a me. Defo.5 The basc dcaors of relably are: () he mea me o falure (MTTF) :MTTF=E(T), () he mea me o repar (MTTR):MTTR=E(Y),

6 340 Chaper 8 () he po-wse (or saaeous) avalably: PZ ( ( ) = ), (v) he seady-sae avalably: A= lm P( Z( ) = ), (v) he average avalably o he erval [,+u]: + u A (, + u) = Audu ( ), u (.4) (v) he lm average avalably o he erval [,+u]: A (, + u) = lm Audu ( ). 0 (.5).5 Complex ad Cohere rucures I geeral, a srucure s composed of several smple compoes ad he eve falure of hs complex srucure depeds o he way hese compoes are workg. Theorecally, hs s gve wh he so-called srucure fuco. Le us suppose ha all he compoes of he complex srucure C are smple, so ha, for each =,,,, (operag or up sae), x = x() = (.6) 0, (faled or dow sae). The sae x of he srucure C a me, s gve by, ( up sae), x ( ) = ψ ( x ( ),, x ( )) = (.7) 0, (dow sae), where he fuco ψ :{ 0,} { 0,} (.8) s called he srucure fuco of C. Le us ow gve he followg defos. Defo.6 The complex compoe has a moooe srucure ff x y, ψ ( x,, x ) ψ ( y,, y ), =,,, ψ(0,,0) = 0, ψ(,,) =. (.9) Defo.7 A compoe (=,...,) of a complex sysem s rreleva ff he srucure fuco s cosa x ; oherwse, hs compoe s called releva. Defo.8 The complex compoe has a cohere srucure ff s moooe ad each compoe s releva. Parcular cases () A seres srucure fucos ff each compoe does. () A parallel srucure fucos f a leas oe compoe does.

7 Relably ad cred rsk models 34 () A k-ou-of srucure fucos ff a leas k of he compoes fuco. For hese hree ypes of srucure, he srucure fucos are successvely gve by: () ψ ( x,..., x) = x( = m x), =,..., = () ψ ( x x) = x( = x) =,..., () ψ ( x,..., x ) (.30),..., max, (.3) =, x k, = = x < = 0, k. (.3) Remarks. () Barlow ad Proscha (965) have proved ha for ay cohere srucure, we have he uve resul x ψ ( x,..., x) x. (.33) = = () If all he compoes are depede ad f U ( =,..., ) s he relably fuco of compoe, he, we have for a seres sysem: = U () = U (), (.34) ψ for a parallel sysem: Uψ () = ( U ()), (.35) = ad for a k-ou-of sysem: Uψ ( ) = ( U ( )) ( U ( )) ( for U ( ) = U ( ), =,..., ). (.36) = k Example. Le us cosder a complex sysem for whch compoe (=,,) has a egave expoeal dsrbuo of parameer λ ( =,..., ). If he srucure s a seres srucure, follows from resul (.34) ha: λ λ = Uψ () = e e = (.37) =

8 34 Chaper 8 ad so, hs srucure has also a egave expoeal dsrbuo whose parameer λ = λ. Moreover as we have: = T =, =,...,, (.38) λ, T T = (.39) = a resul showg ha MTTF for he complex srucure s gve by he harmoc mea of he MTTF of all he compoes. If all he compoes have he same relably fuco, hs las resul gves: T T =. (.40) Example. Le us cosder he reduda srucure formed by a complex srucure composed of decal compoes parallel, all havg a egave expoeal relably fuco of parameer λ. From resul (.35), we ge: U () ( e λ ), ψ = (.4) ad so from relao (.4): λ λ ( e ) rψ () = λe, (.4) λ ( e ) provg ha here he falure rae s me depede. I s o dffcul o show ha: > rψ ( ) < λ, 0 rψ ( ) λλ ( ), (.43) λ e rψ () λ λ. λ e These las resuls show he effec of redudacy, whch adds o a smple egave expoeal compoe, supplemeary compoes parallel o mprove he relably. Ths effec s mpora a he begg ad of course s me decreasg wh me o coverge o λ. For he MTTF, we have ha: Tψ = T, (.44) k k =

9 Relably ad cred rsk models 343 ad so he raos of he MTTF are gve for example by : N 3 4 MTTF rao Table.: example of MTTF Ths clearly shows ha he effec of redudacy s o proporoal o he umber of added compoes.. TOCHATIC MODELLING IN RELIABILITY THEORY. Maeace ysems I he las subseco, resul (.34) shows ha for a seres srucure, he relably fuco s hghly decreasg wh he umber of compoes. However, f he compoes are reparable, s possble o errup he sysem momearly durg he reparao of he faled compoe ad he o reser he compoe he sysem ad so o. For such a possbly, oe ca cosruc a sochasc model (Moha e al (96)) o compue he ma dcaors gve seco.4. Le us assume ha all he compoes are depede wh egave expoeal dsrbuos, respecvely wh parameers λ,..., λ, ad ha he repar me for compoe (=,,) has a egave expoeal dsrbuo of parameer μ. All he repar mes are also depede ad of oher ad of o he workg mes of he compoes. Moreover here s o me loss o replace he repared compoes he sysem. The evoluo of he sysem ca be see as a successve sequece of workg ad repar mes. For example for =, he radom sequece ( X, Y, X, Y,..., X, Y,... ) (.) represes successvely he workg ad repar mes ad f we roduce a wosae se {0,}, so ha, a me, he sysem sae Z() s sae f s operag ad sae 0 f s uder repar, he he process (.) s a couous Markov process where he raso marx of he mbedded Markov cha s gve by: 0 0 (.) ad for whch he codoal sojour mes are gve by:

10 344 Chaper 8 0, < 0, 0: F0( ) = λ e, 0, (.3) 0, < 0, 0 : F0( ) = μ e, 0. Wh he oao of Chaper 3, we have here: b = 0 η =, b0 η0, λ = = μ (.4) ad for he saoary dsrbuo of he mbedded Markov cha: π0 = π =. (.5) We are eresed he followg wo raso probables: φ0() = P( Z() = 0 Z(0) = ), (.6) φ0() = P( Z() = Z(0) = 0). They are gve by he sysem (0.3) of Chaper 4. For =, usg Laplace rasforms, s possble o show ha λ ( λ+ μ) φ0() = ( e ), λ+ μ (.7) μ μ ( λ+ μ) φ0() = + e. λ + μ λ + μ The asympoc behavour s gve by relao (0.8) of Chaper 3, or here drecly from resul (.7), λ lm φ0( ) =, λ + μ (.8) μ lm φ0( ) =. λ + μ As here, we have: MTTF =, MTTR, λ = μ (.9) relaos (.8) ake he form MTTR lm φ0( ) =, MTTF + MTTR (.0) MTBF lm φ0( ) =. MTTF + MTTR Remark. a) Moha e al (96) also solved he case of a seres sysem wh compoes, depede wh egave expoeal dsrbuos, respecvely

11 Relably ad cred rsk models 345 wh parameers λ,, λ, ad ha he repar me for compoe (=,,) has a egave expoeal dsrbuo of parameer μ. I erms of sem-markov modellg, hs meas ha he sae process ( Z (), 0) has as sae space he se {,,,Op} where sae (=,,), meas ha he sysem s he falure sae due o compoe, ad sae Op ha he sysem s operag a me. Wh he same sem-markov approach, s possble o show ha: λ ( λ+ μ) φ ( ) = ( e ), =,...,, λ + μ ( λ+ μ) μ + λe φop () =, λ + μ (.) ( λ+ μ) λ λe φ0 ()( = φop ()) =, λ + μ λ = λ + + λ. Ad cosequely, he asympoc behavour s gve by he followg relaos: λ lm φ ( ) =, =,...,, λ + μ ( λ+ μ) μ + λe lm φop ( ) =, (.) λ + μ λ φ0 () =. λ + μ b) For he geeral sochasc model of Moha e al (96), here exss a smple form of he asympoc behavour gve by he followg expressos: λ lm φ ( ) =, =,...,, μ λ + = μ lm φop ( ) =, λ + = μ λ = μ φ0 () =. λ + μ = (.3)

12 346 Chaper 8.. The em-markov Model For Maeace ysems Le us cosder a complex sysem havg as sae space he se I={,,m} wh m fe. The se I s paroed wo o-vod subses U ad D where U s he se of all fucog or up saes ad D all he faled or dow saes ad a ay me, he cosdered sysem s oe of hese saes ad rasos are of course possble. Ay durao oe of he saes of U s a operag me ad ay durao a sae of D a o-operag me; a raso from a sae of U o a sae of D meas ha here s a breakdow ad a raso from a sae of D o a sae of U may be see as he ed of a me of reparao. = s a sem-markov process wh kerel Q. As seco., he mmal case s cosdered wh he sem-markov process I = 0, where sae 0 meas breakdow sae ad sae operag sae. The basc assumpo s ha he process Z { Z(), 0} wh { } As before he ma relably dcaors are: () The avalably fucos defed as: A() = P Z() U Z(0) =, I, ( ) ( ) A(,) s = P Z() U Z() s =, I, (.4) respecvely he homogeeous ad he o-homogeeous case. If as Chaper 3, we defe by φ j he raso probably fucos for he Z- process, for boh he cases, we ge: A() = φ (), I, j U j A(,) s = φ (,), s I. j U j (.5) If, he homogeeous case, he process s ergodc, we ca also defe he asympoc avalably as: π jη j A( ) = lm A( ) =. (.6) π η j U k k k I () The relably fucos gvg he probably ha he sysem s always workg o he me erval [0,], R () = P( Z() U Z(0) = ), U, (.7) R (,) s = PZ ( () UZ() s = ), U.

13 Relably ad cred rsk models 347 D To compue hese probables, we wll ow work wh aoher kerel Q for whch all he saes of he subse D are chaged o absorbg saes, meag ha: D pj = δj, D, j I, (.8) D pj () s = δj, D, j I, respecvely for he homogeeous ad he o-homogeeous case. Dog so, he wo cases, we ge: D R() = φj (), U, j U (.9) D R (,) s = φ (,), s U, j U where of course, boh he homogeeous ad he o-homogeeous cases, he D marx Φ gves he probables raso for he sem-markov process of D kerel Q. () he maaably fucos gvg he probably ha he sysem s dow a me 0 he homogeeous case ad a me s he o-homogeeous case ad ha he sysem wll leave he se D wh he me, M() = P( Z( u) D, u ( 0, ] ), (.0) M (,) s = P( Zu () D, u ( s,]). U To compue hese probables, we wll ow work wh aoher kerel Q for whch all he saes of he subse U are chaged absorbg saes, meag ha: U pj = δj, U, j I, (.) U pj () s = δ j, U, j I. Dog so, respecvely, we ge: U M() = φj (), D, j U (.) U M (,) s = φ (,), s D, j U U where of course, he marx Φ gves he probables raso for he sem- U Markov process of kerel Q. I cocluso, we see ha he compug of he ma dcaors for he sem- Markov model s smple from our umercal resuls gve before. j j

14 348 Chaper 8.3 A Classcal Example I hs par he example gve Barlow ad Proscha (965) page 0 wll be developed. I s supposed ha here are wo maches (compuers he orgal example) workg parallel. The sysem s formed by e saes. Wh hree of hese saes D = { 5,7,9} he sysem s dow, he oher sx he sysem ca work. The MC ha descrbes he sysem s gve Table e λ e λ 0 0 e λ 0 0 e e e λ 0 0 e λ 0 e λ 0 0 e e θ λ λ + θ λ + θ θ λ λ + θ λ + θ Table.: embedded MC 0 The embedded MC for he homogeeous case s he same as Table. ad wh parameer values: 35 λ = ; θ = 6 ; γ = ; 0 = 4. (.3) I he o-homogeeous case he parameers are fucos of me; more precsely: 40,,30 λ = ; =,, ; 0.4,,. θ 8 4 γ = ; 0 = 30,,0 (.4) where: λ s he mea me o falure. I he o-homogeeous case, s a decreasg λ fuco of he me he sese ha goes from 40 o 30 seps. s he mea me o perform emergecy repar. I he o-homogeeous case θ θ s a creasg fuco ha goes from o 8 4 seps.

15 Relably ad cred rsk models 349 γ s he me o perform preveve maeace. I he o-homogeeous case s a creasg fuco ha goes from 0.4 o 0. seps. 0 s he scheduled preveve maeace perod. I he o-homogeeous case s a decreasg fuco ha goes from 30 o 0 seps. The followg ables repor he avalably, relably ad maaably fucos boh he homogeeous ad he o-homogeeous cases. Each colum of he ables repors he sarg sae. The rows represe me. I he homogeeous case he rows gve probables afer oe perod, wo perods ad so o. I he o-homogeeous case here are may resuls ad, our opo, was oo edous o repor all of hem. o, he frs eleme of each row gves he sarg me ad he evaluao me Table.: Homogeeous Avalably Fuco Table.3: No-Homogeeous Po-wse Avalably Fuco

16 350 Chaper Table.4: Homogeeous Relably Fuco Table.5: No-Homogeeous Relably Fuco Table.6: Homogeeous Maaably Fuco

17 Relably ad cred rsk models Table.7: No-Homogeeous Maaably Fuco 3 TOCHATIC MODELLING FOR CREDIT RIK MANAGEMENT 3. The Problem Of Cred Rsk A he prese me, he cred rsk problem s oe of he mos mpora coemporary problems ad has bee developed he facal leraure boh from heorecal ad praccal pos of vew. I cosss compug he defaul probably of a frm. Baks ad oher facal ermedares are he ype of frms ha are he mos cocered wh evaluao of cred rsk. There s a very wde rage of leraure o cred rsk models (see for example, Bluhm e al (00), Crouhy e al (000)). I he 990s, Markov models were roduced o sudy cred rsk problems. May mpora papers o hese kds of models were publshed (see Jarrow e al (997), Nckell e al (000), Israel e al (00), Hu e al (00)), maly for solvg he problem of evaluao of raso marces. I he paper by Lado ad kodeberg (00) some problems regardg he durao of he raso are explored, bu oly recely models whch he radomess of me he sae rasos have bee cosruced (see D Amco e al (004), (005a), (005b), Vasleou ad Vasslou (006)).

18 35 Chaper 8 By meas of he sem-markov model, s possble o geeralse he Markov models roducg he radomess of me for rasos bewee he saes. Furhermore we hk ha he cred rsk problem ca be see as a relably facal model for he frm uder sudy for whch we would lke o compue he defaul probably (D Amco e al (004), (005a), (005b)). 3. Cosruco Of A Rag Usg The Mero Model For The Frm I hs seco, we wll develop a elaborao of a rag model usg he classcal Mero model for he frm (974) ad used Credmercs, alsed by J-P Morga as a sequel of he Rskmercs compuer program dedcaed o he VaR mehods. I he Mero model (974), he value V of he frm s modelled wh a Black ad choles sochasc dffereal equao wh red μ ad saaeous volaly σ so ha s value me a s gve by σ ( μ ) + σw( ) 0 V0 beg he value of he frm a me 0 ad W W(), [ 0, T] V () = Ve, (3.) ( ) = a sadard Browa moo defed o he flered probably space ( ΩI I P) If Vdef,,( ),. s he hreshold beyod whch he frm defauls, called he hreshold defaul, he probably P def ha he compay defauls before me s gve by: (, ) = ( ( ) < ) Pdéf Vdéf P V Vdéf σ Vdéf = P μ + σw( ) < l V 0 Vdéf σ = P W( ) < l μ. σ V0 (3.) As, for all posve, W ()/ has a ormal dsrbuo, we ge: Vdef σ Pdef ( Vdef, ) =Φ l μ. (3.3) σ V0 o, f we fx he value of P def, we ca compue he correspodg value of V def usg he quales of he ormal dsrbuo. Le us suppose ha we fx he defaul probably P def o he correspodg quale Z CCC. Ths meas ha f Z s below or equal o, wh Z defed by: Z CCC

19 Relably ad cred rsk models 353 V σ Z = l μ, (3.4) σ V0 we ge: Pdef ( Vdef, ) =Φ( ZCCC ), Vdef σ (3.5) ZCCC = l μ. σ V0 O he corary, f he value of Z s larger ha ZCCC bu before he quale Z B, he rag gve o he frm s oed CCC ad so o. o we oba a scale of creasg hresholds represeed by: ZCCC < ZB < ZBB < ZBBB < ZA < ZAA < ZAAA, (3.6) assgg a cred rag or grade o frms as a esmae of her credworhess. If Z represes he observed value of Z for he cosdered frm, he scale used here s he rag use by he famous cred rag ageces adard & Poor s ad Moody s gve below: Zobs value oao Zobs<ZCCC defaul ZCCC<Zobs<ZB CCC ZB<Zobs<ZBB B ZBB<Zobs<ZBBB BB ZBBB<Zobs<ZA BBB ZA<Zobs<ZAA A ZAA<Zobs<ZAAA AA ZAAA<Zobs AAA Table 3.: rag ageces I s clear ha he cred rags deped o me ad also o he seleco of he probables P def, PZ ( CCC ), PZ ( B ), PZ ( BB ), PZ ( BBB ), PZ ( A), PZ ( AA), PZ ( AAA) chose by he cred rag agecy. We ca also compue he followg probables:

20 354 Chaper 8 Pdéf = P( Zobs < ZCCC ), PCCC = P( ZCCC < Zobs < ZB ), PB = P( ZB < Zobs < ZBB ), PBB = P( ZBB < Zobs < ZBBB ), (3.7) PBBB = P( ZBBB < Zobs < ZA), PA = P( ZA < Zobs < ZAA), PAA = P( ZAA < Zobs < ZAAA), PAAA = P( ZAAA < Zobs ), ad so: PB = Pdéf + PCCC, (3.8) Pdéf + PCCC + PB + L + PAA + PAAA =. Usg relao (3.3), we ge: V CCC σ Pdéf =Φ l μ, σ V0 V B σ Pdéf + PCCC =Φ l μ, σ V0 V BB σ Pdéf + PCCC + PB =Φ l μ, (3.9) σ V0... V AAA σ Pdéf + PCCC + PB + PBB + L + PAA =Φ l μ ; σ V0 V < V < V < V < V < V < V are he frm values correspodg where CCC B BB BBB A AA AAA o he rag ZCCC < ZB < ZBB < ZBBB < ZA < ZAA < ZAAA ad so:

21 Relably ad cred rsk models 355 V CCC σ Pdéf =Φ l μ, σ V0 Z B σ V CCC σ PCCC =Φ lv μ Φ l μ, σ V0 σ V0 V BB σ V B σ PB =Φ l μ Φ l μ σ V0, σ V0 (3.0)... V AAA σ V AA σ PAA =Φ l μ Φ l μ, σ V0 σ V0 V AAA σ PAAA = Φ l μ. σ V0 All hese relaos show how he grades are me depede, whch s why we wll ow sudy he dyamcs of rags 3.3 Tme Dyamc Evoluo Of A Rag 3.3. Tme Couous Model I couous me, he rag process s ohg else ha he sochasc process defed by relao (3.4), Z = { Z,0 T} (3.) where he r.v. Z represes he cred rag a me gve by: Pdéf ( V, ) =Φ( Z ), or (3.) V σ Z = l μ. σ V0 Here, he grade Z represes exacly he value sde oe of he classes defed above ad o loger oly he class. Usg relao (3.) o subsue he value of V (3.), we ge: W () Z =, > 0, (3.3) so ha W ( +Δ) W ( ) P( Z+Δ j Z) = P j =, Δ > 0,, j > ZCCC. (3.4) +Δ

22 356 Chaper 8 As he sadard Browa process has saoary ad depede cremes (see Chaper, seco 9, Defo 9.), we also ge: W ( +Δ) W ( ) P j = +Δ (3.5) W () = P W( +Δ) W() j +Δ W() =, or usg relao (3.4): W ( +Δ) W ( ) j +Δ PZ ( +Δ jz) = P Z = Δ Δ (3.6) j +Δ =Φ, Δ he las equaly comg from he ormaly of he cremes of a sadard Browa moo. We ca also wre hs las resul he form: j s PZ ( s jz = ) = Φ. (3.7) s The correspodg desy fuco s gve by: d s j s ( PZ ( s jz = ) ) = Φ '. (3.8) dj s s Ths las resul s correc oly for Z CCC. O he oher had, for < Z CCC, he defaul sae beg cosdered as a absorbg sae, we have ecessarly for j : PZ ( s jz = ) =. (3.9) I cocluso, as he raso probably gve by (3.7) depeds o boh s ad ad o oly o s, we jus prove ha he Z process s a o-homogeeous Markov process, roduced Chaper Dscree Couous Model Le us defe {,...,m } as he se of he m cred rags raked creasg order wh he Moody scale: = D def (defaul),=z CCC,...,m=Z AAA. Excep for he exreme classes, he rag classes defed below wll ow be represeed by her ceres as follows:

23 Relably ad cred rsk models 357 ( ] -, : (, ] : 3... (, ] : - (3.0)... m- ( m, m] : ( m, ) : m Le Z =, beg a class cere dffere from ; from resul (3.7), we have ha: P( j < Zs j Z = ) j s ( j ) s (3.) =Φ Φ, s >. s s To ge a dscree me, le us suppose ha we gve oaos a mes 0,u,u,,ku represeg for example oe year or a semeser. Now raso probables become: P( j < Zku+ j Zku = ) j ku+ ku ( j ) ku+ ku (3.) =Φ Φ, k = 0,,... u u Of course, f Z ku equals Z Def, we kow from relao (3.9) ha 0, j >, P( j < Zku+ j Zku = ZD ) = (3.3), j. Relaos (3.) ad (3.) defe a sequece of probably raso marces P(k), k=0,,... wh: P ( k) = pj ( k) (3.4) ad pj ( k) = P( j < Zku+ j Zku = ),, j =,..., m, k = 0,,.... (3.5) I follows ha he cred rag process Z dscree me Z=(Z ku,k=0,,...) s wha we call a o-homogeeous Markov cha defed Chaper 3. Of course, he very parcular ad urealsc case where he probably raso marces P(k), k=0,,... are depede of, he process dscree me Z=(Z ku,k=0,,...,) s he a homogeeous Markov cha as defed Chaper.

24 358 Chaper Example I real ecoomc lfe, cred rag ageces play a crucal role; hey comple daa o dvdual compaes or coures o esmae her probably of defaul, represeed by her scale of cred rags a a gve me ad also by he probably of rasos for successve cred rags. A chage he rag s called a mgrao. Mgrao o a hgher rag wll of course crease he value of a compay s bod ad decrease s yeld, gvg wha we call a egave spread, as has a lower probably of defaul, ad he verse s rue wh a mgrao owards a lower grade wh cosequely a posve spread. Here we have a example of a possble raso marx for mgrao from oe year o he successve oe: Toal AAA AA A BBB BB B CCC D Table 3. : Example of raso marx of cred rags We clearly see ha he probables of o mgrao, gve by he elemes of he prcpal dagoal, are he hghes elemes of he marx bu ha hey decrease wh he poor qualy of he rag. Here, we see for example ha a compay wh rak AA has more or less e chaces ou of e o keep s rag ex year, bu wll move o rak AAA wh oly sx chaces oe housad. O he oher had, a compay wh a CCC as rag wll be defaul ex year wh wey chaces ou of a hudred. As a more real example, he ex able gves he raso probably marx of cred rags of adard & Poor s for year 998 (see rags performace, adard & Poor s) for a sample of 404 compaes. Le us po ou he presece of a ew sae called N.R. (rag whdraw) meag ha for a compay such a sae, he rag has bee whdraw ad ha hs eve does o ecessarly lead o defaul he followg year, hus explag he las row of he above marx.

25 Relably ad cred rsk models 359 Effec. N.R. Toal 65 AAA AA A BBB BB B CCC N.R Table 3.3: example wh rag whdraw Here, we see for example ha compaes sae AA wll o be defaul he ex year bu ha 5.7 % of hem wll degrade o smple A ad 8 % o a BBB ad 0.8 wll upgrade o a AAA. Uder he assumpo of a homogeeous Markov cha, we oba he followg resuls: () he probably ha a AA compay defauls afer wo years: P () (D/AA)= =0.0006%, whch s sll very low. () he probably ha a BBB compay defauls oe of he ex wo years : Ths probably s gve by: PD ( / BBB;) = PD ( / BBB) + PBBB ( / BBBPD ) ( / BBB) + PBB ( / BBBPD ) ( / BB) + PB ( / BBBPD ) ( / B) + PCCC ( / BBBPD ) ( / CCC) = 0.34%+(84.93% 0.34%) +(4.46% 0.65%)+(0.67% 4.47%)+(0.% 36.67%) =0.77%. () he probably for a compay BBB o defaul bewee year ad year : Usg he sadard defo of codoal probably ( see Chaper ) we ge P(D a /o-def. a ) = P(D a & o-def. a )/ P(o-def. a ) =(0.77%-0.34%)/(-0.34%) =0.43%. Le us po ou ha hese llusrave resuls are rue uder he homogeeous Markov cha model ad moreover gve smlar resuls for all he compaes of he pael he same cred rag. I fac, real lfe applcaos, cred rag ageces also sudy each compay o s ow accou so ha specfc formao s also deermg for gvg he fal grade.

26 360 Chaper Rag Ad preads O Zero Bods Le us frs recall ha a zero-coupo bod s a corac payg a kow fxed amou called he prcpal, a some gve fuure dae, called he maury dae. o f he prcpal s oe moeary u ad T he maury dae, he value of hs zero-coupo a me 0 s gve by: B(0, T) = e δt (3.6) f δ s he cosdered cosa saaeous esy of eres rae. Of course, he vesor zero-coupos mus ake o accou he rsk of defaul of he ssuer. To do so, we cosder ha, a rsk eural framework, he vesor has o preferece bewee he wo followg vesmes: () o receve almos ceraly a me he amou e δ as couerpar of he vesme a me 0 of oe moeary u, ( δ + s () o receve a me he amou e ) ( s> 0) wh probably ( p) or 0 wh probably p, as couerpar of he vesme a me 0 of oe moeary u, p beg he defaul probably of he ssuer. The posve quay s s called he spread wh respec o he o-rsky saaeous eres rae δ as couerpar of hs rsky vesme zerocoupo bods. From he dfferece gve above, we oba he followg relao: δ ( δ+ s) e = ( p) e (3.7) or s = ( p) e, (3.8) s = l( p); (3.9) s p, (3.30) s p+ p. Le us ow cosder a more posve ad realsc suao whch he vesor ca ge a amou α,(0< α < ) f he ssuer defauls a maury or before. I hs case, he expecao equvalece prcple relao (3.7) becomes: δ δ+ s δ e = ( p) e + pα e, (3.3) or s = ( p) e + pα. (3.3) I follows ha hs case he value of he spread sasfes he equao s pα e = (3.33) p ad so he spread value s

27 Relably ad cred rsk models 36 pα s = l. (3.34) p As above, usg he Mac Laur formula respecvely of order ad, we oba he wo followg approxmaos for he spread: p s ( α), p (3.35) p p s ( α) ( α). p p 4 CREDIT RIK A A RELIABILITY MODEL 4. The em-markov Relably Cred Rsk Model As we already kow, he cred rsk problem ca be see as a relably problem whch he rag process, carred ou by he rag agecy, gves a relably degree of a frm bod ad moreover, he defaul sae ca be see as a dow sae ad a absorbg sae. From relaos (.5) ad (.9) resuls ha hs case he cocep of relably ad avalably cocde. We kow ha rag ageces lke adard & Poor s, Moody s or Fch gve each examed frm a rag. I he precedg subsecos, we used he &P smplfed model gvg egh kds of rags: AAA, AA, A, BBB, BB, B, CCC, D, where he saes are decreasg order depedg o he relably of her debs, ad he sae D meas defaul (for he precse defo of each sae see Crouhy e al (00)). I order o apply relably models a cred rsk evrome s possble o cosder, followg &P classfcao, he frs seve saes as good saes ad he D sae as he oly bad sae ad apply our sem-markov relably models o he cred rsk problem. The sae D wll be a absorbg sae, because oce he sae s reached, he sese ha he frm s o poso o pay s debs ad so herefore defauls, s o possble o ex from he sae. Furhermore hs case we are eresed oly he R() fuco; he A() ad M() fucos are meagless hs evrome. R () gves he probably ha he sysem was always workg up o he me gve ha he sysem was he workg sae a me 0.

28 36 Chaper 8 I hs case he relably model s subsaally smplfed ad o ge all he resuls ha are releva he cred rsk case, suffces o solve he sem- Markov evoluo equao oly oce o ge he followg probables: ) φj ( ) ad φj ( s, ) represeg respecvely he probables o be he sae j afer a me sarg he sae a me 0 he homogeeous case ad sarg a me s he sae he o-homogeeous case. The sem-markov evrome akes o accou he dffere probables of sae chages durg he permaece of he sysem he same sae (durao problem); ) R () = φ () ad R ( s, ) = φ ( s, ), represeg respecvely he j U j j U j probables ha he sysem ever goes o he defaul sae a me he homogeeous case ad from me s o me he o-homogeeous oe; 3) H ( ) ad H ( s, ), represeg he probables ha a me erval, he homogeeous case, ad from me s o me, he o-homogeeous case, o oe ew rag evaluao was doe for he frm; 4) ϕ ( ) ad ϕ ( s, ) represeg he probables o ge he rak j a he ex j j rag f he prevous sae was ad o oe rag evaluao was made up o he me he homogeeous case ad from me s o me he ohomogeeous oe. I hs way, for example, f he raso o he defaul sae s possble ad f he sysem does move for a me from he sae, we kow he probably ha he ex raso he sysem wll go o he defaul sae. They are defed by he followg relaos: pj Qj () ϕj () =, (4.) H ( ) pj () s Qj (,) s ϕj (,) s =. (4.) H ( s, ) 4.. A Homogeeous Case Example Now we gve a example usg he raso marx gve Jarrow e al (997). Ths example wll be chose he homogeeous case. Ths marx was cosruced sarg from he oe year raso marx gve adard & Poor s Cred Revew (993). We repor he marx for he sake of compleeess. AAA AA A BBB BB B CCC D Table 4.: year raso marx

29 Relably ad cred rsk models 363 The marx value for d.f. are o kow ad so we cosruc hese d.f. by meas of radom umber geeraors. We repor he resuls a me 5 ad a me 0 of he marx φ j ( ) respecvely Table 4.. ad Table 4... For example he eleme ha s row AA ad colum A represes he probably ha a frm ha a me 0 has a rag AA wll have rag A a me 5. AAA AA A BBB BB B CCC D Table 4..: probables φ j (5) AAA AA A BBB BB B CCC D Table 4..: probables φ j (0) Table 4.3 gves he relably resuls R (), probables o have o defaul a me (row dex) sarg he sae (colum) a me Table 4.3: probables of o havg a defaul

30 364 Chaper 8 Table 4.4 gves probables H ( ) o rema always he sarg sae whou rasos Table 4.4: probables o rema he sarg sae Lasly, Tables 4.5. ad 4.5. gve probables ϕ j ( ) a 5 years ad 0 years. For example represes he probably ha a frm ha was a me 0 sae AA ad remaed hs sae up o me 5 wll he have he ex raso sae A. AAA AA A BBB BB B CCC D Table 4.5.: probables ϕ j (5) AAA AA A BBB BB B CCC D Table 4.5.: probables ϕ j (0)

31 Relably ad cred rsk models A No-Homogeeous Case Example Now we gve a o-homogeeous example usg as bass he raso marces gve Table 5 of adard & Poor s (00). I hese marces he sae No Rag was prese. arg from he daa repored he &P publcao, he o-homogeeous raso marx was cosruced. Each eleme pj ( s ) of he embedded o-homogeeous Markov cha should be cosruced drecly from he daa. Cosrucg he MC, all possble rasos from sae o sae j sarg from he year s should be ake o accou. Bu we do o have he raw daa ad so we use he oe year raso marces gve he adard & Poor s publcao. The publcao repors 0 years of hsory (oe year raso marces from 98 up 000). The example covers from me 0, correspodg o he year 98, o me 9, correspodg o he year 000. Table 4.6 repors hree years of he o homogeeous embedded MC. TRANITION MATRICE MATRIX AT TIME 0 AAA AA A BBB BB B CCC D MATRIX AT TIME0 AAA AA A BBB BB B CCC D MATRIX AT TIME9 AAA AA A BBB BB

32 366 Chaper 8 B CCC D Table 4.6: Embedded NHMC To apply he model s ecessary o cosruc also he d.f. of he wag mes each sae, gve ha he sae successvely occuped s kow. As we do o have hese daa eher, we cosruc hem by meas of radom umber geeraors. Probables H (,) s o rema he sae from s o whou ay raso are repored Table 4.7. For example he eleme represes he probably ha he rag AA has o oher rag evaluao from he me 0 up o he me 9. Probably o moveme TIME AAA AA A BBB BB B CCC Table 4.7: probables o rema he sarg sae whou rasos

33 Relably ad cred rsk models 367 Tables 4.8. ad 4.8. gve he probables ϕj ( s, ) ha he ex raso from he sae wll be o he sae j gve ha here s o raso from he me s o he me. j (,) s ϕ Prob. Nex ae Whou Trasos from s o TIME 0- AAA AA A BBB BB B CCC D TIME 0-0 AAA AA A BBB BB B CCC D TIME 0-9 AAA AA A BBB BB B CCC D Table 4.8.: probables ϕ (0, ) For example he eleme gves he probably ha he ex raso from he rag AA wll be o he rag A, gve ha from he me 0 up o he me 0 here wll be o real or vrual rasos; by vrual raso we deoe he fac ha he ex raso s he same sae. j (,) s ϕ Prob. Nex ae Whou Trasos from s o TIME 5-6 AAA j

34 368 Chaper 8 AA A BBB BB B CCC D TIME 5-7 AAA AA A BBB BB B CCC D TIME 5-8 AAA AA A BBB BB B CCC D TIME 5-9 AAA AA A BBB BB B CCC D Table 4.8.: probables ϕ (5, ) Tables 4.9. ad 4.9. repor he probables φ ( s, ). φ (,) s EVOLUTION EQUATION MATRICE j TIME 0- AAA AA A BBB j j

35 Relably ad cred rsk models 369 BB B CCC D TIME 0-0 AAA AA A BBB BB B CCC D TIME 0-9 AAA AA A BBB BB B CCC D Table 4.9.: probables φ (0, ) For example represes he probably o be he sae BBB a me 0, gve ha he rag evaluao was A a me 0. φ (,) s EVOLUTION EQUATION MATRICE j TIME 5-6 AAA AA A BBB BB B CCC D TIME 5-7 AAA AA A BBB BB B j

36 370 Chaper 8 CCC D TIME 5-8 AAA AA A BBB BB B CCC D TIME 5-9 AAA AA A BBB BB B CCC D Table 4.9.: probables φ (5, ) The las Table 4.0 repors he relably probables gvg he probables ha a frm beg a gve rag a me s wll o have a defaul up o he me. RELIABILITY TIME AAA AA A BBB BB B CCC j

37 Relably ad cred rsk models Table 4.0: relably probables