Credit Portfolio Modelling, Marginal Risk Contributions, and Granularity Adjustment

Transcription

1 Credt ortolo Modellg, Margal Rsk Cotrbutos, ad Graularty djustmet Revsed: Jue 4, 00 Has Rau-Bredow rv.-doz. Dr. oec. publ. Has Rau-Bredow Leo Wesmatel Str. 4 D Würzburg phoe.: 49( moble: 49( has.rau-bredow@mal.u-wuerzburg.de

2 Credt ortolo Modellg, Margal Rsk Cotrbutos, ad Graularty djustmet bstract Ths paper rst provdes a smple but very geeral ramework or credt portolo modellg whch s based o the dstcto betwee systematc ad usystematc rsk. Usystematc or borrower-specc rsk vashes through dverscato a very large, tely e-graed portolo. The ramework cotas typcal models lke CredtRsk ad CredtMetrcs as specal cases. aalyss o margal rsk cotrbutos s the doe whch also cludes a theoretcal ormula or the graularty adjustmet a "lumpy" credt portolo. JEL classcato: D 8, G, G 8

3 3. Itroducto The stadard tool or credt portolo maagemet s today Value at Rsk (VaR, whch s deed as the quatle o the prot ad loss dstrbuto or a gve codece level: For a codece level o e.g. p99%, oe s 99% certa that at the ed o the plag horzo there wll be o greater loss tha just the VaR. I VaR s completely covered by equty captal, the codece level s the mmum probablty that solvecy wll ot occur. I practce, VaR or a credt loa portolo s calculated wth models lke CredtRsk (997 rom Credt Susse Frst Bosto or CredtMetrcs (997 rom J Morga. Recetly, the Basel Commttee o Bakg Supervso (00 has also adopted VaR the proposals or a ew captal accord. I the past, some researchers have observed the gve smlartes betwee deret credt rsk models. Koyluoglu ad Hckma (998 ad Fger (999 have poted to the act that or gve realzatos o the backgroud actors or systematc rsk actors, deaults ad ratg chages are geerally assumed to be stochastcally depedet. Smlar, Gordy (000 has show that a restrcted two-state verso o CredtMetrcs, whch deretates oly betwee deault ad o-deault, ca be mapped to the CredtRsk ramework ad vce versa. I ths paper, I beg wth a smple but very geeral ramework or credt portolo modellg whch cotas models lke CredtRsk or the urestrcted mult-state verso o CredtMetrcs as specal cases. I ths ramework, the value o each loa at the ed o the plag horzo s a ucto o some systematc rsk actors commo to all borrowers ad a addtoal specc or usystematc rsk actor. s a cosequece o the law o large umbers, usystematc rsk vashes through dverscato a very large, tely e-graed portolo. mportat questo s how much addtoal equty captal s requred a sgle loa s added to the credt portolo. I order to aswer ths questo, the dervatve o the VaR must be calculated. It ca be show mathematcally that the dervatve s gve by the codtoal mea o the margal loa, o codto that the value o the credt portolo ad VaR are exactly detcal. I ths geeral result s appled to a smple oeactor model, the model used by the Basel Commttee ca be obtaed. other result s a theoretcal ormula or the graularty adjustmet, whch s eeded to cover the rema-

4 4 g usystematc rsk. Such a ormula was recetly preseted by Wlde (00. I ths paper, a deret dervato o that ormula through a Taylor expaso wll be gve. Ths paper s orgazed as ollows. Secto troduces a geeral ramework or credt portolo modellg. Secto 3 explas the role o dverscato that ramework. I Secto 4, a geeral ormula or margal rsk cotrbutos wll be preseted ad appled to a smple oe-actor model. Ths wll be doe by assumg a tely egraed credt portolo. Subsequetly, a graularty adjustmet or "lumpy" credt portolos s cosdered.. geeral credt portolo model Cosder a portolo o loas wth exposure szes,..,. s a percetage o the exposure sze, the derece betwee the actual value o each loa ad the value at the ed o the plag horzo (usually oe year s descrbed by a radom loss varable L. Let L L (,ε be gve as a ucto o some systematc rsk actors (,..., k, whch represet the state o the ecoomy ad are commo to all borrowers, ad a specc or usystematc rsk actor ε. Each ε s assumed to be stochastcally depedet rom all other systematc ad usystematc rsk actors. Obvously, such a very geeral approach cotas typcal models lke CredtMetrcs or CredtRsk as specal cases. CredtMetrcs or example s a mark-to-market model whch the value o each loa s a ucto o the borrower`s credt ratg. Note that our model a upgradg would result a ga market value ad cosequetly a egatve value o the loss varable L. CredtMetrcs assumes that ratg chages are drve by a uderlyg asset value process. The retur r o the assets o borrower s explaed as a lear combato o systematc ad usystematc rsk actors: r w... w ŵ k k ( For example, the usual goal o a ratg or the bak requres a codece level o 99,97% (plag horzo oe year.

5 5 The realzato o the asset retur r the determes the ratg o the borrower, ad the respectve ratg dees the value o the loa at the ed o the plag horzo. CredtRsk deretates oly betwee deault ad o-deault. Deault probabltes p p ( are volatle 3 ad geeral gve as a lear combato o some gammadstrbuted backgroud actors (,..., k : 4 p ( w... w k k ( Obvously, the backgroud actors CredtRsk play the same role as the systematc rsk actors CredtMetrcs. To see the smlartes, assume that the backgroud actors determe a certa threshold T ( so that borrower deaults the correspodg usystematc rsk actor ε ulls ε <T (. The threshold T ( has to be chose so that the probaltty or ths s exactly p (. It ollows that both models the value o each loa at the ed o the plag horzo s gve as a ucto o some systematc rsk actors ad a addtoal usystematc rsk actor. 3. Dverscato s a percetage o total exposure, the radom loss o the etre portolo at the ed o the rsk horzo s L L (3 Now assume that the realzatos o the systematc rsk actors (,..., k occur beore the realzatos o the usystematc rsk actors ε. I the values o the systematc rsk actors are take as gve, L s a sum o stochastcally depedet radom vara- The ratg would be r T, T > r T ad so o, where the thresholds T must be chose so that mgrato probabltes are accordace wth the hstorcal trasto matrx. I addto, because systematc rsk actors are commo to all borrowers, the approach also takes the stochastc depedece o ratg mgratos to accout. 3 For gve realzatos o the deault probabltes, deault evets are assumed to be stochastcally depedet. 4 Gordy (000 p..

6 6 bles. Thus, the cetral lmt theorem ca be appled. Codtoal o, the portolo loss varable L s asymptotcally ormal-dstrbuted wth mea (L (4 ad varace ( (L (5 It s easy to show that 0 < < < ad m max < or all wth te max boudares max ad max, the 0 as. For sucetly large, the varace teds to zero ad the probalty or a arbtrary small devato o L rom the codtoal mea gets arbtrary small. Ths s, o course, othg else tha a applcato o the law o large umbers. O codto that the values o the systematc rsk actors are gve, L becomes ostochastc a very large, tely e-graed portolo. Borrower-specc or usystematc rsk ca thus be elmated through dverscato. The oly remag rsk s systematc rsk, that s the rsk that the actual values o the systematc rsk actors (,..., k result a hgher or lower value o the codtoal mea. 4. Margal rsk cotrbutos 4. geeral result I bakg practce, the margal rsk cotrbuto a ew loa s added to a portolo s ote assumed to be proportoal to the margal stadard devato. From a theoretcal perspectve, ths s obvously wrog because credt rsk s by ature hghly skewed ad at taled. The stadard devato s thereore ot a approprate measure or credt

7 7 rsk. So what s eeded s a geeral ormula or margal rsk cotrbutos whch does ot rely o specc assumptos about the loss dstrbuto. I order to ormulate, rst wthout ay reerece to the prevous stated ramework, such a geeral result, suppose that the value o the actual portolo s gve by a radom varable ad that a racto t o aother radom varable Z s added to that portolo. The, the codto r ob( tz > VaR( tz cost. (6 that the actual realzato o tz exceeds VaR(tZ oly wth a costat probablty mplctly dees VaR(tZ as a ucto o t. I appedx, the rst ad secod dervatves o VaR(tZ wth respect to t are calculated 5. The oly assumptos made s that the radom varables ad Z have a jot probablty desty ucto ad that rst ad secod momets exsts. The rst dervatve s smply the codtoal mea: VaR( t tz t 0 (Z VaR( (7 Itutvely, ths result ca be terpreted as ollows: I >VaR( (the bak s already bakrupt or <VaR( (there s a remag equty buer ad or a sucetly low value o t, addg a very small sucetly small racto tz would ot chage the outcome. Thereore, the margal captal requremet or a addtoal rsk s the average value or all crtcal cases wth VaR(. s a specal case, assume that ad Z are bvarate ormal dstrbuted,.e. the case whe the stadard devato s act the rght rsk measure. I ths case, the usual ormula or the lear regresso apples, ad the codtoal mea s exactly equal to: VaR( t tz cov(,z 0 (Z VaR( ( Z (VaR( ( (8 ( t 5 For smlar results see also Goureroux et al. (000, Tasche (999.

8 8 Here, cov(,z / ( s the usual beta-actor kow rom the classcal CM. s VaR s commoly cosdered as the sum o expected ad so-called uexpected loss, the ormula states that margal VaR s gve by expected loss o the margal loa plus beta-actor tmes uexpected loss o the portolo. O course, as already metoed, the uderlyg assumpto o a ormal dstrbuto s problematc whe appled to the loss dstrbuto o a credt loa portolo. 4. Oe-actor model bove, a geeral ormula has bee derved whch states that margal VaR s the codtoal mea o the margal rsk, o codto that the value o the orgal portolo exactly equals VaR. I appled to the credt rsk ramework developed earler, the codto that the portolo value equals VaR would mpose a restrcto o the choce o the rsk actors. For a smple case, assume that the value o each loa at the ed o the plag horzo s a creasg ucto o oly oe systematc rsk actor (the loss varable L s the a decreasg ucto o,.e. s a scalar usystematc rsk s perectly dversed away,.e. L I ths case, the oly remag rsk s that the actual realzato o wll be below the quatle x, wth x mplctly deed by r ob( < x. The restrcto mposed o the rsk actor s smply x. Thereore, as a percetage o total exposure, VaR o the whole credt portolo s gve by: VaR( L (L x x (9 Margal VaR or each Euro borrowed to borrower s the gve by the codtoal mea o the dvdual loa x, wth the codto that the systematc rsk actor equals the quatle x.

9 9 It ollows that such a oe-actor model margal rsk cotrbutos deped oly o the characterstcs o the dvdual loa, ad ot o the characterstcs o the portolo to whch t s added. Ths s the reaso why such a oe-actor model has bee adopted by the Basel Commttee the proposals or a ew captal accord. I stead a multactor model had bee used, the margal rsk cotrbutos o each loa would also deped o how well the credt portolo s dversed over the deret sectors (coutres or dustres, wth the state o each sector beg represeted by oe o the systematc rsk actors. It would be dcult or the regulator to obta such detaled ormato about dvdual bak portolos. The model used by the Basel Commttee s a smpled CredtMetrcs model whch deretates oly betwee deault ad o-deault 6. Deault occurs the asset retur alls below a certa threshold D: r - < D (0 Here, ρ s the correlato coecet o the asset returs ad, ε are depedet stadard ormal dstrbuted radom varables wth mea zero ad varace oe. The, as a cosequece o the choce o the coecets, r s also stadard ormal dstrbuted. The relatoshp betwee the deault threshold D ad the probablty o deault D s D N ( D, where N s the cumulatve dstrbuto ucto or a stadard ormal radom varable. Wth deault resultg a loss gve deault LGD (as a percetage o the exposure, margal VaR s gve as ollows as the codtoal mea, o codt- o that x N ( : x LGD r ob( < N - (D - x ( LGD N N( - (D - x 6 The model s due to Vascek (997. See also Schobucher (00.

10 0 For example, the cosultatve paper rom Jauary 00, the Basel Commttee has set 0. ad. 57 or a corporate loa portolo (Ths wll be probably ot the x 0.5% parameter choce the al accord. Formula ( s the used to calculate the captal charge or a loa wth probablty o deault D. 4.3 Graularty adjustmet a oe-actor model Because o real-world portolo ca be tely e-graed, a graularty adjustmet has to be added to accout or the remag usystematc rsk,.e. or large cocetratos o rsk a "lumpy" portolo. Such a graularty adjustmet has also bee proposed by the Basel Commttee the already metoed cosultatve paper rom Jauary 00. There, the calculato o the graularty adjustmet s based o a theoretcal result o Gordy (00, who shows that the remag usystematc rsk s versely proportoal to the eectve umber o loas. Gordy also estmates the proportoal costat or typcal loa portolos umercally through Mote Carlo smulatos. theoretcal ormula or the graularty adjustmet was recetly gve by Wlde (00. Here I take a deret approach whch leads exactly to the same result as Wlde (00. It has bee show above that VaR s gve as the codtoal mea, o codto that x. The trck s the to develop a secod-order Taylor expaso wth respect to the error term L. Ths results (see appedx B: VaR x ( x / x / x x x x x l ( x where ( deotes the probablty desty ucto o the codtoal mea, whch s a ucto o the systematc rsk actor.

11 The graularty adjustmet cossts o two terms: The sestvty ( / /( / / o the codtoal varace wth respect to the cod- toal mea ad the codtoal varace tmes the dervatve o the logarthmc desty l. The secod term s postve the desty o the codtoal mea slopes dowwards the rght tal,.e. or very hgh average losses. Ths wll usually be the case. Uclear s the sg o the rst term. To get a tuto, ote that the remag usystematc rsk could also lt the value o the credt portolo above the VaR-threshold a volato o that threshold would otherwse occur. I / s postve (the varace s a creasg ucto o average losses, the chace that the remag usystematc rsk prevets a volato o the VaR-threshold s greater tha the correspodg rsk that a volato o the VaR-threshold s trggered oly by usystematc rsk. s a cosequece, t caot be completely ruled out that the graularty adjustmet mght be egatve, at least theoretcally. smple example s a model wth varable deault probabltty p(, ad wth, LGD 00% or all. The ( ad equato (3 reduces to: VaR x x x ( x l x ( x x x (3 Here, the rst part o the graularty adjustmet wll be act egatve most practcal cases where the worst possble deault probablty p ( x x s lower tha 50%. I addto, the graularty adjustmet s versely proportoal to the umber o loas, whch corms the above metoed result o Gordy ( Cocluso I recet years, practtoers have developed may deret credt portolo models. Here, a geeral ramework or credt portolo modellg has bee developed whch s based o the dstcto betwee systematc ad usystematc rsk. s a cosequece o the law o large umbers, usystematc rsk ca be completely dversed away a ve-

12 ry large, tely e-graed portolo. VaR ad margal rsk cotrbutos the deped oly o systematc rsk. smple case s a oe-actor model where the systematc rsk actor s a scalar. The, usystematc rsk s perectly dversed away, the oly remag rsk s that the realzato o the systematk rsk actor wll be below the respectve quatle. However, because o real-world credt portolo s tely e-graed, a addtoal graularty adjustmet has to be added to accout or large cocetratos o rsk "lumpy" credt portolos. s has bee show, the mpact o the remag usystematc rsk ca be added cremetally rather tha calculatg both rsks at oce. Mathematcal appedx. Frst ad secod dervatve o Value at Rsk Cosder two radom varables ad Z wth a jot probablty desty ucto (y,z ad dee VaRVaR(tZ as a ucto o a real varable t by r ob( tz > VaR cost. The: VaR ( Z tz VaR ( Z tz s V ar ( Z tz l s tz ( s s VaR where ( s tz deotes the probablty desty ucto o tz. roo: Note rst that the ormula or the codtoal desty s:

13 3 ( Z z tz VaR ( VaR tz,z (VaR tz The: 0 r ob( tz > VaR ( y,z dy dz VaR tz VaR ( z ( VaR tz,z dz VaR ( ( Z tz VaR (VaR tz Dvdg by (VaR yelds the result or the rst dervatve. The ormula or the tz secod dervate ca be get as ollows: 0 r ob( tz > VaR ( y,z dy dz VaR tz VaR ( z ( VaR tz,z dz VaR ( VaR ( VaR tz,z VaR tz,z ( z dz t

14 4 ar V ( VaR tz,z VaR ( z ( s tz,z s VaR dz VaR ( z Z tz VaR (VaR tz VaR ( z ( ( z Z tz s tz ( s s VaR dz VaR (VaR tz ( ( Z tz VaR z Z ( z tz s s VaR (VaR dz tz ( ( Z tz VaR z ( z Z tz VaR ( s tz s VaR dz ar ( Z tz s V ( Z l tz s ( s tz s VaR (VaR tz q.e.d. B. Graularty djustmet Wth appedx ad the parameter choce (L, t, L Z t (L - (L whch esures, uder the assumptos made secto 3, that the codtoal varace o Z s te, a Taylor expaso aroud t0 drectly leads to the ollowg result:

15 !!!! 5 VaR VaR( tz VaR( VaR( tz t t 0 t VaR( tz t 0 VaR( ( tz VaR( ( tz s ( tz l s ( s s VaR( 0 x / x / x x x x x l ( x wth ( ( x x / x. q.e.d. Reereces: Basel Commttee o Bakg Supervso (00: The New Basel Captal ccord, Jauary 00. Dowload: CredtMetrcs (997: Techcal Documet. J.. Morga. Dowload: (regstrato requred CredtRsk (997: Techcal Documet. Credt Susse Facal roducts. Dowload: Fger, C.C. (999: Codtoal pproaches or CredtMetrcs ortolo Dstrbutos, : CredtMetrcs Motor, pp Dowload: (regstrato requred Gordy, M. B. (00: Rsk-Factor Model or Ratg-Based Captal Rules. Workg aper. Dowload: mgordy.trpod.com Gordy, M. B. (000: Comparatve atomy o Credt Rsk Models, : Joural o Bakg ad Face, Vol. 4, pp Goureroux C., Lauret J.., Scallet O. (000: Sestvty alyss o Values at Rsk, : Joural o Emprcal Face Vol. 7 (3-4 pp

16 6 Dowload: or _Scallet.htm Koyluoglu, H. U., Hckma,. (998: Recoclable Dereces, Rsk, October 998, pp. 56-6; Schobucher,. (00: Factor Models: ortolo Credt Rsks whe Deault are correlated, : Joural o Rsk Face Vol. 3, pp Tasche, D. (999: Rsk Cotrbutos ad erormace Measuremet. Workg paper. Dowload: www-m4.mathematk.tu-mueche.de/m4/pers/tasche Vascek, O. (997: The Loa Loss Dstrbuto. Wlde, T. (00: robg Graularty, : Rsk, ugust 00, pp Dowload: