Multi-factor adjustment

Transcription

1 Portfolo credt rsk l Cttng edge lt-factor adjstment lt-factor erton-type portfolo models of credt rsk have become very poplar among rsk management practtoners. Practcal mplementatons of these models mostly rely on onte Carlo smlatons, whle analytcal methods have been lmted to the one-factor case. Here, chael Pykhtn presents an analytcal method for calclatng portfolo vale-at-rsk and expected shortfall n the mlt-factor erton framework. Ths method s essentally an extenson of the granlarty adjstment techne to a new dmenson ost of the portfolo models of credt rsk sed n the bankng ndstry are based on the condtonal ndependence framework. In these models, defalts of ndvdal borrowers depend on a set of common systematc rsk factors descrbng the state of the economy. erton-type models, sch as Portfoloanager and Credtetrcs, have become very poplar. However, mplementaton of these models reres tme-consmng onte Carlo smlatons, whch sgnfcantly lmts ther attractveness. For one-factor erton-type models, several analytcal technes have been developed. One sch techne s the lmtng loss dstrbton dscovered by Vascek (99) for homogeneos portfolos and extended by Gordy (003) and Vascek (00) to non-homogeneos portfolos. Ths approxmaton replaces the orgnal loss dstrbton wth the loss dstrbton for an nfntely fne-graned portfolo, whose vale-at-rsk and expected shortfall (ES) can be calclated analytcally. However, the dfference between the VARs (or ESs) of the orgnal and the lmtng loss dstrbtons can be sgnfcant f the portfolo s not large enogh. To estmate ths dfference, the granlarty adjstment techne was ntrodced by Gordy (003). Wlde (00) and artn & Wlde (00) have derved a general closed-form expresson for the granlarty adjstment for portfolo VAR. ore specfc expressons for a one-factor defalt-mode erton-type model (known as the Vascek model) have been derved by Wlde (00) and Pykhtn & Dev (00). Emmer & Tasche (003) have developed an analytcal formlaton for calclatng VAR contrbtons from ndvdal exposres. Gordy (004) has derved a granlarty adjstment for ES. For mlt-factor erton-type models, no prely analytcal methods for estmatng portfolo VAR or ES have been reported. Althogh Gordy (003) has shown that the lmtng loss dstrbton s stll applcable to large enogh portfolos, calclaton of the portfolo VAR stll reres onte Carlo smlaton of the systematc rsk factors. oreover, t s not clear how large the portfolo needs to be to ensre applcablty of the lmtng loss dstrbton. In ths artcle, we present an analytcal method for calclatng portfolo VAR and ES n the mlt-factor erton framework. Ths method s essentally an extenson of the granlarty adjstment techne to a new dmenson. odel et s frst set p a mlt-factor defalt-mode erton model. We consder a portfolo of loans to dstnct borrowers. To avod cmbersome notatons, we assme that each borrower has exactly one loan wth prncpal A. We also defne the weght of a loan n the portfolo as the rato of ts prncpal to the total prncpal of the portfolo, w = A /Σ j = A j. Borrower wll defalt wthn a chosen tme horzon (typcally, one year) wth probablty p. Defalt happens when a contnos varable X descrbng the fnancal well-beng of borrower at the horzon falls below a threshold. We assme that varables {X } (whch may be nterpreted as the standardsed asset retrns) have standard normal dstrbton. The defalt threshold for borrower s gven by (p ), where ( ) s the nverse of the cmlatve normal dstrbton fncton. We assme that asset retrns depend lnearly on normally dstrbted systematc rsk factors wth a fll-rank correlaton matrx. These systematc factors represent ndstry, geography, global economy or any other relevant ndexes that may affect borrowers defalts n a systematc way. Borrower s standardsed asset retrn s drven by a certan borrower-specfc combnaton of these systematc factors Y (known as a composte factor): X = ry + r ξ () where ξ s the standardsed normally dstrbted dosyncratc shock. Factor loadng r measres borrower s senstvty to the systematc rsk. Snce t s more convenent to work wth ndependent factors, we assme that orgnal correlated systematc factors are decomposed nto ndependent standard normal systematc factors Z k (k =,..., ). The relaton between {Z k } and the composte factor s gven by: where coeffcents α k mst satsfy the relaton Σ k = α k = to ensre that Y has nt varance. Asset correlaton between dstnct borrowers and j s gven by ρ j = r r j Σ k = α k α jk. If borrower defalts, the amont of loss s determned by ts loss-gven defalt (GD) stochastc varable Q wth mean µ and standard devaton σ. We assme that these GD varables are ndependent between themselves as well as from all the other varables n the model. We do not make any specfc assmptons abot the probablty dstrbton of Q. Fnally, portfolo loss rate can be wrtten as the weghted average of ndvdal loss rates : Y = αkz k = = w = w Q X ( p ) { } where { } s the ndcator fncton. Eaton (3) descrbes the dstrbton of the portfolo losses at the horzon and ths completes the model. A tradtonal approach to estmatng antles of the portfolo loss dstrbton n the mlt-factor framework s onte Carlo smlaton. If the portfolo s large enogh to be consdered fne-graned, most of the dosyncratc rsk n the portfolo s dversfed away and portfolo losses are drven prmarly by the systematc factors. In ths case, eaton (3) can be replaced by the lmtng loss dstrbton of an nfntely fne-graned portfolo. The lmtng loss s gven by the expected loss condtonal on the systematc rsk factors, as can be shown by applyng the law of large nmbers condtonally on the factors (see Gordy, 003, for detals): See Blhm, Overbeck & Wagner (00) for an excellent revew of portfolo credt rsk models Another nterestng extenson of the granlarty adjstment techne s gven n Canabarro, Pcolt & Wlde (003), where the athors se t for analysng dervatves conterparty credt rsk k () (3) ARCH 004 RISK 85

2 Cttng edge l Portfolo credt rsk E Z w p r ( k Z k = k = { k} = ) α µ r (4) Althogh eaton (4) s mch smpler than eaton (3), t stll reres onte Carlo smlaton of the systematc factors {Z k } when the nmber of factors s greater than one. oreover, t s not clear how large the portfolo needs to be for eaton (4) to become accrate. In what follows, we desgn an analytcal method for calclatng tal antles and tal expectatons of the portfolo loss gven by eaton (3). The method s based on dervatves of VAR ntrodced by Gorerox, arent & Scallet (000) and perfected by artn & Wlde (00). Dervatves of VAR We are nterested n calclatng the antle at a confdence level of the portfolo loss. We wll denote ths antle as t (). et s assme that we have constrcted a random varable _ sch that ts antle at level, t ( _ ), can be calclated analytcally and s close enogh to t (). We wll thnk of the portfolo loss as the varable _ pls pertrbaton U defned as U = _. To descrbe the scale of the pertrbaton, let s also ntrodce a pertrbed varable = _ + U. artn & Wlde (00) have shown that, for hgh enogh confdence levels, t ( ) can be calclated va the expanson n powers of arond t ( _ ). By keepng terms p to adratc and settng = n ths Taylor seres, we can hope to calclate the portfolo loss antle as: Gorerox, arent & Scallet (000) have derved the frst two dervatves of VAR. The frst dervatve s gven by the expectaton of the pertrbaton condtonal on _ = t ( _ ): whle the second dervatve s: t t = + dt dt =0 dt d = f l U l () var f () l dl = = 0 l= t ( ) where f _ ( ) s the probablty densty fncton for _ and var[u _ = l] s the varance of U condtonal on _ = l. The problem s now redced to fndng approprate _. Comparable one-factor model We defne _ va the lmtng loss dstrbton for the same portfolo as defned above, bt n the one-factor erton framework: = = l Y wµ p Y where Y _ s the sngle systematc rsk factor havng the standard normal dstrbton, and p(y) s the probablty of defalt of borrower condtonal on Y _ = y, whch s gven by: p y ( p ) a y = (9) a where a s the effectve factor loadng for borrower. Snce _ s a determnstc monotoncally decreasng fncton of Y _, the antle of _ at level can be calclated analytcally smply as the fncton vale at Y = ( ): t l = dt + = EU = t ( ) ( ( )) = 0 = 0 (5) (6) (7) (8) (0) and therefore can be sed as the zeroth-order approxmaton to t (). et s note that the dervatves of VAR n eatons (6) and (7) are gven by expressons condtonal on _ = t ( _ ). Snce _ s a determnstc monotoncally decreasng fncton of Y _, ths condtonng s evalent to condtonng on Y _ = ( ). The frst and second dervatves of VAR can now be stated as: dt ( ) = E U Y = ( ) () and: dt () respectvely, where v( ) s the condtonal varance of U defned as v(y) = var[u Y _ = y], l ( ) s the frst dervatve of l( ) and n( ) s the standard normal densty (t appears here as the probablty densty of Y _ ). To relate random varable _ to the portfolo loss, we need to relate the effectve systematc factor Y _ to the orgnal systematc factors {Z k }. We assme a lnear relaton gven by: (3) where the coeffcents mst satsfy Σ k = b k = to preserve nt varance of Y _. ow, we need to specfy the set of effectve factor loadngs {a } and coeffcents {b k } to complete the specfcaton of _. Or frst step n determnng {a } and {b k } wll be the rerement that _ eals the expected loss condtonal on Y _ (that s, _ = E[ Y _ ]) for any portfolo composton. Apart from beng very appealng nttvely, ths rerement garantees that the frst-order term n the Taylor seres, gven by eaton (), vanshes for any confdence level. To calclate E[ Y _ ], let s represent the composte rsk factor for borrower as: (4) where η s a standard normal varable ndependent of Y _ (bt, n contrast to the tre one-factor case, varables {η } are nter-dependent) and ρ _ s the correlaton between Y and Y _ gven by: ρ = 0 = n ( y ) Y = b k Zk k = Y = ρy + ρ η = d dy n y v y l y = 0 y= cor Y, Y α b k k k = (5) Usng these notatons, we can rewrte the asset retrn gven by eaton () as: X rρy rρ ζ (6) = + where ζ s a standard normal varable ndependent of Y _. Therefore, the condtonal expectaton of s: E Y w p r Y = ρ µ (7) ( rρ ) By comparng eatons (7) and (8), we see that _ eals E[ Y _ ] for any portfolo composton f and only f the effectve factor loadngs are defned as: k k k = a = rρ = r α b (8) From now on, we assme that the effectve factor loadngs {a } are gven by eaton (8) and that the correcton to t ( _ ) s gven by the second dervatve of VAR (eaton ()). Whle eaton (8) s crtcal to the presented method, the choce of the coeffcents {b k } s not. The choce of {b k } specfes the zeroth-order term t ( _ ) n the Taylor seres of eaton (5). Therefore, the method wll work wth many alternatve specfcatons of {b k } that yeld t ( _ ) close enogh to the nknown target fncton vale t (). Ideally, we wold want 86 RISK ARCH 004

3 to fnd a set {b k } that mnmses the dfference between the two antles. However, fndng sch a set s not an easy task, and an alternatve, easy to calclate specfcaton of {b k } s desrable. Inttvely, one wold expect the optmal sngle effectve rsk factor Y _ to have as mch correlaton as possble wth the composte rsk factors {Y }. We can express ths ntton mathematcally by rerng that the set of coeffcents {b k } solve the followng maxmsaton problem: max c (9) ( Y, Y) bk { bk } cor sch that = k = Takng nto accont eaton (5), we can fnd the solton to ths maxmsaton problem gven by: b = ( c / λα ) (0) where postve constant λ s the agrange mltpler chosen so that {b k } satsfy the constrant. Unfortnately, t s not clear how to choose the coeffcents {c }. However, some ntton abot ther possble form can be developed by mnmsaton of the condtonal varance v(y) (more precsely, ts systematc part gven by eaton (8) below). Under an addtonal assmpton that all r are small, ths mnmsaton problem has a closed-form solton gven by eaton (0) wth c = w µ n[ (p )]. 3 Even thogh the assmpton of small r s often nrealstc and the performance of ths solton s sb-optmal, t may serve as a startng pont n a search of optmal {c }. After tryng several dfferent specfcatons, we have fond that the set gven by: () s one of the best-performng choces. We sed t n all examples dscssed below. lt-factor adjstment The remanng task s to derve an explct algebrac form for eaton (). Frst, by takng the dervatve wth respect to y n eaton (), we can wrte the correcton to t ( _ ) de to pertrbaton U as 4 : t t ( ) t ( ) = v ( y) v( y) () The frst and second dervatves of fncton l(y) rered n eaton () are obtaned by dfferentaton of eaton (8): and: (3) (4) where p (y) and p (y) are the frst and second dervatves of the condtonal probablty of defalt. The latter s gven by eaton (9), whose dfferentaton yelds: a p ( y)= n p a y a a and: k c w p r = µ r l ( y) + y l y l y = µ l y w p y + l ( y)= wµ p ( y) = y a p a y p ( y)= n p a y a a a Snce _ s a determnstc fncton of Y _, the condtonal varance of U k s the same as the condtonal varance of, that s, v(y) = var( Y _ = y). If, condtonal on Y _, ndvdal loss contrbtons were ndependent, eaton () wold be evalent to Wlde s granlarty adjstment. However, even thogh the second term n the asset retrn n eaton (6) s ndependent of Y _, t gves rse to a non-zero condtonal asset correlaton between two dstnct borrowers and j. Ths becomes clear f we rewrte eaton (6) as: (5) Wth {a } defned accordng to eaton (8), the second term (gven by the sm over k) s ndependent of Y _. However, ths term s responsble for the condtonal asset correlaton, whch can be obtaned drectly from eaton (5) and takng nto accont eaton (8) along wth the constrant Σ k = b k = : (6) Althogh ρ Y j has the meanng of the condtonal asset correlaton only for dstnct borrowers and j, we extend eaton (6) to nclde the case j =. evertheless, condtonal on {Z k }, the asset retrns are ndependent, and we may decompose the condtonal varance as the sm of systematc and dosyncratc parts: var Y = y var E Z Y y k E var Zk Y y (7) The frst term of the rght-hand sde of eaton (7) s the condtonal on Y _ = y varance of the lmtng portfolo loss gven by eaton (4). It antfes the dfference between the mlt-factor and one-factor lmtng loss dstrbtons (we wll denote ths term as v (y)) and s gven by: (8) where (,, ) s the bvarate normal cmlatve dstrbton fncton. 5 Dfferentatng eaton (8) wth respect to y yelds: (9) The second term of the rght-hand sde of eaton (7) descrbes the effect of the fnte nmber of loans n the portfolo. Ths term, whch we wll denote as v GA (y), descrbes the granlarty adjstment and vanshes n the lmt. 6 It s gven by: vga ( y)= w Y ( ( ) σ p y ) µ p y p y, p y, ρ whle ts dervatve s: X = ay + rαk ab k Zk r ξ ρ k = = = { } rr α α aa k = = a a Y j k jk j j ( ) j v ( y)= ww µµ p y p y,,ρ + = jµµ j v y ww p y j= p j y Y j ρ p ( y) ( Y ρj ) = + { } j j j= Y ( j j) p ( y) p j( y) p j ( y) + 3 The solton s obtaned by sng the tetrachorc expanson of the bvarate normal n eaton (8) (see Vascek, 998, for detals) and expandng the terms of the resltng expresson n powers of r and r j p to the second order 4 The relaton n (y) = yn(y) has been sed 5 Algorthms for evalaton of ths fncton are dscssed n great detal n Vascek (998) 6 Provded that Σ = w 0 whle Σ = w = (see Vascek, 00, or Emmer & Tasche, 003) (30) ARCH 004 RISK 87

4 Cttng edge l Portfolo credt rsk (3) Snce eaton () s lnear n the condtonal varance v(y) = v (y) + v GA (y) and ts frst dervatve, the antle correcton (we wll call t mltfactor adjstment) s also the sm of the systematc and granlarty-adjstment terms: t = t + t GA. Each term n the mlt-factor adjstment s obtaned by sbstttng the correspondng condtonal varance and ts frst dervatve nto eaton (). In the lmt, t GA vanshes, so we can nterpret t ( _ ) + t as the antle of. Expected shortfall The approxmaton developed above allows for calclaton of the portfolo VAR n the mlt-factor erton framework. Whle VAR s stll sed as a measre of rsk by most fnancal nstttons, t s known to have certan shortcomngs (see Szegö, 00, for a dscsson). As an alternatve to VAR, Acerb & Tasche (00) have proposed ES. Ignorng dscontntes of the portfolo loss rate dstrbton at ts antle t (), ES at a confdence level for s defned as the expected loss above the -antle: ES( )= E t( ) (3) Ths defnton s free from the shortcomngs of VAR and s ganng poplarty among practtoners. In ths secton, we extend or mlt-factor adjstment method to ES. Ths extenson s done smlarly to the dervaton of the one-factor granlarty adjstment for ES n Gordy (004). As has been shown by Acerb & Tasche (00), we can rewrte eaton (3) as: ES( )= (33) dst s( ) Snce we know how to calclate t s () for any confdence level s, we can jst ntegrate eaton (33) nmercally to arrve at the ES. However, we can do better than ths. If we sbsttte the antle of n the form t s () = t s ( _ ) + t s () nto eaton (33), we mmedately obtan the expected shortfall n the form: (34) where the frst term s the ES for or comparable one-factor portfolo and the second term s the ES mlt-factor adjstment, whch we wll denote as ES (). If we assme that effectve one-factor loadngs {a } are the same for all confdence levels above, we can fnd both terms n closed form. One mght arge that or defnton of {b k } n eaton (0) nvolves the coeffcents {c } (eaton ()) dependent on the confdence level, whch makes the factor loadngs {a } depend on s. However, ths problem can easly be avoded by redefnng {b k } to be the same for all s above. In the examples below, we sed {b k } defned accordng to eaton (0) wth the confdence level for all s above. To fnd ES ( _ ), we wll se the ES defnton gven by eaton (3). If we recall that _ s the monotonc determnstc fncton l(y _ ), we can wrte t as: ( ) ES ( ) = E l ( Y ) Y ( ) dy n y l y (35) Sbstttng l(y _ = ) from eaton (8) nto eaton (35) and evalatng the ntegral 7 yelds: ES GA = v y w p y ES( )= ES( ) + ds t s ( ) ( ) = w p a µ ( ),, p y Y ρ p µ y + σ Y ( ρ ) (36) To fnd ES (), we wll recall that t s () eals one half of the second dervatve of VAR. Usng the second dervatve of VAR n the form of eaton (), the second term n eaton (34) can be wrtten as: d ES ( )= ( ) ds n ( y ) dy n y v y l ( y) (37) y= ( s) Changng the ntegratng varable from s to y = ( s) yelds: = ( ) ( ) ES n v l ( ) (38) As wth the mlt-factor adjstment to VAR, the adjstment to ES gven by eaton (38) s lnear n the condtonal loss varance. Therefore, t can also be represented as the sm of the systematc and dosyncratc parts: ES () = ES () + ESGA (). As n the case of VAR, ES ( _ ) + ES () can be nterpreted as the ES of. Two-factor examples We wll se a two-factor set-p as a startng pont for or performance testng of the mlt-factor adjstment approxmaton. We assme that the loans n the portfolo are groped nto two bckets: A and B. Bcket (ndex can take vales A or B) contans dentcal loans charactersed by a sngle probablty of defalt p, expected GD µ, standard devaton of GD σ, composte factor Y and composte factor loadng r. The composte factors are correlated wth correlaton ρ. In these notatons, the asset correlaton nsde bcket s r, whle the asset correlaton between the bckets s ρr A r B. We also ntrodce bcket weghts ω defned as the rato of the net prncpal of all loans n bcket to the net prncpal of all loans n the portfolo. Indvdal loan weghts are related to bcket weghts as ω = w. Homogeneos case, =. et s look frst at the performance of the systematc part of the mlt-factor adjstment. Fgre (a) compares t 99.9% ( _ ) + t 99.9% (dashed ble crves) wth the exact 99.9% antle of 8 (sold red crves) for the homogeneos case when loans n both bckets have dentcal characterstcs. In ths example, we assme p A = p B = 0.5%, µ A = µ B = 40%, σ A = σ B = 0% and r A = r B = 0.5. The antle s plotted as a fncton of the correlaton ρ between the composte rsk factors at three dfferent bcket weghts ω A. The method performs very well except for the case of eal bcket weghts (ω A = ω B = 0.5) at low ρ. For all choces of bcket weghts, performance of the method mproves wth ρ. At any gven ρ, performance of the method mproves as one moves away from the ω A = ω B case. Ths behavor s natral becase any of the lmts ρ =, ω A = 0 and ω A = corresponds to the one-factor case where the approxmaton becomes exact. As one moves away from one of the exact lmts, the error of the approxmaton s expected to ncrease. The performance of the approxmaton s the worst when one s as far from the lmts as possble the case of eal bcket weghts and low ρ. on-homogeneos case, =. Fgre (b) compares the performance of the systematc part of the mlt-factor adjstment wth the exact solton for a non-homogeneos case. Bcket A s now charactersed by the PD p A = 0.% and the composte factor loadng r A = 0.5, whle bcket B has p B =.0% and r B = 0.. The GD parameters are left at the same vales as before. Ths choce of parameters (assmng one-year horzon) s reasonable f we nterpret bcket A as the corporate sb-portfolo (lower PD and hgher asset correlaton) and bcket B as the consmer sb-portfolo (hgher PD and lower asset correlaton). From fgre (b), one can 7 Evalatng the ntegral amonts to establshng the valdty of the relaton: z = [ ] dy n y x ay / a x, z, a whch can be verfed by dfferentatng (x, z, a) wth respect to x 8 By exact 99.9% antle of we mean a antle vale calclated nmercally, bt wthot smlatons 88 RISK ARCH 004

5 . VAR n two-factor set-p, nfnte oss antles (%) oss antles (%) ω A = Exact Approx Rsk factor correlaton (a) p A = p B = 0.5% r A = r B = 0.5 ω A = 0. ω A = 0.3 (b) p A = 0.% r A = 0.5 p B =.0% r B = 0. ω A = 0. ω A = 0.3 ω A = 0.5 ω A = 0.7 Exact Approx Rsk factor correlaton see that performance of the systematc part of the mlt-factor adjstment s excellent for all choces of the bcket weghts and the rsk factor correlaton. Ths example llstrates a general observaton that the method performs mch better n non-homogeneos cases than t does n homogeneos ones. on-homogeneos case, fnte. Snce t s mpossble to calclate a antle of the loss dstrbton exactly, we se onte Carlo smlaton as a benchmark for comparson. Table A compares the 99.9% antle calclated wth or method wth the one obtaned va a onte Carlo smlaton at varyng bcket poplaton. The comparson s made for the cases w A = 0.3 and w A = 0.7 assmng the rsk factor correlaton ρ = 0.5. As wth Wlde s one-factor granlarty adjstment, performance of the granlarty adjstment part of or method generally mproves as the nmber of loans n the portfolo ncreases. However, ths mprovement s not nform across all bcket weghts and poplaton choces. lt-factor examples ow we assme that there are more than two systematc rsk factors n the model. We wll consder a mlt-factor set-p, whch s a smplfed verson of the KV/Credtetrcs systematc factor strctre. et s assme that there are ndstry-specfc (that s, ndependent) systematc factors {Z k } k = and one global systematc factor Z. Composte systematc factors have the form: Y = αz + α Zk () (39) where k() denotes the ndstry that borrower belongs to. For llstratonal smplcty, we assme that all the loans n the same ndstry are groped nto a homogeneos bcket. 9 Ths, all loans n bcket are charactersed by the same PD p, expected GD µ, standard devaton of GD σ, composte systematc rsk factor Y and composte factor loadng r. The weght of the global factor s assmed to be the same for all composte factors: α = α. The correlaton between any par of the composte A. VAR n two-factor set-p, fnte Portfolo loss antles w A A B Approxmaton onte Carlo %.57%, %.76% %.67% %.69% %.49% %.04% %.4% 0.3.5%.5%, %.30% %.37% %.68% %.87% % 3.0% % 4.48% ote: 99.9% antles of the loss dstrbton n the two-factor two-bcket nonhomogeneos set-p wth ρ = 0.5 at varyng nmber of loans n each bcket. Accracy of approxmaton for VAR n mltfactor homogeneos set-p, nfnte Accracy of approxmaton = 50 = = 0 = Rsk factor correlaton systematc factors s ρ = α, whle the asset correlaton nsde bcket s r and the asset correlaton between bckets and v s ρr r v. As before, bcket weghts ω are defned as the rato of the net prncpal of all loans n bcket to the net prncpal of all loans n the portfolo. Homogeneos case, =. We assme that the bckets are dentcal and are poplated by a very large nmber of dentcal loans. In fgre, we show the accracy of the approxmaton as a fncton of ρ for several vales of at fxed p = 0.5%, µ = 40%, σ = 0% and r = The accracy s defned as the rato of t 99.9% ( _ ) + t99.9% to the 99.9% antle of obtaned va onte Carlo smlaton. Apart from the nose comng from the smlaton, the accracy ckly mproves as ρ ncreases. Ths behavor s nversal becase n the lmt of ρ = the model s redced to the one-factor framework. A more ntrgng observaton from fgre s that, at any gven ρ, the approxmaton based on a one-factor model works better as the nmber of factors ncreases. Ths happens becase, n the homogeneos case wth composte rsk factor correlaton ρ, the lmt = s evalent to the one-factor set-p wth the factor loadng r ρ. When we ncrease the nmber of the systematc rsk factors, we move towards ths one-factor lmt and the alty of the approxmaton s bond to mprove. on-homogeneos case, VAR and ES. ow we compare 99.9% antles and ESs of the portfolo loss calclated sng the mlt-factor adjstment approxmaton wth the ones obtaned from a onte Carlo smlaton 9 Ths assmpton s not crtcal to the approxmaton performance 0 These are the parameters we sed n the two-factor homogeneos example ARCH 004 RISK 89

6 Cttng edge l Portfolo credt rsk B. odel parameters for -factor nonhomogeneos set-p p 0.% 0.% 0.% 0.5% 0.5%.0%.0%.0%.0% 5.0% µ 50% 30% 50% 30% 50% 30% 50% 30% 50% 30% σ 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% r ω I II III ,000 ote: model parameters for the -factor 0-bcket non-homogeneos setp. The expected loss rate s 0.45% for all portfolos C. VAR and ES n -factor non-homogeneos set-p ρ mtng loss Portfolo I Portfolo II Portfolo III Approx C Approx C Approx C Approx C VAR %.5%.33%.34% 3.06% 3.09%.3%.36% %.9%.%.%.9%.9%.09%.3% %.68%.90%.90%.80%.78%.87%.9% %.47%.7%.7%.75%.65%.66%.73% 0.0.3%.6%.55%.54%.8%.54%.46%.55% Expected shortfall %.57%.76%.77% 3.55% 3.60%.77%.83% %.3%.46%.48% 3.33% 3.35%.46%.5% %.96%.8%.9% 3.5% 3.6%.6%.5% %.67%.93%.94% 3.06%.99%.88%.0% %.43%.7%.7% 3.09%.85%.6%.8% ote: 99.9% antles and expected shortfalls of the loss dstrbton for -factor 0-bcket non-homogeneos portfolos at varyng ρ for the case of 0 ndstres at several vales of the composte rsk factor correlaton ρ. The parameters of the bckets are shown n table B. All bckets have eal weghts ω = 0., so the net exposre s the same for each bcket. The comparson for three portfolos (denoted as I, II and III), whch only dffer by the nmber of loans n the bckets, s gven n table C. The nmber of loans n each bcket for these portfolos (denoted as I, II and III ) s shown n the last three rows of table B. Frst, let s compare the calclated antles and ESs for the asymptotc loss ( s the same for all three portfolos). The performance of the method s excellent even for very low levels of ρ. Smlar to the two-factor set-p, the performance of the approxmaton for n non-homogeneos cases s typcally mch better than t s n homogeneos cases (the approprate homogeneos case for VAR s shown by the ble dash-dotted crve n fgre ). Comparng the calclated antles and ESs of for portfolo I, we see that the method performs as mpressvely as t does for the asymptotc loss at all levels of ρ. Ths s becase the largest exposre n the portfolo s rather small only 0.% of the portfolo exposre. In portfolo II, we have decreased the nmber of loans n each of the bckets nformly by a factor of fve, whch broght the largest exposre to % of the portfolo exposre. The method s performance s stll very good at hgh to medm vales of rsk factor correlaton, bt s rather dsappontng at low ρ. Portfolo III has the same largest exposre as portfolo II, bt mch hgher dsperson of the exposre szes than ether portfolo I or portfolo II. Althogh the resltng loss antle s very close to the one for portfolo I, the approxmaton does not perform as well as t does for portfolo I becase of the hgher largest exposre. Conclson Analytcal methods for credt rsk of loan portfolos have been mostly lmted to one-factor models. In ths artcle, we have presented a techne for calclatng VAR and ES n the mlt-factor erton framework analytcally. Applcaton of ths techne allows one to avod slowly convergng tme-consmng onte Carlo smlatons and, at the same tme, keep all the benefts of a mlt-factor model. The techne s based on fndng a comparable one-factor portfolo whose loss dstrbton has propertes smlar to the ones of the orgnal mlt-factor loss dstrbton. VAR (or ES) for the orgnal portfolo s calclated as the sm of VAR (or ES) for the lmtng loss dstrbton of the comparable portfolo and the mlt-factor adjstment. Calclaton of the mlt-factor adjstment s based on analytcal expressons for the dervatves of VAR and s closely related to the granlarty adjstment method. The performance of the mlt-factor adjstment approxmaton s excellent throghot a wde range of model parameters, as we have llstrated by several examples. Generally, the accracy of the approxmaton mproves as the nmber of the systematc factors and/or the correlaton between the factors ncrease. Addtonally, the accracy of the mlt-factor granlarty adjstment mproves as the sze of the largest exposre n the portfolo becomes a smaller fracton of the entre portfolo exposre. chael Pykhtn s vce-presdent, rsk management, at KeyCorp n Cleveland, Oho. He wold lke to thank Ashsh Dev for valable dscssons and anonymos referees for ther helpfl comments. Emal: chael_v_pykhtn@keybank.com Acerb C and D Tasche, 00 On the coherence of expected shortfall Jornal of Bankng and Fnance 6(7), pages,487,503 Blhm C, Overbeck and C Wagner, 00 An ntrodcton to credt rsk modelng Chapman & Hall/CRC Canabarro E, E Pcolt and T Wlde, 003 Analysng conterparty rsk Rsk September, pages 7 Emmer S and D Tasche, 003 Calclatng credt rsk captal charges wth the onefactor model Workng paper, September Gordy, 003 A rsk-factor model fondaton for ratngs-based bank captal rles REFERECES Jornal of Fnancal Intermedaton (3), Jly, pages 99 3 Gordy, 004 Granlarty In ew Rsk easres for Investment and Reglaton, edted by G Szegö, Wley Gorerox C, J-P arent and O Scallet, 000 Senstvty analyss of vales at rsk Jornal of Emprcal Fnance 7, pages 5 45 artn R and T Wlde, 00 Unsystematc credt rsk Rsk ovember, pages 3 8 Pykhtn and A Dev, 00 Analytcal approach to credt rsk modellng Rsk arch, pages S6 S3 Szegö G, 00 easres of rsk Jornal of Bankng and Fnance 6(7), pages,53,7 Vascek O, 99 mtng loan loss probablty dstrbton KV Corporaton Vascek O, 998 A seres expanson for the bvarate normal ntegral Jornal of Comptatonal Fnance (4), smmer, pages 5 0 Vascek O, 00 oan portfolo vale Rsk December, pages 60 6 Wlde T, 00 Probng granlarty Rsk Agst, pages RISK ARCH 004