Dealing with Idiosyncratic Credit Risk in a Finitely Granulated Portfolio - 2 -

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2 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - -

3 Absrac The cred VaR model ha uders he Basel II Ieral Rags Based IRB aroach assumes ha he orfolo dosycrac rsk s fully dversfed aay ad hus he reured ecoomc caal deeds oly o he sysemac facor. Neverheless ha s o rue mos cases. The mac of remag frm-secfc rsks due o exosure coceraos o he orfolo VaR may be esmaed by he usage of he so called Graulary Adusmes GA. The scoe of hs aer s o mahemacally rese he ees ad mos accurae GA s hle ryg o reserve he same oao ad o summarze hch oe orks bes each arcular suao. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 3 -

4 Table of Coes Absrac Table of Coes I. Iroduco II. Basel II ad he Vascek oe-facor Model The Vascek Model III. Graulary Adusme ad Sem-Asymoc Aroaches Ad Hoc Graulary Adusme Aroach Vascek Graulary Adusme Aroach Emmer ad Tasche Graulary Adusme Aroach Sem-Asymoc Graulary Adusme Aroach SAGA Gordy ad ükebohmer Graulary Adusme GA Aroach IV. Numercal Mehods for Dervao of VaR he Vascek Model Normal Aroxmao NA Mehod Saddleo Aroxmao SA Mehod Smlfed Saddleo Aroxmao SSA Mehod Imorace Samlg IS Mehod V. Numercal Evdece ad Comarso of Exlaed Models The Sem-Asymoc Aroach The Gordy ad ükebohmer Aroach Numercal Mehods VI. Effcecy Evaluao of Exlaed Models VII. Cocluso Refereces: s of Acroyms: Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 4 -

5 I. Iroduco Gve a loa orfolo cred rsk may arse geeral from o sources sysemac ad dosycrac. Sysemac rsk reflecs he effec of uexeced chages he facal markes ad he macroecoomc codos o he borroers erformace. Borroers usually dffer he degree of sesvy o he sysemac facor bu almos o comaes are vulerable by he ecoomc codos hch hey oerae. Tha s he reaso hy he sysemac orfolo rsk s uavodable ad almos o dversfable. O he oher had dosycrac rsk mrrors he effec of borroer-secfc rsks o he borroers erformace. Those rsks are secfc for every dvdual oblgor. The more fe-graed he orfolo s he more he dosycrac rsk s dversfed aay by fe-graed e mea ha eve he larges exosures he orfolo rerese oly a small ar of he oal orfolo exosure. I he frameork of he Vascek model ha e cosder from o o s assumed ha bak orfolos are all erfecly fe-graed fely graulaed. Tha meas ha he dosycrac o-sysemac rsk has bee oally dversfed aay oher ords he caal reuremes deed oly o he sysemac rsks. O he corary he real orld he bak orfolos are ofe fely graulaed ad do o sasfy he asymoc assumo. Exosure ame cocerao he orfolo geeral may be he resul of merfec dversfcao of he dosycrac rsk due o he small sze of he orfolo or he large sze of some dvdual exosures. I case of subsaal ame cocerao he Ieral Rags Based IRB formula meoed he ex chaer oms he corbuo of he remag dosycrac rsk o he reured ecoomc caal. Ths corbuo may be assessed by meas of mehodology called Graulary Adusme GA. I realy baks use a varey of ays o measure cocerao rsk some of hem use sohscaed models ha accou for eracos beee dffere kds of exosure ohers use smler ad-hoc aroaches. Some baks use sress ess ha clude a cocerao rsk comoe. A research of BCBS 006 shos ha for he esed orfolos ame cocerao adds beee % ad 8% o he Value a Rsk VaR. I geeral he Vascek model roduces hgher rsk measures due o he fac ha does o fully accou for dversfcao beee dffere cred-yes he orfolo. O he oher had some suaos oe-facor models roduce loer rsk Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 5 -

6 measures due o he fac ha ame coceraos are o caured. The scoe of hs aer s o mahemacally descrbe he mos accurae ad rece graulary adusme mehods hle smulaeously ryg o kee he oao he same. A he ed of he aer hose mehods are comared ad he varous suaos hch each of hem may be aled are summarzed. II. Basel II ad he Vascek oe-facor Model I 988 s ake he frs se oards roducg sadard caal adeuacy for baks based o rsk measureme he Basel I Caal Accord. I geeral Basel Accords are recommedaos ad regulaos ssued by he Basel Commee of Bakg Suervso BCBS suo cossg of seor rereseaves of ceral baks ad suervsory auhores from hree coures. As he facal orld ad arcular facal markes ad srumes s flourshg ad radly chagg ever sce hs mome fe years laer baks already eeds mroved regulaos o maage her rsks. Because of ha reaso 007 he Basel II Caal Accord s roduced. I ams o mrove he recedg regulaos by meas of makg caal reuremes more rsk-sesve searag oeraoal from cred rsk ad algg ecoomc ad regulaory caal more closely o reduce he level of regulaory arbrage. Pars of he Basel II Caal Accord are he cred rsk regulaos esmag reured caal ha covers he rsk of baks cred orfolos. Ths s accomlshed by meas of he so called Ieral Rags Based IRB Aroach. The model behd IRB s ko as he Asymoc Sgle Rsk Facor ASRF model ad s based o he Vascek model roduced for he frs me 99 ad exeded by ohers lke Fger 999 Gordy 003 ec. I geeral he Vascek model s a oe facor model ha assumes ormal dsrbuo of boh he dosycrac ad sysemac rsk facors of ay cred orfolo.. The Vascek Model I hs ar e brefly rese he Vascek oe-facor model mahemacal erms for a gve cred orfolo of loas h maury T. We assume ha loa hus oblgor Belgum Caada Frace Germay Ialy Jaa uxemburg Neherlads Sa Sede Szerlad U ad US Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 6 -

7 has a robably of defaul arameers:. I geeral e ca characerze ay borroer by hree EAD - Exosure a defaul of loa - measure of oeal exosure currecy calculaed ll maury. GD - oss gve defaul of loa - he erceage of recovered case of defaul of loa. PD - Probably of defaul of loa deoed smly by loa ll o be read ul maury. We defe he defaul dcaor o loa o be EAD ha ll o be he lkelhood ha D ad o follo a Beroull dsrbuo: D h rob. f - h borroer defauls 0 h rob. f - h borroer does o defaul We also defe he effecve exosure loa he orfolo by: of loa ad he relave effecve exosure c of EAD GD ad c EAD GD EAD. Thus he absolue loss due o defaul of loa s calculaed by D. The he absolue loss o he hole orfolo s gve by he eghed average of all searae absolue losses: rel D. O he oher had he relave loss of loa s defed by c D ad he orfolo relave loss s he rel rel c D. I hs ork Vascek assumes ha he GD of each loa s 00% s defed o be loss before recoveres ad because of c hs e ge. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 7 -

8 Weer rocess: A couous-me sochasc rocess W for > 0 h W 0 0 ad such ha he creme W W s s Gaussa h mea 0 ad varace s for ay 0 s < ad cremes for o-overlag me ervals are deede. Broa moo.e. radom alk h radom se szes s he mos commo examle of a Weer rocess. Geomerc Broa Moo GBM: A couous-me sochasc rocess G he logarhm of hch follos a Weer rocess. G follos a GBM f: G µ G G W h µ - drf - volaly ad W - a Weer rocess. We assume ha oblgor defauls f hs asses value A falls belo he amou oes o he bak. The he asses value of comay may be modeled by a GBM: B ha he A µ A A W h W beg a Weer rocesses. The he aalyc soluo of he above saed sochasc dffereal euao s: A A 0 ex µ / W. Fally he asses value of borroer a maury T s gve by: A T ex log A 0 µ T T T X A 0 ex µ T T T X The sadardzed asse log-reurs X are sadard ormal varables ad ρ s he correlao beee he frm s asses ad he commo facor. Because of her o ormaly X ca be decomosed o f a oe-facor model:. X ρ Z ρ. The varables { Z... } are sadard ormal ad muually deede. I hs decomoso ρ sads for he -h loa s sysemac rsk comay s exosure o he commo facor h beg he orfolo commo facor for examle ecoomc dex Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 8 -

9 some macroecoomc facor ec. for he me ll maury. The oher ar Z ρ of he decomoso sads for he frm-secfc dosycrac rsk. The Vascek model s dely used he dusres for rsk maageme ad caal allocao. The mehod calculaes he dsrbuo of cred orfolo losses ad ses he reured caal allocaos o be eual o he 999% ercele of a lmg dsrbuo. The Assumo: Assume ha all loas are of he same amou homogeeous orfolo have he same robably of defaul ad have a commo asse correlao c / PD ρ ρ.... For a fxed commo facor e may exress he robably of orfolo loss codoal o. Thus for ay loa e ge he robably of defaul uder he secfed commo facor: Φ ρ P D [ ] Φ ρ here Φ sads for he cumulave ormal dsrbuo fuco. We may hk of hs aoher maer - every commo facor geeraes a uue ecoomc scearo ad uder every scearo e ca calculae he codoal robably of defaul ad he egh each of hem by s lkelhood. I hs ay e ge he ucodoal defaul robably for each loa. I our case s us he average of all codoal robables over all scearos. Uder he orfolo losses rel are deede ad decally dsrbued h fe varace. The aga uder he commo facor he orfolo loss rel coverges o s execao as he umber of loas he orfolo goes u o fy. I oher ords: lm P[ rel x] lm P[ x] lm P[ x ] lm Φ ρ Φ x Φ x Φ ρ hch shos he cumulave dsrbuo fuco of losses o a very large orfolo. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 9 -

10 Defo of VaR: Accordg o Basel II Accord Value a rsk VaR s he measure o calculae caal reuremes for a gve cofdece erval. The value a rsk corresodg o a cofdece erval s smly he - uale of he loss dsrbuo he dsrbuo of. I oher ords: { x P x } VaR f. Defo of VaR Corbuo: VaRC measures he corbuo of each searae oblgor o he oal VaR of he orfolo. I s mora because ca aly cosras o larger cred exosures. Brefly seakg s he codoal execao of orfolo loss s eual o he corresodg VaR: VaR VaRC E VaR E D VaR. gve ha he I ca be easly derved ha he sum of all searae corbuos s eual o he VaR of he orfolo hch s o rue for he dvdual VaR s. Tha s he ma reaso for hch VaRC s are cosdered: VaRC E[ D VaR ] E D VaR E[ VaR ] VaR. Thus Vascek 99 fally derves he VaR ad VarC formulae uder he Basel II Accord I erms of he relave loss rel sead of he absolue loss : VaR Φ Φ ρφ ρ VaRC Φ Φ ρφ ρ The raso o absolue VaR may be doe by mullyg he relave oe by he oal orfolo exosure EAD. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 0 -

11 The geeral case: I he case of a homogeeous orfolo he above descrbed covergece sll holds h oe ecessary ad suffce resrco. The he orfolo loss rel uder he codo coverges o s execao as before f he so called Herfdahl-Hrschma Idex exlaed deal laer goes o zero as goes o fy: EAD HHI 0 as. EAD Thus e eed o cosder orfolos hou ay ame coceraos fely graulaed orfolos.e. all he searae loas are of smlar sze ad oe of hem s much larger ha he ohers. For hose orfolos e ca assume ha he lmg dsrbuo gve above s a good aroxmao of he orfolo loss dsrbuo. I geeral reroduces ell he fa al ad skeess of he loss dsrbuo. Assume o ha all loas have dffere robables of defaul PD as saed he begg of he chaer ad dffere correlaos ρ. I hs case Huag e al. 007 derve he geeral VaR ad VarC formulae aga hs me erms of he absolue loss : VaR VaRC Φ Φ Φ Φ ρ ρ ρ Φ ρ Φ III. Graulary Adusme ad Sem-Asymoc Aroaches There are varous mehodologes roosed by racoers ad researchers ha deal h ame coceraos cred orfolos. I geeral hose ca be dvded o o grous mehods based o heursc measures of rsk cocerao ad mehods based o more sohscaed rsk modelg. The secod grou s o be referred f here exs feasble mlemeaos of he models. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - -

12 . Ad Hoc Graulary Adusme Aroach The smles graulary adusme s o based o ay cred rsk model. I uses he so called Herfdahl-Hrschma Idex HHI of he loa orfolo: EAD HHI EAD here EAD s as defed he secod chaer. For examle for orfolo cossg of o loas of eual amous he HHI s eual o I s cosdered ha he closer he HHI o he hgher he cocerao of he orfolo ad so he hgher he graulary adusme. The shorfall s ha here s o cosse rule ha ses arorae graulary add-o for a gve HHI. I fac emrcal sudes o sgle ame cocerao sho ha here ca be esablshed almos lear relao see Fgure beee he HHI ad he erceage graulary adusme see Deusche Budesbak 006. Aoher shorfall of hs aroach s ha does o accou for chages he cred ualy of he oblgors such chages have huge mac o he graulary adusme o he reured reserves.. Vascek Graulary Adusme Aroach I 00 Vascek rooses a graulary adusme aroach ha res o crease sysemac rsk order o comesae for he gored due o ame coceraos dosycrac rsk. The GA assumes eual robables of defaul ad correlaos of he loas. The roblem s ha for small orfolos he la of large umbers does o hold ad Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - -

13 he covergece assumo of he Vascek model also does o hold. I oher ords he rel codoal varace of he orfolo loss Var a maury T s o zero. e δ be he HHI of he orfolo. The e ca derve exressos for he codoal varace ucodoal execao ad ucodoal varace of rel a me T for deals o he dervao refer o Vascek 00. Havg hose ad he lmg dsrbuo uderlyg he Vascek model e ca derve he erceage graulary adusme o he dsrbuo of rel a maury: ρ δ ρ Φ x Φ P[ rel x] Φ. ρ δ ρ Ths aroach res o crease he sysemac rsk ad hs ay o comesae for he dosycrac rsk ha s cosdered o be fully dversfed bu s raher vague ad effce due o he fac ha boh rsk yes have dffere dsrbuos. The model s ko o erform oorly racce ad for hs reaso s o mahemacally exlaed hs aer ad o comared o he eer ad more sohscaed graulary adusme aroaches laer o. 3. Emmer ad Tasche Graulary Adusme Aroach Gordy 003 reseed a mehodology for esmag a caal add-o ha s suosed o cover he udversfed dosycrac rsk ha remas he cred orfolo. Emmer ad Tasche 003 mrove furher he aroach by develog a e formula for comug adusmes a he rasaco level. Tha formula calculaes he aral dervaves of he esmaed aroxmao of VaR ad mulles hem h he orfolo eghs. Thus e smly ge aroxmaed caal reuremes for each searae asse he orfolo assumg aga ha every loa has GD00% he so called VaR corbuos VaRC s : Noce ha he VaRC formula saed by Emmer ad Tasche s erms of relave exosures lke he oe defed by Vascek he homogeeous case ulke he VaRC s saed by he auhors of he oher GA aroaches. The raso s doe aga by mullyg by he exosure of he arcular loa. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 3 -

14 Caal Reuremes for loa c VaR c. The mehod has several shorfalls: I assumes ha he orfolo HHI hs case eual o c goes o zero as. Thus s accuracy s o sasfacory he here are very large exosures he orfolo. To deal h hs roblem e refer o he so called Sem-Asymoc Aroach descrbed laer he chaer. I does o maury-adus he u arameers. The formulae are oo comlex ad hard o mleme. I does o accou for dosycrac recovery raes. The loa recovery rae s ormally eual o oe mus he loa GD erceage. The model assumes fxed GD00% ad hus decal execed GD s for each loa. To roduce recovery rsk he GD s should be dra from some roer dsrbuo lke he gamma dsrbuo. I geeral he model may be exeded o cover hs ssue bu he he formulae ge eve more comlex ad hard o mleme. We sk he mahemacal exlaao of hs model because of he above saed shorfalls. We coue h he sem-asymoc graulary adusme aroach hch s much smlfed ad easer o mleme. 4. Sem-Asymoc Graulary Adusme Aroach SAGA The aroach ha e descrbe hs seco s develoed by Emmer ad Tasche 005 ad s a alerave aroach for measurg caal reuremes for loas h hgher coceraos ad hus s aled oly o ar of he orfolo caurg fully he effec of hose coceraos o he orfolo rsk. e us frs roduce he e varables e eed order o mahemacally rese he model. We defe he eve defaul ha loa defauls because s value falls belo some crcal hreshold r : ED eve of Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 4 -

15 ED { ρ Z r } ρ Assume hou ay loss of geeraly ha ad ρ τ ρ ρ for... Z Z r a r r for.... We also assume ha he GD of each loa s 00% c c c ad lm HHI lm c 0. If e defe ED { τ Z a} τ o be he eve of defaul of he frs loa ad r ρ y X P Z e ca he gve he follog formulae for he orfolo loss y ρ ad he so called sem-asymoc erceage loss fuco P : c { } c c τ τ Z a { ρ ρ Z r} P c cd c X A roof of he laer euao s o he scoe of hs hess ad ca be foud he aedx of Emmer ad Tasche 005. I geeral gves us a model o aly searaely for each loa ha may be cosdered a shorfall of he aroach. The sem-asymoc caal reureme a level of he loa h exosure c hs may be ay loa he orfolo s he defed as: cp[ ED P c P c]. Havg hose searae asse charges e ca add hem u ad ge he caal reuremes for he hole orfolo. The good hg abou hs mehod s ha accous for he exosure c oher ords s o orfolo vara. The las hg o do s o derve he soluo for he gve caal charges euao. e us exress he codoal robably of X gve D by: x [ X x D ] F x P f d x R 0 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 5 -

16 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo gve ha F ad f are he codoal dsrbuo ad desy fucos of X resecvely ad D s he defaul dcaor of he frs loa. The e ca easly derve he dsrbuo fuco of he erceage loss: c z F PD c c z F PD z P P o ] [ ad s desy fuco gve by he frs dervave of he dsrbuo fuco: c z f PD c c z f PD c z z P P ] [ 0 h PD PD beg he robably of defaul of he cosdered loa our case he frs loa. Oe ca use hese formulae o umercally derve he -uale of he erceage loss c P. Gve our assumos abou he dsrbuos of he facors ad Z remember e assumed ha hey are boh deede ad sadard ormal e ca fally derve he formula for he erceage caal charges for he loa uder cosderao: c z f PD c c z f PD c c z f PD c z c ED P P c VaRC ] [ 0. Because of he dsrbuo assumos e ca o exlcly derve he codoal deses f ad use hem he above exresso: 0 oherse 0 Φ Φ Φ z r z PD z f PD z f ρ ρ τ τ 0 oherse 0 0 Φ Φ Φ z r z PD z f PD z f ρ ρ τ τ

17 h ρ ex f z ρ 0 oherse Φ z ρ Φ z r ρ z 0. Hoever here s oe maor shorfall of value a rsk VaR as a orfolo rsk measure. I he case PD < e may cocerae all he exosure o he sgle loa h hs lo robably of defaul ad hs s gog o reduce he caal charges. To avod hs e may cosder dffere ad souder measures lke Execed Shorfall ES sead of VaR. More deals abou hs lmao of VaR are gve Emmer ad Tasche 005 ad Tasche ad Theler Gordy ad ükebohmer Graulary Adusme GA Aroach I hs ar e roduce a graulary adusme suable o use uder Pllar of Basel II Basel Commee o Bak Suervso 006 develoed by Gordy ad ükebohmer 006. I s a CredRsk - based aroach smlar o he oe ublshed by Emmer & Tasche 005. The smlary comes from he fac ha boh models are based o he basc coces ad al form of GA roduced by Gordy 000 for alcao Basel II ublshed 003 ad laer refed by Wlde 00b Mar & Wlde 003 ad ohers. I rcal he adusme ca be aled o ay rsk-facor model ad s adaed o he laes chages he defo of regulaory caal falzed Basel II dsgushes beee uexeced loss U caal reuremes ad execed loss E reserves. Ths meas ha he graulary adusme for caal reuremes should be vara o E ad should o be exressed erms of E lus U as he as. To esmae he reured value a rsk for he gve orfolo e eed ha s he - h ercele of he dsrbuo of he orfolo loss. The IRB formula esmaes he -h ercele of he codoal execed loss - E[ ] ad hus e defe he exac graulary adusme o be eual o he dfferece: GA E[ ]. Ths exac adusme cao be obaed bu ca be aroxmaed by a Taylor seres orders of Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 7 -

18 /. e µ E[ ] ad Var[ ] are defed o be he codoal mea ad varace of he orfolo loss ad h s he robably desy fuco of. For ε e ca exress he orfolo loss as: µ ε µ. We use secod order Taylor exaso a ε 0 o aroxmae he - uale of he loss as dd Wlde 00b: ε µ [ µ ε µ ] ε 3 0 [ µ ε µ ] 0 ε ε O ε Goureroux 000 shos ha he frs dervave he exaso vashes ad he secod dervave s exacly he frs-order graulary adusme gve by he formula: y h y GA h y y µ y. We chage oao slghly oly for hs chaer ad se he eghs rooro of he exosure of loa o he oal exosure of he orfolo: The GD D ad aga o be he EAD. EAD become he relave losses of loa ad he hole orfolo resecvely. We ca exress our graulary adusme erms of rue E ad U ad her asymoc aroxmaos as as E ad U : E[ ] U E U E[ E[ ]] U U as as because E E[ ] E[ E[ ]]. The exressos µ ad deed o he model for he orfolo loss. I ould be desrable o calculae he GA based o he Vascek model bu he he codoal mea ad varace are oo comlex for suervsory alcao. Ths mles ha e should use a drec mehodology e ally base he calculaos of he exressos o a dffere model ad he chage he u arameers a secfc ay o resore he cossecy of he overall adusme as much as ossble. The chose al model s he Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 8 -

19 CredRsk. I geeral s sadard verso he recovery raes ad hus he GD s are assumed o be deermsc. Hoever Gordy ad ükebohmer allo her model he GD s o be radom varables. Accordg o he robables of defaul are gve by: CredRsk model he codoal P[ D ] PD y PD l l y here l s defed as a facor loadg ha corols he sesvy of borroer o he commo facor hch e assume s gamma dsrbued h mea ad osve varace dcaor ad ha ξ. Also as ar of he model e assume a Posso dsrbuo of he defaul GD s deede of for every loa. Thus e ge: µ y E[ y] µ y ad y Var[ y] y here: µ y E[ y] EGD PD y EGD PD l l y y Var[ y] E[ GD D y] EGD PD y I he gve exressos devao of dsrbuo so: µ E[ GD ] E[ D y] y. EGD ad VGD sad for he execed value ad he sadard GD resecvely. For he defaul dcaor e have assumed a Posso E[ D ] Var[ D ] PD E[ D ] PD PD. We ca also subsue [ GD ] Var[ GD ] E [ GD ] VGD EGD E ad ge: VGD y VGD EGD PD PD µ y Cµ µ EGD Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 9 -

20 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo here EGD VGD EGD C. By defo of he CredRsk model he E reured reserves for loa are: PD EGD R ad he U reured caal s gve by: ] [ l PD EGD E I he CredRsk frameork e have: R µ µ EGD VGD C µ µ. Usg hose ad gve he graulary adusme formula e ca o subsue everyhg e have ad ge : y y y y h h GA µ µ h y C y y y µ µ µ µ. Thus a he ed e have he follog exresso for he graulary adusme: EGD VGD R C EGD VGD R R C GA δ δ

21 here s he reured caal er u exosure for he hole cred orfolo ad here ξ δ ξ s a regulaory arameer hrough hch he varace arameer ξ of he sysemac facor affecs he GA. Accordg o Basel II e se 999% ad ξ 0 5 hs value deeds o he calbrao mehod used ad hus e ge δ To someha smlfy he GA formula e assume ha all secod-order erms may be omed because he uaes hey add o he GA are oo small. Thus e ge he smlfed formula he accuracy of hch s evaluaed laer: ~ GA * C δ R. I he model dscussed above here s oly oe mlemeao challege he aggregao of mulle exosures o a sgle oe for each borroer. Gordy ad ükebohmer 007 roose a ay o deal h hs comlcao he baks are alloed o calculae he GA based oly o he se of larges exosures he orfolo m ou of. Sll he smaller exosures fluece should also be ake o accou ad hs s accomlshed by seg a uer boud of he GA for hem. From he suervsory o of ve ha s o a very cosse mehod because he uer boud mos cases s bgger ha he real GA ad ever smaller. IV. Numercal Mehods for Dervao of VaR he Vascek Model I hs chaer e descrbe he umercal mehods for comuao of value a rsk VaR he Vascek oe-facor orfolo cred loss model evaluaed by Huag e al I he ex o chaers hose models are comared ad her effcecy ad characerscs are dscussed. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - -

22 . Normal Aroxmao NA Mehod Ths mehod s derved for he frs me by Mar 004. Mos smly u s us a alcao of he Ceral m Theorem CT. CT: Uder some codos he average or sum of a large umber of deede radom varables s ormally dsrbued. Oe of he codos for he CT o hold s ha he radom varables should be decally dsrbued. I our case ha codo s o sasfed bu acually s reured oly o make he roof feasble. I geeral he CT ales o he case of o-decal dsrbuos as log as he se of dsrbuos s bouded erms of mea ad varace for deals - WolframMahorld. Smly u he CT ales for ay dsrbuo of fe varace ad our case he codoal varace used belo fulfls he crero. We o ould lke o use he CT o aroxmae he orfolo loss codoal o he commo facor. e us assume ha s ormally dsrbued h mea ad varace µ. The varables EAD GD ad P[ D ] are gve as he Vascek model chaer I ar. We ca o easly exress he codoal al robably by: µ x P[ > x ] Φ ad hus by egrag over e ca derve he ucodoal al robably: µ x µ y x P [ > x] E Φ Φ φ y dy. x y Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - -

23 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo I he above euaos Φ deoes he cumulave dsrbuo fuco ad φ deoes he robably desy fuco of he sadard ormal dsrbuo. I order o umercally aroxmae he egral ad calculae he reured VaR e cosder he so called Dscree Fourer Trasformao DFT hch geeral urs he egral o a fe sum ha s calculaed sead. For more deals o he dscresao rocedure refer o Mar 004. I order o calculae he VaRC of a gve oblgor e eed o comue he dervaves of he al robably ] [ x P > of he codoal mea µ ad of he codoal varace of he orfolo loss h resec o he asse allocaos. Those dervaves are gve by he exressos: > ] [ x x x E x P µ φ µ µ h µ ad. We a he al robably ] [ VaR P > o be fxed a ad x VaR o vary. Ths meas ha he lef had-sde of he above euao vashes for x VaR : 0 VaR VaR E VaR VaR E µ φ µ φ µ µ x E x x E VaR µ φ µ φ µ µ

24 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo x E x x E µ φ µ φ µ. We fally derve he VaRC for oblgor o be: VaR VaRC.. Saddleo Aroxmao SA Mehod The saddleo aroxmao SA mehod s based o he exsece of he so called aalyc Mome Geerag Fuco MGF hch e use o geerae he loss dsrbuo of he cred orfolo ad s ko o aroxmae small al robables accuraely. Mome Geerag Fuco: For a gve radom varable hch aga rereses he loss of asse e defe he MGF as he uko aalyc fuco ] [ e E M. If... ad s he oal orfolo loss he he MGF of s gve by: dx x f e E M M x ] [ex h f beg he desy fuco of. Cumula Geerag Fuco: We roduce also he Cumula Geerag Fuco CGF of by smly akg he logarhm of he MGF: log M. We use o exress he verse MGF of ad hus s dsrbuo also ko as he Bromch egral or he Fourer-Mell egral hch e derve usg Fourer verso for deals refer o Taras 005: x x d x d e d M e x f ex ex π π π here.

25 The Drec Saddleo Aroxmao: The use of saddleo aroxmao for cred orfolos aeared he leraure for he frs me aroud 00. I he begg as aled by Mar 00a b o he ucodoal MGF of he orfolo loss. The MGF s uue hus he Bromch egral exss bu geeral s o aalycally racable. Tradoal aroaches for solvg lead o ll-codog ad rucao errors. Thus saddleo aroxmao SA arses hs seg o gve a accurae aalyc alerave for f hch avods calculag he egral. For dealed descro of saddleo aroxmaos refer o Jese 995. e ~ be he o a hch x s saoary. I s he called he saddleo ad s gve by he soluo of ~ x he dervave s h resec o. The ma dea of he mehod s ha he desy fuco f ad he ucodoal al robably P [ > x] ca be aroxmaed by he CGF ad s dervaves a ~ u o some order. For ha urose e use Taylor exaso of he fuco x as a fuco of aroud he saddleo ~ : x ~ ~ ~ ~ /3 ~ 3 x / ~.... The -h dervave of he MFG s gve by M E[ e ] ad hus: M M M M ; M M ad so o. There are a fe dffere aroxmaos ha have bee suded already deedg o he order of he Taylor exaso. The formula for he desy e cosder s derved by Daels 987 ad he oe for he al robably - by ugaa & Rce 980. Boh of hem use fourh-order Taylor exaso. These formulae are gve by: f x φ zl ~ ~ 5 4 ~ 3 4 ~ 8 ~ O Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 5 -

26 P[ > x] Φ z l φ z l z z l O 3/ h z sg ~ [ x ~ ~ ] ad z ~ ~. l The Idrec Saddleo Aroxmao: I he revous ar e aled he SA o he ucodoal MGF of he orfolo loss bu he assumo of muual deedece of all s he volaed. Aaer 006 shos ha hs case he mehod s effce ad he resuls are rog for orfolos h exosure sze of hgh skeess ad kuross. Dese hose cos he model s smle ad ca be easly exeded o he oe e meo he ex ar - he Smlfed Saddleo Aroxmao SSA. To avod he shorfalls of he Mar s mehod Huag 006 ales he SA mehod o he codoal MGF of he orfolo loss uder. The all he searae losses are deed deede. The laer aroach s called Idrec SA ad s mosly used racce a he cos of some brue comuaos for every realzao of he commo facor e eed o esmae he saddleo ~ from he euao ~ x. I le h he revous aragrah Huag 006 defes he codoal MGF of o be: M e. No he codoal CGF ad s frs four dervaves ca be defed as: log e e e [ e e ] Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 6 -

27 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo e e e e ] [ ] [ e e e e e e ] [ 6 ] [ 6 ] [. Oce e have calculaed he codoal CGF ad s dervaves e may use saddleo aroxmao o esmae he codoal desy ad al robably of he orfolo loss as exlaed he drec SA. For ] 0 [ x e have a uue saddleo ~. We ge he ucodoal loss desy ad al robably by egrag over for examle > > ] [ ] [ ] [ dp x P x P. We ca calculae he VaR by verg he loss dsrbuo. To calculae VaRC Huag 006 dffereaes he al robably h resec o he effecve exosures : > d x x E x P ex ] [ π. Jus lke for he ormal aroxmao he revous ar f e relace x h VaR he above formula he lef had-sde becomes zero because he al robably > ] [ VaR P. Tha s ho e fally derve he VaRC : d VaR VaR E ex 0 π d VaR E d VaR E VaR VaRC ex ex

28 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo [ ] ] [ ex ex ex VaR f E d VaR E. If e defe he CGF of codoal o ad D as: e e log log ˆ he e ca smlfy he formula for he VaRC o: [ ] ˆ ex VaR f E d VaR E VaR VaRC. Aoher ay o derve he VaRC exlcly s o se D ˆ o be he loss of he orfolo excludg oblgor. The e ge a dffere formula: ]] [ [ ] ] ˆ [ [ ] [ ] ; [ ] [ VaR f E VaR f E VaR f D VaR f VaR D E VaRC h f beg he codoal loss desy of he orfolo excludg oblgor hch ca also be aroxmaed by drec SA. The o formulae for he VaRC are he same he sese ha hey boh are aroxmaed by saddleo aroxmaos h he same saddleos. The deomaors are aroxmaed by he saddleo derved from x X ~. The omaors are aroxmaed usg he saddleo ~ eual o he soluo of he follog euao: k k k k k k k VaR ex ex. Alhough e assume heerogeey of he oblgors s more effce o grou he oblgors o buckes of smlar characerscs as much as ossble esecally for larger orfolos.

29 The he comuao of he CGF ad s dervaves s more smlfed ad he umber of searae VaRC s s loer. 3. Smlfed Saddleo Aroxmao SSA Mehod The smlfcao comes from omg he calculaos of ~ for every realzao of. Isead e calculae ~ oly. I geeral hs s ossble f e assume ha boh values are close o each oher hus savg sgfca amou of calculao me. Of course some cases he model s less accurae f he assumo s srogly volaed. Bascally hs mehod as frs reseed by Mar 00b ad he furher develoed durg he ex years. Mar assumed ha all oblgors are deede ad derved a exlc formula for he value a rsk corbuo of oblgor h beg he CGF of he orfolo loss : VaRC ~ ex ex ~ ~ ~. If he oblgors are codoally deede sead of deede e ca exed he model ad derve he SSA soluo hch f VaR should be comued by drec SA: VaRC ~ ex E f VaR ex ~. E [ f VaR ] For deals abou he dervao of he gve formula refer o Aoov Imorace Samlg IS Mehod The mehod s ally suggesed by Glasserma & 005 ad Glasserma 006 as a mehod for VaR ad VaRC calculao ad s laer adoed by Huag 007. Ths arcular IS s a rocedure of rare eve smulao for cred rsk maageme h ma dea o crease he freuecy of rare eve occurrece by chagg he dsrbuo from hch e samle. I hs frameork caurg he deedece beee oblgors he orfolo s of grea morace. I he laer descrbed aroach hs s accomlshed by usg he ormal Gauss coula model hch s dely used by racoers oadays. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 9 -

30 Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo We assume ha he ucodoal PD s are gve as ell as he effecve exosures. I he Gauss coula model he deedece beee he oblgors s modeled by roducg deedece amogs he defaul dcaors D for deals - Glasserma & 005. Thus he used formula for he codoal defaul robables s he already famlar oe from he revous ars he case of oe commo facor: Φ Φ D ρ ρ ] [. I geeral he IS mehod cosss of o ars ad s called o-se IS: Exoeal Tsg chage of he dsrbuo of he codoal PD s order o crease hem ad make losses less rare eves. Mea Shfg shfg he mea of he commo facor for he same reaso as exoeal sg. Exoeal Tsg: The urose of exoeal sg s o crease s he codoal defaul robables o he e robables gve by he formula belo order o crease he esmaes of our al robables ad make he defauls less rare: ] [ex ex θ θ θ h 0 > θ. Thus e oba a e robably measure Q oce ha he hgher he exosure he hgher he crease he defaul robably. We see ha for 0 θ he e robables are he same as he old oes ad for 0 > θ here s deed a crease. There are may ossbles o crease he PD s bu hs oe has secfc feaures ha make more effecve. Oe of hem s he resulg form of he lkelhood rao ha correcs for chagg hose robables. I s gve by: D D D e e θ θ θ θ.

31 Gve ha he defaul dcaors D are deede uder he lkelhood rao for chagg all PD s s smly he roduc of he dvdual lkelhood raos: e θd h log E[ ex θ y] θ e ex θ θ θ log e θ beg he codoal ~ CFG of. Glasserma & 005 sugges ha θ y be he uue soluo of he euao uue soluo deed exss due o he fac ha he dervave s mooocally creasg θ : θ y y θ x. ~ If x > E[ y] he θ y > 0 ad he e sed PD s are hgher ha he orgal oes. I he case x E[ y] o exoeal sg s eeded. Mea Shfg: I geeral e have assumed ha he commo facor s sadard ormally dsrbued. e us defe a robably measure R uder hch he commo facor s ormally dsrbued h mea µ 0 ad varace sll eual o oe so e have he raso N 0 N µ. The dea s o arfcally crease he mea of he dsrbuo ad hus also he robably P [ > x] order o make he losses less rare. Afer ha mea shf e eed o correc for hs chage of he dsrbuo by mullyg each observao by a gve lkelhood rao. Glasserma 006 saes ha he rao s eual o µ / ex µ ad hus e ge he exresso for he correced al robably uder R gve he orgal PD s: R P[ > x] E [{ > x} ex µ µ / ]. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 3 -

32 The effecveess of he IS deeds o a grea degree o he choce of µ. The value of µ roosed by Glasserma 006 s he soluo ~ µ hch s uue uder he oe-facor ~ ~ ~ model of he euao: max{ θ y y θ y x / y } for θ 0. y Calculag he VaRC s: Usg he o ars exlaed above e ca derve he al robably uder he e robably measures Q ad R usg he e robables of defaul ad he e dsrbuo of he commo facor : I ha case he Q P[ > x] E E { x} θ D D θ > Q { E [ ex θ ]} E > { x} θ [ ex θ θ ] R Q E { ex µ µ / E { > x} }. VaRC of oblgor ca be drecly exressed by he formula: VaRC k D l k k { VaR }. k l k { VaR } k I he gve formula he subscr k deoes he k -h smulao ad l sads for he combed lkelhood rao a mullcao of he raos of he shfg ad of he sg: l ex[ µ µ / θ θ ]. V. Numercal Evdece ad Comarso of Exlaed Models I hs ar of he aer are reseed he umercal resuls derved by he auhors of he descrbed graulary adusmes ha ll hel us o comare hose echues ad ll reare us for he ex ar here e are gog o fally vesgae hch of hose echues are referable ad ha suaos. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 3 -

33 . The Sem-Asymoc Aroach To demosrae he model descrbed ar III.4 Emmer & Tasche referred o as he auhors ll he ed of he chaer cosder a orfolo drve by sysemac rsk h a average cred sadg E X 005 Φ r ad asse correlao ρ. The ma dea s o elarge he orfolo by a addoal loa exressed by he dcaor D he euao gve before - P c cd c. Ths loa has a hgh cred orhess - P ED 000 Φ a ad asse correlao τ. I le h he Basel II roosal for cororae loa orfolos BCBS 005 he auhors se ρ 0 54 ad τ 09. They ould lke o lo he rsk calculaed by meas of VaR a cofdece 999% of he orfolo loss varable P c as defed he above euao as a fuco of he relave egh c of he addoal loa. They lo hree VaR s for comarso: he rue VaR he graulary adusme aroxmao of he VaR ad he VaR accordg o he Basel II Accord for deals o he exac comuao ses refer o Emmer 005. O Fgure. e see ha for u o % boh aroxmaos are ue accurae comared o he rue VaR. For hgher relave egh of he e loa hey sar losg ualy very fas. The Basel II VaR becomes more ad more accurae also comared o he graulary adused VaR. For examle for c of abou 4% s abou 7 mes more accurae. The reaso for ha sudde dro of ualy s of course he assumo of boh Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

34 mehods ha he orfolo has o ame cocerao. Obvously for c hgher ha % hs s o he case. Aoher asec ha he auhors cosder s a comarso of relave VaRC s of he e loa o he orfolo rsk cosdered also as fucos of c. They lo o Fgure. hose corbuos o he rue VaR he GA VaR as ell as o he Basel I ad II VaR s. The Basel I curve crosses he real corbuo curve a he mmal rsk orfolo for relave egh * c of abou 7%. Aga for c u o % he GA corbuo s ue accurae comared o he real corbuo bu for hgher eghs sars losg ualy fas. The Basel II curve s o a bad aroxmao of he rue VaRC as ell for relave egh of he e loa u o 5% % alhough o as good as he GA VaRC bu afer ha becomes ay oo accurae also comared o he GA VaRC. We also oce ha boh he GA VaRC ad he Basel II VaRC curves le belo he Basel I curve. Tha may gve us he rog feelg ha hgher relave egh of he addoal loa may sll mrove he overall rsk of he orfolo. To rovde a reasoable argume agas ha clam e ake a look aga a Fgure. ad e oce ha for a hgher ha 7% he VaR of he orfolo s uacceably hgh. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

35 . The Gordy ad ükebohmer Aroach To demosrae he graulary adusme aroach descrbed seco III.5 Gordy ad ükebohmer referred o as he auhors ul he ed of he chaer aly o a umber of realsc bak orfolos. The daa used s rovded by he Germa cred regser ad cosss of loas greaer ha or eual o 5 Mllo Euros. The orfolos hey use coa more ha 50 exosures ad hey beleve ha s a arorae sze for calculag he GA. Those orfolos are dvded o 4 grous accordg o he umber of exosures ad degree of heerogeey: large more ha 4000 exosures medum exosures small exosures ad very small The PD of each loa s redeermed by he S&P s cred rags. The auhors calculae he GA as erceage of he oal orfolo exosure for he gve orfolo grous he resuls are gve o Fgure 3. ad comare o he HHI. We oce ha for he large orfolos he GA s o 4 bass os bu gros subsaally for he very small orfolos u o 6 bass os. Also a hgh correlao h he HHI s observable. The correlao s o erfec hough due o he fac ha he HHI s o sesve o he PD of he exosures ad hus o he cred rags. I her resuls he auhors also clude a bechmark referece orfolo of 6000 loas each h PD 0 0 ad EGD 0 45 ad of homogeeous EAD. We oce he morace of heerogeey for calculag he GA by comarg he GA for he bechmark ad for he larges orfolo. Alhough he umber of exosures of boh orfolos s bg eough he GA of he bechmark orfolo s abou sx mes smaller due o s homogeey ame coceraos. Aoher hg Gordy ad ükebohmer calculae s he VaR of he large ad medum orfolos by meas of he CredRsk model ad he GA as erceage of hs VaR. For Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

36 he large orfolos he GA add-o s beee 3% ad 4% ad for he medum oes s beee 5% ad 8%. Tha s evdece ha GA s hgher for smaller orfolos as erceage of he VaR. Nex he auhors comue he GA for he large medum ad small orfolo grous as erceage of he Basel II Rsk Weghed Asses RWA used for esmao of IRB caal charges. The resuls are gve Fgure 3.. As sho Fgure 3. he GA for he bechmark orfolo s abou 008%. The IRB caal charge for hs same orfolo he caal reuremes ha accou for he sysemac rsk of he orfolo s 586% of he oal orfolo exosure. Thus he auhors derve ha he relave add-o due o graulary s abou 03% of he RWA. Same mehodology s used o calculae he add-o for he hree real orfolo grous. Aga he heerogeey of he large bak orfolos lays mora role calculag he GA he GA for he large orfolos s more ha 3 mes hgher ha ha of he referece orfolo. Neverheless he GA s ue small for he large orfolos ad also for some of he medum oes bu hghly sgfca for he small orfolos u o 3% of he RWA. Aoher mora ssue s o vesgae he aroxmao ualy of he smlfed GA G A ~ comared o he full GA. For ha reaso Gordy ad ükebohmer cosder sx orfolos of 000 exosures each cosa PD eher % or 4% ad EGD The orfolos dffer by degree of heerogeey P0 s comleely homogeeous ad P50 s h hgh ame cocerao he larges exosure 50 A accous for 5% of he oal exosure. The resuls are gve o Fgure 3.3 belo. I geeral e oce ha he error creases h cocerao ad PD bu s really rval for realsc orfolos. Eve for P0 ad PD of 4% he error s 3 bass os ad for he exreme case P50 h PD4% he error s 4 bass os bu boh are small comared o he oal GA. Tha Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

37 roves he clam ha he smlfed GA s a good aroxmao ha may be used racce. As saed earler oe maor draback of ad hoc GA mehods s he fac ha hey do o accou for he PD s of he loas bu ha s o he case h he model roduced by Gordy ad ükebohmer. Fgure 3.4 shos he deedece of he smlfed GA o he defaul robably. The GA s calculaed erceage of he oal orfolo exosure for homogeeous orfolos of 00 borroers h dffere PD s. We oce ha for crease of he PD from zero o 0% he GA becomes abou 04% hgher. I ar III.5 e exlaed ha he varace arameer ξ of he sysemac rsk facor may dffer accordg o he calbrao mehod e cosder. For ha reaso he auhors exlore he effec of dffere ξ values o he GA. O oe had he GA s creasg h ξ Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

38 for examle for ξ gog u from 0 o 03 he GA creases by 0%. O he oher had he absolue degree of he GA s really sesve o chages ξ bu he relave oe across bak orfolos s fac o ha sesve a all. Tha s hy e coclude ha geeral he recso of he calbrao of ξ s o ha crucal for he effce fucog of he GA as a suervsory ool. 3. Numercal Mehods I hs ar e vesgae he ualy of he mehods descrbed chaer IV by meas of umercal resuls esmaed by Huag Ooserlee ad Mesers referred o as he auhors ul he ed of he chaer. They use a sylzed orfolo of lo graulary meag ha he larges oblgor has a lo hgher exosure ha he smalles oe cossg of 35 oblgors. I s dvded sx buckes of dffere exosure sze see Table 4.. The oal exosure of he orfolo s Exosure coceraos are o sgfca due o he fac ha he egh of he larges oblgor s less ha 5%. The asse correlao ρ s 0% ad he PD s 033% for each loa. Bucke: Exosure: # of Oblgors Table 4.: The sylzed orfolo used o evaluae umercal models for comuao of VaR. I Huag 007 may be observed addoal deals o he comuaos comared laer o. For he res of he ar by Vascek e deoe asymoc aroxmao of he Vascek model. NA ad SA sad for he ormal ad saddleo aroxmaos resecvely. By IS-0 e deoe morace samlg h e housad scearos hch are dvded o 0 samles of eual sze order o esmae VaR ad samle sadard devaos. For a referece bechmark mehod he auhors use a Moe Carlo smulao based o e subsamles h 6 Mllo scearos each. By CI e deoe he 95% cofdece ervals. I Fgure 4. e comare boh 999% ad 9999% VaR s calculaed by he above meoed mehods. Comuao CPU me s reored secods. I brackes ex o he VaR esmaes e sho also he samle sadard devaos. The ma hg e oce by frs Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

39 lookg a he resuls s he fac ha eve for he sylzed orfolo of lo exosure cocerao he Vascek resuls a boh cofdece levels are far from he bechmark resuls relave errors abou 5%. The ormal aroxmao ouus are beer a he cos of small addoal comuao me he relave errors are less ha %. The saddleo aroxmao resuls are eve beer hey are boh he 95% cofdece erval of he bechmark ad he relave errors are less ha 0% ad he mehod remas rey fas. The resuls from he morace samlg are ue smlar o hose of he SA bu he comuaoal coss are sgfcaly hgher. For all he umercal mehods he auhors esmae he VaRC of a oblgor a slghly dffere maer ha exlaed chaer IV scaled by he effecve egh of he oblgor hus he VaRC s are exressed as erceage of he oblgor s effecve exosure ad he values are all beee 0 ad.e. VaR P[ c D VaR]. O Fgure 4. e gve he VaRC s for each bucke a o dffere loss levels ad We see ha he Vascek aroxmao smly does o ork for calculag he VaRC s ad o resuls fall o he 95% cofdece erval. The ormal aroxmao s also ue accurae h 7 esmaes ousde he cofdece erval ad 7% dfferece h he bechmark. Imorace samlg has 5 oucomes ousde he cofdece erval ad 4% dfferece h he bechmark. A good mehod here s he smlfed saddleo aroxmao h oly o resuls ousde he cofdece erval ad 4% dfferece h he bechmark. A ssue h IS s ha he calculaed corbuos do o crease mooocally h he effecve egh ad for ha reaso e coclude ha 0000 scearos are o eough. The mos accurae model s he full saddleo Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

40 aroxmao h all esmaes sde he cofdece erval ad a maxmal error of 033%. O he oher had he SSA s abou seve mes faser ha he full SA. As saed earler he orfolo cosdered ul o had o serous ame cocerao. We o roceed h aoher esg orfolo used by Huag Ooserlee ad Mesers for hch ha s o he case. I cosss of oe bucke of 000 oblgors h exosure ad a secod bucke of oly oe oblgor h a hgher exosure S he auhors cosder o cases - S 0 ad exreme cocerao S 00. We kee ρ 0% ad PD 033%. As a bechmark s used Bomal Exaso Mehod roduced Huag 006. To vesgae he accuracy of he models case of hgh ame cocerao he auhors comare he 9999% VaR erms of relave error h he bechmark. The resuls are rovded Fgure 4.3. For S 0 e oce ha all mehods exce he Vascek aroxmao gve relavely small errors belo %. Thgs chage dramacally he case S 00 : he Vascek ad ormal aroxmaos are hghly accurae hle he SA ad IS mehods rema h he % error boud. The SA s also esed for S u o 000 ad he resuls Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

41 are sll smaller ha %. Tha holds also for he SSA hch s o cluded he resuls bu s as accurae as he SA ad much faser as ell. For he laer orfolo he auhors also vesgae VaRC aroxmaos ad Fgure 4.4 e rese he resuls boh for a small VaRC ad for a large VaRC oblgors; S 0 ad S 00 are cosdered aga. Aga he resuls of he Vasvek mehod are far from he realy. The NA gves relavely accurae resuls for S 0 bu hghly accurae oes for S 00. SA s accurae for VaRC bu o really accurae for VaRC ad S 00 he absolue error exceeds %. The SSA rovdes smlar resuls o SA for VaRC bu for VaRC he error s oo hgh above 5%. The bes mehod here s he IS hch gves absolue errors belo % all cases. VI. Effcecy Evaluao of Exlaed Models I hs fal ar e are gog o alk abou he effcecy ad robusess of he mehods descrbed before ad o meo her ros ad cos. The sem - asymoc aroach exlaed seco III.4 descrbed by Emmer ad Tasche 005 geeral s a exeso of he aroach roduced by Gordy 003 ha accous for he exac eghs of he orfolo asses. Aroxmao of he caal charge for every dvdual asse may be calculaed by esmag he aral dervaves of he orfolo loss uale ad mullyg hem by he asse eghs. The aroach also Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 4 -

42 accous for coceraos bu he resuls are couer-uve case of very small robably of defaul of he cosdered loa. Ths draback ca be avoded by usg cohere rsk measures lke Execed Shorfall. The mehod roduced by Gordy ad ükebohmer 007 s a revso some echcal advaces ad modfcaos ad exeso of he mehodology roduced he Basel II Secod Cosulave Paer. Afer horough sudes of he model o varous orfolos here are o furher oeal sources of accuracy ha e eed o meo. The frs oe s he fac ha he GA formula s asymoc ad due o hs fac may o ork roerly for small orfolos. I geeral oversaes he effec of graulary for ha ye of orfolos bu s ue accurae for medum-szed orfolos of lke 00 lo-ualy oblgors or eve 500 vesme-grade oblgors. The secod ad more serous ssue h hs aroach s he form of bass rsk or model msmach he IRB formulae are based o a dffere cred rsk model. O he oher had he model uder cosderao has a grea advaage s aalycal racably ha erms us o re-arameerze he GA formula erms of he IRB caal charges cludg maury adusme. The maury adusme of he GA s drec he sese ha oly he us are adused raher ha he formula self. A las lmao of he aroach descrbed seco III.5 s he fac ha assumes ha each oso he orfolo s a uhedged loa o a sgle borroer ad ha makes dffcul o cororae cred defaul sas ad loa guaraees he GA. As meoed earler he ormal aroxmao descrbed seco IV. s based o he ceral lm heorem. I ha case e eed o be cocered heher holds for orfolos h severe exosure coceraos. Aarely ha s o he case as e already sa he revous ar here e cosdered he umercal examles of Huag e al. 007 called he auhors ll he ed of he chaer. The reaso s smle he mehod res o aroxmae he loss desy by he ormal dsrbuo bu cao caure he aer ad herefore he CT does o hold more exreme cases cocerao of more ha % of he oal exosure. A roof s gve by he auhors o Fgure 5. here hey comare he al robably gve by he ormal aroxmao codoal o he commo facor. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo - 4 -

43 e us o cosder he GA from seco IV.. A mora ssue o hadle s o vesgae ho he saddleo aroxmao ca hadle ame coceraos. To do ha he auhors comare he hole loss dsrbuo derved by he SA case of very hgh cocerao o he bechmark oe. We easly oce ha he rue dsrbuo s o smooh aroud S 00 refer o Fgure 5.. Recall ha he SA s based o he formulao of he Bromch egral meoed before hch rereses a robably desy fuco. The orfolo loss s dscree he he GD s cosa. The he auhors assume ha ca be closely aroxmaed by a couous radom varable h absoluely couous cumulave dsrbuo fuco. Tha s ho a smoohed verso of he loss dsrbuo s roduced. O Fgure 5. e oce ha he SA o he al robables s correc for almos all uales before he o-smoohess oe aroud 996% bu s accurae for hgher Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

44 uales. I ha sese e may coclude ha h oe or more exceoal exosures he orfolo he SA may o acheve uform accuracy of he loss dsrbuo ad hs may be a roblem f he uale e are eresed recedes he o-smoohess oe. As a soluo o hs roblem Bera 005 suggess he so called Adave Saddleo Aroxmao. The morace samlg seems o erform ell for deermg he VaRC s. A reaso for ha s ha he samle orfolos cosdered he revous ar he oblgors each bucke are assumed decal ad as a resul he calculaed VaRC s are less volale. I geeral IS ca cluser he smulaed losses aroud he reured uale ad hus here s a subsaal crease he robably P [ VaR]. O he oher had ue a large umber of relcaos s eeded o oba such resuls. Huag e al. cosder a orfolo of a hudred oblgors each h dffere exosure asses correlao ad PD are aga 0% ad 033%. O Fgure 5.3 he auhors sho scaerlos of he scaled VaRC a he loss level 700 o he y-axs ad he EAD o he x-axs for he varous umercal mehods. All of hem rovde hgher VaRC s for hgher EAD hch s hghly desrable. Cosse h he resuls from he las seco he NA overesmaes he VaRC of small exosures ad uderesmaes he VaRC of large exosures. The SSA aga almos relcaes he SA. We oce he dfferece he ualy of he IS deedg o he umber of scearos. For Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo

45 0000 scearos he relao beee VaRC ad EAD s o really clear. For scearos e see a huge mroveme he uard red of VaRC s obvous bu s hghly volale. For ha reaso he auhors sugges eve hgher umber of smulaed scearos. As a cocluso abou he umercal mehods from seco IV e ould lke o say ha for orfolos of loer graulary ad medum ame cocerao all mehods rovde relavely good soluos. There s o mehod o be absoluely referred uder all crcumsaces; he choce ll alays be a radeoff beee accuracy robusess ad seed. The NA s he fases mehod acheves a far accuracy ad s caable of hadlg orfolos h hgh exosure coceraos. The SSA s he secod seed erforms beer ha he NA calculag VaR bu may suffer from he same roblem calculag VaRC s. IS s really robus f a large eough umber of scearos are smulaed because makes o assumos o he orfolo srucure. The IS VaR resuls are o alays he bes amogs he oher mehods bu are accurae eough. A draback s he more comuaoal me reured for a good esmao of VaRC as saed he revous aragrah. The SA s more accurae ha he NA ad he SSA ad hadles ell exreme exosure coceraos. Thus s a faser subsue of he IS h sasfacory accuracy level bu oe should be aare of he o-smoohess of he loss dsrbuo meoed earler hs seco. Aga he NA ad SA are based o asymoc aroxmaos ad hus her accuracy mroves sgfcaly by creasg he orfolo sze. O he oher had IS becomes eve more me demadg for large orfolos. VII. Cocluso I hs aer e have deal h a umber of heorecal ad raccal ssues relaed o he measureme of cred rsk ad dealg h cocerao rsk. The umercal resuls are rovded by he auhors of he aers ha roduced he descrbed Graulary Adusmes ad our comarsos ad evaluao of he mehods are based o hose same resuls. A ossble exeso of he aer may be he mlemeao of he models ad her comarso o he same orfolos. Tha ll gve us more soud ad relable umercal resuls o comare he echues ad o fully vesgae her luses ad drabacks. Dealg h Idosycrac Cred Rsk a Fely Graulaed Porfolo