Bank loans pricing and Basel II: a multi-period risk-adjusted methodology under the new regulatory constraints

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1 Baks ad Bak Systems, Volume 4, Issue 4, 2009 Domeco Cuco (Italy, Igo Gafacesco (Italy Bak loas pcg ad Basel II: a mult-peod sk-usted methodology ude the ew egulatoy costats Abstact Ude the ew Basel II egulatoy famewok, the eed fo a effectve sk-usted pcg mechasm has become eve moe cetal bakg tha the past: baks ae spued to develop sk-usted measues, to avod wasteful customes coss-subsdzato ad suppot the value ceato pocess fo the shaeholdes The pape ams at detectg how the Iteal Ratgs-Based appoach affects the bak loa pcg mechasm, by developg a multpeod sk-usted pcg methodology, whch allows us to sepaate the cotbuto of the two compoets of cedt losses (the expected loss ad the uexpected loss, ude the pevalet epaymet schemes Followg Hasa ad Zazzaa (2006, sk-usted pcg ca be splt to two ma pats: a techcal oe, whch s based o Basel IIcosstet sk factos (pobablty of default, loss case of default, exposue at default ad matuty; the secod pat, ot aalyzed ths pape, s defed as commecal ad cludes commssos, opeatoal costs, ad othe subjectvely allocated costs I ths eseach we focus o the emueato fo both the expected ad uexpected losses The ma puts we eed ou pcg fomula ca smply be daw fom a teal atg model ad fom easy-tofd maket data (sk-fee teest ates ad shaeholdes taget etu The pcg fomula we popose s cosstet wth the ew Basel II egulatoy appoach to cedt sk maagemet ad povdes a mmedate suppot fo bak maages makg a loa pce-elated decso Keywods: asset pcg, baks, Basel II, sk maagemet JEL Classfcato: G2, G2, G28, G32 Itoducto Measug ad pcg cedt sk ae cucal bakg Baks ae flueced by dffeet factos the loa pcg decsos: coutepates chaactestcs, summazed by the pobablty of default; faclty chaactestcs, such as the pesece of guaatees, the loa matuty, etc; bak teal factos, such as the dvesfcato degee of ts cedt potfolo o the cost of ts fudg stuctue; sttutoal (exteal elemets, patly elated to the maket (the avalablty of hedgg stumets o the exstece of a actve secoday maket, ad patly coceg the bak egulatoy famewok The paadgm of value ceato fo bak shaeholdes, togethe wth the ew Basel II egme of captal equemets (the Accod, dove bak maages to develop effectve sk-usted pefomace measues (RAPMs dug the past decades Basel II Plla defes the methodologes to calculate captal equemets fo cedt, maket ad opeatoal sks Wth egad to cedt sk, the Accod allows baks to choose betwee two appoaches: the stadadzed appoach, whch bascally efes the old set of sk weghts poposed the 988 Accod; the Iteal Ratgs- Based (IRB appoach, whch allows baks to use the teal estmates of the cedt sk compoets: the coutepates pobablty of default (PD, the loss gve default (LGD, e the loss that the bak would face fo a specfc loa faclty case t defaults, the exposue at default Domeco Cuco, Igo Gafacesco, (EAD ad the loa matuty (M Specfcally, thee ae two vaats of the IRB appoach: the IRB- Foudato, whee baks oly povde estmates of each boowe s PD, ad the IRB-Advaced, whee baks estmate all the cedt sk compoets pevously metoed Cedt sk ca geeate two types of losses, kow as expected loss (EL ad uexpected loss (UL EL depeds o the boowe s PD ad the LGD Assumg the depedece betwee PD ad LGD, the expected loss ate (ELR fo a sgle loa/boowe j s smply gve by the followg poduct: PD j LGD j, wheeas, fo a whole cedt potfolo, t s the sum of each loa s ELR Sce they ae expected, these losses must be hedged by adequate accoutg loa-loss povsos ad epeset a physologcal cost of bak ledg actvty 2 UL s fucto of the PD vaablty ad the coelato betwee the potfolo assets ad must be coveed by a appopate amout of ecoomc captal Ex post, UL equals the dffeece betwee the actual loss ad EL Ex ate, the uexpected loss ca be measued though a potfolo model based o a Value-at-Rsk (VaR methodology 3 Wth the ew egulatoy famewok, baks have to set asde a amout of egulatoy captal to face the sk of uexpected losses, devatg these esouces fom the ledg actvty ad suffeg fom the cosequet oppotuty cost Fo futhe detals o the Accod see BCBS (2006, ad fo a teestg pespectve o the captal adequacy egme, see Hasa et al ( See Sata ( See Hasa ad Zazzaa (2006 ad Rest ad So (2007 fo futhe detals o the models used to measue UL

2 By mplemetg oe of the IRB appoaches, baks ae supposed to acheve captal savgs whe compaed to the stadadzed appoach, takg advatage of a hghe sk-sestvty, to mplemet moe selectve ledg polces ad to take moe sk-sestve pcg decsos I the ed, sce the egulatoy captal equemet affects bak loa pcg, IRB cedt sttutos would be able to bette quatfy ad tasfe both the expected ad the uexpected losses Ths pape detects the bak loa pcg mechasm ude the IRB famewok, showg how the captal absobed by a sgle loa should be take to accout detemg ts pce We popose a fomula to calculate sk-usted pce measues fo loas epad ude dffeet amotzato schemes We develop po eseaches tutos about the techcal pcg methodology ad go futhe by extedg the geeally adopted -yea pespectve, to get a moe accuate ad sk-sestve pce measue The est of the pape s ogazed as follows: secto evews the studes that have aleady aalyzed loa pcg ssues ude the ew egulatoy egme; secto 2 descbes the adopted loa pce methodology; sectos 3 ad 4 show the applcato of ou pcg fomula to calculate skusted teest ates ad speads fo loas wth dffeet epaymet schemes; the last secto cocludes ad povdes suggestos fo futhe eseach developmets Loa pcg ude the ew Basel II famewok: a lteatue evew The mplcatos of Basel II o loa pcg have aleady bee vestgated by pevous lteatue, eve f ot extesvely, due to the ecet publcato of the Accod s fal veso I a pape devoted to cedt sk modellg of small commecal loa potfolos, Detsch ad Petey (2002 assume that a bak has to maxmze ts expected potfolo etu ude the costat that the ecoomc captal equemet must be equal to a exogeous, ceta amout Gve a expected Retu o Equty (RoE, a -yea matuty, a fxed ecovey ate, ad eglectg taxes ad opeatg costs, they deteme the sk-usted pce cosstet wth the expected RoE They show that the pce of loas gated to SMEs depeds o the classfcato as etal o copoate exposues Repullo ad Suaez (2004 aalyze the mpact of the ew captal equemets o the loa pcg a pefectly compettve maket fo busess loas, whee the coelato defaults acoss fms s dve by a sgle systematc sk facto Futhemoe, baks have zeo temedato costs, Baks ad Bak Systems, Volume 4, Issue 4, 2009 ae fuded wth fully sued deposts ad equty captal, emueatg the latte moe tha the fome, though bak shaeholdes ae supposed to be sk-eutal, ad supply loas to a huge umbe of uated fms to fud sky vestmet pojects They fd that, ude pefect competto ad a - yea plag hozo, the ates whch equate the expected paymets of a loa to ts weghted magal fudg cost, ae calculated by maxmzg the expected dscouted value of ts et woth (goss loa etus mus goss depost labltes, holdg the mmum possble amout of egulatoy captal Cosdeg two goups of baks, ledg to hgh-sk fms ad to low-sk fms, espectvely, due to the advatageous teatmet fo low-sk ledg the IRB method elatve to Basel I, the ates of low-sk loas wll be detemed by the captal chages of the IRB appoach ad wll be lowe tha ude Basel I, whle the ates of hgh-sk loas wll be detemed by the captal chages of the stadadzed appoach Fom a quattatve pot of vew, they show that the IRB appoach may mply a educto o a cease loa ates, elatve to Basel I, depedg o the boowes cedtwothess Based o the esults, baks ledg to hgh-sk loas wll adopt the stadadzed appoach, leavg the ates the same as ude Basel I Hasa ad Zazzaa (2006 popose a methodology to estmate sk-usted speads fo bak copoate loas They pce bak loas though a fomula fed by the same puts eeded to calculate the Basel II captal equemets Followg the appoach, the loa spead ca be splt to two potos: the techcal spead, whch s dectly ad fully devable fom a teal atg model, ad the commecal spead, whch accouts fo opeatoal costs, commssos ad othe subjectvely allocated costs They focus o the fome, explag ts lk to some pefomace dcatos, such as the EVA TM ad the RARO ad fdg evdece of a sgfcat elatoshp betwee sk ad loa spead Based o the model of a sk-eutal bak opeatg ude ucetaty a mpefectly compettve loa maket, Ruthebeg ad Ladskoe (2008 detect the mpact of the two ew egulatoy appoaches (IRB ad stadadzed usg the PD dstbuto of a leadg Isael bak s customes They show that bg baks wll attact hgh-qualty boowes, due to the educto loa ates stemmg fom the adopto of the IRB appoach; low-qualty fms wll be fuded by small temedaes, whch ae moe lkely to adopt the stadadzed appoach; etal customes wll ejoy a educto loa ates f they boow fom IRB baks 67

3 Baks ad Bak Systems, Volume 4, Issue 4, The loa pcg methodology I ths secto we descbe the loa pcg methodology adopted hee estmatg sk-usted ates ad speads fo bak cedt exposues Followg Hasa ad Zazzaa (2006, the sk-usted pce fo bak loas ca be splt to two ma compoets: the techcal pat, whch takes to accout both expected ad uexpected losses ad the oppotuty cost fo povdg commtted cedt les; the commecal compoet, whch cludes commssos, opeatoal costs, ad othe subjectvely allocated costs We do t take cae of these latte elemets sce the allocato does t have ay elevace tems of cedt sk maagemet We focus o the two ma compoets of the techcal pce: the emueato fo EL ad UL fo loas wth fxed exposue The ma puts we eed to take both expected ad uexpected losses to accout ou pcg fomula ca smply be daw fom a teal atg model (PD, LGD, EAD ad M ad fom easy-to-fd maket data (sk-fee teest ates ad shaeholdes taget etu Ou fomula s cosstet wth the logc udelyg the ew Basel II egulatoy appoach to cedt sk maagemet ad povdes a mmedate suppot fo bak maages I the ext paagaphs we develop the pcg methodology fo zeocoupo loas (ZCLs, whee both teests ad pcpal ae epad a sgle sum o a set matuty The, we exted the aalyss to the othe pevalet epaymet schemes 2 The cost of the expected loss fo zeo-coupo loas Bak emueato to cove expected losses s calculated wth a sk-eutal famewok: let s assume a bak ssug a -yea matuty loa of to a boowe classfed the -th atg class The expected loa value must be equal to the futue value of a sk-fee vestmet: ( p Rp ( ( s, ( whee: s the sk-fee ate fo a -yea hozo; s s the spead to emueate expected losses fo a -yea matuty loa; p s the pobablty of default wth yea; R s the ecovey ate the evet of default, set flat fo each atg class Wth egad to loas wth a vaable exposue, we have to cosde the oppotuty cost that baks bea to gat to some boowes the possblty to daw moey up to a ceta amout a totally dscetoay way I ths case, Hasa ad Zazzaa (2006 assume that baks apply the sk-usted teest ate (spead of a loa wth fxed exposue o the daw poto, ad chage the udaw poto wth the dffeece betwee the sk-usted ate ad the etu they would get f vested t at the sk-fee teest ate Expected losses must be coveed by addg a spead to the sk-fee ate The left-had sde of equato ( s the loa expected value, equal to the sum of the loa futue value case of suvval, wth pobablty ( p, ad the loa ecoveed amout case of default, wth pobablty p Afte some algebac mapulatos, we ca get the -yea skeutal teest ate, eutal, expessed by the - yea sk-fee teest ate plus the -yea spead fo expected losses, ad the coespodg spead s, eutal s :, p ( R s EL, (2 -p ( R p ( R (2 -p ( R Extedg ths -yea aalyss to a -yea hozo, fomula ( becomes: ( p Rp ( ( s, (3 whee the subscpt eflects the -yea pespectve, p s the cumulatve pobablty of default wth yea, ad R s set costat ove tme Fom equato (3 we ca get the -yea sk-eutal teest ate s, eutal, ad the coespodg spead, both calculated o a aual bass:, eutal s, (4 p ( R s ( (4 p ( R 22 The cost of the uexpected loss fo zeocoupo loas Sce the sk-eutalty assumpto s uealstc fo baks, we clude the sk aveso to ou pcg fomula: fo each loa, the fal teest ate must emueate ot oly the cost of the expected loss, but also that of the uexpected loss The esultg teest ate s geeally defed as sk-usted just because t accouts fo the bude of uexpected losses too, the actual sk fo cedt sttutos UL s fucto of the coelato betwee bak loas ad ca be estmated though potfolo models, whose fal objectve s to calculate a VaR measue fo both the sgle loa ad the oveall potfolo: ths VaR, also amed CaR (Captal at Rsk, sce t s efeed to the bak captal, epesets the amout of sk that must be coveed by equty Followg a VaR 68

4 appoach, the dffeece betwee the maxmum potetal loss, calculated wth a ceta tme teval ad fo a gve cofdece level, ad the expected loss s a measue of the uexpected loss, ad also epesets the bak captal at sk We measue the bak ecoomc captal eeded to face the uexpected loss though the egulatoy captal, whch s calculated though the IRB closed fomula, ad s made up of Te, o coe captal, UL, ( s s ( p B, S, RC RC (, Rp RC B, ( whee, apat fom the pevously defed vaables, sc ad ss ae the costat speads ove the skfee teest ate, equed by the coe captal holdes ad the supplemetay captal holdes, C espectvely; RC, S ad RC, ae the amouts of Baks ad Bak Systems, Volume 4, Issue 4, 2009 ad Te 2, o supplemetay captal 2 The last put we eed to feed ou pcg fomula s the cost of the ecoomc captal, whch s fucto of the etu expected by the captal povdes Fally, ou model each loa s fuded by both debt captal ad equty captal, ulke some othe famewoks whee the equty captal has oly a collateal fucto 3 Icludg the above metoed factos, equato ( ca be modfed as follows: s RC B S, ( s s (5 coe captal equemets ad supplemetay captal equemets, calculated ude the Basel II ules fo a -yea matuty loa, espectvely; Fom equato (5, afte some algebac mapulatos, we ca deve the sk-usted teest ate emueatg both EL ad UL fo ths -yea ZCL: S, RC RC ( - S,, UL, RC ( sb RC ( ss s s - p ( R The spead to compesate fo the uexpected losses ca be smply wtte as follows: RC ( RC ( S, RC RC ( - S, UL, B S s -p ( R s If we take to accout a -yea ZCL, equato (5 becomes: ( UL, s s ( p S, RC RC (, Rp RC ( whee the subscpt eflects the -yea pespectve We have to pot out that usg the aualzed pobabltes of default as puts of the fomula to assess egulatoy captal, we move away fom what poposed by the Basel Commttee sce to feed the sk-weght fuctos, they use the -yea, s EL s RC s UL EL ( s RC UL s B, C ( RC s S, B ( ss p ( R RC S, s B RC S, ( s S (6 (6 (7 pobablty of default, egadless of loa matuty I equatos (8 ad (8 we show how to calculate the -yea sk-usted teest ate ad the spead to emueate the two techcal compoets that we take to accout ou pcg mechasm Both ae calculated o a aual bass: S, RC RC ( - ( ss p ( R B, S, RC RC ( -, (8 2 3 (8 The Basel II IRB sk-weght fuctos, used to assess the egulatoy captal fo uexpected losses, ae based o a specfc model descbed Gody (2003 Fo futhe detals o the egulatoy fomulas, see BCBS ( Fo detals coceg the compoets of the two tes, see BCBS ( See Rest ad So (2007 fo a example of the fst appoach, ad Hasa ad Zazzaa (2006 fo a applcato of the secod oe 69

5 Baks ad Bak Systems, Volume 4, Issue 4, Estmatg sk-usted pce measues fo zeo-coupo loas wth fxed exposues: skusted spead beak-dow I ths paagaph we use the above descbed methodology to estmate the tem stuctue of the techcal sk-usted ates ad speads fo ZCLs wth fxed exposues, wth egad to the Basel II copoate segmet I ou pcg smulatos fo the yea 2009: we use a mult-peod atg maste scale as a souce of cumulatve pobabltes of default fo a 0-yea tme hozo (Table 2, fom whch we calculate the aualzed PDs to feed the egulatoy fomula fo -yea matuty loas; we adopt the tem stuctue of swap teest ates, as of Jauay st 2009, as a poxy fo the tem stuctue of sk-fee teest ates (see the bottom ow of Table ; Table The mult-peod atg maste scale ad the tem stuctue of swap teest ates Aveage cumulatve ssue-weghted global default ( Matuty yea Aaa 00% 002% 002% 005% 009% 04% 09% 09% 09% 09% Aa 002% 006% 00% 07% 025% 029% 032% 035% 037% 04% A 003% 03% 03% 048% 068% 089% % 34% 55% 7% Baa 08% 052% 093% 4% 89% 236% 282% 324% 365% 44% Ba 5% 37% 569% 829% 048% 247% 422% 585% 732% 874% B 433% 983% 527% 2009% 2447% 2867% 3267% 3600% 3893% 445% Caa 373% 235% 370% 384% 4375% 4762% 5036% 5352% 5837% 6478% Ca-C 3295% 4430% 5326% 584% 6393% 6649% 7034% 7499% 7499% 7499% Ivestmet gade 007% 023% 044% 067% 092% 5% 38% 60% 80% 20% Speculatve gade 435% 892% 337% 732% 2069% 2370% 2639% 2869% 307% 3252% Tem stuctue of swap teest ates (as of Jauay st 2009 Swap teest ates 268% 276% 296% 32% 336% 324% 357% 346% 366% 374% Souce: Moody s (2009 ad Datasteam we set the ecovey ate costat ad equal to 55% of the cedt exposue, cosstetly wth the 45% LGD of the IRB-Foudato appoach fo seo usecued clams o copoates, soveegs ad baks; 2 we assume that the ecoomc captal absobed by each loa cocdes wth the egulatoy captal Based o the evdece efeed to the Itala bakg system 3, we hypothesze that 70% of the egulatoy captal s coe captal ad the emag 30% s supplemetay captal; egadless of loa matuty, we hypothesze a costat sk-pemum fo both coe captal holdes ad supplemetay captal holdes, whch must be added to the skfee teest ate to calculate the espectve taget emueato Assumg that the supplemetay captal s made up of oly subodated debt, ad followg some suggestos fom bak maages, we set a sk-pemum of 800 bps ad 200 bps fo Fo detals about the dffeet segmets of bak exposues wth the Accod, see BCBS ( See Moody s ( See Bak of Italy (2009 coe captal ad subodated debt, espectvely Based o fomulas (8 ad (8, fed wth the above lsted puts, we get the tem stuctue of the skusted pce measues (Table 2 O aveage, skusted speads fo vestmet gades cease wth matuty, wheeas they move dowwad fo speculatve gades Spead beak-dow: EL vs UL Hee we calculate the cotbuto of the two compoets (EL ad UL to the total spead, aowg ou aalyss to some matutes, by estmatg the shae of the total spead explaed by EL ad UL, espectvely We do that by calculatg the atos of the spead to cove EL to the total spead, o the oe had, ad the spead to emueate UL to the total spead, o the othe had (see Table 3 As expected, fo each matuty speads of bette atg classes ae chaactezed by a lowe cdece of expected losses, elatve to uexpected oes The EL weght ceases wth the decle of the coutepates cedtwothess, ad becomes lage tha the UL oe fom atg Ba O aveage, the cdece of the uexpected loss ases wth the loa matuty fo speculatve gades, wheeas t dmshes fo the vestmet gades, eve f at a slowe pace 70

6 Baks ad Bak Systems, Volume 4, Issue 4, 2009 Table 2 The tem stuctue of sk-usted ates (R ad speads (S fo the copoate segmet zeo-coupo loa Matuty yea R S R S R S R S R S Aaa 273% 004% 299% 003% 342% 007% 366% 00% 382% 009% Aa 275% 006% 306% 00% 349% 04% 370% 04% 388% 04% A 276% 008% 37% 02% 362% 027% 388% 03% 40% 036% Baa 298% 029% 340% 044% 389% 053% 45% 058% 438% 064% Ba 370% 02% 447% 5% 504% 69% 528% 7% 545% 7% B 545% 277% 628% 33% 670% 334% 69% 334% 695% 32% Caa 060% 79% 954% 657% 97% 58% 863% 506% 870% 497% Ca-C 2202% 933% 423% 26% 203% 867% 079% 722% 956% 582% Ivestmet gade 284% 06% 323% 027% 368% 033% 393% 036% 44% 040% Speculatve gade 547% 278% 59% 295% 624% 288% 634% 277% 634% 260% Souce: Ou elaboatos o data fom Moody s (2009 ad Datasteam TM Table 3 Spead beak-dow: EL* vs UL* Matuty yea UL EL UL EL UL EL UL EL UL EL Aaa 8908% 092% 9205% 795% 883% 87% 8709% 29% 8963% 037% Aa 8622% 378% 8433% 567% 830% 690% 8450% 550% 8644% 356% A 8466% 534% 7734% 2266% 7640% 2360% 7652% 2348% 7786% 224% Baa 726% 2874% 6742% 3258% 6666% 3334% 6752% 3248% 6954% 3046% Ba 4779% 522% 4072% 5928% 4043% 5957% 4252% 5748% 4643% 5357% B 2634% 7366% 2536% 7464% 2698% 7302% 2885% 75% 3259% 674% Caa 453% 8547% 755% 8245% 203% 7969% 2353% 7647% 2675% 7325% Ca-C 753% 9247% 258% 8742% 634% 8366% 986% 804% 25% 7489% Ivestmet gade 786% 239% 7458% 2542% 7393% 2607% 7466% 2534% 7657% 2343% Speculatve gade 2628% 7372% 2708% 7292% 2929% 707% 396% 6804% 3647% 6353% Note: * pecetage of the total spead Souce: Ou elaboatos o data fom Moody s (2009 ad DataSteam TM 4 Estmatg sk-usted pce measues fo dffeet epaymet plas The pcg model peseted above efes to a zeocoupo loa but, pactce, bak loas ae ssued ude dffeet amotzato plas I ths secto we cosde thee dffeet schemes: bullet loa (BL, whee teests ae pad at egula tevals ad captal s epad o the fal matuty; costat captal epaymet (CCR, whee the captal compoet of the stallmet s take costat fo each matuty; ad staght-le amotzato (SLA, whee the stallmet s costat ove tme Fo each of the thee amotzato schemes, we deve the flat tem stuctue of the aualzed skusted teest ates fo a loa, whch s equvalet to the tem stuctue of sk-usted teest ates efeed to the ZCLs case: we decompose each amotzato pla to a sees of ZCLs whose amout equals the sgle stallmet value, ad use the sk-usted teest ates pevously deved fo the ZCLs to calculate the costat sk-usted teest ate fo ay atg class I each of the cases descbed below, ou aalyss gouds o the followg equlbum codto at tme t 0, whe the loa s ssued: CFt ( t t, t (9 fom whch ts value ( has to be equal to the sum of the loa cash-flows peset values (CF t, dscouted at the coespodg sk-usted t, teest ates (, calculated, fo each atg class, usg fomula (8, ad epoted Table 2 Bullet loa (BL Let s suppose a loa to a boowe aked the -th atg class, wth a -yea matuty ad teest epaymet at the ed of each yea Accodg to the equlbum codto, we ca wte: ( BL,, ( BL, 2, 2, ( BL,, (0 7

7 Baks ad Bak Systems, Volume 4, Issue 4, 2009 whee BL, s the aualzed sk-usted ate, whch s costat ove tme The above fomula ca be ewtte as follows: ( ( BL, t t,, ( Sce we calculated the sk-usted teest ates at the deomatos of ( though the methodology descbed paagaphs 2 ad 3, we deve the aualzed sk-usted teest ate as follows: Do C CCR, D C 2 ( ( CCR,, 2, CCR, D ( whee D t s the outstadg debt used to calculate the teest epaymet at tme t+ ad, s the CCR (, BL, t t ( t, (2 Costat captal epaymet (CCR Let s cosde a loa to a boowe aked the -th atg class, wth a -yea matuty ad stallmet epaymet at the ed of each yea, wth a costat pcpal epaymet (C The equlbum codto hee becomes:, C, (3 aualzed sk-usted teest ate, whch s costat ove tme Equato (3 ca be ewtte as: Do D D CCR, C 2 t (, ( 2, (, t ( t, (4 Fom equato (4 we deve the costat aualzed sk-usted teest ate fo ths amotzato scheme: C t t ( t, CCR, Do D D (5 2 (, ( 2, (, Staght-le amotzato (SLA Let s suppose a loa to a boowe wth the -th atg class, wth a -yea matuty, whose costat stallmets (I ae pad at the ed of each yea I ths case, the equlbum codto ca be wtte as follows: I (, I ( 2, 2 I (, (6 Equato (6 ca also be ewtte the followg way: I ( t t, (7 fom whch, by usg the sk-usted ates deved fo the ZCL case, we ca calculate the coespodg stallmet though the followg fomula: I (8 t ( t, Besdes, sce fo ths patcula amotzato pla, the elatoshp betwee the loa value ad the stallmet ca be fomalzed as follows: I, (9 whee a SLA, ( SLA, a SLA, SLA, (20 ad SLA, s the aualzed sk-usted teest ate, whch s costat fo each matuty, we ca eplace equato (20 to equato (9 ad obta the aualzed sk-usted teest ate, by solvg the followg equato usg umecal methods: ( SL, I (2 SL, I Table 4 we epot the sk-usted teest ates fo the thee epaymet plas fo fve matutes Whe the loa expes afte yea we get the same teest ates fo the thee amotzato schedules, ad these ates ae also equal to those calculated fo the ZCL case Fo matutes beyod yea, esults ae dffeet, depedg o the boowe atg class: egadless of the matuty, fo atg classes agg fom Aaa to B, the bullet pla shows hghe teest ates tha the equal stallmet oe, wheeas the costat captal epaymet s chaactezed by the lowe values; vce vesa fo atg classes Caa ad Ca-C Obvously, these dffeeces ae due to 72

8 the dffeet dstbuto of teest ad captal epaymets dug the loa tme hozo ad to the cosequet mpact o the peceved loa skess Wth egad to the sk-usted speads, sce t s ot possble to deve them dectly, we adopt the followg thee-step pocedue: we calculate the tem stuctue of the skeutal teest ates fo the zeo-coupo loas, takg oly the expected loss to accout, va fomula (4; 2 fo each amotzato pla, we estmate the costat aualzed sk-eutal teest ates usg the tem stuctue deved above at pot, to feed fomulas (2, (5 ad (2, espectvely; Baks ad Bak Systems, Volume 4, Issue 4, fo each amotzato pla, we deve the costat aualzed sk-fee teest ate usg the tem stuctue of the sk-fee teest ates epoted at the bottom of Table, to feed fomulas (2, (5 ad (2, espectvely Cosequetly, the spead to cove EL s the dffeece fom what we get at the secod step of the pocedue (the aualzed sk-eutal teest ates ad what we deve at the thd step (the aualzed sk-fee teest ate The spead to emueate UL s the dffeece betwee the skusted teest ates (Table 4 below ad the skeutal teest ates calculated at pot 2 These speads ae show Table 5 Table 4 The tem stuctue of sk-usted ates bullet loa (BL, costat captal epaymet (CCR ad staght-le amotzato (SLA Matuty yea BL CCR SLA BL CCR SLA BL CCR SLA BL CCR SLA BL CCR SLA Aaa 273% 273% 273% 299% 288% 288% 340% 32% 33% 36% 328% 329% 376% 346% 348% Aa 275% 275% 275% 305% 293% 293% 346% 38% 39% 366% 333% 335% 38% 35% 354% A 276% 276% 276% 36% 30% 30% 359% 329% 330% 382% 346% 348% 402% 367% 370% Baa 298% 298% 298% 339% 324% 324% 385% 353% 354% 408% 37% 373% 429% 393% 396% Ba 370% 370% 370% 445% 49% 49% 498% 458% 46% 59% 479% 482% 534% 50% 505% B 545% 545% 545% 624% 599% 600% 663% 63% 633% 68% 647% 65% 685% 662% 666% Caa 060% 060% 060% 959% 985% 983% 925% 953% 950% 878% 92% 95% 877% 896% 888% Ca-C 2202% 2202% 2202% 475% 670% 642% 268% 455% 49% 52% 328% 286% 046% 25% 70% Ivestmet gade 284% 284% 284% 322% 307% 308% 365% 335% 336% 387% 352% 354% 406% 372% 375% Speculatve gade 547% 547% 547% 590% 574% 574% 620% 596% 598% 628% 605% 607% 629% 64% 66% Souce: Ou elaboatos o data fom Moody s (2009 ad Datasteam TM Table 5 The tem stuctue of sk-usted speads bullet loa (BL, costat captal epaymet (CCR ad staght-le amotzato (SLA Matuty yea BL CCR SLA BL CCR SLA BL CCR SLA BL CCR SLA BL CCR SLA Aaa 004% 004% 004% 003% 004% 004% 007% 005% 005% 009% 007% 007% 008% 008% 008% Aa 006% 006% 006% 00% 009% 009% 03% 0% 0% 03% 02% 02% 04% 03% 03% A 008% 008% 008% 02% 06% 07% 026% 02% 022% 030% 025% 025% 034% 029% 030% Baa 029% 029% 029% 044% 039% 039% 052% 046% 046% 056% 050% 05% 06% 055% 055% Ba 02% 02% 02% 49% 34% 35% 65% 5% 53% 67% 58% 60% 67% 62% 64% B 277% 277% 277% 329% 34% 35% 330% 324% 325% 329% 326% 328% 38% 324% 326% Caa 79% 79% 79% 663% 70% 698% 592% 646% 642% 526% 600% 592% 50% 557% 548% Ca-C 933% 933% 933% 79% 385% 358% 935% 48% % 800% 006% 964% 679% 877% 829% Ivestmet gade 06% 06% 06% 026% 023% 023% 032% 028% 028% 035% 03% 03% 038% 034% 035% Speculatve gade 278% 278% 278% 294% 289% 290% 287% 289% 290% 276% 284% 285% 26% 275% 275% Souce: Ou elaboatos o data fom Moody s (2009 ad Datasteam TM As doe befoe fo the ZCL case, fo each amotzato pla we calculated the spead beakdow ode to vestgate the cotbuto of EL ad UL to the whole spead: ou evdece suppots the esults that we foud the zeocoupo loa case sce speads of bette atg classes show a lowe cdece of the expected loss, elatve to the uexpected oe The weght of the expected loss ceases wth the se of the coutepates skess ad becomes lage tha that of the uexpected losses fom atg Ba O aveage, the cdece of the uexpected loss ases wth the loa matuty fo speculatve gades, wheeas t dmshes fo the vestmet gades, eve f at a slowe pace (see Tables 6, 7 ad 8 below 73

9 Baks ad Bak Systems, Volume 4, Issue 4, Table 6 Spead beak-dow: EL* vs UL* bullet loa Matuty yea UL EL UL EL UL EL UL EL UL EL Aaa 8908% 092% 997% 803% 882% 79% 878% 282% 8943% 057% Aa 8622% 378% 8434% 566% 834% 686% 8442% 558% 866% 384% A 8466% 534% 7739% 226% 7644% 2356% 765% 2349% 7767% 2233% Baa 726% 2874% 6745% 3255% 6667% 3333% 6743% 3257% 692% 3079% Ba 4779% 522% 4077% 5923% 4039% 596% 4227% 5773% 4563% 5437% B 2634% 7366% 2532% 7468% 2680% 7320% 2845% 755% 360% 6840% Caa 453% 8547% 748% 8252% 2006% 7994% 2295% 7705% 2555% 7445% Ca-C 753% 9247% 245% 8755% 596% 8404% 909% 809% 2345% 7655% Ivestmet gade 786% 239% 746% 2539% 7395% 2605% 746% 2539% 7629% 237% Speculatve gade 2628% 7372% 2704% 7296% 2909% 709% 350% 6850% 3534% 6466% Note: * pecetage of the total spead Souce: Ou elaboatos o data fom Moody s (2009 ad Datasteam TM Table 7 Spead beak-dow: EL* vs UL* costat captal epaymet Matuty yea UL EL UL EL UL EL UL EL UL EL Aaa 8908% 092% 9077% 923% 8926% 074% 882% 88% 8846% 54% Aa 8622% 378% 8456% 544% 8368% 632% 8389% 6% 848% 59% A 8466% 534% 7872% 228% 7724% 2276% 7683% 237% 7702% 2298% Baa 726% 2874% 683% 369% 6726% 3274% 6724% 3276% 6800% 3200% Ba 4779% 522% 4242% 5758% 4099% 590% 436% 5864% 4290% 570% B 2634% 7366% 2544% 7456% 2603% 7397% 2695% 7305% 2862% 738% Caa 453% 8547% 647% 8353% 83% 887% 983% 807% 2204% 7796% Ca-C 753% 9247% 064% 8936% 30% 8699% 508% 8492% 78% 829% Ivestmet gade 786% 239% 7548% 2452% 7449% 255% 7446% 2554% 753% 2487% Speculatve gade 2628% 7372% 2673% 7327% 278% 729% 299% 708% 342% 6858% Note: * pecetage of the total spead Souce: Ou elaboatos o data fom Moody s (2009 ad Datasteam TM Table 8 Spead beak-dow EL* vs UL* staght-le amotzato Matuty yea UL EL UL EL UL EL UL EL UL EL Aaa 8908% 092% 9080% 920% 8923% 077% 8807% 93% 8847% 53% Aa 8623% 377% 8459% 54% 8364% 636% 8389% 6% 8488% 52% A 8467% 533% 7868% 232% 7720% 2280% 7680% 2320% 7704% 2296% Baa 726% 2874% 6829% 37% 6723% 3277% 6725% 3275% 6808% 392% Ba 4779% 522% 4240% 5760% 4098% 5902% 44% 5859% 430% 5690% B 2634% 7366% 2545% 7455% 2608% 7392% 2707% 7293% 2889% 7% Caa 453% 8547% 650% 8350% 82% 879% 999% 800% 2239% 776% Ca-C 753% 9247% 070% 8930% 34% 8686% 533% 8467% 829% 87% Ivestmet gade 7864% 236% 7545% 2455% 7446% 2554% 7445% 2555% 7520% 2480% Speculatve gade 2628% 7372% 2675% 7325% 2787% 723% 2933% 7067% 373% 6827% Note: * pecetage of the total spead Souce: Ou elaboatos o data fom Moody s (2009 ad Datasteam TM Cocludg emaks Ths pape detects how the Basel II IRB-Foudato appoach affects the bak loa pcg, by developg a mult-peod pcg methodology to estmate skusted ates ad speads fo cedt exposues wth dffeet epaymet schemes The ma puts we eed to feed ou pcg fomula ca be daw fom a teal atg model ad fom easy-to-fd maket data Ou model cosstecy wth the ew egulatoy famewok to cedt sk measuemet povdes a mmedate suppot fo bak maages makg a loa pce-elated decso ad allows us to fd evdece of a sgfcat elatoshp betwee the sk measues ad the techcal speads chaged o copoate loas

10 Nevetheless, though vey smple, ou model eeds to be efed spug futhe studes amg at: emovg the uealstc assumpto of a flat ecovey ate fo all copoate segmets, as equed the IRB-Foudato appoach Ths codto does t cosde that lage copoatos ae moe lkely to offe appopate guaatees o collateals tha small fms; elaxg the hypothess of depedece betwee PD, LGD ad EAD, whch allows to calculate the expected loss as the smple poduct of the thee vaables To do that, we eed to estmate the depedeces betwee the factos detemg loa losses fo both the sgle boowe ad the whole cedt potfolo; Refeeces Baks ad Bak Systems, Volume 4, Issue 4, 2009 cludg to the aalyss a way to accout fo potfolo effects: sce we measue uexpected losses though the egulatoy sk-weght fomulas, asset coelatos caot be dectly estmated because they ae automatcally deved though a algothm whch s based o the vese elatoshp betwee PD ad asset coelato supposed by Basel II How well do the coelato values calbated by the Commttee eflect the sk pofle ad the actual loss expeece of cedt potfolos, ad what ae the cosequeces of the adopto tems of loa pcg ae elevat ssues that must be addessed, gve the stog mpact of coelatos o the IRB captal equemet ad, fally, o the loa pce BCBS (Basel Commttee o Bakg Supevso Iteatoal Covegece of Captal Measuemet ad Captal Stadads A Revsed Famewok Basel: Bak fo Iteatoal Settlemets, pp 2 Bak of Italy, Aual Repot fo 2008 Rome: Detsch M, Petey J (2002 The cedt sk SME loas potfolos: modellg ssues, pcg, ad captal equemets, Joual of Bakg ad Face, N o 26, pp Gody MB (2003 A sk-facto model foudato fo atgs-based bak captal ules, Joual of Facal Itemedato, N o 2, pp Hasa I, Sddque A, Su X (2009 Captal Adequacy Revst: a Maket Pespectve, Wokg Pape Resselae Polytechc Isttute ad Offce of the Comptolle of Cuecy 6 Hasa I, Zazzaa C (2006 Pcg Rsky Bak Loas the New Basel II Evomet, Joual of Bak Regulato, N o 7, pp Moody s Ivestos Sevce Copoate default ad ecovey ates, Specal Commet New Yok: Febuay Repullo R, Suaez J (2004 Loa pcg ude Basel II captal equemets, Joual of Facal Itemedato, N o 3, pp Rest A, So A Rsk Maagemet ad Shaeholdes Value Bakg Fom Rsk Measuemet Models to Captal Allocato Polces Chcheste: Wley & Sos Ltd, pp 0 Ruthebeg D, Ladskoe Y (2008 Loa pcg ude Basel II a mpefectly compettve bakg maket, Joual of Bakg ad Face, N o 32, pp Sata F (2003 Measug sk-usted pefomaces fo cedt sk, Wokg Pape SDA Bocco, N o 89, pp 27 75