A GENERALIZED FRAMEWORK FOR CREDIT RISK PORTFOLIO MODELS

A GENERALIZED FRAMEWORK FOR CREDIT RISK PORTFOLIO MODELS H. UGUR KOYLUOGLU ANDREW HICKMAN Olver, Wyman & Company CSFP Capal, Inc. * 666 Ffh Avenue Eleven Madson Avenue New Yor, New Yor 10103 New Yor, New Yor 10010 e-mal: uoyluoglu@owc.com e-mal: andrew.hcman@csfp.co.u verson: 14 Sepember, 1998 ABSTRACT Sophscaed new cred rs porfolo modelng echnques have been developed recenly, holdng he poenal for subsanal reform of cred rs managemen echnques and regulaory capal gudelnes. Ths paper examnes hree such cred rs porfolo models, placng hem whn a sngle general framewor and demonsrang ha hey are lle dfferen n eher heory or resuls, provded ha npu parameers are harmonzed. Ths resul, ha a srong consensus has emerged n he approach o modelng defaul rs, has sgnfcan mplcaons for he accepance of hese new echnques amongs boh end-users and regulaors. Helpful commens and dscussons were provded by Andrew Cross, Fran Debold, Tom Garsde, Marc Inraer, Andrew Kurzes, Hashem Pesaran, Tl Schuermann, James Wener, and Tom Wlde. All errors are he responsbly of he auhors alone. An abrdged verson of hs paper s forhcomng n Rs magazne. Opnons expressed heren are hose of he auhors alone and do no necessarly reflec he opnons of Cred Susse Fnancal Producs, CSFP Capal, Inc. or Olver, Wyman & Company. * CSFP Capal, Inc. s an ndrec subsdary of Cred Susse Fnancal Producs Copyrgh 1998. Ths documen may no be duplcaed whou he pror wren consen of he auhors. All rghs reserved. prned: 16 Ocober, 1999

In he pas few years, maor advances n cred rs analycs have led o he prolferaon of a new breed of sophscaed cred porfolo rs models. A number of models have been developed, ncludng boh propreary applcaons developed for nernal use by leadng-edge fnancal nsuons, and hrd pary applcaons nended for sale or dsrbuon as sofware. Several have receved a grea deal of publc aenon, ncludng J.P. Morgan s CredMercs/CredManager, Cred Susse Fnancal Producs' CredRs+, McKnsey & Company s CredPorfoloVew, and KMV s PorfoloManager. These new models allow he user o comprehensvely measure and quanfy cred rs a boh he porfolo and conrbuory level, whch was no possble prevously. As such, hey have he poenal o cause profound changes o he lendng busness, accelerang he shf o acve cred porfolo managemen 1, and evenually leadng o an nernal models reform of regulaory cred rs capal gudelnes. Bu before hese models can delver on her promse, hey mus earn he accepance of cred porfolo managers and regulaors. To hese praconers, hs seemngly dsparae collecon of new approaches may be confusng, or may appear as a warnng sgn of an early developmenal sage n he echnology. Whle hese msgvngs are undersandable, hs paper wll demonsrae ha hese new models n fac represen a remarable consensus n he underlyng framewor, dfferng prmarly n calculaon procedures and parameers raher han fnancal nuon. Ths paper explores boh he common ground and he dfferences amongs he new cred rs porfolo models, focusng on hree represenave models: 1. Meron-based e.g. CredMercs and PorfoloManager 3. Economerc e.g. CredPorfoloVew 3. Acuaral e.g. CredRs+ Secon I provdes a quc revew of he basc srucure of each of he models. Secon II descrbes he common underlyng framewor of hese models and how he models and her assumpons can be relaed o he generalzed framewor. Secon III draws lnages beween he equvalen parameers n each model. Secon IV assesses he mpac of her dfferng assumpons usng llusrave examples. Secon V presens conclusons. Noe ha hs paper examnes only he defaul componen of porfolo cred rs. Some of he models ncorporae cred spread (or rangs mgraon) rs, whle ohers advocae a separae model. In hs aspec of cred rs here s less consensus n modelng echnques, and he dfferences need o be explored and resolved n fuure research. The reader should srcly nerpre cred rs o mean defaul rs hroughou hs paper. Addonally, for comparably, he models have been resrced o he case of a sngle-perod horzon, fxed recovery rae, and fxed exposures. Ths should no sgnfcanly affec he conclusons, as defaul domnaes he conrbuon from hese oher effecs, and he mechansms o ncorporae hese effecs are no oo deeply enangled wh he models of defaul. I. Overvew Models The followng secons brefly descrbe he calculaon procedures of each of he models, modfed o he wo-sae (defaul or no), sngle-perod, fxed recovery, and fxed exposure resrcons. CredMercs CredMercs s a Meron-based model, relyng on Meron s model of a frm s capal srucure 4 : a frm defauls when s asse value falls below s lables. A borrower s defaul probably hen depends on he amoun by whch asses exceed lables, and he volaly of hose asses. If changes n asse value are normally dsrbued, he defaul probably can be expressed as he probably of a sandard normal varable fallng below some crcal value. The frs sep n hs model s hen o calculae crcal values correspondng o each borrower s defaul probably (mapped from he borrower s cred rang). Jon defaul evens amongs borrowers n he porfolo are relaed o he exen ha he borrowers changes n 1 for example, see Kurzes (1998). see ISDA (1998). 3 he dscussons whch follow wll focus on CredMercs as he example, bu wll also apply reasonably well o PorfoloManager. 4 see Meron (1974), Kealhoffer (1995), and Gupon, Fnger, and Bhaa (1997). - 1-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

asse value are correlaed (npu n he form of a parwse correlaon marx deermned accordng o counry and ndusry groupngs). The porfolo loss dsrbuon s calculaed by Mone Carlo smulaon, as follows: 1. Draw random correlaed sandard normal varables represenng he change n asse value for each borrower.. Compare hs sandardzed change n asse value o he pre-calculaed crcal value o deermne whch borrowers defaul. 3. Sum he losses resulng from each borrower defaul o arrve a a oal porfolo loss. 4. Repea housands of mes o buld a dsrbuon of porfolo losses. CredPorfoloVew CredPorfoloVew poss an emprcally derved relaonshp whch drves each borrower s (or segmen of borrowers ) defaul rae p, accordng o a normally dsrbued ndex of macroeconomc facors for ha borrower 5. The macroeconomc ndex y, s expressed as a weghed sum of macroeconomc varables, x,, each of whch s normally dsrbued and can have lagged dependency. x, = a,0 + a,1x, 1 + a,x, + K + ε,, and y, b,0 + b,1x1, + b,x, + + υ, where he ε, and, = K, υ are normally dsrbued random nnovaons. The ndex s ransformed o a defaul probably by he Log funcon: 1 p, =. y, 1 + e The facor loadngs b, for he ndex are deermned by he emprcal relaonshp beween sub-porfolo defaul raes and explanaory macroeconomc varables, usng logsc regresson. The coeffcens a, o he macro-economc varables can be deermned by an approprae economerc model 6. The porfolo loss dsrbuon s calculaed by Mone Carlo smulaon, as follows: 1. Draw random nnovaons o each macroeconomc varable and ndex value accordng o her covarance srucure.. Calculae: a) macroeconomc varables oucomes accordng o her lagged pas values and random nnovaons; b) ndex values accordng o he macroeconomc values and he ndex random nnovaons; and c) resulng defaul probables. 3. Calculae he dsrbuon of defaul oucomes for hs eraon by successvely convolung each oblgor s (wo-sae) dsrbuon of oucomes. 4. Repea housands of mes o buld a dsrbuon of porfolo losses. CredRs+ CredRs+ maes use of mahemacal echnques common n loss dsrbuon modelng n he nsurance ndusry 7. Jon-defaul behavor of borrowers s ncorporaed by reang he defaul rae as a random varable common o mulple borrowers. Borrowers are allocaed amongs secors each of whch has a mean defaul rae and a defaul rae volaly. The defaul rae volaly s he sandard devaon whch would be observed on an nfnely dversfed homogeneous porfolo of borrowers n he secor. The defaul rae x for he h secor s assumed o follow a Gamma dsrbuon, wh parameers α and β se o yeld a gven mean defaul rae µ and volaly of defaul rae σ : x ~ Γ [ α, β ], 5 see Wlson (1997). 6 Wlson (1997) suggess several possbles. 7 see Cred Susse Fnancal Producs (1997). - - 1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

where α µ = σ, and σ β =. µ In he sngle-secor case, a borrower s defaul rae s scaled o hs Gamma-dsrbued secor defaul rae. For mulple secors, a borrower s defaul rae s scaled accordng o he weghed average of secor defaul raes: x p X = p ω, µ where p s he uncondonal defaul rae, and ω s he wegh n secor, ω =. 1 For a homogeneous sub-porfolo of borrowers (same secor and same exposure sze) wh ndependen on-defaul behavor, CredRs+ assumes ha he defaul dsrbuon follows he Posson dsrbuon. Snce borrowers on-defaul behavors are ndependen condonal on fxed defaul raes, he uncondonal defaul dsrbuon for he homogeneous sub-porfolo can be obaned by averagng Posson condonal defaul dsrbuons accordng o defaul raes from he Gamma dsrbuon sascally, he convoluon of he Posson dsrbuon wh he Gamma dsrbuon. Ths convoluon leads o an analyc expresson for he resulng uncondonal dsrbuon of porfolo losses. As presened so far, hese models appear o be que dssmlar. CredMercs s a boom-up model (each borrower s defaul s modeled ndvdually) wh a mcroeconomc causal model of defaul (he Meron model). CredPorfoloVew s a boom-up model based on a macroeconomc causal model of subporfolo defaul raes. CredRs+ s an almos enrely op-down model of sub-porfolo defaul raes, mang no assumpons wh regard o causaly. Despe hese apparen dfferences, each of hese models can be demonsraed o f whn a sngle generalzed underlyng framewor, presened n he followng secon. II. Generalzed Underlyng Framewor for Cred Porfolo Modelng The generalzed cred porfolo model consss of hree man componens o calculae he porfolo loss dsrbuon: A. Jon-defaul behavour Defaul raes vary over me, nuvely as a resul of varyng economc condons; when condons are favorable, fewer borrowers defaul, and vce versa. A condonal defaul rae s generaed for each borrower n each sae of he world for he relevan economc condons. The degree of concenraon or correlaon n he porfolo s refleced by he exen o whch he borrowers condonal defaul raes vary ogeher n dfferen saes of he world. B. Condonal dsrbuon of porfolo defaul rae for each sae of he world and s correspondng se of borrowers condonal defaul raes, he condonal dsrbuon of a homogeneous sub-porfolo defaul rae can be calculaed as f ndvdual borrower defauls are ndependen, as all of he ondefaul behavor has been accouned for n generang condonal defaul raes. C. Convoluon / Aggregaon he uncondonal dsrbuon of porfolo defauls s obaned by aggregang homogeneous sub-porfolo s condonal dsrbuon of defaul rae n each sae of he world, and hen by smply averagng across he condonal dsrbuons of porfolo defauls for dfferen saes of he world, weghed by he probably of a gven sae. Ths generalzed framewor (see Fgure 1) allows a srucured comparson of he models. The followng secons explan how each of he hree models approaches hese componens, wheher explcly or mplcly. - 3-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Fgure 1 A. Jon-defaul behavor Meron-based Economerc Sysemc Facors and Transformaon Funcon Acuaral Defaul Rae Dsrbuon B. Condonal Dsrbuon of Porfolo Defauls C. Convoluon / Aggregaon Technque Porfolo Loss Dsrbuon II. A. Condonal Defaul Raes and Probably Dsrbuon of Defaul Rae All hree models explcly or mplcly relae defaul raes o varables descrbng he relevan economc condons ( sysemc facors ). Ths relaonshp can be expressed as a ransformaon funcon, he condonal defaul rae funcon (see Fgure ). Ths funcon s explcly assgned n he economerc model, and can be derved n closed form for boh he Meron-based and acuaral models. The sysemc facors are random, and are usually assumed o be normally dsrbued. Snce he condonal defaul rae s a funcon of hese random sysemc facors, he defaul rae wll also be random. The defaul rae dsrbuon s an explc assumpon n he acuaral model, and an mplc assumpon n he Meronbased and economerc models. For purposes of comparson, he defaul rae dsrbuons mpled by he laer wo models can be derved easly. The relaonshp beween he sysemc facor dsrbuon and he defaul rae dsrbuon s represened graphcally n Fgure. defaul rae Fgure Defaul Rae Transformaon Funcon (unfavorable economc condons) (favorable economc condons) economc condons Sysemc Facors Meron model Snce he Meron model neher assgns he ransformaon funcon, nor assumes a probably dsrbuon for defaul raes explcly, hese relaonshps mus be derved. - 4-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

As explaned n secon I, a borrower s defaul s movaed by a normally dsrbued change n (sandardzed) asse value A, whch s correlaed wh he change n asse value of oher borrowers n he porfolo. Ths change n asse value can be decomposed no a se of normally dsrbued orhogonal sysemc facors x, and a normally dsrbued dosyncrac componen ε : A = b, 1x1 + b, x + + 1 K b, ε, where b, are he facor-loadngs, x, ε ~ d N[0,1 ], and consequenly ~ N[0,1 ]. If he values of he sysemc facors are nown, hen he sandardzed change n asse value wll be normally dsrbued wh a mean gven by he facor loadngs and facor values, and a sandard devaon gven by he wegh of he dosyncrac facor. These sysemc facors and facor loadngs can be seleced n such a way as o exacly replcae he parwse asse correlaon srucure 8 (n general hs wll requre N- 1 facors for N borrowers, fewer f he parwse asse correlaon srucure was creaed from a smaller se of ndusry and counry groupngs). Alernavely, a smaller se of sysemc facors mgh be seleced. Accordng o he Meron model, defaul occurs when A c, where he crcal value c s calbraed o provde he correc uncondonal defaul probably p ; ha s Φ ( c ) = p, where Φ[x] s he cumulave densy funcon of he normal dsrbuon. The defaul rae, condoned on he values of sysemc facors, can hen be expressed as 9 c b, x =Φ p X. 1 b, For he sngle borrower or homogeneous porfolo case, he sysemc facors can be summarzed by a sngle varable m, reducng he ransformaon funcon o c ρ m p m = Φ, 1 ρ where m ~ N[0,1] and ρ b s he asse correlaon n he homogeneous porfolo. = Snce he cumulave normal funcon s bounded [0,1] and concave n he relevan regon, he resulng defaul rae dsrbuon s bounded n [0,1] and sewed o he rgh as n Fgure. The probably densy funcon for he defaul rae f (p) can be derved explcly, as s relaed o he probably densy funcon of sysemc facors ϕ(m) by he followng: ( m( p) ) dm ϕ f ( p) = ϕ ( m( p) ) =. dp dp dm Appled o he Meron-based model s ransformaon funcon and normally-dsrbued sysemc facors, hs yelds 1 c 1 ( p) 1 ρ Φ ρ ϕ f ( p) ρ =, 1 ρ ϕ Φ ( p) ( ) where ϕ(z) s he sandardzed normal densy funcon. A 8 The correlaons mgh be derved by Cholesy decomposon or Prncpal Componens Analyss, for example. 9 Vasce (1987) develops hs represenaon of he Meron model for a sngle facor. - 5-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Economerc model As n secon I, he economerc model explcly defnes he borrower s defaul probably as a funcon of an ndex value: 1 p =., y, 1 + e Ths ndex value n urn depends on macroeconomc varables. The ndex and macroeconomc varable models can be combned o a sngle equaon for he ndex: y, = b,0 + b a a x,,0,, + b, ε, + υ +,. The ndex hus consss of a consan erm (self composed of a consan and he weghed lagged values of macroeconomc varables), and random erms represenng normally dsrbued sysemc and ndexspecfc nnovaons. For he sngle borrower or homogeneous porfolo case, he normally dsrbued sysemc and ndex-specfc nnovaons can be summarzed by a sngle normally dsrbued varable m, so ha y = U + V m,, where U = b 0 +, V = b, a,0 + a, x,, var( υ, ) + ε,, b + +, cov( υ,, ε, ) b, var( ε, ) b, b, m cov( ε,, m ) m and m ~ N[0,1]. The condonal defaul rae funcon can hen be expressed as 1 p m =. 1 U+ V m + e Snce he Log funcon s bounded [0,1] and concave, he resulng dsrbuon s bounded [0,1] and sewed o he rgh as n Fgure. Dervng he mpled probably densy funcon for he defaul rae f (p) proceeds us as n he Meronbased model, yeldng 1 p ln U 1 p f ( p) = ϕ V p p V. (1 ) Acuaral model The acuaral model assumes ha he defaul rae dsrbuon f ( p; µ, σ ) follows a Gamma dsrbuon, or n he mul-secor case, a weghed-average of Gamma dsrbuons 10. The Gamma dsrbuon s bounded a lef by zero bu has nfne posve suppor. I s possble o derve he acuaral model s mpled ransformaon funcon such ha when appled o a normally dsrbued sysemc facor, resuls n a Gamma dsrbuon for he defaul rae. The ransformaon funcon consss of all pons ( χ, ξ) whch sasfy: ξ 0 Γ( p ; α, β) dp = ϕ( m) dm. χ 10 Only he sngle-secor case s consdered n he dscusson whch follows. A weghed-average of Gamma dsrbuons wll no n general be a Gamma dsrbuon self. The equaons hroughou he remander of hs paper can be modfed o he mul-secor case wh moderae addonal complexy. However, all else equal, hs should no mae much dfference. - 6-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Hence, he ransformaon funcon s gven by: 1 p m = Ψ 1 Φ( m); α,β, p where α =, σ Γ. ( z;α,β ) ( ) σ p β =, and ( z;α,β ) Ψ s he cumulave densy funcon of he Gamma dsrbuon Noe ha all hree models are relaed as descrbed o normally dsrbued sysemc facors, bu he normal dsrbuon s no a crcal assumpon. The defaul rae dsrbuons for he Meron-based and economerc models, and he mpled condonal defaul rae funcon n he acuaral model, can easly be calculaed for an arbrary sysemc facor dsrbuon. Whle non-normaly may affec he resuls of model comparsons, does no n prncple render hem ncomparable. The hree condonal defaul rae funcons and defaul rae dsrbuons are compared n secon IV. II. B. Condonal Dsrbuon of Porfolo Defaul Rae Under he condon ha all loans are ndependen, gven fxed or condonal defaul raes, a homogeneous sub-porfolo s dsrbuon of defauls follows he Bnomal dsrbuon B ( ; n, p), whch provdes he probably ha defauls wll occur n a porfolo of n borrowers gven ha each has probably of defaul p: n n B n p =! ( ;, ) p (1 p! ( n )! ). CredMercs mplcly uses he Bnomal dsrbuon by calculang he change n asse value for each borrower and esng for defaul exacly equvalen o he Bnomal case of wo saes wh a gven probably. CredPorfoloVew explcly uses he Bnomal dsrbuon by eravely convolung he ndvdual oblgor dsrbuons, each of whch s Bnomal. CredRs+ approxmaes he Bnomal wh he Posson dsrbuon P ( ; pn), whch provdes he probably ha defauls wll occur n a porfolo of n borrowers gven ha each occurrence has a rae of nensy per un me p: ( pn) pn P( ; pn) = e.! The Bnomal and Posson dsrbuons are que smlar. Indeed, he Posson dsrbuon s easly shown o be he lmng dsrbuon for he Bnomal dsrbuon, as he number of borrowers becomes large and he defaul probably becomes small 11. The Posson dsrbuon does allow for he possbly of mulple defauls for a sngle borrower, bu, for reasonably small defaul raes, he probably of mulple defauls s neglgble. For porfolos whch are heavly concenraed n a few names, or do no have a large enough number of borrowers, he Bnomal and Posson dsrbuons may show dfferences; bu n hese cases, cred rs models may no be relevan anyway, as he queson reduces o wheher or no parcular borrowers defaul. The dfference beween Bnomal and Posson should no be sgnfcan for reasonable porfolos 1. II. C. Aggregaon The uncondonal probably dsrbuon of porfolo defauls s obaned by aggregang he condonal dsrbuons of porfolo defauls under all possble saes of he world for relevan economc condons. Ths s smply calculaed by averagng across he condonal dsrbuons of porfolo defauls for dfferen saes of he world, weghed by he probably of a gven sae, as depced n Fgure 3. Mahemacally, hs s expressed as a convoluon negral. 11 see Freund (199). 1 Suar & Ord (1994) provdes expressons for he maxmum dfference beween he Bnomal and Posson dsrbuons. - 7-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Fgure 3 0% 1% % 3% 4% 5% Porfolo Loss Dsrbuon Condonal Loss Dsrbuons 0% 1% % 3% 4% 5% Defaul Rae Dsrbuon The Meron-based and economerc models are condoned on normally dsrbued sysemc facors, and he ndependen loans are Bnomally dsrbued. Hence, he convoluon negral for a homogeneous subporfolo wh a sngle sysemc facor s expressed as Prob( defauls na sub-porfolo of nborrowers ) = B( ; n, p m ) ϕ ( m) dm. The acuaral model s condoned on he random defaul rae, and he ndependen loans are assumed o be Posson dsrbued. Therefore, he convoluon negral for a homogeneous sub-porfolo s gven by: Prob( defauls n asub-porfolo of nborrowers ) = 0 P( ; np) Γ ( p;, β ) α dp. These negrals are easly evaluaed; n parcular, he convoluon of he Posson dsrbuon and Gamma dsrbuon yelds a closed-form dsrbuon, he Negave Bnomal Dsrbuon. I s he dfferences beween sub-porfolos dfferng exposure sze or defaul probables, or mulple sysemc facors, complex correlaon srucure, ec. ha creae dffculy n aggregaon. CredMercs performs a Mone Carlo smulaon of boh he sysemc facors and he ndvdual borrower defaul oucomes. Mone Carlo smulaon nroduces samplng error whch can be mgaed by ncreasng he number of smulaons. CredPorfoloVew also uses Mone Carlo smulaon of sysemc facors (and hence defaul probables), and hen evaluaes he condonal porfolo dsrbuon wh a convoluon algorhm. Ths algorhm alles he porfolo dsrbuon n a grd, causng an approxmaon error whch can be mgaed by ncreasng he number of grd pons. CredRs+ evaluaes he convoluons wh a numerc algorhm. Ths numerc algorhm nroduces an approxmaon error n he bandng of exposures 13, whch can be mgaed by decreasng he un sze. In all hree cases, he procedures are heorecally exac n he lm. Fgure 4 depcs he models as hey are redefned n relaon o he generalzed framewor, hghlghng he specfc componens of each. 13 See CSFP (1997) for analyss of he magnude of hs approxmaon error. - 8-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Fgure 4 CredMercs CredPorfoloVew CredRs+ Jon-Defaul Behavor Dsrbuon of Sysemc Facors (Normal) Condonal Defaul Rae Meron P m macroeconomc model regresson m m Defaul Rae Dsrbuon (Gamma) P Condonal Defaul Dsrbuon Bnomal Dsrbuon Posson Dsrbuon Convoluon / Aggregaon Mone Carlo Smulaon Numerc Algorhm Porfolo Loss Dsrbuon III. Harmonzaon of on-defaul parameers The precedng dscusson of he models worngs hghlghs ha all hree models crcally depend on wo elemens of nformaon: uncondonal defaul probably and on-defaul behavor. Whle uncondonal defaul probably appears n a relavely conssen, sraghforward manner n each model, on-defaul behavor appears n a varey of forms. The Meron-based model uses parwse asse correlaons; he acuaral model uses secor defaul rae volales and borrower secor weghngs; and he economerc model calculaes regresson coeffcens o macroeconomc facors whch ncorporae correlaons amongs he facors. Alhough hese parameers are very dfferen n naure, hey all are relaed o each oher and should conan equvalen nformaon o characerze on-defaul behavor. Coeffcens and correlaons As descrbed n secon II.A, on-defaul behavor s represened n Meron-based models as a parwse asse correlaon marx, or equvalenly as a se of asse facor-loadngs for each borrower: A = b, 1x1 + b, x + + 1 K b, ε. The sysemc facors are defned o be orhonormal, so ha E[ A A ] E[ A ] E[ A ] correlao n[ A, A ] = = b,1b,1 + b,b, + K ( E[ A ] E[ A ] )( E[ A ] E[ A ] ) Usng hs relaonshp, a parwse correlaon marx s easly calculaed gven asse facor-loadngs, and facor-loadngs can be derved from a parwse correlaon marx (hough he facors wll no be specfed). Hence asse correlaon wll be relaed o oher on-defaul behavor parameers, as s usefully a sngle sasc for any par of borrowers; asse facor-loadngs wll frs need o be ranslaed o asse correlaon n order o mae use of he relaonshps whch wll be derved. The economerc model s logsc regresson coeffcens, whch defne he relaonshp of he defaul rae ndex o macroeconomc varables, bear srong resemblance o he asse facor loadngs of he Meronbased model. In fac, an ndex correlaon s easly defned n a smlar fashon. Frs, as n secon II.1, he ndex s re-saed n erms of ndex coeffcens and random nnovaons as well as macroeconomc varable coeffcens and random nnovaons: y, = b,0 + b a a x,,0,, + b, ε, + υ +,. Then, - 9-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

E[ y correlao n[ y, y ] = V = =, y cov( υ,, var( y, υ ] E[ y,,, )var( y ) + ] E[ y, ), ] b, cov( υ,, ε, ) + b, cov( υ,, ε, ) + b, b, m cov( ε,, εm, ) m V V var( υ, ) + b cov( υ, ε ) + b var( ε ) + b b cov( ε, ε, ). where,,,,,,, m, m m Asse correlaon and ndex correlaon are very smlar n concep, bu wll provde slghly dfferen resuls o he exen ha her respecve condonal defaul rae funcons are dfferen (see secon IV). For he purposes of ranslaon o oher forms of on-defaul parameers, he degree of smlary should be suffcen; accordngly, only asse correlaon wll be consdered n he dscusson whch follows. The neresed reader s encouraged o wor ou he relaonshps for ndex correlaons as dsnc from asse correlaons usng he same echnques. Defaul rae volaly Defaul rae volaly σ s calculaed by he sandard formula for varance: σ = ( p p) f( p) dp. 0 In parcular, for he Meron model, he defaul rae volaly can be expressed as a funcon of ρ and p : 1 Φ [ p] ρ m σ = ( p m p) ϕ( m) dm = ( Φ p) ϕ( m) dm. 1 ρ Fgure 5 shows he resulng relaonshp beween asse correlaon and defaul rae volaly. Fgure 5 Defaul Rae Volaly 10.00% 9.00% 8.00% 7.00% 6.00% 5.00% 4.00% 3.00%.00% p = 5.00% p = 4.00% p = 3.00% p =.00% p = 1.00% p = 0.50% 1.00% 0.00% 0.00 0.10 0.0 0.30 0.40 0.50 Asse Correlaon Defaul correlaon Some applcaons, and ndeed some models, ae a Marowz varance-covarance vew o cred rs porfolo modelng. Each borrower has a varance of defaul gven by he varance for a Bernoull varable: var( defaul ) = p (1 p ). A parwse defaul correlaon marx s specfed, and he porfolo varance (due o defauls) can be T calculaed by he sandard ω Σ ω operaon (ω s he poson vecor n hs case conanng each borrower s exposure ne of recovery and Σ s he covarance marx). - 10-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

For a homogeneous porfolo wh a large number of names, he porfolo varance approaches σ = p (1 p) ρdefaul. As he nfne homogeneous porfolo s defaul rae volaly s precsely he parameer n he acuaral model, hs provdes he relaonshp beween defaul correlaon and defaul rae volaly for wo borrowers wh he same uncondonal defaul rae 14. Ths relaonshp s llusraed for varous levels of uncondonal defaul rae n Fgure 6. Fgure 6 Defaul Rae Volaly 10.00% 9.00% 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% p= 5.00% p= 4.00% p= 3.00% p=.00% p= 1.00% p= 0.50%.00% 1.00% 0.00% 0.00 0.05 0.10 0.15 0.0 Defaul Correlaon In general, defaul correlaon s calculaed by he sandard formula E[defaul and defaul ] E[defaul ] E[defaul ] E[defaul and defaul ] p p ρdefaul (, ) = = var( defaul ) var( defaul ) ( p (1 p ))( p (1 p )) where E [ defaul and defaul ] = p p f ( p, p ) dp dp, 0 0 or, f he defaul rae can be saed n erms of a sysemc facor, E[ defaul and defaul ] = p m p m ϕ ( m) dm. Reverng o he causal model of defaul n he Meron model, a smplfed expresson can be derved for defaul correlaon n he Meron model, as on-defaul occurs only when he wo borrowers (sandardzed) asse values fall below her respecve crcal values: c E [ defaul and defaul ] = f ( A, A ; ρ ) d A d A. c The on dsrbuon of wo borrowers changes n asse values s he bvarae normal dsrbuon: A ρ A A + A 1 (1 ρ ) ( A, A ; ρ) = e f. π 1 ρ Thus calculaed, he relaonshp beween asse correlaon and defaul correlaon s llusraed n Fgure 7., 14 CSFP (1997) provdes a general formula for he defaul correlaon beween wo borrowers wh dfferng defaul probables, gven her secor allocaons and he secor defaul rae volales. - 11-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Defaul Correlaon 0.0 0.18 0.16 0.14 0.1 0.10 0.08 0.06 0.04 0.0 Fgure 7 p = 5.00% p = 4.00% p = 3.00% p =.00% p = 1.00% p = 0.50% 0.00 0.00 0.10 0.0 0.30 0.40 0.50 Asse Correlaon Ths calculaon can be performed for dfferng defaul probables as well. Noe ha also apples o he mul-secor case, as for any par of borrowers, he sysemc facors can be summarzed o a sngle facor;.e. wo borrowers always have us one asse correlaon no maer how many facors here are. These mappngs n Fgures 5, 6, and 7 relae he on-defaul parameers n all hree models as well as he covarance model. Such mappngs are especally useful n parameer esmaon. For nsance, n he absence of equy prces, defaul rae volales can be used o esmae mpled asse correlaons. The mappngs also allow parameer esmaes o be rangulaed by mulple mehods, o he exen ha model dfferences are no sgnfcan. Noe, however, ha hese llusraed mappngs assume he homogeneous porfolo case ( p = p p ); f = he defaul rae dffers beween borrowers, he more general ranslaon formulae wll be requred. IV. Dfferences n he Defaul Rae Dsrbuon The dscusson n secon II demonsraes ha subsanal model dfferences could arse only from he dfferng reamen of on-defaul behavor he condonal defaul dsrbuons are effecvely he same for he relevan cases, and he convoluon and aggregaon echnques are heorecally exac n he lm. Secon III has provded he means o compare he on-defaul behavor on an apples-o-apples bass. Ths comparson wll be llusraed for a homogeneous porfolo wh an uncondonal defaul rae p of 116bp and a sandard devaon of defaul rae σ equal o 90bp 15. Snce each model produces a woparameer defaul rae dsrbuon, he mean and sandard devaon are suffcen sascs o defne he parameers for any of he models. Before he comparsons can be performed, he relevan parameers for each model mus be derved such ha he models mach he uncondonal defaul rae and sandard devaon of defaul rae. Meron-based model The meron model requres wo parameers: he crcal value c, and he asse correlaon ρ. The crcal value s defned n erms of he uncondonal defaul probably: 1 p c = Φ ( ) As n secon III, he defaul rae volaly s calculaed by c ρ m σ = ( p m p ) ϕ( m) dm = Φ p ϕ( m) dm, 1 ρ whch depends only on c and ρ. 15 These parameers were seleced o mach Moody s All Corporaes defaul experence for 1970-1995, as repored n Cary & Leberman (1996). - 1-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

In order o yeld p = 116bp and σ = 90bp, he parameers for he Meron-based model are c = -.7 and ρ = 0.073. Economerc model The economerc model requres a varey of consans and coeffcens whch, as n secon II.A, can be summarzed by wo parameers U and V. The wo parameers are found by solvng he followng sysem of equaons: = 1 p ϕ ( m) + + e dm, and U Vm 1 1 σ = p ϕ( m) dm. + 1 + e U Vm In order o yeld p = 116bp and σ = 90bp, he parameers for he economerc model are U = 4.684 and V = 0.699. Acuaral model The parameers requred by he acuaral model are defned drecly n erms of he uncondonal defaul rae and defaul rae volaly: p σ α =, and β =. σ p In order o yeld p = 116bp and σ = 90bp, he acuaral model s Gamma dsrbuon parameers are α = 1.661 and β = 0.0070. Fgure 8 compares he condonal defaul rae funcons. In hs example, he models are vrually ndsngushable when he sysemc facor s greaer han negave wo sandard devaons, whch accouns for almos 98% of he probably mass. For exremely unfavorable economc condons (sysemc facor less han negave wo sandard devaons), he economerc model predcs a somewha hgher defaul rae, and he acuaral model predcs a somewha lower defaul rae. Fgure 8 Condonal Defaul Rae 1% 10% defaul rae (p m) 8% 6% 4% Meron Economerc Acuaral % 0% -4-3 - -1 0 1 3 4 sysemc facor (m) Fgure 9 compares he defaul rae dsrbuons, whch naurally show a smlar resul. These models mply very smlar probably densy funcons for he defaul rae, wh only mnor dscrepances a he al. Fgure 10 compares he rgh als of hese defaul rae dsrbuons. Noe ha boh of hese fgures show defaul rae dsrbuons, no porfolo loss dsrbuons; he porfolo loss dsrbuon approaches he defaul rae dsrbuon as he number of borrowers becomes very large. - 13-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Fgure 9 Defaul Rae Dsrbuons Fgure 10 Defaul Rae Dsrbuons -- Tal Probably Meron Economerc Acuaral Probably Meron Economerc Acuaral p + σ 0% 1% % 3% 4% 5% Defaul Rae 3% 4% 5% 6% Defaul Rae The degree of agreemen n he als of hese dsrbuons can be assessed wh he followng sasc: f ( x) g( x) dx Ξ ( f, g) = 1, z z z f ( x) dx + z g( x) dx where f(x) and g(x) are probably densy funcons and z defnes he lower bounds of he al. Ths sasc measures he amoun of he probably dsrbuons mass whch overlaps n he relevan regon, normalzed o he oal probably mass of he wo dsrbuons n ha regon. The sasc wll be bounded [0,1], where zero represens wo dsrbuons wh no overlappng probably mass, and one represens exac agreemen beween he dsrbuons. The al has been defned arbrarly as he area more han wo sandard devaons above he mean,.e. z = p + σ. The al-agreemen sascs for he example dsrbuons n Fgures 8-10 are gven n Table 1. Table 1 Ξ(Meron vs. Economerc) 94.90% Ξ(Meron vs. Acuaral) 93.38% Ξ(Economerc vs. Acuaral) 88.65% Whou a credble alernave dsrbuon (he normal dsrbuon, for example, s obvously napproprae and would clearly show ha he hree dsrbuons are relavely very dfferen from normal and very smlar o each oher 16 ), hs al-agreemen sasc provdes a relave raher han absolue measure. However, can be used o es he robusness of he smlary o he parameers, as n he Table : 16 Comparson o he normal dsrbuon would yeld a al-agreemen sasc of less han 70%, gven ha he normal dsrbuon has.3% probably mass n he al above wo sandard devaons whereas he hree cred model dsrbuons have around 4.4% - 4.7% probably mass n her als. - 14-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

0.05% 0.10% 0.5% p 0.50% 1.00%.50% 5.00% 10.00% Table σ / p 0.50 1.00.00 3.00 98.10% 95.94% 91.16% 89.1% 93.53% 88.50% 81.17% 78.99% 91.71% 84.70% 73.13% 69.04% 97.9% 93.75% 91.73% 97.61% 94.11% 91.79% 97.33% 94.4% 91.88% 97.06% 94.93% 9.04% 96.6% 95.8% 9.6% 96.33% 97.0% 93.55% 96.1% 98.55% 95.45% 95.57% 88.94% 84.78% 94.93% 89.73% 84.97% 94.41% 90.56% 85.30% 93.97% 91.69% 85.93% 93.6% 93.94% 87.79% 93.85% 96.70% 90.87% 94.9% 95.57% 95.% 91.15% 8.16% 73.95% 90.35% 83.87% 74.83% 89.91% 85.71% 76.15% 89.89% 88.9% 78.57% 90.77% 93.65% 84.77% 9.79% 95.4% 91.38% 95.79% 7.95% 7.41% 88.40% 80.40% 69.73% 87.86% 8.9% 71.60% 88.09% 85.61% 74.8% 88.97% 89.38% 78.7% 91.33% 94.68% 87.53% 94.59% 81.79% 79.96% N/A* *No a reasonable combnaon of parameers model resuls become unsable Ξ(Meron vs. Economerc) Ξ(Meron vs. Acuaral) Ξ(Economerc vs. Acuaral) The resuls n Table demonsrae ha he smlary of he models holds for a reasonably wde range of parameers. The models begn o dverge a a very hgh rao of defaul rae volaly o defaul probably, parcularly for very low or very hgh defaul probables (upper rgh and lower rgh regons of he able). In he case of very low defaul probables (upper rgh) he condonal defaul rae funcons of he Meron-based and economerc models become much more concave han ha of he acuaral model. In he case of very hgh defaul probables (lower rgh), he gamma dsrbuon s lac of upper bound begns o have a sgnfcan effec. Accordngly, n very hgh qualy (AA or beer) or very low qualy (B or worse) porfolos, model selecon can mae a dfference, hough here s scan daa on whch o base such selecon. In a porfolo where very hgh or very low qualy sub-porfolos have only moderae wegh, hese dfferences should no be sgnfcan n aggregaon. Ths fndng, ha he models are que smlar across a broad range of reasonable parameers, should be aen wh cauon, as hnges on usng harmonzed parameer values. In pracce, he parameer values acually used wll vary by esmaon echnque. The dfferen esmaon echnques approprae o dfferen on-defaul parameers may resul n esmaes whch are nconssen relave o he equvalences n secon III. Even defaul probables may vary consderably dependng on he esmaon echnque, sample, ec. Unsurprsngly, when he parameers do no mply conssen mean and sandard devaon of defaul rae dsrbuon, he resul s ha he models are sgnfcanly dfferen. Such a case s llusraed n Fgure 11, where hree models are compared on he bass of hree hypohecal parameer ses (see Table 3) whch are no conssen, hough plausbly obanable for he same porfolo. Table 4 presens he al-agreemen sascs for all possble combnaons of model and parameer se. - 15-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

parameer se 1 Fgure 11 10% 9% 8% Meron Economerc Acuaral condonal defaul rae 7% 6% 5% 4% 3% % 1% 0% -3 - -1 0 1 3 sysemc facor condonal defaul rae parameer se 10% 9% 8% 7% 6% 5% 4% 3% % Meron Economerc Acuaral condonal defaul rae nconssen parameers 10% 9% 8% 7% 6% 5% 4% 3% % 1% 1-Economerc -Acuaral 3-Meron 1% 0% -3 - -1 0 1 3 sysemc facor condonal defaul rae parameer se 3 10% 9% 8% 7% 6% 5% 4% 3% % 1% Meron Economerc Acuaral 0% -3 - -1 0 1 3 sysemc facor 0% -3 - -1 0 1 3 sysemc facor Table 3 Hypohecal nconssen parameer values: p σ c ρ α β U V model for comparson* 1.6% 1.70% -.00 8.5% 1.767 0.018 4.00 0.70 Economerc 1.5% 1.71% -.16 14.4% 0.790 0.019 4.60 0.95 Acuaral 3 1.54%.63% -.16 6.% 0.343 0.0449 4.95 1.30 Meron *In he nconssen parameer case, he parameer ses models for comparson were seleced arbrarly. Unshaded cells ndcae he parameers approprae o he seleced model. - 16-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

Table 4 Tal-agreemen sascs ** for hypohecal nconssen parameer ses: 1 Meron 1 Economerc 1 Acuaral Meron Economerc Acuaral 3 Meron 3 Economerc 3 Acuaral 1 Meron N/A 1 Economerc 94.78% N/A 1 Acuaral 94.47% 89.43% N/A Meron 75.86% 80.74% 71.11% N/A Economerc 69.53% 74.40% 65.0% 93.03% N/A Acuaral 81.63% 85.00% 78.05% 91.96% 85.7% N/A 3 Meron 73.85% 77.46% 68.90% 77.74% 7.8% 77.85% N/A 3 Economerc 70.87% 75.17% 65.69% 83.36% 80.94% 79.3% 91.8% N/A 3 Acuaral 76.53% 78.77% 7.09% 70.7% 65.55% 73.85% 90.91% 8.49% N/A **To allow for comparson beween parameer ses, he al was conssenly defned as he smalles value of z = p + σ, 494bp (parameer se ) Bold fgures represen he hree possbles llusraed n he nconssen parameers case n Fgure 11. Table 4 shows ha whn a conssen parameer se (3x3 boxes on he dagonal), he models are n close agreemen, wh al-agreemen sascs averagng 91.40% and rangng from 8.49% o 94.78%. Large dfferences are found beween nconssen parameer ses (3x3 boxes off of he dagonal), wh alagreemen sascs averagng 76.18% and rangng from 65.0% o 85.00%. The dfferences n parameers, well whn he ypcal range of esmaon error, have much greaer mpac han model dfferences n hs example. V. Conclusons On he surface, he cred rs porfolo models suded n hs paper seem o be que dfferen he approaches range from a boom-up mcroeconomc model of an ndvdual borrower s defaul o a opdown macroeconomc model of he defaul rae for a sub-porfolo of borrowers. However, deeper examnaon reveals ha he models belong o a sngle general framewor, whch denfes hree crcal pons of comparson he defaul rae dsrbuon, he condonal defaul dsrbuon, and he convoluon / aggregaon echnque. Dfferences were found o be mmaeral n he condonal defaul dsrbuons and he convoluon / aggregaon echnques, so ha any sgnfcan dfferences beween he models mus arse from dfferences n modelng on-defaul behavor whch manfes n he defaul rae dsrbuon. Furher, when he ondefaul parameer values are harmonzed o a conssen expresson of defaul rae and defaul rae volaly, he defaul rae dsrbuons are suffcenly smlar as o cause lle meanngful dfference across a broad range of reasonable parameer values. Any sgnfcan model dfferences can hen be arbued o parameer value esmaes whch have nconssen mplcaons for he observable defaul rae behavor. Parameer nconssency s no a rval ssue. A naïve comparson of he models, wh parameers esmaed from dfferen daa usng dfferen echnques, s que lely o produce sgnfcanly dfferen resuls for he same porfolo. The conclusons of emprcal comparsons of he models wll vary accordng o he degree of dfference n parameers 17. In such comparsons, s mporan o undersand he proporons of parameer varance and model varance f dfferen resuls are produced for he same porfolo. The fndngs n hs paper sugges ha parameer varance s lely o domnae. Fuure sudes should focus on he magnude of parameer dfferences and he sensvy of resuls o hese dfferences. The mplcaons of parameer nconssency range beyond sample porfolo comparsons. Parameer nconssency can arse from wo sources: (1) esmaon error, whch could arse from small sample sze or oher samplng ssues, or () model ms-specfcaon. Whle defaul rae volaly may be mmedaely observable, even long perods of observaon provde small sample sze (e.g. 5 years of defaul rae experence s only 5 daa pons) and rs change n he underlyng behavors. A he oher exreme, asse correlaons can be measured wh reasonable sample sze n much shorer perods, bu rs msspecfcaon n he assumpons wh respec o reurn dsrbuons and defaul causaly n he ranslaon from asse correlaons o defaul rae volaly. Raher han conclude ha parameer nconssency 17 For example, ISDA (1998) and Robers and Wener (1998) compare he resuls of several models on es porfolos. The former fnds ha model resuls are farly conssen, whle he laer fnds ha he models may produce que dfferen resuls for he same porfolo usng parameers ndependenly seleced for each model. - 17-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)

poenally consues rreconclable dfferences beween he resuls of hese models, hs paper concludes ha because he models are so closely relaed, he esmaes are complemenary and should provde mproved accuracy n parameer esmaon whn he generalzed framewor as a whole. The fndngs of hs paper do no n any way sugges ha one of he models s necessarly superor o he ohers, nor do hey sugges ha users should be ndfferen beween he models. Raher, hese fndngs sugges ha relave heorecal correcness need no ran among a user s model selecon crera, whch mgh hen conss prmarly of praccal concerns such as ease of use, daa avalably, speed, flexby, ec. whch are beyond he scope of hs paper. A useful meaphor can be drawn from he success of he Value-a-Rs framewor n modelng mare rs. Value-a-Rs has become he ndusry sandard, wh wdespread use among end-users and accepance as he bass for capal requremens among regulaors. Bu n pracce, Value-a-Rs encompasses a varey of dfferen modelng echnques, as well as dfferen parameer esmaon echnques, some of whch can be que sgnfcan; for example, hsorcal smulaon vs. varancecovarance, dela-gamma vs exac Mone Carlo smulaon, ec. I s he underlyng coherence of he Valuea-Rs concep ha rs s measured by combnng he relaonshp beween he value of radng posons o mare varables wh he dsrbuon of hose underlyng mare varables whch ensures a conssency suffcen for wdespread accepance and regulaory change. In he same way, he underlyng coherence of concep amongs hese new sophscaed cred rs porfolo models should allow hem o overcome dfferences n calculaon procedures and parameer esmaon. Raher han dssmlar compeng alernaves, hese models represen an emergng ndusry sandard for cred rs managemen and regulaon. References Cary, Lea and Dana Leberman Corporae Bond Defauls and Defaul Raes 1938 1995, Moody s Invesors Servce Global Cred Research, January 1996 Cred Susse Fnancal Producs CredRs+ A Cred Rs Managemen Framewor, 1997 Inernaonal Swaps and Dervaves Assocaon Cred Rs and Regulaory Capal, March 1998 Freund, John "Mahemacal Sascs", 5 h edon, Prencel Hall, New Jersey, 199 Gupon, Greg, Chrsopher Fnger, and Mcey Bhaa CredMercs Techncal Documen, Morgan Guarany Trus Co., 1997 Kealhoffer, Sephen Managng Defaul Rs n Dervave Porfolos n Dervave Cred Rs: Advances n Measuremen and Managemen, Renassance Rs Publcaons, London, 1995 Kurzes, Andrew Transformng Porfolo Managemen Banng Sraeges, July-Augus 1998, p. 57-6 Meron, Rober On he Prcng of Corporae Deb: The Rs Srucure of Ineres Raes Journal of Fnance, vol. 9, 1974 Robers, Dylan and James Wener Handle wh Care, Olver, Wyman & Company worng paper, 1998 Vasce, Oldrch Probably of Loss on Loan Porfolo, KMV Corporaon, 1 February 1987 Suar, Alan and J. Keh Ord Kendall s Advanced Theory of Sascs Volume 1: Dsrbuon Theory 6 h edon, John Wley & Sons, New Yor, 1994 Wlson, Tom Porfolo Cred Rs, Rs, Sepember 1997 (par I) and Ocober 1997 (par II) - 18-1998 Koyluoglu (Olver, Wyman & Co.) and Hcman (CSFP Capal, Inc.)