ENHANCING BASEL METHOD VIA CONDITIONAL DISTRIBUTIONS THAT CAPTURE STRONGER CONNECTION AMONG CREDIT LOSSES IN DOWNTURNS

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1 ENHANCING BASEL METHOD VIA CONDITIONAL DISTRIBUTIONS THAT CAPTURE STRONGER CONNECTION AMONG CREDIT LOSSES IN DOWNTURNS Fernando F. Morera Credt Researh Centre, Unversty of Ednburgh Emal: ABSTRACT Ths paper suggests a formula able to apture potental stronger onneton among redt losses n downturns wthout assumng any spef dstrbuton for the varables nvolved. We frst show that the urrent model adopted by regulators (Basel) s equvalent to a ondtonal dstrbuton derved from the Gaussan Copula (whh does not dentfy tal dependene). We then use ondtonal dstrbutons derved from opulas that express tal dependene (stronger dependene aross hgher losses) to estmate the probablty of redt losses n extreme senaros (rses). Next, we present an example based on a spef opula that ndates upper-tal dependene among redt losses. Smulatons show that, for both redt lasses tested (retal and orporate), the alternatve method outperforms the Basel formula whh, n turn, s prone to result n nsuffent aptal when the losses have postvely-skewed dstrbutons (long tal n the rght sde) and are upper-tal dependent (a realst representaton of redt portfolos, aordng to the lterature). The method proposed s extendable to any dfferentable opula famly whh gves flexblty to future pratal applatons of the model. JEL odes: G8, G1, G3, C46, C49 Keywords: Fnanal Regulaton, Basel Aords, redt rsk, ondtonal dstrbutons, opulas.

2 1. INTRODUCTION The model (Basel Aord) adopted by regulators n many ountres to alulate the aptal to over unexpeted redt losses n fnanal nsttutons assumes normally-dstrbuted varables and uses the lnear orrelaton to measure dependene aross losses. However, these assumptons do not allow the dentfaton of possble stronger dependene aross losses n extreme senaros (whh seems to be the ase for several fnanal assets, loans nluded) and, therefore, the Basel method may underestmate jont redt losses n perods of rss. Albet the formula urrently used n Basel Aords has a dervaton not assoated to opula funtons, we show that t turns out to be equvalent to the frst dervatve of the Gaussan Copula (whh denotes symmetr assoaton wthout tal dependene). Moreover, the dstrbuton of one varable ondtonal on another varable an be alulated as the frst dervatve of the opula that represents the dependene between the onsdered varables wth respet to the ondtonng varable. In other words, the Basel formula an be nterpreted as the umulatve dstrbuton of a latent varable (asset returns of oblgors, for nstane) ondtonal on the eonom status. Based on ths nterpretaton of the Basel model, we propose the use of opulas that apture stronger dependene among hgh losses (stronger dependene among low values of debtors asset returns) to generate alternatve ondtonal dstrbutons. So, we keep the bas ntuton of the tradtonal approah but hange the dependene struture suh that we an dentfy hgher probablty of default n adverse senaros. The alternatve model s basally set as the frst dervatve of the opula hosen to represent the relatonshp between the latent varable and the eonom fator wth respet to the latter varable. At ths pont, we fae a hallenge pertanng to the opula parameter that measures the dependene ntensty. For some opulas, ths parameter an be dretly dedued from the rank orrelaton (Kendall s tau) between the varables. Thus we need to fnd the rank orrelaton between the latent varable of eah loan and the eonom fator but we annot alulate t sne we do not have enough nformaton about the seond varable. To overome ths problem, we show that the rank orrelaton between the latent varable of eah debtor and the eonom fator s related to the rank orrelaton between two latent varables (e.g. asset returns of two oblgors) whh an be presumed from past losses (default rates). One we have an estmate for the former rank orrelaton, we wll have all neessary nformaton to alulate the ondtonal probablty by means of the frst dervatve of a opula wth a gven onfdene (unfavorable eonom level). 1

3 As an example, we present a formula orgnated from the Clayton Copula that s able to detet stronger onneton (tal dependene) among low values of latent varables (whh s equvalent to dentfy hgher dependene among hgh redt losses). The resultant formula to alulate unexpeted (extreme) redt losses does not assume any knd of dstrbuton for the varables onsdered and therefore overomes the lmtatons of the exstng method wth regard to the assumpton of normalty and the use of the lnear orrelaton. Smulatons orroborate our hypothess that the Basel formula tends to underestmate the lkelhood of jont extreme defaults when losses present postvely-skewed dstrbutons and rght-tal dependene (whh, aordng to the pertnent lterature 1, haraterze loan portfolos held by fnanal nsttutons) and show that the alternatve approah yelds better estmates of unexpeted losses when ompared to Basel. In short, our ontrbutons are threefold: () we present an alternatve dervaton of the Basel formula and show that t orresponds to the frst dervatve of the Gaussan Copula; () we set up a model able to apture stronger dependene among redt losses n unfavorable senaros whh results n more effent estmatons of potental extreme losses; and () we propose a way to derve the dependene between a latent varable of eah loan and an eonom fator from the dependene observed aross loans default rates. Ths paper s organzed as follows. In Seton, we ntrodue opula funtons and explan how ondtonal dstrbutons an be derved from them. Then, we present two dervatons of the Basel formula used to estmate extreme redt losses. Seton 4 ontans an example of ondtonal dstrbutons that an apture potental asymmetr dependene aross losses. Smulatons are used n Seton 5 to ompare the performane of the proposed model to the performane of the Basel model. Seton 6 onludes.. COPULAS AND CONDITIONAL DISTRIBUTIONS Copulas are multvarate dstrbuton funtons wth unformly dstrbuted margns n (0,1) that lnk margnal (ndvdual) dstrbutons of varables to ther jont dstrbutons: F x,..., x ) C( F ( x ),..., F 1... n ( 1 n 1 1 n ( xn )) 1 Mentoned ahead.

4 where F (.) denotes a umulatve dstrbuton funton and C stands for a opula. Thus, C s an expresson (funton) wth n nputs and, when evaluated at F ( x 1 ),..., F n ( x ), returns the jont umulatve dstrbuton of the n varables evaluated at varables X,...,X 1 n 1 n are onurrently below the respetve values 1 n. x 1,...,x n,.e., the probablty that all x,...,x Aordng to Joe (1996), the umulatve dstrbuton of a random varable ondtonal on other varables s gven by the frst dervatve of the opula that represents the dependene among the varables wth respet to the ondtonng varables (those plaed after the symbol ): C F( x v) xv v j j ( F( x v F( v j j v ), F( v j ) j v j )) where F(x v) s the dstrbuton of x ondtonal on vetor v, Cxv j v s a opula dstrbuton j funton, v j s a omponent of vetor v and v j s the vetor v exludng ths omponent. When v s unvarate, the ondtonal dstrbuton beomes: F( x v) C x v C ( F( x) F( v)) xv ( F( x), F( v)) F( v) where x and v ndate the ondtoned and the ondtonng varables respetvely and the remanng notaton s the same used n the pror formula. The frst dervatve of some bvarate opulas an be found, for example, n Joe (1997, Chapter 5), Aas et al. (009), and Bouyé and Salmon (009). Two famles of partular nterest here are the Gaussan (Normal) and the Clayton that respetvely generate the ondtonal dstrbutons stated n [1] and []: A detaled proof of ths formula s gven n Czado (010). 3

5 Pr[ X 1 x1 X x ] F1 ( x1 X x ) 1 ( F ( x )) ( F ( x )) [ 1 ] where and 1 represent the standard normal dstrbuton and ts nverse respetvely, F(..) s the dstrbuton of X 1 ondtonal on X, F(.) s an unondtonal dstrbuton and 1 s the Gaussan Copula parameter 3 between X 1 and X. Pr[ X1 x1 X x] F1 ( x1 X x) { F ( 1 ) / ( x) [ F1 ( x1 ) 1] 1} [ ] where F(..) s the dstrbuton of X 1 ondtonal on X, F(.) s an unondtonal dstrbuton and s the Clayton Copula parameter between X 1 and X. Note that ths same onept of ondtonal dstrbutons s treated n Bouyé and Salmon (009) as nonlnear quantle regressons. The models proposed here are equvalent to quantle regressons but we do not use ths termnology to keep the dsusson as smple as possble. The strength of the dependene (opula) s expressed by a parameter whh s losely related to rank orrelatons Kendall s tau ( ). For two varables X 1 and X wth dstrbuton funtons evaluated at x 1 and x, F1 ( x1 ) u1 and F ( x ) u orrespondngly, the ntensty of ther representatve opula an be nferred from 4 : [0,1] 4 C( u1, u ) dc( u1, u ) 1 [ 3 ] 3 The parameter of the Gaussan Copula s usually represented by. We adopt the notaton to dstngush the Gaussan Copula parameter from the lnear orrelaton oeffent between the varables studed. These two measures of dependene are dental only when the margnal dstrbutons are normal. 4 The proof s gven n Nelsen (006, hapter 5). 4

6 3. BASEL METHOD: TWO DERIVATIONS 3.1 The alulaton of extreme redt losses For eah homogeneous redt segment, the aptal requred to over unexpeted losses s alulated as the unexpeted losses adjusted by the portfolo maturty. In mathematal terms: [ LGD* KV LGD* PD]* Maturty [ LGD*( KV PD)] * Maturty [ 4 ] where LGD s the loss gven default,.e. the perentage of exposure the lender wll lose f borrowers default and PD stands for probablty of default. Maturty orresponds to the maturty of orporate loans (.e., not appled to onsumer debt) and s added to the alulaton n order to gve hgher weght to long-term oblgatons whh are known to be rsker. For the sake of brevty, the maturty formula s not presented here. See BCBS (005, 006) for more detals. The other term n [4], K V, s the expeted default rate at the 99.9% perentle of the PD dstrbuton ( Vasek Formula ) - see Vasek (1991, 00) - and s alulated as: where: and K V 1 ( PD) 1 1 (0.999) 1 represent the standard normal umulatve dstrbuton and ts nverse, respetvely; PD s the probablty of default of the loan portfolo (average); 1 (0.999), whh s equal to 1 (0.001), s the level of the eonomy (onfdene) hosen to represent an extreme senaro when unexpeted losses may our. Therefore, the systemat fator s assumed to be normally dstrbuted; and [ 5 ] Rho ( ) s the orrelaton between returns of oblgors assets. s the lnear orrelaton between the unobserved systemat fator and those asset returns. In Basel method, the orrelaton between asset returns s alulated as a funton of PD and (n the ase of orporate 5

7 debt) the sze of debtors (measured n terms of annual sales). Thus, for retal loans, the orrelaton s gven by 5 : 1 e 0.03 * 1 e 35* PD e 1 e 35* PD 35 [ 6 ] And, for orporate debts, the orrelaton s alulated as: 1 e.1 * 1 e 50* PD 1 e e 50* PD Sze 0.04 * [ 7 ] where Sze (n mllon) refers to the oblgors sze and s appled for annual sales between 5 mllon and 50 mllon. 3. Dervaton from fator models Some leadng ndustry redt rsk models, suh as CredtMetrs and KMV, rely on the presumptons of strutural models (ntally proposed by Merton, 1974) aordng to whh an oblgor defaults when a latent varable assoated to t (typally nterpreted as the log-returns of ts assets) falls below a threshold (the amount needed to pay the outstandng debt). The dependene aross defaults of dfferent oblgors s estmated n lne wth fator models whh assume that the orrelaton among defaults s drven by the debtors latent varables (see, for nstane, Crouhy et al., 000 and Bluhm et al., 00). Suh underlyng varables are mpated by ommon (systemat) fators that affet all oblgors and spef (dosynrat) fators that have effet only on the respetve borrowers. The dosynrat fators are assumed to be ndependent from one another and therefore do not ontrbute to asset return orrelatons whh are exlusvely determned by the systemat fators. To llustrate ths dea, onsder a ase based on an example gven by Bluhm et al. (00). If two automotve ompanes A and B operatng n ountry C are debtors, the ablty of those frms to 5 Ths formula does not apply to revolvng and mortgage redts, for whh the orrelatons were spefed n Basel as 0.04 and 0.15, respetvely. 6

8 pay ther oblgatons s lkely affeted n the same dreton by the underlyng fator automotve ndustry. That s, f the atvty n that setor falls, the default probablty of A and B nreases smultaneously. Another aspet that ertanly have nfluene on the performane of those ompanes s the ountry C s eonom level. So ths s another systemat fator that may hange the default probablty of A and B n the same way. In ontrast, f the frm A s CEO steps down or one of ts fatores s flooded, ths event wll, n prnple, mpat only the default lkelhood of A (not B s). Hene, ths would be an dosynrat rsk of A. Naturally, there are many ommon fators that at together and nfluene debtors stuaton. However ths model may be smplfed f we onsder that the asset returns of all borrowers are drven by only one ommon fator (the eonom status ). The latent varable (Y ), the sngle systemat fator ( E ), and the spef fator ( ) are assumed to be standardzed normally dstrbuted. Also, eah dosynrat rsk s unorrelated wth the systemat rsk and the spef rsks of all other oblgors. For smplty, all pars of asset returns ( and j) are onsdered to present the same orrelaton ( ). The orrelaton between the systemat fator and the asset return of eah debtor s denoted. Owen and Stek (196) show that equally orrelated and jontly standard normal varables may be expressed as a funton of ther orrelaton oeffent and two other standard normal varables. Thus, onsderng all assumptons of fator models, n the ase of a sngle ommon rsk, the latent varable Y for a debtor may be expressed as a funton of E,, and, namely: Y E 1 [ 8 ] Due to some propertes of jontly standard normal varables, we have: [ 9 ] Ths equalty s essental to the subsequent alulatons sne there s usually no adequate proxy for E (whh s not observable) and, onsequently, annot be dretly estmated from 7

9 empral data. On the other hand, we an nfer the orrelaton between asset returns,, from hstoral losses (default rates). Expresson [9] s often mentoned n the lterature but ts dervaton s rarely presented. In order to fll ths gap, we present one possble dervaton n Appendx A. By replang wth n [8], we get: Y E 1 [ 10 ] where and 1 ndate how muh of the varablty of Y s explaned by E and, respetvely. Apart from the doubtful presumpton of normal behavor for some of these varables, the use of the lnear orrelaton oeffent s a lmtaton gven that t does not apture asymmetr dependene whh ould ndate more or less ntense assoaton aross some varables n ertan senaros (see Embrehts et al., 00). In general, K V (defned n [5]) follows the man presumptons of fator models (see, e.g., Gordy, 003) where eah latent varable ( Y ) s a lnear funton of an unobserved sngle fator (systemat rsk, E ) and spef haratersts of the respetve oblgor (dosynrat rsk, ). The sngle fator s assumed to be standard normally dstrbuted and mpats all oblgors equally (same orrelaton ). The latent varables are onsdered equorrelated (same for all pars) and also follow the standard normal dstrbuton. Ths leads to expresson [10] mentoned above. For eah loan, the probablty of default s the lkelhood that the latent varable Y beomes smaller than the utoff y, that s, PD Pr[ Y y ]. Extreme redt losses happen when the eonomy E reahes an extremely unfavourable level e *. In other words, these hgh losses are the probablty of default ondtonal on a poor eonom status. Representng ths probablty as PD *, we have PD* Pr[ Y y E e*] and usng [10] wth the smplfed notaton for : PD* Pr[ E 1 y E e*] 8

10 Solvng for and replang E wth e *: PD* Pr y e * 1 As mentoned above, Thus, the prevous equaton turns nto: s presumed to be normally dstrbuted wth mean 0 and varane 1. y PD* e* 1 where ndates the df of the standard normal dstrbuton. Sne Y s also normally dstrbuted, PD y ) whh mples that y 1 ( PD),.e. the ( utoff of the latent varable below whh default ours s the nverse of the normal dstrbuton, 1, evaluated at PD. Basel demands onfdene of 99.9% whh means that the aptal s supposed to be suffent to over the losses whenever the eonomy s above (better than) the 0.01 perentle of ts dstrbuton (also assumed to be normal). Hene the extreme adverse senaro e * s gven by 1 (0.001). Due to two propertes of the standard normal dstrbuton (symmetry and mean 0), 1 (0.001) 1 (0.999). Usng ths fat and replang e * wth 1 (0.999) and y wth 1 ( PD ) n the pror equaton, we get the formula presented n Basel Aord (here the extreme loss, PD *, s denoted as K V ): K V 1 ( PD) 1 1 (0.999) Readers nterested n more detals about ths dervaton of Vasek formula ( K ) should onsult, for nstane, Shönbuher (000), Perl and Nayda (004), and Crook and Bellott (010). V 9

11 Some models have been proposed to transform [5] nto another expresson that does not have the lmtaton regardng the assumpton of normalty. Startng from [10], Hull and Whte (004) relax the dstrbutons 6 of Y, E and, suh that they an, for example, present heavy tals (whh tends to nrease the jont ourrenes of extreme realzatons of the latent varables). Representng the dstrbutons of those three varables respetvely by F, G and H and followng the same steps that derved [10] from [5], the expresson to estmate the probablty of default ondtonal on an unfavorable eonom status (the worst 0.1% senaro,.e. wth onfdene of 99.9%) turns nto: Pr[ Y y E e*] F H 1 ( PD) G 1 1 (0.001) where e* ndates an extreme adverse eonom senaro and an be alulated as the nverse dstrbuton of E evaluated at (sne the rtal level was set at 0.1%). PD s the hstoral probablty of default and s the lnear orrelaton between returns of oblgors assets. Obvously, the expresson above annot be solved unless the shapes of the three dstrbutons F, G and H are known. Some studes, suh as Bluhm et al. (00), Kostadnov (005) and Kang (005), have suggested the Student t dstrbuton for E and to haraterze the exstene of more events (than the normal dstrbuton) n the tals. In ths ase, t s not possble to defne the dstrbuton of the latent varable n [10] and the probablty of default n downturns (at the 0.1% worst senaro) s: Pr[ Y y E e*] F T v 1 ( PD) T 1 1 v (0.001) where T v s the Student t dstrbuton wth v degrees of freedom. Gven that the latent varable s dstrbuton F remans unknown, the preedng lkelhood annot be alulated. In vew of the mpossblty of the estmaton of the probablty of default n adverse eonom senaros when 6 Provded that they are saled wth mean zero and varane one. 10

12 one (or more) of the varables n [10] are not normally dstrbuted, we propose a dfferent setup to norporate Copula Theory nto ths analyss and apture potental tal dependene even f we do not know any of the dstrbutons onernng the latent varable, the eonom fator and the dosynrat fators (whh s the realty n fnanal nsttutons). 3.3 Dervaton from the Gaussan Copula Departng from [1], the ondtonal dstrbuton alulated from the Gaussan Copula (restated below for onvenene), onsder that X 1 s a latent varable, x 1 s the level below whh defaults happen and X s the eonom status (sngle fator). So, that formula gves the lkelhood of the latent varable X 1 beng below a spef value x 1 ondtonal on X = x. Assume that both varables follow the standard normal dstrbuton. F 1 ( x 1 X x ) 1 ( F ( x ) ( F ( x ) [ 1 ] restated Therefore F1 ( x1 ) ( x1 ) PD (.e. the probablty of the latent varable X 1 beng below the 1 1 utoff x 1 ) and ( F ( x )) ( PD) returns the latent varable utoff 7. F x ) ( ) s 1 1 ( x the level of the eonom stuaton and the nverse of ts dstrbuton 1 F ( x )) ( 1 ( ( x)) x gves the value of the eonom varable. So, the smaller ( x ) s the worse the eonom status gets and to express adverse senaros n [1] small values for x ) should be used. Basel adopts the onfdene level of 99.9%; as sad before, 1 (0.999) = 1 (0.001). The parameter 1 n [1] refers to the dependene between X 1 and X. If we assume that X 1 and X have ndvdual normal dstrbutons, 1 wll be equal to the lnear orrelaton between the varables (denoted here as 1) whh annot be estmated gven that there s no suffent nformaton on the eonom status. Assume we an assess the lnear orrelaton between the latent varables (based on the observed probabltes of default). Under the ondtons spefed (.e. the latent varables and the eonom fator follow the standard ( 7 represents the standard normal dstrbuton and 1 ndates ts nverse. 11

13 normal dstrbuton) and aordng to [9], 1 an be assoated to the lnear orrelaton between the latent varables (or the probabltes of default) suh that 1. In resume, settng F 1 (x 1 ) = PD and F (x ) = 0.999, replang 1 (0.999) wth 1 (0.001) and notng that 1 1, we see that the frst dervatve of the Gaussan Copula, [1], orresponds to the formula (restated below) used n Basel to alulate the probablty of default ondtonal on an extremely unfavorable eonom stuaton: K V 1 ( PD) 1 1 (0.999) 4. EMPLOYING ALTERNATIVE CONDITIONAL DISTRIBUTIONS TO CAPTURE TAIL DEPENDENCE 4.1 An example to detet hgher dependene aross losses n downturns As ndated n some empral studes (for nstane, D Clemente and Romano, 004 and Das and Geng, 006), hgher redt losses tend to be more assoated than low levels of losses. Reallng redt losses mply the exstene of small values of the latent varables, we an nterpret the stronger onneton among losses n downturns as an effet of the ntensfaton of the dependene aross small latent varables. In other words, ths s evdene that small values of the latent varables tend to be more onneted over adverse perods. Thus the relatonshp between two latent varables, Y and Y j, an be represented by a satterplot lke the one n Fgure 1. [Insert Fgure 1 here] When the eonom fator E s nserted n the analyss, redued levels of ths varable wll present more ntense assoaton wth the latent varables. Fgure shows the dependene between E and eah latent varable n ths ontext. The orrespondene between Fgure 1 and Fgure an be noted by omparng the level of Y and Y j n a downturn (e*, for example) wth the level of those latent varables when the eonomy s boomng (e**, for example). In the frst ase, both Y and Y j tend to be small whlst n the better eonom senaro, e**, a wder range 1

14 of dfferent values of the latent varables are assoated (.e. there s a hgher lkelhood that a small Y and an elevated Y j, for nstane, wll happen at the same tme). So, ths means that the lower-tal dependene haraterzes not only the relatonshp between the underlyng varables but also the lnk between the eonom status and eah latent varable. [Insert Fgure here] Suh dependene struture an be represented by, for example, the Clayton Copula and, n ths ase, the proporton of loans n the portfolo for whh the latent varable, Y, wll be smaller than the utoff y (.e. the probablty of default) when the eonomy falls to an extremely low level (e*) s derved from []: F( y E e*) { F ( e*) E [ F ( y ) 1] 1} ( 1 ) / Y [ 11 ] where F(..) ndates a ondtonal dstrbuton, F E (e*) s the umulatve dstrbuton of the eonom fator (whh ndates adverse senaros when t approahes 0 and booms when t gets lose to 1), F Y (y ) s the average (hstorally observed) probablty of default and opula parameter between Y and E. s the Among the three varables neessary to ompute extreme losses by applyng [11], two, F Y (y ) and F E (e*), are readly avalable; the former s the expeted probablty of default of the homogeneous portfolo and the latter s to be set aordng to the onfdene demanded for the eonom senaro 8. Naturally, t s expeted that the probablty of the latent varable of eah oblgor beng below a partular utoff, gven a spef eonom level, nreases when the dependene among the defaults beomes stronger. In the partular ase of the Clayton Copula, ths monotonally nreasng behavor of F(y E) wth respet to happens only f F E (e*) F Y (y ). When F E (e*) F Y (y ), F(y E) s a quadrat funton of and starts fallng after rsng up to a spef value. Therefore the alulaton of the regulatory aptal wll yeld more onsstent results f the extreme eonom level s restrted to perentles smaller than or equal 8 Sne F E (e*) s trunated n the nterval [0,1] and small values represent adverse senaros, 0.01 ndates the onfdene level of 99%, 0.05 s assoated wth the onfdene of 95% and so on. 13

15 to the perentles of the latent varables,.e. f F E ( e*) PD, where PD s the average default probablty of the portfolo. Ths does not represent any sgnfant onern n ths ontext beause we are nterested n small values of F E (e*) that ndate downturns. One way to fnd the other varable n [11],, s to derve t from the rank orrelaton between Y and E (Kendall s tau, ). As shown n [3], the Kendall s tau between two varables s assoated wth the parameter of the opula that represents ther dependene. For some famles ths assoaton has losed form (see some examples n Nelsen, 006, hapter 5) and the Clayton Copula s one of them suh that ts parameter an be alulated as: 1 [ 1 ] However we do not have enough nformaton on E to estmate. When the Gaussan Copula s used, ths problem s resolved by replang the orrelaton between Y and E wth the orrelaton between the latent varables of debtors (expresson [9]). Thus, assumng the rank orrelaton between the latent varables,, an be nferred from datasets pertanng to redt losses (n the same way the lnear orrelatons aross probabltes of default were estmated n Basel Aords for dfferent loan lasses), we should look for a orrespondene between and so that an be alulated and plugged nto [11]. 4. Relatonshp between rank orrelatons Kendall s tau ( ) s based on the number of onordant and dsordant pars of varables. Assumng ( X 1,Y1 ) and ( X,Y ) are two ndependent pars from a jont dstrbuton, they wll be onordant f X X )( Y Y ) 0,.e., f the two varables move n the same dreton. They ( 1 1 wll be dsordant when X X )( Y Y ) 0. Kendall s tau s the dfferene between the ( 1 1 proporton of onordant and dsordant pars,.e., = Pr[onordane] Pr[dsordane]. Defnng as the number of onordant pars and d as the number of dsordant ones, Kendall s tau s equvalently expressed as: 14

16 d d [ 13 ] Table 1 llustrates the o-movements of the three varables onsdered here: Y, Y j and E. The arrows and ndate the dreton n whh the varables move. So, f two of them have equal arrows, they move n the same dreton and are therefore onordant. Conversely, f one arrow ponts up whle the other one ponts down, the pars of varables are dsordant. Denote the number of onordant pars as, E, and Ej for the pars (Y,Y j ), (E,Y,) and (E,Y j ) respetvely. The related number of dsordant pars wll be represented by d, d E, and d Ej. Let N be the total number of observatons (whh wll be obvously the same for all varables). So, for any par, + d = N and, from [13], - d = N. Combne these two expressons, we get: N( 1) [ 14 ] [Insert Table 1 here] Sne we are assumng that the latent varable of eah debtor has equal dependene n terms of the eonom fator, E = Ej (and also d E = d Ej ). Hene, ths ondton s satsfed whenever Y and Y j are onordant beause the relatonshp between eah of them and E wll be always the same (ths s the ase of all observatons of Y and Y j n Panel B of Table 1 and the two frst observatons of those two varables n Panels C and D). On the other hand, when Y and Y j are dsordant, E wll be neessarly onordant wth one latent varable and dsordant wth the other one. Therefore f E s onordant wth Y (Y j ) when the latent varables are dsordant, E must be onordant wth Y j (Y ) n another perod when the latent varables are dsordant. Panel A represents the only ase n whh (rank orrelaton for eah par of latent varables Y and Y j ) mples a sngle value of (rank orrelaton related to eah Y and E),.e. when = -1. Sne Y and Y j present a ompletely nverse behavor, all pars n the frst two olumns are dsordant. In ths senaro, the ondton E = Ej wll be met only f E s onordant wth Y n half of the observatons and onordant wth Y j n the other half suh that E = Ej = 0.5 N. 15

17 Reallng that + d = N, we have that d E = d Ej = 0.5 N and the Kendall s tau between E and the latent varable of eah oblgor ( and j) wll be: E E d d E E Ej Ej d d Ej Ej 0.5N 0.5N N 0 So, when = -1, we know for sure that, gven the assumpton of equal dependene between eah latent varable and the sngle eonom fator, = 0. Apart from ths speal ase, there s no mappng from to a unque value of. Panel B shows the hghest rank orrelaton between the latent varables ( = 1) where all pars of arrows n the two frst olumns pont n the same dreton and therefore any ombnaton of dretons n E wll omply wth the requrement E = Ej (the thrd olumn of Panel B s an example). Ths means that f the latent varables present the strongest possble onneton, any value for s possble. Fortunately, redt losses tend not to be perfetly orrelated and ths redue the range of feasble values of when an be estmated (or assumed based on some reasonable presumptons). Whenever s dfferent from -1 and 1, there wll be onordant and dsordant pars of Y and Y j. Panels C and D n Table 1 help us to dentfy the mnmum and maxmum possble (.e. ts bounds) for a gven n that nterval. Both panels symbolze pars (Y,Y j ) wth dental observatons: the two frst lnes are onordant and the others are dsordant (the dretons of the arrows are just llustratve). From [13], t s lear that the mnmum n ths senaro wll happen when E (= Ej ) s mnmum and ths happens when E s dsordant wth the onordant pars (Y,Y j ); see the frst two lnes of Y and Y j n Panel C where the arrows of E have the opposte dreton of the respetve arrows of Y and Y j. Furthermore, as explaned above, when the pars are dsordant, E must be onordant wth eah Y half of the observatons (represented n the last four lnes n Panel C). From ths, we dedue that the mnmum number of onordant pars between E and a latent varable s 0. 5d mn E mn Ej j, that s, half of the observatons presentng dsordant 16

18 pars (Y,Y j ). The equvalent dsordant pars wll be therefore d d 0. 5d mn E mn Ej j. In Panel C,, for nstane, s equal to 0.5d 0.5(4) (whh refers to the thrd and the mn E j mn fourth lnes where Y and E are onordant) and d 0.5d 0.5(4) 4 (onernng the frst, the seond, the ffth and the sxth lnes). From ths, t follows that the mnmum Kendall s tau between E and eah latent varable (Y, for example) an be assoated to the onordant and dsordant pars that generated the alulable Kendall s tau between Y and Y j : E j mn mn E mn E d d mn E mn E 0.5d 0.5d ( ( 0.5d 0.5d ) ) d [ 15 ] The maxmum wll happen when E (= Ej ) s maxmum and ths ours when E s onordant wth the onordant pars (Y,Y j ) as demonstrated n the two frst lnes of Panel D n Table 1. As before, E must be onordant wth eah Y half of the dsordant observatons (see the last four lnes n Panel D). In these rumstanes, the hghest number of onordant pars nvolvng E and a latent varable s max E max Ej j 0. 5d and the dsordant pars totalze max E max Ej max d d 0. 5d. In Panel D, (4) 4 (the frst four lnes E d j max n Panel D) and d (4) (the last two lnes). The maxmum Kendall s tau E d j relatng E to eah Y expressed n terms of onordant and dsordant pars between the latent varables s (takng loan as an example): j max max E max E d d max E max E ( ( 0.5d 0.5d ) ) 0.5d 0.5d d [ 16 ] Combnng [14], where + d = N, wth [15] and [16], these two expressons an be rewrtten respetvely as: 17

19 mn d N( N 1) / ( 1) [ 17 ] and max d N( 1) / N ( 1) [ 18 ] Ths means that, when 1 1, the rank orrelaton between E and eah latent varable, (whh s the same for both loans and j due to the assumpton of homogeneous dependene) s always n the range whose lmts are the values dsplayed n [17] and [18],.e.: [( ( 1)) /, ( 1) / ] [ 19 ] suh that the smaller s, the shorter the range of s 9. Note that [17] and [18] are also ompatble wth the extreme ases mentoned earler ( = -1 and = 1) and the maxmum, for example, s respetvely equal to 0 (the only admssble value for when = -1) and 1 (the maxmum theoretal, whh reflets the fat that = 1 allows any rank orrelaton between E and the latent varables). Another nterestng example s the possble range of when the loan defaults (.e. the latent varables) are ndependent. When = 0, may vary between -0.5 and 0.5. In other words, the ndependene between Y and Y j does not mply that eah latent varable (and onsequently, the probablty of default of eah debtor) s free from the nfluene of the eonomy. In prnple, any value n the nterval [ ( ( 1)) /, ( 1) / ] an be used to estmate the parameter of the opula that expresses the dependene between the eonom fator and the latent varable at the portfolo level. However, n the partular ase of the Clayton Copula, the parameter s n the nterval (0, ). Thus [19] beomes: 9 In aordane to what was sad before, the shortest range s assoated wth = -1 (the smallest possble rank orrelaton between the latent varables) whh results n a sngle value for (= 0). 18

20 ( 0, ( 1) / ] [ 0 ] In a prudental regulatory ontext, a reasonable hoe for seems to be ts hghest value (orrespondng to ( 1) / ) sne t denotes the strongest onneton aross the latent varables and represents the hghest possble dependene among redt losses (so, the aptal requred wll be estmated aordng to the worse senaro gven the observed rank orrelaton between defaults). However ths alternatve may lead to the overestmaton of the regulatory aptal and therefore some ntermedary values of an be employed at the dsreton of regulators and prattoners. In the smulatons ahead we wll test three levels of the rank orrelaton between eah latent varable and the eonom fator: ⅓ of the maxmum hghest respetvely gven by:, the average and the. In the nstane of the Clayton Copula, onsderng [0] these three levels are ( 1), 6 ( 1), and 4 ( 1) [ 1 ] If the Clayton Copula s adopted to represent the dependene between the eonom fator and the redt losses, the aptal requred to over unexpeted losses wth hgher dependene n downturns wll be estmated by means of [11], restated below: F( y E e*) { F ( e*) E [ F ( y ) 1] 1} ( 1 ) / Y where the parameter wll be defned aordng to the level of the rank orrelaton between redt losses and the eonom fator. Three optons are ⅓ of the maxmum value, the average value, and the hghest value, alulated respetvely as (by ombnng [1] and [1]): ( 1) 5, ( 1) 3, and ( 1) 1 [ ] 19

21 Reall that s the observable (omputable) rank orrelaton (Kendall s tau) aross probabltes of default and an be determned n the same way the lnear orrelaton n [5] was defned by several redt lasses n Basel Aord. Ths model s flexble and the opula famly an be hanged aordng to dfferent dependene shapes emprally found. One example ould be the ase of redt losses that present tal dependene n a symmetr jont dstrbuton. An opton to express ths stuaton would be the Student t Copula. Thus, [11] would be replaed wth the frst dervatve of the Student t Copula (dsplayed, for example, n Aas et al., 009) where the dstrbuton of the latent varable evaluated at the utoff pont (= PD) and the hosen perentle of the eonom fator would be the ondtoned and the ondtonng varables, respetvely. 5. COMPARISON BETWEEN THE PERFORMANCE OF THE BASEL METHOD AND THE PERFORMANCE OF THE SUGGESTED APPROACH 5.1 Intal omparsons Smulatons were run to hek whether the estmaton of unexpeted losses based on formulas derved from a left-tal-dependent opula outperforms the formula used n the Basel Aord. Followng the evdene presented n the lterature aordng to whh hgh asset losses are prone to be more onneted than low losses 10, redt portfolos wth rght-tal-dependent losses were smulated wth 50 observatons eah 11. Two lasses of loans were onsdered, retal (Table ) and orporate (Table 3), beause the Basel Aord uses dfferent ranges of orrelaton for those two groups: between 0.03 and 0.16 for the former and between 0.1 and 0.4 for the latter 1. In Basel, expressons [6] and [7] refer to orrelatons aross asset returns of oblgors but, gven that these orrelatons are assumed to drve the relatonshp among defaults, we take the values alulated from [6] and [7] as proxes for orrelatons among PDs. In both lasses (retal and orporate), orrelatons were taken as dereasng funtons of PD (see [6] and [7]) followng the dea that rsker oblgors (hgher PD) present more dosynrat rsk 10 See for example, Ang and Bekaert (00), D Clemente and Romano (004), Das and Geng (006), Nng (006), Patton (006) and Rosenberg and Shuermann (006). 11 Ths s equvalent to around 10 years of weekly data or 43 years of monthly data. 1 The retal lass n these smulatons exludes revolvng redt and mortgages gven that they have orrelaton ndes fxed respetvely at 0.04 and 0.15 n Basel Aords. The orrelaton aross orporate loans s also a funton of the sze of the oblgors. In these smulatons, all orporate debtors were set at the maxmum sze (annual sales = 50 mllon) stpulated n Basel. 0

22 and therefore the systemat porton of ther rsk s smaller whh results n lower orrelaton (see, for nstane, BCBS, 005 and Das and Geng, 006). So, to avod poor performane of the Basel approah due to mspefed orrelatons, only portfolos wth the respetve levels of assoaton determned n the Basel Aord were onsdered n the omparson (see seond olumn of Tables and 3 where the orrelaton oeffents dsplayed are onsstent wth the formulas adopted n Basel [6] and [7] 13 ). In order to smplfy the alulatons for orporate portfolos, the maturty term (see [4]) was set equal to one year. As n the Basel approah, the portfolos are assumed to be homogeneous,.e. all pars have the same dependene (shape and ntensty). Whlst Basel mpltly presumes the Gaussan dependene (opula), whh represents a symmetr relatonshp wthout tal dependene, the smulatons run n ths seton follow empral evdene from the lterature aordng to whh hgh redt losses are typally more onneted than low losses. Thus the losses n all smulated portfolos are upper-tal dependent and the Gumbel Copula was used to generate suh relatonshp. Followng Kalyvas et al. (006), aordng to whom the dstrbutons of redt losses are skewed to the rght (postve skewness), we used two dstrbutons (beta and gamma) to represent the loss dstrbutons 14. The shape of suh dstrbutons s generally lke the one shown n Fgure 3. [Insert Fgure 3 here] Ten PD levels (expeted losses) were tested (from 0.01 to 0.10). So, for eah loan lass (retal and orporate), the aptal was estmated n 0 senaros (ten PD levels tmes two dstrbutons). The smulaton of senaro was repeated 1,000 tmes to elmnate potental randomness effets. The alternatve approah s llustrated by a ondtonal dstrbuton derved from the Clayton Copula (as n Seton 4). Sne there s not a unque possble parameter for the opula between eah latent varable and the eonom fator, three parameters were tested. They were based on the average rank orrelaton ( ) between Y and E, ⅓ of the maxmum possble, and the maxmum. The rank orrelaton between the latent varables of two oblgors ( ) and the 13 Dfferene no greater than Both dstrbutons were smulated suh that they were postvely skewed (longer tal n the rght sde),.e., extremely hgh redt losses were farther from the mean than the extremely small losses were. 1

23 three opula parameters ( ) resulted from the three rank orrelatons used are presented n Appendes B (retal redt) and C (orporate redt). [Insert Tables and 3 here] Aordng to what was sad n Seton 4.1, the alulaton of the regulatory aptal va the frst dervatve of the Clayton Copula yelds more onsstent results (aptal nreasng wth the dependene aross defaults) f the extreme eonom level s restrted to perentles smaller than or equal to the perentles of the latent varables (whh represents the average default probablty, PD, of the portfolo),.e. f F E ( e*) PD. As an example, we set the onfdene (perentle of the eonom level) equal to 99% (.e. ( e*) 0. 01), whh omples wth all PD values onsdered. In general, the results ndate that the formula urrently used n Basel Aord tends to underestmate extreme (unexpeted) losses for both retal and orporate portfolos when ther losses present postvely-skewed dstrbutons and are rght-tal dependent (hgher losses more assoated). As for retal redt, Table detals the estmatons for ten PD levels whose orrelatons are onsstent wth those determned n Basel (see the seond olumn). The maxmum unexpeted losses (.e. the maxmum losses mnus the average losses) are dsplayed n the thrd olumn. The next four olumns present the estmates resultant from Basel approah and from the alternatve method based on three possble values of the rank orrelaton, F E, between eah oblgor and the eonom fator. The last four olumns show the absolute dfferene between the maxmum smulated unexpeted losses and the four estmates (the values n eah of these four olumn s added and the total s shown to ndate whh approah gave the overall best approxmaton to the maxmum losses). In all senaros of both dstrbutons tested (beta n Panel A and gamma n Panel B), at least one alternatve estmate was better than the Basel evaluaton. The best approxmaton (smallest dfferene between estmates and the unexpeted losses) n eah senaro s dsplayed n boldfae. Sne we are followng the same presumpton adopted n Basel (namely, that hgh PDs are less onneted than low PDs), the rank orrelaton,, between loans losses s lower for hgher

24 PDs. As a onsequene, the maxmum rank orrelaton,, between eah oblgor and the eonom fator gets smaller as PD nreases 15 and, logally, the same apples to the range of potental values of (see [19] for all opulas and [0] for the Clayton Copula). In omplement to ths fat, our results suggest that, n general, the best value of n a gven nterval follows a smlar relatonshp: low levels of yeld better results for hgh PDs and ve versa. So, n Panel A of Table (beta-dstrbuted losses), the hghest level of tested resulted n the best estmate of unexpeted losses for the lowest PD (0.01). The extreme losses for the next three values of PDs were better aptured by the ntermedary. The lowest onsdered gave the best outomes for the sx hghest PDs. In Panel B (gamma-dstrbuted losses), the three lowest PDs had the better results by means of the average nreases, the best rank orrelaton beomes the smallest one (⅓ of the maxmum ).. As PD Table 3 (pertanng to orporate loans) onfrmed the pattern showed n Table for both dstrbutons. 5. Calbraton of the dependene between the latent varables and the eonom fator Consonant wth the reasonng that supports the Basel model (.e. hgh PDs are less subjet to systemat rsk and therefore less onneted than low PDs), Tables and 3 showed that the rank orrelatons related to the eonom fator and the latent varable of eah loan ( ) that generated the best estmates dereased wth the portfolos PDs. Hene, based on the funton used n Basel to alulate the orrelaton aross asset returns (see [6] and [7]), we assume that s an exponental funton of PD. Nevertheless, we adopt a dfferent expresson to keep the onssteny wth the nterval spefed n [0]: (1 PD) e * PD * K _ MAX [ 3 ] 15 See [1] where nreases monotonally wth for any of the levels hosen (⅓ of the maxmum average, and maxmum )., 3

25 where K s a onstant to be set aordng to the haratersts of the portfolos and _ MAX s the maxmum possble (see [0]). Ths results n the shapes shown n Fgure 4 (for three values of K) where s n the range presented n [0], _ MAX when PD = 0 and 0 when PD = 1. [Insert Fgure 4 here] However, addtonal smulatons (not dsplayed here) revealed that the estmaton of s mproved when K s defned as a dereasng funton of PD, suh that, K = (K1 K*PD). So, [3] beomes: e (1 PD) * PD *( K1 K * PD ) _ MAX [ 4 ] and, for relatvely small PDs, ths new funton has a smlar shape to the funton represented n Fgure 4. Conernng the smulated portfolos (Gumbel-dependent wth beta and gamma dstrbutons), the best approxmatons were obtaned when K1= 30 and K = 00. Tables 4 (retal redt) and 5 (orporate redt) ompare the aptal estmated va the alternatve approah (where the opula parameter,, s derved from the rank orrelaton omputed aordng to [4] 16 ) to the aptal alulated n lne wth the Basel method. [Insert Tables 4 and 5 here] The results show that the alternatve model outperforms Basel method n all senaros gven that, for all PD levels tested, the absolute dfferene between the true (smulated) unexpeted losses and the alternatve estmate s smaller than the absolute dfferene between the true unexpeted losses and the Basel estmate (ompare the seventh olumn to the last one). 16 We used [1] to estmate. 4

26 6. CONCLUSIONS We show that the formula used n Basel Aord to estmate unexpeted redt losses orresponds to a ondtonal dstrbuton derved from the Gaussan Copula. Sne ths opula famly does not apture tal dependene, the model largely used by regulators may underestmate the aptal neessary to fae redt losses n downturns (when the onneton aross defaults tends to be more ntense than n perods of normal eonom atvty). Based on ths fndng, we propose a model that keeps the bas struture of the urrent method but uses dfferent ondtonal dstrbutons able to detet possble tal dependene among losses. The suggested method s flexble and an apture several dependene shapes sne t an be adapted to any dfferentable opula famly. Its mplementaton s as smple as the mplementaton of the exstent model and has the advantage of dentfyng potental hgher assoaton between losses n downturns. As an example, we set a formula derved from the Clayton Copula that an apture the supposed stronger dependene aross defaults n downturns. There are typally several possble rank orrelatons between the eonom fator and the latent varable of eah loan (alled n ths paper) for eah rank orrelaton aross loans (named here as ) and followng the assumpton present n the Basel method (aordng to whh, hgh PDs are less onneted than low PDs), we proposed a losed-form expresson to estmate as a dereasng funton of the portfolos PD. We are urrently workng on smplfed formulas to alulate. If the losses have small rank orrelaton, the model proposed gets more aurate beause the range of possble assoatons between the eonom fator and eah latent varable tends to be shorter than ntervals resulted from hgh rank orrelaton between the latent varables. So, the varaton of potental outomes s redued for low rank orrelatons aross defaults and we move towards a unque soluton. By smulatng redt losses that potentally represent defaults observed n real loan portfolos (.e. wth postvely skewed dstrbutons and upper-tal dependene 17 ) and omply wth the dependene levels spefed n the Basel Aord, we onfrm that the urrent model tends to underestmate jont extreme losses. We also demonstrate that the alternatve formula outperforms Basel n all senaros tested. 17 See, for example, Kalyvas et al. (006), Rosenberg and Shuermann (006), D Clemente and Romano (004), and Das and Geng (006). 5

27 It s possble that many trals to nsert opulas n ths Basel framework have faled due to the lak of a lnk between the dependene measure we need ( ) and the dependene we an nfer from empral data ( ). Therefore the relatonshp between those two measures found n ths study wll ertanly ontrbute to the applaton of opulas to many models dealng wth dependene among varables mpated by systemat (unobservable) fators. REFERENCES Aas, K., C. Czado, A. Frgess, H. Bakken (009). Par-opula onstrutons of multple dependene. Insurane Mathemats & Eonoms, 44, pp Ang, A., Bekaert, G. (00). Internatonal Asset Alloaton Wth Regme Shfts. Revew of Fnanal Studes, 15, pp Basel Commttee on Bankng Supervson - BCBS (005). An Explanatory Note on the Basel II IRB Rsk Weght Funtons. Bank for Internatonal Settlements. Basel Commttee on Bankng Supervson - BCBS (006). Internatonal Convergene of Captal Measurement and Captal Standards: A Revsed Framework. Bank for Internatonal Settlements. Bluhm, C., L. Overbek, C. Wagner (00). An Introduton to Credt Rsk Modelng. London: Chapman & Hall/CRC. Bouyé, E., M. Salmon (009). Dynam opula quantle regressons and tal are a dynam dependene n Forex markets. European Journal of Fnane, Vol. 15, Issue 7, pp Crook, J., T. Bellott, (010). Tme Varyng and Dynam Models for Default Rsk n Consumer Loans. Journal of Statstal Soety Seres A (Statsts n Soety), Vol. 173, Issue, pp Crouhy, M., D. Gala, R. Mark (000). A omparatve analyss of urrent redt rsk models. Journal of Bankng & Fnane, 4, pp Czado, Clauda (010). Par-opula onstrutons for multvarate opulas. In: Jawork, P., F. Durante, W. Härdle, T. Ryhlk (eds.) Workshop on Copula Theory and ts Applatons. Dortreht: Sprnger, pp Das, S., G. Geng (006). Correlated Default Proesses: A Crteron-Based Copula Approah. In: Fong, H. (ed.) The Credt Market Handbook. Advaned Modelng Issues. Hoboken, New Jersey: John Wley & Sons, pp

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