Basel II Credit Loss Distributions under Non-Normality

Transcription

1 Basel II Credt Loss Dstrbutons under Non-Normalty Enrque Batz-Zuk, 1 George Chrstodoulaks and Ser-Huang Poon Abstract In the context of Vascek (1987, 00) sngle factor model, we examne the mpact of skewness and excess kurtoss n the asset return process on the shape of the credt loss dstrbuton and, consequently, over the Basel II requrements. We use Skew Normal and Skew Student s t denstes to develop a Maxmum Lkelhood estmator of the credt loss densty for aggregate charge-off rates publshed by the Federal Reserve Board for ten U.S. sectors. We show that, the non-gaussan modellng of the common factor provdes a better characterzaton than ts Gaussan counterpart, and has a sgnfcant mpact on the captal requrement dependng on the sgn and magntude of the skew-related coeffcent. On the other hand, the non-gaussan modellng of the dosyncratc factor does not provde a sgnfcantly better characterzaton than the Gaussan base case. The latter could be due to the fact that the sector portfolos are large and the dosyncratc component has been fully dversfed away. JEL Classfcaton: C13,C16, C, G1, G3 Keywords: Vascek loan loss dstrbuton, sngle factor model, Basel II, non-gaussan dstrbutons, Skew Normal dstrbuton, Skew Student s t dstrbuton. Acknowledgements: The authors wsh to thank Professor Adelch Azzaln for hs helpful comments and gudance on estmatng the Skew Normal and the Skew Student s t dstrbutons. 1 A set of Matlab functons to compute the Skew Student s t dstrbuton, densty, quantles, cumulants and a pseudo ST random number generator were ported from Azzaln s Skew Student s t R lbrary. The Matlab functons of these routnes, created by the frst author, are now avalable on Azzaln s web page ( All authors are at Manchester Busness School, Unversty of Manchester, Crawford House, Oxford Road, Manchester M13 9PL, Unted Kngdom, Tel: (secretary), Fax: , Emal: Enrque.Batz(at)postgrad.mbs.ac.uk, George.Chrstodoulaks(at)mbs.ac.uk, Ser- Huang.Poon(at)mbs.ac.uk. 1

2 Basel II Credt Loss Dstrbuton under Non-Normalty 1. Introducton It s a regulatory requrement that fnancal nsttutons should reserve suffcent captal because of ther exposure to credt and other rsks. Whlst the adequacy of such reserves s crucal for ther survval as well as the systemc fnancal stablty, the ongong credt crss has placed a serous doubt on the way these reserves are calculated. Pllar I of Basel II (004) provdes the regulatory framework for determnng bank captal requrements for takng credt rsk. Under ths regulatory framework, a bank may choose the nternal ratngs-based approach that utlzes rsk weghts derved mplctly from Vascek s (1987, 00) sngle factor model (SFM). In the Vascek s model, changes to asset value are drven by a common and an dosyncratc rsk factors both of whch are assumed to be Gaussan. However, the mposed Gaussanty assumpton can be a poor proxy of the true and unobservable dstrbuton, leadng to a hgher captal charge when the rght tal of the dstrbuton s underrepresented and a lower captal charge when the left tal s underrepresented and vce cersa when the rght tal s overrepresented. The latter s partcularly mportant as rsk s not adequately covered. In ths paper we relax the Gaussanty assumpton and estmate wth maxmum lkelhood generalzed Vascek credt loss dstrbutons that are based on asset processes that feature skewness and excess kurtoss. Our data concern quarterly charge-off rates n ten US sectors from 1985 to 007. Our fndngs provde overwhelmng evdence n favour of non-normalty and leads to sgnfcantly dfferent captal charge calculaton as compared to those n Basel II.

3 Snce the publcaton of Vascek (1987), there have been a number of theoretcal extensons (see Batz, Chrstodoulaks and Poon (008) for a comprehensve survey) for the credt loss dstrbuton. In practce, the common factor s unobservable and there s no emprcal methodology avalable to study and test the Gaussan assumpton. Ths departure from Normalty could exert a large mpact on the loan portfolo loss dstrbuton and, thus, the regulatory captal charges (see Schönbucher (001)). In ths paper we model and assess the mpact on Vascek s captal charges due to non-gaussan common or dosyncratc factors separately. We consder Skew Normal (SN) and Skew Student s t (ST) as alternatves to Gaussan. These two non-gaussan denstes encompass the normal as a specal case and both have the property of beng analytcally tractable. Moreover, both are very flexble for controllng the amount of skewness and excess kurtoss n the dstrbuton. The SN acheves ths, to a moderate degree, through a sngle addtonal parameter. The ST, whch ncludes SN as a lmtng case, provdes a much greater flexblty over the degree of skewness and excess kurtoss through two addtonal parameters (see Azzaln (005)), over the Normal. To compare Vascek s Gaussan model aganst these alternatves we study the followng two cases; () the common factor has a non-gaussan dstrbuton, and () the dosyncratc factor has a non-gaussan dstrbuton. We estmate the parameters of each modellng choce through Maxmum Lkelhood. Snce both non-gaussan specfcatons nclude Gaussan as a specal case, we use the lkelhood rato test to assess the ft of the unrestrcted (non-gaussan) over the restrcted (Gaussan) model. Addtonally, we assess the mpact of these dstrbutonal assumptons on the captal requrements for ten portfolos of dfferent loan types for the entre US Bankng System. The results show that both non-gaussan alternatves provde a better ft for case (). 3

4 Moreover, the Skew Student s t specfcaton does not provde a better ft over the Skew Normal. We quantfy the mpact of the non-gaussan modellng choces on the captal charges and fnd that the degree of under(over) estmaton depends on the sgn and the absolute value of the shape parameter estmate of the Skew Normal. To our knowledge ths s the frst emprcal paper that develops a methodology and examnes the mpact of non-gaussanty on the dstrbuton of portfolo credt losses and on captal charges. Non-Gaussan process has not been studed before possbly due to the fact that t s techncally challengng to mplement, and for the case of the Skew Student s t dstrbuton the estmaton s computatonally ntensve. In partcular, as explaned later n Secton 4, the estmaton loss dstrbuton nvolves the use of nonstandard quadrature functons wthn the optmzaton routne. The remander of ths paper s organzed as follows. Secton brefly revews Vascek s orgnal (1987, 00) sngle factor model and the generalzed verson derved by Schönbucher (001). Secton 3 dscusses the statstcal propertes of the Skew Normal and the Skew Student s t denstes. Secton 4 presents the estmaton framework whch s based on maxmum lkelhood. Secton 5 descrbes the data sets. In Secton 6, we present the estmaton results and assess the mpact of non-gaussanty on captal requrement calculaton. Fnally, Secton 7 provdes some concludng remarks.. A Revew of Vascek (1987) and ts Generalzaton In ths secton, we frst revew the dervaton of Vascek s (1987, 00) sngle factor lmtng loss dstrbuton and ts underlyng assumptons. Then, we descrbe the extensons to non-gaussan dstrbutons made by Schönbucher (001) whch opens the way for our emprcal specfcatons through SN and ST as we show later n the text. 4

5 Fnally, we show how the captal requrements are computed under the generalzed dstrbuton. The ndvdual loss due to oblgor s defned as the product of the Exposure At Default (EAD), the Loss Gven Default (LGD) and a default ndcator varable (D ) as follows: L = EAD LGD D (.1) The varable D s a Bernoull random varable that takes the value one f the oblgor defaults and zero otherwse. Ths setup mplctly assumes that EAD and LGD are tme nvarant for each oblgor. 3 Then, the portfolo loss rate, L, can be calculated as n where w ( EAD EAD = 1 ) n L n = 1 L = = n w LGD D EAD = 1 = 1 (.) = s the portfolo weght for the th loan. Vascek (1987) assumes that the sze of EAD and LGD are the same for all oblgors, and, moreover, that the recovery rate s equal to zero such that LGD = 1. Further assumpton that EAD = EAD leads to w = w = 1/ n, and a homogeneous portfolo wth loss rate: L N D = 1 = (.3) Note that the N default random varables D have been treated as ndependent of each other. To allow for correlaton among the default varables D n (.3), let the asset return, R, for oblgor n the portfolo be drven by a sngle common factor Y and an dosyncratc nose component ε : n R = ρy + 1 ρε (.4) where Y and ε are assumed to be mutually and serally ndependent random varables that follow a standardzed Gaussan dstrbuton N ( 0,1), and ρ and 1 ρ are the 3 The LGD can be treated as a stochastc varable wthout changng the model results as long as t s assumed to be ndependent of D. 5

6 correspondng factor loadngs. Note that under ths specfcaton, the asset returns of all frms are multvarate Normal wth the same parwse correlaton ρ. Vascek (1987) assumes that the credt portfolo s fne gran,.e. t conssts of a large number of relatvely small exposures. If ths assumpton holds, the dosyncratc rsk assocated wth the ndvdual exposures wll cancel out and only systematc rsks that affect all the exposures wll have an mpact on the portfolo value and loss rate. So far the default process has been treated as exogenous. Followng Merton (1974), Vascek (1987) assumes that the th oblgor defaults f the value of ts assets, A,T, at loan maturty, falls below the debt contractual value, B,T. In the context of the credt portfolo model and assumng that all oblgors have the same default probablty,.e. pd = pd, the endogenous default process based on Merton s (1974) can be ntroduced f D s defned as: ( ) ( ) D = 1 f R Φ pd and D = 0 f R > Φ pd (.5) 1 1 where Φ( ) s the cumulatve Gaussan dstrbuton functon and pd s the uncondtonal default probablty. The default process defned n eq.(.5) depends on the latent random varable Y that drves the asset returns R as follows: ( ) = ( = 1 = ) 1 = P( R Φ ( pd ) Y = y) 1 = P( ρ Y + 1 ρ ε Φ ( pd ) Y = y) p y P D Y y ( pd ) ( pd ) 1 Φ ρ Y = P ε Y = y 1 ρ 1 Φ ρ y = Φ 1 ρ (.6) whch s the probablty of default condtonal on the value of Y. Condtonal on the realzaton y of Y, the ndvdual defaults happen ndependently of each other. Thus, the uncondtonal probablty of observng exactly k defaults s the average of the 6

7 condtonal probabltes of k defaults, averaged over the possble realzatons of Y and weghted wth the probablty densty functon φ ( y) : n n = = = = = 1 = 1 ( ) P D k P D k Y y φ y dy n k n k = ( p( y) ) ( 1 p( y) ) φ ( y) dy k (.7) Vascek (1987) showed that f the portfolo s large, then the law of large numbers ensures that the fracton of oblgors that actually default s (almost surely) exactly equal to the ndvdual default probablty. From eq.(.7), Vascek shows that the lmtng loss dstrbuton for the homogeneous portfolo loss rate s: ( l ) ( pd ) 1 ρφ ( ) [ ] 1 Φ 1 FL l; pd, ρ = Pr L l = Φ ρ (.8) The loan portfolo loss dstrbuton s fully determned by two parameters: the probablty of default (pd) and the asset correlaton (ρ). The former fxes the expected loss rate of the portfolo, whle the latter controls the shape of the loss dstrbuton. The densty of F ( ;, ) equal to: L l PD ρ can be derved by usng the nverse functon theorem and ths s f L ( l pd ρ ) ( 1 ρφ ( l) Φ ( pd )) Φ ( l) ( ) 1 ρ ;, = exp + ρ ρ (.9) Schönbucher (001) extends the sngle factor results of Vascek to cases where the common and the dosyncratc factors have non-gaussan dstrbutons. In partcular, Schönbucher (001) showed that f Y ~ G( ), and ε ~ H ( ) for all, and Y and ε (both centred and standardzed) are ndependent, then the lmtng loan loss dstrbuton s equal to: 7

8 K 1 ρ 1 FL ( l; pd, ρ ) = 1 G H ( l ) ρ ρ (.10) where K s the default barrer whch s gven by the nverse of the asset return dstrbuton. Notce that ths extenson adds flexblty and could potentally be very mportant for modellng real data. However, ths extenson comes at the cost of mplementaton complextes. Ths s because the default barrer K s no longer equal to the Gaussan nverse of pd. In partcular, K s now equal to the nverse of the functon that arses from the sum of the assumed dstrbutons for Y and ε n eq.(.4). Accordng to the advanced and foundaton approach contaned n Basel II, the captal requrements are computed as the dfference between the unexpected loss (UL) and the expected loss (EL) scaled by the LGD and the effectve remanng maturty. In ths paper, we omt the effect of the tme to maturty (see Kjerst (005)). In the Basel II framework, banks are expected to cover ther EL on an ongong bass, because t represents just another cost component of the lendng busness. Therefore, under ths methodology, captal s only needed for coverng unexpected losses. Hence, banks are requred to hold captal aganst UL and ths corresponds to the CredtVaR of the portfolo. The captal requrements per unt of exposure are computed n ths paper as: ( ) CR = LGD UL EL α ( α ) K ρg 1 1 ULα = H 1 ρ EL = pd α (.11) where UL s obtaned by nvertng Schönbuchers (001) generalzed lmtng loss dstrbuton for a sgnfcance level α whch, under Basel II, s set equal to 99.9%. Note 8

9 that n eq.(.11), the value of LGD depends on the type of loan under analyss and ths value s prescrbed n Basel II. 3. Specfyng Non- Normalty for the Generalzed Vascek Dstrbuton Ths secton presents densty specfcatons for functons G( ) and H ( ) n equaton (.10) usng the Skew Normal and the Skew Student s t denstes. It provdes a bref account of ther man propertes to facltate the understandng of ther applcaton n the context of credt loss estmaton. 3.1 The Skew Normal The Skew Normal dstrbuton proposed by Azzaln (1985, 1986) has densty functon: where φ ( ) and ( ) ( α ) φ ( ) ( α ) f z; = z Φ z, < z < (3.1) Z Φ are the standard normal densty and cumulatve normal dstrbuton functons, respectvely, and α s the shape parameter wth functon n eq.(3.1), we wrte Z ~ ( ) the skew normal densty. If ~ ( ) locaton parameter and σ ( 0, ) (,, ) W SN µ σ α and W has densty: < α <. Wth the densty SN α. In practce, we rarely work wth ths form of Z SN α and W µ σ Z µ s the = +, where (, ) s the scale parameter, then we shall wrte ( ) w µ w µ fw ( w; µ, σ, α ) = φ ( ) Φ α ( ), < w <, (3.) σ σ σ The Skew Normal densty has four specal propertes: () SN ( 0) s N ( 0,1) α, ( ; ) f z α tends to the half-normal densty; () f Z s a ( ) Z ; () as SN α random varable, then Z s a SN ( α ) ; and (v) f ( z; α ) s strongly unmodal,.e. log f ( z; α ) s a concave functon of z. Z Z 9

10 Azzaln (1985) shows that the frst four moments of the standardzed Skewed Normal random varable Z are: E[ Z ] = bδ (3.3) [ Z ] ( bδ ) var = 1 (3.4) γ 1 ( Z ) = ( 4 π ) { E ( Z )} ( Z ) sgn( α ) var γ ( Z ) = ( π 3) { E ( Z )} var ( Z ) 3 (3.5) (3.6) where b =, π δ α α = 1 +, ( ) sgn s a functon that returns the sgn of ts argument and γ1, γ denote the thrd and fourth standardzed cumulants. 4 Azzaln (1985) shows that, for the Skewed Normal dstrbuton, the maxmum value of skewness s about 0.995, whle that for kurtoss s Fgure 1 shows the effect of ncreasng the magntude of the shape parameter value on the shape of the SN densty. In Panel A, that: V ( Z µ Z ) σ Z V are functons of Z SN( α ) ~ such Z 1 =. Therefore, t s possble to show that V ~ SN (,, α ), where { 0,4,10,+ } α for = 1,,3, 4 for the four cases. For ncreasng postve values of α, the Skew Normal densty s rght skewed, the mass of the densty concentrates on the left and the rght tal wll always be heaver than that of the Normal. Conversely, Panel B shows the effect of ncreasng the magntude of α when α s negatve. The * random varables V,( = 1,,3,4 ) are exactly the same as V, but µ σ * V are defned for the Z σ Z 4 The skewness s defned as the thrd standardzed moment, whle the kurtoss can be defned as the fourth standardzed moment. Alternatvely, the skewness and the kurtoss can be defned as the thrd and fourth standardzed cumulants. 10

11 correspondng negatve α values. In ths case, the Skew Normal densty s left skewed, the mass of the densty concentrates on the rght and the rght tal wll always be thnner than that of the Normal. The Skew Normal s more flexble than the Normal because we can regulate the skewness and the excess kurtoss through the shape parameter α, albet to a moderate degree. Snce there s only one varable, we cannot regulate skewness and excess kurtoss at the same tme. In ths regards, the Skew Student s t s more flexble as skewness and excess kurtoss are separately controlled by two ndependent parameters. 3. Skew Student s t Dstrbuton The Skew Student s t-dstrbuton 5, ST ( α, v), has densty functon: v + 1 f X ( x; α, v) = tv ( x) Tv + 1 α x, x x + v (3.7) where t ( ) v s the densty of the standard Student s t-dstrbuton wth v degrees of freedom wth 0 < v < ; T + ( ) s the dstrbuton functon of the standard Student s t- v 1 dstrbuton wth v + 1 degrees of freedom and α s the shape parameter wth < α <. If X ~ ST (, v) α and M µ σ X = +, where (, ) σ ( 0, ) s the scale parameter, then M ST ( µ, σ, α, v) µ s the locaton parameter and and M has densty: 1 m µ m µ v + 1 fm ( m; µ, σ, α, v) = tv ( ) T v 1 ( ), m σ + α σ m µ σ ( σ ) + v (3.8) The Skew Student s t densty has sx specal propertes: () ST ( 0, ) standard Student s t-dstrbuton T ( x) v 1 v s the + ; () as v, the Skewed Student s t- densty 5 Branco and Dey (001) provde the orgnal specfcaton. The notatons here follow those of Azzaln and Captano (003). See Kjerst (006) for a survey of dfferent ST specfcatons. 11

12 converges to the skew-normal densty; () f 0 α = and v, then X s a N ( 0,1) random varable; (v) as α, the Skew Student s t tends to the folded-t dstrbuton; (v) f X s a ST ( α, v) random varable, then X s a ST ( α, v) ; (v) f v = 1, then the Skewed Student s t-densty becomes a Skew-Cauchy densty. Azzaln and Captano (003) provde expressons for the standardzed moments of a Skewed Student s t random varable X: [ ] δ, 1 E X = c for v > (3.9) Var X c for v [ ] v =, > ( v ) ( δ ) (3.10) ( δ ) 3 v, 3 3 3v γ 1 ( X ) = δc + ( δc) ( δc) for v > v 3 v v ( δc) v( δ ) ( δ ) (3.11) 3v c v 4 v γ ( X) = + 3( δc) ( δc) 3, for v> 4 ( v )( v 4) v 3 v v (3.1) where c ( v π ) (( v 1) ) ( v ) = Γ Γ and = 1 +. The appealng modellng δ α α advantage of the Skew Student s t s that t allows the tal thckness to be controlled separately va the degrees of freedom parameter, v. The most mportant theoretcal dfference between the ST and the SN s that the former has no restrcton on the range of values for skewness and kurtoss (see Azzaln (005, pp.180)). Fgure shows the shape of four ST denstes for the same shape parameter value (.e. α = 9) and for dfferent degrees of freedom parameter values. The varable P s a functon of the random varable X ~ ST (, v ) α, such that: P ( X µ X ) σ X = for { 1,,3,4 }. It s possble to show that P ~ ST ( µ σ 1 σ, α, v X X, X ) v { 3,5,30, } where for the four cases. As the degrees of freedom parameter ncreases, the ST 1

13 converges to the SN. In fact, when v=30, the shape of the ST densty approxmates that of SN. Snce the four denstes are postvely skewed, the rght tal wll always be heaver than that of Normal. Moreover, the tals of the ST wll always be heaver than that of SN for any gvenα. In summary, to model the non-gaussanty of the common and the dosyncratc factors densty, t s useful f the densty has the followng three propertes: () strct ncluson 6 of the normal densty; () mathematcal tractablty; and () cover a wde range of skewness and kurtoss values. The Skew Normal densty fulfls the frst two requrements n havng some tractablty, and n capturng skewness and kurtoss through ts shape parameter. The Skew Student s t-densty fulfls the second and thrd propertes and has a great control over the skewness and kurtoss range through the shape and the degrees of freedom parameters. 4. Maxmum Lkelhood Estmaton The parameters of Schönbucher s (001) generalzed loss densty can be estmated usng maxmum lkelhood. The estmaton s based on observed portfolo default rates. Followng Düllmann and Trapp (004), we assume that the systematc and dosyncratc rsk factors have no autocorrelaton. The generalzed probablty densty for the observed default rates l t s gven by: f Lt ( l ; θ) t FL = l ( l ; θ) t t ( ) 1 1 ρ K 1 ρ H l t 1 = g 1 ρ ρ h( H ( lt )) (3.13) 6 A densty wll strctly nclude the normal densty f for a partcular value of one or more of ts parameters we obtan the normal specfcaton. A densty wll not have the strct ncluson property f the normal densty results as one or more of ts parameters tend to the lmt. 13

14 where g ( ) s the densty for the common factor and g G( lt ) = ; ( ) lt h s the densty of the dosyncratc factor and ( ) H lt lt h = ; θ s the vector of parameters and K s the default barrer whch s gven by the nverse functon of the asset return dstrbuton as a functon of θ. The objectve s to maxmze the followng constraned log-lkelhood functon: = T max ln L ; l,..., l ln f ; l θ ( 1 T ) ( L ( t )) θ θ (3.14) The θ set may contan addtonal parameters dependng on the choce for g ( ) and h( ). t= 1 Düllmann and Trapp (004) show that for the Vascek (1987) loss densty, 7 the value of the parameters ( ml ml, ) pd ρ that maxmze eq.(3.14) has a closed form soluton. Therefore, the maxmzaton problem can be solved analytcally. Moreover, for ths Gaussan case, Düllmann and Trapp (004) derve a closed-form soluton of the asymptotc Cramer-Rao lower bounds for the standard devaton of the estmators. The man challenge n ntroducng non-gaussanty s the computaton of the default barrer, whch s equal to the nverse functon of the asset return dstrbuton. Compared wth the Skew Student s t, the Skew Normal case s a relatvely manageable task. We collect our analytcal results for the Skew Normal case n the followng proposton. Proposton: For an asset return process of the form (a) f Y SN ( α ) and ε N ( 0,1) R = ρy + 1 ρε (3.15), then 7 Vascek (1987) assumes that g(.) and h(.) are both standard normal denstes. 14

15 R (b) f Y N ( 0,1) and ε SN ( α ) ρ α SN (3.16) 1+ α ( 1 ρ ), then R 1 ρ α SN (3.17) 1+ α ρ Proof: Usng the method of moment generatng functons we can show, for any real number a and b, that the proposal stated by Azzaln (005) s au + bz b ~ SN α a + b a ( 1+ α ) + b (3.18) where U ~ N ( 0,1), Z ~ ( ) SN α, and U and Z are mutually ndependent. Then, defnng a = 1 ρ, b = ρ, we obtan result n eq.(3.16). Also, defnng a = ρ, b = 1 ρ, we obtan result n eq.(3.17). To approxmate the value of the default barrer K,.e. the pd-quantle of the asset return dstrbuton gven n eq.(3.16) and eq.(3.17), we use the Cornsh Fsher Expanson (see Cornsh and Fscher (1960)). For the Skew Student s t case, the default barrer does complcate the computaton consderably. Ths s because the dstrbuton followed by the sum of a Gaussan and a Skew Student s t s not known. Here, we compute the dstrbuton va numercal quadrature (for a revew on ths topc, see Gander and Gautsch (000) or Moler (004)). Then, one can compute the quantle by mnmzng the dstance between the approxmated dstrbuton at a gven probablty level, F R ( k) and the correspondng pd value. 15

16 We analyze the case where Y ~ ST ( α, v) and ~ N ( 0,1) ε to llustrate how ths approach works. Frst, rewrte the SFM as R = C + D, where C = ρ Y and D = 1 ρ ε. Next, gven that Y ~ ST ( 0,1,, v) ( ) α, we have C ~ ST ( 0,,, v) ρ α and D ~ N 0,1 ρ for all. Thrd, snce R s the sum of two ndependent random varables, then the convoluton f C f D of fc and fd s the functon gven by: ( ) = ( ) ( ) = f ( c) f ( r c) f r f c f d R C D C D ( ) ( ) = f c f r c dc C D 1 r c c c v ρ = tv Tv 1 α e + dc ρ ρ ρ ( c ρ ) ρ π ( 1 ρ ) + (3.19) Note that once we have ntegrated eq.(3.19), we got rd of c, and the remanng expresson for the densty s a functon of r. Fnally, the dstrbuton functon of be obtaned by ntegratng eq.(3.19) wrt r as follows: R can ( ) = Pr[ ] = ( ) F k R k f r dr R R k k ( ) ( ) = f c f r c dcdr C D (3.0) Now, combne the two steps, where F k t T e dcdr ( c ρ ) + ρ ( 1 ) 1 r c k c c v ρ R ( ) = v v+ 1 α ρ ρ ρ π ρ (3.1) We use an adaptve Smpson quadrature to numercally evaluate the double ntegral. The Smpson quadrature was computed such that t approxmates the ntegral to wthn an error of To compute the pd-quantle, we smply solve the followng nonlnear problem wrt k: 16

17 R ( ) 0 F k pd = (3.) The tolerance level was set equal to 1e -8. All computatons were performed n Matlab. 5. The Federal Reserve Aggregate Loss Data Our data sample conssts of quarterly sector aggregate charge-off rates (not seasonally adjusted) for all US commercal banks startng from Q1:1985 to Q3:007. The chargeoff rates are publshed by the Federal Reserve Board on a quarterly bass. The chargeoff rates for any category of loan are defned as the flow of a bank s net charge-offs (gross charge-offs mnus recoveres) durng a quarter dvded by the average level of ts loans outstandng n that quarter. 8 The charge-off seres s reported at three aggregate levels. At the top level, we have the Commercal Bankng System whch conssts of Busness, Consumer, Secured by Real Estate, Agrcultural and Leases. The Consumer loans, n turn, conssts of Credt Cards and Other Consumer loans, whle the Secured by Real Estate conssts of Mortgages 9 and Commercal Real Estate. 10 Snce the charge-off rates are net of recoveres whch can be from any perod n the past, charge-off rates can sometme have negatve or zero values. 11 Ths happens whenever the recovery amount for a quarter s greater than or equal to gross charge-off of that quarter. We replace all non-postve charge-off rates by the seres As publshed, these ratos are multpled by 4x100 to convert to annual percentage rates. Mortgage loans nclude loans secured by one-to four-famly propertes, ncludng home equty lnes of credt. Commercal real estate loans nclude constructon and land development loans, loans secured by multfamly resdences, and loans secured by nonfarm, non-resdental real estate. The data for the Mortgage and Commercal Real Estate are avalable from Q1:1991. There are other problems assocated wth the use of banks charge-off rates. As Lamb and Perraudn (006) noted, when a new manager takes over a dvson of a bank, he or she may wsh to wrte off delnquent and sem-delnquent loans n order to be able to demonstrate a better performance subsequently. Nevertheless, followng Lamb and Perraudn s (006), we assume that the aggregaton of many banks charge-offs wll help to remove any possble bas due to such actons. 17

18 mnmum postve hstorcal value. Followng Lamb and Perraudn (006), we scaled the seres by ( 1 LGD ). Ths s because the charge-off rates are publshed net of recoveres. The respectve LGD for each loan portfolo was taken from Basel Commttee on Bankng Supervson (004). The tme seres plots of the ten charge-off rate seres are shown n Fgure 3. Panel A shows the relatonshp between the scaled charge-off rate for the Bankng System and ts man sectors; Panel B and C show the Consumer and Real Estate sectors wth ther respectve sub-components whle Panel D shows the non-scaled Bankng System and ts consttuent sectors. In Table 1 we present some descrptve statstcs for the scaled (by ( 1 LGD ) ) and non-scaled charge-off rate seres. Gven the scale of the sub-prme fnancal crss, some readers mght be surprsed by the relatvely low level of the real estate charge-off seres. The observed levels are relatvely low because the sub-prme crss s largerly related to fnancal nstruments (.e. CDO s) that do not form part of the bank s balance sheet. Moreover, t s not lkely to detect sgns of deteroraton n the bankng system from our hstorcal data set snce t wll take some tme before the bad loans are charge-off from the system. From the tme seres plot of the charge-off rate n Fgure 3, Panel A, one can observe that the Bankng System seres remans stable even though there s a clear deteroraton n Busness and Lease that starts n Moreover, the Consumer seres exhbts hgh levels of charge off compared to all the other seres. Snce only Bankng System and Secured by Real Estate share the same tme seres propertes but not the others, t suggests that Secured by Real Estate must be the largest component of Bankng System. It s mportant to note that the Bankng 18

19 System seres s the only that was not scaled by the LGD. The unscaled seres s presented n Panel D. Table 1 reports n ts frst two rows the uncondtonal mean and standard devatons of the unscaled loss rates. Ths table s useful for comparng the entre Bankng System aganst ts subcomponents. The table reveals that Credt Cards has the hghest loss rate at.16% whch s more than doubles that of the Bankng System (0.84%). By contrast, Mortgages exhbt a small loss rate of just 0.15%. Agrcultural loans have the hghest volatlty (1.06%), measured as the standard devaton of the seres, and ths represents approxmately three tmes that of the Bankng System (0.36%). Ths s not too surprsng gven the hgh levels of loss rate n the Agrcultural sector at the begnnng of the sample perod. Volatlty of the other seres, eg Credt Cards (1.0%), Commercal Real Estate (0.71%), Consumer (0.58%) and Busness (0.56%), s relatvely hgh compared wth the entre Bankng System. Mortgages has the lowest volatlty (0.07%), much smaller than the volatlty of the other seres. Overall, the statstc suggests that Mortgages s less rsky, and Credt Cards are most rsky. Nevertheless, as remarked by Lamb and Perraudn (006), a more mportant aspect of the rskness of a loan type s ts asset correlaton wthn the sector. Ths and other sources of rsk wll be analyzed n the next secton. Snce we estmate the model for the scaled seres t s mportant to study the skewness and kurtoss of the scaled seres that are reported n the bottom panel of Table 1. Note that the mean and standard devaton for the scaled Consumer seres, the Credt Card loan n partcular, are much hgher than the other seres. Ths s partly because the LGD set by Basel II for these sectors s also hgher. 19

20 6. Emprcal Results on US Credt Portfolo Losses In ths secton we report the parameter estmates for Vascek s (1987) Gaussan case for the ten sets of observed charge-off rates. Then, we show the results for the case where the common factor follows a Skew Normal or a Skew Student s t-dstrbuton. We assess the ft of these non-gaussan models to the observed charge-off rates and compare them aganst that of the Gaussan case. Moreover, we also compare the ft of the ST aganst that of the SN, and compare the mpact of these two alternatve non- Gaussan specfcatons on captal charges. Fnally, we repeat the analyss for the case where the dosyncratc factor s ether SN or ST dstrbuted. 6.1 The Vascek s (1987) Gaussan Model The estmaton results of the base case Vascek model wth Gaussan common and dosyncratc rsk factors are presented n Table. Ths model assumes that the pd, the probablty of default, and ρ, the correlaton between the asset returns of any par of frms, are constant for all frms and across all tme perods. The pd parameter s an estmator of the expected charge-off rate. Therefore, ts value s close to the mean reported n the bottom panel of Table 1. Credt cards (6.5%), Consumer (3.3%), Busness (1.88%) and Other Consumer (1.56%) have the hghest estmated default probabltes. The correlaton parameter ρ determnes the shape of the loss densty, and consequently, ts quantles. The square root of ρ measures the correlaton between the asset return and the sngle common factor. The hgher the ρ, the stronger s the sector s exposure to fluctuatons n the common factor whch s beleved to be drven by the busness cycle. Accordng to Table, ρ for Commercal Real Estate (8.37%), 0

21 Agrcultural (16.61%), Secured by Real Estate (11.11%), Busness (8.31%) and Leases (6.74%) are among the hghest suggestng that these sectors are the most senstve to changes n the economc condtons. All these portfolos have a hgher ρ when compared to that of the Bankng System (.07%). Note that Busness s the only portfolo that appears n the hgh pd and hgh ρ group, whereas none of the three consumpton seres has hgh ρ s. Ths result s not too surprsng gven that the consumer portfolos typcally represented by a large number of small heterogenous loans, whereas the Busness portfolos tends to be domnated by a smaller number of large loans. Under some regularty condtons (see Greene (000, pp.17)), the maxmum lkelhood estmator follows an asymptotc Gaussan dstrbuton. The asymptotc standard devaton of the ML parameters can be estmated wth: () the nverse of the Hessan; () the outer product of gradents (OPG), whch s also known as the Berndt, Hall, Hall and Haussman estmator; and () the Sandwch or Quas-Maxmum- Lkelhood Estmator (see Whte (198)). All three estmates are computed and reported n Table whch shows that all parameters estmates are sgnfcant at the 1% level regardless of the choce of standard error estmator. We also drew 1,000 bootstrap data samples for each of the charge-off seres. Ths s because the pd and ρ parameters are constraned to the nterval [0,1] and ths mght cause the asymptotc dstrbuton of the estmates not to be Normal. However, as shown by Table, the bootstrap estmator s of the same magntude as the asymptotc estmators. 1

22 Table, Panel B reports the Jarque-Bera normalty test 1 f the samplng dstrbuton of pd and ρ follows a standard normal dstrbuton. For the case of the pd, the Jarque-Bera test rejected normalty for the case of Agrcultural and Commercal Real Estate. Regardng ρ, the Jarque Bera rejected normalty for the case of the Bankng System, Busness, Credt Cards and Mortgages. We can conclude that t s not clear that the MLE wll be normally dstrbuted for all seres. Gven that the standard error for the parameter estmates s almost dentcal between the bootstrap and the QMLE estmator, we wll report only the QMLE standard errors. 6. Non-Gaussan Common Factor Tables 3 and 4 report the results, respectvely, for the cases where the common factor follows a SN and a ST dstrbuton whle the dosyncratc factor follows the standard Gaussan dstrbuton. The pd and ρ estmates for the Gaussan Vascek base case reported n Table are repeated here for ease of comparson. From Table 3 one can see that: () all pd and ρ parameters are statstcally sgnfcant at the 1% level; () the pd estmates are very smlar to the Vascek s base case; () the ρ parameters ncrease wth respect to the Vascek s base case, the ncrease s greater whenever there s a sgnfcant shape parameter α ; (v) the shape parameter s not statstcally sgnfcant for three portfolos, vz. Credt Card, Other Consumer and Commercal Real Estate, and ths suggests that the SN specfcaton does not provde a sgnfcantly better ft than that of the Gaussan case for these three portfolos; (v) negatve shape parameters were observed for Secured by Real Estate (-9.5), Mortgages (-7.6), Bankng System (-3.) and Agrcultural (-.9); (v) the correspondng skewness 1 We also performed Lllefors normalty tests for pd and ρ, but we omtted the results gven that these were smlar to the Jarque Bera.

23 coeffcents for the sectors lsted n (v), as shown n Table 3, are -0.95, -0.9, and - 0.7, respectvely. Snce the SN dstrbuton contans the normal as a partcular case, we can assess the ft of Y~SN(α) aganst that of Y~N(0,1) wth the log-lkelhood rato test (LR). Table 3 shows that the LR test s sgnfcant at the 10% level for 7 out of ten seres ( Bankng System, Busness, Consumer, Secured by Real Estate, Mortgages, Agrcultural and Leases ). Note that the three seres for whch the SN does not provde a better ft are also the ones that dd not have a statstcally sgnfcant shape parameter. Fgure 4 plots the dstrbuton of charge-off rates ftted under Vascek s Gaussan densty and that of the SN(α) common factor aganst the hstorcal observed rates. It s clear that the SN(α) provdes a marked mprovement n the ft especally n cases where α s large. Regardng the ST results shown n Table 4, we provde the followng observatons: () the magntudes of the pd, ρ and α parameters are almost dentcal to those for the SN case, except for Agrcultural loans ; () the same three portfolos that dd not have a sgnfcant shape parameter n the SN case reman nsgnfcant under the ST (vz. Credt Card, Other Consumer, Commercal Real Estate ); () the relatonshp reported n pont () of the SN case between α and ρ also holds here for the ST ; (v) the estmated degrees of freedom parameter v s very large n all sectors except for the agrcultural seres. Regardng the ft of the ST, we can clearly see from the LR lsted at the bottom of Table 4 that: () the ST provdes a better ft than Vascek s Gaussan alternatve for exactly the same sectors where SN also provded a better ft; () the ST does not provde 3

24 a statstcally better ft than the SN. Ths result corresponds to the large estmated values for v and the fact that the ST collapses to SN as v. The only excepton to pont () s the Agrcultural Loan portfolo, where the estmated v s Non-Gaussan Idosyncratc Nose The results n Tables 5 and 6 show the parameter estmates for the case where the dosyncratc factor s SN and ST dstrbuted, respectvely, whle the common factor s normally dstrbuted. The log-lkelhood rato shows that n none of the ten seres, the non-gaussan alternatve for the dosyncratc factor provdes a better ft than the Normal dstrbuton. 6.4 Impact on Captal Charge Table 7 compares the captal charge per unt of exposure for the cases where the common factor s Gaussan, SN or ST. Note that the captal charges for the SN and the ST cases are hgher for negatve shape parameter values (vz. Commercal Bankng System, Secured by Real Estate, Mortgages and Agrcultural ), and lower for postve shape parameter (vz. Busness loan, Consumer loan, Credt Card, Other Consumer loan, Commercal Real Estate and Lease ). The dfference n captal charge estmates becomes smaller the closer the shape parameter s to zero. Note that the skewness has a large mpact on the captal charges. For example, the captal requrement for Real Estate s more than double under SN than under Gaussan. It s clear that hgher negatve skewness n asset returns leads to hgher captal requrements. For the case of the SN, the estmated skewness s close to the maxmum admssble skewness of SN, whch corresponds to roughly double the captal requrements. It s 4

25 mportant to recall that as the degrees of freedom parameter tends to nfnte, then the ST converges to the SN. Thus, lower estmates for the degrees of freedom parameter lead to hgher captal requrements. The only portfolo wth a small degrees of freedom parameter estmate s Agrcultural. In ths partcular case, the estmated degrees of freedom parameter s 7.3 (see Table 4) and ths explans the dfference between the captal requrement for the ST common factor (0.09) and that of the SN (0.05). The last two panels of Table 7 show that the captal requrements under the SN and the ST dosyncratc nose assumpton are smlar and do not dffer sgnfcantly form those of the Vascek Gaussan assumpton. Ths result mght be due to the fact that wthn each portfolo, the dosyncratc rsk s well dversfed, and there s no sgnfcant exposure to dosyncratc rsk assocated wth ndvdual exposures. Ths s plausble snce our data represent large portfolos. Nevertheless, ths result should be taken wth care. If we were to repeat ths exercse on the loan portfolo of small or medum szed commercal banks, then the result and the concluson mght change. 7. Concludng Remarks Vascek (1987, 00) derve a lmtng loan portfolo loss dstrbuton whch s founded on a stochastc asset return process that s drven by a common and an dosyncratc factor both of whch are Gaussan. Schönbucher (001) extends the Vascek model to nclude cases where the common and the dosyncratc factors are non-gaussan. Despte ts analytcal tractablty and the rch theoretcal nsghts whch have heavly nfluenced Basel II, the lterature lacks drect emprcal support. In ths paper, we have developed a methodology to model emprcally the mpact of non-gaussan rsk factors on credt loss dstrbutons and captal charge. We allow 5

26 the underlyng common and dosyncratc factors to be Skewed Normal and Skewed Student s t ndvdually. The man nterest of usng non-gaussan dstrbutons s to control for the effect of the asset return skewness and excess kurtoss on the shape of the loss dstrburon. The maxmum lkelhood of our models requre further analytcal results of functons of non-normal varables whch was then performed to offcal charge-off rates publshed by the Federal Reserve Board for ten U.S. sector charge-off rates. The man fndng of our paper s that non-gaussan modellng provdes a sgnfcantly mproved ft n the loss densty for seven out of the ten portfolos analyzed. The most conclusve fndng s that the common factor should be best modelled as Skewed Normal. Allowng the common factor to be Skewed Student s t or the dosyncratc factor to be non-gaussan, does not provde notceably sgnfcant mprovement to the emprcal ft. Our fndngs confrm that non-gaussan modellng of the common factor s very mportant, and hghlght the nadequacy of the exstng Basel II framework. The captal requrements obtaned by assumng a Gaussan dstrbuton for the asset return could over-or underestmate the captal requrements. Ths degree of over-or underestmaton depends on the sgn and the magntude of the skew parameter of the Skew Normal. Large negatve skew parameter value leads to an underestmated captal requrement, whle large postve skew parameter leads to an overestmaton. The non-gaussan modellng of the dosyncratc factor dd not produce any nsgnfcant mpact possbly because the ten sectors analyzed here are large portfolos, and the dosyncratc rsks mght have already been cancelled out. Due dlgence should be observed when the loan portfolo s small or not well dversfed. 6

27 Our emprcal evdence suggests that Skew Normal s an adequate representaton of the dstrbutonal propertes of the latent common factor, snce the estmated degrees of freedom for the Skew Student s t-dstrbuton takes very hgh values, approachng a Skew Normal. However, we propose usng Skew Student s t as a modellng choce because ths dstrbuton adds extra flexblty and has the potental to accommodate both heavy tals and skewness, whch mght prove useful for modellng the losses due to the credt crss when the data becomes avalable. 7

28 References [1] Azzaln, A., (1985). A class of dstrbutons whch ncludes the Normal ones. Scandnavan Journal of Statstcs 1, pp [] Azzaln, A., (1986). Further results on a class of dstrbutons whch ncludes the Normal ones. Statstca XLVI, pp [3] Azzaln, A. and A. Captano (003). Dstrbutons generated by perturbaton of symmetry wth emphass on a multvarate skew t dstrbuton. Journal of the Royal Statstcal Socety B 65, pp [4] Azzaln, A. (005). The skew-normal dstrbuton and related multvarate famles. Scandnavan Journal of Statstcs 3, pp [5] Basel Commttee on Bankng Supervson (004). Internatonal convergence of captal measurement and captal standards. A revsed framework. Consultatve document. [6] Batz, E., Chrstodoulaks, G., and Poon, S., (008). Credt loss dstrbutons: Vascek and beyond. Workng Paper. [7] Branco, M. D. and Dey, D.K. (001). A general class of multvarate skewellptcal dstrbutons. Journal of Multvarate Analyss 79, pp [8] Cornsh, E. A. and Fsher, R. A. (1960). Technometrcs,, pp [9] Düllmann, K., and Trapp, M. (004). Systematc rsk n recovery rates- An emprcal analyss of U.S. corporate credt exposures. Deutsche Bundesbank, pp [10] Gander, W. and Gautsch, W. (000). Adaptve quadrature-revsted. BIT Numercal Mathematcs 40, pp Avalable at [11] Greene, W. (000). Econometrc Analyss. Upper saddle rver, New Jersey, Prentce-Hall, Inc. [1] Kjerst, A. (005). The Basel II IRB approach for credt portfolos: A survey. Norsk Regnesentral (Norwegan Computng Center, NR); avalable at [13] Kjerst, A. (006). The generalzed hyperbolc skew student s t-dstrbuton. Journal of Fnancal Econometrcs 4, pp

29 [14] Merton, R. (1974). On the prcng of corporate debt: The rsk structure of nterest rates. Journal of Fnance 9, [15] Moler, C. (004). Numercal computng wth Matlab. Phladelpha, USA, SIAM (Socety for Industral and Appled Mathematcs). Avalable at [16] Perraudn, W. and Lamb, R. (006). Dynamc loan loss dstrbuton: estmaton and mplcatons. Workng paper. Imperal College, Tanaka Busness School, London. [17] Schönbucher, P. (001). Factor models: Portfolo credt rsks when defaults are correlated. The Journal of Rsk Fnance, 3, [18] Vascek, O. (1987). Probablty of loss on loan portfolo. KMV Corp.; avalable at [19] Vascek, O. (00). Loan portfolo value. Rsk 15, [0] Whte, H. (198). Maxmum lkelhood estmaton of msspecfed models. Econometrca, 53, pp

30 Charge-Off Seres a Commercal Bankng System Table 1. Descrptve statstcs for sector charge-off rates (not seasonally adjusted) Publshed by the Federal Reserve Board for the perod Q1:1985 to Q3:007 Busness Consumer Credt Card Other Consumer Secured by Real Estate Commercal Real Estate Mortgages Agrcultural Mean Std. Dev Scaled Charge-Off Seres by (1/LGD) LGD (from Basel II) 1 b Mean Std. Dev Skewness Kurtoss No of observatons Lease (a) The sector charge-off rates (not seasonally adjusted) are defned as the flow of a bank s net charge-offs (gross charge-offs mnus recoveres) durng a quarter dvded by the average level of loans outstandng n that quarter. (b) The Commercal Bankng System comprses Busness, Consumer, Secured by Real Estate, Agrcultural and Lease. The share of losses for the fve sectors s not dsclosed by the Fed and Basel II does not provde any value for the LGD of Commercal Bankng System. For smplcty, we assume that the LGD for the Commercal Bankng System s equal to 1. ( ) The Fed started reportng the charge-off rate for the Commercal Real Estate and Mortgages only from Q1:

31 PANEL A Commercal Bankng System Table. Parameter estmates for Vascek loss dstrbuton of sector charge-off rates (not seasonally adjusted) Publshed by the Federal Reserve Board for the perod Q1:1985 to Q3:007 Busness Consumer Credt Card Other Consumer Secured by Real Estate Commercal Real Estate Mortgages Agrcultural PD Inv. Hessan Std Error QMLE Std Error OPG Std Error Bootstrap Std Error ρ Inv. Hessan Std Error QMLE Std Error OPG Std Error Bootstrap Std Error Log-Lkelhood PANEL B: ρ JBera Statstc ρ Crtcal Value PD JBera Statstc PD Crtcal Value The Fed started reportng the charge-off rate for the Commercal Real Estate and Mortgages only from Q1:1994. The Jarque-Bera test s a two sded goodness of ft test sutable when a fully-specfed null dstrbuton s unknown and ts parameters must be estmated. The test statstc s JB = ( n 6) s + ( k 3) 4 where n s the sample sze, s s the sample skewness, and k s the sample kurtoss. For large sample szes, the test statstc has a ch-square dstrbuton. ( ) In Matlab, the Jarque-Bera test uses a table of crtcal values computed usng Monte-Carlo smulaton for sample szes less than 000 and sgnfcance levels between and Crtcal values for a test are computed by nterpolatng nto the table, usng the analytc ch-square approxmaton only when extrapolatng for larger sample szes. Lease 31

32 Table 3. Parameter estmates for Vascek loss dstrbuton where Y ~ SN( α ) and ε ~ N( 0,1) on sector charge-off Rates (not seasonally adjusted) publshed by the Federal Reserve Board for the perod Q1:1985 to Q3:007 Commercal Bankng System Base case results from Table 3: Busness Consumer Credt Card Other Consumer Secured by Real Estate Commercal Real Estate j Mortgages Agrcultural PD ρ PD QMLE Std Error ρ QMLE Std Error α QMLE Std Error Log-Lkelhood Log-Lk Vascek LR rato p-value * * # # # # CF Skewness CF Exc. Kurtoss Notes: The Fed started reportng the charge-off rate for the Commercal Real Estate and Mortgages only from Q1:1994. CF refers to the Common Factor # ndcates cases where the skew normal provdes a sgnfcant better ft than the normal at the 1% level. * ndcates cases where the skew normal provdes a sgnfcant better ft than the normal at the 5% level. + ndcates cases where the skew normal provdes a sgnfcant better ft than the normal at the 10% level. Gven that the results by usng dfferent estmators are smlar, we report only the nference based on the QMLE standard error. Lease 3