Loss Distribution Generation in Credit Portfolio Modeling

Transcription

1 Loss Dstrbuto Geerato Credt Portfolo Modelg Igor Jouravlev, MMF, Walde Uversty, USA Ruth A. Maurer, Ph.D., Professor Emertus of Mathematcal ad Computer Sceces, Colorado School of Mes, USA Key words: Loss dstrbuto, Copula, Credt Portfolo, Large Homogeeous Portfolo, Mote Carlo Smulatos, Importace Samplg, Saddle-pot Approxmato Itroducto I the curret paper we aalyze several methods for geerato of loss dstrbuto for credt portfolos. Loss dstrbutos play a mportat role prcg of credt dervatves ad credt portfolo optmzato. A loss dstrbuto s a fucto of the umber of ettes the portfolo, ther credt ratgs, the otoal amout ad recovery of each etty, default probabltes, loss gve defaults, ad the correlato/depedece structure betwee ettes corporated the portfolo. Drect geerato of loss dstrbuto may requre Mote Carlo smulato whch s tme cosumg ad s ot effectve whe appled for credt portfolo optmzato. To overcome computatoal complexty a umber of approaches were udertae based o assumptos mposed o the put parameters, goals of loss dstrbutos geerato, ad the accepted level of tolerace ad computatoal errors. Lterature revew A wde rage of lterature was dedcated to geerato of loss dstrbutos for credt portfolos. Vasce (987, 00) developed a large homogeeous portfolo approxmato that played a mportat role oe of the frst sythetc CDOs prcg methods developed by J.P. Morga. Ths method was geeralzed by cosderg a fte umber of oblgors the portfolo. Hull ad Whte (004) troduced a bucetg approach, ad Aderse, Sdeus ad Basu

2 (003) offered a recursve method vald more geeral cases. Glasserma ad L (005) proposed the two-step-mportace samplg method whch they appled Mote Carlo smulato methods wth varace reducto techques. A Fourer aalytcal approach loss dstrbuto geerato was aalyzed by Mero ad Nyfeler (00), Ress (003), ad Grude (007) whch was used for prcg of CDO by Greogry ad Lauret (004), Lauret (004), ad Lauret ad Gregory (005). Ths method requres a good mplemetato of the fast Fourer trasform. Saddle-pot approxmato was aalyzed by Arvats ad Gregory (00), Gordy (00), Mart (006), Glasserma (008). We wll aalyze ad compare these methods, ther advatages ad dsadvatages. Large homogeeous approxmatos of loss dstrbutos Large homogeeous portfolo (LHP) approxmato was the frst method developed by Oldrch Vasce 987 at KMV Corporato. I ths model, Vasce assumed that the portfolo cotas a fte umber of ettes. Each etty has the same otoal amout, default probablty, ad recovery rate (or loss gve default). Loss gve default ca be calculated as recovery rate. The correlato structure s preseted usg oe-factor Gaussa copula. Although these assumptos were very restrctve, the closed formula derved by Vasce was easy to mplemet ad the model was much faster tha the oe used Mote Carlo smulatos. Accordg to ths model, the rs eutral portfolo cumulatve loss dstrbuto ad probablty desty fucto for a large portfolo wth uderlyg Gaussa copula ca be expressed as follows (Vasce, 00): ρ Φ ( x) Φ ( p) P ( L x) = Φ () ρ

3 3 f ρ ( x) = exp ( ρφ ( x) Φ ( p) ) + ( Φ ( x) ) ρ ρ () where L [0,] portfolo loss estmated as a fracto of the whole portfolo, Φ cumulatve ormal dstrbuto, Φ s the verse of cumulatve ormal dstrbuto, p the probablty of default of a loa the portfolo; ths case we assume that all credts have the same probablty of default, ad the umber of credts s fte, ρ - correlato corporated to the credt portfolo usg oe-factor Gaussa copula. Although the portfolo cotas loas wth the same default characterstcs; the loas are fact dfferet, ad, moreover, correlated wth correlato ρ. The advatage of usg ths formula s that t ca be used for fast approxmato of loss dstrbuto for ay rage [ a, b], where a < b, ad a, b [0,]. The above metoed formulas () ad () deped o very mportat parameters correlato ad probablty of default of a loa the portfolo. A credt portfolo wth hgh value of correlato (say 90%) wth very small value of probablty of default of a loa (say < 0 bps) wll have loss dstrbuto cocetrated aroud 0%, ad ca be approxmately cosdered as a credt portfolo rsless. Oe of the extesos of the Gaussa LHP approach s to use double-t oe factor model proposed by Hull ad Whte (004). They assumed that commo ad dvdual factors are t- dstrbuted ad derved a formula that gves fast approxmato of the loss cumulatve dstrbuto fucto. The LHP approxmato ca be exteded to the case of the Studet-t copula. Ths approach also allows oe to obta aalytcal formulas for desty ad the cumulatve dstrbuto fucto of the portfolo loss dstrbuto (Schloegl & O'Kae, 005). Schloegl ad

4 4 O Kae (005) also compared the VaR mpled by the Studet-t copula to that obtaed usg the Gaussa, Calyto, ad Gumbel copulas. Accordg to Schloegl ad O Kae (005), the returs ζ of each oblgor follow a multvarate Studet-t dstrbuto, so that (p. 578) ζ βz + β ε = (3) W ν ζ where Z, ε,..., ε ~ N(0,), W ~ χ ( ν ), β M ]0,[. Issuer defaults f ad oly f D where D s a certa threshold. Ths equalty s equvalet to: Codtoally o, the default dcator fuctos W β ε D βz =. (4) ν ζ ] are all depedet ad the codtoal default probablty ca be wrtte as a fucto of the stadard ormal cumulatve desty fucto [ ] P ζ D = Φ. By the law of large umbers, the fracto of β [ D defaultg ssuers wll ted to Φ β for each realzato of as M. Assumg that exactly ths fracto of ssuer defaults for each realzato of, thereby replacg the radom varable L by [ L ] E, we have that h( ) [ ] ( ) ( θ [0,] ) P θ = F ( θ ) L, where h( x = ( R) x ) Φ, ad β h, where F s the cumulatve dstrbuto fucto of. Based o ths logc, aalytc formulas for cumulatve dstrbuto fucto ad probablty desty fucto ca be expressed usg the followg formulas (Schloegl & O Kae (005), pp ):

5 5 t u t ν ν ( t + βu) F( t) P[ t], e du D = = Φ + β Γ (5) β π f ( t) = β π ν + ν Γ w t D w + e ν w β ν e 0 dw. (6) As was oted by O Kae (008), there are approaches that approxmately estmate loss dstrbutos ad ca serve as acceptable compromses to the trade-off betwee speed ad accuracy. These approaches are the Gaussa approxmato, bomal ad adjusted bomal dstrbutos (pp ). The Gaussa approxmato uses a Gaussa desty whch fts the frst two momets of the codtoal loss dstrbuto. It s the possble to obta a closed-form expresso for the expected trache loss codtoal o the maret factor (O Kae, 008, p. 354). The dea behd the bomal approxmato s to approxmate the exact multomal dstrbuto wth a bomal dstrbuto. The reaso for ths s that the shape of bomal dstrbuto s a better ft to the multomal dstrbuto tha Gaussa. However, ths approach we match the frst momet of the exact codtoal loss dstrbuto. The further mprovemet s based fdg a way to ft the varace. For ths purpose, a adjusted bomal approxmato was proposed by O Kae (008) to esure that we match frst two momets of the exact homogeeous loss dstrbutos (pp ). The LHP approxmato ca be exteded by usg ormal verse Gaussa dstrbuto (NIG). The ormal verse Gaussa dstrbuto s a mxture of ormal ad verse Gaussa dstrbuto. Accordg to Kalemaova, Schmd, ad Werer (007), a o-egatve radom varable Y has verse Gaussa dstrbuto wth postve parameters α ad β f ts desty fucto ca be represeted usg the followg form:

6 6 f IG α 3 y e ( y; α, β ) = πβ 0, otherwse ( α βy ) βy,f y > 0 (7) A radom varable X follows the NIG dstrbuto wth parameters α, β, μ, δ ( X ~ NIG( α, β, μ, δ ) ) f X Y = y ~ N( μ + βy, y) Y ~ IG( δγ, γ ), γ = α β (8) formula: where The parameters should satsfy the followg codtos: 0 β < α ad δ > 0. The desty of the radom varable X ~ NIG( α, β, μ, δ ) s gve by the followg ( δγ + β ( x μ) ) + ( x μ) αδ exp ( ) f = + ( ) NIG x; α, β, μ, δ K α δ x μ (9) π δ K 0 ( w) = exp w( t + t ) dt (0) s the modfed Bessel fucto of the thrd d. Kalemaova et al. (007) suggested usg the followg parameters the LHP model wth NIG copula (wth the depedecy parameter defed as a square root of correlato assumed a credt portfolo): 3 βγ γ M ~ NIG α, β,, α α V ~ NIG α, β, βγ α, 3 γ α () where M s a systematc maret rs factor, ad are dosycratc factors. V The, asset returs wll also follow NIG dstrbuto wth the followg parameters:

7 7 3,,, ~ α γ α βγ β α NIG M () The thrd ad fourth parameters were chose to get expected value of zero ad varace of. Usg the followg otato = 3 ) (,,, ; ) ( α γ α βγ β α s s s s x F x F NIG s NIG, the default threshold ca be computed as follows: ) ( p F C NIG = ; (3) the default probablty of each credt codtoal o maret factor s gve by = ) ( M C f M p NIG (4) ad the loss dstrbuto of the LHP ca be estmated usg the followg formula (Kalemaova et al., 007): = ) ( ) ( () x F C F x F NIG NIG (6) These formulas use specal fuctos; however, the advatage of usg them s that computato of loss dstrbutos usg Gaussa copula, Studet-t, double-t, or NIG copula s much less tme cosumg tha usg Mote Carlo smulato for geerato loss dstrbutos. Oe of the advatages of usg NIG copula s the possblty of estmatg four parameters of ths dstrbuto gve observed frst four momets. Schöbucher (00) used a algorthm from the theory of Archmedea copula fuctos to estmate lmtg loss dstrbutos whch are drve by radom varable wth dfferet

8 8 depedecy structures. Ths approach allowed presetg smple ad realstc formulas for the loa portfolo dstrbuto. The jot dstrbutos a credt portfolo are modeled dfferet ways tha just usg a varat of the multvarate ormal dstrbuto fucto, ad ths approach proved to be feasble. I obtag loss dstrbutos, t s mportat to vestgate the effect of the mplct assumpto of a Gaussa depedecy structure o the rs measures ad the returs dstrbuto of the portfolo, as well as the cosequeces of extreme evets ad lac of avalable data o credt rs modelg. Schöbucher (00) showed that the credt rs case, ths effect ca be ether mor (whe the Vasce model s compared to the Clayto-depedet model) or sgfcat (whe oe ths that the Gumbel copula s a realstc alteratve). Fte homogeeous approxmato of loss dstrbutos I fte homogeeous approxmato (ofte called exact computato), the fty assumpto s dropped ad the model uses assumptos of a sgle systematc factor ad homogeety; ths case the umercal procedure s stll easy to mplemet. I most cases, however, the portfolos are ot homogeeous, but f we assume that the portfolo s large, graularty adjustmet developed by Gordy (003), Pyht ad Dev (003), ad Gordy (004) ca be appled. I both large homogeeous ad fte homogeeous portfolos the loss dstrbuto ca be geerated based o assumpto of several systematc rs factors usg approprate radom umber geeratos for each factors. For a credt portfolo cosstg of ettes, the Gaussa copula case, the probablty of defaults (or ucodtoal loss dstrbuto dscrete case) ca be expressed usg the followg formula (Vasce, 00):

9 9 P = Φ ρ ( ( p) u) Φ Φ ρ Φ ( p) ρ ( ρu) dφ( u) (7) where p s the probablty of default, ρ s correlato corporated to the credt portfolo usg Gaussa copula, the tegrad s the codtoal probablty of the portfolo loss gve the maret factor u whch s assumed to be ormally dstrbuted. If we cosder m maret factors ths approach, the the tegrato would be over these m maret factors. Codtoal ad ucodtoal loss dstrbutos geerato The prevous formula developed by Vasce (00) gves a dea o how to obta the ucodtoal loss dstrbuto the geeral case. Frst of all, oe has to compute a codtoal loss dstrbuto codtoal o a set of uderlyg factors whch defaults are depedet, ad the tegrate the codtoal loss dstrbuto over the dstrbuto of the uderlyg factors. I mathematcal otato, we eed to compute codtoal probabltes codtoal o maret factors frst: M M M [ ] ( ) (,,..., l = L M, M,..., M = p P L M, M,..., M l l t pt ) (8) ad the tegrate over these factors: where L ( R [ L L M = m,..., M l = ml ] df( m ) df( ml )... P =... (9) = )s the percetage portfolo loss, P [ L L M, M,..., ] = s the probablty M l that exactly out of ssuers default codtoal o maret factors M M,..., ; ad M M p,..., l t, s the codtoal default probablty of oblgor at tme t. I case of Gaussa copula ad case whe we cosder oly oe maret factor our model, we have: M l

10 0 p M t Φ = Φ ( p ) ρ M. (0) ρ Edgeworth expaso ca be used as oe of the methods for geeratg of loss dstrbuto (Arvats & Gregory, 00). Ths expaso uses the hgher-order momets of the dstrbutos of the costtuet varables (such as umber of defaults) ad formato cotaed cumulats. It s well ow that for the ormal dstrbuto, the frst two cumulats are the mea ad varace ad the others are equal to zero. The hgher cumulats gve quattatve formato about the o-ormalty of a dstrbuto. Edgeworth expaso states that f the umber of defaults are depedet radom varables wth meas p, stadard devato σ ad cumulats κ, the probablty desty fucto of the radom varable Y = d = whch represets umber of defaults, ca be gve by f Y ( y) dy = t / e dt κ 3H 3( t) + 3 π 3! σ κ 4H 4 ( t) κ 3 H 6 ( t) ! σ!3! σ +... () wth y p t =, p = p, σ = σ, κ r = κ, σ = = = r () where H r (t) s the rth Hermte polyomal, whch ca be obtaed by successve dfferetato of the fucto t e usg followg equato: t t r d H r ( t) = ( ) e e (3) r dt Cumulats ca be obtaed from the power-seres expaso of the logarthm of the momet geeratg fucto of the radom varable. The leadg term ths expaso s the

11 ormal dstrbuto, ad the frst ad secod terms adjust the sewess ad urtoss. These adjustmets gve better approxmato to the loss dstrbuto by corporatg of the sew (whch correspods to the asymmetry of the dstrbuto) ad urtoss (whch correspods to the fat tals of dstrbuto). As was show by Arvats ad Gregory (00), the Edgeworth approxmato wth four momets s better tha the ormal approxmato, but t becomes less accurate further to the tal. The approxmato s poor to the left of the org, where true probablty desty fucto vashes. For example, the estmato of the uexpected loss at the 99.9 th percetle correspods to a tal probablty of The ormal approxmato uderestmates the true value by 40% ad the Edgeworth approxmato (wth four terms) uderestmates t by 4% (pp ). Hull ad Whte (004) preseted two approaches geeratg loss dstrbutos. They cosdered a umber of maret factors M,..., M l cosdered codtoal o these maret factors. Defg ad the codtoal default probabltes were π T () the probablty that exactly defaults occur the portfolo before tme T, codtoal o the default tmes t are depedet. The codtoal default probablty that all the ames wll survve beyod tme T s π T ( 0 V,..., Vm ) = S ( T M,..., M l ) where S ( T M,..., M l ) s the survval probablty of the = oblgor. Hull ad Whte (004) showed that the codtoal probablty of exactly defaults by tme T s T ( M,..., M l ) π T ( 0 M,..., M l ) π (4) = wz() wz()... wz( )

12 where w ( T M,..., M l ) ( T M,..., M ) S = ad z( ),... z() s the set of dfferet umbers from the fte S l set{,,}. Hull ad Whte (004) provded a fast algorthm for computg the codtoal losses. By tegratg over the maret factors, oe ca obta ucodtoal loss dstrbuto. Hull ad Whte (004) proposes also a bucetg approach whle Aderso et al. (003) proposed a recursve method for geeratg loss dstrbuto. I both approaches, loss gve default ad otoal values ca vary betwee the ettes, but they are stll assumed to be determstc. Hull ad Whte (004) dvded potetal losses to the followg rages: { 0 }{ } {, b, b, b,..., b, 0 0 K b } where { b, b } s referred as th bucet. The loss dstrbuto s bult by cludg oe debt strumet at a tme. The procedure eeps trac of both the probablty of the cumulatve loss beg a bucet ad the mea cumulatve loss codtoal that the cumulatve loss s the bucet. The approach offered by Hull ad Whte (004) does ot assume bucets of detcal sze; t allows the aalyst accommodatg stuatos where extra accuracy s eeded some regos of the loss dstrbuto. The latter ca be acheved by cosderg smaller bucet szes. The loss dstrbuto ca be trucated at some level so that the aalyst eed ot sped extra computatoal tme o large losses that have oly a very small chace of occurrg. Hull ad Whte (004) reported that ther approach s comparable wth the Fourer-aalytcal approach terms of computatoal tme ad accuracy ad s umercally stable. The ma dea of the approach suggested by Aderse et al. (003) was to compute some u as a commo dvsor of all potetal losses, ad the cosder losses * 0, u,u,..., u rouded to the earest dscrete pot as the loss dstrbuto s bult up. * u here s the maxmum possble

13 3 loss. It s mportat to ote here that the speed of ths algorthm depeds o the statstcal characterstcs of the credt spreads, so that some cases, the value of the commo dvsor of all potetal losses ca be very small substatally affectg the speed of the algorthm. The ext step was to mplemet a recursve algorthm to determe the portfolo loss dstrbuto that s used for the codtoal probabltes ad case whe default evets are depedet. It should be oted here that the value for u ca be very small some credt portfolos ad cur computatoally extesve loss dstrbuto geerato. Suppose we ow the loss dstrbuto P K ( l; t), l = 0,..., l max, K for a referece pool of some sze K 0, where s the sum of all the loss weghts such referece pool. Suppose we add aother compay to the pool wth loss weght w K + ad ow default probablty p K loss dstrbuto of the larger baset (Aderse et al., 003, p.67): ( t) l max, K +. The usg depedece of defaults we fd for the K ( pk + ( t) ) + P ( l wk + ; t) p K + ( t), l = max, K + w K + K P ( l; t) = P ( l; t)! 0,..., l + (5) Ths recursve relato starts wth a empty baset, ad crease baset sze leads to the same relatve crease the maxmal loss. The cost of buldg the codtoal loss dstrbuto grows as roughly the square of the baset sze (Aderse et al., 003, p.67). The resultg codtoal loss dstrbuto s trasformed to the ucodtoal loss dstrbuto by tegratg over commo factor. Fourer aalytcal approach s aother approach of codtoal ad ucodtoal loss dstrbuto geerato. Ths approach s alteratve to recurso techques ad cosders a map of the orgal problem to aother space where the problem s more aalytcally tractable. Oce the problem ths space s solved, we eed to map the soluto bac to the orgal space. Ths approach depeds o the successful mplemetato of fast Fourer techques. Ths approach was used, for example, by Gregory ad Lauret (003, 004) for prcg CDOs. Reβ (003)

14 4 provded detaled formato o how loss dstrbuto ad ts frst momets ca be obtaed usg Fourer aalytcal approach, descrbed the CredtRs+ model terms of characterstc fuctos stead of probablty fuctos. Sce costructo of loss dstrbuto very much depeds o the basc loss ut, Reβ preseted a alteratve approach where o basc loss ut had to be troduced. Mero ad Nyfeler (00) descrbed a algorthm that combes techques from umercal mathematcs ad actuaral scece. Ther approach geerato of loss dstrbutos was groupg all potetal losses to exposure bucets. Approxmatg the Beroull default dcators by Posso radom varables allowed reducg the umber of radom varables whch correspod to the umber of credts the portfolo to the umber of bucets. Applyg the fast Fourer trasform, umercal quas Mote Carlo methods allowed geeratg loss dstrbutos of credt portfolos cotag 500,000 couterpartes wth four hours wth adequate accuracy. It was show that t was ot ecessary to smplfy the credt rs model or portfolo structure to calculate the body ad the tal of the portfolo loss dstrbuto ad that the algorthm was useful for aalyzg ad desgg CDO structures. Grude (007) aalyzed whether a Fourer-based approach could be a effcet for calculato of rs measures the cotext of a credt portfolo model wth tegrated maret rs factors. He appled ths approach to CredtMetrcs credt portfolo model exteded by correlated terest rate ad credt spread rs. He showed that Fourer-base methods beg superor to Mote-Carlo smulatos could t be superor case of the tegrated maret ad credt portfolo model eve after applyg stadard mportace samplg techques for mprovg the performace of Fourer-based approach. Ths s because the hgher the cofdece level of the VaR, the larger the asset retur correlato, or the larger the umber of systematc rs factors. For the tegrated maret ad credt portfolo methods, oe should combe the Mote Carlo

15 5 smulato wth a mportace samplg techque. Combato of Mote-Carlo ad mportace samplg techques would be approprate for those cases where the Fourer-based approach performs badly, for example, for the estmato of small percetles whch are eeded credt rs maagemet. The FFT approach was the frst used loss dstrbuto costructo methodology. However, t s slower approach compared to the recursve approach (but stll faster tha the Mote Carlo approach) of geeratg loss dstrbutos due to the followg reasos (O Kae, 008): recursos are faster tha Fourer methods, recursos are easer to mplemet ad do t requre access to ay specalzed umercal lbrares; usg recursos the researcher ca buld the loss dstrbuto for a specfc trache; ths s ot possble to do whe usg FFT. The saddle-pot approxmato for geeratg loss dstrbuto proved to be very accurate practce. Whe cosderg sums of depedet radom varables, t s coveet to cosder the momet geeratg fucto (MGF). The MGF of a radom varable ca be represeted as follows (Arvats & Gregory, 00): M st ( s) = e pv ( t dt (6) V ) The well ow property of the MGF of a radom varables s that whe depedet varables are added, ther dstrbutos are covolved, but MGFs are multpled. Multplcato operato s easy to perform tha covoluto. O the other sde, oe has to obta the loss dstrbuto of from MGF usg verso tegrato. By sutably approxmatg the shape of the tegrad oe obtas a aalytcal approxmato of the probablty desty fucto. Ths

16 6 techque s ow as the method of steepest descets or saddle-pot method. The method also allows obtag aalytcal approxmatos to the probablty wthout havg to tegrate the desty fucto. The saddle-pot approxmato does ot mae ay pror assumptos about the shape of the loss dstrbuto. As was suggested by Arvats ad Gregory (00), saddle-pot approxmato method s a fast method obtag approxmate loss dstrbuto ad tal probabltes, ca be used for approxmato to derve expressos for loss dstrbutos that clude varable exposures ad default probabltes, ad allows corporatg correlato to the loss dstrbuto (p. 85). If we deote dvdual losses V,...,V be characterzed through ts cumulat geeratg fucto (Glasserma, 008): Each V, the dstrbuto of each loss ca ( exp( θ )) θ & ( =,..., ) Λ ( θ ) = log E (7) V s a radom fracto of a largest possble loss upo default of oblgor, all oblgors cosdered here are depedet. As a cosequece of depedece, the momet geeratg fucto ca be represeted as L E (8) = = ( θ ) = expθ YV = E[ exp( θyv )] = ( + p ( exp( Λ ( θ )) ) ) ad the cumulat geeratg fucto of L as (Glasserma, 008) ψ L ( θ ) = log[ L ( θ )] = log ( + p ( exp( Λ ( θ )) ) ) = ( log( + p ( exp( Λ ( θ )) ) )) (9) = = Aalyss of ths fucto shows that t s creasg, covex, ftely dfferetable ad taes zero whe = 0 θ. For a loss level x > 0, the saddle-pot s the root θ x of the equato ψ ' ( ) L θ x = x whch s uque. Cosder the cumulat geeratg fucto ψ L ( θ ). The dervatves of ( θ ) cumulats of L. The frst cumulat whe θ = 0 s the mea ψ L gve

17 7 ' L = = = ' ( 0) = p Λ ( 0) = p E[ V ] E[ L] ψ. (30) Whe losses are assumed to be costat, V = v, we have ( θ ) = v = ' ψ L p. (3) I other words, the dervatve of the cumulat geeratg fucto s the expected loss where orgal default probabltes are replaced wth p I geeral, the formula for dervatve s p = + p exp( v) ( exp() v ). ' ( θ ) ( θ ) ( θ ) ' ψ = p Λ = (3) whch ca be terpreted as the expected loss whe the orgal default probabltes are replaced p exp( Λ ( θ )) by p ( θ ) = ad the expected losses [ V ] + p ( exp( Λ ( θ )) ) ( ) ' E are replaced wth θ where Λ orgal expected loss [ V ] ' E cocdes wth Λ ( 0) because Λ s the cumulat geeratg fucto of V.Thus, each value of θ determes a modfed set of default probabltes ad a modfed loss gve default for each oblgor (Glasserma, 008, pp ). Due to complexty of the formulas provded, t s mportat to ether develop a fast algorthm or approxmato formulas. A saddle-pot approxmato ca be represeted the followg way (Glasserma, 008): P ( θ ) θ ψ " Φ ( θ ) ( L x ) ψ ( θ ) ψ " exp ( θ ) x x L x L x ( x L x ) > (33) ad the closely related Lugaa-Rce approxmato s P (34) λ( x) r( x) ( L > x) Φ( r( x) ) + ( r( x) )

18 8 where r( x) = ( θ ψ ( θ )) ad " λ x) θ ψ ( θ ) x x A modfcato of the prevous formula s P L x ( =. ( L > x) Φr( x) + log x r( x) L x λ. (35) r( x) Ths modfcato has the advatage that t always produces a value betwee 0 ad (Glasserma, 008, p. 456). To geerate the ucodtoal loss dstrbuto, as t was show before, oe has to compute a codtoal loss dstrbuto based o the assumpto of oblgor depedece ad the tegrate over the possble values of maret factors. Factor models are popular sce oblgors are codtoally depedet such models. Whe tegratg over the possble values of maret factors, fast tegrato procedures should be used; these procedures deped o the probablty dstrbuto of the chose maret factors ad the umber of maret factors. Glasserma (008) proposed ad algorthm for geerato of ucodtoal loss dstrbuto. The model specfes two sets of parameters correspodg to a hgh-default regme ad a low-default regme, wth depedet oblgors each regme. The the model uses a mxture of the two sets of parameters. I ths case, the uderlyg factor s the regme ad the ucodtoal loss dstrbuto may be computed as a mxture of the two codtoal loss dstrbutos (p. 457). Ths approach s qute useful whe aalyzg the credt portfolo cosstg of over 00 0 credts that ca perform dfferetly over the specfed tme perod. The credt portfolo ca be parttoed to the clusters (or bucets) of credts (as well as possble outlers) ad for each cluster low default regme ad hgh default regme ca be detfed. Glasserma ad Ruz-Mata (006) compared the computatoal effcecy of ordary Mote Carlo smulato wth methods that combe smulato for the factors wth the

19 9 techques used to geerate codtoal loss dstrbutos. They determed that umercal trasform verso ad saddle-pot approxmato volve some error the calculato of codtoal default probabltes. Because each replcato usg covoluto, trasform verso, or saddle-pot approxmato taes loger tha each replcato usg ordary smulato, these methods complete fewer replcatos a fxed amout of computg tme. The recursve covoluto method computes the full codtoal loss dstrbuto o each replcato whle trasform verso ad saddle-pot approxmato must practce be lmted to a smaller umber of loss thresholds. Usg the saddle-pot approxmato requres solvg for multple saddle-pot parameters o each replcato. The computato tme requred usg recursve covoluto grows qucly wth the umber of oblgors. As a cosequece of these, wth the total computg tme held fxed, ordary Mote Carlo ofte produces a smaller mea square error (Glasserma, 008, p. 458). The umber of factors plays a substatal role choosg whether Mote Carlo or aother method such as saddle-pot or recursve approach should be used. Whe the umber of factors s small, Mote Carlo smulatos ca be replaced by tegrato. For the moderate umber of dmesos, a quas-mote Carlo samplg ca be appled. O the other sde, oe ca use approxmatos where a sgle most mportat value of the factors s used (Glasserma, 004). Zheg (007) suggested approxmato of the codtoal loss dstrbuto usg a ormal dstrbuto by matchg two momets ad the computg the ucodtoal dstrbuto through umercal tegrato assumg small umber of factors. Specfcally Zheg (007) suggest computg mea ad varace depedg o maret factor Z: μ( Z) = p ( Z) ad σ Z = p ( Z) p ( Z) (36) = ( ) ( ) so that the 0 th ad frst order approxmato ca be represeted respectvely the followg way (Edgworth expaso): =

20 0 P P [ L x] x E Φ σ μ( Z ) ( Z ) x μ( z) x μ( z) [ L x Z] E Φ + H σ ( z) σ ( z) (37) (38) where H ( x) = (/ 6) σ ( z) γ ( z) = = 3 γ ( z)( x ) ( x) p ( z)( p ( z))( p ( z)) (39) The hgher order expaso ca be expressed usg Chebyshev-Hermte polyomals ad / momets of up to order +. The approxmato error was estmated as ( ) o. The ucodtoal loss dstrbuto the ca be obtaed usg tegrato. Glasserma ad L (005) proposed a two-step-mportace samplg techque that helps compute the upper tal of the loss dstrbuto. Such techques clude Mote-Carlo smulato methods combed wth adequate varace reducto approach, ad are applcable to a wde rage of applcatos. Importace samplg s a specal varace reducto techque ad s used to chage the dstrbutos of the relevat rs factors such a way that more realzatos of the loss radom varable are the upper tal. The each realzato s weghted by the lelhood rato to correct for the chage dstrbuto. Dembo et al. (004) provded a large devato approxmato of the tal dstrbuto of total facal losses o a portfolo cosstg of may postos. Quattatve aalyss of large losses s helpful structurg large portfolo so as to wthstad severe losses. Applcatos of ths approach clude the total default losses o a ba portfolo or the total clams agast a surer. A ey assumpto was that codtoal o a commo correlatg factor, posto looses are depedet. For large losses, facal dstress costs are more severe f the losses occur over

21 a relatvely short perod of tme. Sudde losses may cause extreme cash-flow stress, ad vestors may requre more favorable terms whe offerg ew les of facg over short tme perods, wth whch they may have a lmted opportuty to gather formato about the credt qualty ad log-term prospect of a dstressed facal sttutos. The results provded by the authors clude codtos uder whch a large-devatos estmate of the lelhood of a falurethreateg loss durg some sub-terval of tme durg a gve plag horzo ca be calculated from the lelhood of the same sze loss a certa fxed ey tme horzo. The codtoal dstrbuto of losses o each type of posto ca be estmated gve the large portfolo loss of cocer. The authors provded some aalytcal gudace o the depedece of large-loss probabltes o the structure of a portfolo wth a large umber of postos ad the most lely way that a large loss ca occur. Gve a large loss, the codtoal lelhood of loss o each type of posto ad codtoal dstrbuto of exposure the evet of loss were calculated. These codtoal calculatos ca be terpreted the asymptotc sese of Gbbs codtog prcple. Sdeus et al. (008) preseted the SPA framewor whch models are specfed by a two-layer process. The frst layer models the dyamcs of portfolo loss dstrbutos the absece of formato about default tmes. Ths bacgroud process ca be explctly calbrated to the full grd of margal loss dstrbutos as mpled by tal CDO trache values dexed o maturty, as well as to the prces of sutable optos. The authors gave suffcet codtos for cosstet dyamcs. The secod layer models the loss process tself as a Marov process codtoed o the path tae by the bacgroud process. The choce of loss process s ouque. Sdeus et al. (008) preseted a umber of choces, ad dscussed ther advatages ad dsadvatages. Several cocrete model examples were gve, ad valuato the ew

22 framewor was descrbed detal. Amog the specfc securtes for whch algorthms are preseted were CDO trache optos ad leveraged super-seor traches. I practce, oe ca estmate the mpled loss dstrbuto from observed maret data. For example, Kreel ad Partehemer (006) descrbed how to determe the mpled loss surface of a credt portfolo from CDO trache quotes. The approach ca be appled for prcg of CDO traches ad N th -to-default swaps ad for rs maagemet of CDO traches. It also ca serve as a tal dstrbuto for dyamc loss models. The calbrato ca be performed umercally by solvg a olear optmzato problem usg sequetal quadratc programmg method. Aalyss, cocluso ad recommedatos The most mportat ssue geeratg loss dstrbutos for credt portfolos s to fd a fast ad accurate algorthm whch ca be easly mplemeted. There s always trade-off betwee these characterstcs of the descrbed algorthms. Some of the aalyzed algorthms ca be mplemeted easly for solvg practcal problems; o the other sde, the dsadvatage of mplemetg them s a umber of assumptos that ca mae the model urealstc currg accurate decso mag. For a gve credt portfolo, t s advatageous cosderg varous loss dstrbutos models such as bucetg approach by Hull ad Whte (004), recursve method by Aderse et al. (003), Fourer aalytcal approach, ad saddle-pot approxmato ad use the oes that provde better soluto to the specfc problem ad the gve credt portfolo. Amog the algorthms descrbed above, the fastest algorthm s LHP, but the accuracy of ths algorthm depeds o the umber of oblgors the portfolo ad the credt spread dstrbuto. The more cocetrated the credt spreads are ad the more oblgors the credt

23 3 portfolo has, the more applcable LHP s. If we have a smaller umber of oblgors (say, less tha 00 oblgors) we ca use fte homogeeous approach. Fte homogeeous approach s slower tha LHP sce we eed to tegrate codtoal loss dstrbuto over the maret factor to obta ucodtoal loss dstrbuto. Ufortuately, practce, homogeety assumpto ad assumpto that the portfolo s large are ot the case ad LHP ca be rejected by practtoers seeg more accurate loss dstrbuto geerato. LHP as well as fte homogeeous portfolo methods ca t be used as a approxmato of heterogeeous portfolos a loss dstrbuto geerated usg a recursve method for a heterogeeous portfolo ca be qute dfferet from the loss dstrbuto geerated usg LHP whe the put parameter for the LHP s the weghted average spread of the heterogeeous portfolo. O the other extreme, whe more accurate geerato of loss dstrbuto s requred, the best method s Mote Carlo smulato whch s, at the same tme, the slowest. The Mote Carlo smulato s especally valuable cases whe the level of heterogeety s very hgh, other words whe the credt spread dstrbuto s very dverse. Mote Carlo smulatos (as well as other loss geerato methods) ca be made faster ad wll requre less memory f we cluster credt default spreads to homogeeous parttos of credt spreads. Gve a heterogeeous credt portfolo, the tas s to dstrbute to a group of clusters such a way that the objects wth each cluster are homogeeous ad hghly correlated whle the clusters themselves are dfferet. I fact, ths tas has two dffcultes frstly, we eed to detfy a optmal partto of the credt spreads to dfferet clusters; secodly the spread clusters should be explctly determed. There are a umber of algorthms that allows clusterg of credt spreads K-meas, Daa, Clara, fuzzy aalyss, self-orgazg maps, model-based clusterg. There are also

24 4 teral ad stablty valdato based o teral ad stablty measures. Tag the credt spreads data ad clusterg partto as a put, teral measures are used to assess the qualty of clusterg based o trsc formato. Stablty measures assess clusterg cosstecy by comparg t wth the clusters obtaed after each colum s removed. However, clusterg methods should be alged wth the goals pursued by researcher-practtoer sce clusterg methods aloe may ot gve approprate umber of clusters for accurate prcg of credt dervatves. For example, f a estmato of sum of squares wth groups usg -meas cluster aalyss ad valdato measures suggest 4 clusters of credt spreads, ths umber of clusters may ot eough to estmate CDO far spreads for each trache gve the partcular level of tolerace. Ths s because we stll may have hgh values for the wth-group sum of squares, low level of homogeety each cluster or, equvaletly, hgh level of heterogeety wth the clusters. Hgh level of heterogeety requres further crease umber of clusters to acheve covergece of estmated CDO far spreads to the real oe. The optmal umber of clusters depeds of the trache beg prced, credt spread dstrbuto of the credt portfolos, ad the recovery rates of the oblgors. I practce, the umber of clusters may be greater tha the oe suggested by, for example, the -meas cluster aalyss; however ths umber of clusters stll wll be several tmes less tha the umber of oblgors the credt portfolo thereby creasg the speed of loss dstrbuto geerato. The dsadvatage of usg cluster aalyss s that addtoal tme s requred to perform such aalyss. The movemets of credt spreads should be motored, ad the cluster aalyss mght be requred to be performed after each swtch regme chage. I rs maagemet of traches, loss dstrbutos for the credt portfolos eed to be extesvely recalculated for estmato of credt rs measures such as, for example, dosycratc

25 5 delta of a credt or a group of credts. Oe of the methods to speed up the buldg of loss dstrbuto s perturbato method suggested by O Kae (008, pp ). The perturbato method coupled wth parttog of credt spreads ca help further speed up the loss dstrbuto buldg ad allows more effcet estmato of sestvty of CDO trache prces to the stataeous movemet credt spreads of the cluster. Ths s because such approach s more cosstet wth what we usually observe the maret whe chages spreads occur a group of hghly correlated spreads rather tha oe partcular credt. Not all methods may be easly mplemeted stuatos whe the portfolo s heterogeeous. For example, the well ow algorthm proposed by Aderse et al. (003) oe has to estmate the value of commo dvsor of all potetal losses. The algorthm depeds o the statstcal characterstcs of the credt spreads (whch ca be dverse wth hgh level of heterogeety) ad some cases the value of the commo dvsor ca be very small substatally affectg the speed of the algorthm ad requrg more memory for estmato of the loss dstrbuto. A tolerace level of the commo dvsor of all potetal losses ca be chose to speed up the process, but ths case the resultg loss dstrbuto depeds o the chose tolerace level, whch may cur dscreteess of the loss dstrbuto shape ad eve may ot be accurate affectg credt rs maagemet ad credt dervatves prcg. Easess of mplemetato of specfc algorthms as well as accuracy depeds o the access of specalzed mathematcal lbrares. For example, to use a complex FFT algorthm for loss dstrbuto geerato, a aalyst would eed a access to a effectve FFT algorthm. Moreover, ths algorthm requres more tme for geerato of loss dstrbuto tha the recursve algorthms descrbed above ad ths oe of the ma reasos why FFT algorthms are ot wdely used practce. Whe estmatg a loss dstrbuto usg fte homogeeous portfolo, a

26 6 aalyst eeds to choose a effectve tegrato method as well as the tegrato rages to reach the requred tolerace. Geerato of loss dstrbuto should be alged wth other quattatve ad qualtatve aalyss of the oblgors ad formed decso should be made order to properly estmate the put parameters. For example, credt qualty of each credt ca chage over the specfed perod of tme, ad, therefore, t ca affect the loss dstrbuto that wll chage durg ths perod of tme due to chages credt qualty of the oblgors. The aalyzed methods assumed costat recovery rates. The value for recovery rates s ofte assumed to be equal to 40%, durg facal crss eve less 0 to 0%. Moreover, these values are ofte assumed to be the same for all the oblgors wth dfferet ratgs ad credt spread term structure. Estmato of recovery rates for each oblgor based o facal statemet aalyss, for example, estmato of recovery rates for a portfolo cosstg of 40 oblgors s tme cosumg; o the other sde, the geerated loss dstrbuto based o the carefully estmated recovery rates would be more realstc. Sce recovery rates are chagg over the partcular tme perod, the loss dstrbuto geerato methods ca be mproved by assumg that recovery rates for each oblgor (ad, therefore, loss gve default) are radom varables. The probabltes of default ad recovery rates are egatvely correlated ad they ca be exogeously modeled as stochastc processes whch the ca be corporated to exstg loss dstrbuto geerato models. The descrbed methods do t cosder credt spread dstrbuto of the credt portfolo. Loss dstrbuto of the portfolo or CDO trache deped o the credt spread dstrbutos observed the maret. Aalyss of credt spread dstrbutos ca help detfy whch credt spreads are most lely affect specfc CDO trache, ad whch credt spreads would t. For

27 7 example, for a gve teor, say 3 years, a credt portfolo may have credts wth low credt spreads cocetratg aroud the value gvg tme to default whch are more tha 0 years. Such credt spreads do t affect loss dstrbuto of the cosdered CDO trache, ad, therefore ca be excluded from cosderato. If a umber of such credt spreads s very hgh, the by excludg these credts from cosderato we ca substatally decrease computatoal tme of the loss dstrbuto geerato. I other words, loss dstrbuto geerato ca be faster ad substatally mproved, f we perform prelmary statstcal aalyss of credt spread dstrbuto, recovery rates ad default probabltes of the oblgors.

28 8 Refereces Aderse, L. J., Sdeus, J. & Basu, S. (003, November). All your hedges oe baset. Rs, Arvats, A. & Gregory, J. (999). A credt rs toolbox. I D. Shmo (ed.), Credt rs. Models ad maagemet. pp Lodo: Rs boos. Arvats, A. & Gregory, J. (00). Credt: The complete gude to prcg, hedgg ad rs maagemet. Lodo: Rs Boos. Dembo, A., Deuschel, J., & Duffe, D. (004). Large portfolo losses. Face & Stochastcs, 8(), 3-6. Glasserma, P. & L, J. (003). Importace samplg for a mxed Posso model of portfolo credt rs. I: S. Chc, P.J. Sachez, D. Ferr, ad D.J. Morrse (Eds.), Proceedgs of the 003 Wter Smulato Coferece, pp Glasserma, P. & Ruz-Mata, J. (006). A comparso of approxmato techques for portfolo credt rs. Joural of Credt Rs, Glasserma, P. (004). Tal approxmatos for portfolo credt rs. Joural of Dervatves, 4-4. Gordy, M. (003, July). A rs-factor model foudato for ratgs-based ba captal rules. Joural of Facal Itermedato, (3), Gordy, M. (004). Graularty adjustmet portfolo rs measuremet. I G. Szegö (ed.), Rs Measures for the st Cetury. pp Joh Wley & Sos. Gordy,M.B. (00). Saddlepot approxmato of CredtRs+. Joural of Bag ad Face 6(7), Gregory, J., & Lauret, J. (003, Jue). I wll survve. Rs, 6(6), Gregory, J., & Lauret, J. (004, October). I the core of correlato. Rs 7(0), 87-9 Grude, P.(007). Computatoal aspects of tegrated maret ad credt portfolo models. OR Spectrum, 9(), Hull, J. & Whte, A. (004). Valuato of a CDO ad a th to default CDS wthout Mote Carlo smulatos. Joural of Dervatves, (), 8-3 Kalemaova, A., Schmd, B. & Werer, R. (007, September). The ormal verse Gaussa dstrbuto for sythetc CDO Prcg. Joural of Dervatves, 4(3), pp Kreel, M. & Partehemer, J. (006). The mpled loss surface of CDOs. Retreved Jauary 6, 009, from

29 9 Lauret, J., & Gregory, J. (005, Summer). Baset default swaps, CDOs ad factor copulas. Joural of Rs, 7(4), -0. Lauret, J.-P. (004). Fast aalytc techques for prcg sythetc CDOs. Presetato at Credt Rs Summt Europe. October 3, 004. Mart, R. (006, December). The saddlepot approxmato ad portfolo optoaltes. Rs, Mero, S. & Nyfeler, M. (00, August). Calculatg portfolo loss. Rs 5(8), 8-86 O Kae, D. (008). Modelg sgle-ame ad mult-ame credt dervatves. New Yor, NY: Joh Wley & Sos Pyht, M. (004, March). Mult-factor adjustmet. Rs 7(3) Reβ, O. (003). Mathematcal methods for the effcet assessmet of maret ad credt rs. PhD Thess. Departmet of Mathematcs, Uversty of Kaserslauter. Schloegl, L., & O'Kae, D. (005, October). A ote o the large homogeeous portfolo approxmato wth the Studet-t copula. Face & Stochastcs, 9(4), Schöbucher, P.J. (00). Tae to the lmt: Smple ad ot so smple loa loss dstrbutos. Worg Paper, Departmet of Statstcs, Bo Uversty. Sdeus, J., Pterbarg, V., & Aderse, L. (008, March). A ew framewor for dyamc credt portfolo loss modelg. Iteratoal Joural of Theoretcal & Appled Face, (), Vasce, O. (987). Probablty of loss o loa portfolo. Worg paper, KMV Corporato. Vasce, O. (00, December). Probablty of loss o a loa portfolo. Rs, Zheg, H. (007). Portfolo dstrbuto modelg ad computato. Worshop o Fast Facal Algorthms. Taaa Busess School & Imperal College. Retreved Jauary 0, 009, from