an introduction to credit risk modeling phần 6 pptx

FIGURE 3.2 Asset-Equity relation,Equation (3. 20), for parameter sets (δ, r, γ, µ, σ A ) and D = 1: (-) solid (0.1, 0.05, 1, 0.0, 0.1), (–) dashed (0.1, 0.05, 1, 0.03, 0.1), ( ) dashed-dotted (0.0, 0.05, 1, 0.03, 0.1). 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 asset equity ©2003 CRC Press LLC If we observe E at time t and know the estimate σ E,t for the equity volatility, then A and σ A have to solve the equations E = D  A D − δ r − µ D −  γ − δ r − µ D  A/D γ  λ(σ A )  (3. 23) σ E,t = σ A A   γ − δ r − µ D  1 Dγ  λ(σ A ) λ(σ A )A λ(σ A )−1  . (3. 24) As a further simplification it is often assumed that E locally evolves like a geometric Brownian motion, which leads to σ E,t = σ E E for some σ E . In the implementation one usually starts with some σ A = σ 0 A . For ex- ample, the equity volatility is used to generate two time series (A s ) s≥0 and (E s ) s≥0 . Then, the volatility of E is estimated, and the param- eter σ 1 A is adjusted to a higher or lower level, trying to best match the estimated volatility of E with the observed equity volatility. One proceeds that way until the σ n E , implied by σ n A , is close to the observed σ E . Observe also that the set of equations (3. 23) and (3. 24) can be generalized to any contingent claim approach for the asset values, once a functional relationship E = E(A, D, σ A , t) is specified between assets A, debt D, and equity E. Conceptually, they look like E = E(A, D, σ A ) , σ E E = σ A AE  (A, D, σ A ) . This concludes are discussion of asset value models. ©2003 CRC Press LLC Chapter4 TheCreditRisk + Model InSection2.4.2wealreadydescribedtheCreditRisk + modelasaPois- sonianmixturewithgamma-distributedrandomintensitiesforeach sector.InthissectionwewillexplainCreditRisk + insomegreater detail.Thejustificationforanotherandmoreexhaustivechapteron CreditRisk + isitsbroadacceptancebymanycreditriskmanaginginsti- tutes.EveninthenewCapitalAccord(somereferencesregardingthe BaselIIapproachareGordy[52],Wilde[126],andtheIRBconsultative document[103]),CreditRisk + wasoriginallyappliedforthecalibration oftheso-calledgranularityadjustmentinthecontextoftheInternal Ratings-basedApproach(IRB)ofregulatorycapitalriskweights.The popularityofCreditRisk + hastwomajorreasons: •Itseemseasiertocalibratedatatothemodelthanisthecasefor multi-factorassetvaluemodels.Hereweintentionallysaid“it seems”becausefromourpointofviewthecalibrationofbank- internalcreditdatatoamulti-sectormodelisingeneralneither easiernormoredifficultthanthecalibrationofamulti-factor modelonwhichanassetvaluemodelcanbebased. •Thesecondandmaybemostimportantreasonforthepopularity ofCreditRisk + isitsclosed-formlossdistribution.Usingprob- abilitygeneratingfunctions,theCreditRisk + modeloffers(even incaseofmorethanonesector)afullanalyticdescriptionofthe portfoliolossofanygivencreditportfolio.Thisenablesusers ofCreditRisk + tocomputelossdistributionsinaquickandstill “exact”manner.Formanyapplicationsofcreditriskmodels,this isa“nice-to-have”feature,e.g.,inpricingorABSstructuring. BeforegoingintothedetailsoftheCreditRisk + model,weliketo presentaquotationfromtheCreditRisk + TechnicalDocument[18]on page 8. There we find that CreditRisk + focuses on modeling and managing credit default risk. 145 ©2003 CRC Press LLC Inotherwords,CreditRisk + helpstoquantifythepotentialriskof defaultsandresultinglossesintermsofexposureinagivenportfolio. Althoughitincorporatesatermstructureofdefaultrates(moreex- plicitlyyearlymarginaldefaultrates)forimplementingmulti-yearloss distributions(see[18],A5.2),itisnotanappropriatechoiceifoneis interestedinamark-to-marketmodelofcreditrisk. 4.1TheModelingFrameworkofCreditRisk + CrucialinCreditRisk + istheuseofprobability-generatingfunc- tions 1 .RecallthatthegeneratingfunctionofaPoissonrandomvariable L  withintensityλisgivenby G(z)= ∞  k=0 P[L  =k]z k =e −λ ∞  k=0 λ k k! z k =e λ(z−1) .(4.1) Inordertoreducethecomputationaleffort,CreditRisk + groupsthe individualexposuresoftheobligorsinaconsideredportfoliointoex- posurebands.Thisisdoneasfollows: ChooseanexposureunitamountE.AnalogouslytoChapter1,de- note for any obligor i its Expected Loss by EL i , its Exposure At Default by EAD i , and its Loss Given Default by LGD i . The exposure that is subject to be lost after an obligor’s default is then E i = EAD i × LGD i , (4. 2) assuming a nonrandom LGD. The exposure ν i respectively the Ex- pected Loss ε i of obligor i in multiples of the exposure unit E is given by ν i = E i E , ε i = EL i E . From this point on, CreditRisk + “forgets” the exact exposures from the original portfolio and uses an approximation by means of exposure 1 In probability theory there are three concepts of translating a probability distribution into a functional context, namely the Fourier transform, the Laplace transform (which is in case of distributions on R d + often more convenient), and the probability-generating function (often preferably used for distributions on Z + ).Thelatterisdefinedbythefunctionz→E[z X ] forarandomvariableX.Regardingbasicpropertiesofgeneratingfunctionswereferto[53]. ©2003 CRC Press LLC bands by rounding the exposures ν i to the nearest integer number. In other words, every exposure E i is replaced by the closest integer mul- tiple of the unit exposure E. Already one can see that an appropriate choice of E is essential in order to end up at an approximation that is on one hand “close” enough to the original exposure distribution of the portfolio in order to obtain a loss distribution applicable to the origi- nal portfolio, and on the other hand efficient enough to really partition the portfolio into m E exposure bands, such that m E is significantly smaller than the original number of obligors m. An important “rule- of-thumb” for making sure that not too much precision is lost is to at least take care that the width of exposure bands is “small” compared to the average exposure size in the portfolio. Under this rule, large portfolios (containing many loans) should admit a good approximation by exposure bands in the described manner. In the sequel we write i ∈ [j] whenever obligor 2 i is placed in the exposure band j. After the exposure grouping process, we have a par- tition of the portfolio into m E exposure bands, such that obligors in a common band [j] have the common exposure ν [j] E, where ν [j] ∈ N 0 is the integer multiple of E representing all obligors i with min{|ν i − n| : n ∈ N 0 } = |ν i − ν [j] | where i = 1, , m; i ∈ [j]; j = 1, , m E . In cases where ν i is an odd-integer multiple of 0.5, the above minimum is not uniquely defined. In such cases (which are obviously not very likely) one has to make a decision, if an up- or down-rounding would be appropriate. Now let us discuss how to ass ign a default intensity to a given ex- posure band. Because CreditRisk + plays in a Poissonian world, every obligor in the portfolio has its individual (one-year) default intensity λ i , which can be calibrated from the obligor’s one-year default probability DP i by application of (2. 12), λ i = −log(1 − DP i ) (i = 1, , m). (4. 3) 2 Here we make the simplifying assumption that the number of loans in the portfolio equals the number of obligors involved. This can be achieved by aggregating different loans of a single obligor into one loan. Usually the DP, EAD, and LGD of such an aggregated loan are exposure-weighted average numbers. ©2003 CRC Press LLC BecausetheexpectationofL  i ∼Pois(λ i )isE[L  i ]=λ i ,theexpected numberofdefaultsinexposureband[j](usingtheadditivityofexpec- tations)isgivenby λ [j] =  i∈[j] λ i .(4.4) TheExpectedLossinband[j]willbedenotedbyε [j] andiscalculated bymultiplyingtheexpectednumberofdefaultsinband[j]withthe band’sexposure, ε [j] =λ [j] ν [j] .(4.5) Here,theCreditRisk + TechnicalDocumentsuggestsmakinganadjust- mentofthedefaultintensitiesλ i (whichsofarhavenotbeenaffected bytheexposurebandapproximationprocess)inordertopreservethe originalvalueoftheobligor’sExpectedLosses.Thiscouldbedoneby defininganadjustmentfactorγ i foreveryobligoriby γ i = E i ν [j] E (i∈[j],j=1, ,m E ).(4.6) Replacingforeveryobligoritheoriginaldefaultintensityλ i byγ i λ i withγ i asdefinedin(4.6)preservestheoriginalELsafterapproximat- ingtheportfolio’sexposuredistributionbyapartitionintoexposure bands.Inthefollowingweassumewithoutlossofgeneralitythatthe defaultintensitiesλ i alreadyincludetheadjustment(4.6).From(4. 4)respectively(4.5)itisstraightforwardtowritedowntheportfolio’s expectednumberofdefaultevents(respectivelytheportfolio’soverall defaultintensity),namely λ PF = m E  j=1 λ [j] = m E  j=1 ε [j] ν [j] .(4.7) Afterthesepreparationswearenowreadytodescribetheconstruction oftheCreditRisk + lossdistribution.Wewillproceedintwosteps, startingwithaportfolioofindependentobligorsandthenmixingthe involvedPoissondistributionsbymeansofasectormodelasindicated inSection2.4.2. ©2003 CRC Press LLC 4.2ConstructionStep1:IndependentObligors Webeginwithaportfolioofmindependentobligorswhosedefault riskismodeledbyPoissonvariablesL i .AsalreadymentionedinSec- tion2.2.1,Poissonmodelsallowformultipledefaultsofasingleobligor. This is an unpleasant, but due to the small occurrence probability, mostly ignored feature of all Poisson approaches to default risk. Involving the (nonrandom) exposures E i as defined in (4. 2), we obtain loss variables E i L  i where L  1 ∼ P ois(λ 1 ) , , L  m ∼ P ois(λ m ) (4. 8) are independent Poisson random variables. Grouping the individual exposures E i into exposure bands [j] and assuming the intensities λ i to incorporate the adjustments by the factors γ i as described in the intro- duction, we obtain new loss variables ν [j] L  i , where losses are measured in multiples of the exposure unit E. Because obligors are assumed to be independent, the number of defaults L  in the portfolio respectively L  [j] in exposure band j also follow a Poisson distribution, because the con- volution of independent Poisson variables yields a Poisson distribution. We obtain L  [j] =  i∈[j] L  i ∼ P ois(λ [j] ) , λ [j] =  i∈[j] λ i , (4. 9) for the number of defaults in exposure band [j], j = 1, , m E , and L  = m E  j=1  i∈[j] L  i ∼ P ois  m E  j=1 λ [j]  = P ois(λ P F ) (4. 10) (see (4. 7)), for the portfolio’s number of defaults. The corresponding losses (counted in multiples of the exposure unit E) are given by ˜ L  [j] = ν [j] L  [j] respectively ˜ L  = m E  j=1 ν [j] L  [j] = m E  j=1 ˜ L  [j] . (4. 11) Due to grouping the exposures ν [j] ∈ N 0 together, we can now conve- niently describe the portfolio loss by the probability-generating func- ©2003 CRC Press LLC tion of the random variable ˜ L  defined in (4. 11), applying the convo- lution theorem 3 for generating functions, G ˜ L  (z) = m E  j=1 G ˜ L  [j] = m E  j=1 ∞  k=0 P[ ˜ L  [j] = ν [j] k] z ν [j] k (4. 12) = m E  j=1 ∞  k=0 P[L  [j] = k] z ν [j] k = m E  j=1 ∞  k=0 e −λ [j] λ k [j] k! z ν [j] k = m E  j=1 e −λ [j] +λ [j] z ν [j] = exp  m E  j=1 λ [j] (z ν [j] − 1)  . So far we assumed independence among obligors and were rewarded by the nice closed formula (4. 12) for the generating function of the portfolio loss. In the next section we drop the indepe ndence assump- tion, but the nice feature of CreditRisk + is that, nevertheless, it yields a closed-form loss distribution, even in the case of correlated defaults. 4.3 Construction Step 2: Sector Model A key concept of CreditRisk + is sector analysis. The rationale un- derlying sector analysis is that the volatility of the default intensity of obligors can be related to the volatility of certain underlying factors incorporating a common systematic source of credit risk. Associated with every such background factor is a so-called sector, such that every obligor i admits a breakdown into sector weights w is ≥ 0,  m S s=1 w is = 1, expressing for every s = 1, , m S that sector s contributes with a frac- tion w is to the default intensity of obligor i. Here m S denotes the num- ber of involved sectors. Obviously the calibration of sectors and sector weights is the crucial challenge in CreditRisk + . For example, sectors could be constructed w.r.t. industries, countries, or rating classes. 3 For independent variables, the generating function of their convolution equals the product of the corresponding single generating functions. ©2003 CRC Press LLC InordertoapproachthesectormodelofCreditRisk + werewrite Equation(4.12): G ˜ L  (z)=exp  m E  j=1 λ [j] (z ν [j] −1)  (4.13) =exp  λ PF  m E  j=1 λ [j] λ PF z ν [j] −1  , whereλ PF isdefinedasin(4.7).Definingfunctions G L  (z)=e λ PF (z−1) andG N (z)= m E  j=1 λ [j] λ PF z ν [j] ,(4.14) weseethatthegeneratingfunctionoftheportfoliolossvariable ˜ L  can bewrittenas G ˜ L  (z)=G L  ◦G N (z)=e λ PF (G N (z)−1) .(4.15) Therefore,theportfolioloss ˜ L  hasaso-calledcompounddistribution, essentiallymeaningthattherandomnessinherentintheportfoliolossis duetothecompoundeffectoftwoindependentsourcesofrandomness. Thefirstsourceofrandomnessarisesfromtheuncertaintyregarding thenumberofdefaultsintheportfolio,capturedbythePoissonrandom variableL  withintensityλ PF definedin(4.10).ThefunctionG L  (z) isthegeneratingfunctionofL  ;recall(4.1).Thesecondsourceof randomnessisduetotheuncertaintyabouttheexposurebandsaffected bytheL  defaults.ThefunctionG N (z)isthegeneratingfunctionofa randomvariableNtakingvaluesin{ν [1] , ,ν [m E ] }withdistribution P[N=ν [j] ]= λ [j] λ PF (j=1, ,m E ).(4.16) Forsomemorebackgroundoncompound 4 distributions,refertothe literature.Forexamplein[53]thereaderwillfindtheoryaswellassome 4 Compoun d distributions arise very naturally as follows: Assume X 0 , X 1 , X 2 , be i.i.d. random variables with generating function G X . Let N ∈ N 0 be a random variable, e.g., N ∼ P ois(λ), independent of the sequence (X i ) i≥0 . Denote t he generating function of N by G N . Then, the generating function of X 1 + ···+ X N is given by G = G N ◦G X . In the case where the distribution of N is degenerate, e.g., P[N = n] = 1, we obtain G N (z) = z n and therefore G(z) = [G X (z)] n , confirming the convolution theorem for generating functions in its most basic form. ©2003 CRC Press LLC interestingexamples.Lateronwewillobtainthegeneratingfunction ofsectorlossesinformofanequationthat,conditionalonthesector’s defaultrate,replicatesEquation(4.15). Letusassumethatwehaveparametrizedourportfoliobymeansof m S sectors.CreditRisk + assumesthatagamma-distributedrandom variable Λ (s) ∼Γ(α s ,β s )(s=1, ,m S ) isassignedtoeverysector;seeFigure2.2foranillustrationofgamma densities.Thenumberofdefaultsinanysectorsfollowsagamma- mixedPoissondistributionwithrandomintensityΛ (s) ;seealsoSection 2.2.2.HerebyitisalwaysassumedthatthesectorvariablesΛ (1) , ,Λ (m S ) are independent. For a calibration of Λ (s) recall from (2. 38) that the first and second moment of Λ (s) are E[Λ (s) ] = α s β s , V[Λ (s) ] = α s β 2 s . (4. 17) We denote the expectation of the random intensity Λ (s) by λ (s) . The volatility of Λ (s) is denoted by σ (s) . Altogether we have from (4. 17) λ (s) = α s β s , σ (s) =  α s β 2 s . (4. 18) Knowing the values of λ (s) and σ (s) determines the parameters α s and β s of the sector variable Λ (s) . For every sector we now follow the approach that has taken us to Equation (4. 15). More explicitly, we first find the generating func- tion of the number of defaults in sector s, then obtain the generating function for the distribution of default events among the exposures in sector s, and finally get the portfolio-loss-generating function as the product 5 of the compound sector-generating functions. 4.3.1 Sector Default Distribution Fix a sector s. The defaults in all sectors are gamma-mixed Poisson. Therefore, conditional on Λ (s) = θ s the sector’s conditional generating function is given by (4. 1), G s | Λ (s) =θ s (z) = e θ s (z−1) . (4. 19) 5 Recall that we assumed independence of sector variables. ©2003 CRC Press LLC [...]... Proposition 2.5.7, the difference between the ECs is quite significant std ECVaR (0.99) ECTCE (0.99) rel.diff (%) t(3) 1.73 4.54 6. 99 54 N(0,1.73) 1.73 4.02 4 .61 15 LN(0,1) 2. 16 8. 56 13.57 58 std ECVaR (0.99) ECTCE (0.99) rel.diff (%) N(1 .64 ,2. 16) 2. 16 5.02 5. 76 15 Weil(1,1) 1 3 .6 4 .6 27 N(1,1) 1 2.32 2 .66 14 F0.003,0.12 0.0039 0.0 162 0.0237 46 This table highlights the sensitivity of the determination of... portion” of credit i in a way such that all weighted risks sum-up to the portfolio’s UL It is straightforward to show that the quantity RCi corresponds to the covariance of credit (business unit) i and the total portfolio loss, divided by the portfolio’s volatility respectively UL The definition of RCi obviously is in analogy to beta-factor models used in market risk Furthermore, RCi is equal to the partial... in CreditRisk+ the portfolio loss distribution can be described in an analytical manner by means of a closed-form generating function Remarkable is the fact that this nice property even holds in the most general case of a sector model where complex dependence structures are allowed In the general sector model of CreditRisk+ leading to Formula (4 36) , obligors i1 and i2 will be correlated if and only... formulas in [18] can be easily implemented so that everyone is free to program his or her “individual” version of CreditRisk+ There is much more to say about CreditRisk+ , but due to the introductory character of this book we will not go any further The Technical Document [18] contains some more detailed information on the management of credit portfolios, the calibration of the model, and the technical... tails, the “classical” approach can not be applied to credit risk in a straightforward manner without “paying” for the convenience of the approach by allowing for inconsistencies Fortunately, the var/covar approach for capital allocation, adapted to credit risk portfolios, in many cases yields acceptable results and is, due to its simplicity, implemented in most standard software packages The following... details 5.2.1 Variance/Covariance Approach At the core of var/covar is the question of what an individual credit or business unit contributes to the portfolio’s standard deviation ULP F To answer this question, the classical var/covar approach splits the ©2003 CRC Press LLC portfolio risk ULP F into risk contributions RCi in a way such that m wi × RCi = ULP F i=1 In this way, the weighted risk contributions... question of risk contributions (see also Chapter 5) and the role of correlations in CreditRisk+ Risk contributions in CreditRisk+ are also extensively studied in Tasche [121] In [18], A12.3, the introduction of a sector for incorporating specific risk is discussed As a last remark we should mention that because the sector distributions are Poisson mixtures, the general results from Section 2.2 can also... intensities λi can be ©2003 CRC Press LLC calibrated to one-year default probabilities by application of Formula (4 3) Due to the additivity of expectations it is then very natural to define the expected sector intensity λ(s) by mE m λ(s) = wis λi = i=1 wis λi , (4 30) j=1 i∈[j] where the right side expresses the grouping into exposure bands Note that an exposure band j takes part in sector s if and only if... by Frey and McNeil in [ 46] , because we want to think about X in terms of a portfolio loss and about γ(X) as the amount of capital required as a cushion against the loss X, according to the credit management policy of the bank In the original approach by Artzner et al., X was interpreted as the future value of the considered portfolio Let us now briefly explain the four axioms in an intuitive manner:... portfolio with loss X and scale all exposures by a factor λ Then, of course, the loss X changes to a scaled loss λX Accordingly, the originally required risk capital γ(X) will also change to λγ(X) Translation invariance: If x is some capital which will be lost/gained on a portfolio with certainty at the considered horizon, then the risk capital required for covering losses in this portfolio can be increased/reduced . LLC Chapter4 TheCreditRisk + Model InSection2.4.2wealreadydescribedtheCreditRisk + modelasaPois- sonianmixturewithgamma-distributedrandomintensitiesforeach sector.InthissectionwewillexplainCreditRisk + insomegreater detail.Thejustificationforanotherandmoreexhaustivechapteron CreditRisk + isitsbroadacceptancebymanycreditriskmanaginginsti- tutes.EveninthenewCapitalAccord(somereferencesregardingthe BaselIIapproachareGordy[52],Wilde[1 26] ,andtheIRBconsultative document[103]),CreditRisk + wasoriginallyappliedforthecalibration oftheso-calledgranularityadjustmentinthecontextoftheInternal Ratings-basedApproach(IRB)ofregulatorycapitalriskweights.The popularityofCreditRisk + hastwomajorreasons: •Itseemseasiertocalibratedatatothemodelthanisthecasefor multi-factorassetvaluemodels.Hereweintentionallysaid“it seems”becausefromourpointofviewthecalibrationofbank- internalcreditdatatoamulti-sectormodelisingeneralneither easiernormoredifficultthanthecalibrationofamulti-factor modelonwhichanassetvaluemodelcanbebased. •Thesecondandmaybemostimportantreasonforthepopularity ofCreditRisk + isitsclosed-formlossdistribution.Usingprob- abilitygeneratingfunctions,theCreditRisk + modeloffers(even incaseofmorethanonesector)afullanalyticdescriptionofthe portfoliolossofanygivencreditportfolio.Thisenablesusers ofCreditRisk + tocomputelossdistributionsinaquickandstill “exact”manner.Formanyapplicationsofcreditriskmodels,this isa“nice -to- have”feature,e.g.,inpricingorABSstructuring. BeforegoingintothedetailsoftheCreditRisk + model,weliketo presentaquotationfromtheCreditRisk + TechnicalDocument[18]on page 8. There we find that CreditRisk + focuses on modeling and managing credit. LLC Chapter4 TheCreditRisk + Model InSection2.4.2wealreadydescribedtheCreditRisk + modelasaPois- sonianmixturewithgamma-distributedrandomintensitiesforeach sector.InthissectionwewillexplainCreditRisk + insomegreater detail.Thejustificationforanotherandmoreexhaustivechapteron CreditRisk + isitsbroadacceptancebymanycreditriskmanaginginsti- tutes.EveninthenewCapitalAccord(somereferencesregardingthe BaselIIapproachareGordy[52],Wilde[1 26] ,andtheIRBconsultative document[103]),CreditRisk + wasoriginallyappliedforthecalibration oftheso-calledgranularityadjustmentinthecontextoftheInternal Ratings-basedApproach(IRB)ofregulatorycapitalriskweights.The popularityofCreditRisk + hastwomajorreasons: •Itseemseasiertocalibratedatatothemodelthanisthecasefor multi-factorassetvaluemodels.Hereweintentionallysaid“it seems”becausefromourpointofviewthecalibrationofbank- internalcreditdatatoamulti-sectormodelisingeneralneither easiernormoredifficultthanthecalibrationofamulti-factor modelonwhichanassetvaluemodelcanbebased. •Thesecondandmaybemostimportantreasonforthepopularity ofCreditRisk + isitsclosed-formlossdistribution.Usingprob- abilitygeneratingfunctions,theCreditRisk + modeloffers(even incaseofmorethanonesector)afullanalyticdescriptionofthe portfoliolossofanygivencreditportfolio.Thisenablesusers ofCreditRisk + tocomputelossdistributionsinaquickandstill “exact”manner.Formanyapplicationsofcreditriskmodels,this isa“nice -to- have”feature,e.g.,inpricingorABSstructuring. BeforegoingintothedetailsoftheCreditRisk + model,weliketo presentaquotationfromtheCreditRisk + TechnicalDocument[18]on page. LLC 4.3.3SectorConvolution TheportfoliolossL  =L  1 +···+L  m isamixedPoissonvariable withrandomintensityΛ=Λ 1 +···+Λ m .Groupedintosectors,the intensityΛofL  canalsobewrittenasthesumofsectorintensities, Λ=Λ (1) +···+Λ (m S ) . ThisfollowsfromFormulas(4.29)and(4.30).Becausesectorsare assumedtobeindependent,thedistributionofdefaultsintheportfolio istheconvolutionofthesector’sdefaultdistributions.Therefore,due to( 4.25)thegeneratingfunctionofL  isgivenby G L  (z)= m S  s=1  1− β s 1+β s 1− β s 1+β s z  α s .(4.35) Thegeneratingfunctionoftheportfoliolossesisdeterminedbythe convolutionofcompoundsectordistributionsaselaboratedin(4.33), G ˜ L  (z)= m S  s=1   1− β s 1+β s 1− β s 1+β s 1 λ (s)  m E j=1  i∈[j] w is λ i z ν [j]   α s .(4. 36) SoweseethatinCreditRisk + theportfoliolossdistributioncanbe describedinananalyticalmannerbymeansofaclosed-formgenerating function.Remarkableisthefactthatthisnicepropertyevenholdsin themostgeneralcaseofasectormodelwherecomplexdependence structuresareallowed.InthegeneralsectormodelofCreditRisk + leadingtoFormula(4. 36) ,obligorsi 1 andi 2 willbecorrelatedifand onlyifthereexistsatleastonesectorssuchthatw i 1 s >0andw i 2 s >0. Becauseprobabilitydistributionsandgeneratingfunctionsareunique- lyassociatedwitheachother,Formula(4. 36) allowsforveryquick computationsoflossdistributions.IntheCreditRisk + TechnicalDoc- ument[18]thereaderwillfindacalculationschemeforthelossamount distributionofportfoliosbasedonacertainrecurrencerelation.The formulasareslightlytechnicalbutallowthecalculationoflossdis- tributionsinfractionsofasecond,usingastandardpersonalcom- puterwithsomesuitablenumericalsoftware.Alternativelyademo versionofCreditRisk + informofaspreadsheetcanbedownloaded fromwww.csfb.com/creditrisk.Forreadersinterestedinworkingwith CreditRisk + we