an introduction to credit risk modeling phần 4 docx

account that we are in a Bernoulli framework, E[L (m) |Y ] = m  i=1 w i E[L i |Y ] = N  N −1 [p] − √  Y √ 1 −   =: p(Y ) , such that Proposition 2.5.4 guarantees that L (m) m→∞ −→ p(Y ) = N  N −1 [p] − √  Y √ 1 −   almost surely. (2. 54) So for portfolios with a sufficiently large portfolio size m satisfying Assumption 2.5.2, the percentage quote of defaulted loans for a given state of economy Y = y is approximately equal to the conditional default probability p(y). In the limit we obtain a portfolio loss variable p(Y ) describing the fraction of defaulted obligors in an infinitely fine- grained credit portfolio. We now want to derive the cumulative distribution function and the probability density of the limit loss variable p(Y ), Y ∼ N (0, 1), with p(·) as in (2. 54). Denote the portfolio’s percentage number of defaults in an infinitely fine-grained portfolio (again assuming constant LGDs of 100%) by L. We then have for every 0 ≤ x ≤ 1 P[L ≤ x] = P[p(Y ) ≤ x] (2. 55) = P  −Y ≤ 1 √   N −1 [x]  1 −  − N −1 [p]   = N  1 √   N −1 [x]  1 −  − N −1 [p]   . In the sequel we will denote this distribution function by F p, (x) = P[L ≤ x] (x ∈ [0, 1]). (2. 56) The corresponding probability density can be derived by calculating the derivative of F p, (x) w.r.t. x, which is f p, (x) = ∂F p, (x) ∂x =  1 −  × (2. 57) × exp  − 1 2  (1 −2)  N −1 [x]  2 − 2  1 −N −1 [x]N −1 [p] +  N −1 [p]  2   ©2003 CRC Press LLC FIGURE 2.5 The probability density f p, for different combinations of p and  (note that the x-axes of the plots are differently scaled). 0 0.005 0.01 0.015 0.02 1000 2000 3000 4000 5000 0 0.00 5 0.01 0.01 5 0.02 0 50 100 150 200 250 bpsbpsp 130 == ρ %530 == ρ bpsp 0 0.005 0.01 0.015 0.02 0 200 400 600 800 1000 %2030 == ρ bpsp 0 0.2 0. 4 0.6 0.8 1 100 200 300 400 500 %99.9930 == ρ bpsp 0 0.01 0.02 0.03 0.04 0 20 40 60 80 %5%1 == ρ p 0 0.05 0.1 0.1 5 0.2 0 5 10 15 2 0 %5%5 == ρ ©2003 CRC Press LLC =  1−  exp  1 2  N −1 [x]  2 − 1 2  N −1 [p]−  1−N −1 [x]  2  . Figure2.5showsthelossdensitiesf p, fordifferentvaluesofpand. ItcouldbeguessedfromFigure2.5thatregardingtheextremecases w.r.t.pandsomereasonablelimitoff p, shouldexist.Indeed,one caneasilyprovethefollowingstatement: 2.5.7PropositionThedensityf p, admitsfourextremecasesin- ducedbytheextremevaluesoftheparameterspand,namely 1.=0: Thisisthecorrelation-freecasewithlossvariables L i =1 {r i =Z i <N −1 [p]} ∼B(1;p), taking(2.48)intoaccount.Inthiscase,theabsolute(size-m) portfolioloss  L i followsabinomialdistribution,  m i=1 L i ∼ B(m;mp),andthepercentageportfoliolossL m convergesbyar- gumentsanalogoustoProposition2.5.4(orjustbyanapplication oftheLawofLargeNumbers)topalmostsurely.Therefore,f p,0 isthedensity 22 ofadegeneratedistribution 23 concentratedinp. ThisisillustratedbythefirstplotinFigure2.5,whereanalmost vanishingcorrelation(=1bps)yieldsanf p, ,whichisalmost justapeakinp=30bps. 2.=1: Inthiscaseonehasperfectcorrelationbetweenalllossvariables intheportfolio(seealsoSection1.2,wheretheterm“perfect correlation”wasmentionedthefirsttime).Inthiscasewecan replacethepercentageportfoliolossL m byL 1 ∼B(1;p),which isnolongerdependentonm.Therefore,thelimit(m→∞) percentageportfoliolossLisalsoBernoulliB(1;p),suchthat P[L=1]=pandP[L=0]=1−p.Thecaseof(almost)perfect correlationisillustratedinthefourthplot(p=30bps,=99.99%) ofFigure2.5,clearlyshowingtheshapeofadistributionconcen- trated in only two points, yielding an “all or nothing” loss. 3. p = 0 : All obligors survive almost surely, such that P[L = 0] = 1. 22 More precisely, it is a delta distribution. 23 More explicitly, we are talking about a Dirac measure. ©2003 CRC Press LLC 4.p=1: Allobligorsdefaultalmostsurely,suchthatP[L=1]=1. Proof.Aproofisstraightforward.✷ Fortheinfinitelyfine-grainedlimitportfolio(encodedbytheportfo- lio’spercentagelossvariableL)itisveryeasytocalculatequantilesat anygivenlevelofconfidence. 2.5.8PropositionForanygivenlevelofconfidenceα,theα-quantile q α (L)ofarandomvariableL∼F p, isgivenby q α (L)=p  −q α (Y)  =N  N −1 [p]+ √ q α (Y) √ 1−  whereY∼N(0,1)andq α (Y)denotestheα-quantileofthestandard normaldistribution. Proof.Thefunctionp(·)isstrictlydecreasing,asillustratedbyFigure 2.3.Thereforeitfollowsthat P[L≤p(−q α (Y))]=P[p(Y)≤p(−q α (Y))] =P[Y≥−q α (Y)]=P[−Y≤q α (Y)], taking(2.55)intoaccount.Thisprovestheproposition.✷. Bydefinition(seeSection1.2)theUnexpectedLoss(UL)isthestan- dard deviation of the portfolio loss distribution. In the following propo- sition the UL of an infinitely fine-grained uniform portfolio is calcu- lated. 2.5.9 Proposition The first and second moments of a random vari- able L ∼ F p, are given by E[L] = p and V[L] = N 2  N −1 [p], N −1 [p];   − p 2 , where N 2 is defined as in Proposition 2.5.1. Proof. That the first mome nt equals p follows just by construction of F p, . Regarding the second moment, we write V[L] = E[L 2 ] −E[L] 2 . We already know E[L] 2 = p 2 . So it only remains to show that E[L 2 ] = N 2 [N −1 [p], N −1 [p]; ]. For proving this, we use a typical “conditioning ©2003 CRC Press LLC trick.” For this purpose, le t X 1 , X 2 ∼ N(0, 1) denote two indepen- dent standard normal random variables, independent from the random variable X = N −1 [p] − √  Y √ 1 −  ∼ N(µ, σ 2 ) with µ = N −1 [p] √ 1 −  , σ 2 =  1 −  . We write g µ,σ 2 for the density of X. Then, we can write E[L 2 ] as E[L 2 ] = E[p(Y ) 2 ] = E[N(X) 2 ] =  R P[X 1 ≤ X | X = x] P[X 2 ≤ X | X = x] dg µ,σ 2 (x) =  R P[X 1 ≤ X, X 2 ≤ X | X = x] dg µ,σ 2 (x) = P[X 1 − X ≤ 0, X 2 − X ≤ 0] . The variables X i − X are normally distributed with expectation and variance E[X i − X] = − N −1 [p] √ 1 −  and V[X i − X] = 1 +  1 −  . The correlation between X 1 −X and X 2 −X equals . Standardizing 24 X 1 − X and X 2 − X, we conclude E[L 2 ] = N 2 [N −1 [p], N −1 [p]; ]. ✷ The next proposition reports on higher moments of F p, . 2.5.10 Proposition The higher moments of L ∼ F p, are given by E [L m ] = N m  (N −1 [p], , N −1 [p]), C   where N m [···] denotes the m-dimensional normal distribution function and C  ∈ R m×m is a matrix with 1 on the diagonal and  off-diagonal. Proof. The proof relies on the same argument as the proof of Propo- sition 2.5.9. A generalization to m ≥ 2 is straightforward. ✷ 24 Shifting and scaling a random variable in order to achieve me an zero and standard devi- ation one. ©2003 CRC Press LLC TABLE 2.2: EconomicCapitalEC α foraninfinitelyfine-grainedportfolio(portfolioloss L ∼ F p, ) w.r.t. p and , for α = 99.5%. ©2003 CRC Press LLC TABLE 2.3: EconomiccapitalEC α foraninfinitelyfine-grainedportfolio(portfolioloss L∼F p, )w.r.t.pand,forα=99.98%. ©2003 CRC Press LLC TABLE 2.4: UnexpectedlossULforaninfinitelyfine-grainedportfolio(portfolioloss L ∼ F p, ) w.r.t. p and . ©2003 CRC Press LLC FIGURE 2.6 Dependence of economic capital EC α on the chosen level of confidence α. 0,00% 2,00% 4,00% 6,00% 8,00% 10,00% 12,00% 14,00% 16,00% 99,0% 99,2% 99,4% 99,6% 99,8% 100,0% α EC α %2050 == ρ bpsp 0,00% 2,00% 4,00% 6,00% 8,00% 10,00% 12,00% 14,00% 16,00% 99,0% 99,2% 99,4% 99,6% 99,8% 100,0% α EC %2050 == ρ bpsp ©2003 CRC Press LLC Givenauniformone-yearaveragedefaultprobabilitypandauni- formassetcorrelation,Tables2.2and2.3reportontheEconomic Capital(EC)w.r.t.confidencelevelsofα=99,5%andα=99,98%for aninfinitelyfine-grainedportfolio(describedbythedistributionF p, ), herebyassuminganLGDof100%(seeSection1.2.1forthedefinition ofEC).Analogously,Table2.4showstheUnexpectedLossforagiven pair(p,). Figure2.6illustratesthesensitivityoftheECw.r.t.thechosencon- fidencelevel.Itcanbeseenthatathighlevelsofconfidence(e.g.,from 99,9%on)theimpactofeverybasispointincreaseofαontheportfolio ECisenormous. Anothercommonportfolio-dependentquantityistheso-calledcap- italmultiplier(CM α );seealsoChapter5oncapitalallocation.Itis definedastheECw.r.t.confidenceαinunitsofUL(i.e.,inunitsof theportfoliostandarddeviation).InpricingtoolstheCMissometimes assumedtobeconstantforaportfolio,evenwhenaddingnewdealsto it.ThecontributionofthenewdealtothetotalECoftheenlarged portfolioisthengivenbyamultipleoftheCM.Ingeneral,theCM heavilydependsonthechosenlevelofconfidenceunderlyingtheEC definition.BecauseforgivenpandtheCMisjusttheECscaledby theinverseoftheUL,Figure2.6additionallyillustratestheshapeof thecurvedescribingthedependencyoftheCMfromtheassumedlevel ofconfidence. Forexample,forp=30bps(aboutaBBB-rating)and=20%(the BaselIIsuggestionfortheassetcorrelationofthebenchmarkrisk weightsforcorporateloans)the(rounded!)CMofaportfoliowithloss variableL∼F p, isgivenbyCM 99% ≈4,CM 99,5% ≈6,CM 99,9% ≈10, andCM 99,98% ≈16(inthisparticularsituationwehaveanULof59 bps,ascanbereadfromtheFigure2.4). Now,asalastremarkinthissectionwewanttoreferbacktoSec- tion1.2.2.2,wheretheanalyticalapproximationofportfoliolossdistri- butionsisoutlined.ThedistributionL p, ,eventuallycombinedwith somemodifications(e.g.,randomordeterministicLGDs),isextremely wellsuitedforanalyticalapproximationtechniquesinthecontextof assetvalue(ormoregenerallylatentvariable)models. 2.5.2TheCreditRisk + One-SectorModel WealreadydiscussedCreditRisk + inSection2.4.2andwillcomeback toitinChapter4.Thereforethisparagraphisjust a brief “warming- ©2003 CRC Press LLC [...]... impossible A small and by no means exhaustive selection of papers providing the reader with a good introduction as well as with a valuable source of ideas how to apply the copula concept to standard problems in credit risk is Li [78,79], Frey and McNeil [45 ], Frey, McNeil, and Nyfeler [47 ], Frees and Valdez [44 ], and Wang [125] However, the basic idea of copulas is so simple that it can be easily introduced:... 2.6351% 4. 5287% 20% 4. 3060% 6.3 342 % 10.18 64% Std.Dev 0.5% 0.8% 1.5% 5% 0.3512% 0.5267% 0.8976% 20% 0.8926% 1.2966% 2.1205% Std.Dev 0.5% 0.8% 1.5% 5% 0.3522% 0.5283% 0.89 64% 20% 0.8 946 % 1.3 045 % 2.0988% T-Copula with df = 40 simulated, 100,000 scenarios Mean 5% 20% 0.5% 0 .49 59% 0 .49 92% 0.8% 0.8006% 0.8009% 1.5% 1.5030% 1 .49 70% T-Copula with df = 10 simulated, 100,000 scenarios Mean 5% 20% 0.5% 0 .49 90% 0 .49 73%... 40 , and 10 Quantiles are calculated w.r.t a confidence of 99% Gaussian Copula (not simul.) not simulated Mean 5% 20% 0.5% 0.5000% 0.5000% 0.8% 0.8000% 0.8000% 1.5% 1.5000% 1.5000% T-Copula with df = 10,000 simulated, 100,000 scenarios Mean 5% 20% 0.5% 0.5002% 0 .49 83% 0.8% 0.8028% 0.8037% 1.5% 1.50 34% 1 .49 44% Quantile 0.5% 0.8% 1.5% 5% 1. 747 0% 2.6323% 4. 5250% 20% 4. 3017% 6.2997% 10.3283% Quantile 0.5%... 0 .49 73% 0.8% 0.7999% 0.8051% 1.5% 1.5023% 1.5003% Quantile 0.5% 0.8% 1.5% 5% 2.96 74% 4. 2611% 6.6636% 20% 5.38 14% 7. 640 5% 11.9095% Quantile 0.5% 0.8% 1.5% 5% 6.0377% 8.0921% 11.7 042 % 20% 7.9295% 11. 243 4% 16.5620% Std.Dev 0.5% 0.8% 1.5% 5% 0.6 145 % 0.8802% 1.3723% 20% 1.1201% 1.5726% 2 .46 53% Std.Dev 0.5% 0.8% 1.5% 5% 1.2535% 1.65 74% 2.3889% 20% 1.7135% 2.31 04% 3. 347 5% ©2003 CRC Press LLC Figure 2.9 clearly illustrates... uniform portfolio model of CreditMetricsTM and KMV as a parametric framework Table 2.7 includes R1 =Aaa, R2 =Aa, , and R6 =B, altogether six rating grades For every rating class Ri we can calculate the mean pi and the corresponding volatility from the historic default frequencies of class Ri over the years from 1970 to 2000 The result is shown in Tables 2.8 and 2.9 in the mean and standard deviation column... portfolio statistics compared to a normal copula is very small and just due to stochastic fluctuations in the simulation But with decreasing n the portfolio statistics significantly changes For example, there is a multiplicative difference of almost a factor of 2 between the 99%-quantiles w.r.t (p, ) = (0.8%, 5%) and degrees of freedom of 40 and 10 If we would calculate the quantiles in Table 2.6 w.r.t higher... distributions of CreditMetricsTM respectively KMV with the corresponding distribution in the CreditRisk+ world Assuming infinitely many obligors and only one sector, we obtain a situation comparable to the uniform portfolio model of CreditMetricsTM and KMV Under these assumptions, the portfolio loss is distributed according to a negative binomial distribution N B(α, β) due to a gammadistributed random intensity... in the CreditRisk+ framework is extensively discussed in Chapter 4 Denoting the portfolio loss by L ∼ N B(α, β), the loss distribution is determined by P[L = n] = n+α−1 n 1− β 1+β α β 1+β n , (2 58) where α and β are called the sector parameters of the sector; see Formula (4 26) The expectation and the variance of L are given by E[L ] = αβ and V[L ] = αβ(1 + β) , (2 59) as derived in Formula (4 27)... detailed comparison of the KMV-Model and CreditRisk+ can be found in [12] 2.6 Loss Distributions by Means of Copula Functions Copula functions have been used as a statistical tool for constructing multivariate distributions long before they were re-discovered as a valuable technique in risk management Currently, the literature on the application of copulas to credit risk is growing every month, so that... %69.0 %68.0 %95.0 %53.0 %60.0 %40 .0 %20.0 portfolio 3. 14% 22.0 24. 0 90.1 21.0 62.0 57.0 30.0 80.0 73.0 86.119 58.6 74 23 .48 1 40 .6 94 99.232 52.08 71.87 53.52 83.5 9 8 7 6 5 4 3 2 1 applying Proposition 2.5.9 As a last step we solve (2 59) for α and β One always has α= ˜ m × E[L ]2 , ˜ ˜ m × V[L ] − E[L ] β= ˜ ˜ m × V[L ] − E[L ] , ˜ E[L ] (2 60) e.g., for p=30 bps, =20%, and m=20,000 we apply 2.5.9 for . LLC thePoissonmixturemodel,herebyconfirmingourtheoreticalresults fromSection2.3.AmoredetailedcomparisonoftheKMV-Modeland CreditRisk + canbefoundin[12]. 2.6LossDistributionsbyMeansofCopulaFunctions Copulafunctionshavebeenusedasastatisticaltoolforconstruct- ingmultivariatedistributionslongbeforetheywerere-discoveredasa valuabletechniqueinriskmanagement.Currently,theliteratureon theapplicationofcopulastocreditriskisgrowingeverymonth,so thattrackingeverysinglepaperonthisissuestartsbeingdifficultif notimpossible.Asmallandbynomeansexhaustiveselectionofpa- persprovidingthereaderwithagoodintroductionaswellaswitha valuablesourceofideashowtoapplythecopulaconcepttostandard problemsincreditriskisLi[78,79],FreyandMcNeil [45 ],Frey,McNeil, andNyfeler [47 ],FreesandValdez [44 ],andWang[125].However,thebasic ideaofcopulasissosimplethatitcanbeeasilyintroduced: 2.6.1DefinitionAcopula(function)isamultivariatedistribution (function)suchthatitsmarginaldistributionsarestandarduniform. Acommonnotationforcopulaswewilladoptis C(u 1 ,. LLC Givenauniformone-yearaveragedefaultprobabilitypandauni- formassetcorrelation,Tables2.2and2.3reportontheEconomic Capital(EC)w.r.t.confidencelevelsofα=99,5%andα=99,98%for aninfinitelyfine-grainedportfolio(describedbythedistributionF p, ), herebyassuminganLGDof100%(seeSection1.2.1forthedefinition ofEC).Analogously,Table2.4showstheUnexpectedLossforagiven pair(p,). Figure2.6illustratesthesensitivityoftheECw.r.t.thechosencon- fidencelevel.Itcanbeseenthatathighlevelsofconfidence(e.g.,from 99,9%on)theimpactofeverybasispointincreaseofαontheportfolio ECisenormous. Anothercommonportfolio-dependentquantityistheso-calledcap- italmultiplier(CM α );seealsoChapter5oncapitalallocation.Itis definedastheECw.r.t.confidenceαinunitsofUL(i.e.,inunitsof theportfoliostandarddeviation).InpricingtoolstheCMissometimes assumedtobeconstantforaportfolio,evenwhenaddingnewdealsto it.ThecontributionofthenewdealtothetotalECoftheenlarged portfolioisthengivenbyamultipleoftheCM.Ingeneral,theCM heavilydependsonthechosenlevelofconfidenceunderlyingtheEC definition.BecauseforgivenpandtheCMisjusttheECscaledby theinverseoftheUL,Figure2.6additionallyillustratestheshapeof thecurvedescribingthedependencyoftheCMfromtheassumedlevel ofconfidence. Forexample,forp=30bps(aboutaBBB-rating)and=20%(the BaselIIsuggestionfortheassetcorrelationofthebenchmarkrisk weightsforcorporateloans)the(rounded!)CMofaportfoliowithloss variableL∼F p, isgivenbyCM 99% 4, CM 99,5% ≈6,CM 99,9% ≈10, andCM 99,98% ≈16(inthisparticularsituationwehaveanULof59 bps,ascanbereadfromtheFigure2 .4) . Now,asalastremarkinthissectionwewanttoreferbacktoSec- tion1.2.2.2,wheretheanalyticalapproximationofportfoliolossdistri- butionsisoutlined.ThedistributionL p, ,eventuallycombinedwith somemodifications(e.g.,randomordeterministicLGDs),isextremely wellsuitedforanalyticalapproximationtechniquesinthecontextof assetvalue(ormoregenerallylatentvariable)models. 2.5.2TheCreditRisk + One-SectorModel WealreadydiscussedCreditRisk + inSection2 .4. 2andwillcomeback toitinChapter4.Thereforethisparagraphisjust. LLC FIGURE2.7 Negativebinomialdistributionwithparameters(α,β)=(1,30). up”forthenextparagraphwherewecomparetheuniformportfolio lossdistributionsofCreditMetrics TM respectivelyKMVwiththecor- respondingdistributionintheCreditRisk + world. Assuminginfinitelymanyobligorsandonlyonesector,weobtaina situationcomparabletotheuniformportfoliomodelofCreditMetrics TM andKMV. Undertheseassumptions,theportfoliolossisdistributedaccord- ingtoanegativebinomialdistributionNB(α,β)duetoagamma- distributedrandomintensity.Thederivationofthenegativebinomial distributionintheCreditRisk + frameworkisextensivelydiscussedin Chapter4.DenotingtheportfoliolossbyL  ∼NB(α,β),theloss distributionisdeterminedby P[L  =n]=  n+α−1 n  1− β 1+β  α  β 1+β  n ,(2.58) whereαandβarecalledthesectorparametersofthesector;seeFor- mula (4. 26).TheexpectationandthevarianceofL  aregivenby E[L  ]=αβandV[L  ]=αβ(1+β),(2.59) asderivedinFormula (4. 27).Figure2.7illustratestheshapeof the