assumedtobeconditionallyindependent.Thejointdistributionofthe L i ’sisgivenby P[L 1 =l 1 , ,L m =l m ]= ∞ 0 e −mλ λ (l 1 +···+l m ) l 1 !···l m ! dF(λ).(2.17) Because(seethebeginningofSection2.2)conditionalonΛ=λthe portfolio loss is again a Poisson distribution with intensity mλ, the probability of exactly k defaults equals P[L = k] = ∞ 0 P[L = k | Λ = λ] dF (λ) (2. 18) = ∞ 0 e −mλ m k λ k k ! dF (λ) . Again, note that due to the unbounded support of the Poisson dis- tribution the absolute loss L can exceed the number of “physically” possible defaults. As already mentioned at the beginning of this sec- tion, the probability of a multiple-defaults event is small for typical parametrizations. In the Poisson framework, the uniform default prob- ability of borrowers in the portfolio is defined by p = P[L i ≥ 1] = ∞ 0 P[L i ≥ 1 | Λ = λ] dF (λ) (2. 19) = ∞ 0 (1 − e −λ ) dF (λ) . The counterpart of Formula (2. 16) is Corr[L i , L j ] = V[Λ] V[Λ] + E[Λ] (i = j). (2. 20) Formula (2. 20) is especially intuitive if seen in the c ontext of dis- persion, where the dispersion of a distribution is its variance to mean ratio D X = V[X] E[X] for any random variable X. (2. 21) The dispersion of the Poisson distribution is equal to 1. Therefore, the Poisson distribution is kind of a benchmark when deciding about overdispersion (D X > 1) respectively underdispersion (D X < 1). In ©2003 CRC Press LLC general, nondegenerate 8 Poisson mixtures are overdispersed due to (2. 15). This is a very important property of Poisson mixtures, because before using such a model for credit risk measurement one has to make sure that overdispersion can be observed in the data underlying the calibration of the model. Formula (2. 20) can be interpreted by saying that the correlation between the number of defaults of different coun- terparties increases with the dispersion of the random intensity Λ. For proving this statement we write Formula (2. 20) in the form Corr[L i , L j ] = D Λ D Λ + 1 (i = j). (2. 22) From (2. 22) it follows that an increase in dispersion increases the mix- ture effect, which, in turn, strengthens the dependence between obligor’s defaults. 2.3 Bernoulli Versus Poisson Mixture The law of small numbers 9 implies that for large m and small p B(m; p) ≈ Pois(pm). Setting λ = pm, this shows that under the assumption of independent defaults the portfolio absolute gross loss L = L i of a Bernoulli loss statistics (L 1 , , L m ) with a uniform default probability p can be ap- proximated by a Poisson variable L ∼ P ois(λ). But the law of small numbers is by no means an argument strong enough to support the un- fortunately widespread opinion that Bernoulli and Poisson approaches are more or less compatible. In order to show that both approaches have significant systematic differences, we turn back to the default cor- relations induced by the models; see (2. 6), combined with (2. 4), and (2. 16). In the Bernoulli case we have Corr[L i , L j ] = (2. 23) = Cov[P i , P j ] V[P i ] + E[P i (1 − P i )] V[P j ] + E[P j (1 − P j )] , 8 The random intensity Λ is not concentrated in a single point, P Λ = ε λ . 9 That is, approximation of binomial distributions by means of Poisson distributions. ©2003 CRC Press LLC whereasinthePoissoncaseweobtain Corr[L i ,L j ]= Cov[Λ i ,Λ j ] V[Λ i ]+E[Λ i ] V[Λ j ]+E[Λ j ] .(2.24) LookingonlyatthedrivingrandomvariablesP i ,P j respectivelyΛ i ,Λ j , weseethatinthedenominatorsof(2.23)and(2.24)wecompare V[P i ]+E[P i (1−P i )]=V[P i ]+E[P i ]−E[P 2 i ](2.25) withV[Λ i ]+E[Λ i ]. Now,analogoustothedeterministiccase(2.12),wecan–eveninthe randomcase–expectP i andΛ i tobeofthesameorderofmagnitude. Tokeepthingssimple,letusforamomentassumethatP i andΛ i havethesamefirstandsecondmoments.InthiscaseEquation(2. 25)combinedwith(2.23)and(2.24)showsthattheBernoullimodel alwaysinducesahigherdefaultcorrelationthanthePoissonmodel.But higherdefaultcorrelationsresultinfattertailsofthecorrespondingloss distributions.Inotherwordsonecouldsaythatgivenequalfirstand secondmomentsofP i andΛ i ,theexpectationsofL i andL i willmatch, butthevarianceofL i willalwaysexceedthevarianceofL i ,thereby inducinglowerdefaultcorrelations. SothereisasystematicdifferencebetweentheBernoulliandPoisson mixturemodels.Ingeneralonecanexpectthatforagivenportfolio theBernoullimodelyieldsalossdistributionwithafattertailthana comparably(e.g.,byafirstandsecondmomentmatching)calibrated Poissonmodel.Thisdifferenceisalsoreflectedbytheindustrymodels fromCreditMetrics TM /KMVCorporation(PortfolioManager)and CreditSuisseFinancialProducts(CreditRisk + ).InSection2.5.3we come back to this issue. 2.4 An Overview of Today’s Industry Models In the last five years, several industry models for measuring credit portfolio risk have been developed. Besides the main commercial mod- els we find in large international banks various so-called internal mod- els, which in most cases are more or less inspired by the well-known commercial products. For most of the industry mo dels it is easy to ©2003 CRC Press LLC FIGURE2.1 Today’sBest-PracticeIndustryModels. findsomekindoftechnicaldocumentationdescribingthemathemati- calframeworkofthemodelandgivingsomeideaabouttheunderlying dataandthecalibrationofthemodeltothedata.Anexceptionis KMV’sPortfolioManager TM ,wheremostofthedocumentationispro- prietaryorconfidential.However,evenfortheKMV-Modelthebasic ideabehindthemodelcanbeexplainedwithoutreferencetononpublic sources.InSection1.2.3wealreadybrieflyintroducedCreditMetrics TM andtheKMV-Modelinthecontextofassetvaluefactormodels.In Chapter3wepresentamathematicallymoredetailedbutnontechnical introductiontothetypeofassetvaluemodelsKMVisincorporating. Beforelookingatthemainmodels,wewanttoprovidethereader withabriefoverview.Figure2.1showsthefourmaintypesofin- dustrymodelsandindicatesthecompaniesbehindthem.Table2.1 summarizesthemaindifferencesbetweenthemodels. CreditRisk + couldalternativelybeplacedinthegroupofintensity models,becauseitisbasedonaPoissonmixturemodelincorporat- ingrandomintensities.NeverthelessinFigure2.1weprefertostress thedifferencebetweenCreditRisk + andthedynamicintensitymodels basedonintensityprocessesinsteadofonastaticintensity. DynamicintensitymodelswillbebrieflydiscussedinSection2.4.4 Credit Risk Models Asset Value Models Macroecon. Models Actuarian Models Intensity Models Portfolio Manager (by KMV) CreditMetrics (by RiskMetrics Group) CreditRisk+ (by Credit Suisse Financial Products) CreditPortfolioView (by McKinsey & Company) Jarrow/Lando/ Turnbull–Model (Kamakura) Duffie/Singleton- Model ©2003 CRC Press LLC TABLE 2.1: Overview:MainDi erencesbetweenIndustryModels. Intensity Models CreditRisk + Credit Portfolio View CreditMetricsKMV-Model Deterministic LGD Deterministic LGD, Stoch. Modifications Stochastic, Empirically Calibrated Stochastic (Beta-Distr.) and Fixed Stochastic (Beta-Distr.) and Fixed Severity Correlated Intensity Proc. Implicit by Sectors Implicit by Macroeconomy Equity Value Factor Model Asset Value Factor Model Correlations Not Implemented Not Implemented Stochastic, via Macrofactors Historic Rating Changes, e.g. from S&P EDF-Concept, high migration probabilities Transition Probabilities DefaultDefaultDown/Upgrade and Default Down/Upgrade and Default DtD on contin. Scale Risk Scale Default Risk only Default Risk only Mark-to-Model of Loan Value Mark-to-Model of Loan Value Distance to Default (DtD) Definition of Risk Intensity Process Default Intensity Macro- economic Factors Asset Value Process Asset Value Process Risk Driver Intensity Models CreditRisk + Credit* Portfolio View CreditMetricsKMV- Deterministic LGD Deterministic LGD, Stoch. Modifications Stochastic, Empirically Calibrated Stochastic (Beta-Distr.) and Fixed Stochastic (Beta-Distr.) and Fixed Severity Correlated Intensity Proc. Implicit by Sectors Implicit by Macroeconomy Equity Value Factor Model Asset Value Factor Model Correlations Not Implemented Not Implemented Stochastic, via Macrofactors Historic Rating Changes, e.g. from S&P EDF-Concept, high migration probabilities Transition Probabilities DefaultDefaultDown/Upgrade and Default Down/Upgrade and Default DtD on contin. Scale Risk Scale Default Risk only Default Risk only Mark-to-Model of Loan Value Mark-to-Model of Loan Value Distance to Default (DtD) Definition of Risk Intensity Process Macro- economic Factors Asset Value Process Asset Value Process Risk Driver * Credit Portfolio View in the CPV-Macro mode. In the CPV-Direct mode, segment-specific default probabilities are drawn from a gamma distribution instead of simulating macroeconomic factors as input into a logit function representing a segments conditional default probability. ©2003 CRC Press LLC andtosomeextentinthecontextofsecuritizations.Fromamath- ematician’spointofviewtheyprovidea“mathematicallybeautiful” approachtocreditriskmodeling,butfromtheintroductorypointof viewweadoptedforwritingthisbookwemustsaythatanappropri- atepresentationofdynamicintensitymodelsisbeyondthescopeof thebook.Wethereforedecidedtoprovidethereaderonlywithsome referencestotheliteraturecombinedwithintroductoryremarksabout theapproach;seeSection2.4.4 Inthisbook,ourdiscussionofCreditPortfolioView 10 (CPV)iskept shorterthanourpresentationofCreditMetrics TM ,theKMV-Model, andtheactuarianmodelCreditRisk + .Thereasonfornotgoingtoo muchintodetailsisthatCPVcanbeconsideredasageneralframework forcreditriskmodeling,whichisthentailor-madeforclient’sneeds.In ourpresentationwemainlyfocusonthesystematicriskmodelofCPV. 2.4.1CreditMetrics TM andtheKMV-Model ForsomebackgroundonCreditMetrics TM andKMVwerefertoSec- tion1.2.3.Notethatforbothmodelswefocusontheir“default-only mode”, hereby ignoring the fact that both models incorporate a mark- to-model approach. In the default-only mode, both models are of Bernoulli type, deciding about default or survival of a firm by com- paring the firm’s asset value at a certain horizon with some critical threshold. If the firm value at the horizon is below this threshold, then the firm is considered to be in default. If the firm value is above the threshold, the firm survived the considered time period. In more mathematical terms, for m counterparties denote their asset value at the considered valuation horizon t = T by A (i) T . It is assumed that for every company i there is a critical threshold C i such that the firm defaults in the period [0, T ] if and only if A (i) T < C i . In the framework of Bernoulli loss statistics A T can be viewed as a latent variable driving the default event. This is realized by defining L i = 1 {A (i) T < C i } ∼ B 1; P[A (i) T < C i ] (i = 1, , m). (2. 26) In both mo dels it is assumed that the asset value process is depen- dent on underlying factors reflecting industrial and regional influences, thereby driving the economic future of the firm. For the convenience 10 By McKinsey & Company. ©2003 CRC Press LLC ofthereaderwenowrecallsomeformulasfromSection1.2.3.The parametrizationw.r.t.underlyingfactorstypicallyisimplementedat thestandardized 11 log-returnlevel,i.e.,theassetvaluelog-returns log(A (i) T /A (i) 0 )afterstandardizationadmitarepresentation 12 r i =R i Φ i +ε i (i=1, ,m).(2.27) HereR i isdefinedasin(1.28),Φ i denotesthefirm’scompositefac- tor,andε i isthefirm-specificeffector(asitisalsooftencalled)the idiosyncraticpartofthefirm’sassetvaluelog-return.Inbothmodels, thefactorΦ i isasuperpositionofmanydifferentindustryandcountry indices.Assetcorrelationsbetweencounterpartiesareexclusivelycap- turedbythecorrelationbetweentherespectivecompositefactors.The specificeffectsareassumedtobeindependentamongdifferentfirms andindependentofthecompositefactors.ThequantityR 2 i reflects howmuchofthevolatilityofr i canbeexplainedbythevolatilityof thecompositefactorΦ i .Becausethecompositefactorisasuperposi- tionofsystematicinfluences,namelyindustryandcountryindices,R 2 i quantifiesthesystematicriskofcounterpartyi. InCreditMetrics TM aswellasinthe(parametric)KMVworld,asset valuelog-returnsareassumedtobenormallydistributed,suchthat duetostandardizationwehave r i ∼N(0,1),Φ i ∼N(0,1),andε i ∼N 0,1−R 2 i . Wearenowinapositiontorewrite(2.26)inthefollowingform: L i =1 {r i <c i } ∼B(1;P[r i <c i ])(i=1, ,m),(2.28) wherec i isthethresholdcorrespondingtoC i afterexchangingA (i) T by r i .Applying(2.27),theconditionr i <c i canbewrittenas ε i <c i −R i Φ i (i=1, ,m).(2.29) Now,inbothmodels,thestandardvaluationhorizonisT=1year.De- notingtheone-yeardefaultprobabilityofobligoribyp i ,wenaturally havep i =P[r i <c i ].Becauser i ∼N(0,1)weimmediatelyobtain c i =N −1 [p i ](i=1, ,m),(2.30) 11 Shiftedandscaledinordertoobtainarandomvariablewithmeanzeroandstandard deviationone. 12 Notethatforreasonsofasimplernotationweherewriter i forthestandardizedlog- returns,incontrasttothenotationinSection1.2.3,wherewewrote˜r i . ©2003 CRC Press LLC whereN[·]denotesthecumulativestandardnormaldistributionfunc- tion.Scalingtheidiosyncraticcomponenttowardsastandarddeviation ofone,thischanges(2.29)into ˜ε i < N −1 [p i ]−R i Φ i 1−R 2 i ,˜ε i ∼N(0,1).(2.31) Takingintoaccountthat˜ε i ∼N(0,1),wealtogetherobtainthefol- lowingrepresentationfortheone-yeardefaultprobabilityofobligori conditionalonthefactorΦ i : p i (Φ i )=N N −1 [p i ]−R i Φ i 1−R 2 i (i=1, ,m).(2.32) Theonlyrandompartof(2.32)isthecompositefactorΦ i .Conditional onΦ i =z,weobtaintheconditionalone-yeardefaultprobabilityby p i (z)=N N −1 [p i ]−R i z 1−R 2 i .(2.33) Combinedwith(2.28)thisshowsthatweareinaBernoullimixture settingexactlythesamewayaselaboratedinSection2.1.1.More formally we can – analogously to (2. 2) – specify the portfolio loss distribution by the probabilities (here we assume again l i ∈ {0, 1}) P[L 1 = l 1 , , L m = l m ] (2. 34) = [0,1] m m i=1 q l i i (1 − q i ) 1−l i dF (q 1 , , q m ), where the distribution function F is now explicitly given by F (q 1 , , q m ) = N m p −1 1 (q 1 ), , p −1 m (q m ); Γ , (2. 35) where N m [ · ; Γ] denotes the cumulative multivariate centered Gaussian distribution with correlation matrix Γ, and Γ = ( ij ) 1≤i,j≤m means the asset correlation matrix of the log-returns r i according to (2. 27). In case that the composite factors are represented by a weighted sum of industry and country indices (Ψ j ) j=1, ,J of the form Φ i = J j=1 w ij Ψ j (2. 36) ©2003 CRC Press LLC (seeSection1.2.3),theconditionaldefaultprobabilities(2.33)appear as p i (z)=N N −1 [p i ]−R i (w i1 ψ 1 +···+w iJ ψ J ) 1−R 2 i ,(2.37) withindustryandcountryindexrealizations(ψ j ) j=1, ,J .Byvarying theserealizationsandthenrecalculatingtheconditionalprobabilities (2.37)onecanperformasimplescenariostresstesting,inorderto studytheimpactofcertainchangesofindustryorcountryindiceson thedefaultprobabilityofsomeobligor. 2.4.2CreditRisk + CreditRisk + isacreditriskmodeldevelopedbyCreditSuisseFinan- cialProducts(CSFP).Itismoreorlessbasedonatypicalinsurance mathematicsapproach,whichisthereasonforitsclassificationasan actuarianmodel.Regardingitsmathematicalbackground,themain referenceistheCreditRisk + TechnicalDocument[18].Inlightofthis chapteronecouldsaythatCreditRisk + isatypicalrepresentativeof thegroupofPoissonmixturemodels.Inthisparagraphweonlysum- marizethemodel,focussingondefaultsonlyandnotonlossesinterms ofmoney,butinChapter4amorecomprehensiveintroduction(taking exposuredistributionsintoaccount)ispresented. AsmixturedistributionCreditRisk + incorporatesthegammadistri- bution.Recallthatthegammadistributionisdefinedbytheprobability density γ α,β (x)= 1 β α Γ(α) e −x/β x α−1 (x≥0), whereΓ(·)denotes 13 thegammafunction.Thefirstandsecondmo- mentsofagamma-distributedrandomvariableΛare E[Λ]=αβ,V[Λ]=αβ 2 ;(2.38) seeFigure2.2foranillustrationofgammadensities. 13 We will also write X ∼ Γ(α, β) for any gamma-distributed random variable X with pa- rameters α and β. Additionally, we use Γ to denote the correlation matrix of a multivariate normal distribution. However, it should be clear from the context which current meaning the symbol Γ has. ©2003 CRC Press LLC FIGURE2.2 Figure 2.2: Shape of Gamma Distributions for parameters (α,β)∈ {(2,1/2),(5,1/5)}. Insteadofincorporatingafactormodel(aswehaveseenitinthecase ofCreditMetrics TM andKMV’sPortfolioManager TM inSection1.2.3), CreditRisk + implementsaso-calledsectormodel.However,somehow onecanthinkofasectorasa“factor-inducing”entity,or–asthe CreditRisk + TechnicalDocument[18]saysit–everysectorcouldbe thoughtofasgeneratedbyasingleunderlyingfactor.Inthisway,sec- tors and factors are somehow comparable objects. From an interpreta- tional point of view, sectors can be identified with industries, countries, or regions, or any other systematic influence on the economic pe rfor- mance of counterparties with a pos itive weight in this sector. Each sector s ∈ {1, , m S } has its own gamma-distributed random intensity Λ (s) ∼ Γ(α s , β s ), where the variables Λ (1) , , Λ (m S ) are assumed to be independent. Now let us assume that a credit portfolio of m loans to m different obligors is given. In the sector model of CreditRisk + , every obligor i admits a breakdown into sector weights w is ≥ 0 with m S s=1 w is = 1, such that w is reflects the sensitivity of the default intensity of obligor i to the systematic default risk arising from sector s. The risk of sector s is captured by two paramete rs: The first driver is the mean default 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ©2003 CRC Press LLC [...]... mean µ and admitting jump times according to an independent Poisson process with mean jump arrival rate l The parameter set (κ, θ, σ, µ, l) can be adjusted to control the ©20 03 CRC Press LLC manner in which default risk changes over time, e.g., one can vary the mean reversion rate κ, the long-run mean m = θ + lµ/κ, or the relative ¯ contributions to the total variance of λ(t) that are attributed to. .. namely one-factor respectively one-sector models 2.5.1 The CreditMetricsTM /KMV One-Factor Model The one-factor model in the context of CreditMetricsTM and KMV is completely described by specializing equations (2 27) and (2 32 ) to the case of only one single factor common to all counterparties, hereby assuming that the asset correlation between obligors is uniform More explicitly, this means that the... defining the sector’s mean intensity by m λ(s) = wis λi ; i=1 ©20 03 CRC Press LLC see also Formula (4 30 ) in Section 4 .3. 2 Now, on the portfolio level, the “trick” CreditRisk+ uses in order to obtain a nice closed-form distribution of portfolio defaults is sector analysis Given that we know distribution of defaults in every single sector, the portfolio’s default distribution then just turns out to be the... Xs(i) , ©20 03 CRC Press LLC where the sector factor Xs(i) is common to all obligors in that sector Here s(i) denotes the sector in which obligor i takes place A possible simulation algorithm to generate default times τ1 , , τn up to some time horizon T with given intensities λi , , λn is the multicompensator method In this method it is assumed that the comt pensator Λi (t) = 0 λi (u)du can be simulated... and Singleton [31 , 32 ] In Duffie and Gˆrleanu [29] intensity models are applied to the valuation of collata eralized debt obligations The theory underlying intensity models has much in common with interest rate term structure models, which are mathematically complex and beyond the scope of this book For readers interested in the theory we refer to the already mentioned papers by Duffie et al and also to. .. that the factor Y and the residuals Zi are normally distributed, (Ω, F, P ) turns out to be an infinite dimensional Gaussian space, but due to the more generally applicable proof we can use the same convergence argument for distributions other than normal For example, the t-distribution is a natural candidate to replace the normal distribution; see Section 2.6.1 Now let us apply our findings to uniform... coefficients αij have to be calibrated by the user of CPV, although CPV contains some standard values that can be chosen in case a user does not want to specify the shift factors individually The shift factors depend on the considered migration path Ri → Rj , hereby expressing the sensitivity of P[Ri → Rj ] w.r.t a change in the segment’s risk index rs Because they are intended to reflect rating class... , γs,2 ) have to be calibrated to each segment A main issue is the calibration of the correlation matrix of Γ In general these challenges are much easier to master than calibrating the macroeconomic indices by means of an autoregression as it is suggested by CPV Macro The parameters of the gamma distribution of a segment are calibrated by specifying the mean and the volatility of the random variable... ©20 03 CRC Press LLC (γs,1 , γs,2 ) are then determined by a moment matching based on (2 38 ) Note that the support of the gamma distribution is R+ , so that it theoretically can happen to draw a number ps > 1, which obviously can not be interpreted as a probability This is an unpleasant side effect when drawing a random number that is supposed to be a probability from a gamma distribution However, to. .. constitute an independent family, also independent of the factor Y Under the assumption of a single factor and a uniform , Equation (2 32 ) turns into √ Y N −1 [pi ] − √ (i = 1, , m) (2 49) pi (Y ) = N 1− Figures 2 .3 illustrates the dependence of the conditional default probability pi (y) on realizations y of the single factor Y Figure 2.4 shows for three fixed states of economy Y = 3, 0, 3 the conditional . LLC andtosomeextentinthecontextofsecuritizations.Fromamath- ematician’spointofviewtheyprovidea“mathematicallybeautiful” approachtocreditriskmodeling,butfromtheintroductorypointof viewweadoptedforwritingthisbookwemustsaythatanappropri- atepresentationofdynamicintensitymodelsisbeyondthescopeof thebook.Wethereforedecidedtoprovidethereaderonlywithsome referencestotheliteraturecombinedwithintroductoryremarksabout theapproach;seeSection2.4.4 Inthisbook,ourdiscussionofCreditPortfolioView 10 (CPV)iskept shorterthanourpresentationofCreditMetrics TM ,theKMV-Model, andtheactuarianmodelCreditRisk + .Thereasonfornotgoingtoo muchintodetailsisthatCPVcanbeconsideredasageneralframework forcreditriskmodeling,whichisthentailor-madeforclient’sneeds.In ourpresentationwemainlyfocusonthesystematicriskmodelofCPV. 2.4.1CreditMetrics TM andtheKMV-Model ForsomebackgroundonCreditMetrics TM andKMVwerefertoSec- tion1.2 .3. Notethatforbothmodelswefocusontheir“default-only mode”,. (α,β)∈ {(2,1/2),(5,1/5)}. Insteadofincorporatingafactormodel(aswehaveseenitinthecase ofCreditMetrics TM andKMV’sPortfolioManager TM inSection1.2 .3) , CreditRisk + implementsaso-calledsectormodel.However,somehow onecanthinkofasectorasa“factor-inducing”entity,or–asthe CreditRisk + TechnicalDocument[18]saysit–everysectorcouldbe thoughtofasgeneratedbyasingleunderlyingfactor.Inthisway,sec- tors. LLC whereasinthePoissoncaseweobtain Corr[L i ,L j ]= Cov[Λ i ,Λ j ] V[Λ i ]+E[Λ i ] V[Λ j ]+E[Λ j ] .(2.24) LookingonlyatthedrivingrandomvariablesP i ,P j respectivelyΛ i ,Λ j , weseethatinthedenominatorsof(2. 23) and(2.24)wecompare V[P i ]+E[P i (1−P i )]=V[P i ]+E[P i ]−E[P 2 i ](2.25) withV[Λ i ]+E[Λ i ]. Now,analogoustothedeterministiccase(2.12),wecan–eveninthe randomcase–expectP i andΛ i tobeofthesameorderofmagnitude. Tokeepthingssimple,letusforamomentassumethatP i andΛ i havethesamefirstandsecondmoments.InthiscaseEquation(2. 25)combinedwith(2. 23) and(2.24)showsthattheBernoullimodel alwaysinducesahigherdefaultcorrelationthanthePoissonmodel.But higherdefaultcorrelationsresultinfattertailsofthecorrespondingloss distributions.Inotherwordsonecouldsaythatgivenequalfirstand secondmomentsofP i andΛ i ,theexpectationsofL i andL i willmatch, butthevarianceofL i willalwaysexceedthevarianceofL i ,thereby inducinglowerdefaultcorrelations. SothereisasystematicdifferencebetweentheBernoulliandPoisson mixturemodels.Ingeneralonecanexpectthatforagivenportfolio theBernoullimodelyieldsalossdistributionwithafattertailthana comparably(e.g.,byafirstandsecondmomentmatching)calibrated Poissonmodel.Thisdifferenceisalsoreflectedbytheindustrymodels fromCreditMetrics TM /KMVCorporation(PortfolioManager)and CreditSuisseFinancialProducts(CreditRisk + ).InSection2.5.3we come back to this issue. 2.4 An Overview of Today’s Industry