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Joint Distribution at Horizon
20 03 CRC Press LLC
PCA.Infact,theindustryandcountryindiceshaveacleareconomic
meaning,whereastheglobalfactorsarisingfromaPCAareofsynthetic
type.AlthoughtheyadmitsomevagueinterpretationasshowninFig-
ure1.7,theirmeaningisnotasclearasisthecasefortheindustryand
country ... LLC
agingcreditportfolios.TheirtoolsarebasedonamodificationofMer-
ton’sassetvaluemodel,seeChapter3,andincludeatoolforestimating
defaultprobabilities(CreditMonitor
TM
)frommarketinformationand
atoolformanagingcreditportfolios(PortfolioManager
TM
).Thefirst
tool’smainoutputistheExpectedDefaultFrequency
TM
(EDF),which
cannowadaysalsobeobtainedonlinebymeansofanewlydeveloped
web-basedKMV-toolcalledCreditEdge
TM
.Themainoutputofthe
PortfolioManager
TM
isthelossdistributionofacreditportfolio.Of
course,bothproductshavemanymoreinterestingfeatures,andtous
itseemsthatmostlargebanksandinsuranceuseatleastoneofthe
majorKMVproducts.AreferencetothebasicsoftheKMV-Modelis
thesurveypaperbyCrosbie[19].
CreditMetrics
TM
isatrademarkoftheRiskMetrics
TM
Group,acom-
panywhichisaspin-offoftheformerJPMorganbank,whichnow
belongstotheChaseGroup.Themainproductarisingfromthe
CreditMetrics
TM
frameworkisatoolcalledCreditManager
TM
,whichin-
corporatesasimilarfunctionalityasKMV’sPortfolioManager
TM
.Itis
certainlytruethatthetechnicaldocumentation[54]ofCreditMetrics
TM
waskindofapioneeringworkandhasinfluencedmanybank-internal
developmentsofcreditriskmodels.Thegreatsuccessofthemodelun-
derlyingCreditMetrics
TM
isinpartduetothephilosophyofitsauthors
Gupton,Finger,andBhatiatomakecreditriskmethodologyavailable
toabroadaudienceinafullytransparentmanner.
Bothcompaniescontinuetocontributetothemarketofcreditrisk
modelsandtools.Forexample,theRiskMetrics
TM
Grouprecentlyde-
velopedatoolforthevaluationofCollateralizedDebtObligations,and
KMVrecentlyintroducedanewreleaseoftheirPortfolioManager
TM
PM2.0,herebypresentingsomesignificantchangesandimprovements.
Returningtothesubjectofthissection,wenowdiscussthefac-
tormodelsusedinKMV’sPortfolioManager
TM
andCreditMetrics
TM
CreditManager
TM
.Bothmodelsincorporatetheideathateveryfirm
admitsaprocessofassetvalues,suchthatdefaultorsurvivalofthefirm
dependsonthestateoftheassetvaluesatacertainplanninghorizon.
Iftheprocesshasfallenbelowacertaincriticalthreshold,calledthe
defaultpointofthefirminKMVterminology,thenthecompanyhas
defaulted.Iftheassetvalueprocessisabovethecriticalthreshold,the
firmsurvives.AssetvaluemodelshavetheirrootsinMerton’sseminal
paper[86]andwillbeexplainedindetailinChapter3andalsotosome
extentinSection2.4.1.
20 03 ... },p
i
=P[L
i
≥1], (2. 11)
wherep
i
againdenotesthedefaultprobabilityofobligori.Notethat
(2. 11)allowsformultipledefaultsofasingleobligor.Thelikelihood
oftheeventthatobligoridefaultsmorethanonceisgivenby
P[L
i
2] =1−e
−λ
i
(1+λ
i
),
whichistypicallyasmallnumber.Forexample,inthecaseofλ
i
=0.01
wewouldobtainP[L
i
2] =0.5basispoints.Inotherwords,when
simulatingaPoisson-distributeddefaultvariablewithλ
i
=0.01wecan
expectthatonly1outof20,000scenariosisnotapplicablebecauseof
amultipledefault.Ontheotherside,forobligorswithgoodcredit
quality(forexample,aAAA-borrowerwithadefaultprobabilityof
2basispoints),amultiple-defaultprobabilityof0.5basispointsisa
relativelyhighnumber.
Theintensityλ
i
istypicallyquiteclosetothedefaultprobabilityp
i
,
dueto
p
i
=P[L
i
≥1]=1−e
−λ
i
≈λ
i
(2. 12)
forsmallvaluesofλ
i
.Equation (2. 12) showsthattheone-yeardefault
probabilityequalstheprobabilitythatanexponentialwaitingtimewith
intensityλ
i
takesplaceinthefirstyear.
Ingeneral,thesumofindependentvariablesL
1
∼Pois(λ
1
),L
2
∼
Pois(λ
2
)hasdistribution
7
Pois(λ
1
+λ
2
).Assumingindependence,the
portfolio’stotalnumberoflosseswouldbegivenby
L
=
m
i=1
L
i
∼Pois
m
i=1
λ
i
. (2. 13)
Correlationisintroducedintothemodelbyagainfollowingamixture
approach,imthiscasewithPoissonvariables(seealsoJoe[67],Section
7 .2.
7
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