毕博上海银行Credit Risk Mgmt Sys Analytics Credit metrics monitor outline.docx
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毕博上海银行CreditRiskMgmtSysAnalyticsCreditmetricsmonitoroutline
AOne-ParameterRepresentationofCreditRiskandTransitionMatrices
LawrenceR.Forest,Jr.,KPMGPeatMarwickLLP
BarryBelkinandStephanJ.Suchower,DanielH.WagnerAssociates
1.Overview
Thispaperpresentsaone-parameterrepresentationofcreditriskandtransitionmatrices.WestartwiththeCreditMetricsviewthatratingstransitionmatricesresultfromthe“binning”ofastandardnormalrandomvariableXthatmeasureschangesincreditworthiness.WefurtherassumeherethatXsplitsintotwoparts:
(1)anidiosyncraticcomponentY,uniquetoaborrower,and
(2)asystematiccomponentZ,sharedbyallborrowers.Broadlyspeaking,Zmeasuresthe“creditcycle”,meaningthevaluesofdefaultratesandofend-of-periodriskratingsnotpredicted(usinghistoricalaveragetransitionrates)bytheinitialmixofcreditgrades.IngoodyearsZwillbepositive,implyingforeachinitialcreditrating,alowerthanaveragedefaultrateandahigherthanaverageratioofupgradestodowngrades.Inbadyears,thereversewillbetrue.WedescribeawayofestimatingZfromtheseparatetransitionmatricestabulatedeachyearbyStandard&Poor(S&P)andMoody’s.Conversely,wedescribeamethodofcalculatingtransitionmatricesconditionalonanassumedvalueforZ.
ThehistoricalpatternofZdepictspastcreditconditions.Forexample,Zremainsnegativeformostof1981-89.Thismirrorsthegeneraldeclineincreditratingsoverthatperiod.In1990-91,ZdropswellbelowzeroastheUSsuffersthroughoneofitsworstcreditcrisessincetheGreatDepression.Therelativelyhighproportionoflowergradecreditsinheritedfromthe1980’stogetherwiththe1990-91slump(Z<0)accountsforahighnumberofdefaults.Over1992-97,Zhasstayedpositiveandcreditconditionshaveremainedbenign.ThemovementsofZoverthepast10yearscorrelatecloselywithloanpricing.
OurfocushereisonhowZaffectscreditratingmigrationprobabilities.However,onecanalsomodeltheeffectofZontheprobabilitydistributionoflossintheeventofdefault(LIED),creditparspreads,andultimatelythevalueofacommercialloan,bond,orotherinstrumentsubjecttocreditrisk.ByparametricallyvaryingZ,onecanperformstresstestingtoassessthesensitivityofthevalueofanindividualcreditinstrumentoranentirecreditportfoliotochangingcreditconditions.OnecanalsoquantifyhowvolatilityinZtranslatesintotransactionandportfoliovaluevolatility.
2.DefiningZRisk
FollowingtheCreditMetricsapproachdescribedbyGupton,Finger,andBhatia(1997),weassumethatratingstransitionsreflectanunderlying,continuouscredit-changeindicatorX.OnefurtherassumesthatXhasastandardnormaldistribution.Then,conditionalonaninitialcreditratingGatthebeginningofayear,onepartitionsXvaluesintoasetofdisjointbins
.Tosimplifyreferences,theindicesGandgrepresentsequencesofintegersratherthanlettersorothersymbols.OnedefinesthebinssothattheprobabilitythatXfallswithinagivenintervalequalsthecorrespondinghistoricalaveragetransitionrate(seeExhibit1).
Exhibit1:
RelationshipBetweenContinuousCreditIndexXandRatingTransitions
Wewritetheconditionsdefiningthebinsasfollows:
(1)
inwhichP(G,g)denotesthehistoricalaverageG-to-gtransitionprobabilityand()representsthestandardnormalcumulativedistributionfunction.Thedefaultbinhasalowerthresholdof-.TheAAAbinhasanupperthresholdof+.Theremainingthresholdsarefittotheobservedtransitionprobabilities.
SupposethereareNratingscategoriesincludingdefault.ThenthereareN-1initialgrades,whichrepresentalltheratingsexcludingdefault.Foreachofthoseinitialgrades,weobserveN-1historicalaveragetransitionrates.(Theother(Nth)valuederivesfromtheconditionthattheprobabilitiessumto1.)WemustdetermineN-1thresholdvaluesdefiningthebins.Thus,wecansolveforallofthebinboundaries.
Weillustratethisprocessbelow.Thestartingpointisthesmoothedversionofthe1981-97historicalaveragetransitionmatrixtabulatedbyS&Pfor8grades,includingdefault(seeExhibit2).Thecorrespondingbinsarecomputedusingtheaboveformula.
ConsidertransitionsfromBBB.Weobservea15bpsdefaultrate.Usingtheinverseprobabilityfunctionforastandardnormaldistribution,wecomputeavalueofabout–2.97fortheupperthresholdforthedefaultbin.NextconsidertheCCCbin.Wegetavalueofabout25bpsforthesumoftransitionratestoCCCortodefault.Againapplyingtheinverseprobabilityfunction,wegetanupperthresholdvalueforCCCofabout–2.81.NowconsiderB.Wecomputeaprobabilityofabout1.3percentfortransitionstoBortolowergrades.Onceagainapplyingtheinverseprobabilityfunction,wegetanupperthresholdvalueof-2.23.Continuinginthiswayforeachterminalandeachinitialgrade,wederiveallofthebinvalues.
Exhibit2:
SmoothedHistoricalAverageTransitionRatesandAssociatedBins
Initial
EndofYearCreditRating
Rating
AAA
AA
A
BBB
BB
B
CCC
D
SmoothedHistoricalAverageTransitionRates
AAA
91.13%
8.00%
0.70%
0.10%
0.05%
0.01%
0.01%
0.01%
AA
0.70%
91.03%
7.47%
0.60%
0.10%
0.07%
0.02%
0.01%
A
0.10%
2.34%
91.54%
5.08%
0.61%
0.26%
0.01%
0.05%
BBB
0.02%
0.30%
5.65%
87.98%
4.75%
1.05%
0.10%
0.15%
BB
0.01%
0.11%
0.55%
7.77%
81.77%
7.95%
0.85%
1.00%
B
0.00%
0.05%
0.25%
0.45%
7.00%
83.50%
3.75%
5.00%
CCC
0.00%
0.01%
0.10%
0.30%
2.59%
12.00%
65.00%
20.00%
BinsCorrespondingtoSmoothedHistoricalAverageTransitionRates
AAA
(,-1.35)
[-1.35,-2.38)
[-2.38,-2.93)
[-2.93,-3.19)
[-3.19,-3.54)
[-3.54,-3.72]
[-3.72,-3.89)
[-3.89,-)
AA
(,2.46)
[2.46,-1.39)
[-1.39,-2.41)
[-2.41,-2.88)
[-2.88,-3.09)
[-3.09,-3.43)
[-3.43,-3.72)
[-3.72,-)
A
(,3.10)
[3.10,1.97)
[1.97,-1.55)
[-1.55,-2.35)
[-2.35,-2.73)
[-2.73,-3.24)
[-3.24,-3.29)
[-3.29,-)
BBB
(,3.50)
[3.50,2.73)
[2.73,1.56)
[1.56,-1.55)
[-1.55,-2.23)
[-2.23,-2.81)
[-2.81,-2.97)
[-2.97,-)
BB
(,3.89)
[3.89,3.05)
[3.05,2.48)
[2.48,1.38)
[1.38,-1.29)
[-1.29,-2.09)
[-2.09,-2.33)
[-2.33,-)
B
(,4.11)
[4.11,3.29)
[3.29,2.75)
[2.75,2.43)
[2.43,1.42)
[1.42,-1.36)
[-1.36,-1.64)
[-1.64,-)
CCC
(,4.27)
[4.27,3.72)
[3.72,3.06)
[3.06,2.64)
[2.64,1.88)
[1.88,1.04)
[1.04,-0.84)
[-0.84,-)
AsinBelkin,Suchower,andForest(1998),wedecomposeXintotwoparts:
(1)a(scaled)idiosyncraticcomponentY,uniquetoaborrower,and
(2)a(scaled)systematiccomponentZ,sharedbyallborrowers.Thus,wewrite
.
(2)
WeassumethatYandZareunitnormalrandomvariablesandmutuallyindependent.Theparameter(assumedpositive)representsthecorrelationbetweenZandX.ThusZexplainsafractionofthevarianceofX.
Inanyyear,theobservedtransitionrateswilldeviatefromthenorm(Z=0).WecanthenfindavalueofZsothattheprobabilitiesassociatedwiththebinsdefinedabovebestapproximatethegivenyear’sobservedtransitionrates(seeExhibit3).WelabelthatvalueofZforyeart,Zt.WedetermineZtsoastominimizetheweighted,mean-squareddiscrepanciesbetweenthemodeltransitionprobabilitiesandtheobservedtransitionprobabilities.
Forthiswedefine
.(3)
ThisisthemodelvaluefortheG-to-gtransitionrateinyeart.Thenforafixed
andafixedt,theleast-squaresproblemtakestheform
(4)
wherePt(G,g)representstheG-to-gtransitionrateobservedinyeartandnt,GisthenumberoftransitionsfrominitialgradeGobservedinthatyear.Inthisformula,weweightobservationsbytheinversesoftheapproximatesamplevariancesofPt(G,g).
Exhibit3:
IllustrationoftheZValueforaParticularInitialRatinginaGivenYear
Wedon’tknowthevalueofapriori.Weestimateasfollows.Weapplytheminimizationin(4)for1981-97usinganassumedvalueof.WethenobtainatimeseriesforZt,conditionalon.Wecomputethemeanandvarianceofthisseries.Werepeatthisprocessformanyvaluesof,anduseanumericalsearchproceduretofindtheparticularvalueforwhichtheZttimeserieshasvarianceofone.
WeillustratethisprocessofsolvingforZtatasingletimet.WestartwiththeS&Ptransitionmatrixobservedfor1982(seeExhibit4).Weholdfixedthebinsdeterminedfromthehistoricalaveragematrixandfixatthevalue(.0163)determiningbythesearchprocess.TheindicatedvalueforZtof–0.89providesthebestfittotheobserved1982transitionrates.
Exhibit4:
S&PTransitionMatrixfor1982andCalculationsLeadingtoZEstimate
Initial
#
EndofYearCreditRating
Rating
Obs.
AAA
AA
A
BBB
BB
B
CCC
D
ObservedTransitionMatrix
AAA
85
92.94%
4.71%
2.35%
0.00%
0.00%
0.00%
0.00%
0.00%
AA
220
0.46%
92.52%
6.08%
0.47%
0.47%
0.00%
0.00%
0.00%
A
480
0.00%
4.45%
84.95%
9.54%
0.64%
0.00%
0.00%
0.42%
BBB
298
0.37%
0.37%
3.26%
85.52%
9.78%
0.37%
0.00%
0.34%
BB
168
0.00%
0.68%
0.00%
2.68%
82.42%
10.05%
0.00%
4.17%
B
161
0.00%
0.00%
0.72%
0.72%
2.89%
87.50%
5.06%
3.11%
CCC
16
0.00%
0.00%
0.00%
0.00%
0.00%
7.39%
73.86%
18.75%
FittedTransitionMatrix
AAA
85
89.34%
9.54%
0.89%
0.13%
0.07%
0.01%
0.01%
0.01%
AA
220
0.48%
89.56%
8.93%
0.77%
0.13%
0.09%
0.03%
0.01%
A
480
0.06%
1.72%
90.88%
6.14%
0.78%
0.34%
0.01%
0.07%
BBB
298
0.01%
0.20%
4.39%
88.03%
5.72%
1.33%
0.13%
0.20%
BB
168
0.00%
0.07%
0.38%
6.19%
81.63%
9.39%
1.06%
1.29%
B
161
0.00%
0.03%
0.17%
0.32%
5.56%
83.41%
4.38%
6.14%
CCC
16
0.00%
0.01%
0.06%
0.20%
1.94%
10.09%
64.53%
23.16%
Z
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