AIM4 1.4: Describe hawa.loss distribution isobtained from frequency andseverity distributions.
Througha process calledconvolution,Lhefrequencyand severity distributionscan be combinedinto asingleoperationallossdistribution. Constructingdie loss distribution for theentireinstitutionisaccomplished through MonteCarlo simulation acrossthe lossfrequencyandseverity data,beginningwith each loss typein each businessline,and thenaggregatingacrossall business lines/eventtypes, thissingleoperationalloss distribution,wecan thenidentifythemean annual loss and thelossatanygivenconfidence
level,and determine theamountofeconomiccapital requiredtosupportdiepotential risk.
Note thaL whileit isunlikelythat lossesareperfectly correlated,it maybe unreasonableto
assumenocorrelation. Therefore,analysis of the correladons between business line/event
typedata maybe required.
Once the lossdistribution(thatcombines severity andfrequency) is constructed,wecan examine thedifferencebetween theoperational valueatrisk (OpVaR) measureand expected losstodeterminetheamountofeconomiccapital needed tocoverpotential operational
risklosses (i.e., unexpectedlosses).Ademonstrationof how tocreate this aggregateloss distribution and correspondingly calculateeconomiccapital will heprovidedshordy.
Insurance
Thelossdistribution approach allowsfor arisk profilingofan institution,whichcan
includethe riskreducingeffect ofinsurance,whichalterstheaggregateloss distribution.
Typicallythis isdonebyreducingtheseverityof the losses that exceedagivendeductible
in theinsurance policy.In otherwords,insurance typicallylowersdie severity butnot tbe frequency.
DeterminingEconomic Capital
Economiccapitalisa measureof thecapital thatwill be abletoabsorb unexpected losses witha high degreeofcertainly. Itis populartouse theaveragedefaultEateassociated with
the rating of theinstitutionfor the‘'degreeof certainty.” In that case,the ruledefines
economiccapitalintermsofoperadonal valueatrisk(e.g.,foraAA+ rating the confidence level is 99.98%).Thislinkagehetween ratitigsand valueat risk basedeconomiccapitalis a reason for the popularityof dieeconomiccapitalmethodology.Clearly,aconfidence level of 99.98% placesagreatdealofimportanceonmodelingdie tailsof die lossdistributions correcdyand requires theuse of both internal andexternaldata.
Thestandard measure foroperational riskeconomiccapital is:
economiccapital= (operational valueatrisk)-(expected loss) = unexpected loss
Practitionersoftenadjust theexpectedloss indifferentcases.Forexample, inthecaseof operadonalrisk, toaccommodate the existenceof fattails,themedianlossmight replacethe expectedloss.
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CapitalallocaLion techniquesforbusiness line/event typecells useexpectedshortfall contributions,whichis theaverage expected loss giventhe lossexceeds theVaRlevelof lossfor die confidencelevel, [i.e„ E(loss|loss >VaR)].Thisaverage loss isalsoknownas
conditionalVaR.Sinceanalyticalrepresentations of suchlossesat theaggregatelevelarenot
possibles practitionersusesimulations.
ThefollowingisanoutlineofaMonteCarlosimulation thatcan produceasampleof the
aggregate lossdistribution:
1. Simulate loss distributionsinindividualcells.
* Generateasample(n1 ...nm)of the m-dimensionalfrequency distribution specified byaGaussiancopula
* Foreach cell k,computenÿsamplessÿ,...sÿ of the severitydistribution.
2. Incorporate the effect of insurance.
* Randomize the lossesinthe cells.
* Applythecorrespondinginsurancemodeltoeach lossandcompute the correspondingnetlosses.
3. Compose dieaggregate loss distribution.
* The sum of thenetlosses Isonesampleof theaggregateoperational riskloss distribution.
Thesestepsshouldhe repeatedtocompute alargegroupof losssamples.Theeconomic capitalisthen computedwiththefollowingsteps.
1. TheeconomiccapitalisoperationalVaRminus dieexpected loss.
2. Theeconomiccapitalis then allocated to die cellsof the husiness line/eventtype matrix.Thisisdoneusing expectedshortfall techniques.
3- Theeconomiccapitalof each business lineiscomputed by summingdie capital allocated toitscells with respect tospecificevent types.
Validationof Loss DistributionApproaches
Toshowthat theestimatesforregulator}'andeconomiccapitalare reasonable, there should heavalidation process.Given that confidence levelsareoftenhigh, thiscan hechallenging.
Data problemsand die lack ofarisk-sensitivemeasurefor operational risk modeling complicatedieproblem.Thus,thereIsinherent uncertaintyin the validation process, and
expertjudgmentshouldcomplementquantitativemeasures.
Thevalidationofan operationalrisk model involvesa reviewof data inputs, model mediodology,and modeloutputs.Thereview of themodel would includeexamining the assumedfrequencies,severities,dependencestructure,andinsurance, aswellasassumptions in themodelsuchascorrelations.Thevalidationwould includeanassessmentof the impact ofdifferent modelingassumptions and the sensitivity ofcapital requirements to parameter
changes.
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KEY CONCEPTS
AIM41.1
The lossdistribution approach (LDA) isthe natural waytomeetthe soundness standards foreconomicandregulatory capital by explicitlymodelingdieoperational risk loss distribution of the bankoveraone-year period.
The LDA has severalsteps,whichlargelyinvolveorganizing the datatomodel the
losses whhfrequencyand severity distributions and ultimately determiningthecapital requirements.
AIM41.2
Adequatelymodelingthe lossesusuallyrequires using both internal andexternaldata.
Principles forusingdata include: usingall internal data,adjustingforinflation, usinggross lossesafter recoveries,correct useof externaldata,convertinglossesinforeigncurrenciesto
domesticcurrencies,and minimizing the possibilityofdoublecounting losses.
In modelingthedistribution,dieanalystwill assignanequal weight toalldata points.
Therearethree exceptions:splitlosses,oldlosses,and external losses andscenarios.
AIM41.3
LDAmodelsfrequencyof loss and severity of loss separately. For the frequency,LDA models use thePoissondistribution, thenegativebinomialdistribution,or thebinomial distribution.
Assumptions concerning the severitydistributionsaregenerallyconsideredmoreimportant than thefrequencydistributions,and usingexternal dataisusuallynecessary. Also, the analyst may wish toextrapolate the observed lossestoestimatevaluesbeyond those observed.
Tobuildamodelofadistribution,ananalystcanbreak down thedistributioninto die body,anintermediaterange,andatail.In the threeranges, theanalystwoulduse internal dataonly, internalandexternal data thatapplytothe given cellin thebusinessline/evenc
typematrix,and all available losseslargeenoughtofoilin thetail, respectively.
Tocalibrate the parametrictail,theanalystwouldestimateshapeand scale parametersof thegeneralizedParetodistributionfor increasingexcess thresholds using the
Peaks-Over-Thresholdmethod.
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CrossReferencetoGARPAssignedReading—CJiaiidlmry AJM4L.4
Insurancealters the aggregateloss distributionand,wheninsurance is used,itshould be includedinLDAmodels.Insurance typicallylowersthe severity but notthefrequencyof losses.
The levelofconfidencefor theoperationalvalueat riskcan he usedasthe‘'degreeof certaintyÿ for dieeconomiccapital. Rigorousstandards(e.g., 99.9%)canincrease the necessity of modelingthe lossdistributionscorrectly.Thestandardmeasureforeconomic capitalis:
economiccapital= (operationalvalueatrisk)- (expectedloss)
Practitionersoftenadjust theexpectedlossby usingthe median in placeof the expected loss.MonteCarlo simulationcan produceasampleof theaggregateloss distribution.
AJM41.5
Although manydependenciescan exist,thetypical practiceistofocus onlyon the correlationsoffrequency distributions betweenbusinessline/evenc typecells.
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CONCEPT CHECKERS
Thelossdistributionapproach typicallymodels:
A. frequencyof loss and severity of lossseparately.
B. frequencyof lossandseverity oflossinasinglebivariatedistribution.
C. frequencyof loss butnotseverity ofloss.
D. severity of loss but notfrequencyof loss.
In modelingadistribution, the usual practiceis toassignanequal weightnoalldata points.Excepcionsto thispractice include all of thefollowingexcept:
A. when using old losses.
B. when usingsplitlosses.
C. whenmodelingthefrequencydistribution.
D. when usingexternal losses.
1.
2.
Whichofdiefollowingstatementsiscorrectregardinginsurance and the loss distributionapproach?
I. Theprobaliility thaL theinsurerwillnotpay ismodeled.
II. Iftheinsurerappliesahaircuttotheamountpaid, the loss will begreaterafter severity has been modeled in the lossdistributionmodel.
3.
A. Ionly.
B. IIonly C. Both IandII.
D. Neither I norII.
Economiccapital is a measureof diecapitalthat will be able toabsorb unexpected losses widiahigh degreeof certainty. For diedegreeof certainty anappropriate measureis:
A. theprobabilityvalueassociatedwith theexpectedlossinthe tail.
B. 10%inallcases.
C. theprobabilityvalueassociated withthelargest observedloss.
D. the confidence level associated with theOpVaR for theinstitution'screditrating.
In modelingriskfrequency, it iscommon to:
A. usea Poissondistribution.
B. assume that risksarehighlycorrelated.
C. assume risk frequencyand severityare the same.
D. usestraight-lineprojection from themost recentlossdata.
Foradditional BookX Topic 41practicequestionssee:
4.
5.
PastFRM Exam Questions:#iff (page283)
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CONCEPT CHECKER ANSWERS
1L A TheLDAmodelsfrequencyof loss andseverityofloss separatelyForthefrequency, LDA modelsusethePoisson distribution, the negative binomial distribution,orthe binomial
distribution,
2. C Splitlossesatelosseventsthatcannotbe assignedtoasinglecell. Old lossesshouldreceivea lowerweight.External dataandscenariosmayrequirescalingbecause ofinherent biasesfrom
reporting practices.
3. C Regardinginsuranceand the lossdisrributtonapproach,theprobabilitythat theinsurerwill
notpayismodeled.Also,iftheinsurerappliesahaircuttotheamountpaid,theJosswill be greaterafter severity has beenmodeledinthe loss distribution model.
4. D Thislinkagebetweenratingsand valueatriskbasedeconomiccapital isareasonfor the popularityof theeconomiccapital methodology.
5. A Itiscommon touseaPoisson distributiontomodel loss frequenty.A Poissondistribution hasasingleparameterthatcanbe variedtoaccurately describe loss data.
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Thefollowingisiwiiewof(lieOperationalandJoiegmled RiskManagementprinciplesdesignedLoaddress LheATMstatementssetforth hyGART®.Thistopicisalso oaveredin: