Deutsche Bank’s AMA ModelChallenges for Operational Risk Measurement and ManagementFederal Reserve Bank of Boston, May 14, 2008Michael KalkbrenerLRC, RiskAnalytics& Instruments
RiskAnalytics& at components of an LDA : model validation For more information:F. Aueand M. Kalkbrener(2006). LDA at work: Deutsche Bank’s approach to quantifying operationalrisk. J. Operational Risk, 1(4), Bank’s AMA Model · page 2
RiskAnalytics& InstrumentsAMA Model Development at Deutsche BankTimeline1999 Systematic collection of loss data2000 Economic capital with LDA-Top-down model: loss distribution at Group level, capital allocationwith risk indicators-Internal and external loss data-Qualitative adjustment with Incentive Scheme2001 AMA project2002 Development of AMA model2003 Implementation of prototype 2004 EC test calculations with AMA model2005 Official EC calculation with AMA model (starting Q2 05)2006 Implementation of production engine AMA application submitted in September2007 Regulatory approvalDeutsche Bank’s AMA Model · page 3
RiskAnalytics& InstrumentsRisk Capital Allocation(before QA)(via model)AMA at DB: Group-LevelCalculation flowCorrelation / Aggregate distributionGroup-LevelDiversificationRC before QANet LossesBusiness DivisionsBusiness DivisionsInsuranceXGross LossesQualitative Adjustment (BE&ICF)SeverityFrequency/ DependencyBusiness DivisionsScenario after QADataDataDeutsche Bank’s AMA Model · page 4EventTypes
RiskAnalytics& InstrumentsDB’s Business Line / Event Type MatrixBusiness LinesBasel Level 1Internal Event TypesBL1BL2BL3BL4BL5BL6GroupInternal FraudFraudExternal FraudDamage to physical assetsInfrastructureBusiness disruption …Clients, Products, Clients, Products, Business Practices Business Practices Execution, delivery, Execution, delivery, process managementprocess managementEmployment practices, Employment practices, workplace safety workplace safety Design criteriaTreatment of losses that cannot be –comparable loss profileassigned to a single cell–same insurance type–Group losses–same management responsibilities–Split losses–availability of data –relative importance of cellsDeutsche Bank’s AMA Model · page 5
RiskAnalytics& at components of an LDA : model validationDeutsche Bank’s AMA Model · page 6
RiskAnalytics& InstrumentsRC before QADataCorrelation / Group-LevelAggregate distributionDiversificationNet LossesBusiness DivisionInsuranceXGross LossesSeverityFrequencyScenario Bank’s AMA Model · page 7EventType
RiskAnalytics& InstrumentsData for Modelling Loss DistributionsData sourcesInternal loss data Consortium dataCommercial loss database ScenariosInternal loss data is the most important data sourceEach firm’s operational losses are a reflection of its underlying operational risk exposureInternal losses are used for–modelling frequencies (exclusively)–modelling severities–estimating correlationsMotivation for using external data and scenariosAdditional information on severity profile, in particular on risk of unexpected losses (tails of severity distributions)Deutsche Bank’s AMA Model · page 8
RiskAnalytics& InstrumentsCreating a Relevant Loss Data SetExclude non-Exclude Loan Exclude DB fin. inst. and Map all Fraud datalossesinsurance dataOR losses Publicto BLET MatrixConsortiumBL review and approval of external pointsDB Access To be used for Databasescenario analysisRC EngineScenarios are added as individual data points to relevant external lossesDeutsche Bank’s AMA Model · page 9
RiskAnalytics& InstrumentsScenario Analysis Process and MethodologyOpVardataExisting RLD process for OpVarevents is integral part of scenario analysisEach relevant OpVardata point is considered a scenarioRelevantOpVardataOpVardata is enriched to close gaps through following processes:Divisional Expert driven processControl function driven process (. BCM, Outsourcing)Regions for region specific eventsEnriched scenario dataSame treatment of scenarios and OpVardata points in RC calculationRCDeutsche Bank’s AMA Model · page 10
RiskAnalytics& InstrumentsBiased External Loss DataScale BiasOperational risk is dependent on the size of the bank, . the scale of operationsThe actual relationship between the size of the institution and the frequency and severity may be stronger or weaker depending on the particular OR category Truncation Bias and Data Capture Bias Collection thresholds are not uniform for different data setsData is often captured with a systematic bias. This problem is particularly pronounced with publicly available data: there exists a positive relationship between the loss amount and the probability that the loss is reportedThe disproportionate number of large losses could lead to an estimate that overstates a bank’s exposure to operational riskScaling in AMA at DBNo correction of Scale Bias since it is considered less relevantfor severity modelingCorrection of Truncation Bias and Data Capture BiasDeutsche Bank’s AMA Model · page 11
RiskAnalytics& InstrumentsRC before QAFrequency ModellingCorrelation / Group-LevelAggregate DiversificationdistributionNet LossesBusiness DivisionInsuranceXGross LossesFrequencySeverityScenario Bank’s AMA Model · page 12EventType
RiskAnalytics& InstrumentsFrequencies in AMA at DBDataOnly internal loss data is used for calibrating frequency distributions:Internal loss data reflects DB’s loss profile most accuratelyDifficult to ensure completeness of external data (essential forapplication in frequency calibration)Lower data requirements in frequency modeling (compared to severity modeling)Implemented distributionsPoisson (no dependence between occurrence of events in a cell)Negative Binomial (positive dependence)Selection algorithm based on statistical testsFrequency distributions in official capital calculationsPoisson in all cells Reason: negligible difference to combination of Poisson and Negative Binomial cellsDeutsche Bank’s AMA Model · page 13
RiskAnalytics& InstrumentsRC before QASeverity ModellingCorrelation / Group-LevelAggregate distributionDiversificationNet LossesBusiness DivisionInsuranceXGross LossesSeverityFrequencyScenario Bank’s AMA Model · page 14EventType
RiskAnalytics& InstrumentsModelling DecisionsRange of distributionOne distribution for the entire severity range or different distributions for small, medium and high losses?Choice of distribution family Two-parametric distributions like lognormal, GPD or more flexible distribution families, . three-or four-parametric, or even empirical distributions?One distribution family for all cells or selection of “best” distribution based on quality of fit?Mixing internal and external dataHow much weight is given to internal and external data?How to combine internal and external data?Deutsche Bank’s AMA Model · page 15
RiskAnalytics& InstrumentsSeverities in AMA at DBRange of distribution and choice of distribution familyIn many cells, data characteristics are different for small and big lossesDifferent distributions for body and tail–Body: non-parametric (empirical) distribution –Tail: modified technique from Extreme Value Theory for tail modellingEmpirical and parametric distributions are combined via a weighted sum applied to the cumulative distribution functionsMixing internal and external dataInternal data for calibrating body of distributionInternal and external data for calibrating tailDeutsche Bank’s AMA Model · page 16
RiskAnalytics& InstrumentsCore Idea: Piecewise Defined Severity ,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,00010,000,000,000x (log scale)First section: given by empiric distribution of cell specific internal dataMid section: given by weighted average ofempiric distribution of cell specific internal dataempiric distribution of cell specific external and scenario dataTail section: given by weighted average ofempiric distribution of cell specific internal dataempiric distribution of cell specific external and scenario dataparametric distribution calibrated on all data >= 50mnDeutsche Bank’s AMA Model · page 17P( Loss >= x )
RiskAnalytics& InstrumentsRC before QAModelling InsuranceCorrelation / Group-LevelAggregate distributionDiversificationNet LossesBusiness DivisionInsuranceXGross LossesSeverityFrequencyScenario Bank’s AMA Model · page 18EventType
RiskAnalytics& InstrumentsInsurance in AMA at DBNet losses Net LossesInsuranceGross LossesMappingPoliciesInsurance calculation processOR event types are mapped to insurance policiesInsurance policies are modelled individually, . by specifying deductible, limits and haircutSimulated Insurance payment is calculated for each of the gross losses simulated gross losses separatelyDeutsche Bank’s AMA Model · page 19
RiskAnalytics& InstrumentsInsurance MappingOR eventtypesInsurancepoliciesFidelityFraudBurglary, Theft, RobberyPropertyDamage80%Infrastructure10%Service & Elec. Break-Down10%Execution, Delivery & Process ManagementGeneral LiabilityClients, Products & Business PracticesProfessional LiabilityEmployersPracticeLiabilityEmployment Practices & Workplace SafetyNot insuredDeutsche Bank’s AMA Model · page 20
RiskAnalytics& InstrumentsModelling Insurance ContractsDeductible: amount the bank has to cover by itselfCap: maximum amount compensated by the insurerCompensation,Net lossCompensationCap (c)Additional featuresAggregate capsNet Haircuts (regulatory lossrequirements)Deductible(d)Gross loss(x)Cap (c)Deductible(d)Deductible + Capmin(c,max(x−d,0))Deutsche Bank’s AMA Model · page 21
RiskAnalytics& InstrumentsRC before QAModelling DependenceCorrelation / Group-LevelAggregate distributionDiversificationNet LossesBusiness DivisionInsuranceXGross LossesSeverityFrequencyScenario Bank’s AMA Model · page 22EventType
RiskAnalytics& InstrumentsAnalyzing DependenceDependence in a bottom-up LDAWithin cells–Dependence between the occurrence of loss events –Dependence between the frequency distribution and the severity distribution–Dependence between the severity samplesBetween cells–Dependence between the frequency distributions–Dependence between the severity distributionsStatistical analyses performed at Deutsche BankBased on internal loss dataIdentification of dependence between–occurrence of loss events within a cell => Frequency distribution not Poisson–frequency distributions in different cells=> Copula applied to frequenciesDeutsche Bank’s AMA Model · page 23
RiskAnalytics& InstrumentsDependence in AMA at DBFrequenciesGaussian copula applied to frequency distributionsSeverities Sum of split losses Severities of different loss events are independentExample: Gaussian copula applied to a Poisson and a Negative Binomial factor in uncorrelatedGaussian copulaDeutsche Bank’s AMA Model · page 24
RiskAnalytics& InstrumentsRC before QACalculation of CapitalCorrelation / Group-LevelAggregate distributionDiversificationNet LossesBusiness DivisionInsuranceXGross LossesSeverityFrequencyScenario Bank’s AMA Model · page 25EventType
RiskAnalytics& InstrumentsCalculation and Allocation of Risk CapitalRisk CapitalUnexpected LossExpected ShortfallAverageExpected LossAggregate Loss DistributionAggregate loss distribution:Monte Carlo simulationEconomic Capital:% Quantileminus Expected LossRegulatory Capital:% Quantileminus Expected LossCapital allocationCell level: Expected Shortfall allocationDivisional level: Aggregation of EC in divisional cells plus proportional contributions of Group cellsDeutsche Bank’s AMA Model · page 26ProbabilityLoss ThresholdQuantile
RiskAnalytics& InstrumentsQualitative RC before QAAllocationAdjustment(by model)Group-LevelAggregate distributionCorrelationRC before QANet LossesInsuranceBusiness DivisionsKRIGross LossesDataQualitative SATAdjustment Data(BE&ICF)SeverityFrequency/ DependencyBusiness DivisionsScenario after QADataDataDeutsche Bank’s AMA Model · page 27
RiskAnalytics& InstrumentsQualitative Adjustment in DB’s OR ModelQualitative adjustment applied to contributory capital of business lines–QA is separated from the quantitative capital calculation–Simple and transparent but difficult to justify with statisticalmeansMain facts on QA–Risk indicators and self assessment are main components–QA score (applied on BL /ET level) plus penalty component (inappropriate loss data collection, KRI/SAT minimum standards, etc.)–Insurance OR capital may be adjusted by +40% to -40% –Measurement of risk sensitivity and coverage determines rangeKey risk indicators–Global KRIs: HR, BCM, open issues (Audit, SOX, db-Track), NPA, Technology Risk–Business specific KRIs, . nostroreconciliations, outstanding confirmations, average processing time of customer complaintsDeutsche Bank’s AMA Model · page 28
RiskAnalytics& at components of an LDA : model validationDeutsche Bank’s AMA Model · page 29
RiskAnalytics& InstrumentsValidationBasic properties of LDA modelVariance analysisLoss distributions for heavy-tailed severitiesSensitivity analysis of basic components of LDA modelsFrequenciesSeveritiesDependenceInsuranceImpact analysis of stress scenariosBacktestingand benchmarking Benchmarkingthe tail of the aggregate loss distribution against individual data pointsDeutsche Bank’s AMA Model · page 30
RiskAnalytics& InstrumentsVariance AnalysisCell levelVariance analysis–does not provide information on quantilesof loss distribution –but: quantifies impact of frequencies and severities on volatility of aggregate losses–is independent of specific distribution assumptionsVariance of aggregate losses (Fand S: frequency and severity distribution):2E(F)⋅Var(S)+Var(F)⋅E(S)ConclusionImportance of frequency distribution depends on relationship of Var(F)/E(F)2(frequency vol) and Var(S)/E(S)(severity vol)In high impact cells, the volatility of severities dominates andthe actual form of the frequency distribution is of minor importance:2E(F)⋅Var(S)+Var(F)⋅E(S)Deutsche Bank’s AMA Model · page 31
RiskAnalytics& InstrumentsVariance AnalysisGroup levelFrequency correlationsVariance of loss distribution at Group levelmm2E(F)⋅Var(S)+Var(F)⋅E(S)+Cov(F,F)⋅E(S)⋅E(S) ƒ ƒjjjjjkjkj=1j,k=1,j≠kVariance in the homogeneous model (c:homogeneous correlation coefficient)2m⋅(E(F)⋅Var(S)+Var(F)⋅E(S)⋅(c⋅(m−1)+1))Impact of frequency correlations depends onnumber of (relevant) cells mand 2relationship of Var(F)/E(F)(frequency vol) and Var(S)/E(S)(severity vol)In general, the impact of frequency correlations is rather limited and less significant than the impact of correlations of severities or loss distributions Deutsche Bank’s AMA Model · page 32
RiskAnalytics& InstrumentsLoss Distributions for Heavy-Tailed SeveritiesSubexponentialdistributionsHeavy-tailed: tail decays to 0 slower than any exponential Exp[a*x], a<0Tail of the sum of subexponentialvariables has the same order of magnitude as tail of the maximum:P(X+...+X>x)1nlim=1x→∞P(max(X,...,X)>x)1nAggregate loss distributions of subexponentialseveritiesLet Fbe a frequency distributionSthe distribution function of a subexponentialseverityGthe distribution function of the aggregate loss distributionUnder general conditions on F (satisfied by Poisson and Negative Binomial):G(x)lim=E(F), where S(x):=1−S(x)x→∞S(x)Deutsche Bank’s AMA Model · page 33
RiskAnalytics& InstrumentsSensitivity Analysis of Basic LDA ComponentsBased on theoretical results and experience with Deutsche Bank’sLDA modelFrequency distributions–Mean of frequency distribution is important–Shape has limited impact on capital in cells with fat-tailed severities–Shape has limited impact on Group capitalSeverity distributions–Weights and techniques for combining different data sources are important–Significant impact of distribution assumptions for severity tails and tail probabilitiesDependence–Impact depends on the level where dependence is modelled, . frequencies, severities or aggregate losses–Limited impact of frequency correlationsDeutsche Bank’s AMA Model · page 34
RiskAnalytics& InstrumentsSensitivity Analysis of Insurance ModelOR eventtypesInsurancepoliciesFidelityFraudBurglary, Theft, RobberyPropertyDamageInfrastructureService & Elec. Break-DownExecution, Delivery & Process ManagementGeneral LiabilityProfessional LiabilityClients, Products & Business PracticesEmployersPracticeLiabilityEmployment Practices & Workplace SafetyNot insuredClients, Products & Business Practices consumes most of the capital–Impact of mapping percentages to insurance contracts–Most severe losses fall under Professional Liability: single limit of PL is particularly importantHigher reduction (in percentage) for median (EL) than for high quantiles(EC and RC)Insurance may cause reallocation of capital between different event typesDeutsche Bank’s AMA Model · page 35
RiskAnalytics& InstrumentsStressing Loss DataMethodology: Add (remove) internal and/or external losses and analyze impacton capitalImpact on cell capitalStress 200mn loss in a Fraud / BL on / BL 4: +-BL 4:+:+ on divisional capital40,00BL 435,0030,0025,0020,0015,0010,005,00-BL 5-5,00BL 3BL 2BL 1-10,00BL 6Deutsche Bank’s AMA Model · page 36
RiskAnalytics& InstrumentsBacktestingand BenchmarkingBacktesting–Sequential testing of a model against reality to check the accuracy of the predictions–Backtestingis frequently used for the validation of market risk models–In credit and operational risk, the inherent shortage of loss data severely restricts the application of backtestingtechniques to capital modelsBenchmarking–Comparison of a bank's operational risk capital charge against abank's close peers–Comparison of the AMA capital charge against the BIA or TSA capital charges–Comparison of the LDA model outputs against adverse extreme, butrealistic, scenariosThese tests help to provide assurance over the appropriateness of the level of capital but there are obvious limitationsDeutsche Bank’s AMA Model · page 37
RiskAnalytics& InstrumentsBenchmarkingTail of aggregate loss distribution versus individual data pointsBased on assumption that these tails have the same order of magnitude:–Tail of aggregate loss distribution calculated in a bottom-up LDA model–Tail of loss distribution directly specified at Group levelLoss distribution specified at Group level:–Take all losses (across business lines and event types) above a high threshold, say 1m, for the specification of a severity distribution S–Calculate the bank's average annual loss frequency nabove 1mUnder the assumption that Sis subexponential, identifyα−quantiles of the loss distribution S+...+S with1nα−quantiles of the maximum distribution max(S,...,S) with1n1−((1−α)/n)−quantiles of the severity distribution SDeutsche Bank’s AMA Model · page 38
RiskAnalytics& InstrumentsBenchmarking Result1-((1-alpha)/n) –quantilesof the severity distribution correspond to individual losses for appropriate alpha and nThe amount of loss data provides a limit for the confidence level that can be derived directly from the dataApplication of this method to DB's LDA modelDeutsche Bank’s AMA Model · page 39
