CAS Annual MeetingCAS Annual Meeting New Orleans, LANew Orleans, LA

November 10, 2003November 10, 2003Jonathan Hayes, ACAS, MAAAJonathan Hayes, ACAS, MAAA

UNCERTAINTY AROUND UNCERTAINTY AROUND MODELED LOSS ESTIMATESMODELED LOSS ESTIMATES

AgendaAgenda ModelsModels

Model ResultsModel Results Confidence BandsConfidence Bands

DataData Issues with DataIssues with Data Issues with InputsIssues with Inputs Model OutputsModel Outputs

Company ApproachesCompany Approaches Role of JudgmentRole of Judgment ConclusionsConclusions

Florida HurricaneFlorida Hurricane

Amounts in Millions USD

Return Period A B C 20 249 217 233 50 593 467 545 100 1,056 757 820 250 1,924 1,148 1,197

Annual Average 58.9 46.8 44.3

Mean, unitized 126 100 95 100, unitized 139 100 108 250, unitized 168 100 104

100/20 424% 349% 352%250/100 182% 152% 146%

Model

Florida HurricaneFlorida Hurricane

Event Loss A B C

250 95.0% 95.4% 95.2%500 97.2% 98.4% 97.6%750 98.3% 99.0% 98.7%

1000 98.9% 99.4% 99.3%

Non-Exceedance Probability (Approx)

Amounts in Millions USD

Modeled Event LossModeled Event LossSample Portfolio, Total EventSample Portfolio, Total Event

Modeled Event LossModeled Event LossBy State DistributionBy State Distribution

Modeled Event LossModeled Event LossBy County Distribution, State SBy County Distribution, State S

AgendaAgenda ModelsModels

Model ResultsModel Results Confidence BandsConfidence Bands

DataData Issues with DataIssues with Data Issues with InputsIssues with Inputs Model OutputsModel Outputs

Company ApproachesCompany Approaches Role of JudgmentRole of Judgment ConclusionsConclusions

Types Of UncertaintyTypes Of Uncertainty(In Frequency & Severity)(In Frequency & Severity)

Uncertainty (not randomness)Uncertainty (not randomness) Sampling ErrorSampling Error

100 years for hurricane100 years for hurricane Specification ErrorSpecification Error

FCHLPM sample dataset (1996) 1 in 100 OEP of 31m, 38m, FCHLPM sample dataset (1996) 1 in 100 OEP of 31m, 38m, 40m & 57m w/ 4 models40m & 57m w/ 4 models

Non-sampling ErrorNon-sampling Error El Nino Southern OscillationEl Nino Southern Oscillation

Knowledge UncertaintyKnowledge Uncertainty Time dependence, cascading, aseismic shift, Time dependence, cascading, aseismic shift,

poisson/negative binomialpoisson/negative binomial Approximation ErrorApproximation Error

Res Re cat bond: 90% confidence interval, process risk Res Re cat bond: 90% confidence interval, process risk only, of +/- 20%, per modeling firmonly, of +/- 20%, per modeling firm

Source: Major, Op. Cit..

Frequency-Severity UncertaintyFrequency-Severity UncertaintyFrequency Uncertainty (Miller)Frequency Uncertainty (Miller)

Frequency UncertaintyFrequency Uncertainty Historical set: 96 years, 207 hurricanesHistorical set: 96 years, 207 hurricanes Sample mean is 2.16Sample mean is 2.16 What is range for true mean?What is range for true mean?

Bootstrap methodBootstrap method New 96-yr sample sets: Each sample set New 96-yr sample sets: Each sample set

is 96 draws, with replacement, from is 96 draws, with replacement, from originaloriginal

Review ResultsReview Results

Frequency BootstrappingFrequency Bootstrapping Run 500 resamplings and graph relative Run 500 resamplings and graph relative

to theoretical t-distributionto theoretical t-distribution

Source: Miller, Op. Cit.

Frequency Uncertainty StatsFrequency Uncertainty Stats

Standard error (SE) of the mean:Standard error (SE) of the mean:

0.159 historical SE 0.159 historical SE 0.150 theoretical SE, assuming 0.150 theoretical SE, assuming

Poisson, i.e., (lambda/n)^0.5Poisson, i.e., (lambda/n)^0.5

Hurricane Freq. UncertaintyHurricane Freq. UncertaintyBack of the EnvelopeBack of the Envelope

Frequency Uncertainty OnlyFrequency Uncertainty Only 96 Years, 207 Events, 3100 coast miles96 Years, 207 Events, 3100 coast miles 200 mile hurricane damage diameter200 mile hurricane damage diameter 0.139 is avg annl # storms to site0.139 is avg annl # storms to site SE = 0.038, SE = 0.038, assuming Poisson frequencyassuming Poisson frequency

90% CI is loss +/- 45%90% CI is loss +/- 45% i.e., (1.645 * 0.038) / 0.139i.e., (1.645 * 0.038) / 0.139

Frequency-Severity UncertaintyFrequency-Severity UncertaintySeverity Uncertainty (Miller)Severity Uncertainty (Miller)

Parametric bootstrapParametric bootstrap Cat model severity for some portfolio Cat model severity for some portfolio Fit cat model severity to parametric modelFit cat model severity to parametric model Perform X draws of Y severities, where X Perform X draws of Y severities, where X

is number of frequency resamplings and Y is number of frequency resamplings and Y is number of historical hurricanes in setis number of historical hurricanes in set

Parameterize the new sampled severitiesParameterize the new sampled severities Compound with frequency uncertaintyCompound with frequency uncertainty Review confidence bandsReview confidence bands

OEP Confidence BandsOEP Confidence Bands

Source: Miller, Op. Cit.

Model 1 in 50 1 in 100 1 in 250

A 127 139 168B 100 100 100C 117 104 108

FL HURRICANE EXAMPLE, REVISITED

OEP Confidence BandsOEP Confidence Bands At 80-1,000 year return, range fixes to 50% to At 80-1,000 year return, range fixes to 50% to

250% of best estimate OEP250% of best estimate OEP Confidence band grow exponentially at Confidence band grow exponentially at

frequent OEP points because expected loss frequent OEP points because expected loss goes to zerogoes to zero

NotesNotes Assumed stationary climateAssumed stationary climate Severity parameterization may introduce errorSeverity parameterization may introduce error Modelers’ “secondary uncertainty” may overlap Modelers’ “secondary uncertainty” may overlap

here, thus reducing rangehere, thus reducing range Modelers’ severity distributions based on more Modelers’ severity distributions based on more

than just historical data setthan just historical data set

AgendaAgenda ModelsModels

Model ResultsModel Results Confidence BandsConfidence Bands

DataData Issues with DataIssues with Data Issues with InputsIssues with Inputs Model OutputsModel Outputs

Company ApproachesCompany Approaches Role of JudgmentRole of Judgment ConclusionsConclusions

Data Collection/InputsData Collection/Inputs Is this all the subject data?Is this all the subject data?

All/coastal statesAll/coastal states Inland Marine, Builders Risk, APD, Dwelling FireInland Marine, Builders Risk, APD, Dwelling Fire Manual policiesManual policies

General level of detailGeneral level of detail County/zip/streetCounty/zip/street Aggregated dataAggregated data

Is this all the needed policy detail?Is this all the needed policy detail? Building location/billing locationBuilding location/billing location Multi-location policies/bulk dataMulti-location policies/bulk data Statistical Record vs. policy systemsStatistical Record vs. policy systems Coding of endorsementsCoding of endorsements

Sublimits, wind exclusions, IMSublimits, wind exclusions, IM Replacement cost vs. limitReplacement cost vs. limit

More Data IssuesMore Data Issues

Deductible issuesDeductible issues Inuring/facultative reinsuranceInuring/facultative reinsurance Extrapolations & defaultsExtrapolations & defaults Blanket policiesBlanket policies HPRHPR Excess policiesExcess policies

Model OutputModel Output Data Imported/Not ImportedData Imported/Not Imported Geocoded/Not GeocodedGeocoded/Not Geocoded VersionVersion Perils RunPerils Run

Demand SurgeDemand Surge Storm SurgeStorm Surge Fire FollowingFire Following

DefaultsDefaults Construction MappingsConstruction Mappings Secondary CharacteristicsSecondary Characteristics

Secondary UncertaintySecondary Uncertainty DeductiblesDeductibles

AgendaAgenda ModelsModels

Model ResultsModel Results Confidence BandsConfidence Bands

DataData Issues with DataIssues with Data Issues with InputsIssues with Inputs Model OutputsModel Outputs

Company ApproachesCompany Approaches Role of JudgmentRole of Judgment ConclusionsConclusions

Company ApproachesCompany ApproachesAvailable ChoicesAvailable Choices

Output From:Output From: 2-5 Vendor Models2-5 Vendor Models

Detailed & Aggregate ModelsDetailed & Aggregate Models ECRA FactorsECRA Factors Experience, ParameterizedExperience, Parameterized

Select (weighted) AverageSelect (weighted) Average

Company ApproachesCompany ApproachesLoss CostsLoss Costs

Arithmetic averageArithmetic average Subject to changeSubject to change Significant u/w flexibilitySignificant u/w flexibility

Weighted averageWeighted average Weights by region, peril, class et al.Weights by region, peril, class et al. Weights determined by:Weights determined by:

Model reviewModel review Consultation with modeling firmsConsultation with modeling firms Historical event analysisHistorical event analysis JudgmentJudgment

Weight changes require formal sign-offWeight changes require formal sign-off

ConclusionsConclusions Cat Model Distributions VaryCat Model Distributions Vary

More than one point estimate usefulMore than one point estimate useful Point estimates may not be Point estimates may not be significantlysignificantly different different Uncertainty not insignificant but not insurmountableUncertainty not insignificant but not insurmountable What about uncertainty before cat models?What about uncertainty before cat models?

Data Inputs MatterData Inputs Matter Not mechanical processNot mechanical process Creating model inputs requires many decisionsCreating model inputs requires many decisions User knowledge and expertise criticalUser knowledge and expertise critical

Loss Cost Selection Methodology MattersLoss Cost Selection Methodology Matters # Models used more influential than weights used# Models used more influential than weights used

Judgment UnavoidableJudgment Unavoidable Actuaries already well-versed in its useActuaries already well-versed in its use

ReferencesReferences Bove, Mark C. et al.., “Effect of El Nino on US Landfalling Bove, Mark C. et al.., “Effect of El Nino on US Landfalling

Hurricanes, Revisited,”Hurricanes, Revisited,” Bulletin of the American Meteorological Bulletin of the American Meteorological SocietySociety, June 1998., June 1998.

Efron, Bradley and Robert Tibshirani, Efron, Bradley and Robert Tibshirani, An Introduction to the An Introduction to the BootstrapBootstrap, New York: Chapman & Hall, 1993., New York: Chapman & Hall, 1993.

Major, John A., “Uncertainty in Catastrophe Models,” Major, John A., “Uncertainty in Catastrophe Models,” Financing Financing Risk and ReinsuranceRisk and Reinsurance, International Risk Management Institute, , International Risk Management Institute, Feb/Mar 1999.Feb/Mar 1999.

Miller, David, “Uncertainty in Hurricane Risk Modeling and Miller, David, “Uncertainty in Hurricane Risk Modeling and Implications for Securitization,” Implications for Securitization,” CAS Forum,CAS Forum, Spring 1999. Spring 1999.

Moore, James F., “Tail Estimation and Catastrophe Security Moore, James F., “Tail Estimation and Catastrophe Security Pricing: Cat We Tell What Target We Hit If We Are Shooting in the Pricing: Cat We Tell What Target We Hit If We Are Shooting in the Dark”, Dark”, Wharton Financial Institutions CenterWharton Financial Institutions Center, 99-14., 99-14.