Report of the Tail Factor Working Party

Steven C. Herman, FCAS, MAAASteven C. Herman, FCAS, MAAA

San Diego, CaliforniaSan Diego, CaliforniaSeptember 10-11, 2007September 10-11, 2007

Tail Factor Working Party

Co-ChairsCo-ChairsSteven C. Herman Steven C. Herman

Mark R. Mark R. ShaplandShapland MembersMembers

Mohammed Q. AshabMohammed Q. Ashab Richard KollmarRichard KollmarJoseph A. BoorJoseph A. Boor Rasa V. McKeanRasa V. McKeanAnthony R. BustilloAnthony R. Bustillo Michael R. MurrayMichael R. MurrayDavid A. ClarkDavid A. Clark Bernard A. Bernard A.

PelletierPelletierRobert J. FoskeyRobert J. Foskey Anthony J. PipiaAnthony J. PipiaSejal HariaSejal Haria F. Douglas RyanF. Douglas RyanBertram A. HorowitzBertram A. Horowitz Scott G. SobelScott G. SobelGloria A. HubermanGloria A. Huberman

Tail Factor Working Party

Disclaimer:Disclaimer:While this paper is the product of a CAS working While this paper is the product of a CAS working party, its findings do not represent the official view of party, its findings do not represent the official view of the Casualty Actuarial Society. Moreover, while we the Casualty Actuarial Society. Moreover, while we believe the approaches we describe are very good believe the approaches we describe are very good examples of how to address the issue of estimating examples of how to address the issue of estimating development of loss and loss adjustment expense development of loss and loss adjustment expense payments from a given evaluation to ultimate payments from a given evaluation to ultimate disposition, we do not claim they are the only disposition, we do not claim they are the only acceptable ones, nor do they represent all possible acceptable ones, nor do they represent all possible applications of the specific methods presented.applications of the specific methods presented.

The MotivationThe Motivation• Tail factors used to estimate additional Tail factors used to estimate additional

development occurring development occurring – after the eldest maturity in a given loss development after the eldest maturity in a given loss development

triangle, or triangle, or – after the eldest credible link ratio. after the eldest credible link ratio.

• Over the years, many valuable contributions have Over the years, many valuable contributions have been made to the CAS literature describing been made to the CAS literature describing methods for calculating tail factors. methods for calculating tail factors.

• However no overall paper compiling these However no overall paper compiling these contributions existed.contributions existed.

The PaperThe Paper• The CAS Tail Factor Working Party The CAS Tail Factor Working Party

prepared this paper on the methods prepared this paper on the methods currently used by actuaries in estimating currently used by actuaries in estimating loss development tail factors. loss development tail factors.

• Standard terminology for discussing Standard terminology for discussing aspects of link ratios and tail aspects of link ratios and tail development is communicated within development is communicated within the paper. the paper.

The PaperThe Paper

• Included are Included are – Descriptions of advantages and Descriptions of advantages and

disadvantages of each method and disadvantages of each method and – Identification of what entities Identification of what entities

(companies, rating bureaus, or (companies, rating bureaus, or consulting firms) typically use each consulting firms) typically use each method.method.

Paper DescriptionPaper Description

• Organized by “Type” of MethodOrganized by “Type” of Method• Sections Describe:Sections Describe:

– Mechanics of each method,Mechanics of each method,– Examples for most methods,Examples for most methods,– Results of our Testing, andResults of our Testing, and– Results of our SurveysResults of our Surveys

Paper DescriptionPaper Description

• Standard Notation:Standard Notation:– Consistency,Consistency,– Started with Notation from Reserve Started with Notation from Reserve

Variability Working PartyVariability Working Party– Added new notation where lackingAdded new notation where lacking

• Summarize Areas for Future ResearchSummarize Areas for Future Research

Section DescriptionSection Description

• Bondy-Type MethodsBondy-Type Methods• Algebraic MethodsAlgebraic Methods• Benchmark MethodsBenchmark Methods• Open Claim MethodsOpen Claim Methods• Curve Fitting MethodsCurve Fitting Methods• Lifespan MethodsLifespan Methods• Miscellaneous MethodsMiscellaneous Methods

Bondy-Type Methods Bondy-Type Methods DescriptionDescription• Bondy MethodBondy Method

– Use last link ratio:Use last link ratio:• Modified Bondy MethodModified Bondy Method

– Double or square:Double or square:• Generalized Bondy Method (Weller) Generalized Bondy Method (Weller)

– For 0<B<1: For 0<B<1: • Fully Generalized Bondy Method (Gile)Fully Generalized Bondy Method (Gile)

– Let Vary by Accident YearLet Vary by Accident Year

)1/()()()()1(2 BBBB dfdfdfdF

2)()1( dfdF

Bondy-Type Methods Bondy-Type Methods DescriptionDescription• AdvantagesAdvantages

– Simple to ImplementSimple to Implement– Pattern Described with One FactorPattern Described with One Factor– Only Requires Cumulative Paid DataOnly Requires Cumulative Paid Data

• DisadvantagesDisadvantages– Not Always Useful for Incurred DataNot Always Useful for Incurred Data– Will Fail with Increasing DevelopmentWill Fail with Increasing Development– May Fail with “More Complicated” PatternsMay Fail with “More Complicated” Patterns

Algebraic Methods DescriptionAlgebraic Methods Description

• Equalizing Paid & Incurred Loss Equalizing Paid & Incurred Loss EstimatesEstimates– Use Cumulative Incurred / Cumulative PaidUse Cumulative Incurred / Cumulative Paid

• Boor’s MethodBoor’s Method– Adjust Case ReservesAdjust Case Reserves

• Mueller’s MethodMueller’s Method– Adjust Incremental FactorsAdjust Incremental Factors

• NCCI MethodNCCI Method

Algebraic Methods DescriptionAlgebraic Methods Description

• AdvantagesAdvantages– Simple to ImplementSimple to Implement– Only Requires Cumulative Data Only Requires Cumulative Data – Statistically UnbiasedStatistically Unbiased

• DisadvantagesDisadvantages– May Not be Sophisticated EnoughMay Not be Sophisticated Enough– Subject to Case Reserve DistortionsSubject to Case Reserve Distortions– Some Methods Not Generally Well KnownSome Methods Not Generally Well Known

Benchmark Methods Benchmark Methods DescriptionDescription• Benchmark Development / Link Benchmark Development / Link

RatiosRatios• Adjusted Benchmark Development / Adjusted Benchmark Development /

Link RatiosLink Ratios– Use Link Ratios to Adjust Tail FactorUse Link Ratios to Adjust Tail Factor

• Benchmark Average SeverityBenchmark Average Severity• Benchmark Adjusted by Claims AuditBenchmark Adjusted by Claims Audit

Benchmark Methods Benchmark Methods DescriptionDescription• Advantages:Advantages:

– Supplement when Little DataSupplement when Little Data– Adds CredibilityAdds Credibility– Various Degrees of SophisticationVarious Degrees of Sophistication

• Disadvantages:Disadvantages:– Need Similar DataNeed Similar Data– Claim Handling ProceduresClaim Handling Procedures– Relative Case Reserve StrengthRelative Case Reserve Strength

Open Claim Methods Open Claim Methods DescriptionDescription• Maximum Possible LossMaximum Possible Loss• Judgment of Open Claim Costs / AuditJudgment of Open Claim Costs / Audit

Open Claim Methods Open Claim Methods DescriptionDescription• Advantages:Advantages:

– Incorporates Particulars of Open ClaimsIncorporates Particulars of Open Claims– Uses Knowledge of Claim StaffUses Knowledge of Claim Staff– Can Provide BoundsCan Provide Bounds

• Disadvantages:Disadvantages:– Requires Access to Individual ClaimsRequires Access to Individual Claims– Subject to Judgment/Availability of AuditorsSubject to Judgment/Availability of Auditors– May Underestimate Severe or IBNR/Reopened May Underestimate Severe or IBNR/Reopened

ClaimsClaims

Curve Fitting Methods Curve Fitting Methods Description Description • Exponential DecayExponential Decay

– Constant Rate of Factor DecayConstant Rate of Factor Decay• McClenahan’s MethodMcClenahan’s Method

– Constant Monthly Incremental Paid DecayConstant Monthly Incremental Paid Decay• Skurnick’s MethodSkurnick’s Method

– Simplify Using Annual DecaySimplify Using Annual Decay• Sherman’s MethodSherman’s Method

– Use “Inverse Power” CurvesUse “Inverse Power” Curves• England-Verrall MethodEngland-Verrall Method

– Smooth & Extrapolate Incremental DataSmooth & Extrapolate Incremental Data

Curve Fitting Methods Curve Fitting Methods Description Description • Advantages:Advantages:

– Straightforward & IntuitiveStraightforward & Intuitive– Extrapolate Beyond End of DataExtrapolate Beyond End of Data– Various Levels of SophisticationVarious Levels of Sophistication

• Disadvantages:Disadvantages:– May Underestimate Tail for Long-Tail LinesMay Underestimate Tail for Long-Tail Lines– Sub-Optimal If Pattern Not ConsistentSub-Optimal If Pattern Not Consistent– Sometimes No Closed Form SolutionSometimes No Closed Form Solution

Lifespan Methods Description Lifespan Methods Description

• Static Mortality MethodStatic Mortality Method– Frequency / Severity Using Mortality RatesFrequency / Severity Using Mortality Rates

• Trended Mortality MethodTrended Mortality Method– Greatest Impact on “Distant” YearsGreatest Impact on “Distant” Years

• Sherman-Diss MethodSherman-Diss Method– Separate Impact of Inflation & MortalitySeparate Impact of Inflation & Mortality

• Corro’s MethodCorro’s Method– Modeling of “Pension” ClaimsModeling of “Pension” Claims

Lifespan Methods Description Lifespan Methods Description

• Advantages:Advantages:– Extrapolate “Very-Long” TailExtrapolate “Very-Long” Tail– Can Include “Increasing” FactorsCan Include “Increasing” Factors– Detailed Assumptions/Some Non-Detailed Assumptions/Some Non-

SubjectiveSubjective• Disadvantages:Disadvantages:

– More ComplexMore Complex– Need “Very Old” Data to ParameterizeNeed “Very Old” Data to Parameterize– Need Specific Mortality RatesNeed Specific Mortality Rates

Miscellaneous Methods Miscellaneous Methods Description Description • Restating Historical Experience via a Restating Historical Experience via a

Claims AuditClaims Audit– Adjust for ChangesAdjust for Changes

Miscellaneous Methods Miscellaneous Methods Description Description • Advantages:Advantages:

– Improves “Other” MethodsImproves “Other” Methods– Adjustments Readily UnderstoodAdjustments Readily Understood– Add Claim Professional JudgmentAdd Claim Professional Judgment

• Disadvantages:Disadvantages:– Difficult to Reconstruct Old Claim FilesDifficult to Reconstruct Old Claim Files– Auditor Must Ignore Prior DevelopmentAuditor Must Ignore Prior Development– Auditor Must Evaluate at Multiple PointsAuditor Must Evaluate at Multiple Points