2010RPMBasic RatemakingWorkshop Session3:Introductionto - - PDF document

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2010RPMBasic RatemakingWorkshop Session3:Introductionto IncreasedLimitFactors PatThorpe,ACAS, PatThorpe,ACAS,MAAA MAAA Manager&AssociateActuary

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2010RPMBasic RatemakingWorkshop

Session3:Introductionto IncreasedLimitFactors

PatThorpe,ACAS, PatThorpe,ACAS,MAAA MAAA

Manager&AssociateActuary Manager&AssociateActuary

IncreasedLimits&RatingPlansDivision IncreasedLimits&RatingPlansDivision

InsuranceServicesOffice,Inc. InsuranceServicesOffice,Inc.

Agenda

  • Increasedvs.BasicLimitsRatemaking

Increasedvs.BasicLimitsRatemaking

  • LossSeverityDistributions

LossSeverityDistributions

  • EffectsofTrend

EffectsofTrend

  • ByLimitandLayer

ByLimitandLayer

  • ComponentsofILFCalculation

ComponentsofILFCalculation

  • MixedExponentialMethodology

MixedExponentialMethodology

  • DeductibleandLayerPricing

DeductibleandLayerPricing

CASExam5Reference:

BasicRatemaking Chapter11:SpecialClassification*

GeoffWerner, GeoffWerner,FCAS FCAS, ,MAAA MAAA Claudine ClaudineModlin Modlin, ,FCAS FCAS, ,MAAA MAAA EMB EMB AmericaLLC AmericaLLC

*CandidatesstudyingforExam5shouldrefertothe *CandidatesstudyingforExam5shouldrefertotheCAS CAS text,ratherthanthisworkshoppresentation. text,ratherthanthisworkshoppresentation.

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2

PriorCASExam5Paper:

IncreasedLimitsRatemaking ForLiabilityInsurance*

JosephM.Palmer, JosephM.Palmer,FCAS FCAS, ,MAAA MAAA, ,CPCU CPCU AssistantVicePresident&Actuary AssistantVicePresident&Actuary

IncreasedLimits&RatingPlansDivision IncreasedLimits&RatingPlansDivision

InsuranceServicesOffice,Inc. InsuranceServicesOffice,Inc.

*ThispaperprovidesadescriptionoftheMixedExponential *ThispaperprovidesadescriptionoftheMixedExponential MethodologyusedbyISO,andisreferencedinChapter11oftheBasic MethodologyusedbyISO,andisreferencedinChapter11oftheBasic Ratemakingtextmentionedinthepreviousslide. Ratemakingtextmentionedinthepreviousslide.

LiabilityLinesofBusiness

  • Premises/Operations

Premises/Operations andProducts(GL) andProducts(GL)

  • MedicalProfessional

MedicalProfessional

  • CommercialAuto

CommercialAuto

  • PersonalAuto

PersonalAuto

  • Farm

Farm

  • Personal(Individualor

Personal(Individualor withinHomeowner withinHomeowner Policy) Policy)

  • Management

Management Protection(D&O) Protection(D&O)

  • E

E:Commerce Commerce

  • LawyersProfessional

LawyersProfessional

  • BusinessOwners

BusinessOwners

  • Employment

Employment:Related Related Practices Practices

  • OtherProfessional

OtherProfessional

BasicLimitsRatemaking

  • Uselargevolumeoflossescappedatbasic

Uselargevolumeoflossescappedatbasic limitfordetailed,experience limitfordetailed,experience:based based analysis. analysis.

  • Abletoproducerelativitiesby

Abletoproducerelativitiesby

  • Class

Class

  • Territory

Territory

  • Tiers

Tiers

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SLIDE 3

3 IncreasedLimitsRatemaking

  • Needbroaderexperiencebase

Needbroaderexperiencebase

  • lowclaimvolumeathigherlimits

lowclaimvolumeathigherlimits

  • Grouplossexperienceforcredibility

Grouplossexperienceforcredibility

  • ClassGroups

ClassGroups

  • StateGroups

StateGroups

  • Countrywide

Countrywide

IncreasedLimitFactorDefinition

ExpectedCostsatthedesiredpolicylimit ExpectedCostsatthedesiredpolicylimit

_____________________________________________________________________________________________________________ _____________________________________________________________________________________________________________

ExpectedCostsattheBasicLimit ExpectedCostsattheBasicLimit

KEYASSUMPTION: KEYASSUMPTION:

ClaimFrequencyis ClaimFrequencyisindependent independent of

  • f

ClaimSeverity ClaimSeverity

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SLIDE 4

4

ThisallowsforILFstobedevelopedby ThisallowsforILFstobedevelopedby anexaminationoftherelative anexaminationoftherelative severitiesONLY severitiesONLY

) ( ) ( ) ( ) (

  • ×

× = ) ( ) (

  • =

LimitedAverageSeverity(LAS)

  • Definedastheaveragesizeofloss,where

Definedastheaveragesizeofloss,where alllossesarelimitedtoaparticularvalue. alllossesarelimitedtoaparticularvalue.

  • Thus,theILFcanbedefinedastheratioof

Thus,theILFcanbedefinedastheratioof twolimitedaverageseverities. twolimitedaverageseverities.

  • ILF(k)=LAS(k)

ILF(k)=LAS(k)÷ ÷ LAS(B) LAS(B)

Example

Losses Losses @100,000Limit @100,000Limit @1MillLimit @1MillLimit 50,000 50,000 75,000 75,000 150,000 150,000 250,000 250,000 1,250,000 1,250,000

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Example(cont’d)

Losses Losses @100,000Limit @100,000Limit @1MillLimit @1MillLimit 50,000 50,000 50,000 50,000 75,000 75,000 75,000 75,000 150,000 150,000 100,000 100,000 250,000 250,000 100,000 100,000 1,250,000 1,250,000 100,000 100,000

  • Example(cont’d)

Losses Losses @100,000Limit @100,000Limit @1MillLimit @1MillLimit 50,000 50,000 50,000 50,000 50,000 50,000 75,000 75,000 75,000 75,000 75,000 75,000 150,000 150,000 100,000 100,000 150,000 150,000 250,000 250,000 100,000 100,000 250,000 250,000 1,250,000 1,250,000 100,000 100,000 1,000,000 1,000,000

  • Example– CalculationofILF

TotalLosses TotalLosses $1,775,000 $1,775,000 Limitedto$100,000 Limitedto$100,000 (BasicLimit) (BasicLimit) $425,000/5 $425,000/5 =$85,000 =$85,000 Limitedto$1,000,000 Limitedto$1,000,000 $1,525,000/5 $1,525,000/5 =$305,000 =$305,000 IncreasedLimitsFactor IncreasedLimitsFactor (ILF) (ILF) $305,000/85,000 $305,000/85,000 =3.588 =3.588

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6 EmpiricalData: ILFs

Lower Lower Upper Upper Losses Losses Occs. Occs. Average Average 1 100,000 100,000 25,000,000 25,000,000 1,000 1,000 25,000 25,000 100,001 100,001 250,000 250,000 75,000,000 75,000,000 500 500 150,000 150,000 250,001 250,001 500,000 500,000 60,000,000 60,000,000 200 200 300,000 300,000 500,001 500,001 1Million 1Million 30,000,000 30,000,000 50 50 600,000 600,000 1Million 1Million : 15,000,000 15,000,000 10 10 1,500,000 1,500,000

EmpiricalData: ILFs

LAS@100,000 LAS@100,000 (25,000,000+760 (25,000,000+760× × 100,000) 100,000)÷ ÷ 1760 1760 =57,386 =57,386 LAS@1,000,000 LAS@1,000,000 (190,000,000+10 (190,000,000+10× × 1,000,000) 1,000,000)÷ ÷ 1760 1760 =113,636 =113,636 EmpiricalILF=1.98 EmpiricalILF=1.98

InsuranceLossDistributions

  • LossSeverityDistributionsareSkewed

LossSeverityDistributionsareSkewed

  • ManySmallLosses/FewerLargerLosses

ManySmallLosses/FewerLargerLosses

  • YetLargerLosses,thoughfewerinnumber,

YetLargerLosses,thoughfewerinnumber, areasignificantamountoftotaldollarsof areasignificantamountoftotaldollarsof loss. loss.

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7

LossDistribution: PDF

  • )

(

  • LossSize

ClaimDistribution: CDF

  • )

(

  • 1

Claims

Claimsvs.CumulativePaid$

  • $

$

) (

  • 1

Liability Property Claims

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8

AnovelapproachtounderstandingIncreased AnovelapproachtounderstandingIncreased LimitsFactorswaspresentedbyYoongS. LimitsFactorswaspresentedbyYoongS. LeeintheCASExam9paper LeeintheCASExam9paper– “TheMathematicsofExcessofLoss “TheMathematicsofExcessofLoss CoveragesandRetrospectiveRating CoveragesandRetrospectiveRating: A A GraphicalApproach” GraphicalApproach”

AGraphicalApproach LeeFigure

  • LimitedAverageSeverity

)] ( 1 [ ) (

  • −

+

∫

∫

−

  • )]

( 1 [

Sizemethod;‘vertical’ Layermethod;‘horizontal’

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SLIDE 9

9

SizeMethod

k

∫

× +

  • )

( ) (

) ( 1 ) (

  • −

= ∗

) (

  • 1
  • LossSize

LayerMethod

k

∫

  • )

( ) ( 1 ) (

  • −

= ∗

1

) (

  • LossSize

“Consistency”ofILFs

  • AsPolicyLimitIncreases

AsPolicyLimitIncreases

  • ILFsshouldincrease

ILFsshouldincrease

  • Butatadecreasingrate

Butatadecreasingrate

  • ExpectedCostsperunitofcoverageshould

ExpectedCostsperunitofcoverageshould notincreaseinsuccessivelyhigherlayers. notincreaseinsuccessivelyhigherlayers.

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10

Illustration:Consistency

1

) (

  • k3

k2 k1

  • LossSize

“Consistency”ofILFs: Example

Limit Limit ILF ILF Diff.Lim. Diff.Lim. Diff.ILF Diff.ILF Marginal Marginal 100,000 100,000 1.00 1.00 : : : 250,000 250,000 1.40 1.40 500,000 500,000 1.80 1.80 1Million 1Million 2.75 2.75 2Million 2Million 4.30 4.30 5Million 5Million 5.50 5.50

“Consistency”ofILFs: Example

Limit Limit ILF ILF Diff.Lim. Diff.Lim. Diff.ILF Diff.ILF Marginal Marginal 100,000 100,000 1.00 1.00 : : : 250,000 250,000 1.40 1.40 150 150 0.40 0.40 500,000 500,000 1.80 1.80 250 250 0.40 0.40 1Million 1Million 2.75 2.75 500 500 0.95 0.95 2Million 2Million 4.30 4.30 1,000 1,000 1.55 1.55 5Million 5Million 5.50 5.50 3,000 3,000 1.20 1.20

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11

“Consistency”ofILFs: Example

Limit Limit ILF ILF Diff.Lim. Diff.Lim. Diff.ILF Diff.ILF Marginal Marginal 100,000 100,000 1.00 1.00 : : : 250,000 250,000 1.40 1.40 150 150 0.40 0.40 .0027 .0027 500,000 500,000 1.80 1.80 250 250 0.40 0.40 .0016 .0016 1Million 1Million 2.75 2.75 500 500 0.95 0.95 .0019 .0019 2Million 2Million 4.30 4.30 1,000 1,000 1.55 1.55 .00155 .00155 5Million 5Million 5.50 5.50 3,000 3,000 1.20 1.20 .0004 .0004

“Consistency”ofILFs: Example

Limit Limit ILF ILF Diff.Lim. Diff.Lim. Diff.ILF Diff.ILF Marginal Marginal 100,000 100,000 1.00 1.00 : : : 250,000 250,000 1.40 1.40 150 150 0.40 0.40 .0027 .0027 500,000 500,000 1.80 1.80 250 250 0.40 0.40 .0016 .0016 1Million 1Million 2.75 2.75 500 500 0.95 0.95 .0019* .0019* 2Million 2Million 4.30 4.30 1,000 1,000 1.55 1.55 .00155 .00155 5Million 5Million 5.50 5.50 3,000 3,000 1.20 1.20 .0004 .0004

Inflation– LeveragedEffect

  • Generally,trendsforhigherlimitswillbe

Generally,trendsforhigherlimitswillbe higherthanbasiclimittrends. higherthanbasiclimittrends.

  • Also,ExcessLayertrendswillgenerally

Also,ExcessLayertrendswillgenerally exceedtotallimitstrends. exceedtotallimitstrends.

  • Requiressteadilyincreasingtrend.

Requiressteadilyincreasingtrend.

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12

k2

EffectofInflation

k1

  • 1

) (

  • Example:Effectof+10%Trend

@$100,000Limit

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) 50,000 100,000 100,000 100,000 100,000 100,000 550,000 55,000 100,000 100,000 100,000 100,000 100,000 555,000 +0.9% Pre:Trend($) Post:Trend($) @$100,000Limit

Example:Effectof+10%Trend @$250,000Limit

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) 50,000 250,000 250,000 250,000 250,000 250,000 1,300,000 55,000 250,000 250,000 250,000 250,000 250,000 1,305,000 +0.4% Pre:Trend($) Post:Trend($) @$250,000Limit

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13 Example:Effectof+10%Trend @$500,000Limit

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) 50,000 250,000 490,000 500,000 500,000 500,000 2,290,000 55,000 275,000 500,000 500,000 500,000 500,000 2,330,000 +1.7% Pre:Trend($) Post:Trend($) @$500,000Limit

Example:Effectof+10%Trend @$1,000,000Limit

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) 50,000 250,000 490,000 750,000 925,000 1,000,000 3,465,000 55,000 275,000 539,000 825,000 1,000,000 1,000,000 3,694,000 +6.6% Pre:Trend($) Post:Trend($) @$1,000,000Limit

ExampleSummary TrendEffectbyLimit

  • $100,000:+0.9%

$100,000:+0.9%

  • $250,000:+0.4%

$250,000:+0.4%

  • $500,000:+1.7%

$500,000:+1.7%

  • $1,000,000:+6.6%

$1,000,000:+6.6%

  • Overall:+10.0%

Overall:+10.0% Trends Trends increasewiththelimit. increasewiththelimit.

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14 Example:Effectof+10%Trend $150,000xs$100,000

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) : 150,000 150,000 150,000 150,000 150,000 750,000 : 150,000 150,000 150,000 150,000 150,000 750,000 0.0% Pre:Trend($) Post:Trend($) $150,000excessof$100,000layer

Example:Effectof+10%Trend $250,000xs$250,000

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) : : 240,000 250,000 250,000 250,000 990,000 : 25,000 250,000 250,000 250,000 250,000 1,025,000 +3.5% Pre:Trend($) Post:Trend($) $250,000excessof$250,000layer

Example:Effectof+10%Trend $500,000xs$500,000

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) : : : 250,000 425,000 500,000 1,175,000 : : 39,000 325,000 500,000 500,000 1,364,000 +16.1% Pre:Trend($) Post:Trend($) $500,000excessof$500,000layer

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15 Example:Effectof+10%Trend $1,000,000xs$1,000,000

50,000 250,000 490,000 750,000 925,000 1,825,000 Total RealizedTrend LossAmount($) : : : : : 825,000 825,000 : : : : 17,500 1,000,000 1,017,500 +23.3% Pre:Trend($) Post:Trend($) $1,000,000excessof$1,000,000layer

ExampleSummary TrendEffectbyExcessLayer

Layer Layer NetTrend NetTrend 150xs100 150xs100 +0.0% +0.0% 250xs250 250xs250 +3.5% +3.5% 500xs500 500xs500 +16.1% +16.1% 1,000xs1,000 1,000xs1,000 +23.3% +23.3% Overall Overall +10.0% +10.0%

CommercialAutomobile

ISOAggregateData: BITrends

CalendarYearDataThrough3/31/2008 CalendarYearDataThrough3/31/2008 (Quarterlyyear (Quarterlyyear:endingpoints) endingpoints)

  • $50,000

2.4% 3.0% $100,000 3.1% 3.6% $250,000 3.9% 4.5% $500,000 4.5% 5.3% $1,000,000 5.1% 5.9% Total 4.8% 6.3%

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SLIDE 16

16 ComponentsofILFs

  • ExpectedLoss

ExpectedLoss

  • AllocatedLossAdjustmentExpense

AllocatedLossAdjustmentExpense (ALAE) (ALAE)

  • UnallocatedLossAdjustmentExpense

UnallocatedLossAdjustmentExpense (ULAE) (ULAE)

  • ParameterRiskLoad

ParameterRiskLoad

  • ProcessRiskLoad

ProcessRiskLoad

ALAE

  • ClaimSettlementExpensethatcanbe

ClaimSettlementExpensethatcanbe assignedtoagivenclaim assignedtoagivenclaim::: ::: primarily primarily DefenseCosts DefenseCosts

  • LoadedintoBasicLimit

LoadedintoBasicLimit

  • ConsistentwithDutytoDefendInsured

ConsistentwithDutytoDefendInsured

  • ConsistentProvisioninAllLimits

ConsistentProvisioninAllLimits

ALAEProvisionDetermination

  • EstimateALAE/TotalLimitLossRatio

EstimateALAE/TotalLimitLossRatio

  • FindAverageLAS(LimitedAverage

FindAverageLAS(LimitedAverage Severity)AcrossLimits Severity)AcrossLimits

  • Multiply

Multiply

  • 0.062*10,941=678

0.062*10,941=678

  • UseALAEProvisionateachlimit

UseALAEProvisionateachlimit

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SLIDE 17

17 UnallocatedLAE– (ULAE)

  • AverageClaimsProcessingOverheadCosts

AverageClaimsProcessingOverheadCosts

  • e.g.SalariesofClaimsAdjusters

e.g.SalariesofClaimsAdjusters

  • PercentageLoadingintoILFsforAllLimits

PercentageLoadingintoILFsforAllLimits

  • AverageULAEasapercentageofLossesplus

AverageULAEasapercentageofLossesplus ALAE ALAE

  • LoadingBasedonFinancialData

LoadingBasedonFinancialData

  • RatioofULAEtoIncurredLoss+ALAE

RatioofULAEtoIncurredLoss+ALAE

  • 7.5%LoadinginUpcomingExample

7.5%LoadinginUpcomingExample

ProcessRiskLoad

  • ProcessRisk

ProcessRisk::: ::: theinherentvariabilityof theinherentvariabilityof theinsuranceprocess,reflectedinthe theinsuranceprocess,reflectedinthe differencebetweenactuallossesand differencebetweenactuallossesand expectedlosses. expectedlosses.

  • Chargevariesbylimit

Chargevariesbylimit

ParameterRiskLoad

  • ParameterRisk

ParameterRisk::: ::: theinherentvariabilityof theinherentvariabilityof theestimationprocess,reflectedinthe theestimationprocess,reflectedinthe differencebetweentheoretical(truebut differencebetweentheoretical(truebut unknown)expectedlossesandtheestimated unknown)expectedlossesandtheestimated expectedlosses. expectedlosses.

  • Chargevariesbylimit

Chargevariesbylimit

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18 IncreasedLimitsFactors(ILFs)

ILF@PolicyLimit(k)isequalto: ILF@PolicyLimit(k)isequalto: LAS(k)+ALAE(k)+ULAE(k)+RL(k) LAS(k)+ALAE(k)+ULAE(k)+RL(k)

____________________________________________________________________________________________________________ ____________________________________________________________________________________________________________

LAS(B)+ALAE(B)+ULAE(B)+RL(B) LAS(B)+ALAE(B)+ULAE(B)+RL(B)

ComponentsofILFs

1.74 1.74 135 135 1,432 1,432 974 974 678 678 12,308 12,308 2,000 2,000 1.55 1.55 123 123 803 803 905 905 678 678 11,392 11,392 1,000 1,000 1.37 1.37 108 108 419 419 821 821 678 678 10,265 10,265 500 500 1.19 1.19 94 94 193 193 723 723 678 678 8,956 8,956 250 250 1.00 1.00 79 79 76 76 613 613 678 678 7,494 7,494 100 100

ILF ILF PaRL PaRL PrRL PrRL ULAE ULAE ALAE ALAE LAS LAS Limit Limit

IssueswithConstructingILFTables

  • PolicyLimitCensorship

PolicyLimitCensorship

  • ExcessandDeductibleData

ExcessandDeductibleData

  • Dataisfromseveralaccidentyears

Dataisfromseveralaccidentyears

  • Trend

Trend

  • LossDevelopment

LossDevelopment

  • DataisSparseatHigherLimits

DataisSparseatHigherLimits

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SLIDE 19

19 UseofFittedDistributions

  • Mayaddresstheseconcerns

Mayaddresstheseconcerns

  • EnablescalculationofILFsforallpossible

EnablescalculationofILFsforallpossible limits limits

  • Smoothestheempiricaldata

Smoothestheempiricaldata

  • Examples:

Examples:

  • TruncatedPareto

TruncatedPareto

  • MixedExponential

MixedExponential

MixedExponentialMethodology

  • Trend

Trend

  • ConstructionofEmpiricalSurvival

ConstructionofEmpiricalSurvival Distributions Distributions

  • PaymentLagProcess

PaymentLagProcess

  • TailoftheDistribution

TailoftheDistribution

  • FittingaMixedExponentialDistribution

FittingaMixedExponentialDistribution

  • FinalLimitedAverageSeverities

FinalLimitedAverageSeverities

Trend

  • MultipleAccidentYearsareUsed

MultipleAccidentYearsareUsed

  • EachOccurrenceistrendedfromthe

EachOccurrenceistrendedfromthe averagedateofitsaccidentyeartooneyear averagedateofitsaccidentyeartooneyear beyondtheassumedeffectivedate. beyondtheassumedeffectivedate.

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20

EmpiricalSurvivalDistributions

  • PaidSettledOccurrencesareOrganizedby

PaidSettledOccurrencesareOrganizedby AccidentYearandPaymentLag. AccidentYearandPaymentLag.

  • Aftertrending,asurvivaldistributionis

Aftertrending,asurvivaldistributionis constructedforeachpaymentlag,usingdiscrete constructedforeachpaymentlag,usingdiscrete losssizelayers. losssizelayers.

  • ConditionalSurvivalProbabilities(CSPs)are

ConditionalSurvivalProbabilities(CSPs)are calculatedforeachlayer. calculatedforeachlayer.

  • SuccessiveCSPsaremultipliedtocreateground

SuccessiveCSPsaremultipliedtocreateground: upsurvivaldistribution. upsurvivaldistribution.

ConditionalSurvivalProbabilities

  • Theprobabilitythatanoccurrenceexceeds

Theprobabilitythatanoccurrenceexceeds theupperboundofadiscretelayer,given theupperboundofadiscretelayer,given thatitexceedsthelowerboundofthelayer thatitexceedsthelowerboundofthelayer isaCSP. isaCSP.

  • AttachmentPointmustbelessthanorequal

AttachmentPointmustbelessthanorequal tolowerbound. tolowerbound.

  • PolicyLimit+AttachmentPointmustbe

PolicyLimit+AttachmentPointmustbe greaterthanorequaltoupperbound. greaterthanorequaltoupperbound.

EmpiricalSurvivalDistributions

  • SuccessiveCSPsaremultipliedtocreate

SuccessiveCSPsaremultipliedtocreate ground ground:upsurvivaldistribution. upsurvivaldistribution.

  • Doneseparatelyforeachpaymentlag.

Doneseparatelyforeachpaymentlag.

  • Uses52(ormore)discretesizelayers.

Uses52(ormore)discretesizelayers.

  • Allowsforeasyinclusionofexcessand

Allowsforeasyinclusionofexcessand deductiblelossoccurrences. deductiblelossoccurrences.

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SLIDE 21

21

PaymentLagProcess

  • PaymentLag=

PaymentLag= (PaymentYear (PaymentYear– AccidentYear)+1 AccidentYear)+1

  • LossSizetendstoincreaseathigherlags

LossSizetendstoincreaseathigherlags

  • PaymentLagDistributionisConstructed

PaymentLagDistributionisConstructed

  • UsedtoCombineBy

UsedtoCombineBy:LagEmpiricalLoss LagEmpiricalLoss Distributionstogenerateanoverall Distributionstogenerateanoverall Distribution Distribution

  • ImplicitlyAccountsforLossDevelopment

ImplicitlyAccountsforLossDevelopment

PaymentLagProcess

  • PaymentLagDistributionusesthreeparameters

PaymentLagDistributionusesthreeparameters R1,R2,R3 R1,R2,R3 (Notethatlags5andhigherarecombined (Notethatlags5andhigherarecombined– C.Auto) C.Auto)

R3 = Expected%ofOcc.Paidinlag(n+1) Expected%ofOcc.Paidinlag(n+1) Expected%ofOcc.Paidinlag(n) Expected%ofOcc.Paidinlag(n) R2 = Expected%ofOcc.Paidinlag3 Expected%ofOcc.Paidinlag3 Expected%ofOcc.Paidinlag2 Expected%ofOcc.Paidinlag2 R1 = Expected%ofOcc.Paidinlag2 Expected%ofOcc.Paidinlag2 Expected%ofOcc.Paidinlag1 Expected%ofOcc.Paidinlag1

PaymentLagProcess

Acc.Year Acc.Year Lag1 Lag1 Occ Occ Lag2 Lag2 Occ Occ Ratioof Ratioof Lag2/1 Lag2/1 2002 2002 2,850 2,850 2003 2003 10,000 10,000 3,000 3,000 0.300 0.300 2004 2004 11,000 11,000 3,100 3,100 0.282 0.282 2005 2005 12,000 12,000 3,500 3,500 0.292 0.292 2006 2006 13,000 13,000 3,750 3,750 0.288 0.288 2007 2007 14,000 14,000 Total03 Total03:06 06 46,000 46,000 13,350 13,350 0.290 0.290

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SLIDE 22

22

LagWeights

  • Lag1wt.=1

Lag1wt.=1÷ ÷ k

  • Lag2wt.=R1

Lag2wt.=R1÷ ÷ k

  • Lag3wt.=R1

Lag3wt.=R1× × R2 R2÷ ÷ k

  • Lag4wt.=R1

Lag4wt.=R1× × R2 R2× × R3 R3÷ ÷ k

  • Lag5wt.=R1

Lag5wt.=R1× × R2 R2× × [R3 [R32 ÷ ÷ (1 (1: R3)] R3)]÷ ÷ k

  • Wherek=1+R1+[R1

Wherek=1+R1+[R1× × R2] R2]÷ ÷ [1 [1: R3] R3]

LagWeights

  • Represent%ofground

Represent%ofground:upoccurrencesin upoccurrencesin eachlag eachlag

  • Umbrella/Excesspoliciesnotincluded

Umbrella/Excesspoliciesnotincluded

  • R1,R2,R3estimatedviamaximum

R1,R2,R3estimatedviamaximum likelihood. likelihood.

TailoftheDistribution

  • Dataissparseathighlosssizes

Dataissparseathighlosssizes

  • Anappropriatecurveisselectedtomodel

Anappropriatecurveisselectedtomodel thetail(e.g.aTruncatedPareto). thetail(e.g.aTruncatedPareto).

  • Fittomodelthebehaviorofthedatainthe

Fittomodelthebehaviorofthedatainthe highestcredibleintervals highestcredibleintervals– thenextrapolate. thenextrapolate.

  • Smoothesthetailofthedistribution.

Smoothesthetailofthedistribution.

  • AMixedExponentialisthenfittothe

AMixedExponentialisthenfittothe resultingSurvivalDistributionFunction resultingSurvivalDistributionFunction

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SLIDE 23

23

  • Μeanparameter:X

Μeanparameter:X

  • PolicyLimit:PL

PolicyLimit:PL

SimpleExponential

) ( 1 ) (

  • −

= =

−

  • ]

1 [ ) (

  • −

− =

MixedExponential

  • WeightedAverageofExponentials

WeightedAverageofExponentials

  • EachExponentialhasTwoParameters

EachExponentialhasTwoParameters mean( mean(X Xi)andweight(w )andweight(wi)

  • Weightssumtounity

Weightssumtounity

*PL:PolicyLimit *PL:PolicyLimit

] [ ) (

∑

−

=

  • ]

1 [ ) (

∑

−

− =

  • 2008MethodologyChanges
  • ExpandedNumberofLayersEvaluated

ExpandedNumberofLayersEvaluated

  • SDFsandCSPsfor68

SDFsandCSPsfor68– 75layers, 75layers, VaryingbyLineofBusiness(was52) VaryingbyLineofBusiness(was52)

  • ProvidesEnhancedInformationandFlexibility

ProvidesEnhancedInformationandFlexibility forSmoothingtheTailoftheDistribution forSmoothingtheTailoftheDistribution

  • Highestmeannowlimitedto100M

Highestmeannowlimitedto100M

  • Allowssmoothfitsthroughthe100Mlimit

Allowssmoothfitsthroughthe100Mlimit

  • Previousmaximummeanwas10M(mostlines)

Previousmaximummeanwas10M(mostlines)

slide-24
SLIDE 24

24

MixedExponential

2007CommercialAutoI/LReview 2007CommercialAutoI/LReview

  • Numberofindividualexponentialsvaryby

Numberofindividualexponentialsvaryby stategroup/table stategroup/table

  • Rangebetweenfourandsevenexponentials

Rangebetweenfourandsevenexponentials

  • Highestmeanlimitedto10,000,000

Highestmeanlimitedto10,000,000

MixedExponential

2008CommercialAutoI/LReview 2008CommercialAutoI/LReview

  • Numberofindividualexponentialsvaryby

Numberofindividualexponentialsvaryby stategroup/table stategroup/table

  • Rangebetweennineandeleven

Rangebetweennineandeleven exponentials exponentials

  • Highestmeanlimitedto100,000,000

Highestmeanlimitedto100,000,000

  • AdditionalCSPlayersevaluated(68vs.52)

AdditionalCSPlayersevaluated(68vs.52)

SampleofActualFitted Distribution

Mean Mean Weight Weight 2,763 2,763 0.824796 0.824796 24,548 24,548 0.159065 0.159065 275,654 275,654 0.014444 0.014444 1,917,469 1,917,469 0.001624 0.001624 10,000,000 10,000,000 0.000071 0.000071

slide-25
SLIDE 25

25

Calculationof“Raw”ILF

] 1 [ ) (

∑

−

− =

  • *PL:PolicyLimit

*PL:PolicyLimit

494 , 7 ) 000 , 100 ( =

  • 392

, 11 ) 000 , 000 , 1 ( =

  • 52

. 1 494 , 7 392 , 11 ) 000 , 100 ( ) 000 , 000 , 1 ( = = =

  • LASCalculationDetails

Mean Mean 100KLAS 100KLAS 1MLAS 1MLAS Weight Weight 2,763 2,763 2,763 2,763 2,763 2,763 0.824796 0.824796 24,548 24,548 24,130 24,130 24,548 24,548 0.159065 0.159065 275,654 275,654 83,869 83,869 268,328 268,328 0.014444 0.014444 1,917,469 1,917,469 97,437 97,437 779,227 779,227 0.001624 0.001624 10,000,000 10,000,000 99,502 99,502 951,626 951,626 0.000071 0.000071

Wtd.Average Wtd.Average

7,494 7,494 11,392 11,392 1.000000 1.000000

Deductibles

  • TypesofDeductibles

TypesofDeductibles

  • LossEliminationRatio

LossEliminationRatio

  • ExpenseConsiderations

ExpenseConsiderations

slide-26
SLIDE 26

26

TypesofDeductibles

  • ReductionofDamages

ReductionofDamages

  • Insurerisresponsibleforlossesinexcessofthe

Insurerisresponsibleforlossesinexcessofthe deductible,uptothepointwhereaninsurer deductible,uptothepointwhereaninsurer paysanamountequaltothepolicylimit paysanamountequaltothepolicylimit

  • Aninsurermaypayforlossesinlayersabove

Aninsurermaypayforlossesinlayersabove thepolicylimit(But,totalamountpaidwillnot thepolicylimit(But,totalamountpaidwillnot exceedthelimit) exceedthelimit)

  • ImpairmentofLimits

ImpairmentofLimits

  • Themaximumamountpaidisthepolicylimit

Themaximumamountpaidisthepolicylimit minusthedeductible minusthedeductible

ImpairmentofLimitsExample

LossSize LossSize #of #of Claims Claims Total Total Losses Losses Average Average Loss Loss LossesNetofDeductible LossesNetofDeductible $100 $100 $200 $200 $500 $500 0to100 0to100 500 500 30,000 30,000 60 60 101to200 101to200 350 350 54,250 54,250 155 155 19,250 19,250 201to500 201to500 550 550 182,625 182,625 332 332 127,625 127,625 72,625 72,625 501+ 501+ 335 335 375,125 375,125 1120 1120 341,625 341,625 308,125 308,125 207,625 207,625 Total Total 1,735 1,735 642,000 642,000 370 370 488,500 488,500 380,750 380,750 207,625 207,625 LossEliminated LossEliminated 153,500 153,500 261,250 261,250 434,375 434,375 L.E.R. L.E.R. 0.239 0.239 0.407 0.407 .677 .677

Deductibles(example1)

  • Example1:

PolicyLimit:$100,000 Deductible:$25,000 OccurrenceofLoss:$100,000 Paymentis$75,000 ReductionduetoDed.is$25,000 Paymentis$75,000 ReductionduetoDed.is$25,000 !"!"# $!%&

slide-27
SLIDE 27

27

Deductibles(example2)

  • Example2:

PolicyLimit:$100,000 Deductible:$25,000 OccurrenceofLoss:$300,000 Paymentis$100,000 ReductionduetoDed.is$0 Paymentis$75,000 ReductionduetoDed.is$25,000 ' !"!"# $!"& !"

LiabilityDeductibles

  • ReductionofDamagesBasis

ReductionofDamagesBasis

  • Applytothirdpartyinsurance

Applytothirdpartyinsurance

  • Insurerhandlesallclaims

Insurerhandlesallclaims

  • LossSavings

LossSavings

  • NoLossAdjustmentExpenseSavings

NoLossAdjustmentExpenseSavings

  • DeductibleReimbursement

DeductibleReimbursement

  • RiskofNon

RiskofNon:Reimbursement Reimbursement

  • DiscountFactor

DiscountFactor

DeductibleDiscountFactor

  • TwoComponents

TwoComponents

  • LossEliminationRatio(LER)

LossEliminationRatio(LER)

  • CombinedEffectofVariable&Fixed

CombinedEffectofVariable&Fixed Expenses Expenses

  • ThisisreferredtoastheFixed

ThisisreferredtoastheFixed ExpenseAdjustmentFactor(FEAF) ExpenseAdjustmentFactor(FEAF)

slide-28
SLIDE 28

28 LossEliminationRatio

  • NetIndemnityCostsSaved

NetIndemnityCostsSaved– dividedby dividedby TotalBasicLimit/FullCoverageIndemnity TotalBasicLimit/FullCoverageIndemnity &LAECosts &LAECosts

  • DenominatorisExpectedBasicLimitLoss

DenominatorisExpectedBasicLimitLoss Costs Costs

LossEliminationRatio(cont’d)

  • Deductible(i)

Deductible(i)

  • PolicyLimit(j)

PolicyLimit(j)

  • Consider[LAS(i+j)

Consider[LAS(i+j): LAS(i)] LAS(i)]÷ ÷ LAS(j) LAS(j)

  • Thisrepresentscostsunderdeductibleasa

Thisrepresentscostsunderdeductibleasa fractionofcostswithoutadeductible. fractionofcostswithoutadeductible.

  • Oneminusthisquantityisthe(indemnity)LER

Oneminusthisquantityisthe(indemnity)LER

  • Equalto

Equalto [LAS(j) [LAS(j): LAS(i+j)+LAS(i)] LAS(i+j)+LAS(i)]÷ ÷ LAS(j) LAS(j)

PricingLiabilityDeductibles

  • CanUseFittedIndemnityDistributions

CanUseFittedIndemnityDistributions

  • EstimateCostinCoveredLayer

EstimateCostinCoveredLayer

  • RelatetoCostWithoutDeductible

RelatetoCostWithoutDeductible

slide-29
SLIDE 29

29

LimitedAverageSeverity: Layer

) ( ) ( ) (

1 1 2 2

2 1

  • ×

− × +

∫

∫

2 1

) (

  • )

( 1 ) (

  • −

= ∗

Sizemethod;‘vertical’ Layermethod;‘horizontal’

SizeMethod&LAS

) ( ) ( ) (

1 1 2 2

2 1

  • ×

− × +

∫

) ( 1 ) (

  • −

= ∗       × +

∫

2

2 2

) ( ) (

     × +

∫

1

1 1

) ( ) (

  • −

isequalto ) ( ) ( ) (

1 1 2 2

2 1

  • ×

− × +

∫

) ( 1

1

  • ×

−

∫

2 1

) (

  • )

(

2 2

  • ×

+

SizeMethod– LayerofLoss

) ( 1 ) (

  • −

= ∗

) (

  • 1
  • LossSize

k2 k1

slide-30
SLIDE 30

30

“LayerMethod”– LayerofLoss

∫

2 1

) (

  • )

( 1 ) (

  • −

= ∗

  • LossSize

1

) (

  • k1

k2

Summary

  • Increasedvs.BasicLimitsRatemaking

Increasedvs.BasicLimitsRatemaking

  • LossSeverityDistributions

LossSeverityDistributions

  • EffectsofTrend

EffectsofTrend

  • ByLimitandLayer

ByLimitandLayer

  • ComponentsofILFCalculation

ComponentsofILFCalculation

  • MixedExponentialMethodology

MixedExponentialMethodology

  • DeductibleandLayerPricing

DeductibleandLayerPricing PatThorpe PatThorpe Manager&AssociateActuary Manager&AssociateActuary InsuranceServicesOffice,Inc. InsuranceServicesOffice,Inc. 201 201:469 469:2537 2537 pthorpe@iso.com pthorpe@iso.com