3/19/2012 BasicRatemakingWorkshop: IntrotoIncreasedLimitFactors - - PDF document
3/19/2012 BasicRatemakingWorkshop: IntrotoIncreasedLimitFactors JaredSmollik FCAS,MAAA,CPCU IncreasedLimits&RatingPlansDivision,ISO March19,2012 Agenda BackgroundandNotation
JaredSmollik FCAS,MAAA,CPCU IncreasedLimits&RatingPlansDivision,ISO March19,2012 BackgroundandNotation OverviewofBasicandIncreasedLimits IncreasedLimitsRatemaking DeductibleRatemaking MixedExponentialProcedure(Overview)
BackgroundandNotation
ProbabilityDensityFunction(PDF)– fx
describestheprobabilitydensityofthe
theoreticalequivalentofahistogramof
empiricaldata Lossseveritydistributionsareskewed
afewlargelossesmakeupasignificant
portionofthetotallossdollars
CumulativeDistributionFunction(CDF)
describestheprobabilitythatarandom
variableX takesonvalueslessthanor equaltox
x
moment)
averagevalueofarandomvariable
∞ ∞
expectedvalueoftherandomvairable
limitedtoamaximumvalueofk
severity(LAS)whenworkingwithlosses
k k
OverviewofBasicandIncreasedLimits
Differentinsureds havedifferentcoverage needs,sothird7partyliabilitycoverageis
Typically,thelowestlevelofinsurance
andhigherlimitsarereferredtoas increasedlimits.
BasicLimitlosscostsarereviewedandfiledona regularbasis(perhapsannually)
alargervolumeoflossescappedatthebasic
limitcanbeusedforadetailedexperience analysis
experienceismorestablesincelarge,volatile
lossesarecappedandexcludedfromthe analysis Higherlimitsarereviewedlessfrequently
requiresmoredatavolume fewerpoliciesarewrittenathigherlimits largelossesarehighlyvariable
IncreasedLimitsRatemaking
BasicLimitdataaggregation
lossesarerestatedasifallpolicieswere
purchasedatthebasiclimit
basiclimitisusuallythefinancial
responsibilitylimitoracommonlyselected limit
ALAEisgenerallyuncapped
IncreasedLimitsdataaggregation
lossesarelimitedtoahigherlimit ALAEgenerallyremainsuncapped
theprocessofdevelopingchargesfor
expectedlossesathigherlimitsofliability
usuallyresultsinamultiplicativefactorto
beappliedtothebasiclimitlosscost,i.e. theincreasedlimitfactor(ILF)
AkeyassumptionofILratemakingisthat claimfrequencyisindependentofclaim severity
claimfrequencydoesnotdependon
policylimit
calculateILFs
b k b k b b k k
Forpracticalpurposes,theexpectedcosts includeafewcomponents:
limitedaverageseverity allocatedlossadjustmentexpenses unallocatedlossadjustmentexpenses riskload
WewillfocusmostlyonLAS,withsome discussionofALAE.
Thebasiclimitis$100k.Calculate ILF($1000k)giventhefollowingsetof ground7up,uncappedlosses. RecallILF(k)=E[X^k]/E[X^b].
Losses $50,000 $75,000 $150,000 $250,000 $1,250,000 Losses min{,$100k} min{,$1000k} $50,000 $50,000 $50,000 $75,000 $75,000 $75,000 $150,000 $100,000 $150,000 $250,000 $100,000 $250,000 $1,250,000 $100,000 $1,000,000
ILF(k)=E[X^k]/E[X^b] E[X^$100k]=$425,000/5=$85,000 E[X^$1000k] =$1,525,000/5=$305,000 ILF($1000k) =E[X^$1000k]/E[X^$100k]=3.59 Thebasiclimitis$25k.CalculateILF($125k) giventhefollowingsetoflosses.
Losses $5,000 $17,500 $50,000 $162,500 $1,250,000
Losses $5,000 $17,500 $50,000 $162,500 $1,250,000
SizeofLossmethod
individuallossesaregroupedbysizeinto
predeterminedintervals
theaggregatelosswithineachintervalis
limited,ifnecessary,tothelimitbeing reviewed
ALAEisaddedtotheaggregatelimited
loss
k
k
1
Loss Size
Layermethod
individuallossesareslicedintolayers
basedonpredeterminedintervals
foreachloss,theamountofloss
correspondingtoeachlayerisaddedto theaggregateforthatlayer
theaggregatelossforeachlayerupto
thelimitisaddedtogether
ALAEisaddedtotheaggregatelimited
loss
k
k
1
Loss Size
SizeMethod LayerMethod Advantages
conceptually straightforward datacanbeusedin calculationsimmediately morecomplicatedintegralis actuallygenerallyeasierto calculate computationallysimplefor calculating setsofincreasedlimit factors nointegrationdisadvantage whendataisgivennumerically, whichisgenerallythepractical case
Disadvantages
computationally intensivefor calculatingsetsofincreasedlimit factors unintuitive datamustbeprocessedsothat itcanbeusedincalculations S(x) isgenerallyamoredifficult functiontointegrate
IndividualLossIntervals (basiclimitis$100k) Aggregate LossesinInterval Numberof Claims in Interval LowerBound UpperBound $1 $100,000 $25,000,000 1,000 $100,001 $250,000 $75,000,000 500 $250,001 $500,000 $60,000,000 200 $500,001 $1,000,000 $30,000,000 50 $1,000,001 ∞ $15,000,000 10
[ ]
k k k X E × + = ∧
IndividualLossIntervals (basiclimitis$100k) Aggregate LossesinInterval Numberof Claims in Interval LowerBound UpperBound $1 $100,000 $25,000,000 1,000 $100,001 $250,000 $75,000,000 500 $250,001 $500,000 $60,000,000 200 $500,001 $1,000,000 $30,000,000 50 $1,000,001 ∞ $15,000,000 10
CalculateILF($1000k).
IndividualLossIntervals (basiclimitis$100k) Aggregate LossesinInterval Numberof Claims in Interval LowerBound UpperBound $1 $50,000 $8,400,000 200 $50,001 $100,000 $46,800,000 600 $100,001 $250,000 $64,000,000 400 $250,001 $500,000 $38,200,000 100 $500,001 ∞ $17,000,000 20
CalculateILF($250k)andILF($500k).
[ ]
× + = ∧ k k k k X E
IndividualLossIntervals (basiclimitis$100k) Aggregate Lossesin Interval Agg.ALAE
Interval Numberof Claimsin Interval L.Bound U.Bound $1 $100,000 $16,000,000 $100,000 200 $100,001 $300,000 $42,000,000 $500,000 350 $300,001 $500,000 $36,000,000 $800,000 90 $500,001 ∞ $3,000,000 $200,000 5
IndividualLossIntervals (basiclimitis$100k) Aggregate Lossesin Interval Agg.ALAE
Interval Numberof Claimsin Interval L.Bound U.Bound $1 $100,000 $16,000,000 $100,000 200 $100,001 $300,000 $42,000,000 $500,000 350 $300,001 $500,000 $36,000,000 $800,000 90 $500,001 ∞ $3,000,000 $200,000 5
CalculateILF($500k).
[ ]
k X E = ∧
LossLayer (basiclimitis$50k) Aggregate LossesinLayer Claims Reaching Layer LowerBound UpperBound $1 $50,000 $3,800,000 100 $50,001 $100,000 $2,000,000 50 $100,001 $250,000 $2,500,000 25 $250,001 ∞ $4,000,000 10
CalculateILF($250k).
LossLayer (basiclimitis$50k) Aggregate LossesinLayer Claims Reaching Layer LowerBound UpperBound $1 $50,000 $3,800,000 100 $50,001 $100,000 $2,000,000 50 $100,001 $250,000 $2,500,000 25 $250,001 ∞ $4,000,000 10
[ ]
= ∧ k k X E
LossLayer (basiclimitis$50k) Aggregate LossesinLayer (ALAE=$1.1M) Claims Reaching Layer LowerBound UpperBound $1 $50,000 $39,500,000 1,000 $50,001 $100,000 $32,000,000 800 $100,001 $250,000 $9,500,000 100 $250,001 ∞ $14,200,000 10
CalculateILF($250k).
LossLayer (basiclimitis$50k) Aggregate LossesinLayer (ALAE=$1.1M) Claims Reaching Layer LowerBound UpperBound $1 $50,000 $39,500,000 1,000 $50,001 $100,000 $32,000,000 800 $100,001 $250,000 $9,500,000 100 $250,001 ∞ $14,200,000 10
ConsistencyRule
Themarginalpremiumperdollarofcoverage shoulddecreaseasthelimitofcoverage increases.
ILFs shouldincreaseatadecreasingrate expectedcostsperunitofcoverageshould
notincreaseinsuccessivelyhigherlayers Inconsistencycanindicatethepresenceof anti7selection
higherlimitsmayinfluencethesizeofasuit,
award,orsettlement
Limit($000s) ILF 1ILF/1limit 25 1.00 – 50 1.60 0.0240 100 2.60 0.0200 250 6.60 0.0267 500 10.00 0.0136
k1 k2 k3 Loss Size
additionalmarginalcostforhigher limitsandcannotbelargerthan anylowerlayers.
Limit($000s) ILF 10 1.000 25 1.195 35 1.305 50 1.385 75 1.525 100 1.685 125 1.820 150 1.895 175 1.965 200 2.000 250 2.060 300 2.105 400 2.245 500 2.315
DeductibleRatemaking
Deductibleratemakingiscloselyrelatedto increasedlimitsratemaking
basedonthesameideaoflosslayers differenceliesinthelayersconsidered
Wewillfocusonthefixeddollardeductible
mostcommon simplest sameprinciplescanbeappliedtoother
typesofdeductibles
LossEliminationRatio(LER)
savingsassociatedwithuseofdeductible equaltoproportionofground7uplosseseliminatedby
deductible Expectedground7uploss
fullvaluepropertyortotallimitsliability=E[X]
Expectedlossesbelowdeductiblej
limitedexpectedloss=E[X^j]
Example:LER(j)=E[X^j]/E[X]
TheLER isusedtoderiveadeductible relativity(DR)
deductibleanalogofanILF factorappliedtothebasepremiumto
reflectadeductible Factordependson:
LER ofthebasedeductible LER ofthedesireddeductible
Example:
basedeductibleisfullcoverage(i.e.no
deductible)
insurancepolicywithdeductiblej
benefitsfromasavingsequaltoLER(j)
inthiscase,DR(j)=1– LER(j)
Ifthefullcoveragepremiumforauto physicaldamageis$1,000andthe customerwantsa$500deductible,we candeterminethe$500deductible premiumifweknowLER($500).Assume LER($500)=31%.
DR($500)=1– 0.31=0.69 $500deductiblepremium=0.69' $1,000
=$690
Calculatethe$5,000and$10,000 deductiblerelativitiesusingthefollowing ground7uplossesforunlimitedpolicies withnodeductibles.
Losses $2,000 $9,500 $18,000 $30,500 $75,000
Losses $2,000 $9,500 $18,000 $30,500 $75,000
Thepriorexamplesweresimplisticbecause thebasedeductibleswerefullcoverage. Amoregeneralizedformulacanbeusedto calculatedeductiblerelativitieswherethe basesdeductibleisnon7zero. Wedivideouttheeffectofthebase deductibleandmultiplybytheeffectofthe desireddeductible.Inotherwords,goback tothefullcoveragecaseandworkfrom there.
Thedeductiblerelativityfromthebase deductibled toanotherdeductiblej canbe expressedas: Example:
basedeductibleis$500andLER($500)=0.24 $250deductibleisdesiredandLER($250)=0.19 DR$500($250)=(1– 0.19)/(1– 0.24)=1.066
Thebasedeductibleforthiscoverageis $500andtheunlimitedaverageseverity is$5,000.Calculatethe$0,$250,$500, and$1000deductiblerelativities.
DR$500() $0 $0 $250 $240 $500 $470 $1,000 $900
MixedExponentialProcedure censorship– lossamountsareknownbut
theirvaluesarelimited
rightcensorship(fromabove)occurswhena lossexceedsthepolicyamount,butitsvalue isrecordedasthepolicylimitamount truncation– eventsareundetectedand
theirvaluesarecompletelyunknown
lefttruncation(frombelow)occurswhena lossbelowthedeductibleisnotreported
datasourcesincludeseveralaccident
years
trend lossdevelopment dataissparseathigherlimits
Datacanbeusedtofittheseverityfunction toaprobabilitydistribution Addressessomeconcerns
ILFs canbecaluclated forallpolicylimits empiricaldatacanbesmoothed trend paymentlag
ISOhasuseddifferentdistributions,but currentlyusesthemixedexponentialmodel
Usepaid(settled)occurrencesfrom
statisticalplandataandexcessand umbrelladata
Fitamixedexponentialdistributiontothe
lag7weightedoccurrencesizedistribution fromthedata
Producesthelimitedaverageseverity
componentfromtheresultingdistribution
AdvantagesoftheMixedExponentialModel:
continuousdistribution calculationofLASforallpossiblelimits smootheddata simplifiedhandlingoftrend calculationofhighermomentsusedinriskload providesagoodfittoempiricaldataovera
widerangeoflosssizes,isflexible,andeasy touse
trend constructionoftheempiricalsurvival
distribution
paymentlagprocess tailofthedistribution fittingamixedexponentialdistribution finallimitedaverageseverities
JaredSmollik
FCAS,MAAA,CPCU Manager7Actuarial IncreasedLimits&RatingPlansDivision InsuranceServicesOffice,Inc.
201746972607 jsmollik@iso.com