pricing a portfolio of large commercial risks€¦ · final curve using combined data from segments...
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PricingaPortfolioofLargeCommercialRisks
AdityaKhanna,FCASHeadofPricing
RSAInsurance,India
• COMMERCIALINSURANCELANDSCAPE• PROBLEMSETUP
• FROMDATATODECISIONS• RESTATETHEPROBLEM
• APOSSIBLESOLUTION• ASIMPLEEXAMPLE
AGENDA
SIZEOFRISK
Commercial Insurance Landscape
Smallrisks
JumboAccounts
LargeRisks
• Lossrated(Experience)
• Nosegmentation
Anythingthatliesinbetween
Portfoliorating(exposurerate)followedbyindividualriskmodification
• Portfoliorating(Exposure)
• Segmentedrates
Portfolioratinginthiscontextisparticularlychallenging-considerableheterogeneity,needforsomesegmentation.Subjectofthispresentation
Large risk portfolio rating – Problem Set up
• Yourassignmentistocalculatetheratefortheaverageinsuredinthetwosegmentsoftheportfolio
• Policieshavedifferentstructures
• Policystructurescanchangeovertime
SegmentB
SegmentA
Differentstructures
Analyzeallclaimsatacommonlevel
Datavolumedecreasesasclaimssizeincreases
Splitthedata
Attritionalrate Largelossrate
Largeclaimsdominatetheaggregateamounts
IndividualclaimsStatisticaldistribution
From Data to Decisions (1)
Restating the problem
A
CAttritionalpartofsegmentAAttritionalpartofsegmentBLargepartofsegmentAandBcombined
A
B
CB
Plan of Action
q Analyzelossesatcommonlevel• share,deductible,trend,maturity
q Determinecappingthresholdsforeachsegment
q Developlargelossrateabovethreshold• Fitadistributiontocombinedlosses
q Allocatethelargelossratetoeachsegment
q Developattritionalratebelowthresholds
q Monitor,Validate,Refine
Widerangeoflossamounts
Piece-wisedistributionsMixedcurves
Allocatelargelossratetosegments
Allocateportfolioratetopolicies
From Data to Decisions (2)
Methodallowsacomparisonofattritionalandlarge
Easytodeterminelayeredamounts
From Data to Decisions (3)
NumberofParameters/Difficultytofit
Limita
tions
SingleParameterPareto
Log-Normal
• Deterministicequationtosolveforparameters• Appliedtoexcesslosses–notapplicableforgross(GU)losses
• CanbeappliedtoGUlosses• Deterministicequationtocalculatelayeredlosses• Semi-parametric–balanceoffitandsmoothing
MixedCurves
MIXED EXPONENTIAL CURVES
FinalcurveusingcombineddatafromSegmentsAandB
Fitted Curve (combined data)
Attritional(AreaAandB)–Segmentwiseexperience
Largeclaims(AreaC)–Morevaluederivedusingthedistribution
Slicing the curve
Layer1
Layer2
Layer3
Layer4
AttritionalArea
LargeclaimsArea
A Simple Example Asimpleformulationcouldbeasfollows(motivationfromreinsurancepricing)
Layer SegA-experiencerate SegB-experiencerate Fitted(combined)rate SegAtoFitted SegBtoFitted1 80 120 100 0.80 1.202 68 92 80 0.85 1.153 25 35 40
Layer Fitted(combined)rate SegAtoFitted SegBtoFitted PolicyA PolicyB1 80 1202 68 92
Comparingthefittedresultswiththesegmentspecificexperiencestartstoprovideabasistoallocatethefittedratesathigherlayerstodifferentsegments– ExperienceAdjustedExposureRate
4 10 15 20
3 40 0.90 1.10 36 444 20 0.95 1.05 19 21
Constraint:Aggregateofallocatedlargelossesequalsthefittedlargelosses
Some useful references • ModellingLosseswiththeMixedExponentialDistribution
-CliveLKeatinghttps://www.casact.org/pubs/proceed/proceed99/99578.pdf
• ApracticalGuidetotheSingleParameterParetoDistribution
-StephenWPhilbrick
https://www.casact.org/pubs/proceed/proceed85/85044.pdf
• BasicsofReinsurancePricing-DavidRClarkhttps://www.casact.org/library/studynotes/clark6.pdf
APPENDIX
Pareto - fitting & some useful properties • SingleParameterPareto• F(x)=1–(k/x)^q,xrepresentstherandomvariableforsizeofloss
• UsuallydenotedasF(y)=1–(1/y)^q,wherey=Normalizedsizeofloss,y=X/k
• Riskcharacteristicsdefinedbyonerealparameter:q[actuallythisisatwoparameterdistribution“disguised”assingleparameter]
• AssumingLossesL1,L2,…..,LNabovethelossthresholdk(i.e.excesslosses)
• PDFf(y)=q 𝑦↑−(𝑞+1) • Likelihood=Productofallpdfs=∏𝑖=1↑𝑖=𝑁▒𝑓(𝑦=𝐿𝑖 )
=∏𝑖=1↑𝑖=𝑁▒q𝐿𝑖↑−(𝑞+1) = 𝑞↑𝑛 (∏𝑖=1↑𝑖=𝑁▒𝐿𝑖 ) ↑−(𝑞+1)
• LogLikelihood=nlogq–(q+1)∑𝑖=1↑𝑖=𝑁▒𝐿𝑜𝑔 𝐿 𝑖• MaximizeLogLikelihoodtofindtheparameterq
• 0=n/q-∑𝑖=1↑𝑖=𝑁▒𝐿𝑜𝑔 𝐿 𝑖• q=(∑𝑖=1↑𝑖=𝑁▒𝐿𝑜𝑔 𝐿 𝑖) /𝑛
• Fittingthedistribution:MaximumLikelihoodcanbeusedtofit“most”distributions.
• Heavilyusedinthecommercialpropertyandliabilitymarketpricings–benchmarkrangesfortheParetoparameterwellsocialized.
Mixed Distributions • Whatwejustsawiswhatisknownasa“parametric”distribution.Howeveritisnotalwayspossibletouseonedistributionandexpectittotrackallthelosseswell.
• AlternativeOne:Empiricaldistributiontrackingthelossset–Notenoughsmoothing
• AlternativeTwo:Useamid-way.MixedDistributions-aweightedmixtureoftwoormoredistributions
• Advantageofbothfittingthedatawellandsmoothing
• Moregeneralinnature-allowsapiece-wisefittingofdataatdifferentpartsofthelosscurve
• MLEisnotthatstraightforward
• Likelihoodcanbeformulated:Someusefulpointstokeepinmind:o PDFofamixeddistributionisjusttheweightedPDFofmemberdistributionso CDFofamixeddistributionisjusttheweightedCDFofmemberdistributionso Byextension–anymomentwillbetheweightedmomentofmemberdistributions
• MaximizingtheLikelihoodrequiresaniterativealgorithmduetomultipleparameters
• MixedExponentialsheavilyusedintheUSmarketforSpecialtyCasualtyLines
• MixedLog-Normalshavebeenexperimentedforsimilarlinesandaggregatedata
Fitting Mixed Exponentials Spreadsheetsetup• Usually10curvescovermostoftheinsuranceapplications• 𝑓(𝑥)= 1/µ 𝑒↑−𝑥/µ ;µisthemeanofthedistribution• Let‘y’denotethelossrandomvariableandkdistributionsbe“mixed”• LossesL1,L2,L3…..,LN–grounduplosses• Formixedexponentials;f(y)=∑𝑖=1↑𝑖=𝑘▒𝑊𝑖 𝑓(𝑥 ) = ∑𝑖=1↑𝑖=𝑘▒𝑊𝑖1/µ𝑖 𝑒↑−𝑥 /µ𝑖 • Likelihood= ∏𝑗=1↑𝑗=𝑁▒𝑓(𝑦=𝐿𝑗 ) • LogLikelihood=Ln(likelihood)=Ln{∏𝑗=1↑𝑗=𝑁▒𝑓(𝐿𝑗 ) }=∑𝑗=1↑𝑗=𝑁▒𝐿𝑛 {𝑓(𝐿𝑗 )} = ∑𝑗=1↑𝑗=𝑁▒𝐿𝑛∑𝑖=1↑𝑖=𝑘▒𝑊𝑖1/µ𝑖 𝑒↑−𝐿𝑗 /µ i• Tobemaximizedusinganiterativealgorithm
Ø Parameterstobeestimated–weightsandmeansoftheexponentialdistributionØ Usejudgmentforthemeans–eyeballforthefirstiterationandrefinefurtherØ Constraint:∑𝑖=1↑𝑖=𝑘▒𝑊𝑖 =1
• DeterministiccalculationforanymomentforunlimitedandlimitedseveritiesUsingR• package‘mixdist’
•
Simulating the Mixed Exponential • Step 1: Generate a random number to select which curve is to be used • Step 2: Generate a random number for the Cumulative probability and work the loss backwards Since F(x)= 1−𝑒↑−𝑥/µ ;𝑥=µ 𝑋−log�[1−𝐹(𝑥) ] Let's say we have a mixed distribution of 5 exponential curves with the means and weights as in the table below
Example: Random numbers 0.7 and 0.3 are generated • Step 1: random number is 0.7, select curve 4 (mean = 40K). • Step 2: random number is 0.3, work the loss backwards = 40K X [ - log (1-0.3)] = 14,267
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