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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