Non-life insurance mathematics

1. Non-lifeinsurancemathematics Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

2. Overview done done done done done done

3. Credibilitytheorywas used before computers arrived Motivation Linear credibility Example Optimal credibility

4. The need for credibilitytheory • So far we have studiedhowGLMsmay be used to estimaterelativities for ratingfactorswith a moderate numberoflevels. • The ratingfactorswereeithercategoricalwithfewlevels (e.g., gender), or • formed by groupingof a continuous variable (e.g., policy holder age) • In case ofinsufficient data levelscan be merged • However, for categoricalratingfactorswith a largenumberoflevelswithout an inherent orderingthere is no simple way to form groups to solvethe problem withinsufficient data Motivation Linear credibility Example Optimal credibility

5. Example 1: carinsurance • Carmodel is an importantratingfactor in motor insurance • Exlusivecarsare more attractive to thieves and more expensive to repairthancommoncars • Sports carsmay be driven differently (more recklessly?) thanfamilycars • In Norway thereare more than 3000 carmodelcodes • How shouldthese be grouped? Motivation Linear credibility Example Optimal credibility

6. Example 2: Experiencerating • The customer is used as a ratingfactor • The credibility estimatorsare used to calibratethe pure premium • The credibility estimators areweightedaveragesof • the pure premiumbasedontheindividualclaimsexperience • The pure premium for theentireportfolioofinsurancepolicies Motivation Linear credibility Example Optimal credibility

7. Example 3: Geographiczones • How shouldthepremium be affected by the area ofresidence? (parallel to thecarmodelexample) • Example: postcode. Thereare over 8000 postcodes in Norway • Wecanmergeneighbouring areas • Buttheymay have different risk characteristics Motivation Linear credibility Example Optimal credibility

8. The credibilityapproach • The basicassumption is that policy holders carry a list ofattributeswithimpacton risk • The parameter could be howthecar is used by thecustomer (degreeofrecklessness) or for example driving skill • It is assumedthatexists and has beendrawnrandomly for eachindividual • X is the sum ofclaims during a certainperiodof time (say a year) and introduce • Weseektheconditional pure premiumofthe policy holder as basis for pricing • On grouplevelthere is a commonthatapplies to all risks jointly • Wewillfocusontheindividuallevelhere, as thedifferencebetweenindividual and group is minor from a mathematicalpointofview Motivation Linear credibility Example Optimal credibility

9. Omega is random and has beendrawn for each policy holder Motivation Linear credibility (driving skill) Example (poor driver) (excellent driver) Optimal credibility

10. Omega is random and has beendrawn for each policy holder Motivation Linear credibility (Riskynessofcar, measured by carmodel) Example (carwithlarger risk (sports car)) (carwithlittle risk (familycar)) Optimal credibility

11. Omega is random and has beendrawn for each policy holder Motivation Linear credibility (Riskynessofgeographical area, measured by post code) Example (area withgoodclimate) (area withpoorclimate ) Optimal credibility

12. Goal Wewant to estimate Motivation Linear credibility • Thesearecalledstructural parameters. • If X1,…,XKarerealizations from X, where X is the sum ofclaims during a yearwewantthefollowingexpression • Wewant to findwhat w is in twosituations. • The weight w defines a compromisebetweentheaverage pure premiumofthepopulation and thetrackrecordofthe policy holder. Example Optimal credibility

13. Linear credibility • The standard method in credibility is the linear onewithestimatesof pi ofthe form • where b0,…,bKarecoefficientstailored to minimizedthemeansquarederror • The factthat X,X1,…,XKareconditionally independent withthe same distributionforces b1=…=bK, and if w/K is theircommonvalue, theestimatebecomes Motivation Linear credibility Example Optimal credibility

14. Linear credibility • To proceedweneedthe so-calledstructural parameters • where is theaverage pure premium for theentirepopulation. • It is alsotheexpectation for individualssince by theruleof double expectation • Bothrepresentvariation. The former is caused by diversitybetweenindividuals and the latter by thephysicalprocessesbehindtheincidents.Theirimpacton var(X) can be understoodthroughtheruleof double variance, i.e., • and representuncertaintiesofdifferentoriginthatadd to var(X) Motivation Linear credibility Example Optimal credibility

15. Linear credibility • The optimal linear credibilityestimatenowbecomes • which is proved in Section 10.7, where it is alsoestablishedthat • The estimate is ubiased and its standard deviationdecreaseswith K • The weight w defines a compromisebetweentheaverage pure premium pi bar ofthepopulation and thetrackrecordofthe policy holder • Note that w=0 if K=0; i.e. withouthistoricalinformationthe best estimate is thepopulationaverage Motivation Linear credibility Example Optimal credibility

16. Estimationofstructural parameters • Credibilityestimation is basedon parameters that must be determined from historical data. • Historical data for J policiesthatbeen in thecompany for K1,…,KJ yearsarethenofthe form • wherethej’throwaretheannualclaims from client j and • theirmean and standard deviation • The followingestimatesareessentially due to Sundt (1983) and Bühlmann and Straub (1970): Motivation Linear credibility Example Optimal credibility Policies Annualclaims mean sd

17. Estimationofstructural parameters Motivation Linear credibility Example Optimal credibility • For verificationseeSection 10.7. • The expression for may be negative. If it is let and assumethatthe underlying variation is toosmall to be detected.

18. The most accurateestimate • Let X1,…,XK (policy level) be realizationsofXdating K years back • The most accurateestimateof pi from suchrecords is (section 6.4) theconditionalmean • where x1,…,xKaretheactualvalues. • A naturalframework is thecommonfactormodelofSection 6.3 where X,X1,…,XKareidentically and independentlydistributed given omega • This won’t be true when underlying conditionschangesystematically • A problem withtheestimateabove is that it requires a joint model for X,X1,…,XK and omega. • A more naturalframework is to break Xdownonclaimnumber N and losses per incident Z. • First linear credibility is considered Motivation Linear credibility Example Optimal credibility

19. Optimal credibility • The precedingestimatesarethe best linearmethodsbuttheBayesianestimate is optimal amongallmethods and offers an improvement. • Break thehistoricalrecord x1,…,xKdownonannualclaimnumbers n1,…nK and losses z1,…,znwhere n=n1+…+nKand assumeindependence. • The Bayesestimateofnowbecomes • and therearetwo parts. • Oftenclaimintensitymuhfluctuates more strongly from one policy holder to anotherthandoestheexpected loss . • Weshalldisregard all suchvariation in . • Then is independentlyof z1,…,zn and theBayesianestimatebecomes Motivation Linear credibility Example Optimal credibility

20. Optimal credibility • A model for past and futureclaimnumbers is needed. The naturalone is thecommonfactormodelwhere N,N1,…,Nkareconditionallyindependent and identicallydistributed given muhwith N and all Nkbeing Poisson (mu T). • For muassumethe standard representation • where E(G)=1. It is shown in Section 10.7 thattheestimatenowbecomes • Note thatthepopulationaverage pi bar is adjusted up or downaccording to whethertheaverageclaimnumber n bar is larger or smallerthanitsexpectation . The error (proved in Section 10.7) is Motivation Linear credibility Example Optimal credibility