STK 4540 Lecture 4

1. STK 4540 Lecture 4 Randomintensities in theclaimfrequency and Claimfrequencyregression

2. Overviewpricing (1.2.2 in EB) Premium Individual Insurance company Claim Due to thelawof large numberstheinsurancecompany is cabableofestimating theexpectedclaimamount Distributionof X, estimatedwithclaims data Risk premium Expectedclaimamount given an event • Probabilityofclaim, • Estimatedwithclaimfrequency • Weareinterested in thedistributionoftheclaimfrequency • The premiumcharged is the risk premiuminflatedwith a loading (overhead and margin) Expectedconsequenceofclaim

3. The world ofPoisson Numberofclaims Ik Ik+1 Ik-1 tk-2 tk tk+1 tk=T t0=0 tk-1 • What is rare can be describedmathematically by cutting a given time period T into K small pieces ofequallengthh=T/K • Assumingno more than 1 event per intervalthecount for theentireperiod is • N=I1+...+IK , whereIj is either 0 or 1 for j=1,...,K Poisson Somenotions Examples • Ifp=Pr(Ik=1) is equal for all k and eventsare independent, this is an ordinaryBernoulli series Random intensities • Assumethat p is proportional to h and set where is an intensitywhichapplies per time unit In the limit N is Poissondistributedwith parameter

4. The Poissondistribution • Claimnumbers, N for policies and N for portfolios, arePoissondistributedwith parameters Policy level Portfoliolevel The intensity is an average over time and policies. Poisson Poissonmodels have usefuloperationalproperties. Mean, standard deviation and skewnessare Somenotions Examples The sums of independent Poisson variables must remainPoisson, if N1,...,NJ are independent and Poissonwith parameters then Random intensities ~

5. Carinsuranceclient Policies and claims Carinsurance policy Poisson Somenotions Insurableobject (risk), car Claim Third part liability Insurance cover third party liability Examples Legal aid Driver and passenger acident Fire Random intensities Insurance cover partial hull Theft from vehicle Theftofvehicle Rescue Accessoriesmountedrigidly Insurance cover hull Ownvehicledamage Rentalcar

6. Key ratios – claimfrequency TPL and hull • The graph shows claimfrequency for third part liability and hull for motor insurance Poisson Somenotions Examples Random intensities

7. Howvaries over theportfoliocanpartially be described by observablessuch as age or sex oftheindividual (treated in Chapter 8.4) Therearehoweverfactorsthat have impactonthe risk whichthecompanycan’t know muchabout • Driver ability, personal risk averseness, This randomenesscan be managed by making a stochastic variable This extensionmay serve to captureuncertaintyaffecting all policy holders jointly, as well, such as alteringweatherconditions The modelsareconditionalonesofthe form Let which by double rules in Section 6.3 imply Now E(N)<var(N) and N is no longer Poissondistributed Randomintensities (Chapter 8.3) Poisson Somenotions Examples Policy level Portfoliolevel Random intensities

8. Randomintensities Specificmodels for arehandledthroughthemixingrelationship Gamma modelsaretraditionalchoices for and detailedbelow Poisson Estimatesofcan be obtained from historical data withoutspecifying . Let n1,...,nn be claims from n policy holders and T1,...,TJtheirexposure to risk. The intensityofindividual j is thenestimated as . Somenotions Uncertainty is huge butpooling for portfolioestimation is still possible. One solution is Examples (1.5) Random intensities and (1.6) Bothestimatesareunbiased. SeeSection 8.6 for details. 10.5 returns to this.

9. The negative binomial model The most commonlyappliedmodel for muh is the Gamma distribution. It is thenassumedthat Here is the standard Gamma distributionwithmeanone, and fluctuatesaroundwithuncertaintycontrolled by . Specifically Poisson Since , the pure Poissonmodelwithfixedintensityemerges in the limit. Somenotions The closed form ofthedensityfunctionof N is given by Examples Random intensities for n=0,1,.... This is the negative binomial distribution to be denoted . Mean, standard deviation and skewnessare (1.9) Where E(N) and sd(N) follow from (1.3) when is inserted. Note thatif N1,...,NJareiidthen N1+...+NJ is nbin (convolutionproperty).

10. Fitting the negative binomial Moment estimationusing (1.5) and (1.6) is simplest technically. The estimateof is simply in (1.5), and for invoke (1.8) right whichyields Poisson If , interpret it as an infinite or a pure Poissonmodel. Likelihoodestimation: the log likelihoodfunctionfollows by insertingnj for n in (1.9) and addingthelogarithm for all j. This leads to thecriterion Somenotions Examples Random intensities whereconstantfactors not dependingon and have beenomitted.

11. Claimfrequencyregression

12. Overviewofthissession What is a fair priceofan insurancepolicy? The model (Section 8.4 EB) An example Why is a regressionmodelneeded? Repetitionofimportantconcepts in GLM

13. Before ”Fairness” wassupervised by theauthorities (Finanstilsynet) • To someextentcommon tariffs betweencompanies • The market wascontrolled During 1990’s: deregulation Now: free market competitionsupposed to give fairness According to economictheorythere is noprofit in a free market (in Norway general insurance is cyclical) • Thesearethedaysof super profit • 15 yearsagoseveral general insurerswerealmostbankrupt Hence, thepriceequalstheexpectedcost for insurer Note: costofcapitalmay be includedhere, butnoadditionalprofit Ethical dilemma: • Original insuranceidea: One price for all • Today: thedevelopment is heading towardsmicropricing • Thesetworepresentextremes What is a fair priceof an insurance policy? The fair price The model An example Whyregression? Repetitionof GLM

14. Main component is expected loss (claimcost) The average loss for a large portfoliowill be close to themathematicalexpectation (by thelawof large numbers) So expected loss is the basis oftheprice Variesbetweeninsurancepolicies Hencethe market pricewillvarytoo Thenaddotherincome (financial) and costs, incl administrative cost and capitalcost Expectedcost The fair price The model An example Whyregression? Repetitionof GLM

15. Toohighpremium for somepoliciesresults in loss ofgoodpolicies to competitors Toolowpremium for somepoliciesgivesinflowofunprofitablepolicies This willforcethecompany to charge a fair premium In practicethethreatofadverseselection is constant Adverseselection The fair price The model An example Whyregression? Repetitionof GLM

16. How to findtheexpected loss ofeveryinsurance policy? Wecannotpriceindividualpolicies (why?) • Thereforepoliciesaregrouped by rating variables Rating variables (age) aretransformed to ratingfactors (age classes) Ratingfactorsare in most cases categorical Ratingfactors The fair price The model An example Whyregression? Repetitionof GLM

17. The model (Section 8.4) • The idea is to attributevariation in to variations in a setofobservable variables x1,...,xv. Poissonregressjon makes useofrelationshipsofthe form The fair price (1.12) • Why and not itself? • The expectednumberofclaims is non-negative, where as thepredictoronthe right of (1.12) can be anythingonthe real line • It makes more sense to transform so thattheleft and right side of (1.12) are more in line witheachother. • Historical data areofthefollowing form • n1 T1 x11...x1x • n2 T2 x21...x2x • nnTn xn1...xnv • The coefficients b0,...,bvareusuallydetermined by likelihoodestimation The model An example Whyregression? Repetitionof GLM Claimsexposurecovariates

18. The model (Section 8.4) • In likelihoodestimation it is assumedthatnj is Poissondistributedwhere is tied to covariates xj1,...,xjv as in (1.12). The densityfunctionofnj is then The fair price • or The model An example • log(f(nj)) above is to be added over all j for thelikehoodfunction L(b0,...,bv). • Skip themiddle terms njTj and log (nj!) sincetheyareconstants in thiscontext. • Thenthelikelihoodcriterionbecomes Whyregression? Repetitionof GLM (1.13) • Numerical software is used to optimize (1.13). • McCullagh and Nelder (1989) provedthat L(b0,...,bv) is a convexsurfacewith a single maximum • Thereforeoptimization is straight forward.

19. Poisson regression: an example, bus insurance The fair price The model An example Whyregression? Repetitionof GLM • The modelbecomes • for l=1,...,5 and s=1,2,3,4,5,6,7. • To avoidover-parameterizationputbbus age(5)=bdistrict(4)=0 (thelargestgroup is often used as reference)

20. Take a look at the data first The fair price The model An example Whyregression? Repetitionof GLM

21. Then a model is fittedwithsomesoftware (sas below) The fair price The model An example Whyregression? Repetitionof GLM

22. Zonneedssomere-grouping The fair price The model An example Whyregression? Repetitionof GLM

23. Zon and bus age arebothsignificant The fair price The model An example Whyregression? Repetitionof GLM

24. Model and actualfrequenciesarecompared • In zon 4 (marked as 9 in thetables) thefit is ok • There is much more data in thiszonthan in theothers • Wemaytry to re-groupzon, into 2,3,7 and other The fair price The model An example Whyregression? Repetitionof GLM

25. Model 2: zonregrouped • Zon 9 (4,1,5,6) still has the best fit • The otherarebetter – butaretheygoodenough? • Wetry to regroup bus age as well, into 0-1, 2-3 and 4. The fair price The model An example Whyregression? Bus age Bus age Repetitionof GLM Bus age Bus age

26. Model 3: zon and bus age regrouped • Zon 9 (4,1,5,6) still has the best fit • The otherare still better – butaretheygoodenough? • May be there is not enoughinformation in thismodel • May be additionalinformation is needed • The final attempt for now is to skip zon and relysolelyon bus age The fair price The model An example Whyregression? Repetitionof GLM Bus age Bus age Bus age Bus age

27. Model 4: skip zon from themodel (only bus age) • From thegraph it is seenthatthefit is acceptable • Hypothesis 1: Theredoes not seem to be enoughinformation in the data set to provide reliable estimates for zon • Hypothesis 2: there is anothersourceofinformation, possiblyinteractingwithzon, thatneeds to be takenintoaccountifzon is to be included in themodel The fair price The model An example Whyregression? Repetitionof GLM Bus age

28. The variables in themultiplicativemodelareassumed to work independent ofoneanother This may not be the case Example: • Auto model, Poissonregressionwith age and gender as explanatory variables • Young males drive differently (worse) thanyoungfemales • There is a dependencybetween age and gender This is an exampleof an interactionbetweentwo variables Technicallytheissuecan be solved by forming a newratingfactorcalledage/genderwithvalues • Young males, youngfemales, older males, older femalesetc Limitationofthemultiplicativemodel The fair price The model An example Whyregression? Repetitionof GLM

29. There is not enough data to pricepoliciesindividually What is actually happening in a regressionmodel? • Regressioncoefficientsmeasuretheeffectceterisparibus, i.e. when all other variables are held constant • Hence, theeffectof a variable can be quantifiedcontrolling for theother variables Whytakethe trouble ofusing a regressionmodel? Why not pricethepoliciesonefactor at a time? Why is a regressionmodelneeded? The fair price The model An example Whyregression? Repetitionof GLM

30. Claimfrequencies, lorry data from Länsförsäkringer (Swedish mutual) The fair price The model An example • ”One factor at a time” gives 6.1%/2.6% = 2.3 as themileagerelativity • But for eachVehicle age, theeffect is close to 2.0 • ”One factor at a time” obviouslyoverestimatestherelativity – why? Whyregression? Repetitionof GLM

31. Claimfrequencies, lorry data from Länsförsäkringer (Swedish mutual) The fair price The model • New vehicles have 45% oftheirduration in lowmileage, while old vehicles have 87% • So, the old vehicles have lowerclaimfrequenciespartly due to less exposure to risk • This is quantified in theregressionmodelthroughthemileagefactor • Conclusion: 2.3 is right for High/Lowmileageif it is theonlyfactor • Ifyou have bothfactors, 2.0 is the right relativity An example Whyregression? Repetitionof GLM

32. Hull coverage (i.e., damagesonownvehicle in a collisionor othersudden and unforeseendamage) Time period for parameter estimation: 2 years Covariates: • Driving length • Car age • Region ofcarowner • Tariff class • Bonus ofinsuredvehicle Log Poisson is fitted for claimfrequency 120 000 vehicles in theanalysis Example: carinsurance

33. The model is evaluatedwithrespect to fit, result, validationofmodel, type 3 analysis and QQ plot Fit: ordinaryfitmeasuresareevaluated Results: parameter estimatesofthemodelsarepresented Validationofmodel: the data material is split in two, independentgroups. The model is calibrated (i.e., estimated) onone half and validatedontheother half Type 3 analysisofeffects: Doesthefitofthemodelimprovesignificantly by includingthespecific variable? QQplot: Evaluationofmodel

34. Fitinterpretation

35. Resultpresentation

36. Resultpresentation Tariff class

37. Resultpresentation Bonus

38. Resultpresentation Region

39. Resultpresentation Driving Length

40. Resultpresentation Car age

41. Validation

42. Type 3 analysis Type 3 analysisofeffects: Doesthefitofthemodelimprovesignificantly by includingthespecific variable?

43. QQ plot

44. Somerepetitionofgeneralizedlinear models (GLMs) ExponentialdispersionModels (EDMs) • FrequencyfunctionfYi (eitherdensity or probabilityfunction) • For yi in the support, else fYi=0. • c() is a function not dependingon • C • twicedifferentiablefunction • b’ has an inverse • The setofpossible is assumed to be open The fair price The model An example Whyregression? Repetitionof GLM

45. Claimfrequency • ClaimfrequencyYi=Xi/Tiwhere Ti is duration • NumberofclaimsassumedPoissonwith • LetC • Then • EDM with The fair price The model An example Whyregression? Repetitionof GLM

46. ...is not a parametricfamilyofdistributions (like Normal, Poisson) ...is rather a classofdifferentsuch families The function b() speficieswhichfamilywe have The idea is to derive general results for all families withintheclass – and use for all Note that an EDM... The fair price The model An example Whyregression? Repetitionof GLM

47. By usingcumulant/moment-generatingfunctions, it can be shown (seeMcCullagh and Nelder (1989)) that for an EDM • E • Ee This is why b() is calledthecumulantfunction Expectation and variance The fair price The model An example Whyregression? Repetitionof GLM

48. Recallthat is assumed to exist Hence The variancefunction is defined by Hence The variancefunction The fair price The model An example Whyregression? Repetitionof GLM

49. Commonvariancefunctions Distribution Normal Poisson Gamma Binomial The fair price The model An example Note: Gamma EDM has stddeviationproportional to , which is much more realisticthanconstant (Normal) Whyregression? Repetitionof GLM

50. Theorem Withinthe EDM class, a familyofprobabilitydistributions is uniquelycharacterized by itsvariancefunction The fair price The model Proof by professor Bent Jørgensen, Odense An example Whyregression? Repetitionof GLM