Non-life insurance mathematics

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

2. Insurance mathematics is fundamental in insuranceeconomics The result drivers ofinsuranceeconomics:

3. The world ofPoisson (Chapter 8.2) 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 • On shortintervalsthechanceof more thanoneincident is remote • Assumingno more than 1 event per intervalthecount for theentireperiod is • N=I1+...+IK , whereIj is either 0 or 1 for j=1,...,K • Ifp=Pr(Ik=1) is equal for all k and eventsare independent, this is an ordinaryBernoulli series where • Assumethat p is proportional to h and set is an intensitywhichapplies per time unit

4. Client Policies and claims Policy Insurableobject (risk) Claim Insurance cover Cover element /claim type

5. Key ratios – claimfrequency • The graph shows claimfrequency for all covers for motor insurance • Noticeseasonalvariations, due to changingweatherconditionthroughouttheyears

6. 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) Policy level Portfoliolevel

7. Overview

8. Overviewofthissession What is a fair priceofinsurance policy? The model (Section 8.4 EB) An example Why is a regressionmodelneeded? Repetitionofimportantconcepts in GLM

9. 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) Hence, thepriceequalstheexpectedcost for insurer Note: costofcapitalmay be includedhere, butnoadditionalprofit What is a fair priceof an insurance policy? The fair price The model An example Whyregression? Repetitionof GLM

10. 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 The addotherincome and costs, incl administrative cost and capitalcost Expectedcost The fair price The model An example Whyregression? Repetitionof GLM

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

12. 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

13. 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

14. 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.

15. 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)

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

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

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

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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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 2 modelsaretested and compared – Gamma and lognormal Example: carinsurance Non parametric Log-normal, Gamma The Pareto Extreme value Searching

29. 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 includingthespecificvariable? QQplot: Evaluationofmodel Non parametric Log-normal, Gamma The Pareto Extreme value Searching

30. Fitinterpretation

31. Resultpresentation

32. Resultpresentation Tariff class

33. Resultpresentation Bonus

34. Resultpresentation Region

35. Resultpresentation Driving Length

36. Resultpresentation Car age

37. Validation

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

39. Type 3 analysis

40. 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

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

42. ...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

43. 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

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

45. 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

46. 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

47. Let c>0 IfcYbelongs to same distributionfamily as Y, thendistribution is scale invariant Example: claimcostshouldfollowthe same distribution in NOK, SEK or EURO Scaleinvariance The fair price The model An example Whyregression? Repetitionof GLM

48. If an EDM is scale invariant then it has variancefunction This is alsoproved by Jørgensen This definestheTweediesubclassofGLMs In pricing, suchmodelscan be useful TweedieModels The fair price The model An example Whyregression? Repetitionof GLM

49. OverviewofTweedieModels The fair price The model An example Whyregression? Repetitionof GLM

50. Link functions • A general link function g() • Linear regression: identity link • Multiplicativemodel: log link • Logisticregression: logit link The fair price The model An example Whyregression? Repetitionof GLM