Non-life insurance mathematics

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

2. Main issues so far • Whydoesinsurancework? • How is risk premiumdefined and why is it important? • Howcanclaimfrequency be modelled? • Poissonwithdeterministicmuh • Poissonwithstochasticmuh • Poissonregression

3. Insurance worksbecause risk can be diversifiedawaythroughsize • The coreideaofinsurance is risk spreadonmanyunits • Assumethat policy risks X1,…,XJarestochastically independent • Mean and variance for theportfolio total arethen which is averageexpectation and variance. Then • The coefficientofvariationapproaches 0 as J grows large (lawof large numbers) • Insurance risk can be diversifiedawaythroughsize • Insurance portfoliosare still not risk-freebecause • ofuncertainty in underlyingmodels • risks may be dependent

4. Risk premiumexpressescost per policy and is important in pricing • Risk premium is defined as P(Event)*ConsequenceofEvent • More formally • From aboveweseethat risk premiumexpressescost per policy • Goodpricemodelsrelyon sound understandingofthe risk premium • We start by modellingclaimfrequency

5. The world ofPoisson (Chapter 8.2) Numberofclaims Ik Ik+1 Ik-1 tk-2 tk tk+1 tk=T t0=0 tk-1 Poisson • 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 Somenotions Examples • Ifp=Pr(Ik=1) is equal for all k and eventsare independent, this is an ordinaryBernoulli series Random intensities where • Assumethat p is proportional to h and set is an intensitywhichapplies per time unit

6. The world ofPoisson Poisson Somenotions Examples Random intensities In the limit N is Poissondistributedwith parameter

7. The world ofPoisson • It followsthattheportfolionumberofclaimsN is Poissondistributedwith parameter • Whenclaimintensitiesvary over theportfolio, onlytheiraveragecounts Poisson Somenotions Examples Random intensities

8. 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 Randomintensities (Chapter 8.3) Poisson Somenotions Examples Random intensities 0 1 2 3 0 1 2 3

9. 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 Policy level Portfoliolevel Somenotions Examples Random intensities

10. The Poissonregressionmodel (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

11. 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.

12. Main steps • How to build a model • How to evaluate a model • How to use a model

13. How to build a regressionmodel • Select detail level • Variable selection • Groupingof variables • Remove variables thatarestronglycorrelated • Modelimportantinteractions

14. Select detail level Client Policies and claims Policy Poisson Somenotions Insurableobject (risk) Claim Examples Random intensities Insurance cover Cover element /claim type Step 1 in modelbuilding: Select detail level

15. Groupingof variables PP: Select detaillevel PP: Reviewpotential risk drivers PP: Select groups for each risk driver PP: Select largeclaimsstrategy PP: identifypotentialinteractions PP: construct final model Price assessment

16. Groupingof variables PP: Select detaillevel PP: Reviewpotential risk drivers PP: Select groups for each risk driver PP: Select largeclaimsstrategy PP: identifypotentialinteractions PP: construct final model Price assessment

17. Groupingof variables PP: Select detaillevel PP: Reviewpotential risk drivers PP: Select groups for each risk driver PP: Select largeclaimsstrategy PP: identifypotentialinteractions PP: construct final model Price assessment

18. Modelimportantinteractions • Definition: • Consider a regressionmodelwithtwoexplanatory variables A and B and a response Y. If theeffectof A (on Y) dependsonthelevelof B, wesaythatthere is an interactionbetween A and B • Example (house owner): • The risk premiumofnewbuildingsarelowerthanthe risk premiumofoldbuildings • The risk premiumofyoung policy holders is higherthanthe risk premiumofold policy holders • The risk premiumofyoung policy holders in oldbuildings is particularlyhigh • Thenthere is an interactionbetweenbuilding age and policy holder age PP: Select detaillevel PP: Reviewpotential risk drivers PP: Select groups for each risk driver PP: Select largeclaimsstrategy PP: identifypotentialinteractions PP: construct final model Price assessment

19. Modelimportantinteractions PP: Select detaillevel PP: Reviewpotential risk drivers PP: Select groups for each risk driver PP: Select largeclaimsstrategy PP: identifypotentialinteractions PP: construct final model Price assessment

20. How to evaluate a regressionmodel • QQplot • AkaikeInformationCriterion (AIC) • Scaleddeviance • Cross validation • Type 3 analysis • Resultsinterpretations

21. QQplot • The quantilesofthemodelareplottedagainstthequantiles from a standard Normal distribution • Ifthemodel is good, thepoint in the QQ plot willlieapproximatelyonthe line y=x

22. AIC • AkaikeInformationcriterion is a measureofgoodnessoffitthatbalancesmodelfitagainstmodelsimplicity • AIC has the form • AIC=-2LL+2p where • LL is the log likelihoodevaluated at thevaluefotheestimated parameters • p is thenumberof parameters estimated in themodel • AIC is used to comparemodel alternatives m1 and m2 • If AIC of m1 is less than AIC of m2then m1 is betterthan m2

23. Scaleddeviance • Scaleddeviance = 2(l(y,y)-l(y,muh)) where • l(y,y) is themaximumachievable log likelihood and l(y,muh) is the log likelihood at themaximumestimatesoftheregression parameters • Scaleddeviance is approximatelydistributed as a Chi Square random variables with n-p degreesoffreedom • Scaleddevianceshould be close to 1 ifthemodel is good

24. Cross validation Estimation Validation • Model is calibratedon for example 50% oftheportfolio • Model is thenvalidatedontheremaining 50% oftheportfolio • For a goodmodelthatpredictswellthereshould not be toomuchdifferencebetweenthemodellednumberofclaims in a group and theobservednumberofclaims in the same group

25. Type 3 analysis • Doesthedegreeofvariationexplained by themodelincreasesignificantly by includingthe relevant explanatory variable in themodel? • Type 3 analysis tests thefitofthemodelwith and withoutthe relevant explanatory variable • A low p valueindicatesthatthe relevant explanatory variable improvesthemodelsignificantly

26. Resultsinterpretation • Is theinterceptreasonable? • Are the parameter estimatesreasonablecompared to theresultsoftheoneway analyses?

27. How to use a regressionmodel • Smoothingofestimates