Non-life insurance mathematics

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

2. Exame 2011 problem 1, 2 and 3 (1.5-2h) Repetition, highlightingofimportanttopics from pensum and advice for exame (0.5-1h) Somebriefwordsabouttheassignment (0.5h) Last lecture….

3. Abouttheexame 1 • 4th ofDecember, 1430-1830 • Bring an approvedcalculator • Bring nobooks or notes

4. Abouttheexame 2 • The exameaims to reflectthefocusofthecourse, which has beenpractical and focusedontheapplicationofstatisticaltechniques in general insurance • However, sincethis is a course at the Department ofMathematics, thereshould be somemathematics in theexame • The exameaims to be comprehensive, i.e., cover as manytopics from the pensum as possible • Therewill be 3 practical and 1 theoreticaltask • The exameaims to test understandingofimportantconcepts from thecourse

5. STK 4540 - mainissues • The conceptofdiversification and risk premium • Howcanclaimfrequency be modelled? • Howcanclaimsize be modelled? • Howcansolvency be simulated? • Pricing in general insurance by regression • Pricing in general insurance by credibilitytheory • Reductionof risk in general insuranceusingre-insurance

6. 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

7. 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

8. 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

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

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

11. 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

12. 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

13. 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

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. Repetitionclaimsize The concept Non parametricmodelling Scale families ofdistributions Fitting a scalefamily Shifteddistributions Skewness Non parametricestimation Parametricestimation: the log normal family Parametricestimation: the gamma family Parametricestimation: fitting the gamma

16. Claimseveritymodelling is aboutdescribingthevariation in claimsize The concept • The graphbelow shows howclaimsizevaries for fire claims for houses • The graph shows data up to the 88th percentile • Howdoesclaimsizevary? • Howcanthisvariation be modelled? • Truncation is necessary (large claimsare rare and disturbthepicture) • 0-claims canoccur (becauseofdeductibles) • Two approaches to claimsizemodelling – non-parametric and parametric

17. Non-parametricmodellingcan be useful Non parametricmodelling • Claimsizemodellingcan be non-parametricwhereeachclaimziofthepast is assigned a probability 1/n ofre-appearing in thefuture • A newclaim is thenenvisaged as a random variable for which • This is an entirely proper probabilitydistribution • It is known as theempiricaldistribution and will be useful in Section 9.5.

18. Non-parametricmodellingcan be useful Scale families ofdistributions • All sensible parametricmodels for claimsizeareofthe form • and Z0 is a standardizedrandom variable corresponding to . • The large thescale parameter, the more spreadoutthedistribution

19. Fitting a scalefamily Fitting a scalefamily • Models for scale families satisfy • wherearethedistributionfunctionsof Z and Z0. • Differentiatingwithrespect to z yieldsthefamilyofdensityfunctions • The standard wayof fitting suchmodels is throughlikelihoodestimation. If z1,…,znarethehistoricalclaims, thecriterionbecomes • which is to be maximizedwithrespect to and other parameters. • A usefulextension covers situationswithcensoring.

20. Fitting a scalefamily Fitting a scalefamily • Full valueinsurance: • The insurancecompany is liablethattheobject at all times is insured at its true value • First loss insurance • The object is insured up to a pre-specified sum. • The insurancecompanywill cover theclaimiftheclaimsizedoes not exceedthepre-specified sum • The chanceof a claim Z exceeding b is , and for nbsucheventswithlowerbounds b1,…,bnb theanalogous joint probabilitybecomes • Takethelogarithmofthisproduct and add it to the log likelihoodofthefullyobservedclaims z1,…,zn. The criterionthenbecomes completeinformation (for objectsfullyinsured) censoring to the right (for first loss insured)

21. Shifteddistributions Shifteddistributions • The distributionof a claimmay start at sometreshold b insteadoftheorigin. • Obviousexamplesaredeductibles and re-insurancecontracts. • Models can be constructed by adding b to variables starting at theorigin; i.e. where Z0 is a standardized variable as before. Now • Example: • Re-insurancecompanywillpayifclaimexceeds 1 000 000 NOK The payoutoftheinsurancecompany Currency rate for example NOK per EURO, for example 8 NOK per EURO Total claimamount

22. Skewness as simple descriptionofshape Skewness • A major issuewithclaimsizemodelling is asymmetry and the right tailofthedistribution. A simple summary is thecoefficientofskewness Negative skewness Positive skewness Negative skewness: thelefttail is longer; themassofthedistribution Is concentratedonthe right ofthefigure. It has relativelyfewlowvalues Positive skewness: the right tail is longer; themassofthedistribution Is concentratedontheleftofthefigure. It has relativelyfewhighvalues

23. Non-parametricestimation Non parametricestimation • The random variable that attaches probabilities 1/n to all claimsziofthepast is a possiblemodel for futureclaims. • Expectation, standard deviation, skewness and percentilesare all closelyrelated to theordinary sample versions. For example • Furthermore, • Third order moment and skewnessbecomes

24. Parametricestimation: the log normal family The log-normal family • A convenientdefinitionofthe log-normal model in the present context is • as where • Mean, standard deviation and skewnessare • seesection 2.4. • Parameter estimation is usuallycarriedout by notingthatlogarithmsareGaussian. Thus • and whenthe original log-normal observations z1,…,znaretransformed to Gaussianonesthrough y1=log(z1),…,yn=log(zn) with sample mean and variance , theestimatesofbecome

25. Parametricestimation: the gamma family The Gamma family • The Gamma family is an importantfamily for whichthedensityfunction is • It wasdefined in Section 2.5 as is the standard Gamma withmeanone and shapealpha. The densityofthe standard Gamma simplifies to • Mean, standard deviation and skewnessare • and there is a convolutionproperty. Suppose G1,…,Gnareindependentwith • . Then

26. Parametricestimation: fitting the gamma The Gamma family • The Gamma family is an importantfamily for whichthedensityfunction is • It wasdefined in Section 2.5 as is the standard Gamma withmeanone and shapealpha. The densityofthe standard Gamma simplifies to

27. Parametricestimation: fitting the gamma The Gamma family

28. Solvency • Financial controlofliabilities under nearlyworst-case scenarios • Target: thereserve • which is theupperpercentileoftheportfolioliability • Modelling has beencovered (Risk premiumcalculations) • The issuenow is computation • Monte Carlo is the general tool • Some problems can be handled by simpler, Gaussianapproximations

29. 10.2 Portfolioliabilities by simple approximation • The portfolio loss for independent risks becomeGaussian as J tends to infinity. • Assumethat policy risks X1,…,XJarestochastically independent • Mean and variance for theportfolio total arethen which is averageexpectation and variance. Then • as J tends to infinfity • Note that risk is underestimated for smallportfolios and in brancheswith large claims

30. Normal approximations

31. The ruleof double variance Let X and Y be arbitraryrandom variables for which Thenwe have theimportantidentities Poisson Somenotions Ruleof double expectation Ruleof double variance Examples Random intensities

32. The ruleof double variance Portfolio risk in general insurance Poisson Elementaryrules for random sums imply Somenotions Examples Random intensities

33. The ruleof double variance This leads to the true percentileqepsilonbeingapproximated by Where phi epsilon is theupperepsilonpercentileofthe standard normal distribution Poisson Somenotions Examples Random intensities

34. Fire data from DNB

35. Portfolioliabilities by simulation • Monte Carlo simulation • Advantages • More general (norestrictiononuse) • More versatile (easy to adapt to changingcircumstances) • Better suited for longer time horizons • Disadvantages • Slowcomputationally? • Dependingonclaimsizedistribution?

36. An algorithm for liabilitiessimulation • Assumeclaimintensities for J policiesarestoredon file • Assume J differentclaimsizedistributions and paymentfunctions H1(z),…,HJ(z) arestored • The program can be organised as follows (Algorithm 10.1)

37. Experiments in R 1. Log normal distribution 2. Gamma on log scale 3. Pareto 4. Weibull 5. Mixeddistribution 1 6. Monte Carlo algorithm for portfolioliabilities 7. Mixeddistribution 2

38. Comparisonofresults

39. Monte Carlo theory Suppose X1, X2,… are independent and exponentiallydistributedwithmean 1. It canthen be proved (1) • for all n >= 0 and all lambda > 0. • From (1) weseethattheexponentialdistribution is thedistributionthatdescribes time betweenevents in a Poissonprocess. • In Section 9.3 welearntthatthedistributionof X1+…+Xn is gamma distributedwithmean n and shape n • The Poissonprocess is a process in whicheventsoccurcontinuously and independently at a constantaverage rate • The Poissonprobabilitiesonthe right definethedensityfunction • which is thecentralmodel for claimnumbers in propertyinsurance. • Mean and standard deviationare E(N)=lambda and sd(N)=sqrt(lambda)

40. Monte Carlo theory It is thenutilizedthatXj=-log(Uj) is exponentialifUj is uniform, and the sum X1+X2+… is monitoreduntil it exceeds lambda, in otherwords

41. Monte Carlo theory (1)

42. 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 Model Example Combiningcredibilitytheory and GLM

43. Omega is random and has beendrawn for each policy holder Motivation Model (driving skill) Example (poor driver) (excellent driver) Combiningcredibilitytheory and GLM

44. 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 Model Example Combiningcredibilitytheory and GLM

45. Linear credibility • The standard method in credibility is the linear onewithestimatesof pi ofthe form • where b0,…,bKarecoefficients so thatthemeansquarederror is as small as possible. • The factthat X,X1,…,XKareconditionallyindependentwiththe same distributionforces b1=…=bK, and if w/K is theircommonvalue, theestimatebecomes • To proceedweneedthe so-calledstructural parameters • where is theaverage pure premium for theentirepopulation. • It is alsotheexpectation for individualssince by theruleof double expectation Motivation Model Example Combiningcredibilitytheory and GLM

46. Linear credibility • Bothrepresentvariation. The former is caused by diversitybetweenindividuals and the latter by thephysicalprocessesbehindtheincidents.Theirimpacton var(X) can be understoodthroughtheruleof double variance, i.e., • and representuncertaintiesof different originthatadd to var(X) • The optimal linear credibilityestimatenowbecomes • which is proved in Section 10.7, where it is alsoestablishedthat • The estimate is ubiased and its standard deviationdecreaseswith K Motivation Model Example Combiningcredibilitytheory and GLM

47. Linear credibility • 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