Recap (1.2.1 in EB)

2. Recap (1.2.1 in EB) • Property insurance is economicresponsibility for incidentssuch as fires and accidentspassedon to an insureragainst a fee • The contract, known as a policy, releasesclaimswhensucheventsoccur • A centralquantity is the total claim X amassed during a certainperiodof time (typically a year)

3. 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

4. Control (1.2.3 in EB) • Companiesareobliged to aside funds to cover futureobligations • Suppose a portfolioconsistsof J policieswithclaims X1,…,XJ • The total claim is then Portfolioclaimsize • Weareinterested in as well as itsdistribution • Regulators demandsufficient funds to cover withhighprobability • The mathematicalformulation is in term of , which is thesolutionoftheequation • where is small for example 1% • The amount is known as thesolvencycapital or reserve

5. Insurance worksbecause risk can be diversifiedawaythroughsize (3.2.4 EB) • 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 coefficientofvariation (shows extentofvariability in relation to themean) approaches 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

6. Howarerandom variables sampled? • Inversion (2.3.2 in EB): • Let F(x) be a strictlyincreasingdistributionfunctionwith inverse and let • Considerthespecificationontheleft for which U=F(X) • Note that since 1 F(x) U[0,1]

7. Outlineofthecourse Binomial models Monte Carlo simulation

8. Course literature Curriculum: Chapter 1.2, 2.3.1, 2.3.2, 2.5, 3.2, 3.3 in EB Chapter 8,9,10 in EB Note onChain Ladder Lecture notes by NFH Exercises Assignment must be approved to be able to participate in exame The following book will be used (EB): Computation and Modelling in Insurance and Finance, Erik Bølviken, Cambridge University Press (2013) • Additions to the list abovemayoccur during thecourse • Final curriculum will be postedonthecourse web site in due time

9. Overviewofthissession The Poisson model (Section 8.2 EB) Someimportantnotions and somepracticetoo Examplesofclaimfrequencies Random intensities (Section 8.3 EB)

10. Introduction • Actuarialmodelling in general insurance is broken downonclaimfrequency and claimsize • This is natural due to definitionof risk premium: • The Poissondistribution is often used in modellingthedistributionofclaimnumbers • The parameter is lambda = muh*T (single policy) and lambda = J*muh*T (portfolios • The modellingcan be made more sophisticated by extendingthemodel for muh, either by making muhstochastic or by linking muh to explanatory variables Poisson Somenotions Risk premium Expectedclaimamount given an event Examples • Probabilityofclaim, • Estimatedwithclaimfrequency Expectedconsequenceofclaim Random intensities

11. 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

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

13. The world ofPoisson • Let usproceedremovingthezero/onerestrictiononIk. A more flexiblespecification is Where o(h) signifies a mathematicalexpression for which Poisson Somenotions It is verified in Section 8.6 that o(h) does not count in the limit Examples Consider a portfoliowith J policies. Therearenow J independent processes in parallel and if is theintensityof policy j and Ikthe total numberofclaims in period k, then Random intensities No claims Claims policy i only

14. The world ofPoisson • Bothquanitiessimplifywhentheproductsarecalculated and thepowersof h identified Poisson Somenotions Examples Random intensities • It followsthattheportfolionumberofclaimsN is Poissondistributedwith parameter • Whenclaimintensitiesvary over theportfolio, onlytheiraveragecounts

15. Whentheintensityvaries over time • A time varyingfunction handles themathematics. The binary variables I1,...Ikarenowbasedondifferentintensities • When I1,...Ikareadded to the total count N, this is the same issue as if K differentpoliciesapplyon an intervaloflength h. In otherwords, N must still be Poisson, nowwith parameter Poisson Somenotions Examples wherethe limit is how integrals aredefined. The Poisson parameter for N canalso be written Random intensities And theintroductionof a time-varyingfunctiondoesn’tchangethingsmuch. A time averagetakes over from a constant

16. 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 ~

17. Client Policies and claims Policy Poisson Somenotions Insurableobject (risk) Claim Examples Random intensities Insurance cover Cover element /claim type

18. 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

19. Some notes onthe different insurance covers ontheprevious slide:Third part liability is a mandatory cover dictated by Norwegian lawthat covers damagesonthird part vehicles, propterty and person. Someinsurancecompaniesprovideadditionalcoverage, as legal aid and driver and passengeraccidentinsurance.Partial Hull covers everythingthatthethird part liability covers. In addition, partial hull covers damagesonownvehiclecaused by fire, glass rupture, theft and vandalism in associationwiththeft. Partial hull alsoincludesrescue. Partial hull does not cover damageonownvehiclecaused by collision or landing in theditch. Therefore, partial hull is a more affordable cover thanthe Hull cover. Partial hull also cover salvage, home transport and helpassociatedwithdisruptions in production, accidents or disease. Hull covers everythingthatpartial hull covers. In addition, Hull covers damagesonownvehicle in a collision, overturn, landing in a ditch or other sudden and unforeseendamage as for example fire, glass rupture, theft or vandalism. Hull mayalso be extended to cover rentalcar. Some notes onsomeimportantconcepts in insurance: What is bonus?Bonus is a reward for claim-free driving. For everyclaim-freeyearyouobtain a reduction in theinsurancepremium in relation to the basis premium. This continuesuntil 75% reduction is obtained. What is deductible?The deductible is theamountthe policy holder is responsible for when a claimoccurs. Doesthedeductibleimpacttheinsurancepremium?Yes, by selecting a higherdeductiblethanthedefaultdeductible, theinsurancepremiummay be significantlyreduced. The higherdeductibleselected, thelowertheinsurancepremium. How is thedeductibletakenintoaccountwhen a claim is disbursed?The insurancecompanycalculatesthe total claimamountcaused by a damageentitled to disbursement. Whatyouget from theinsurancecompany is thenthecalculated total claimamount minus theselecteddeductible. Poisson Somenotions Examples Random intensities

20. Key ratios – claimfrequency • The graph shows claimfrequency for all covers for motor insurance • Noticeseasonalvariations, due to changingweatherconditionthroughouttheyears Poisson Somenotions Examples Random intensities

21. Key ratios – claimseverity • The graph shows claimseverity for all covers for motor insurance Poisson Somenotions Examples Random intensities

22. Key ratios – pure premium • The graph shows pure premium for all covers for motor insurance Poisson Somenotions Examples Random intensities

23. Key ratios – pure premium • The graph shows loss ratio for all covers for motor insurance Poisson Somenotions Examples Random intensities

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

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

26. 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

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

28. wikipedia tellsushowtheruleof double variancecan be proved Poisson Somenotions Examples Random intensities

29. The ruleof double variance Var(Y) willnow be proved from theruleof double expectation. Introduce which is simplytheruleof double expectation. Clearly Poisson Somenotions Passing expectations over thisequalityyields Examples where Random intensities whichwill be handledseparately. First note that and by theruleof double expectationapplied to The second term makes useofthefactthat by theruleof double expectation so that

30. The ruleof double variance The final term B3 makes useoftheruleof double expectationonceagainwhichyields where Poisson Somenotions And B3=0. The secondequality is true becausethefactor is fixed by X. Collectingtheexpression for B1, B2 and B3 proves the double varianceformula Examples Random intensities

31. 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 intensityifindividual j is thenestimated as . Somenotions Uncertainty is huge. One solution is Examples (1.5) Random intensities and (1.6) Bothestimatesareunbiased. SeeSection 8.6 for details. 10.5 returns to this.

32. 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).

33. 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.