STK 4540 Lecture 3

1. STK 4540 Lecture3 Uncertaintyondifferentlevels And Randomintensities in theclaimfrequency

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. 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

4. 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

5. 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

6. 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 ~

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

8. 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

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

10. 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

11. 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

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

13. 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

14. 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

15. 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.

16. 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).

17. 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.