Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

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1 No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig

2 Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy be modelled? Poisso with determiistic muh Poisso with stochastic muh Poisso regressio

3 Isurace works because risk ca be diversified away through size The core idea of isurace is risk spread o may uits Assume that policy risks X1,,XJ are stochastically idepedet Mea ad variace for the portfolio total are the E ( ) 1... J ad var( ) 1... ad E( X ) ad 1 (... J ) J sd( X ). Itroduce 2 1 ad ( 1... J 1 J which is average expectatio ad variace. The sd( ) / E( ) J ad sd( ) J so that E( ) J The coefficiet of variatio approaches 0 as J grows large (law of large umbers) Isurace risk ca be diversified away through size Isurace portfolios are still ot risk-free because of ucertaity i uderlyig models risks may be depedet J )

4 Risk premium expresses cost per policy ad is importat i pricig Risk premium is defied as P(Evet)*Cosequece of Evet More formally Risk premium P(evet)*Cosequece of Claim frequecy Number of claims Total claim amout * Number of risk years Number of claims Total claim amout Number of risk years *Claim severity evet From above we see that risk premium expresses cost per policy Good price models rely o soud uderstadig of the risk premium We start by modellig claim frequecy

5 The world of Poisso (Chapter 8.2) Number of claims Poisso Some otios Examples Radom itesities I k-1 I k I k+1 t 0 =0 t k-2 t k-1 t k t k+1 t k =T What is rare ca be described mathematically by cuttig a give time period T ito small pieces of equal legth h=t/ O short itervals the chace of more tha oe icidet is remote Assumig o more tha 1 evet per iterval the cout for the etire period is N=I I,where I is either 0 or 1 for =1,..., If p=pr(i k =1) is equal for all k ad evets are idepedet, this is a ordiary Beroulli series Pr( N )! p!( )! (1 p), for 0,1,..., Assume that p is proportioal to h ad set is a itesity which applies per time uit p h where 5

6 The world of Poisso 6 T T T T T p p N ) ( 1) (! ) ( 1 )!!(! ) (1 )!!(! ) Pr( 1 1 T e T e T N! ) ( ) Pr( I the limit N is Poisso distributed with parameter T Some otios Examples Radom itesities Poisso

7 The world of Poisso Poisso Some otios Examples Radom itesities It follows that the portfolio umber of claims N is Poisso distributed with parameter ( J ) T JT, where (... J ) / J Whe claim itesities vary over the portfolio, oly their average couts 7

8 Poisso Radom itesities (Chapter 8.3) Some otios Examples Radom itesities How varies over the portfolio ca partially be described by observables such as age or sex of the idividual (treated i Chapter 8.4) There are however factors that have impact o the risk which the compay ca t kow much about Driver ability, persoal risk averseess, This radomeess ca be maaged by makig a stochastic variable 1 2 N ~ Poisso ( 2 2 T ) N ~ Poisso ( 1 1 T )

9 Poisso Radom itesities (Chapter 8.3) Some otios Examples Radom itesities The models are coditioal oes of the form N ~ Poisso ( T ) ad Ν ~ Poisso ( JT ) Let E( ) Policy level ad sd( ) ad recall Portfolio level that E( N ) var( N ) T which by double rules i Sectio 6.3 imply E 2 2 ( N) E( T ) T ad var( N) E( T ) var( T ) T T Now E(N)<var(N) ad N is o loger Poisso distributed 9

10 The fair price The Poisso regressio model (Sectio 8.4) The model A example Why regressio? The idea is to attribute variatio i to variatios i a set of observable variables x1,...,xv. Poisso regresso makes use of relatioships of the form log() log( ) b b x... b x v v Why ad ot itself? The expected umber of claims is o-egative, where as the predictor o the right of (1.12) ca be aythig o the real lie It makes more sese to trasform so that the left ad right side of (1.12) are more i lie with each other. Historical data are of the followig form (1.12) Repetitio of GLM 1 T1 x11...x1x 2 T2 x21...x2x T x1...xv Claims exposure covariates The coefficiets b0,...,bv are usually determied by likelihood estimatio 10

11 The fair price The model (Sectio 8.4) The model A example Why regressio? I likelihood estimatio it is assumed that is Poisso distributed where is tied to covariates x1,...,xv as i (1.12). The desity fuctio of is the or ( T ) f ( ) exp( T )! T Repetitio of GLM log( f ( )) log( ) log( T ) log(!) T log(f()) above is to be added over all for the likehood fuctio L(b0,...,bv). Skip the middle terms T ad log (!) sice they are costats i this cotext. The the likelihood criterio becomes L ( b0,..., bv ) { log( ) T} where log( ) b0 b1 x b Numerical software is used to optimize (1.13). McCullagh ad Nelder (1989) proved that L(b0,...,bv) is a covex surface with a sigle maximum Therefore optimizatio is straight forward. x v (1.13) 11

12 Mai steps How to build a model How to evaluate a model How to use a model

13 How to build a regressio model Select detail level Variable selectio Groupig of variables Remove variables that are strogly correlated Model importat iteractios

14 Cliet Select detail level Policies ad claims Poisso Some otios Examples Radom itesities Policy Isurable obect (risk) Claim Isurace cover Cover elemet /claim type Step 1 i model buildig: Select detail level

15 PP: Select detail level PP: Review potetial risk drivers Groupig of variables PP: Select groups for each risk driver PP: Select large claims strategy PP: idetify potetial iteractios PP: costruct fial model Price assessmet 250 Number of water claims buildig age

16 PP: Select detail level PP: Review potetial risk drivers Groupig of variables PP: Select groups for each risk driver PP: Select large claims strategy PP: idetify potetial iteractios PP: costruct fial model Price assessmet Exposure i risk years buildig age

17 PP: Select detail level PP: Review potetial risk drivers Groupig of variables PP: Select groups for each risk driver PP: Select large claims strategy PP: idetify potetial iteractios PP: costruct fial model Price assessmet 12,00 Claim frequecy water claims buildig age 10,00 8,00 6,00 4,00 2,00 0,00

18 PP: Select detail level PP: Review potetial risk drivers Model importat iteractios PP: Select groups for each risk driver PP: Select large claims strategy PP: idetify potetial iteractios PP: costruct fial model Price assessmet Defiitio: Cosider a regressio model with two explaatory variables A ad B ad a respose Y. If the effect of A (o Y) depeds o the level of B, we say that there is a iteractio betwee A ad B Example (house ower): The risk premium of ew buildigs are lower tha the risk premium of old buildigs The risk premium of youg policy holders is higher tha the risk premium of old policy holders The risk premium of youg policy holders i old buildigs is particularly high The there is a iteractio betwee buildig age ad policy holder age

19 PP: Select detail level PP: Review potetial risk drivers Model importat iteractios PP: Select groups for each risk driver PP: Select large claims strategy PP: idetify potetial iteractios PP: costruct fial model Price assessmet

20 How to evaluate a regressio model QQplot Akaike Iformatio Criterio (AIC) Scaled deviace Cross validatio Type 3 aalysis Results iterpretatios

21 QQplot The quatiles of the model are plotted agaist the quatiles from a stadard Normal distributio If the model is good, the poit i the QQ plot will lie approximately o the lie y=x

22 AIC Akaike Iformatio criterio is a measure of goodess of fit that balaces model fit agaist model simplicity AIC has the form AIC=-2LL+2p where LL is the log likelihood evaluated at the value fo the estimated parameters p is the umber of parameters estimated i the model AIC is used to compare model alteratives m1 ad m2 If AIC of m1 is less tha AIC of m2 the m1 is better tha m2

23 Scaled deviace Scaled deviace = 2(l(y,y)-l(y,muh)) where l(y,y) is the maximum achievable log likelihood ad l(y,muh) is the log likelihood at the maximum estimates of the regressio parameters Scaled deviace is approximately distributed as a Chi Square radom variables with -p degrees of freedom Scaled deviace should be close to 1 if the model is good

24 Cross validatio Estimatio Validatio Model is calibrated o for example 50% of the portfolio Model is the validated o the remaiig 50% of the portfolio For a good model that predicts well there should ot be too much differece betwee the modelled umber of claims i a group ad the observed umber of claims i the same group

25 Type 3 aalysis Does the degree of variatio explaied by the model icrease sigificatly by icludig the relevat explaatory variable i the model? Type 3 aalysis tests the fit of the model with ad without the relevat explaatory variable A low p value idicates that the relevat explaatory variable improves the model sigificatly

26 Results iterpretatio Is the itercept reasoable? Are the parameter estimates reasoable compared to the results of the oe way aalyses?

27 How to use a regressio model Smoothig of estimates

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