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 Solvecy Fiacial cotrol of liabilities uder early worst-case scearios Target: the reserve which is the upper percetile of the portfolio liability Modellig has bee covered (Risk premium calculatios) The issue ow is computatio Mote Carlo is the geeral tool Some problems ca be hadled by simpler, Gaussia approximatios

3 0. Portfolio liabilities by simple approximatio The portfolio loss for idepedet risks become Gaussia as J teds to ifiity. Assume that policy risks X,,XJ are stochastically idepedet Mea ad variace for the portfolio total are the E ( )... J ad var( )... ad j E( X j ) ad (... J ) J j sd( X j ). Itroduce ad (... J J which is average expectatio ad variace. The J d X N(, i J i as J teds to ififity Note that risk is uderestimated for small portfolios ad i braches with large claims ) J )

4 where Normal approximatios Let be claim itesity ad ad of the idividual losses.if they are thesame for all policy holders, the mea ad stadard deviatio of Χ over a period of legth T become E( Χ) a J, sd( Χ) a 0 J mea ad stadard deviatio a 0 T ad a T

5 Poisso The rule of double variace Some otios Examples Radom itesities Let X ad Y be arbitrary radom variables for which ( x) E( Y x) ad var( Y x) The we have the importat idetities E( Y) E{ ( X )} Rule of double expectatio ad var(y) E{ ( X )} var{ ( X )} Rule of double variace 5

6 Poisso The rule of double variace Some otios Examples Radom itesities Portfolio risk i geeral isurace Z Let E( Z Z )... Z Z where, Z where var( ) Elemetary rules for radom sums imply, Z,... Z are stochastically idepedet. E ( N) N ad var( N) N Let Y ad x i the formulas o the previous slide var( ) var{ E( N)} E{ sd( N)} var( ) E( var( ) E( ) JT ( ) ) 6

7 Poisso The rule of double variace Some otios Examples Radom itesities This leads to the true percetile qepsilo beig approximated by q NO a 0 J a J Where phi epsilo is the upper epsilo percetile of the stadard ormal distributio 7

8 Desity Fire data from DNB desity.default(x = log(y)) N = 75 Badwidth = 0.366

9 Normal approximatios i R =sca("c:/users/weche_adm/desktop/nils/uio/exercises/bratest.txt"); # removes egatives y=ifelse(>,,.00); mu=0.0065; T=; ksiz=mea(y); sigmaz=sd(y); a0 = mutksiz; a = sqrt(mut)sqrt(sigmaz^+ksiz^); J=5000; qepsno95=a0j+aqorm(.95)sqrt(j); qepsno99=a0j+aqorm(.99)sqrt(j); qepsno9997=a0j+aqorm(.9997)sqrt(j); c(qepsno95,qepsno99,qepsno9997);

10 The ormal power approximatio y3hat = 0; =legth(y); for (i i :) { y3hat = y3hat + (y[i]-mea(y))3 } y3hat = y3hat/; LargeKsihat=y3hat/(sigmaZ3); a = (LargeKsihatsigmaZ3+3ksiZsigmaZ+ksiZ3)/(sigmaZ^+ksiZ^); qepsnp95=a0j+aqorm(.95)sqrt(j)+a(qorm(.95)-)/6; qepsnp99=a0j+aqorm(.99)sqrt(j)+a(qorm(.99)-)/6; qepsnp9997=a0j+aqorm(.9997)sqrt(j)+a(qorm(.9997)-)/6; c(qepsnp95,qepsnp99,qepsnp9997); Percetile 95 % 99 % 99.97% Normal approximatios Normal power approximatios

11 Portfolio liabilities by simulatio Mote Carlo simulatio Advatages More geeral (o restrictio o use) More versatile (easy to adapt to chagig circumstaces) Better suited for loger time horios Disadvatages Slow computatioally? Depedig o claim sie distributio?

12 A algorithm for liabilities simulatio Assume claim itesities,..., J for J policies are stored o file Assume J differet claim sie distributios ad paymet fuctios H(),,HJ() are stored The program ca be orgaised as follows (Algorithm 0.) Iput: T ( j,.., J ), claim sie models, 0 Retur j For j,..., J do Draw U ~ Uiform ad Repeat while Draw claim sie Z j Draw U S H j j ( ) S ~ Uiform ad log( U S S ) H log( U ( ),..., H ) J ( )

13 Frequecy No parametric Log-ormal, Gamma The Pareto Extreme value Searchig 700 Fire up to 88 percetile Bi

14 Frequecy No parametric Log-ormal, Gamma The Pareto Extreme value Searchig 80 Fire above 88th percetile Bi

15 More Frequecy No parametric Log-ormal, Gamma The Pareto Extreme value Searchig Fire above 95th percetile Bi

16 Searchig for the model How is the fial model for claim sie selected? Traditioal tools: QQ plots ad criterio comparisos Trasformatios may also be used (see Erik Bølvike s material) No parametric Log-ormal, Gamma The Pareto Extreme value Searchig 6

17 No parametric Log-ormal, Gamma The Pareto Extreme value Searchig Descriptive Statistics for Variable Skadeestimat Number of Observatios 85 Number of Observatios Used for Estimatio 85 Miimum Maximum Mea Stadard Deviatio

18 Results from top % modellig No parametric Log-ormal, Gamma The Pareto Extreme value Searchig Weibull is best i top % modellig Model Selectio Table Distributio Coverged - Log Likelihood Selected Burr Yes 5808 No Log Yes 5807 No Exp Yes 587 No Gamma Yes 5799 No Igauss Yes 5804 No Pareto Yes 5874 No Weibull Yes 5799 Yes

19 Experimets i R. Log ormal distributio. Gamma o log scale 3. Pareto 4. Weibull 5. Mixed distributio 6. Mote Carlo algorithm for portfolio liabilities 7. Mixed distributio 9

20 Check out bimodal distributios o wikipedia

21 Compariso of results Percetile 95 % 99 % 99.97% Normal approximatios Normal power approximatios Mote Carlo algorithm log ormal claims Mote Carlo algorithm gamma model for log claims Mote Carlo algorithm mixed empirical ad Weibull Mote Carlo algorithm empirical distributio

22 Or check out mixture distributios o wikipedia

23 Solvecy day

24 Solvecy Fiacial cotrol of liabilities uder early worst-case scearios Target: the reserve which is the upper percetile of the portfolio liability Modellig has bee covered (Risk premium calculatios) The issue ow is computatio Mote Carlo is the geeral tool Some problems ca be hadled by simpler, Gaussia approximatios

25 Structure Normal approximatio Mote Carlo Theory Mote Carlo Practice a example with fire data from DNB

26 where Normal approximatios Let be claim itesity ad ad of the idividual losses.if they are thesame for all policy holders, the mea ad stadard deviatio of Χ over a period of legth T become E( Χ) a J, sd( Χ) a 0 J mea ad stadard deviatio a 0 T ad a T

27 Poisso Normal approximatios Some otios Examples Radom itesities This leads to the true percetile qepsilo beig approximated by q NO a 0 J a J Where phi epsilo is the upper epsilo percetile of the stadard ormal distributio 7

28 Mote Carlo theory Suppose X, X, are idepedet ad expoetially distributed with mea. It ca the be proved Pr( X... X X... X ) e! for all >= 0 ad all lambda > 0. From () we see that the expoetial distributio is the distributio that describes time betwee evets i a Poisso process. I Sectio 9.3 we leart that the distributio of X+ +X is gamma distributed with mea ad shape The Poisso process is a process i which evets occur cotiuously ad idepedetly at a costat average rate The Poisso probabilities o the right defie the desity fuctio Pr( N ) e!, 0,,,... which is the cetral model for claim umbers i property isurace. Mea ad stadard deviatio are E(N)=lambda ad sd(n)=sqrt(lambda) ()

29 Mote Carlo theory It is the utilied that Xj=-log(Uj) is expoetial if Uj is uiform, ad the sum X+X+ is moitored util it exceeds lambda, i other words Algorithm.4 Poisso geerator Iput: Y 0 For Draw U If,,... Y do ~ Uiform the stopad retur N ad Y Y log( U )

30 Proof of Algorithm.4 Let commo desity fuctio f is But ( ) ( s), the ( ) (sectio 9.3). This meas that ( ) Pr( S Pr( s X as was to be proved. Mote Carlo theory ( x) e The Poisso geerator of Algorithm.4 is based o the probability f p p S S is Gamma distributed with mea 0 0 e ( s) s e s ) f,..., X -x s) f ( s) s /( )! ds which ca be evaluated by coditioig o S p S X ( s) ds e ( )! be stochastically idepedet with for x 0 ad let S e s. If its desity fuctio ( s) ds. ad shape /( )!ad 0 s 0 e ( s) f X ds e!... X ().

31 A algorithm for liabilities simulatio Assume claim itesities,..., J for J policies are stored o file Assume J differet claim sie distributios ad paymet fuctios H(),,HJ() are stored The program ca be orgaised as follows (Algorithm 0.) Iput: T ( j,.., J ), claim sie models, 0 Retur j For j,..., J do Draw U ~ Uiform ad Repeat while Draw claim sie Z j Draw U S H j j ( ) S ~ Uiform ad log( U S S ) H log( U ( ),..., H ) J ( )

32 Experimets i R. Log ormal distributio. Gamma o log scale 3. Pareto 4. Weibull 5. Mixed distributio 6. Mote Carlo algorithm for portfolio liabilities 7. Mixed distributio 3

33 Compariso of results Percetile 95 % 99 % 99.97% Normal approximatios Normal power approximatios Mote Carlo algorithm log ormal claims Mote Carlo algorithm gamma model for log claims Mote Carlo algorithm mixed empirical ad Weibull Mote Carlo algorithm empirical distributio

Simulation. Two Rule For Inverting A Distribution Function

Simulation. Two Rule For Inverting A Distribution Function Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump