Non-life insurance mathematics

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

2. 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

3. 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

4. Normal approximations

5. 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

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

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

8. Fire data from DNB

9. Normal approximations in R z=scan("C:/Users/wenche_adm/Desktop/Nils/uio/Exercises/Branntest.txt"); # removes negatives nyz=ifelse(z>1,z,1.001); mu=0.0065; T=1; ksiZ=mean(nyz); sigmaZ=sd(nyz); a0 = mu*T*ksiZ; a1 = sqrt(mu*T)*sqrt(sigmaZ^2+ksiZ^2); J=5000; qepsNO95=a0*J+a1*qnorm(.95)*sqrt(J); qepsNO99=a0*J+a1*qnorm(.99)*sqrt(J); qepsNO9997=a0*J+a1*qnorm(.9997)*sqrt(J); c(qepsNO95,qepsNO99,qepsNO9997);

10. The normal powerapproximation ny3hat = 0; n=length(nyz); for (i in 1:n) { ny3hat = ny3hat + (nyz[i]-mean(nyz))**3 } ny3hat = ny3hat/n; LargeKsihat=ny3hat/(sigmaZ**3); a2 = (LargeKsihat*sigmaZ**3+3*ksiZ*sigmaZ**2+ksiZ**3)/(sigmaZ^2+ksiZ^2); qepsNP95=a0*J+a1*qnorm(.95)*sqrt(J)+a2*(qnorm(.95)**2-1)/6; qepsNP99=a0*J+a1*qnorm(.99)*sqrt(J)+a2*(qnorm(.99)**2-1)/6; qepsNP9997=a0*J+a1*qnorm(.9997)*sqrt(J)+a2*(qnorm(.9997)**2-1)/6; c(qepsNP95,qepsNP99,qepsNP9997);

11. 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?

12. 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)

13. Non parametric Log-normal, Gamma The Pareto Extreme value Searching

14. Non parametric Log-normal, Gamma The Pareto Extreme value Searching

15. Non parametric Log-normal, Gamma The Pareto Extreme value Searching

16. Searching for themodel • How is the final model for claimsizeselected? • Traditionaltools: QQ plots and criterioncomparisons • Transformationsmayalso be used (see Erik Bølviken’s material) Non parametric Log-normal, Gamma The Pareto Extreme value Searching

17. Non parametric Log-normal, Gamma The Pareto Extreme value Searching

18. Results from top 12% modelling Weibull is best in top 12% modelling Non parametric Log-normal, Gamma The Pareto Extreme value Searching

19. 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

20. Checkout bimodal distributionsonwikipedia

21. Comparisonofresults

22. Or…checkoutmixturedistributionsonwikipedia

23. Solvency – day 2

24. 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

25. Structure • Normal approximation • Monte Carlo Theory • Monte Carlo Practice – an examplewith fire data from DNB

26. Normal approximations

27. Normal approximations This leads to the true percentileqepsilonbeingapproximated by Where phi epsilon is theupperepsilonpercentileofthe standard normal distribution Poisson Somenotions Examples Random intensities

28. 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)

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

30. Monte Carlo theory (1)

31. 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)

32. 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

33. Comparisonofresults