CAS Seminar on Ratemaking Las Vegas, Nevada March 11-13, 2001

1. CAS Seminar on RatemakingLas Vegas, NevadaMarch 11-13, 2001 Fitting to Loss Distributions with Emphasis on Rating Variables Farrokh Guiahi, Ph.D., F.C.A.S, A.S.A. 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

2. Fitting distributions to insurance data serves an important function for the purpose of pricing insurance products.The effect of the rating variables upon loss distributions has important implications for underwriting selection.It also provides for a more differentiated rating system. Why? 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

3. Data MethodologyKnowledge/Experience of “Curve Fitter”Time Purpose Process of fitting distributions to losses: 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

4. Data – Situation 1# Loss 1 112 2 107 3 100,000 4 5,000,000 5 430 6 4,500 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

5. Ask questions about the data: Losses in excess of deductible? Losses capped by policy limit? etc. Insurance Data are usually “Incomplete” Left truncated Right Censored Data – Situation 1 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

6. Policy# Deductible Limit Loss1 0 100,000 1122 100,000 10,000,000 117 3 0 100,000 100,0004 1,000 5,000,000 5,000,0005 0 250,000 4306 10,000 1,000,000 4,500 Data – Situation 2 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

7. “The” distribution!Ranking alternative distributionsAn“overall” measure of fitAkaike’s Information Criterion, AIC AIC = - 2 (maximized log-likelihood) + 2 (number of parameters estimated) Selection of a parametric distribution 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

8. Incomplete data Proper specification of the LikelihoodFunction for data that is “Incomplete” Maximum Likelihood Estimation, MLE Estimation of Model Parameters 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

9. yi : ith loss amount (incurred value) Di : deductible for the ith lossPLi : policy limit for the ith lossf(yi ;,): density function  : primary parameter of interest: nuisance parameter F(yi ;,): cumulative distribution function Notations 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

10. Case 1: No deductible, and loss below policy limit (neither left truncated nor right censored data) The complete sample case. 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

11. Case 2:A deductible, and loss below policy limit (left truncated data) 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

12. Case 3:No deductible, and loss capped by policy limit (right censored data) 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

13. Case 4:A deductible, and loss capped by policy limit (left truncated and right censored data) 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

14. Likelihood Function 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

15. Iterative solution, “Solver” Initial Parameter Values Convergence Uniqueness Robustness Maximum Likelihood Estimation 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

16. Incorporating rating variables into fitting process Data – Situation 3 Policy # Deduct. Limit Loss Constr. Prot. Occupancy 1 0 100,000 112 1 2 23 2 100,000 10M 117 2 1 33 3 0 100,000 100,000 1 6 16 4 1,000 5M 5M 3 3 8 5 0 250,000 430 1 4 70 6 10,000 1M 4,500 2 2 40 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

17. Incorporating rating variables into fitting process Approaches: Subdividing data Using all of data to estimate model parameters simultaneously. 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

18. Relating rating variables to a parameter of the selected loss distribution Rating variables: Quantitative Qualitative Generalized Linear Modeling 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

19. An example: Commercial Loss Fire Data Rating variables: Construction Building Value -- Exposure 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

20. Linear Predictors4 linear predictors; 4 statistical models: A, B, C, D 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

21. 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

22. Estimation of parameters: Lognormal:  and  From:  and  to beta_0, beta-1, beta_2, beta_3 &  2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

23. Assessing the effect of Rating Variables Nested models 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

24. Nested Hypotheses based on Model D Test of Hypothesis 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

25. mydata<-TableA m<-data.frame(mydata) lognormal.model.D <- function(b0,b1,b2,b3,sigma, data=data.matrix) { D <- data.matrix[,1] PL <- data.matrix[,2] y <- data.matrix[,3] z <- D+(y*(y<PL)+PL*(y>=PL)) cnst <- data.matrix[,4] C1 <- cnst == 1 C2 <- cnst == 2 d <-D+(D == 0)*1 mu <- b0+b1*log(PL)+b2*C1+b3*C2 Appendix B - Exhibit 2A 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

26. delta1 <- (D == 0)*(y < PL) delta2 <- (D > 0)*(y < PL) delta3 <- (D == 0)*(y >= PL) delta4 <- (D > 0)*(y >= PL) L1 <- dlnorm(z,mu,sigma) L2 <- dlnorm(z,mu,sigma)/(1-plnorm(d,mu,sigma)) L3 <- 1-plnorm(z,mu,sigma) L4 <- (1-plnorm(z,mu,sigma))/(1-plnorm(d,mu,sigma)) logL <-delta1*log(L1)+delta2*log(L2)+delta3*log(L3)+delta4*log(L4) -logL }min.model.D<-ms(~lognormal.model.D(b0,b1,b2,b3,sigma), data=m, start=list(b0=4.568, b1=0.238, b2=1.068, b3=0.0403, sigma=1.322))min.model.Dvalue: 892.7099 parameters: b0 b1 b2 b3 sigma 1.715296 0.3317345 2.154994 0.4105021 1.898501formula: ~ lognormal.model.D(b0, b1, b2, b3, sigma) 100 observationscall: ms(formula = ~ lognormal.model.D(b0, b1, b2, b3, sigma), data=m, start =list(b0=4.568, b1=0.238, b2=1.068, b3=0.0403, sigma=1.322)) Appendix B - Exhibit 2B 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada