CAS Seminar on Ratemaking Las Vegas Nevada March

CAS Seminar on Ratemaking Las Vegas, Nevada March 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 1

Why? 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. 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada 2

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

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

Data – Situation 1 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 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada 5

Data – Situation 2 # Deductible Policy Limit 1 2 3 4 5 6 0 100, 000 0 10, 000 100, 000 5, 000 250, 000 1, 000 Loss 112 117 100, 000 5, 000 430 4, 500 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada 6

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

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

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

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 10

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

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

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

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

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

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

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

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

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

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

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada 21

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

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

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

Appendix B - Exhibit 2 A mydata<-Table. A m<-data. frame(mydata) lognormal. model. D <- function(b 0, b 1, b 2, b 3, 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] C 1 <- cnst == 1 C 2 <- cnst == 2 d <-D+(D == 0)*1 mu <- b 0+b 1*log(PL)+b 2*C 1+b 3*C 2 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada 25

Appendix B - Exhibit 2 B delta 1 <- (D == 0)*(y < PL) delta 2 <- (D > 0)*(y < PL) delta 3 <- (D == 0)*(y >= PL) delta 4 <- (D > 0)*(y >= PL) L 1 <- dlnorm(z, mu, sigma) L 2 <- dlnorm(z, mu, sigma)/(1 -plnorm(d, mu, sigma)) L 3 <- 1 -plnorm(z, mu, sigma) L 4 <- (1 -plnorm(z, mu, sigma))/(1 -plnorm(d, mu, sigma)) log. L <-delta 1*log(L 1)+delta 2*log(L 2)+delta 3*log(L 3)+delta 4*log(L 4) -log. L } min. model. D<-ms(~lognormal. model. D(b 0, b 1, b 2, b 3, sigma), data=m, start=list(b 0=4. 568, b 1=0. 238, b 2=1. 068, b 3=0. 0403, sigma=1. 322)) min. model. D value: 892. 7099 parameters: b 0 b 1 b 2 b 3 sigma 1. 715296 0. 3317345 2. 154994 0. 4105021 1. 898501 formula: ~ lognormal. model. D(b 0, b 1, b 2, b 3, sigma) 100 observations call: ms(formula = ~ lognormal. model. D(b 0, b 1, b 2, b 3, sigma), data=m, start =list(b 0=4. 568, b 1=0. 238, b 2=1. 068, b 3=0. 0403, sigma=1. 322)) 2001 CAS Seminar on Ratemaking - Las Vegas, Nevada 26