2001 CAS Seminar on Ratemaking - Las Vegas, Nevada1 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, Nevada2 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, Nevada3 Data Methodology Knowledge/Experience of “Curve Fitter” Time Purpose Process of fitting distributions to losses:

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada4 Data – Situation 1 # Loss , ,000, ,500

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada5 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, Nevada6 Policy #DeductibleLimit Loss , ,000 10,000, , ,000 41,000 5,000,000 5,000, , ,000 1,000,0004,500 Data – Situation 2

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada7 “The” distribution! Ranking alternative distributions An“overall” measure of fit Akaike’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, Nevada8 Likelihood Incomplete data Proper specification of the Likelihood Function for data that is “Incomplete” Maximum Likelihood Estimation, MLE Estimation of Model Parameters

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada9 y i : i th loss amount (incurred value) D i : deductible for the i th loss PL i : policy limit for the i th loss f(y i ; ,  ): density function  : primary parameter of interest  : nuisance parameter F(y i ; ,  ): cumulative distribution function Notations

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada10 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, Nevada11 Case 2: A deductible, and loss below policy limit (left truncated data)

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada12 Case 3: No deductible, and loss capped by policy limit (right censored data)

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada13 Case 4: A deductible, and loss capped by policy limit (left truncated and right censored data)

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada14 Likelihood Function

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada15 Iterative solution, “Solver” Initial Parameter Values Convergence Uniqueness Robustness Maximum Likelihood Estimation

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada16 Incorporating rating variables into fitting process Data – Situation 3 Policy #Deduct. Limit Loss Constr. Prot. Occupancy , ,000 10M , , ,000 5M 5M , ,000 1M 4,

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada17 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, Nevada18 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, Nevada19 An example: Commercial Loss Fire Data Rating variables: Construction Building Value -- Exposure

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada20 Linear Predictors 4 linear predictors; 4 statistical models: A, B, C, D

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada21

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada22 Estimation of parameters: Lognormal:  and  From:  and  to beta_0, beta-1, beta_2, beta_3 & 

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada23 Assessing the effect of Rating Variables Nested models

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada24 Nested Hypotheses based on Model D Test of Hypothesis

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada25 mydata =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, Nevada26 delta1 0)*(y = PL) delta4 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.D value: parameters: b0 b1 b2 b3 sigma formula: ~ lognormal.model.D(b0, b1, b2, b3, sigma) 100 observations call: 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