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

2001 CAS Seminar on Ratemaking - Las Vegas, Nevada

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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.


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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?


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Data

Methodology

Knowledge/Experience of “Curve Fitter”

Time

Purpose

Process of fitting distributions to losses:


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Data – Situation 1# Loss1 1122 1073 100,0004 5,000,0005 4306 4,500


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Ask questions about the data:Losses in excess of deductible?Losses capped by policy limit?etc.

Insurance Data are usually “Incomplete”Left truncatedRight Censored

Data – Situation 1


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Policy# Deductible Limit Loss

1 0 100,000 1122 100,000 10,000,000 1173 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


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“The” distribution!Ranking alternative distributionsAn“overall” measure of fit

Akaike’s Information Criterion, AIC

AIC = - 2 (maximized log-likelihood) + 2 (number of parameters

estimated)

Selection of a parametric distribution


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Incomplete data

Proper specification of the LikelihoodLikelihood Function for data that is “Incomplete”

Maximum Likelihood Estimation, MLE

Estimation of Model Parameters


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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

Notations


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Case 1: No deductible, and loss below policy limit (neither left truncated nor right censored data) The complete sample case.

L f yi i1

( ; , )

Otherwise ,0

PL and 0D If ,1 ii1

ii

y


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Case 2: A deductible, and loss below policy limit (left truncated data)

L f yFi

i2 1

( ; , )( ; , )

DD

i

i

Otherwise ,0

PL and 0D If ,1 ii2

ii

y


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Case 3: No deductible, and loss capped by policy limit (right censored data)

L Fi i3

1 ( ; , )PL

Otherwise ,0

PL and 0D If ,1 ii3

ii

y


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Case 4: A deductible, and loss capped by policy limit (left truncated and right censored data)

LFFi

i4

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( ; , )

( ; , )D PL

Di

i

Otherwise ,0

PL and 0D If ,1 ii4

ii

y


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Likelihood Function

L L L L Li i i i ii i i i 1 2 3 41 2 3 4

L Lii


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Iterative solution, “Solver”

Initial Parameter Values

Convergence

Uniqueness

Robustness

Maximum Likelihood Estimation


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Incorporating rating variables into fitting process

Data – Situation 3 Policy

# Deduct. Limit Loss Constr. Prot. Occupancy1 0 100,000 112 1 2 232 100,000 10M 117 2 1 333 0 100,000 100,000 1 6 16

4 1,000 5M 5M 3 3 85 0 250,000 430 1 4 70 6 10,000 1M 4,500 2 2 40


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Incorporating rating variables into fitting process

Approaches:Subdividing data

Using all of data to estimate model parameters simultaneously.


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Relating rating variables to a parameter of the selected loss distribution

Rating variables:QuantitativeQualitative

Generalized Linear Modeling


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An example:Commercial Loss Fire Data

Rating variables:ConstructionBuilding Value -- Exposure


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Linear Predictors4 linear predictors; 4 statistical models: A, B, C, D

i 0

i i iC C 0 1 1 2 2

i 0 1 ilog(BV )

iii CC 012132 i log(BV)


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Otherwise 0,

frame a isrisk i theIf 1, th

1iC

Otherwise 0,

masonry a isrisk i theIf 1, th

2iC


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Estimation of parameters: Lognormal: and

From: and tobeta_0, beta-1, beta_2, beta_3 &


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Assessing the effect of Rating Variables Nested models

H0 1 2 3 0:

H0 2 3 0:

H0 1 0:


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Nested Hypotheses based on Model D

Test of Hypothesis

H0 1 2 3 0:

2(log log )L LA D


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mydata<-TableAm<-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


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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

Documents

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