Claims Reserving using GAMLSS

Gian Paolo Clemente Giorgio Spedicato

Università Cattolica di Milano

XXXVII Convegno AmasesStresa, 5 Settembre 2013

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Aim of the paper

- To propose an alternative methodology for a valuation of claims reserve

- To use Generalized Additive Models for location, shape and scale (GAMLSS)

- To derive both a point estimate and a measure of uncertainty

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Claims reserve General framework

• For non-life insurance companies, estimating reserves is an essential andrecurring task. The estimate of claims reserve plays indeed a key role todetermine insurance liabilities and several methods have been developed inthe past, in order to obtain an estimated value of claims reserve.

• Stochastic models for outstanding claims valuation have been recentlydeveloped with the aim to obtain at least a variability coefficient related tothe point estimate (also known as best estimate) of the reserve.

• Nowadays both the evaluation of the accuracy of claims reserve and thequantification of capital requirement appear key issues in Solvency IIframework.

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Claims reserve Claims reserve evaluation: main references

• Mack (1993) introduced a distribution free model which yields the samereserve estimates as the chain ladder method and which allows to estimate ameasure of accuracy of estimation (the mean squared error).

• Renshaw and Verrall (1994) casted the chain ladder method into theframework of generalized linear models (GLM) with an overdispersed Poissonmodel for incremental payments.

• Furthermore, England and Verrall (2001) applied generalized additive models(GAM) in order to incorporate smoothing of parameter estimates overaccident years, while leaving the model describing the run-off pattern.

• An alternative way to derive the estimation of prediction error is based on theuse of bootstrapping analysis where the scaled Pearson residuals arecommonly used (England and Verrall (1999) and England (2002))

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Claims reserve the ODP structure

• Focusing on a claims triangle of a single LoB with rows (accident years)i = 0, ..., I and columns (development years) j = 0, ...,J, England and Verrall(2002) proposed the following log-linear model for the incremental paymentsPi,j

E [Pij ] = mijvar [Pij ] = φmijmij = xiyjln(mij) = ηij = c +αi +βj

(1)

• This approach has been developed under an over-dispersed Poissonframework. Here, xi is the expected ultimate claims and yj is the proportionof ultimate claims to emerge in each development year (with the constraint∑J

j=1 yj = 1). Over-dispersion is introduced through the parameter φ, whichis unknown and estimated from the data.

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Claims reserve A generalization

• A flexible framework, within which previous model could be regarded as aspecial case, is reported in (2) (see England and Verrall). The first two itemsin (2) bundle the claim reserving within the GAM framework.{

E [Pij ] = mijvar [Pij ] = φmρij

(2)

• The value of the power function ρ dictates the choice of error distribution,with normal, Poisson, Gamma and Inverse Gaussian specified by 0, 1, 2, and3, respectively.

• Both approaches allow to derive a measure of uncertainty via a closed formulaor a two-step methodology based on bootstrap and monte-carlo simulation

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GAMLSS methodology GAMLSS

• GAMLSS is a general class of univariate regression models where theexponential family assumption is relaxed and replaced by a generaldistribution family.

• The systematic part of the model allows in this framework that all theparameters of the conditional distribution of the response variable Pi,j can bemodelled as parametric or non-parametric functions of explanatory variables.

• In particular, it implies that moments of response variable in each cell can bedirectly expressed as a function of covariates after a convenientparametrization.

• Considering now the claims reserve framework, we can identify the theincremental payments Pi,j as response variables and derive the followingstructure: {

E [Pi,j ] = g−11 (η1,i,j)

var [Pi,j ] = g−12 (η2,i,j)

(3)

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GAMLSS methodology Prediction error with GAMLSS

• The prediction error is here derived by adapting the boostrapping–simulation methodologyproposed by England and Verrall.

• The normalized randomized quantile residuals (see Dunn and Smyth, 1996) are usuallyused to check the adequacy of a GAMLSS model and, in particular, its distributioncomponent. These residuals are given by ri,j =Φ−1(ui,j ) where Φ−1 is the inversecumulative distribution function of a standard normal distribution and ui,j = F (Pi,j |θi,j ) isderived by the assumed cumulative distribution for the cell (i , j).

• We adapt then the procedure proposed by the literature for GLM models to GAMLSSfollowing the next steps:

1 choose and fit the GAMLSS model M;2 evaluate the residuals ri,j =Φ−1[F (Pi,j |θi,j ];3 generate N upper triangles of residuals rk

i,j with k = 1, ...N through a sample withreplacement;

4 derive N upper triangles of pseudo-incremental payments from the gamlss modelthrough the inverse relation: Pk

i,j = F−1[Φ(rki,j )|θi,j ];

5 refit the gamlss model M;6 for each cell of the lower part of the triangle simulate from the process distribution

with mean and variance depending by the fitted gamlss;7 sum the simulated payments in the future triangle by origin year and overall to give

respectively the origin year and total reserve estimates.

• In this way we derive the full distribution of claims reserve and we can quantify both theprocess and the estimation error.

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A Numerical application Loss triangle and classical methodology

• Following the examples in England, Verrall (1999), the data from Taylor-Ashe (1983),available in the ChainLadder package (see GenIns data in [12]), are used.

• This triangle, with size 10x10, has been used here in the incremental form in order toderive both the estimation and the distribution of claims reserve in order to compare theproposed GAMLSS approach to the classic ODP methodology.

• We report in in Table main results derived by applying two classical approaches (Mack andODP) based on Chain-Ladder method. Furthermore the comparison is extended to a GLMbased on a Gamma distribution.

model BE CV QuantMack 18680855.61 0.13 25919050.29ODP GLM 18680856.00 0.16 28243801.41Gamma GLM 18085805.00 0.15 27241493.24

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A Numerical application GAMLSS results

• We try now, to apply to the same triangle several GAMLSS model by evaluating theconditional distribution assumption through a comparison of GAIC indices.

• At the moment, GAMLSS have been applied by assuming to model only the expected valueof the incremental payments and by testing a wide range of conditional distributions, muchmore beyond the classical exponential family.

• Several distributions provide almost the same GAIC.

models df GAIC Best EstimateWeibull 20.00 1495.04 19,939,326NegativeBinomial_TypeII 20.00 1495.25 18,995,459NegativeBinomial 20.00 1500.77 18,085,841Gamma 20.00 1500.77 18,085,822Gumbel 20.00 1515.18 23,467,287InverseGaussian 20.00 1515.69 17,364,127Exponential 19.00 1599.88 18,085,822

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A Numerical application GAMLSS results

• The greater advantage of GAMLSS for reserving is that we can model explicitely more thanone distribution parameter. So we assume to model the variance of incremental paymentsas a function of development year in order to assure a better fitting on data

• The evaluation has been computed again under the same distributional assumptions, butthe best fitting is derived now by using a Gamma distribution.

• In particular, we report in Table GAIC values determined by assuming that the dispersionparameter is varying by development year or by accident year. Results confirm a betterbehaviour according to development year.

model GAIC BEorigin, factor 1380.79 20387740.73development, factor 1241.41 20277355.31

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A Numerical application Main results

model BE CV QuantMack 18680855.61 0.13 25919050.29ODP GLM 18680856.00 0.16 28243801.41Gamma GLM 18085805.00 0.15 27241493.24Gamma GAMLSS 20458149.23 0.12 28906116.96

Claims Reserve (Gamma GLM)

Reserve

Fre

quen

cy

1.0e+07 1.5e+07 2.0e+07 2.5e+07 3.0e+07

020

040

060

080

0

Claims Reserve (Gamma GAMLSS)

Reserve

Fre

quen

cy

1.0e+07 1.5e+07 2.0e+07 2.5e+07 3.0e+07

020

040

060

080

0

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Conclusions

• GAMLSS approach appears more flexible than classical GLM aiming to describe thevariance effect as a function of accident or development year.

• Furthermore gamlss methodology leads to overcome the exponential family restrictionallowing the use of a variety of distribution.

• Numerical results shows an improvement of GAIC and a lower variability respect to classicalGLM

• Main weakness could be the overparameterization of the model leading to the need of agreater quantity of data (larger triangles).

• Further development will regard an analysis of the behaviour of GAMLSS on severaltriangles and the identification of a closed formula for the prediction error evaluation

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References

Main references

England P.D., Addendum to analytic and bootstrap estimates of prediction errors in claimreserving, Insurance Mathematics and Economics, 31:461–466, 2002.

England P.D., Verrall R.J., Analytic and bootstrap estimates of prediction errors in claimsreserving, British Actuarial Journal, 8:443–544,2002.

England P.D., Verrall R.J., Stochastic claim reserving in general insurance, InsuranceMathematics and Economics, 25:281–293, 1999.

Gesmann M., Zhang Y., ChainLadder: Mack, Bootstrap, Munich andMultivariate-chain-ladder Methods, R package version 0.1.5-1, 2011.

Mack T., Distribution-free calculation of the standard error of chain ladder reserveestimates, Astin Bulletin, 23(2):213–225, 1993.

McCullagh P., Nelder J.A., Generalized Linear Models, Chapman and Hall 1989, London.

Rigby R.A., Stasinopoulos D. M., Generalized additive models for location, scale andshape,(with discussion), Applied Statistics, 54:507–554, 2005.

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