P & C Reserving Using GAMLSS


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P & C Reserving Using GAMLSS

  1. 1. Claims Reserving using GAMLSS Gian Paolo Clemente Giorgio Spedicato Università Cattolica di Milano XXXVII Convegno Amases Stresa, 5 Settembre 2013 1 / 14
  2. 2. 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 2 / 14
  3. 3. Claims reserve General framework • For non-life insurance companies, estimating reserves is an essential and recurring task. The estimate of claims reserve plays indeed a key role to determine insurance liabilities and several methods have been developed in the past, in order to obtain an estimated value of claims reserve. • Stochastic models for outstanding claims valuation have been recently developed with the aim to obtain at least a variability coefficient related to the point estimate (also known as best estimate) of the reserve. • Nowadays both the evaluation of the accuracy of claims reserve and the quantification of capital requirement appear key issues in Solvency II framework. 3 / 14
  4. 4. Claims reserve Claims reserve evaluation: main references • Mack (1993) introduced a distribution free model which yields the same reserve estimates as the chain ladder method and which allows to estimate a measure of accuracy of estimation (the mean squared error). • Renshaw and Verrall (1994) casted the chain ladder method into the framework of generalized linear models (GLM) with an overdispersed Poisson model for incremental payments. • Furthermore, England and Verrall (2001) applied generalized additive models (GAM) in order to incorporate smoothing of parameter estimates over accident years, while leaving the model describing the run-off pattern. • An alternative way to derive the estimation of prediction error is based on the use of bootstrapping analysis where the scaled Pearson residuals are commonly used (England and Verrall (1999) and England (2002)) 4 / 14
  5. 5. 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 payments Pi,j    E [Pij ] = mij var[Pij ] = φmij mij = xi yj ln(mij ) = ηij = c +αi +βj (1) • This approach has been developed under an over-dispersed Poisson framework. Here, xi is the expected ultimate claims and yj is the proportion of ultimate claims to emerge in each development year (with the constraint J j=1 yj = 1). Over-dispersion is introduced through the parameter φ, which is unknown and estimated from the data. 5 / 14
  6. 6. Claims reserve A generalization • A flexible framework, within which previous model could be regarded as a special case, is reported in (2) (see England and Verrall). The first two items in (2) bundle the claim reserving within the GAM framework. E [Pij ] = mij var[Pij ] = φmρ ij (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, and 3, respectively. • Both approaches allow to derive a measure of uncertainty via a closed formula or a two-step methodology based on bootstrap and monte-carlo simulation 6 / 14
  7. 7. GAMLSS methodology GAMLSS • GAMLSS is a general class of univariate regression models where the exponential family assumption is relaxed and replaced by a general distribution family. • The systematic part of the model allows in this framework that all the parameters of the conditional distribution of the response variable Pi,j can be modelled as parametric or non-parametric functions of explanatory variables. • In particular, it implies that moments of response variable in each cell can be directly expressed as a function of covariates after a convenient parametrization. • Considering now the claims reserve framework, we can identify the the incremental payments Pi,j as response variables and derive the following structure: E [Pi,j ] = g−1 1 (η1,i,j ) var[Pi,j ] = g−1 2 (η2,i,j ) (3) 7 / 14
  8. 8. GAMLSS methodology Prediction error with GAMLSS • The prediction error is here derived by adapting the boostrapping–simulation methodology proposed by England and Verrall. • The normalized randomized quantile residuals (see Dunn and Smyth, 1996) are usually used to check the adequacy of a GAMLSS model and, in particular, its distribution component. These residuals are given by ^ri,j = Φ−1(^ui,j ) where Φ−1 is the inverse cumulative distribution function of a standard normal distribution and ^ui,j = F(Pi,j |^θi,j ) is derived by the assumed cumulative distribution for the cell (i,j). • We adapt then the procedure proposed by the literature for GLM models to GAMLSS following 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 with replacement; 4 derive N upper triangles of pseudo-incremental payments from the gamlss model through the inverse relation: Pk i,j = F−1[Φ(^rk i,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 the process and the estimation error. 8 / 14
  9. 9. 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 to derive both the estimation and the distribution of claims reserve in order to compare the proposed GAMLSS approach to the classic ODP methodology. • We report in in Table main results derived by applying two classical approaches (Mack and ODP) based on Chain-Ladder method. Furthermore the comparison is extended to a GLM based on a Gamma distribution. model BE CV Quant Mack 18680855.61 0.13 25919050.29 ODP GLM 18680856.00 0.16 28243801.41 Gamma GLM 18085805.00 0.15 27241493.24 9 / 14
  10. 10. A Numerical application GAMLSS results • We try now, to apply to the same triangle several GAMLSS model by evaluating the conditional distribution assumption through a comparison of GAIC indices. • At the moment, GAMLSS have been applied by assuming to model only the expected value of the incremental payments and by testing a wide range of conditional distributions, much more beyond the classical exponential family. • Several distributions provide almost the same GAIC. models df GAIC Best Estimate Weibull 20.00 1495.04 19,939,326 NegativeBinomial_TypeII 20.00 1495.25 18,995,459 NegativeBinomial 20.00 1500.77 18,085,841 Gamma 20.00 1500.77 18,085,822 Gumbel 20.00 1515.18 23,467,287 InverseGaussian 20.00 1515.69 17,364,127 Exponential 19.00 1599.88 18,085,822 10 / 14
  11. 11. A Numerical application GAMLSS results • The greater advantage of GAMLSS for reserving is that we can model explicitely more than one distribution parameter. So we assume to model the variance of incremental payments as 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, but the best fitting is derived now by using a Gamma distribution. • In particular, we report in Table GAIC values determined by assuming that the dispersion parameter is varying by development year or by accident year. Results confirm a better behaviour according to development year. model GAIC BE origin, factor 1380.79 20387740.73 development, factor 1241.41 20277355.31 11 / 14
  12. 12. A Numerical application Main results model BE CV Quant Mack 18680855.61 0.13 25919050.29 ODP GLM 18680856.00 0.16 28243801.41 Gamma GLM 18085805.00 0.15 27241493.24 Gamma GAMLSS 20458149.23 0.12 28906116.96 Claims Reserve (Gamma GLM) Reserve Frequency 1.0e+07 1.5e+07 2.0e+07 2.5e+07 3.0e+07 0200400600800 Claims Reserve (Gamma GAMLSS) Reserve Frequency 1.0e+07 1.5e+07 2.0e+07 2.5e+07 3.0e+07 0200400600800 12 / 14
  13. 13. Conclusions • GAMLSS approach appears more flexible than classical GLM aiming to describe the variance effect as a function of accident or development year. • Furthermore gamlss methodology leads to overcome the exponential family restriction allowing the use of a variety of distribution. • Numerical results shows an improvement of GAIC and a lower variability respect to classical GLM • Main weakness could be the overparameterization of the model leading to the need of a greater quantity of data (larger triangles). • Further development will regard an analysis of the behaviour of GAMLSS on several triangles and the identification of a closed formula for the prediction error evaluation 13 / 14
  14. 14. References Main references England P.D., Addendum to analytic and bootstrap estimates of prediction errors in claim reserving, Insurance Mathematics and Economics, 31:461–466, 2002. England P.D., Verrall R.J., Analytic and bootstrap estimates of prediction errors in claims reserving, British Actuarial Journal, 8:443–544,2002. England P.D., Verrall R.J., Stochastic claim reserving in general insurance, Insurance Mathematics and Economics, 25:281–293, 1999. Gesmann M., Zhang Y., ChainLadder: Mack, Bootstrap, Munich and Multivariate-chain-ladder Methods, R package version 0.1.5-1, 2011. Mack T., Distribution-free calculation of the standard error of chain ladder reserve estimates, 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 and shape,(with discussion), Applied Statistics, 54:507–554, 2005. 14 / 14