1. Claims Reserving using GAMLSS
Gian Paolo Clemente Giorgio Spedicato
Università Cattolica di Milano
XXXVII Convegno Amases
Stresa, 5 Settembre 2013
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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
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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.
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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))
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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.
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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
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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)
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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.
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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
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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
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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
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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
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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
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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.
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