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15
Hierarchical insurance claims modeling
 Journal of the American Statistical Association
, 2008
"... This work describes statistical modeling of detailed, microlevel automobile insurance records. We consider 19932001 data from a major insurance company in Singapore. By detailed microlevel records, we refer to experience at the individual vehicle level, including vehicle and driver characterist ..."
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This work describes statistical modeling of detailed, microlevel automobile insurance records. We consider 19932001 data from a major insurance company in Singapore. By detailed microlevel records, we refer to experience at the individual vehicle level, including vehicle and driver characteristics, insurance coverage and claims experience, by year. The claims experience consists of detailed information on the type of insurance claim, such as whether the claim is due to injury to a third party, property damage to a third party or claims for damage to the insured, as well as the corresponding claim amount. We propose a hierarchical model for three components, corresponding to the frequency, type and severity of claims. The
rst is a negative binomial regression model for assessing claim frequency. The drivers gender, age, and no claims discount as well as vehicle age and type turn out to be important variables for predicting the event of a claim. The second is a multinomial logit model to predict the type of insurance claim, whether it is third party injury, third party property damage, insureds own damage or some combination. Year, vehicle age and vehicle type turn out to be important predictors for this component. Our third model is for the severity component. Here, we use a generalized beta of the second kind longtailed distribution for claim amounts and also incorporate predictor variables. Year, vehicle age and a persons age turn out to be important predictors for this component. Not surprisingly, we show that there is a signi
cant dependence among the di¤erent claim types; we use a tcopula to account for this dependence. The three component model provides justi
cation for assessing the importance of a rating variable. When taken together, the integrated model allows an actuary to predict automobile claims more e ¢ ciently than traditional methods. Using simulation, we demonstrate this by developing predictive distributions and calculating premiums under alternative reinsurance coverages. Keywords: Longtail regression and copulas. yThe authors acknowledge the research assistance of Mitchell Wills, Shi Peng and Katrien
Discussion of ‘A Bayesian generalized linear model for the BornhuetterFerguson method of claims reserving
 North American Actuarial Journal
, 2005
"... Professor Verrall nicely illustrates how Bayesian models can be applied to claims reserving within the framework of Generalized Linear Models (GLMs) and how they lead to posterior predictive distributions of quantities of interest. In this discussion we apply a Bayesian model in the context of disco ..."
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Cited by 2 (2 self)
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Professor Verrall nicely illustrates how Bayesian models can be applied to claims reserving within the framework of Generalized Linear Models (GLMs) and how they lead to posterior predictive distributions of quantities of interest. In this discussion we apply a Bayesian model in the context of discounted loss reserves. The outcomes of this approach are compared with results on approximations for the distribution of the discounted loss reserve when the runoff triangle is modelled by a GLM. These approximations are based on the concepts of comonotonicity and convex order and are explained in full details in Hoedemakers et al. (2003) (for lognormal claims reserving models) and Hoedemakers et al. (2004) (for claims reserving using GLMs). The comonotonicity approach has been shown to provide elegant solutions to various other actuarial and financial problems involving the distribution of a sum of dependent random variables (check www.kuleuven.ac.be/insurance research papers for more illustrations). We realize that the Bayesian posterior predictive distribution is a very general workhorse, which takes into account all sources of uncertainty in the model formulation and is applicable to different statistical domains, whereas our approximations originate from a specific actuarial context. We want to illustrate however that the predictive distribution based on the comonotonic bounds provides results that are close to the results obtained via MCMC. The main advantage of the bounds is that several risk measures such as percentiles (VaRs), expected shortfalls (stoploss premia) and TailVaRs can be calculated easily from it. Generalized linear models for claims reserving: the likelihoodbased approach We use the notation from Verrall (2004). An insurer is interested in the aggregated value � n i=2 � n j=n+2−i Cij corresponding with the future part of a classical runoff triangle (as shown in Table 1 of Verrall’s paper). In a (likelihoodbased) GLM framework this reserve will be predicted by n � n� reserve =
Issues in claims reserving and credibility: a semiparametric Approach with Mixed Models
, 2006
"... Verrall (1996) and England & Verrall (2001) first considered the use of smoothing methods in the context of claims reserving. They applied two smoothing procedures in a likelihoodbased way, namely the locally weighted regression smoother (‘loess’) and the cubic smoothing spline smoother. Using ..."
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Verrall (1996) and England & Verrall (2001) first considered the use of smoothing methods in the context of claims reserving. They applied two smoothing procedures in a likelihoodbased way, namely the locally weighted regression smoother (‘loess’) and the cubic smoothing spline smoother. Using the statistical methodology of semiparametric regression and its connection with mixed models (see e.g. Ruppert et al., 2003), this paper revisits smoothing models for loss reserving and credibility. Apart from the flexibility inherent to all semiparametric methods, advantages of the semiparametric approach developed here are threefold. Firstly, a Bayesian implementation of these smoothing models is relatively straightforward and allows simulation from the full predictive distribution of quantities of interest. Since the main interest of actuaries lies in prediction, this is a major advantage. Secondly, because the constructed models have an interpretation as (generalized) linear mixed models ((G)LMMs), standard statistical theory and software for (G)LMMs can be used. Thirdly, more complicated data sets, dealing for example with quarterly development in a reserving context, heavytails, semicontinuous data, or extensive longitudinal data, can be modelled within this framework. Throughout this article, data examples illustrate these different aspects. Several comments are included regarding model specification, estimation and selection.
Semiparametric regression models for claims reserving and credibility: the mixed model approach. Working Paper (online at www.econ.kuleuven.be/katrien.antonio
"... Verrall (1996) and England & Verrall (2001) considered the use of smoothing methods in the context of claims reserving, by applying two smoothing procedures in a likelihoodbased way, namely the locally weighted regression smoother (‘loess’) and the cubic smoothing spline smoother. Using the sta ..."
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Cited by 1 (1 self)
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Verrall (1996) and England & Verrall (2001) considered the use of smoothing methods in the context of claims reserving, by applying two smoothing procedures in a likelihoodbased way, namely the locally weighted regression smoother (‘loess’) and the cubic smoothing spline smoother. Using the statistical methodology of semiparametric regression and its connection with mixed models (see e.g. Ruppert et al., 2003), this paper revisits smoothing models for loss reserving and considers their use in an example from credibility. Next to the flexibility of a semiparametric regression model, advantages of the presented approach are threefold. Firstly, because the constructed semiparametric models have an interpretation as (generalized) linear mixed models ((G)LMMs), standard statistical theory and software for (G)LMMs can be used. Secondly, a Bayesian implementation of these smoothing models is relatively straightforward and allows simulation from the full predictive distribution of quantities of interest. Since actuaries are interested in predictions, this is a major advantage. Thirdly, more complicated statistical models, dealing for example with semicontinuous data or extensive longitudinal data, can be handled within the same framework. Throughout this work, data examples illustrate these different aspects. Evidently, the methodology is not restricted to the problems discussed in this paper, but is relevant for other kinds of actuarial regression problems. Keywords: loss reserving, credibility, generalized additive mixed models, Psplines, Bayesian statistics.
1. THE VARIABILITY PROBLEM The Challenge of Reserving
"... Property/casualty reserves are estimates of losses and loss development and as such will not match the ultimate results. Sources of error include model error (the methodology used does not accurately reflect the development process), parameter error (model parameters are calibrated from the data), a ..."
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Property/casualty reserves are estimates of losses and loss development and as such will not match the ultimate results. Sources of error include model error (the methodology used does not accurately reflect the development process), parameter error (model parameters are calibrated from the data), and process error (future development is random). This paper provides a comprehensive and practical methodology for quantifying risk that includes all three sources. The key feature is that variability is captured by examining historical changes in ultimate values rather than examining the underlying claim distribution. We present both the conceptual framework as well as practical examples.
Multilevel NonLinear Random Effects Claims Reserving Models And Data Variability Structures
"... Characteristic of many reserving methods designed to analyse claims data aggregated by contract or sets of contracts, is the assumption that features typifying historical data are representative of the underwritten risk and of future losses likely to affect the contracts. Kremer (1982), Bomheutter a ..."
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Characteristic of many reserving methods designed to analyse claims data aggregated by contract or sets of contracts, is the assumption that features typifying historical data are representative of the underwritten risk and of future losses likely to affect the contracts. Kremer (1982), Bomheutter and Ferguson (1972), de Alba (2002), and many others, consider models with development patterns common to all underwriting years and known meanvariance relationships. Data amenable to such assumptions are indeed rare. More usual are large variations in settlement speeds, exposure and claim volumes. Also typifying many published models are Incurred But Not Reported (IBNR) predictions limited to periods with known claims, frequently adjusted with "tail factors " generated from market statistics. Of concern could be analytical approach inconsistencies behind reserves for delay periods before and after the last known claims, under reserving and unfair reserve allocation at underwriting year, array or contract levels. As applications of Markov Chain Monte Carlo (MCMC) methods, the models proposed in this paper depart from the neat assumptions of quasilikelihood and extended quasilikelihood, and introduce random effects models. The primary focus is the close dependency of the 1BNR on data variability structures and variance models, built with reference to the generic model derived in Vera (2003). The models have been implemented in BUGS
Gary Blumsohn, Chairperson
"... The Spring 2009 Edition of the CAS EForum is a cooperative effort between the Committee for the CAS EForum and the Committee on Reinsurance Research. The CAS Committee on Reinsurance Research presents for discussion four papers prepared in response to their 2009 Call for Papers. ..."
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The Spring 2009 Edition of the CAS EForum is a cooperative effort between the Committee for the CAS EForum and the Committee on Reinsurance Research. The CAS Committee on Reinsurance Research presents for discussion four papers prepared in response to their 2009 Call for Papers.
A Multivariate Bayesian Claim Count Development Model With Closed Form Posterior and Predictive Distributions
"... We present a rich, yet tractable, multivariate Bayesian model of claim count development. The model combines two conjugate families: the gammaPoisson distribution for ultimate claim counts and the Dirichletmultinomial distribution for emergence. We compute closed form expressions for all distribut ..."
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We present a rich, yet tractable, multivariate Bayesian model of claim count development. The model combines two conjugate families: the gammaPoisson distribution for ultimate claim counts and the Dirichletmultinomial distribution for emergence. We compute closed form expressions for all distributions of actuarial interest, including the posterior distribution of parameters and the predictive multivariate distribution of future counts given observed counts to date and for each of these distributions give a closed form expression for the moments. A new feature of the model is its explicit sensitivity to ultimate claim count variability and the uncertainty surrounding claim count emergence. Depending on the value of these parameters, the posterior mean can equal the BornhuetterFerguson or chainladder reserve. Thus the model provides a continuum of models interpolating between these common methods. We give an example to illustrate use of the model.
THREE ESSAYS IN FINANCE AND ACTUARIAL SCIENCE
, 2013
"... The first and most important thank goes to my supervisor Elisa Luciano, for her constant and invaluable support throughout all my Ph.D. years, fruitful discussions and suggestions. I am deeply indebted to Fabrizio Restione, Alberto Fasano and the Risk Management Department at FondiariaSai s.p.A. fo ..."
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The first and most important thank goes to my supervisor Elisa Luciano, for her constant and invaluable support throughout all my Ph.D. years, fruitful discussions and suggestions. I am deeply indebted to Fabrizio Restione, Alberto Fasano and the Risk Management Department at FondiariaSai s.p.A. for providing me with the data I use in the first Part of my dissertation. I would also like to thank Carmelo Genovese for helpful discussions on the first Chapter. The work benefited of comments from participants to the IV MAF Conference
ACTUARIAL STATISTICS AND MIXED MODELS: APPLICATIONS AND OPPORTUNITIES
"... The purpose of this paper is twofold. On the one hand, it is a short overview of our recent work on the use of mixed model methodology in actuarial statistics, which covers topics from credibility, claims reserving and nonlife ratemaking. On the other hand, opportunities and challenges for future r ..."
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The purpose of this paper is twofold. On the one hand, it is a short overview of our recent work on the use of mixed model methodology in actuarial statistics, which covers topics from credibility, claims reserving and nonlife ratemaking. On the other hand, opportunities and challenges for future research are sketched. 1.