Spatial modelling of claim frequency and claim size in insurance

In this paper models for claim frequency and claim size in non-life insurance are considered. Both covariates and spatial random e ects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model going back to Lundberg (1903), we allow for dependencies between claim size and claim frequency. Both models for the individual and average claim sizes of a policyholder are considered. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly addressed. Besides the deviance information criterion suggested by Spiegelhalter et al. (2002), the predictive model choice criterion (Gelfand and Ghosh (1998)) and proper scoring rules (Gneiting and Raftery (2005)) based on the posterior predictive distribution are investigated. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial e ects significantly improves the models for both claim frequency and claim size and also leads to more accurate predictions of the total claim sizes. Further we quantify the significant number of claims e ects on claim size

Calculation of solvency capital requirements for non-life underwriting risk using generalized linear models

The paper presents various GLM models using individual rating factors to calculate the solvency capital requirements for non-life underwriting risk in insurance. First, we consider the potential heterogeneity of claim frequency and the occurrence of large claims in the models. Second, we analyse how the distribution of frequency and severity varies depending on the modelling approach and examine how they are projected into SCR estimates according to the Solvency II Directive. In addition, we show that neglecting of large claims is as consequential as neglecting the heterogeneity of claim frequency. The claim frequency and severity are managed using generalized linear models, that is, negative-binomial and gamma regression. However, the different individual probabilities of large claims are represented by the binomial model and the large claim severity is managed using generalized Pareto distribution. The results are obtained and compared using the simulation of frequency-severity of an actual insurance portfolio.Web of Science26446645

Gradient Boosting in Motor Insurance Claim Frequency Modelling

Modelling claim frequency and claim severity are topics of great interest in property-casualty insurance for supporting underwriting, ratemaking, and reserving actuarial decisions. This paper investigates the predictive performance of Gradient Boosting with Decision Trees as base learners to model the claim frequency in motor insurance using a private cross-country large insurance dataset. The Gradient Boosting algorithm combines many weak base learners to tackle conceptual uncertainty in empirical research. The findings show that the Gradient Boosting model is superior to the standard Generalised Linear Model in the sense that it provides closer predictions in the claim frequency model. The finding also shows that Gradient Boosting can capture the nonlinear relation between the claim counts and feature variables and their complex interactions being, thus, a valuable tool for feature engineering and the development of a data-driven approach to risk management

Modelling Motor Insurance Claim Frequency and Severity Using Gradient Boosting

Clemente, C., Guerreiro, G. R., & Bravo, J. M. (2023). Modelling Motor Insurance Claim Frequency and Severity Using Gradient Boosting. Risks, 11(9), 1-20. [163]. https://doi.org/10.3390/risks11090163 ---This research was funded by national funds through the FCT—Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020—Center for Mathematics and Applications—(G.R. Guerreiro) and grants UIDB/04152/2020—Centro de Investigação em Gestão de Informação (MagIC) and UIDB/00315/2020—BRU-ISCTE-IUL—(J.M. Bravo).Modelling claim frequency and claim severity are topics of great interest in property-casualty insurance for supporting underwriting, ratemaking, and reserving actuarial decisions. Standard Generalized Linear Models (GLM) frequency–severity models assume a linear relationship between a function of the response variable and the predictors, independence between the claim frequency and severity, and assign full credibility to the data. To overcome some of these restrictions, this paper investigates the predictive performance of Gradient Boosting with decision trees as base learners to model the claim frequency and the claim severity distributions of an auto insurance big dataset and compare it with that obtained using a standard GLM model. The out-of-sample performance measure results show that the predictive performance of the Gradient Boosting Model (GBM) is superior to the standard GLM model in the Poisson claim frequency model. Differently, in the claim severity model, the classical GLM outperformed the Gradient Boosting Model. The findings suggest that gradient boost models can capture the non-linear relation between the response variable and feature variables and their complex interactions and thus are a valuable tool for the insurer in feature engineering and the development of a data-driven approach to risk management and insurance.publishersversionpublishe

Estimation of MTPL claim frequency using GLM, GAM and XGBoost techniques

The purpose of this master’s thesis is to provide an overview of the XGBoost algorithm and examine its suitability to model the claim frequency of motor third party liability insurance. The first three chapters introduce generalized linear models, generalized additive models and the algorithms of gradient boosting and XGBoost. In the fourth chapter, the aforementioned methods are applied on the data of Estonian Motor Insurance Bureau to predict claim frequency

Smoothed Quantiles for Claim Frequency Models, with Applications to Risk Measurement

Statistical models for the claim severity and claim frequency variables are routinely constructed and utilized by actuaries. Typical applications of such models include identification of optimal deductibles for selected loss elimination ratios, pricing of contract layers, determining credibility factors, risk and economic capital measures, and evaluation of effects of inflation, market trends and other quantities arising in insurance. While the actuarial literature on the severity models is extensive and rapidly growing, that for the claim frequency models lags behind. One of the reasons for such a gap is that various actuarial metrics do not possess ``nice\u27\u27 statistical properties for the discrete models whilst their counterparts for the continuous models do. The objectives of this dissertation to addressing the issue described above are the following: • Generalize the definitions of ``smoothed quantiles\u27\u27 for samples and populations of claim counts to vectors of smoothed quantiles. This is motivated by the fact that multiple quantiles are needed for better understanding of insurance risks. • Investigate large- and small-sample properties of smoothed quantile estimators for vectors, when the underlying claim count distribution has finite support. • Extend the definition of smoothed quantiles for discrete distributions with infinite support, and study asymptotic and finite-sample properties of the associated estimators. • Illustrate the appropriateness and flexibility of such tools in solving risk measurement problems. Smoothed quantiles are defined using the theory of fractional or imaginary order statistics, which was originated by Stigler (1977). To prove consistency and asymptotic normality of sample estimators of smoothed quantiles, we utilize the results of Wang and Hutson (2011) and generalize them to vectors of smoothed quantiles. Further, we thoroughly investigate extensions of this methodology to discrete populations with infinite support (e.g., Poisson and zero-inflated Poisson distributions). Furthermore, large- and small-sample properties of the newly designed estimators are investigated theoretically and through Monte Carlo simulations. Finally, applications of smoothed quantiles to risk measurement (e.g., estimation of distortion risk measures such as value-at-risk, conditional tail expectation, and proportional hazards transform) are discussed and illustrated using actual insurance data

Efisiensi Sistem Bonus Malus Sebagai Model Rantai Markov

. Developed bonus malus system (BMS) is to make the premium paid by insured will be as closed as possible with expected occurrence of claim in every year basis. To study the efficiency of a BMS, we must previously observe the effect of claim frequency on value of premium. The efficiency of Bonus Malus System can be found through its Markov model; that is by found a stationary distribution in form of line vectors of its BMS Markov chain with its components as a function of claim frequency. In this paper, the BMS used is that of Brazil

The Use of Monte Carlo Method to Model the Aggregate Loss Distribution

Based on Law Number 24 of 2011, a state program was established to provide social protection and welfare for everyone, one of which is health insurance by the Social Insurance Administration Organization (BPJS). In its implementation, several important evaluations are needed. One that requires accurate evaluation is claim frequency and claim severity in determining premiums and reserved funds. This thesis provides one form of a method for selecting the distribution of claim frequency and claim severity. The data used in this study was taken from BPJS Health in the City of Tangerang in 2017. The distribution of opportunities chosen had been adjusted to the participant's claim data and parameter estimated using the Maximum Likelihood Estimation method. The chi-square test was used to check the goodness of fit for claim frequency distributions whereas the Anderson Darling tests were applied to claim severity distributions. The results of the chi-square test and the Anderson-Darling test showed that the model that matched the claim frequency distribution was the Z12M–NBGE distribution while the model that matched the claim severity was lognormal. The Z12M–NBGE distribution and the lognormal formed the aggregate loss distribution using the Monte Carlo method. Furthermore, the simulation results were obtained to the measurement of the Value in Risk (VaR) and Shortfall Expectations (ES). So, the Monte Carlo method is simple to implement the aggregate loss distributions and can easily handle various risks with dependency.