Motor Third-party Liability Research Articles

Nagging Predictors

We define the nagging predictor, which, instead of using bootstrapping to produce a series of i.i.d. predictors, exploits the randomness of neural network calibrations to provide a more stable and accurate predictor than is available from a single neural network run. Convergence results for the family of Tweedie’s compound Poisson models, which are usually used for general insurance pricing, are provided. In the context of a French motor third-party liability insurance example, the nagging predictor achieves stability at portfolio level after about 20 runs. At an insurance policy level, we show that for some policies up to 400 neural network runs are required to achieve stability. Since working with 400 neural networks is impractical, we calibrate two meta models to the nagging predictor, one unweighted, and one using the coefficient of variation of the nagging predictor as a weight, finding that these latter meta networks can approximate the nagging predictor well, only with a small loss of accuracy.

Motor Third-Party Liability of Personal Transportation Devices and Its Insurance

The aim of the article is to describe the principles of liability of autonomous and dependent possessors as well as of users of personal transportation devices for damage caused by their movement. The liability of holders of such devices is independent of their fault and is based on the principle of risk. However, in the event of a collision between two such transporters or a personal transportation device with a car or a motorcycle, the liability shall be based on the principle of guilt. The liability of the user of a personal transportation device is also fault-based. Owners of personal transportation devices are not subject to the motor third-party liability insurance. As long as these devices are not legally required to be registered, there is no effective mechanism to control the fulfilment of the third-party liability insurance obligation. It is therefore not possible to introduce compulsory liability insurance. However, it is necessary and important to popularize and strengthen the insurance coverage under liability insurance in private life, including liability for damage resulting from the usage of personal transportation devices.

Combined tail estimation using censored data and expert information

We study tail estimation in Pareto-like settings for datasets with a high percentage of randomly right-censored data, and where some expert information on the tail index is available for the censored observations. This setting arises for instance naturally for liability insurance claims, where actuarial experts build reserves based on the specificity of each open claim, which can be used to improve the estimation based on the already available data points from closed claims. Through an entropy-perturbed likelihood, we derive an explicit estimator and establish a close analogy with Bayesian methods. Embedded in an extreme value approach, asymptotic normality of the estimator is shown, and when the expert is clair-voyant, a simple combination formula can be deduced, bridging the classical statistical approach with the expert information. Following the aforementioned combination formula, a combination of quantile estimators can be naturally defined. In a simulation study, the estimator is shown to often outperform the Hill estimator for censored observations and recent Bayesian solutions, some of which require more information than usually available. Finally we perform a case study on a motor third-party liability insurance claim dataset, where Hill-type and quantile plots incorporate ultimate values into the estimation procedure in an intuitive manner.

On the Validation of Claims with Excess Zeros in Liability Insurance: A Comparative Study

In this study, we consider the problem of zero claims in a liability insurance portfolio and compare the predictability of three models. We use French motor third party liability (MTPL) insurance data, which has been used for a pricing game, and show that how the type of coverage and policyholders’ willingness to subscribe to insurance pricing, based on telematics data, affects their driving behaviour and hence their claims. Using our validation set, we then predict the number of zero claims. Our results show that although a zero-inflated Poisson (ZIP) model performs better than a Poisson regression, it can even be outperformed by logistic regression.

Concordance-based predictive measures in regression models for discrete responses

ABSTRACTDependence measures are often used in practice in order to assess the quality of a regression model. This is for instance the case with Kendall's tau and other association coefficients based on concordance probabilities. However, in case the response variable is discrete, correlation indices are often bounded and restricted to a sub-interval of . Hence, in this context, small positive values of Kendall's tau may actually support goodness of prediction when getting close to its highest attainable value. In this paper, we derive the best-possible upper bounds for Kendall's tau when the response variable is discrete. Two cases are considered, depending on whether the score is continuous or discrete. Also, we illustrate the obtained upper bounds on a motor third-party liability insurance portfolio.

AGGREGATE CLAIM ESTIMATION USING BIVARIATE HIDDEN MARKOV MODEL

AbstractIn this paper, we propose an approach for modeling claim dependence, with the assumption that the claim numbers and the aggregate claim amounts are mutually and serially dependent through an underlying hidden state and can be characterized by a hidden finite state Markov chain using bivariate Hidden Markov Model (BHMM). We construct three different BHMMs, namely Poisson–Normal HMM, Poisson–Gamma HMM, and Negative Binomial–Gamma HMM, stemming from the most commonly used distributions in insurance studies. Expectation Maximization algorithm is implemented and for the maximization of the state-dependent part of log-likelihood of BHMMs, the estimates are derived analytically. To illustrate the proposed model, motor third-party liability claims in Istanbul, Turkey, are employed in the frame of Poisson–Normal HMM under a different number of states. In addition, we derive the forecast distribution, calculate state predictions, and determine the most likely sequence of states. The results indicate that the dependence under indirect factors can be captured in terms of different states, namely low, medium, and high states.

Case Study: French Motor Third-Party Liability Claims

We provide a tutorial that compares a classical generalized linear model for claims frequency modeling to regression tree, boosting machine and neural network approaches. We explore these methods, discuss their calibration and study their predictive power on an explicit motor third-party liability insurance data set. The results of the case study show that a simple generalized linear model does not capture interactions of feature components appropriately, whereas the other methods are able to address these interactions.

Telematika i noviji trendovi u osiguranju od autoodgovornosti i drugim vidovima osiguranja

Rapid development, innovations and the blend of different technologies have brought about the development of new IT-communication products. Cloud computing, the Internet of things, mobile devices, telematics etc. are only some of the products of modern technology that provide swift and highly efficient data processing. The possibility of free access to the Internet and cheaper technology for the determination of geographical position contribute to the development of an unlimited number of smartphone applications for use in telematics. Simultaneously, IT communication makes a decisive impact on improving the way in which society and businesses operate. In this paper, we examine the general significance of telematics, its notion, history and the advantages it brings to insurer operations in the motor third-party liability (MTPL) and a few other insurance types. Statistical data about the telematics market, the conditions for the provision of telematics services and some other resources were used in this paper.

Beyond the Tweedie Reserving Model: The Collective Approach to Loss Development

This article proposes a new loss reserving approach, inspired from the collective model of risk theory. According to the collective paradigm, we do not relate payments to specific claims or policies, but we work within a frequency-severity setting, with a number of payments in every cell of the run-off triangle, together with the corresponding paid amounts. Compared to the Tweedie reserving model, which can be seen as a compound sum with Poisson-distributed number of terms and Gamma-distributed summands, we allow here for more general severity distributions, typically mixture models combining a light-tailed component with a heavier-tailed one, including inflation effects. The severity model is fitted to individual observations and not to aggregated data displayed in run-off triangles with a single value in every cell. In that respect, the modeling approach appears to be a powerful alternative to both the crude traditional aggregated approach based on triangles and the extremely detailed individual reserving approach developing each and every claim separately. A case study based on a motor third-party liability insurance portfolio observed over 2004–2014 is used to illustrate the relevance of the proposed approach.

Profitability and Growth in Motor Insurance Business: Empirical Evidence from Germany

Over recent years, the German motor insurance business has faced significant changes, including a growing importance of direct insurance offerings. Motor insurance products are offered by a wide range of insurers, with companies differing in terms of legal status, size, product portfolio, distribution strategy and operational efficiency. Furthermore, one distinguishes between two main products, namely motor third-party liability (MTPL) and motor own damage (OD). In our research, we analyse to what extent the characteristics of the companies can explain the premiums, the total claims costs and the operating expenses per contract in MTPL and OD. For our analysis, we use panel data of insurance companies, offering motor insurance products in Germany, for the years 2002–2014. The panel data provide almost full market coverage. In our study, we apply different statistical tests and multilinear regression models. We show that mutuals relate to lower premiums, lower total claims costs and lower operating expenses per contract when compared to listed companies. In addition, direct insurance companies get along with lower premiums and lower operating expenses per contract compared to traditional companies selling via agents or brokers. Furthermore, we find major differences related to the range of the product portfolio, the size of the motor business, the dominance of the motor business within the non-life business, and the calendar year. Our results are relevant to academics and practitioners alike and help to better understand the German motor insurance business.

Confidence intervals of the premiums of optimal bonus malus systems

In view of the economic importance of motor third-party liability insurance in developed countries the construction of optimal BMS has been given considerable interest. However, a major drawback in the construction of optimal BMS is that they fail to account for the variability on premium calculations which are treated as point estimates. The present study addresses this issue. Specifically, nonparametric mixtures of Poisson laws are used to construct an optimal BMS with a finite number of classes. The mixing distribution is estimated by nonparametric maximum likelihood (NPML). The main contribution of this paper is the use of the NPML estimator for the construction of confidence intervals for the premium rates derived by updating the posterior mean claim frequency. Furthermore, we advance one step further by improving the performance of the confidence intervals based on a bootstrap procedure where the estimated mixture is used for resampling. The construction of confidence intervals for the individual premiums based on the asymptotic maximum likelihood theory is beneficial for the insurance company as it can result in accurate and effective adjustments to the premium rating policies from a practical point of view.

Zastosowanie modelu Bühlmanna-Strauba do estymacji stawek składki netto w systemach bonus-malus ubezpieczeń odpowiedzialności cywilnej posiadaczy pojazdów mechanicznych

One of the elements used in the process of tariff calculation of premiums in motor liability insurance is a bonus-malus system. This systems takes into account the “claims ratio” by means of increases and discounts of the base premium called net premium rates. The aim of this work is to propose an estimation method of the net premium rates in the groups of the motor third-party liability insurance portfolio of individuals. One of the maximum likelihood models, called the Buhlmann-Straub model was used for the premium estimation.

Stochastic model to evaluate the fair value of motor third-party liability under the direct reimbursement scheme and quantification of the capital requirement in a Solvency II perspective

As commonly known, to evaluate the claims reserve (otherwise known as the provision for outstanding claims), the loss adjuster uses as a first component the claims reserve given by the sum of the estimated provision for each outstanding claim (known as case reserve). Traditional statistical-actuarial methods are used to control and/or asseverate the evaluation and recent developments have tended to ensure that these enable an independent assessment of the claims reserve. In some European countries, a large subset of motor liability claims is managed within a direct reimbursement (DR) scheme. In Italy, the introduction of the direct compensation CARD system for third-party liability insurance has resulted in greater attention in the use of these traditional methods due to the heterogeneity of the data available for evaluations. This paper presents the results of a study undertaken to define a calculation method that, using the different assumptions that describe the evolution of settlement mechanisms, is able to quantify the claims reserve. The proposed methodology reduces the loss adjuster’s discretion in applying statistical methods since all assumptions must be made explicit and can hence be monitored and controlled. Furthermore, a stress test can be performed on all the parameters that influence the settlement and thus the claims reserve. Finally, in a backtesting perspective, the proposed methodology enables an ex-post analysis of actual cash flow deviations compared to expected values, identifying the variables that lead to these differences. In particular, a calculation method is presented that using the different assumptions describing the evolution of the settlement of different claim handling procedures (Non-Card, Card, Handler Forfait and Debtor Forfait) is able to quantify the claims reserve. Via simulations, future payments, the expected value of the claims reserve and some indicators of variability are also estimated. The numerical application allows comparing the results obtained with those deriving from the application of traditional statistical methods.The problem of aggregating the four claim handling procedures is dealt with by using copulas with the goal of building a single distribution of the aggregate claims reserve. Furthermore, in a Solvency II perspective, the capital requirement is determined against the reserve risk based on the internal model proposed and is compared with the standard formula proposed by EIOPA.

Non-life insurance markets in CEE countries– part II: Polarisation and concentration

(ProQuest: ... denotes formulae omitted.)IntroductionSince the European Union accession in May, 2004, the non-life insurance markets in CEE countries faced many opportunities for growth. In the academic debate, however, an attention was given to the performance of insurance sector as a whole or to other segments of financial system (banking and capital markets in particular). Studies addressing the tendencies on selectively the non-life segment of insurance market remain scarce. In this context, there is an obvious research gap. Moreover, there are virtually no studies that compare the insurance markets performance in cross-country perspective, in this with a focus on CEE countries. This study may contribute to fill in the identified research gap, as it aims at evaluating the general tendencies of the development of non-life insurance markets in the eight CEE countries, that overcame transition at similar period of time (the 1990s) and accessed European Union on the 1st May 2004. Thus, these countries are regarded as a relatively unified sample of European Union member countries.The first part of this study1 gave the background of the general tendencies observed on non-life markets of the CEEs countries. In the overview of previous studies, the available works and researches were discussed and led to the distinction of three important periods of non-life market development in these countries. The first was pre-transition period when the markets were operating under the regime of command economy. The second was the transition period when the rapid changes began, mostly due to the high pace of foreign direct investments. The third period was the accession period which began with the European Union accession in 2004. This period was in focus of the empirical evaluations of nonlife market tendencies in the first part of this study, where the problem of products' structure of the non-life markets was addressed. It was found that in many cases there was still a strong domination of MTPL (motor third party liability) insurance. In this paper (as the continuation of the study and its second part) the trends within non-life markets polarisation and concentration were analysed.The paper is structured as follows. The first section explains the adopted research design and methodology. The second section presents and discusses the results and is broken to subsections referring the polarisation of gross premiums written, insurance penetration and density ratios, and finally market concentrations ratios. The fourth section concludes the study.1. Research design and methodologyThe study of concentration and polarisation of non-life insurance markets presented in this paper refers to the situation of eight CEE countries: Czech Republic, Hungary, Slovak Republic, Slovenia, Poland and three Baltic states: Latvia, Lithuania and Estonia (further referred to as the EU-8 countries). These countries formerly performed in terms of command economy and overcame transition in similar period (in 1990s). Moreover, these countries joined together the European Union on 1st May 2004, which created certain conditions of further insurance market development (due to the required harmonisation).The evaluation of concentration and polarisation performed in this paper was focused on observations for four years: 2004, 2008, 2012 and 20152. The four-year time frame was adopted with the aim to capture 2004 as the initial observation and the beginning of accession, then 2008 to capture the year of the global financial crisis emergence and 2012 to capture the post-crisis state. In order to show the recent trends, 2015 was added to observations.The data analysed in this paper was derived from the official statistics provided by the EU-8 national supervisory bodies or national associations of insurers. In some cases the statistics offered by Swiss Re were used as well. Additionally, some aspects of the research were based on the data provided by the federation of European insurers' associations. …

Optimization Approaches to Multiplicative Tariff of Rates Estimation in Non-Life Insurance

We focus on rating of non-life insurance contracts. We employ multiplicative models with basic premium levels and specific surcharge coefficients for various levels of selected risk/rating factors. We use generalized linear models (GLM) to describe the probability distribution of total losses for a contract during one year. We show that the traditional frequency–severity approaches based only on GLM with logarithmic link function can lead to estimates which do not fulfill business requirements. For example, a maximal surcharge and monotonicity of coefficient can be desirable. Moreover, our approach can handle total losses, which are based on arbitrary loss distributions, possibly decomposed into several classes, e.g., small and large or property and bodily injury. Various costs and loadings can be also incorporated into the tariff rates. We propose optimization problems for rate estimation which enable hedging against expected losses and taking into account a prescribed loss ratio and other business requirements. Moreover, we introduce stochastic programming problems with reliability type constraints which take into account individual risk of each rate cell or collective risk. In the numerical study, we apply the approaches to Motor Third Party Liability (MTPL) policies.

Estimation and Calculation Procedures of the Technical Provisions for Outstanding Insurance Claims

Abstract The paper presents algorithms for insurance technical provisions taking into account losses, which are incurred but not reported. Evaluation of insurance technical provisions for the kinds of insurance, such as Motor Third Party Liability (MTPL) Insurance, Property Insurance and some others, have difficulties in assessing the impact of the losses from insurance claims incurred requiring a longer time for the settlement of insurance claims. These insurance requirements are mainly associated with health insurance in the MTPL Insurance, losses related to compensation for moral injuries, as well as on life care and life-long pension. To run these payments, you need to know the financial indicators for the period of settlement of loss (such as the effective interest rate, investment income, etc.) In the article the procedures for the most accurate forecast possible losses for the expected excess of loss amount for a treaty year are provided, using the loss experience of the previous years of the occurrence with their development. However, certain adjustments should be made to take account of the impact of losses from previous years for the current period. This article describes how outstanding losses have to be projected on a year of reporting, so that they are correspond to the current values

Economic capital modelling for the MTPL man-made catastrophe risk

AbstractWe undertake a mathematical clarification of the QIS5 proposal for the calculation of the Motor Third Party Liability (MTPL) man-made catastrophe risk capital in terms of two more general models. The QIS5 model assumption implies that the total loss consists of a single catastrophe claim in case it occurs during the next one-year insurance time period. However, the total loss should instead be dynamically modelled by a sequence of claims of varying size that follow a compound Poisson Pareto model, which is our first alternative model. A second possibility also takes into account the effect of investments, whose financial return process follows a Black-Scholes-Merton model. If one excludes limits of coverage, then asymptotically as the total loss increases without limits the first model is equivalent to the model assumption obtained from the QIS5 assumption by replacing a single catastrophe claim by the total loss. In other words, the QIS5 simple model is justified as limiting asymptotic approximation to the classical compound Poisson Pareto model. Conversely, an asymptotic approximation to the VaR economic capital from this model identifies with a modified QIS5 CAT formula. The inclusion of limits of coverage is also analyzed. In this situation we obtain new simple closed-form implementations of the economic capital formulas.

Development of Motor Third Party Liability Insurance Market in Terms of Changing Regulation

Motor third party liability (MTPL) is one of the most important lines of insurance business, in developed and in developing countries, as well. The legal framework in the European Union's Member States is determined with five Directives for motor insurance. Countries aspiring to join the European Union (EU) need to harmonise their national legislation in accordance with the requirements of these directives. The topic of this paper is to present an analysis of motor third party liability insurance market in terms of changing regulation. The main emphasis is directed towards the implementation of the Directives from the EU in the legislation of the Republic of Macedonia, regarding the question of the pricing for this class of insurance. There will be an analysis of the advantages and disadvantages as a result of turning from a state-regulated system to a liberal tariff regime for MTPL.