平衡损失函数下具有共同效应的信度保费(英文)

在经典的信度理论中,一个保单组合的各风险之间是相互独立的,同时从二次损失函数中推导出信度保费.(Wen et al.,2009)给出了风险间具有共同效应的特殊的相关结构的信度保费表达式.本文在平衡损失函数下考虑此种风险结构的信度理论,特别地得到了Bühlmann和Bühlmann-Straub模型的信度保费表达式.     

平衡指数损失函数下的信度保费

在经典的信度保费模型中,得到的信度保费估计均是考虑的是纯保费,然而在保险实务中,保险公司收取的保费不可能是纯保费,必须具有正的安全负荷.在平衡指数损失函数下,研究了多合同的信度保费模型.利用正交投影方法,得到了未来保费的信度估计.最后对估计进行了数值模拟.     

Credibility premiums for the zero-inflated Poisson model and new hunger for bonus interpretation

The purpose of this paper is to explore and compare the credibility premiums in generalized zero-inflated count models for panel data. Predictive premiums based on quadratic loss and exponential loss are derived. It is shown that the credibility premiums of the zero-inflated model allow for more flexibility in the prediction. Indeed, the future premiums not only depend on the number of past claims, but also on the number of insured periods with at least one claim. The model also offers another way of analysing the hunger for bonus phenomenon. The accident distribution is obtained from the zero-inflated distribution used to model the claims distribution, which can in turn be used to evaluate the impact of various credibility premiums on the reported accident distribution. This way of analysing the claims data gives another point of view on the research conducted on the development of statistical models for predicting accidents. A numerical illustration supports this discussion.     

Multivariate negative binomial models for insurance claim counts

It is no longer uncommon these days to find the need in actuarial practice to model claim counts from multiple types of coverage, such as the ratemaking process for bundled insurance contracts. Since different types of claims are conceivably correlated with each other, the multivariate count regression models that emphasize the dependency among claim types are more helpful for inference and prediction purposes. Motivated by the characteristics of an insurance dataset, we investigate alternative approaches to constructing multivariate count models based on the negative binomial distribution. A classical approach to induce correlation is to employ common shock variables. However, this formulation relies on the NB-I distribution which is restrictive for dispersion modeling. To address these issues, we consider two different methods of modeling multivariate claim counts using copulas. The first one works with the discrete count data directly using a mixture of max-id copulas that allows for flexible pair-wise association as well as tail and global dependence. The second one employs elliptical copulas to join continuitized data while preserving the dependence structure of the original counts. The empirical analysis examines a portfolio of auto insurance policies from a Singapore insurer where claim frequency of three types of claims (third party property damage, own damage, and third party bodily injury) are considered. The results demonstrate the superiority of the copula-based approaches over the common shock model. Finally, we implemented the various models in loss predictive applications.     

加权平衡指数损失函数下的广义信度保费估计

在非寿险中,在索赔经历虽然相互独立,但有时会服从不同的分布.通过考虑保费的目标估计来对风险保费进行了研究,并采用正交投影的方法得到了目标问题的最优解,从而得到了加权平衡指数损失函数下的信度估计.此外,给出了结构参数的无偏估计,并给出了模拟.结果表明,在考虑目标保费的情况下,当选取一个合适的权重,可以得到未来保费的最优估计.     

The credibility premiums for models with dependence induced by common effects

In classical Bühlmann credibility models, claims are assumed to be independent between different risks. In many practical situations, however, this assumption may be violated because there are situations that could drive possible relationship among the insured individuals. This paper aims to extend the Bühlmann and Bühlmann-Straub credibility models to account for a special type of dependence between risks induced by common stochastic effects. By means of the projection method, the corresponding credibility premiums are obtained, which generalize some well known existing results in credibility theory.     

Duration dependence models for claim counts

The aim of the paper is to introduce new claim count distributions constructed from different waiting time assumptions, such as the Exponential, Gamma and Weibull distributions. These models are then fitted to panel data with Gamma distributed random effects. The random effects allow for serial dependence and take residual heterogeneity into account. Predictive distributions are obtained with the help of Markov Chain Monte Carlo simulations. The approach is illustrated on the basis of a Belgian motor third party liability insurance portfolio observed for three years.     

Statistical risk evaluation applied to (Belgian) car insurance

Before applying actuarial techniques to determine different subportfolios and adjusted insurance premiums for contracts that belong to a more or less heterogeneous portfolio, e.g. using credibility theory, it is worthwhile performing a statistical analysis on the relevant factors influencing the risk in the portfolio. Also the distributional behaviour of the portfolio should be examined. In this paper such a programme is presented for car insurance data using logistic regression, correspondence analysis, and statistical techniques from survival analysis. The specific mechanisms governing large claims in such portfolios will also be described. This work is based on a representative sample from Belgian car insurance data from 1989.     

Calculation of Property Insurance Premiums from Statistical Data

A method is proposed for calculation of premiums for a class of insurance risks in an inhomogeneous portfolio. The premiums are calculated using statistical data collected over time for different risk classes. The data represent various contract amounts and various levels of insurance claims. The calculations are carried out by the maximum likelihood method.     

A generalization of the credibility theory obtained by using the weighted balanced loss function

In this paper an alternative to the usual credibility premium that arises for weighted balanced loss function is considered. This is a generalized loss function which includes as a particular case the weighted quadratic loss function traditionally used in actuarial science. From this function credibility premiums under appropriate likelihood and priors can be derived. By using weighted balanced loss function we obtain, first, generalized credibility premiums that contain as particular cases other credibility premiums in the literature and second, a generalization of the well-known distribution free approach in [Bühlmann, H., 1967. Experience rating and credibility. Astin Bull. 4 (3), 199-207].