Stochastic Claims Reserving Models And Methods In Non-life Insurance:a Statistical Perspective

Claims reserve is typically one of the largest items of liabilities in the balance sheet of non-life insurance companies. Assessment of the operating performance and solvency of non-life insurance companies depends on accurate evaluation of claims reserve liabilities. On the one hand, the accuracy of claims reserving is based on to reflect the operating outcome, and is also a scientific decision making basis for the operation and management of non-life insurance companies. On the other hand, the adequacy of claims reserves is of fundamentally importance to the solvency position and risk profile of non-life insurance companies, as well as a basic requirement for solvency regulations stipulated by insurance regulatory authority. Therefore, scientific and appropriate assessment of the liabilities is a matter of vital importance to the operation and regulatory for non-life insurance companies.Currently within the international non-life actuarial science community, there exist two categories of claims reserving models, one is aggregate claims models, and the other is individual claims models. The aggregate claims reserving models have been popularized over many decades and are still the mainstream. For both categories of claims reserving models, estimation of claims reserve is no longer confined to a point estimate, the estimation has gradually involved in quantifying variability in reserve estimates, such as the concepts of best estimates and interval estimates. In order to demonstrate these concepts theoretically, it is imperative to investigate thoroughly a variety of stochastic models and methods of claims reserving. However, stochastic models are relatively more complex than deterministic models, and the available two monographs both domestic and overseas focus on aggregate claims models to consider stochastic claims reserving methods, and both monographs rely heavily on the mean square error of prediction to measure the volatility of reserve estimation. In addition, requirement of stochastic methods has become an issue recently in China’s non-life actuarial practices, and stochastic methods have been increasingly discussed. Considering these reasons, this dissertation is designed to further explore the issues with respect to estimated mean square error of prediction and simulated predictive distribution of stochastic claims reserving methods, advancing our previous research results, within the framework of aggregate claims models, from the perspective of more rigorous statistical models and methods. Below is a summary of the content of the dissertation:First, considerations are regards to some univariate stochastic models and methods of claims reserving based on a single runoff triangle, including stochastic chain ladder methods with distribution-free assumptions, distributional methods, and hierarchical models taking account on longitudinal characteristics of runoff triangle data. They are expanded in chapter2,3and4respectively.Second, discussions are on two types of multivariate stochastic models and methods of claims reserving based on multiple runoff triangles within dependence structures. One type is based on the correlation between the paid payments and the incurred payments claims data, anther type is based on several correlated lines of business or payment types, as well as hierarchical structures. They are investigated in chapter5and6respectively.Third, in the previous chapters it is assumed that the aggregated runoff triangle data are correct, i.e. without the outlying or abnormal claims amount. For the issue of possible outlying values, it is proposed to consider the robust inference tools for a variety of claims reserving methods in chapter7. In the case of univariate claims reserving, two basic stochastic models and methods of robust claims reserving are discussed, i.e. robust chain ladder method and robust generalized linear models.Finally, the dissertation further extends the issues of statistics diagnostics and testing, and model selection in claims reserving, based on robustness in data and models. The contents mainly include the inspection of whether outlying values exist in the claims data; how statistical diagnostics and testing are incorporated in stochastic models and methods, including the utilization of more intuitive and rigorous statistical methodologies, such as residual analysis, graphical diagnostics, and comparison of various statistical indexes, which test model assumptions and compare the fitting effects of different models. With the aforementioned study, the dissertation points out clearly some ideas and directions for further research.In summary, the dissertation has made a systematical study on measuring variability of claims reserve from the perspective of simulating its predictive distribution with respect to various stochastic models and methods, within the analytical framework of aggregate claims models. As a complete description, predictive distribution contains many more distributional characteristics. These studies will further expand the scope of the uncertainty risk measure of claims reserving.The main contribution of the dissertation is that in domestic non-life actuarial science community it proposes innovatively researches into four topics as follows, multivariate claims reserving methods, hierarchical models for claims reserving in both univariate framework and multivariate framework, several robust claims reserving methods with outlying values, and statistical diagnostics regards to various claims reserving models. Some new ideas also are proposed and discussed. Especially, concerning use of rigorous statistical models and methods to measure the volatility and simulate predictive distribution of claims reserves, the dissertation explores stochastic claims reserving methods from a new perspective of hierarchical models, in combination with the structure of non-life insurance data, i.e. some hierarchical structure and dependence, incorporating Bayesian methods, stochastic simulation, credibility theory, data analysis techniques, and scientific computing into the hierarchical framework. These studies not only have an important scientific significance to enhance the statistical analysis system of the non-life actuarial science and promote the development of non-life actuarial science in China, but also provide theoretical support and practical reference for stochastic claims reserving of domestic property insurance companies.It is worth pointing out that the dissertation contains a large amount of complex numerical calculation, which is largely attributed to the support of increasingly sophisticated computer technology and statistical software. R language is free development software, which is increasingly popular in current international community and has many packages. For various stochastic models and methods of claims reserving, the dissertation applies R for complete programming. All algorithms are modular and have a high level of flexibility and portability. R software is increasingly widespread with applications in financial engineering, quantitative risk management, statistics and actuarial science and so on. In the international actuarial academia, it has become a development trend to solve actuarial numerical problems using R. In addition, in some chapters WinBUGS software specifically for Bayesian statistical analysis are also applied.