Research on Estimation Methods of Loss Reserve Based on Generalized Linear Models
|Tianjin University of Finance and Economics
|loss reserve generalized linear models comonotonicity generalized linear mixed models Hoerl curve
As an important item of debts, the estimate of loss reserve have been paid moreattention by Non-insurance company and regulatory authorities. On the one hand,a scientific estimate of loss reserve ensure exact price of insurance product so as toimprove the competence of insurance company. on the other hand, it can strengthenthe solvency and debase the risk of the company as well. So the estimate of loss reserveis a important item in the non-insurance actuary study.In last two decades, stochastic models for the estimate of loss reserve have been ahotspot in actuary by virtue of the rapid development of computer and statistics. As anatural generalization of the classical linear models, generalized linear models(GLMs)have much adaptive characteristics for non-life actuary and have been applied widely.At present, GLMs have been the standard model for automobile, individual and prop-erty insurance of small company in Europe.In this paper, we study estimate methods of loss reserve based on GLM in virtueof recent research. The paper is divided into six chapters and is structured as follows.Chapter one is an introduction of all the paper. Firstly, the basic concept and themodels for the estimate of loss reserve in practice are introduced. The motivation fordrawing of loss reserve is described. Secondly, the relations among loss reserve, profit,solvency of insurance company and price of production are analyzed. The deficiencyof conventional models are discussed, meanwhile the background and evolution ofstochastic models are shown. At last the basic ideas and frame of this research areproposed in this study.Chapter two introduce the basic assumption, estimate of parameters and testof GLM. Based on this the main characteristics of GLM which are fit for the non-life actuary are analyzed. Then the applications of GLM in estimate of loss reserve issummarized . The emphases are the basic idea, structure and the estimate of predictionerror of the models.Champer three propose two-stage GLMs to estimate claim number and paymentsper claim respectively so as to estimate the loss reserve of insurance company. Themodels in this chapter include two types. The one is two-stage GLMs based on PPCImethod and the other is based on PPCF method. Firstly, the necessity for adoptingtwo-stage GLMs is analyzed. Secondly, the two types of two-stage GLMs are estab-lished. Meanwhile the formulas of predictive error are deduced. Finally, a practice example is shown to illustrate the model.The aim of chapter four is to estimate the distribution function of loss reservebased on the comonotonicity theory. Firstly, the basic theory of comonotonicity is in-troduced. Then the stochastic bounds of loss researve based on GLMs for four diflerentcases are discussed and the stochastic bounds and formulas of stop-loss premium areobtained in the sense of convex order. Meanwhile, two kinds of convex combinationsare proposed to appropriate the estimate of loss reserve. The characteristics (distribu-tion or stop-loss premiums etc.)of stochastic bounds are more simple to be calculated,so their distribution (or stop-loss premiums) can act as a kind of approximation ofdistribution (or stop-loss premiums) of loss reserve so as to provide more informationof loss reserve. Two examples are shown to illustrate the conclusion.Chapter five discuss the application of extended models of GLM in estimationof loss reserve, including generalized linear mixed models and generalized nonlinearmodels. In first section, the estimate of loss reserve in case of multiple excess layers arediscussed. Stochastic eflects are introduced so as to reflect the correlation among thepayment in same unit, so generalized linear mixed models are established to estimatethe loss reserve in case of multiple excess layers. In second section, the Hoerl curve isimproved in order to overcome its deficiency. The improved models include nonlinearitem in covariates so that the generalized nonlinear models is established. The newmodels can be used to fit diflerent development patterns of payments. Two numericexamples are shown and the results between new model and the common model arecompared so as to illustrate applicability of the new models.Chapter six conclude all the paper. The innovation and deficiency of this researchare pointed out and some items for future research are proposed.