A Study On Trivariate Copula Model In IBNR Reserve

Non-life insurance actuarial is the comprehensive application of statistics and mathematics in non-life insurance field, it is a cross subject of the study of the relationship between rating determination and reserve assessment. Non-life insurance outstanding claims reserve is the largest amount of the debt for insurance companies operating non-life insurance business. In the non-life insurance business, the retention and assessment of the outstanding claims reserve is an important content, including incurred but not reported(IBNR) reserve, reported claim reserve and loss adjustment expenses reserve, and also top priority for healthy development of the insurance industry. In fact, the IBNR reserve is the most immature research, and its supervision is the weakest, means that the accidents have occurred but the insured have not reported to the insurance company, so the insurance company should hold reserve for them, which involves the issue of the delay of the claim. If we do not more accurate assessment IBNR reserve, the management level will greatly affected. For this reason, the regulators in Europe and the United States clearly stipulate that the insurance company must independently assess IBNR reserve, rather than the outstanding claims reserve.Due to the later start of non-life insurance industry, evaluated more using the basic method, which mostly based on aggregate traffic from the run-off triangle. But the aggregate data discard a lot of useful information, reducing the accuracy and reliability of the estimate, thus,how to use individual data establish the probability model to assessment outstanding claim reserve has important theoretical and practical significance.Meanwhile, the copula construction has been popular in the financial and actuarial literature, which turns out to be very useful to model the dependence in finance,actuarial science and survival analysis.A distinct property of Archimedean copulas is that they are fully specified by some generator function. It is important for modeling purposes that Archimedean copulas are flexible to capture various dependence structures, even tail dependence. In addition,another advantage of Copula theory is that its marginal distribution and the dependence structure can treat separately, so when we need modeling variety of vitiate from different distribution and characterization the dependence structure, Copula’s role is self-evident. This paper uses the theory of actuarial science and statistics, simulation thedependence of IBNR problem, and given the corresponding policy recommendation according to the estimation results.Model of this paper is trivariate Copula based on random right censor data, which three marginal are event occurrence time, delay and amount of claim. First, we use non-parametric method estimate cumulative baseline function and regression coefficient of the event time, then use partial likelihood and Nelson-Aalen estimate the distribution of delay. On this basis, a pseudo-maximum likelihood function is employ to estimate the distribution of claim amount, which assumed to be log-normal, and the dependence parametric of copula. Last, we give the numerical simulation and the asymptotic properties of estimator.The first chapter is introduction, mainly including research background, research meaning and purpose, reviews on the reserve and Copula theory. Next mainly introduction the trivariate Copula model. The third chapter is the theoretical part of this article, focus on the estimate method of marginal distribution and the dependence parametric, and given the numerical simulation and the asymptotic properties of estimator. The fourth chapter is the empirical part of our paper. The last chapter concluding the contents of this paper, points out the deficiency and the further research direction.