The Estimation Of IBNR Reserves On The Individual Claim Loss Models

Estimation of adequate reserves for IBNR (Incurred But Not Reported) claims is one of the main activities of actuaries in property insurance companies and a major topic in actuarial science, and it is the most uncertain and challenging task. The process of this estimation is complicated because the observed event time and the reporting time of the IBNR claims have a long delay, for year-end audits, which means that the claims have not been reported. So for sufficient reserves of the insurance companies, we need to estimate the IBNR claims solely. In addition, a systematic overestimation or underestimation of the IBNR reserves would lead to premiums too high or too low. Therefore, although the actuaries have put forward many models for IBNR claims, it is clear that we should carefully attempt to find an estimator of the IBNR reserves accurately.In this paper, we first introduce the definition and the background and the development of the IBNR claims. And then we review the existing claim models. The most important part of this paper is the two models we suggested. For seeking for the accurate and useful methods of the estimation of the IBNR reserves, we analysis the problem from two aspects, the compensation of the IBNR claims and the number of the IBNR claims. We choose the unbiased estimation and the model randomly truncated on the right, and the simulation studies indicates that both of the proposed models can produce efficient results.The paper is organized as follows. The first section introduces the meaning and the background of the IBNR claims, we know the research in our country is backward, especially the research in the individual models. Then this section presents some major results about the IBNR claims from a large number of papers. The research of the IBNR claims in our country starts later than the other countries, so we summarize the results in our country generally. We summarize the results in other countries from aggregated claim loss models to individual claim models. At the end of this section, we present the main thought and the basic frame of this paper, and emphasize the innovative parts. In this paper, we choose two different models from two different point of view, the compensation of the IBNR claims and the number of the IBNR claims. These models have been put forward recently and can get more precise results. In the following sections, we prove that the unbiased estimation model can attain practical use, and it can be applicable in different types of insurance. For the model randomly truncated on the right, the time-independent situation has been discussed particularly, and the proportion of the IBNR claims has been figured out according to the computer simulation. This proportion can be used to the insurance business directly.The second section introduces the definition and the elementary theory of the IBNR reserves. From this section we can know the reason of estimating the IBNR claims and the important meaning to the insurance supervision. The IBNR claims can influence the price of the insurance product, the competition between the insurance companies, and the normal running of the insurance company. If the estimation of the IBNR claims is not precise, the shareholders will have a weaker confidence in the insurance company, so the reputation of the insurance company will be injured, and the supervision will be harder. At the end of this section we introduce the methods which are in common use, and summarize their character.Section three presents the two models in this paper. For the unbiased estimation of the compensation of the IBNR claims, we assume that development factors are lognormally distributed. It comes from the chain-ladder method. It gets the unbiased estimation of the IBNR claims and the standard errors by the lognormally distributed of the development factors. The strong point of this method is that it can remove some errors in computation, so the result of the estimation is more precise. For the model randomly truncated on the right, this method usually be used in survival analyses. First we introduce the basic form of the model randomly truncated on the right. Then we consider two situations, the time-independent and the time-dependent, the Clayton copula is used in the time-dependent situation to describe the dependence of them.Section four applies the obtained results to some data sets. For the unbiased estimation of the compensation of the IBNR claims, after deriving an unbiased estimate of the IBNR reserves, we use different types of insurance data (Commercial Auto Insurance and Injury Insurance) to prove that it is suitable for different types of insurance. From these examples we know the unbiased estimation method is easy and direct, and the errors are minor, so it can be widely used in many types of insurance. For the model randomly truncated on the right, first estimate the parameters in joint distribution, then write out the subsection functions of the parameters, so we get the joint distribution. After the simulation results of the parameters, we figure out the proportion of the IBNR claims according to the computer simulation. This proportion can be used in insurance companies directly. So it also attains practical use.Section five presents some concluding remarks, the main advantage, and the shortcomings. This paper wants to find an easy and practical method to estimate the IBNR claims, use the unbiased estimation model and the model randomly truncated on the right separately to analyses the problem. After the analysis by practice data and the computer simulation, we know that these methods have many advantages according to other methods. They are easy in computation, the errors are small, and can be widely used. But how to add the judgment of the practice people is still a problem, it will be studied afterwards. At last, this paper gives some suggestions about the IBNR reserves to the insurance business of our country. The insurance companies should strengthen the study of the theory, choose suitable models cautiously, and draw lessons from foreign insurance business.