THE ESTIMATION OF OUTSTANDING CLAIMS LIABILITY IN LONG-TAIL INSURANCE BUSINESS BASED ON CORRELATED RANDOM SUM - KOMARUDIN MUTAQIN (NIM : 30107005); Tim Pembimbing: Prof. Dr. Sutawanir Darwis, Dumaria R. Tam, ACENG
In long-tail insurance business, there are many different statistical methods are available for estimating outstanding claim liability (OCL). In general, there are two different approaches to estimate OCL. One of these approaches is based on the aggregate claim payments data which summarized in a run-off triangle (aggregate method), and the other one is based on the individual claim payments data (individual method). The aggregate claim payments data analysis is often reasonable where there are large numbers of small claims, and the run-off pattern is stable. On the other side, the individual claim payments data analysis most useful for large claims, unstable run-off pattern, a few number of claims, and if there are continuous covariates. In long-tail insurance business, a claim can be settled by more than one payment. The existing individual methods are based on modeling claim data. This dissertation develops estimation method of OCL by involving payment frequency, payments amount, and correlation between payments based on correlated random sum. The major contributions of the dissertation are to construct the theory of correlated random sum (CRS), and to construct the new estimation method of OCL using theory of CRS. The distribution, mean, variance, and percentile of CRS are discussed. The special case of CRS is correlated lognormal random sum (CLNRS), where the conditional joint distribution of the random variables of CRS given number random variables is multivariate lognormal. Approximate distribution of CLNRS is derived, where the resulting distribution is mixture of lognormal distribution. The estimated approximation distribution using plug-in principle is obtained. It is statistically not different from its exact distribution for any sample size and correlation coefficient. Two methods are used to estimate parameters of CLNRS distribution, i.e., separated maximum likelihood estimation (SMLM) and generalized moment method (GMM). Simulation result shows that both methods are consistent. Another result shows that SMLM method is better than continuous-updating GMM method. One moment method (MM) and two parametric methods (MP1 and MP2) are used to estimate mean and variance of CLNRS. This dissertation shows that estimated mean of compound binomial-correlated lognormal and compound Poisson correlated lognormal distributions are asymtotically unbiased estimators. All methods are consistent except MP1 to estimate variance. The Monte Carlo simulation shows that for almost cases MP1 is the best method to estimate mean and variance of CLNRS. MM is the best method to estimate variance of CLNRS for 30 and the distribution of N is binomial. When the distribution of N is Poisson, MM is the best method to estimate variance of CLNRS, on the other side there is no one best method to estimate mean of CLNRS. Two new methods to estimate OCL are constructed, i.e., the methods based on expectation of CLNRS, and parametric method based on conditional distribution of total claim payments. In general, parametric method with covariate is more accurate to estimate OCL. Chain ladder method more accurate than others for total number of claims more than 2.000 and the probability function of payment frequency is increasing. In theory, parametric method with conditional distribution is better than others. The simulation also shows that the method is more accurate than others. But it need more simulation to validate the theory. The application on personal injury insurance claims shows that chain ladder method is more accurate for large number of claims (22.036). On the other side, for small number of claims (500),parametric method with involving legal representative is more accurate.