Stochastic Loss Reserving: Individual Data Vs Aggregate Data Models

Loss reserving is one of the most important issues for actuaries in general insurance companies and an accurate assessment is of great significance to their operational man-agement and solvency. Traditionally, the loss reserving is computed by some deterministic methods and to measure the variability of its result, stochastic models were introduced into the framework of claim reserving. Stochastic models for claim reserving can be di-vided into aggregate data models and individual data models. The aggregate models evaluate the reserving with data in a runoff triangle, such models are of little calculation and easy to operate but the drawback is there is a waste of information. Individual data models can make use of full information but the computation tends to be large and the modelling is complex. Although many scholars confirmed by some numerical method that the individual data models is superior to aggregation data model, theoretical results are almost empty.In this dissertation, we investigate the individual loss reserving problem in discrete time models under micro-level data structure. Also, the asymptotic behaviors and MSE of individual loss reserving and aggregate loss reserving with the aggregate data deduced from the individual data are compared. The contents of this dissertation are as follows:(1) we investigate the efficiency of individual loss reserve compared to aggregate loss reserve and the finite sample as well as asymptotic behavior of individual loss reserving and aggregate loss reserving under the assumptions that each claim is paid only once in a discrete time model. Given the observed individual data set and the corresponding aggregate data set, the analytical expressions of individual reserve and aggregate reserve as well as the efficiency of individual loss reserve compared to aggregate loss reserve are studied. Then we obtain the expression of individual loss reserving via individual loss reserve and the asymptotic behavior of which is studied. The asymptotic behavior of aggregate loss reservings is also studied. Finally, we compared the individual method and aggregate method via simulation.(2) we consider the MSE and asymptotic behavior of individual loss reserving and aggregate loss reserving under the assumptions that each claim is paid only once at its set-tlement and the settlement process do not depends on reporting delay. We firstly obtain the analytical expression of individual reserve under model assumption and then a table showing the computing process is given. Then expression for individual loss reserving is obtained by substituting the unknown parameters in the expression of individual loss re-serve with their estimates. Thereafter, the asymptotic behaviors of the difference between the two classes of loss reserving and individual loss reserve is studied and compared. In the end, we compared the MSE of the two methods in finite sample case by Monte Carlo simulation. The result shows that in both the finite case and asymptotic meaning, the individual method outperforms the aggregate methods.(3) The RBNS claim reserving problem in the framework of individual data structure is studied. Firstly the analytical form of RBNS reserve under data structure is obtained, then the individual RBNS reserving is obtained by substituting the unknown quantities in RBNS reserve with their estimates, where the parameters concerned to the settlement delay are estimated by the method of maximum likelihood and the conditional mean is estimated by Watson-Nadaraya estimate. Meanwhile, the asymptotic distribution of aggregate RBNS reserving is also investigated. Finally, we compared the two classes of reserving by Monte Carlo simulation and the result shows that the individual method is better than the aggregate methods in the MSE criterion.