Individual Claims Reserving Models

The adequacy and accuracy of the reserve affect the insurance company much in income calculation and its solvency capacity; as main liabilities, the non-life loss reserve can be twice or even more than the annual net premiums of the company. The principal concern of loss reserving is appropriate prediction of reserves for IBNR (Incurred But Not Reported), RBNS (Reported But Not Settled) and NI (Not Incurred) Losses.Traditional claims reserving approaches are all based on the aggregated data that might fail to preserve all the information contained in the individual claims. The main body of this paper (including Chapter 3,4 and 5) provides certain individual claims model, trying to improve the accuracy of loss reserving.Chapter 3 describes an individual claims model via marked Poisson process with a random scaling factor. In this model, claims are assumed to be generated following a marked Poisson process with a random scaling factor, and the marks represent the individual claims' progress (from occurrence to settlement, including the information of report delay and settlement delay). The reserve prediction is more accurate than the traditional aggregated model and the marked Poisson process in the literature because the proposed model make use of more information than aggregated model and is more flexible and realistic than marked Possion model due to the introduction of the stochastic intensity. By means of a decomposition of the claims process, this chapter obtains the posterior distribution of IBNR and NI via conditioning approaches, which can in turn be used to predict loss reserves under arbitrarily selected loss reserving functional.Chapter 4 treats a similar individual claims model with marked Cox process via shot noise claim intensity; unlike in Chapter 3, the intensity of the claims arrival process now respects a shot noise process. With a decomposition of the claims process similar that in Chapter 3 the problem of predicting IBNR is then reduced to the problem of obtaining the best estimate of the claim intensity of time t,0≤t≤τby the assistance of reported claims, which can be achieved by means of smoothing method of processes. It is proved that, if the primary event arrival rate of the shot noise process is sufficiently large, the intensity process of reported claims process and reported claims arrival process can be approximated by a linear system represented by a two-dimensional Gaussian process, and hence, the IBNR can be predicted via the Kalman-Bucy filtering approach and the smoothing theory of linear systems. A third linear prediction model for loss reserving is proposed in Chapter 5, in which the time horizon is discretized as the one used in chain-ladder method and only the existence of the first two moments of the data is required. In this model, the data structure falls into the framework of monotone missing data, thus EGLS (Estimated Generalized Least Square) procedure is used to estimate the mean and the covariance matrix. Via a simulation study, the proposed method behaves much more effective than traditional chain ladder method. Moreover, a continuous version of this linear prediction model has also been studied and parallel results are obtained.Sometimes it is quite natural to subdivide a non-life run-off portfolio into sub-portfolios so as to make each sub-portfolio satisfy certain homogeneity property, and thus there exist several run-off triangles running at the same time. Traditionally, every loss reserving method is always applied to each single triangle to obtain its reserving prediction separately, and then the portfolio reserve is obtained by summing up all of the sub-portfolios reserves. Many scholars have pointed out the inappropriateness of these methods because correlations among different triangles are not incorporated into the forecasting procedure. Therefore, it is valuable to propose a model with the correla-tions included, i.e. multivariate claims model. Chapter 6, the other part of this paper, proposes a more realistic multivariate claims model by means of the hierarchical credibility theory with consideration in not only the payment personality of each triangle, but also the payment commonalities between different triangles caused by the similar insurance environment. In this model, the reserve in each single run-off triangle is obtained as a function of the data in all related run-off triangles.