Chain Ladder Method For Generalized Run-off Triangle And Related Stochastic Models

For the property and casualty companies, sufficient loss reserving is an important approach to keep the company in enough business solvency, and is also a significant aspect of checking corporate profits. The observable aggregate claims can form an upper left corner triangle, i.e. run-off triangle. Our aim is to predict the lower right corner triangle, i.e. non-observable claims.Up to now most discussions are based on full and simple run-off triangles. 3ut in pratice, the true data stream we generally observed is not a full triangle, i.e. we can only observe claims by m years after claims' occuring. It's easy to understand because the occurrence years continue with the insurance liabilities, but for cases occured in an occurrence year, its' incremental claims can't continue infinitely. Therefore, we extend the previous triangle into a generalized one, and make some research on it.Because of the property of simplicity and distribution-free, the chain ladder method has become the most popular method of claims reserving. But due to the implicit stochastic nature of the data, it deserve research on whether the chain ladder method can be justified by a stochastic model and a statistical method related to the model and under which conditions the chain ladder method should be applied or not.Based on generalized run-off triangle, we introduce multiplicative model and its marginal-sum estimation in chapter 2. We get that the marginal-sum estimation is the chain ladder estimation. In chapter 3, we make further assumptions on distribution of the incremental claims and discuss the parameter estimation when Poisson distribution or multinomial distribution is fulfilled. We prove that MLE is just the marginal-sum estimation in such circumstances. Furthermore, we discuss the testing of the data's Poisson distribution property and its steadiness. We extend the test to the over-dispersed Poisson model.In chapter 4, we discuss Mack model and Schnus model of the run-off triangle in sense of n≠m, and make paralleled conclusions with the former simple triangle, that is, for the first non-observable claims, its optimal quadratic loss prediction is the chain ladder prediction, and it's unbiased.Finally, we make a summary of the models and respective methods of estimation or prediction in this paper, which can be figured out clearly by our chart.