Insurance loss model and surplus analysis

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An integral part of the financial risk management activities of insurers is the accurate modelling of claims costs both at a specific juncture and over time. Such endeavors necessarily involve the use of sophisticated mathematical tools and techniques. In turn, these models are needed in order to predict and assess quantities necessary for the ongoing financial viability of the insurer. The insured public will also benefit as accurate analysis of the insurer's liabilities helps to reduce the overall costs associated with claim causing events. One goal of the proposed research involves analysis of the so-called 'mixed Poisson' models for claim counts, and 'mixed Erlang' models for the costs of individual claim sizes. Both of these classes of models enjoy extremely useful tractable analytic and computational properties, while at the same time possessing the ability to accurately reproduce the corresponding real world insurance quantities. However, their applications in insurance, while far-reaching, are only partially understood. The proposed research will further examine their use in insurance risk management and in particular, their implications for risk related quantities such as premiums, claim liabilities, contingency reserves, etc. An important feature to be explored further with respect to mixed Poisson claim count models is their ability to accommodate risk heterogeneity of the insured population. Other useful yet less obvious consequences of their use in an insurance context will also be explored, including their ability to accommodate claims inflation (often a very costly component of claim costs), as well as their use in situations involving per claim deductibles. Similarly, mixed Erlang claim severity models are very convenient for tractable quantification of the size of losses, yet their use in situations with 'long-tailed' data inherently arising in some types of insurance has not yet been explored. Further analysis of the incorporation of dependencies between various claim sizes will also be done. An equally important aspect of the analysis of claim costs involves their development over time, particularly in situations involving contracts of a long term nature. Historically, the somewhat distinct 'Gerber-Shiu' and 'Seal' approaches have been used. Both approaches have strengths and weaknesses insofar as the analysis of insurance solvency is concerned, and a second goal of the proposed research is to study further the uses of each of these approaches. However, it is also planned to identify and explore underlying relationships between the two approaches, thus allowing for the strengths of each approach to be carried over to the other. The anticipated outcomes of the proposed research are more accurate and better understood modelling of the liabilities resulting from claim costs. Both insurers and the insured public will benefit financially from this additional knowledge.