Insurance loss model and surplus analysis

保险公司财务风险管理活动的一个组成部分是对特定时刻和一段时间内的索赔费用进行准确建模。这种努力必然涉及使用复杂的数学工具和技术。反过来，需要这些模型来预测和评估保险公司持续财务可行性所需的数量。投保公众也将受益，因为对保险公司责任的准确分析有助于降低与索赔引发事件相关的总体成本。 拟议的研究的一个目标是分析所谓的“混合泊松”模型的索赔计数，和“混合Erlang”模型的个人索赔规模的成本。这两类模型都具有非常有用的易处理的分析和计算特性，同时具有准确再现相应的真实的世界保险量的能力。然而，它们在保险中的应用虽然意义深远，但只被部分理解。拟议的研究将进一步研究其在保险风险管理中的使用，特别是其对风险相关的数量，如保费，索赔责任，应急准备金等的影响。一个重要的功能，以进一步探讨混合泊松索赔计数模型是他们的能力，以适应风险异质性的投保人群。还将探讨它们在保险方面使用的其他有用但不太明显的后果，包括它们适应索赔通货膨胀（索赔成本中通常是非常昂贵的组成部分）的能力，以及它们在涉及每次索赔免赔额的情况下的使用。类似地，混合Erlang索赔严重性模型对于易于处理的损失规模的量化非常方便，但它们在某些类型的保险中固有的“长尾”数据的情况下的使用尚未被探索。还将进一步分析各种索赔额之间的相互依存关系。 索赔费用分析的另一个同样重要的方面涉及索赔费用随时间的发展，特别是在涉及长期合同的情况下。在历史上，有些不同的“Gerber-Shiu”和“Seal”方法已经被使用。就保险偿付能力的分析而言，这两种方法都有优点和缺点，拟议研究的第二个目标是进一步研究每种方法的用途。不过，还计划查明和探讨这两种办法之间的基本关系，从而使每种办法的长处能够相互借鉴。 拟议研究的预期结果是更准确和更好地理解索赔费用产生的负债模型。保险公司和被保险公众都将从这一额外的知识中获益。

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.