New directions in insurance loss modeling, ruin theory and their applications

本研究项目继续在数学风险理论中寻找新的理论和实践成果，将该领域与其他保险或金融领域联系起来，如保险损失建模、索赔准备金和操作风险管理。当两家或更多的保险公司同时运作时，要考虑实际情况。在再保险公司承担保险公司部分损失的情况下，或者两个或两个以上的保险公司试图确定一个最佳的风险分担策略，以最小化某些风险措施的情况下，将详细讨论。已发生而未报备的损失准备和已报而未结算的损失准备是索赔准备中的重要数量，将通过破产时间、破产前盈余和破产时亏损等经典破产相关指标来分析保险公司的盈余。在更实际的方面，对保险损失建模是精算师最具挑战性的问题之一。本建议中要解决的一个重要问题是，在处理截断和删减的数据集时，确定可用于拟合损失金额的潜在候选数据。由于其非常吸引人的特性（密集性、闭包性等），混合Erlang类分布将在开发几种基于EM方法的拟合算法中发挥主要作用。详细研究了基于Erlang的索赔保留方法在索赔保留中的潜在应用。金融机构面临着战略风险、声誉风险、市场风险、信用风险、经营风险、操作风险、流动性风险等多种风险。在这些风险中，市场风险、信用风险和操作风险在实践中比较容易量化。在这个项目中，我们感兴趣的是根据巴塞尔II指令中提出的高级测量方法（AMA）来测量操作风险。为了避免模拟，我们感兴趣的是寻找与操作风险相关的总体损失分布的封闭形式解析解，同时考虑到不同度量单位之间可能出现的某些依赖结构。

This research project continues the search for new theoretical and practical results in mathematical risk theory, connecting the area with other insurance or financial areas, such as insurance loss modeling, claim reserving and operational risk management. Realistic scenarios are considered when two or more insurance companies are at play at the same time. Situations where a reinsurer covers parts of the insurer losses, or where two or more insurers try to determine an optimal risk-sharing strategy that minimizes certain risks measures, will be pursued in details. The incurred but not reported loss reserve, and the reported but not settled reserve are important quantities in claim reserving, and will be included in the analysis of the surplus of an insurance company via its classical ruin related measures - the time to ruin, the surplus prior to ruin and the deficit at ruin. On a more practical side, modeling insurance losses is one of the most challenging problems for actuaries. An important question to be addressed in this proposal is the identification of potential candidates which can be used for fitting loss amounts when dealing with truncated and censored data sets. Due to its very appealing properties (denseness, closure etc.), the mixed Erlang class of distributions will play the main role in developing several fitting algorithms based on the well-known EM method. Potential applications of the proposed Erlang based methods in claim reserving will be studied in detail. Financial institutions are exposed to a variety of risks such as, strategy risk, reputation risk, market risk, credit risk, business risk, operational risk, liquid risk, and so on. Among these risks, market, credit and operational risks can be easily quantified in practice. In this project we are interested in measuring the operational risk according to the Advanced Measurement Approach (AMA) proposed in the Basel II directive. In order to avoid simulation, we are interested in finding closed form analytic solutions for the distribution of the overall losses associated to the operational risk, taking into consideration certain dependence structures that may occur among the different units of measurement.