Decision making problems in Actuarial Science

2017年，全球巨灾(CAT)事件的保险损失为1440亿美元(瑞士再保险)，是有记录以来最高的一年。在加拿大，这一数字约为13.3亿美元(CatIQ)。这些损失给保险公司造成了严重的资本短缺甚至偿付能力问题。因此，对这些风险进行套期保值对保险公司来说至关重要。如果没有适当的对冲安排，保险公司可能会资不抵债，暴露在风险之下的公众可能得不到赔偿，或者政府机构可能需要救助一家或多家保险公司，最终费用由纳税人承担。 这项研究的目标是研究整合现有对冲机制的最佳方法，以建立一个有效的巨灾风险管理体系，该体系考虑到所有利益相关者的利益，包括投保人、保险公司、再保险公司、投资者和政府。 保险公司常用的两种对冲CAT风险的机制是(1)向再保险人购买传统的再保险，其中弥偿是主保险人承保损失的函数；以及(2)从金融市场购买巨灾损失指数证券，其中弥偿是某些CAT损失指数的函数，例如整个行业的损失。要问的问题是如何最大限度地利用这些机制。 最优再保险问题是一个古老的问题。关于保险公司应该购买什么样的再保险，以及应该花多少钱购买这些再保险，文献中有很深的结果。K.Borch和K.J.Arrow等著名经济学家认为，为了最小化保险公司的风险，或最大化其预期效用，最优再保险政策应该具有止损形式，超过一定阈值的部分损失由再保险公司支付。研究领域仍然非常活跃，目前的重点是最优再保险的形式，当风险是用更现代的偿付能力相关风险衡量指标，如在险价值(VaR)和在险尾值(TVAR)来衡量的时候。 在文献中，关于基于CAT损失指数的证券的最优合约的结果要少得多。然而，世界各地都在进行积极的研究。 一个关键的见解是，研究人员一直在分别研究最优再保险合同或基于指数的证券。整合这些不同的对冲机制来构建一个风险管理体系可能要有效得多。因此，研究这样做的最优方法是本研究计划的目标。 为了达到这一目标，我们将把经典的决策理论与先进的自然灾害工程模型结合起来，例如加拿大的地震。这一结合是由于我们与西部的土木工程团队的密切合作而实现的。我们相信，这项研究计划将有助于建立一个有效的加拿大猫风险管理系统。

In 2017, global insured losses from catastrophe (CAT) events were USD 144 billion (swiss Re), the highest-ever recorded in a single year. In Canada, the figure was about $1.33 billion (CatIQ). These losses cause serious capital shortage or even solvency problems for insurance companies. Therefore, it is extremely important for insurance companies to hedge these risks. Without proper hedging arrangements, insurance companies may become insolvent, the public exposed risks may not have their claims paid, or the government agencies may need to bail out one or more insurers at the ultimate expenses of the tax payers. The goal of this research program is to study the optimal approach to integrate the available hedging mechanisms to build an effective catastrophe risk management system that considers the interests of all stakeholders, including insureds, insurance companies, reinsurance companies, investors, and the government. Insurers' two commonly used mechanisms to hedge CAT risks are (1) purchasing conventional reinsurance coverages from reinsurers, where the indemnity is a function of the primary insurer's covered losses; and (2) purchasing catastrophe loss index securities from the financial market, where the indemnity is a function of certain CAT loss index, such as industry-wide losses. The question to ask is how to make best uses of such mechanisms. The optimal reinsurance problem is an old one. There are deep results in the literature about what kind of reinsurance coverages insurance companies should purchase and how much it should spend on such coverages. Famous economists such as K. Borch and K. J. Arrow argued that to minimize an insurer's risks, measured by variance, or to maximize its expected utility, the optimal reinsurance policy should have a stop-loss form, where the portion of losses above certain threshold is paid by the reinsurance company. The research area is still very active, with current focus on the format of optimal reinsurance when risks are measured by more modern solvency related risk measures such as Value at Risk (VaR) and Tail Value at Risk (TVaR). There are much fewer results on optimal contracts for CAT loss index-based securities in the literature. However, active researches are being performed around the world. A key insight is that researchers have been studying the optimal reinsurance contracts or index-based securities separately. It is likely much more effective to integrate these different hedging mechanisms to build a risk management system. Thus, studying the optimal approach to do so is the goal of this research program. To reach the goal, we will combine classical decision-making theory with advanced engineering models of natural hazards, such as earthquakes, in Canada. This combination was made possible by our close collaborations with Western's Civil Engineering team. We believe that this research program will contribute to building an effective Canadian CAT risk management system.