Decision making problems in Actuarial Science

2017年，全球巨灾（CAT）事件造成的保险损失为1440亿美元（瑞士再保险），是有史以来最高的一年。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金短缺甚至偿付能力问题。因此，保险公司对这些风险进行对冲就显得尤为重要。如果没有适当的对冲安排，保险公司可能会资不抵债，公众暴露的风险可能无法得到赔偿，或者政府机构可能需要以纳税人的最终费用来救助一家或多家保险公司。本研究计划的目的是研究整合现有套期保值机制的最佳方法，以建立一个有效的巨灾风险管理系统，考虑所有利益相关者的利益，包括被保险人、保险公司、再保险公司、投资者和政府。保险公司对冲CAT风险的两种常用机制是：(1)从再保险公司购买常规再保险保险，其中赔偿是主保险人承保损失的函数；(2)从金融市场购买巨灾损失指数证券，其中赔偿是特定的巨灾损失指数的函数，例如全行业损失。要问的问题是如何最好地利用这些机制。最优再保险问题是一个老问题。关于保险公司应该购买何种再保险险种以及应该在再保险险种上花费多少，文献中有深刻的结论。波奇（K. Borch）和阿罗（K. J. Arrow）等著名经济学家认为，为了使保险人的风险（以方差衡量）最小化，或使其预期效用最大化，最优再保险政策应该具有止损形式，即超过一定阈值的损失部分由再保险公司支付。这一研究领域仍然非常活跃，目前的重点是当风险用更现代的偿付能力相关风险指标（如风险价值（VaR）和风险尾部价值（TVaR））来衡量时，最优再保险的格式。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，世界各地正在进行积极的研究。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，以建立一个风险管理系统，可能会有效得多。因此，研究这样做的最佳方法是本研究计划的目标。为了实现这一目标，我们将把经典的决策理论与先进的自然灾害工程模型（如加拿大的地震）结合起来。我们与西方土木工程团队的密切合作使这种组合成为可能。我们相信这个研究项目将有助于建立一个有效的加拿大CAT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.