Measures and Models for Dependent Actuarial Risks

我建议的研究领域是精算学中的风险管理。投资组合管理的一大缺陷是对风险缓解的系统性考虑。由于监管或会计规则的原因，投资组合的所有风险都不能总是完全聚合。此外，即使它们可以做到这一点，为了进行风险比较，或者为了更准确和保守的风险保护，可能也最好是单独或以同类小组的形式考虑它们。在我的五年研究计划中，我将不会通过多元框架来研究这个问题。后者最近被引入精算和科学领域，用于风险衡量。它在加拿大和大多数发达国家的学者和从业者中越来越受欢迎。 我计划开发封闭形式的表达式，并研究多元风险度量的性质。将开发一种新的多变量风险度量--多变量截尾风险值(MTTVaR)，以衡量特定封闭概率范围内的损失。*将重新调查基于多变量数据的自然风险统计，因为基于总风险的对排序限制了它的使用。我将研究不同的顺序，以提供更易处理、更稳健、更可靠和更实用的统计数据。我将开发多变量置信域的估计器，并将它们与其他多变量接受集进行比较。我们将区分具有非常相似定义的实际多变量风险度量，除了在考虑多变量集合的方式上的差异，提供不同的统计数据。我还打算提供具有精算科学中常见的多变量随机过程和分布的多变量风险度量的闭合形式表达式。我们将开发和研究基于多变量尾部风险值和mTTVaR的精算模型适用性的统计测试。 我的计划的第二部分致力于多元风险衡量的应用。我将基于财务和精算的角度，研究使用多元风险衡量标准对保险、再保险和其他金融产品定价的影响。我将研究一些特定的原则，以进一步优化联合互惠再保险合同，例如，当配额份额和调整因素同时设定时。优化将以联合生存和盈利分配为基础。最后，我们还将推广多元再保险合同的结果。保护风险不受依赖风险的同质投资组合类别的影响有几个目的。它允许为每个风险分配价值，进行比较，重新评估每个业务线，并为风险管理目的或偿付能力要求分配资本。 挑战是创新和原创的。他们将为研究生、学者和实践者提供有趣的研究项目。

My proposed research lies in the area of risk management in actuarial science. A major drawback of portfolio management is the systematic consideration of risk mitigation. All risks of a portfolio cannot always be aggregated, because of regulation or accounting rules. Moreover, even if they can be, it might be desirable to consider them individually or in homogeneous groups, for risk comparison, or for more accurate and conservative protections. In my five-year research program, I will investigate this issue through a multivariate framework. The latter has recently been introduced in the actuarial science field for risk measurement. It is gaining popularity, both with academics and practitioners in Canada, and across most developed countries.  I plan on developing closed-form expressions and studying properties of multivariate risk measures. A new multivariate risk measure, multivariate Truncated Tail Value-at-Risk (mTTVaR) will be developed, to measure losses in a particular closed range of probabilities. Multivariate data-based natural risk statistics will be reinvestigated, since couple ordering based on aggregate risks restricts its use. I will study different ordrings to provide statistics that are more tractable, robust, reliable and practical. I will develop estimators for multivariate confidence regions and compare them with multivariate acceptance sets. We will distinguish actual multivariate risk measures that have very similar definitions, exept for differences in the way multivariate sets are being considered, providing different statistics. I also intend provide closed-form expressions of multivariate risk measures with common multivariate stochastic processes and distributions in actuarial science. A statistical test for the suitability of actuarial models, based on multivariate Tail Value-at-Risk and mTTVaR will be developed and studied. A second part of my program is dedicated to the application of multivariate risk measures. I will study the impact of using multivariate risk measures on the pricing of insurance, reinsurance and financial products, based on financial and actuarial points of view. I will investigate some particular principles, to optimize joint reciprocal reinsurance contracts, e.g. when both the quota-share and adjustment factors are set simultaneously. The optimization will be based on the joint survival and profitability distributions. Finally, we will extend the results for a multivariate reinsurance contract. Protecting risks from homogeneous classes of portfolios of dependent risks is useful for several purposes. It allows to allocate values to each risk, to compare them, to evaluate each business line and to allocate capital for risk management purposes or solvency requirements. The challenges are innovative and original. They will provide interesting research projects for graduate students, academics and practitionners.