Quantitative methods for risk management

“他们做这些假设是为了做他们的幻想数学！”这是我们所有精算师都从从业人员那里听到的指责，不管这些从业人员有多精通数学。就在几天前，在风险与保险研究中心/舒立克商学院的联合研讨会上，精算师兼数学博士斯蒂芬·理查兹（Stephen Richards）以一种尖刻的方式批评了“与现实毫无联系的花哨数学模型”。不管Stephen的话是否有一定道理，作为负责任的学者，我们必须意识到这种相当普遍的观点，它加剧了象牙塔和工作场所现实之间已经很深的差距。本文试图提出深层次的数学方法，解决风险专业人员在实践中遇到的问题。三个不同而又相互联系的研究线索，即风险聚合、风险分配和变异性评估，在本提案中通过一个可疑的风险从业者的镜头进行了批判性的探索。具体来说，我们解决了一些不方便但突出的问题，这些问题都是这样的：“如果假设XYZ是不合适的怎么办？”，其中各种XYZ是由加拿大保险公司和银行的许多精算师和定量风险专业人员确定的。简要地说：在第1节中，我们提出了一种有效的算法，它能够快速准确地逼近具有不同参数族分布的相关风险的和的分布，而不考虑求和的数量和涉及的重尾性；在第2节中，我们主张如何对待风险资本配置的范式转变，作为副产品，我们统一了自上而下和自下而上的方法来配置风险资本；在第3节中，我们阐明了用方差leu衡量可变性的途径，特别是，我们建议开发著名的Hattendorff定理的重尾变体，该定理将可变性分配到未来几年——当用一致的和共同的可变性累加度量来衡量时——用于几乎任何支付流和利息积累规则。拟议的研究将产生的方法是真正的行业驱动的，因此具有直接的翻译重要性，以及严谨和最高的学术水平。套用史蒂夫·乔布斯的话，我可以真诚地说，美丽的数学与社会公益相结合，让我的心在歌唱。我真的很兴奋有能力的翻译研究，这是一个例子。

``They are making those assumptions in order to do their fancy maths!'' is an accusation that all of us, academic actuaries, have heard from practitioners, irrespective of how mathematically savvy these practitioners are. Just a few days ago, Stephen Richards, a practicing actuary (FFA) and a Ph.D. in Mathematics criticized in an acerbic manner ``fancy mathematical models that have zero connection with reality'' at a joint Risk and Insurance Studies Centre / Schulich School of Business colloquium talk. Whether or not there is some truth to Stephen's words, as responsible academic we have to be aware of this rather common opinion, which aggravates the already profound gap between the ivory tower and the workplace reality. This proposal is an effort to put forward deep mathematical methods, which solve those problems that are brought up by practicing risk professionals. Three distinct and yet connected research threads, which are risk aggregation, risk allocation and variability assessment, are explored in this proposal critically through the lens of a dubious risk practitioner. Specifically, we address somewhat inconvenient but salient questions that all go like this: "What if assumption XYZ is inappropriate?", where the various XYZs were identified by a number of actuaries and quantitative risk professionals in Canadian insurance companies and banks. Speaking briefly: In Section 1, we put forward an efficient algorithm, which is able to approximate - fast and accurately irrespective of the number of summands and involved heavy-tailedness - the distributions of sums of dependent risks having distributions in distinct parametric families; In Section 2, we argue for a paradigm shift as to how risk capital allocations should be treated and, as a by product, we unify the top-down and the bottom-up approaches to allocate risk capital; In Section 3, we shed light on the routes to measure variability in leu of variance, and in particular, we propose to develop heavy-tailed variations of the celebrated Hattendorff's theorem that allocate the variability - when measured by coherent and comonotonically additive measures of variability - to future years for virtually any payment stream and rule for accumulation of interest. The methodologies that will be coming out of the proposed research are truly industry-driven and so of direct translational importance, as well as rigorous and of the highest academic caliber. Paraphrasing Steve Jobs, I can genuinely say that it is beautiful mathematics married with social good that makes my heart sing. I am really thrilled about the capable translational research, of which this proposal is an example.