Quantitative methods for risk management

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``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.