Weekly seminars: 2023
NovemberNov 30: Qinyu Wu (University of Waterloo)
Title: Model Aggregation for Risk Evaluation and Robust Optimization
Speaker: Qinyu Wu (Postdoc Fellow, University of Waterloo)
Time: 14:00 - 15:30 pm, Nov 30, 2023 (Thu)
Location: M3 3127
Abstract: We introduce a new approach for prudent risk evaluation based on stochastic dominance, which will be called the model aggregation (MA) approach. In contrast to the classic worst-case risk (WR) approach, the MA approach produces not only a robust value of risk evaluation but also a robust distributional model, independent of any specific risk measure. The MA risk evaluation can be computed through explicit formulas in the lattice theory of stochastic dominance, and under some standard assumptions, the MA robust optimization admits a convex-program reformulation. The MA approach for Wasserstein and mean-variance uncertainty sets admits explicit formulas for the obtained robust models. Via an equivalence property between the MA and the WR approaches, new axiomatic characterizations are obtained for the Value-at-Risk (VaR) and the Expected Shortfall (ES, also known as CVaR). The new approach is illustrated with various risk measures and examples from portfolio optimization.
Nov 23: Zachary John Van Oosten (University of Waterloo)
Title: Coherent Extensions
Speaker: Zachary John Van Oosten (PhD Student, University of Waterloo)
Time: 14:00 - 15:30 pm, Nov 23, 2023 (Thu)
Location: M3 3127
Abstract: In this talk, I will discuss some preliminary results on extending coherent risk measures defined on the subspace of random variables that are measurable with respect to a coarser-sigma-algebra. These extensions will be called coherent extensions and this concept is motivated by my earlier project on partially law-invariant coherent risk measures. I will include some questions I am currently working on and a brief description of what I hope to get out of the project.
Nov 16: Samuel Solgon Santos (University of Waterloo)
Title: The (possibly) undesirable aspects of comonotonic additive risk measures: a literature review
Speaker: Samuel Solgon Santos (Postdoc Fellow, University of Waterloo)
Time: 14:00 - 15:30 pm, Nov 16, 2023 (Thu)
Location: M3 3127
Abstract: The theory of risk measures has grown enormously in the last twenty years. In particular, risk measures satisfying the axiom of comonotonic additivity were extensively studied, arguably because of the plethora of results indicating interesting aspects of such risk measures. Recent research, however, has shown that this axiom is incompatible with properties that are central in specific contexts. In this paper we present a literature review of these incompatibilities.
Nov 9: Zhaowen Wang (Fudan University)
Title: CoVaR Under Asymptotic Independence
Speaker: Zhaowen Wang (PhD Student, Fudan University)
Time: 14:00 - 15:30 pm, Nov 9, 2023 (Thu)
Location: M3 3127
Abstract: There has been increasing interest in measuring systemic risk since the global financial crisis of 2007-2009. Accurate assessment of systemic risk would enable regulators to introduce suitable policies to mitigate the risk as well as allow individual institutions to monitor their vulnerability to market movements. One popular measure of systemic risk is the conditional value-at-risk (CoVaR), proposed by Adrian and Brunnermeier (2011). We develop a methodology to estimate CoVaR semi-parametrically under asymptotic independence within the framework of bivariate extreme value theory. There is growing attention on asymptotic independent pairs in quantitative risk management. We use parametric modelling of the bivariate extremal structure to address data sparsity in the joint tail regions and prove consistency of the proposed estimator. The performance of the estimator is illustrated via simulation studies and a real data example.
Nov 2: Zhiqiao Song (University of Waterloo)
Title: Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances
Speaker: Zhiqiao Song (PhD Student, University of Waterloo)
Time: 14:00 - 15:30 pm, Nov 2, 2023 (Thu)
Location: M3 3127
Abstract: This presentation will delve into various financial models related to portfolio selection. We will revisit the Markowitz’s mean-variance model, Fama-French model, as well as Black-Litterman model, and introduce a distributionally robust model newly proposed in the paper “Blanchet, J., Chen, L., and Zhou, X. Y. (2022). Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances.” This paper studies the robust portfolio selection problem with distributional uncertainty centered around the empirical measure and dictate the discrepancy between probability measures by Wasserstein distance. The problem is reformulated as an empirical variance minimization problem with an additional regularization term. Recently developed inference methodology is applied in order to define the size of the distributional uncertainty and the associated robust target return. Finally, a brief comparison would be provided to illustrate the performance of the models.
October
Oct 19: Yimiao Zhao (University of Waterloo)
Title: Catastrophe Risk Pooling
Speaker: Yimiao Zhao (PhD Student, University of Waterloo)
Time: 14:00 - 15:30 pm, Oct 19, 2023 (Thu)
Location: M3 3127
Abstract: The ongoing climate change causes natural disaster losses are on the rise. The huge gap between actual and insured losses, however, is growing. Insurance-based solutions are receiving increasing attention to mitigate disaster and climate risks. Due to the features of catastrophic risks, catastrophe insurance market has failed to provide sufficient protections. In this paper we discuss how pooling catastrophe risks from various geographies and perils can be an effective tool for diversifying catastrophic risks and reducing risk premiums. Simulations are provided to illustrate our results.
Oct 12: Emma Kroell (University of Toronto)
Title: Optimal Robust Reinsurance with Multiple Insurers
Speaker: Emma Kroell (PhD Candidate, University of Toronto)
Time: 14:00 - 15:30 pm, Oct 12, 2023 (Thu)
Location: M3 3127
Abstract: We study a reinsurer who faces multiple sources of model uncertainty. The reinsurer offers contracts to n insurers whose claims follow different compound Poisson processes. As the reinsurer is uncertain about the insurers’ claim severity distributions and frequencies, they design reinsurance contracts that maximise their expected wealth subject to an entropy penalty. Insurers meanwhile seek to maximise their expected utility without ambiguity. We solve this continuous-time Stackelberg game for general reinsurance contracts and find that the reinsurer prices under a distortion of the barycentre of the insurers’ models. We apply our results to proportional reinsurance and excess-of-loss reinsurance contracts and illustrate the solutions numerically.
Oct 5: Qinghua Ren (University of Waterloo)
Title: Risk sharing under heterogeneous beliefs without convexity
Speaker: Qinghua Ren (PhD Student, University of Waterloo)
Time: 14:00 - 15:30 pm, Oct 5, 2023 (Thu)
Location: M3 3127
Abstract: This talk delves into the paper "Liebrich, F. B. (2021). Risk sharing under heterogeneous beliefs without convexity. " The paper aims to find Pareto-optimal allocations of risk among finitely many agents. The associated individual risk measures are assumed to be consistent with the respective second-order stochastic dominance relations, but with respect to potentially heterogeneous reference probability measures. The paper provides a simple sufficient condition for the existence of Pareto optima. Furthermore, the results are extended to capital requirements with multidimensional security markets.
September
Sep 28: Yuling Max Chen (University of Waterloo)
Title: Reinforcement Learning Approaches to Stochastic Control Problems: A Bidirectional Introduction
Speaker: Yuling Max Chen (PhD Student, University of Waterloo)
Time: 14:00 - 15:30 pm, Sep 28, 2023 (Thu)
Location: M3 3127
Abstract: Reinforcement Learning (RL) is the Computer Science counterpart of the statisticians’ Stochastic Control (SC) problem, both seeking optimal solutions for sequential decision-making problems modeled by Markov Decision Processes (MDPs). This presentation explores RL in a bidirectional manner. Starting gently from the SC perspective, we demonstrate how RL enhances exploration and exploitation in classical control problems, comparing RL-facilitated optimal control with non-RL solutions, under the Markowitz portfolio optimization setting. Conversely, Inverse Reinforcement Learning (IRL) has garnered substantial attention within the Deep Learning community, expanding the boundaries of traditional control problems. While traditional control problems (e.g., Linear Quadratic Control) optimize known objective functions within dynamic systems, IRL acknowledges the limitations of this approach in real-world scenarios, as individual’s objective (utility) evolve with accumulating experience. Thereby, we introduce a scalable Bayesian IRL model that addresses control problems with indeterminate objectives. Our method employs Variational Inference for a priori objective function learning and utilizes exploration-exploitation strategies to identify optimal controls. Empirical results underscore the model's efficacy, showcasing its promising performance in solving a toy maze problem.
Sep 21: Ziyue Shi (University of Waterloo)
Title: Robustness in the optimization of risk measures
Speaker: Ziyue Shi (PhD Student, University of Waterloo)
Time: 14:00 - 15:30 pm, Sep 21, 2023 (Thu)
Location: M3 3127
Abstract: This talk delves into the paper "Embrechts, P., Schied, A., & Wang, R. (2022). Robustness in the optimization of risk measures. Operations Research, 70(1), 95-110." A risk measure as a function may have some robustness properties but may fail to be robust within an optimization context. Robustness properties on the procedure of optimization are investigated in this paper. The key concern of the paper is to investigate the influence of the selection of the underlying objective on the robustness in risk optimization. Specifically, it places particular emphasis on the two popular regulatory risk measures, the Value-at-Risk and Expected Shortfall, and their robustness properties in optimization problems. In this talk, I will focus on the motivation, model setup, results, and main conclusions of this paper.
Sep 14: Christopher Blier-Wong (University of Waterloo)
Title: FGM copulas, generalizations, and actuarial applications
Speaker: Christopher Blier-Wong (Postdoc Fellow, University of Waterloo)
Time: 14:00 - 15:30 pm, Sep 14, 2023 (Thu)
Location: M3 3127
Abstract: This talk will outline recent advances on the Farlie-Gumbel-Morgenstern (FGM) family copulas. At its core is a stochastic representation of FGM copulas based on multivariate symmetric Bernoulli distributions. This one-to-one correspondence lets us derive useful properties for this family. We study subfamilies and generalizations of FGM copulas and construct high-dimensional copulas with few dependence parameters. Then, we will explain some actuarial applications, focusing on risk aggregation (including risk-sharing and capital allocation) and collective risk models. Finally, we will conclude with current avenues for research, including high-dimensional estimation strategies and generalizing the results to Baker copulas. This research is based on joint work with Hélène Cossette, Etienne Marceau, and other collaborators.
April
Apr 20: Yixuan Fan (Jilin University)
Title: Safe Testing
Speaker: Yixuan Fan (PhD Candidate, Jilin University)
Time: 15:30 - 17:00 pm, Apr 20 (Thu)
Location: M3 4206
Abstract: This paper develops the theory of hypothesis testing based on the e-value, a notion of evidence that, unlike the p-value, allows for effortlessly combining results from several studies in the common scenario where the decision to perform a new study may depend on previous outcomes. Tests based on e-values are safe, i.e. they preserve Type-I error guarantees, under such optional continuation. This paper defines growth-rate optimality (GRO) as an analogue of power in an optional continuation context and shows how to construct GRO e-variables for general testing problems with composite null and alternative, emphasizing models with nuisance parameters. GRO e-values take the form of Bayes factors with special priors. The theory is also illustrated by using several classic examples including a one-sample safe t-test and the 2×2 contingency table. Sharing Fisherian, Neymanian and Jeffreys-Bayesian interpretations, e-values may provide a methodology acceptable to adherents of all three schools.
Apr 13: Qinyu Wu (University of Waterloo)
Title: Generalization and Regularization via Wasserstein Distributionally Robust Optimization
Speaker: Qinyu Wu (Postdoc Fellow, University of Waterloo)
Time: 15:30 - 17:00 pm, Apr 13 (Thu)
Location: M3 4206
Abstract: Wasserstein distributionally robust optimization (DRO) has found success in operations research and machine learning applications as a powerful means to obtain solutions with favorable out-of-sample performances. Two compelling explanations for the success are the generalization bounds derived from Wasserstein DRO and the equivalency between Wasserstein DRO and the regularization scheme commonly applied in machine learning. Existing results on generalization bounds and the equivalency to regularization are largely limited to the setting where the Wasserstein ball is of a certain type and the decision criterion takes certain forms of an expected function. In this paper, we show that by focusing on Wasserstein DRO problems with affine decision rules, it is possible to obtain generalization bounds and the equivalency to regularization in a significantly broader setting where the Wasserstein ball can be of a general type and the decision criterion can be a general measure of risk, i.e., nonlinear in distributions. This allows for accommodating many important classification, regression, and risk minimization applications that have not been addressed to date using Wasserstein DRO. Our results are strong in that the generalization bounds do not suffer from the curse of dimensionality and the equivalency to regularization is exact. As a byproduct, our regularization results broaden considerably the class of Wasserstein DRO models that can be solved efficiently via regularization formulations.
Apr 6: Jean-Gabriel Lauzier (University of Waterloo)
Title: Risk sharing, measuring variability, and distortion riskmetrics
Speaker: Jean-Gabriel Lauzier (Postdoc Fellow, University of Waterloo)
Time: 15:30 - 17:00 pm, Apr 6 (Thu)
Location: M3 4206
Abstract: We address the problem of sharing risk among agents with preferences modelled by a general class of comonotonic additive and law-based functionals that need not be either monotone or convex. Such functionals are called distortion riskmetrics, which include many statistical measures of risk and variability used in portfolio optimization and insurance. The set of Pareto-optimal allocations is characterized under various settings of general or comonotonic risk sharing problems. We solve explicitly Pareto-optimal allocations among agents using the Gini deviation, the mean-median deviation, or the inter-quantile difference as the relevant variability measures. The latter is of particular interest, as optimal allocations are not comonotonic in the presence of inter-quantile difference agents; instead, the optimal allocation features a mixture of pairwise counter-monotonic structures, showing some patterns of extremal negative dependence.
March
Mar 30: Zachary John Van Oosten (University of Waterloo)
Title: Conditional and Dynamic Convex Risk Measures
Speaker: Zachary John Van Oosten (University of Waterloo)
Time: 15:30 - 17:00 pm, Mar 30 (Thu)
Location: M3 4206
Abstract: The paper extends the definition of convex risk measures to the conditional setting. The framework can be seen as a generalization of classical risk measures. Much like the non-conditional case, the notion of acceptable sets is introduced and the interplay between such sets and measures is discussed. A representation result is proved in terms of conditional expectation. Afterward, a regularity property is defined and conditional convex risk measures are shown to have it. The paper finishes by defining dynamic convex risk measures and characterizes those satisfying some natural time consistency properties.
Mar 23: Prof. Georgios Pitselis (University of Piraeus)
Title: Weighted Credibility Distribution Estimation with Applications to Insurance and Finance
Speaker: Georgios Pitselis (University of Piraeus)
Time: 15:30 - 17:00 pm, Mar 23 (Thu)
Location: M3 3127
Abstract: This paper extends the results of Jewell (1974) of forecasting the distribution of individual risk in cases where the observations are weighted or are grouped in intervals. The credibility estimation is obtained by restricting the class of admissible functions, leading to the so-called linearized credibility results and the optimal projection theorem is also applied for credibility estimation. In addition, distribution credibility estimators are also established, and numerical illustrations are herein presented. Two examples of distribution credibility estimation are given, one with insurance loss data and the other with industry financial data.
Mar 16: Qinghua Ren (University of Waterloo)
Title: Distributionally robust reinsurance with Value-at-Risk and Conditional Value-at-Risk
Speaker: Qinghua Ren (PhD Student, University of Waterloo)
Time: 15:30 - 17:00 pm, Mar 16 (Thu)
Location: M3 3127
Abstract: This paper introduced the model uncertainty into the classic reinsurance framework and focused on distributionally robust reinsurance problems by minimizing the worst-case VaR(CVaR) of the total retained loss. The authors obtained closed-form solutions for the worst-case VaR over an uncertainty set containing distributions with common first two moments. The optimal deductibles of the optimization problems were also obtained. The paper showed that the worst-case CVaR is equal to the worst-case VaR of the insurer’s total retained loss under stop-loss reinsurance contract. Several numerical experiments were conducted to compare this model with classic stop-loss reinsurance framework.
Mar 9: Ziyue Shi (University of Waterloo)
Title: The design of an optimal retrospective rating plan
Speaker: Ziyue Shi (PhD Student, University of Waterloo)
Time: 15:30 - 17:00 pm, Mar 9 (Thu)
Location: M3 4206
Abstract: This talk is based on the work of Chen et al. (2016), which studies an optimal retrospective rating plan by minimizing the risk exposure of the insured under the framework of convex order. This paper adopts a little-studied premium principle called the retrospective rating plan premium. This premium principle considers the insured's actual losses during the policy year and, in fact, it is widely used in insurance practice. In this talk, we will introduce the retrospective rating plan premium, elaborate on the model setup, and discuss the main results of the background paper.
Mar 2: Yudong Luo (University of Waterloo & Vector Institute)
Title: Mean-Variance Reinforcement Learning
Speaker: Yudong Luo (PhD Candidate, University of Waterloo & Vector Institute)
Time: 15:30 - 17:00 pm, Mar 2 (Thu)
Location: M3 4206
Abstract: Restricting the variance of a policy's return is a popular choice in risk-averse reinforcement learning (RL) literature due to its clear mathematical definition and easy interpretability. In this presentation, the fundamental concepts of traditional risk-neutral RL are first introduced, followed with the risk-averse mean-variance RL problem. The existing mean-variance RL methods are described, and their limitations are discussed.
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