## Weekly seminars: 2022

**May**

**May 5: Zhanyi Jiao (University of Waterloo) **

**May 5: Zhanyi Jiao (University of Waterloo)**

Title: A reverse Expected Shortfall optimization formula

Speaker: Zhanyi Jiao (PhD Candidate, University of Waterloo)

Time: 9:00am-10:00am ET, May 5 (Thu)

Location: Online via Zoom

Abstract: The celebrated Expected Shortfall (ES) optimization formula implies that ES at a fixed probability level is the minimum of a linear real function plus a scaled mean excess function. We establish a reverse ES optimization formula, which says that a mean excess function at any fixed threshold is the maximum of an ES curve minus a linear function. Despite being a simple result, this formula reveals elegant symmetries between the mean excess function and the ES curve, as well as their optimizers. The reverse ES optimization formula is closely related to the Fenchel-Legendre transforms, and our formulas are generalized from ES to optimized certainty equivalents, a popular class of convex risk measures. We analyze worst-case values of the mean excess function under two popular settings of model uncertainty to illustrate the usefulness of the reverse ES optimization formula, and this is further demonstrated with an application using insurance datasets.

**April**

**Apr 21: Dr. Yang Liu (University of Waterloo) **

**Apr 21: Dr. Yang Liu (University of Waterloo)**

Title: Uncertainty aversion and equity improvement

Speaker: Yang Liu (Postdoc Fellow, University of Waterloo)

Time: 9:00am-10:00am ET, Apr 21 (Thu)

Location: Online via Zoom

Abstract: We address the relationship between risk equity and aggregate uncertainty avoidance. We find a fundamental conflict between the two concepts in a setting where the model of Keeney [Keeney (1980): Equity and public risk. {\em Operations Research}, 28(3), 527--534] is equipped with uncertainty. Further, we characterize these concepts with the ambiguity and risk attitudes of the decision maker in the framework of the $\alpha$-maxmin preference. Finally, we extend the result to some general aggregation function.

**Apr 7: Prof. Peng Liu (University of Essex)**

**Apr 7: Prof. Peng Liu (University of Essex)**

Title: One axiom to rule them all: A minimalist axiomatization of quantiles

Speaker: Peng Liu (Lecturer/Assistant Professor, University of Essex)

Time: 9:00am-10:00am ET, Apr 7 (Thu)

Location: Online via Zoom

Abstract: We offer a minimalist axiomatization of quantiles among all real-valued mappings on a general set of distributions through only one axiom. This axiom is called ordinality: quantiles are the only mappings that commute with all increasing and continuous transforms. Other convenient properties of quantiles, monotonicity, semicontinuity, comonotonic additivity and elicitability in particular, follow from this axiom. Furthermore, on the set of convexly supported distributions, the median is the only mapping that commutates with all monotone and continuous transforms. On a general set of distributions, the median interval is pinned down as the unique minimal interval-valued mapping that commutes with all monotone and continuous transforms. Finally, our main result, put in a decision-theoretic setting, leads to a minimalist axiomatization of quantile preferences. In banking and insurance, quantiles are known as the standard regulatory risk measure Value-at-Risk (VaR), and thus, an axiomatization of VaR is obtained with only one axiom among law-based risk measures.

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