## Weekly seminars: 2021

**December**

**Dec 9: Prof. Max Nendel (Bielefeld University) **

**Dec 9: Prof. Max Nendel (Bielefeld University)**

Title: A decomposition of general premium principles into risk and deviation

Speaker: Max Nendel (Assistant Professor, Bielefeld University)

Time: 9:00am-10:00am EST, Dec 9 (Thu)

Location: Online via Zoom

Abstract: We provide an axiomatic approach to general premium principles in a probability-free setting that allows for Knightian uncertainty. Every premium principle is the sum of a risk measure, as a generalization of the expected value, and a deviation measure, as a generalization of the variance. One can uniquely identify a maximal risk measure and a minimal deviation measure in such decompositions. We show how previous axiomatizations of premium principles can be embedded into our more general framework. We discuss dual representations of convex premium principles, and study the consistency of premium principles with a financial market in which insurance contracts are traded.

**Dec 2: Prof. Giovanni Puccetti (University of Milan) **

**Dec 2: Prof. Giovanni Puccetti (University of Milan)**

Title: We love swapping (when simple necessary conditions are close-to-optimal)

Speaker: Giovanni Puccetti (Professor, University of Milan)

Time: 9:00am-10:00am EST, Dec 2 (Thu)

Location: Online via Zoom

Abstract: In two relevant examples taken from the fields of mass transportations and decision theory, we show how two simple swapping conditions deliver excellent and fast results.

**November**

**Nov 25: Dr. Kelvin Shuangjian Zhang (University of Waterloo)**

**Nov 25: Dr. Kelvin Shuangjian Zhang (University of Waterloo)**

Title: The Monopolist’s Problem: Old and New

Speaker: Kelvin Shuangjian Zhang (Postdoc Fellow, University of Waterloo)

Time: 9:00am-10:00am EST, Nov 25 (Thu)

Location: Online via Zoom

Abstract: The principal-agent problem is one of the central problems in microeconomics and has many applications in actuarial science. Existence, uniqueness, convexity/concavity, regularity/continuity, and characterization of the solutions have been widely studied after Mirrlees and Spence in the 1970s. Recently, the generalizations of results for the utility functions being quasi-linear to fully nonlinear serve as the major breakthrough in the past 20 years. In this talk, I will first introduce the historical work on the principal-agent framework under the context of the monopolist problem before moving to the recent progress. The results profoundly connect with the Optimal Transport theory, a powerful tool with potential applications in many areas (including actuarial science). This talk contains my joint work with Professor Robert J. McCann.

**Nov 18: Mehdi Tomas (Ecole Polytechnique) **

**Nov 18: Mehdi Tomas (Ecole Polytechnique)**

Title: Measuring portfolio liquidity risk

Speaker: Mehdi Tomas (PhD Candidate, Ecole Polytechnique)

Time: 9:00am-10:00am EST, Nov 18 (Thu)

Location: Online via Zoom

Abstract: Expected Shortfall is the standard risk measure to assess market risk and design capital requirements of financial institutions. In extreme scenarios, institutions may be forced to liquidate their positions. However, institutions cannot buy or sell assets outright at the current market price. Instead, they face liquidity risk and must pay a liquidation premium (also known as execution cost or slippage). Since this risk increases potential losses, measuring it is of particular interest to asset managers and regulators. We present an ongoing study on the measure of the liquidity risk of portfolios. We use databases of private investor orders to measure liquidity risk on stock and future markets and propose a formula which accounts for concentration and diversification. Stress-testing it on data, we find it is more accurate than formulas which ignore these effects.

**Nov 11: Qiuqi Wang (University of Waterloo) **

**Nov 11: Qiuqi Wang (University of Waterloo)**

Title: E-backtesting risk measures

Speaker: Qiuqi Wang (PhD Candidate, University of Waterloo)

Time: 9:00am-10:00am EST, Nov 11 (Thu)

Location: Online via Zoom

Abstract: Abstract: In the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most important risk measure in financial regulation. A crucial issue on ES is its backtesting. In this paper, we produce a model-free backtesting method of general risk measures using the notion of universal e-statistics based on the newly developed concept of e-values. In particular, we demonstrate detailed backtesting procedures for backtesting VaR and ES. Simulation studies of backtesting VaR and ES are also performed based on the proposed backtesting method.

**Nov 4: Zhanyi Jiao (University of Waterloo) **

**Nov 4: Zhanyi Jiao (University of Waterloo)**

Title: On optimal reinsurance treaties in cooperative game under heterogeneous beliefs

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

Time: 9:00am-10:00am EST, Nov 4 (Thu)

Location: Online via Zoom

Abstract: In this talk, I will share a paper written by Jiang et al. (2019). This paper aims to characterize optimal reinsurance treaties in a two-party cooperative game where both parties are assumed to be expected-utility maximizers with divergent beliefs regarding the distribution of underlying loss. The authors mainly consider two scenarios: 1. reinsurance premium is fully negotiable; 2. reinsurance premium is determined by actuarial fair premium principle. For both scenarios, they derive a set of Pareto efficient frontiers and identify the unique and fair reinsurance treaty by applying Nash bargaining model and Kalai-Smorodinsky bargaining model.

**October**

**Oct 28: Dr. Yang Liu (University of Waterloo) **

**Oct 28: Dr. Yang Liu (University of Waterloo)**

Title: A Unified Formula of the Optimal Portfolio for Piecewise HARA Utilities

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

Time: 9:00am-10:00am EST, Oct 28 (Thu)

Location: Online via Zoom

Abstract: We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utility, including many non-standard utilities as examples. A typical application is the composition of an HARA preference and a piecewise linear payoff in hedge fund management. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability.

**Oct 21: Nazem Khan (University of Warwick) **

**Oct 21: Nazem Khan (University of Warwick)**

Title: Sensitivity to large losses and \rho-arbitrage for convex risk measures

Speaker: Nazem Khan (PhD Candidate, University of Warwick)

Time: 9:00am-10:00am EST, Oct 21 (Thu)

Location: Online via Zoom

Abstract: We study mean-risk portfolio selection in a one-period financial market, where risk is quantified by a superlinear risk measure \rho. We introduce two new axioms: weak and strong sensitivity to large losses. We show that the first axiom is key to ensure the existence of optimal portfolios and the second one is key to ensure the absence of \rho-arbitrage. This leads to a new class of risk measures that are suitable for portfolio selection. We show that \rho belongs to this class if and only if \rho is real-valued and the smallest positively homogeneous risk measure dominating \rho is the worst-case risk measure. We then turn to the case that \rho is convex and admits a dual representation. We derive necessary and sufficient dual characterisations of (strong) \rho-arbitrage as well as the property that \rho is suitable for portfolio selection. Finally, we introduce the new risk measure of "Loss Sensitive Expected Shortfall", which is similar to and not more complicated to compute than Expected Shortfall but suitable for portfolio selection – which Expected Shortfall is not.

**Oct 14: Liyuan Lin (University of Waterloo) **

**Oct 14: Liyuan Lin (University of Waterloo)**

Title: Mathematical properties of PELVE

Speaker: Liyuan Lin (PhD Candidate, University of Waterloo)

Time: 9:00am-10:00am EST, Oct 14 (Thu)

Location: Online via Zoom

Abstract: In quantitative risk management, value at risk (VaR) and expected shortfall (ES) are known as the major risk measures. Li and Wang (2020) introduced a new risk measure called Probability Equivalent Level of VaR-ES (PELVE) to derive the equivalent probability level when replacing ES with VaR. We study monotonicity and convergence of PELVE as a function of probability level. The PELVE is not always monotone and does not always converge at 0. We find some conditions for PELVE to be monotone or convergent at 0. Furthermore, we characterize random variables from given PELVEs by the delayed differential equation. Especially, we find some explicit solutions when PELVE is constant.

**July**

**Jul 21: Prof. Niushan Gao (Ryerson University) **

**Jul 21: Prof. Niushan Gao (Ryerson University)**

Title: Automatic Fatou property of law-invariant risk measures

Speaker: Niushan Gao (Assistant Professor, Ryerson University)

Time: 9:00am-10:00am EST, Jul 21 (Wed)

Location: Online via Zoom

Abstract: Automatic continuity has long been an interesting topic and possibly has its roots in the well-known fact that a real-valued convex function on an open interval is continuous. In infinite-dimensional spaces, Birkhoff’s Theorem states that a positive linear functional on a Banach lattice is norm continuous. This result was later extended to the following celebrated theorem for real-valued convex functionals:

Theorem (Ruszczynski and Shapiro ’06). A real-valued, convex, decreasing functional on a Banach lattice is norm continuous.

A natural question is whether law invariance leads to continuity properties. The following result is striking.

Theorem (Jouini et al ’06). A real-valued, convex, decreasing, law-invariant functional on L^\infty has the Fatou property and is thus \sigma(L^\infty, L^1) lower semicontinuous.

In this talk, we show that, on nearly all classical model spaces including Orlicz spaces, every real-valued, law-invariant, coherent risk measure automatically has the Fatou property at every point whose negative part has a “thin tail”. The result is also sharp in the sense that automatic Fatou property cannot be expected at other points.

The talk is based on joint work with Shengzhong Chen, Denny Leung & Lei Li.

**Jul 14: Dr. Felix-Benedikt Liebrich (Leibniz University Hannover) **

**Jul 14: Dr. Felix-Benedikt Liebrich (Leibniz University Hannover)**

Title: Is "star-shaped" the new convex? Collapse to the mean and risk sharing with heterogeneous reference probabilities

Speaker: Felix-Benedikt Liebrich (Research Fellow, Leibniz University Hannover)

Time: 9:00am-10:00am EST, Jul 14 (Wed)

Location: Online via Zoom

Abstract: Over the past decade, the debate of the role that convexity plays in risk measure theory has intensified. We follow two recent contributions of Mao & Wang (2020) and Castagnoli et al. (2021) and try to explore the mathematical power of assumptions like star-shapedness of a risk measure and consistency with second-order stochastic dominance. The first part of the talk focuses on the „collapse of the mean“. The latter refers to the fundamental tension existing between law invariance of functionals and suitable “linearity” properties; in many cases, the expectation turns out to be the only functional with both properties. We shall discuss this phenomenon for consistent risk measures and a broad class of nonconvex Choquet integrals. In the second part, we approach the classical problem of finding Pareto-optimal allocations of risk among finitely many agents. The associated individual risk measures are assumed to be consistent, but with respect to agent-dependent and potentially heterogeneous concordant reference probability measures. Moreover, convexity is replaced by star-shapedness. We provide a simple sufficient condition for the existence of Pareto optima and sketch its proof, which is based on local comonotone improvement combined with a Dieudonné-type argument based on the „collapse to the mean“ established in the first part. The talk is partially based on joint work with Cosimo Munari.

**June**

**Jun 30: Mingren Yin (University of Waterloo)**

**Jun 30: Mingren Yin (University of Waterloo)**

Title: Optimal robust reinsurance policy measured by TVaR with model uncertainty

Speaker: Mingren Yin (PhD Candidate, University of Waterloo)

Time: 9:00am-10:00am EST, Jun 30 (Wed)

Location: Online via Zoom

Abstract: In the context of reinsurance, the insurer and the reinsurer are sharing one underlying risk X, and they both need to take the existence of uncertainty into consideration. Since the insurer and the reinsurer make assessment on X and possible scenarios separately, they may have different choices of uncertainty sets. In this work, we seek to determine the reinsurance policy optimizing a linear combination of the two parties' risk exposure in the corresponding worst-case scenarios measured by TVaR, an important coherent risk measure for insurance and reinsurance companies. Through this paper, an uncertainty set is assumed to include all the distributions with fixed mean and variance that are "close enough" to a reference distribution, in the metric of Wasserstein distance. We considered the optimization problem on the set of all stop-loss insurance contracts with free budget constraint and derived both some theoretical and numerical results based on Bernal et al. (2020).

**Jun 16: Dr. Jean-Gabriel Lauzier (Bocconi University) **

**Jun 16: Dr. Jean-Gabriel Lauzier (Bocconi University)**

Title: Ex-post moral hazard and manipulation-proof contracts

Speaker: Jean-Gabriel Lauzier (PhD, Bocconi University)

Time: 9:00am-10:00am EST, Jun 16 (Wed)

Location: Online via Zoom

Abstract: We examine the trade-off between the provision of incentives to exert costly effort (ex-ante moral hazard) and the incentives needed to prevent the agent from manipulating the profit observed by the principal (ex-post moral hazard). Formally, we build a model of two-stage hidden actions where the agent can both influence the expected revenue of a business and manipulate its observed profit. We show that manipulation-proofness is sensitive to the interaction between the manipulation technology and the probability distribution of the stochastic output. The optimal contract is manipulation-proof whenever the manipulation technology is linear. However, a convex manipulation technology sometimes leads to contracts for which there is manipulation in equilibrium. Whenever the distribution satisfies the monotone likelihood ratio property we can always find a manipulation technology for which this is the case.

**Jun 9: Dr. Xia Han (University of Waterloo)**

**Jun 9: Dr. Xia Han (University of Waterloo)**

Title: On the no reward for concentration axiom

Speaker: Xia Han (Postdoc Fellow, University of Waterloo)

Time: 9:00am-10:00am EST, Jun 9 (Wed)

Location: Online via Zoom

Abstract: Expected Shortfall (ES) is the most important coherent risk measure in both industry practice and academic research in finance, insurance, risk management, and engineering. In Wang and Zitikis (2020) , they put forward four intuitive economic axioms for portfolio risk assessment that provides the first economic axiomatic foundation for the family of ES. In this paper, we incorporate the notion of $p$-concentration aversion ($p$-CA) to present a new axiomatic characterization of risk measures. The risk measure characterized can be regarded as the functionals of ES and expectation. We further show that if the risk measure also satisfies the axioms -- monotonicity, translation-invariance and prudence, then it uniquely characterizes the family of ES. In contrast to Wang and Zitikis (2020), we use the $p$-CA to replace the no reward for concentration axiom NRC, which makes the axiomatic foundation for ES more natural to be illustrated for portfolio risk assessment.

**May**

**May 26: Yang Liu (Tsinghua University)**

**May 26: Yang Liu (Tsinghua University)**

Title: A framework for measures of risk under uncertainty

Speaker: Yang Liu (PhD Candidate, Tsinghua University)

Time: 9:00am-10:00am EST, May 26 (Wed)

Location: Online via Zoom

Abstract: A risk analyst assesses potential financial losses based on multiple sources of information. In particular, the assessment does not only depend on the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. As the most practical choice, the worst-case generalized risk measure is characterized via a few intuitive axioms. We reveal the relationship between a few natural forms of law invariance, under which we further pin down particular forms of the worst-case generalized risk measures. Some connections to decision theory are discussed, and many open questions remain.

**May 19: Qiuqi Wang (University of Waterloo)**

**May 19: Qiuqi Wang (University of Waterloo)**

Title: Optimal reinsurance contracts and the Expected Shortfall

Speaker: Qiuqi Wang (PhD Candidate, University of Waterloo)

Time: 9:00am-10:00am EST, May 19 (Wed)

Location: Online via Zoom

Abstract: The Expected Shortfall (ES) is one of the most important risk measures widely applied in the field of finance, insurance, statistics, and risk management. In light of recent results characterizing ES in the context of financial regulation and statistics, we examine the implication of ES in insurance and actuarial science. In this paper, we study a reinsurance contract design problem focusing on the risk measure used by an overseer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of the overseer, where the optimal contracts are within the common set of ceded loss functions with a deductible form. Characterization results of other classes of risk measures including the mean and the distortion risk measures are demonstrated as the optimal set of ceded loss functions changes. Extension to the case with multiple reinsurers and alternative explanations of the optimal contract condition from the perspective of the insurer are also discussed.

**April**

**Apr 28: Prof. Haiyan Liu (Michigan State University)**

**Apr 28: Prof. Haiyan Liu (Michigan State University)**

Title: Distributionally robust reinsurance with Value-at-Risk and Conditional Value-at-Risk

Speaker: Haiyan Liu (Professor, Michigan State University)

Time: 9:00am-10:00am EST, April 28 (Wed)

Location: Online via Zoom

Abstract: A basic assumption of the classic reinsurance model is that the distribution of the loss is precisely known. In practice, only partial information is available for the loss distribution due to the lack of data and estimation error. We study a distributionally robust reinsurance problem by minimizing the maximum Value-at-Risk (or the worst-case VaR) of the total retained loss of the insurer for all loss distributions with known mean and variance. Our model handles typical stop-loss reinsurance contracts. We show that a three-point distribution achieves the worst-case VaR of the total retained loss of the insurer, from which the closed-form solutions of the worst-case distribution and optimal deductible are obtained. Moreover, we show that the worst-case Conditional Value-at-Risk of the total retained loss of the insurer is equal to the worst-case VaR, and thus the optimal deductible is the same in both cases. This talk is based on joint work with Tiantian Mao.

**Apr 21: Prof. Matteo Burzoni (University of Milan)**

**Apr 21: Prof. Matteo Burzoni (University of Milan)**

Title: Viability and Arbitrage under Knightian Uncertainty

Speaker: Matteo Burzoni (Professor, University of Milan)

Time: 9:00am-10:00am EST, April 21 (Wed)

Location: Online via Zoom

Abstract: We reconsider the microeconomic foundations of financial economics. Motivated by the importance of Knightian Uncertainty in markets, we present a model that does not carry any probabilistic structure ex ante, yet is based on a common order. We derive the fundamental equivalence of economic viability of asset prices and absence of arbitrage. We also obtain a modified version of the Fundamental Theorem of Asset Pricing using the notion of sublinear pricing measures. Different versions of the Efficient Market Hypothesis are related to the assumptions one is willing to impose on the common order.

**Apr 7: Prof. Takaaki Koike (Institute of Statistical Mathematics)**

**Apr 7: Prof. Takaaki Koike (Institute of Statistical Mathematics)**

Title: Tail concordance measures: A fair assessment of tail dependence

Speaker: Takaaki Koike (Project Assistant Professor, Institute of Statistical Mathematics)

Time: 9:00am-10:00am EST, April 7 (Wed)

Location: Online via Zoom

Abstract: In this talk, we propose a new class of measures of bivariate tail dependence called tail concordance measure (TCM), which is defined as the limit of a measure of concordance of the underlying copula restricted to the tail region of interest. The proposed measures capture the extremal relationship between random variables not only along the diagonal but also along all angles weighted by a tail generating measure. Axioms of tail dependence measures are introduced, and the TCMs are shown to characterize linear tail dependence measures. The infimum and supremum of the TCMs over all generating measures are presented to investigate the issue of under- and overestimation of the degree of extreme co-movements. The infimum is shown to be attained by the classical tail dependence coefficient, and thus the classical notion always underestimates tail dependence. A formula for the supremum TCM is derived and shown to overestimate the degree of extreme co-movements. Estimators of the proposed measures are studied, and their performance is demonstrated in numerical experiments. For a fair assessment of tail dependence and stability of the estimation under small sample sizes, TCMs weighted over all angles are suggested, with tail Spearman's rho and tail Gini's gamma being novel special cases of TCMs.

**March**

**Mar 31: Prof. Hirbod Assa (Kent Business School)**

**Mar 31: Prof. Hirbod Assa (Kent Business School)**

Title: COVID 19, how it would change the insurance, the way we know it

Speaker: Hirbod Assa (Professor, Kent Business School)

Time: 9:00am-10:00am EST, Mar 31 (Wed)

Location: Online via Zoom

Abstract: In this talk, I will discuss some practical and theoretical issues related to insurance as a risk management practice at the time of macro-economic systematic events like COVID-19. First, I will briefly discuss the modeling, decision-making, economic and political issues. The investigation of these issues would suggest revisiting the insurance as a general approach to risk management of systematic events. Particularly, we will revisit the so-called “insurance principle” and the ex-ante insurance policies in the presence of the common shocks, by introducing the ex-post policies. From a mathematical standpoint, I will be spending some time reviewing central limit theorems as well as the asymptotic equivalence of value at risk. We see how new concepts like contingent premiums, and VaR conditional on an event, naturally arise from our discussions and will be used for risk management of the systematic risk.

**Mar 24: Yuyu Chen (University of Waterloo)**

**Mar 24: Yuyu Chen (University of Waterloo)**

Title: Aggregation of Two Ordered Risks with Dependence Uncertainty

Speaker: Yuyu Chen (PhD Candidate, University of Waterloo)

Time: 9:00am-10:00am EST, Mar 24 (Wed)

Location: Online via Zoom

Abstract: In this talk, we study the aggregation of two risks when the marginal distributions are known, and one risk is assumed to be smaller than the other. The concave ordering of the aggregate position is discussed. In particular, the largest aggregate risk in concave order is attained when the two risks are directionally lower (DL) coupled. These results are used to calculate the best-case and worst-case values of risk measures. Moreover, we derive an analytical solution for Value-at-Risk. Our numerical results suggest that the new bounds on risk measures with the extra order constraint can greatly improve those where only the marginal distributions are known.

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