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Keywords = comonotonic additivity

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20 pages, 313 KB  
Article
Strong Comonotonic Additive Systemic Risk Measures
by Heyan Wang, Shuo Gong and Yijun Hu
Axioms 2024, 13(6), 347; https://doi.org/10.3390/axioms13060347 - 23 May 2024
Cited by 2 | Viewed by 1778
Abstract
In this paper, we propose a new class of systemic risk measures, which we refer to as strong comonotonic additive systemic risk measures. First, we introduce the notion of strong comonotonic additive systemic risk measures by proposing new axioms. Second, we establish a [...] Read more.
In this paper, we propose a new class of systemic risk measures, which we refer to as strong comonotonic additive systemic risk measures. First, we introduce the notion of strong comonotonic additive systemic risk measures by proposing new axioms. Second, we establish a structural decomposition for strong comonotonic additive systemic risk measures. Third, when both the single-firm risk measure and the aggregation function in the structural decomposition are convex, we also provide a dual representation for it. Last, examples are given to illustrate the proposed systemic risk measures. Comparisons with existing systemic risk measures are also provided. Full article
(This article belongs to the Special Issue Advances in Financial Mathematics)
21 pages, 557 KB  
Article
Bidual Representation of Expectiles
by Alejandro Balbás, Beatriz Balbás, Raquel Balbás and Jean-Philippe Charron
Risks 2023, 11(12), 220; https://doi.org/10.3390/risks11120220 - 15 Dec 2023
Cited by 5 | Viewed by 2628
Abstract
Downside risk measures play a very interesting role in risk management problems. In particular, the value at risk (VaR) and the conditional value at risk (CVaR) have become very important instruments to address problems such as risk optimization, capital requirements, portfolio selection, pricing [...] Read more.
Downside risk measures play a very interesting role in risk management problems. In particular, the value at risk (VaR) and the conditional value at risk (CVaR) have become very important instruments to address problems such as risk optimization, capital requirements, portfolio selection, pricing and hedging issues, risk transference, risk sharing, etc. In contrast, expectile risk measures are not as widely used, even though they are both coherent and elicitable. This paper addresses the bidual representation of expectiles in order to prove further important properties of these risk measures. Indeed, the bidual representation of expectiles enables us to estimate and optimize them by linear programming methods, deal with optimization problems involving expectile-linked constraints, relate expectiles with VaR and CVaR by means of both equalities and inequalities, give VaR and CVaR hyperbolic upper bounds beyond the level of confidence, and analyze whether co-monotonic additivity holds for expectiles. Illustrative applications are presented. Full article
(This article belongs to the Special Issue Optimal Investment and Risk Management)
8 pages, 323 KB  
Article
Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof
by Chuancun Yin and Dan Zhu
Risks 2016, 4(4), 34; https://doi.org/10.3390/risks4040034 - 29 Sep 2016
Viewed by 3962
Abstract
It is well known that a random vector with given marginals is comonotonic if and only if it has the largest convex sum, and that a random vector with given marginals (under an additional condition) is mutually exclusive if and only if it [...] Read more.
It is well known that a random vector with given marginals is comonotonic if and only if it has the largest convex sum, and that a random vector with given marginals (under an additional condition) is mutually exclusive if and only if it has the minimal convex sum. This paper provides an alternative proof of these two results using the theories of distortion risk measure and expected utility. Full article
14 pages, 241 KB  
Article
Discrete Integrals Based on Comonotonic Modularity
by Miguel Couceiro and Jean-Luc Marichal
Axioms 2013, 2(3), 390-403; https://doi.org/10.3390/axioms2030390 - 23 Jul 2013
Cited by 4 | Viewed by 5707
Abstract
It is known that several discrete integrals, including the Choquet and Sugeno integrals, as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of the class of comonotonically modular functions, we axiomatically identify more general families of [...] Read more.
It is known that several discrete integrals, including the Choquet and Sugeno integrals, as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of the class of comonotonically modular functions, we axiomatically identify more general families of discrete integrals that are comonotonically modular, including signed Choquet integrals and symmetric signed Choquet integrals, as well as natural extensions of Sugeno integrals. Full article
(This article belongs to the Special Issue Axiomatic Approach to Monotone Measures and Integrals)
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