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Norms Based on Generalized Expected-Shortfalls and Applications

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  • Shuyu Gong
  • Taizhong Hu
  • Zhenfeng Zou

Abstract

This paper proposes a novel class of generalized Expected-Shortfall (ES) norms constructed via distortion risk measures, establishing a unified analytical framework for risk quantification. The proposed norms extend conventional ES methodology by incorporating flexible distortion functions. Specifically, we develop the mathematical duality theory for generalized-ES norms to support portfolio optimization tasks, while demonstrating their practical utility through projection problem solutions. The generalizedES norms are also applied to detect anomalies of financial time series data.

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  • Shuyu Gong & Taizhong Hu & Zhenfeng Zou, 2025. "Norms Based on Generalized Expected-Shortfalls and Applications," Papers 2507.09444, arXiv.org.
    • Handle: RePEc:arx:papers:2507.09444
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    References listed on IDEAS

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    1. Kremer, Philipp J. & Lee, Sangkyun & Bogdan, Małgorzata & Paterlini, Sandra, 2020. "Sparse portfolio selection via the sorted ℓ1-Norm," Journal of Banking & Finance, Elsevier, vol. 110(C).
    2. Yongjae Lee & Min Jeong Kim & Jang Ho Kim & Ju Ri Jang & Woo Chang Kim, 2020. "Sparse and robust portfolio selection via semi-definite relaxation," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 71(5), pages 687-699, May.
    3. Alois Pichler, 2024. "Connection between higher order measures of risk and stochastic dominance," Computational Management Science, Springer, vol. 21(2), pages 1-28, December.
    4. Ruodu Wang & Yunran Wei & Gordon E. Willmot, 2020. "Characterization, Robustness, and Aggregation of Signed Choquet Integrals," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 993-1015, August.
    5. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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