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Bidual Representation of Expectiles

Author

Listed:
  • Alejandro Balbás

    (Department of Business Administration, University Carlos III of Madrid, C/Madrid, 126, 28903 Getafe, Madrid, Spain)

  • Beatriz Balbás

    (Department of Economics and Business Administration, University of Alcalá, Pl. de la Victoria, 2, 28802 Alcalá de Henares, Madrid, Spain)

  • Raquel Balbás

    (Department of Financial and Actuarial Economics and Statistics, University Complutense of Madrid, Somosaguas, 28223 Pozuelo de Alarcón, Madrid, Spain)

  • Jean-Philippe Charron

    (Department of Finance and Commercial Research, Autonomous University of Madrid, C/Francisco Tomás y Valiente, 5, 28049 Madrid, Spain)

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 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.

Suggested Citation

  • Alejandro Balbás & Beatriz Balbás & Raquel Balbás & Jean-Philippe Charron, 2023. "Bidual Representation of Expectiles," Risks, MDPI, vol. 11(12), pages 1-21, December.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:12:p:220-:d:1301234
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    References listed on IDEAS

    as
    1. Lejeune, Miguel A. & Shen, Siqian, 2016. "Multi-objective probabilistically constrained programs with variable risk: Models for multi-portfolio financial optimization," European Journal of Operational Research, Elsevier, vol. 252(2), pages 522-539.
    2. Balbás, Alejandro & Balbás, Beatriz & Balbás, Raquel & Heras, Antonio, 2022. "Risk transference constraints in optimal reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 27-40.
    3. S. V. Stoyanov & S. T. Rachev & F. J. Fabozzi, 2007. "Optimal Financial Portfolios," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(5), pages 401-436.
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    Cited by:

    1. Alois Pichler, 2024. "Higher order measures of risk and stochastic dominance," Papers 2402.15387, arXiv.org.
    2. Alois Pichler, 2024. "Connection between higher order measures of risk and stochastic dominance," Computational Management Science, Springer, vol. 21(2), pages 1-28, December.
    3. Hakim, Arief & Salman, A.N.M. & Syuhada, Khreshna, 2025. "Conditional generalized quantiles as systemic risk measures: Properties, estimation, and application," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 235(C), pages 60-84.

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