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Haezendonck–Goovaerts risk measures and Orlicz quantiles

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  • Bellini, Fabio
  • Rosazza Gianin, Emanuela

Abstract

In this paper, we study the well-known Haezendonck–Goovaerts risk measures on their natural domain, that is on Orlicz spaces and, in particular, on Orlicz hearts. We provide a dual representation as well as the optimal scenario in such a representation and investigate the properties of the minimizer xα∗ (that we call Orlicz quantile) in the definition of the Haezendonck–Goovaerts risk measure. Since Orlicz quantiles fail to satisfy an internality property, bilateral Orlicz quantiles are also introduced and analyzed.

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  • Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:1:p:107-114
    DOI: 10.1016/j.insmatheco.2012.03.005
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    Cited by:

    1. Niushan Gao & Cosimo Munari & Foivos Xanthos, 2019. "Stability properties of Haezendonck-Goovaerts premium principles," Papers 1909.10735, arXiv.org, revised Aug 2020.
    2. Ling, Chengxiu & Peng, Zuoxiang, 2016. "Tail asymptotics of generalized deflated risks with insurance applications," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 220-231.
    3. Mao, Tiantian & Hu, Taizhong, 2012. "Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 333-343.
    4. Belles-Sampera, Jaume & Guillen, Montserrat & Santolino, Miguel, 2016. "What attitudes to risk underlie distortion risk measure choices?," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 101-109.
    5. Canna, Gabriele & Centrone, Francesca & Rosazza Gianin, Emanuela, 2021. "Haezendonck-Goovaerts capital allocation rules," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 173-185.
    6. Asimit, Alexandru V. & Badescu, Alexandru M. & Verdonck, Tim, 2013. "Optimal risk transfer under quantile-based risk measurers," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 252-265.
    7. Xun, Li & Zhou, Yangzhi & Zhou, Yong, 2019. "A generalization of Expected Shortfall based capital allocation," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 193-199.
    8. Asimit, Alexandru V. & Badescu, Alexandru M. & Cheung, Ka Chun, 2013. "Optimal reinsurance in the presence of counterparty default risk," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 690-697.
    9. Ahn, Jae Youn & Shyamalkumar, Nariankadu D., 2014. "Asymptotic theory for the empirical Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 78-90.
    10. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
    11. Bellini, Fabio, 2012. "Isotonicity properties of generalized quantiles," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 2017-2024.
    12. Qinyu Wu & Fan Yang & Ping Zhang, 2023. "Conditional generalized quantiles based on expected utility model and equivalent characterization of properties," Papers 2301.12420, arXiv.org.
    13. Bellini, Fabio & Laeven, Roger J.A. & Rosazza Gianin, Emanuela, 2021. "Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures," European Journal of Operational Research, Elsevier, vol. 291(2), pages 438-446.
    14. Felix-Benedikt Liebrich & Max Nendel, 2020. "Separability vs. robustness of Orlicz spaces: financial and economic perspectives," Papers 2009.09007, arXiv.org, revised May 2021.
    15. Tang, Qihe & Yang, Fan, 2014. "Extreme value analysis of the Haezendonck–Goovaerts risk measure with a general Young function," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 311-320.
    16. Leipus, Remigijus & Paukštys, Saulius & Šiaulys, Jonas, 2021. "Tails of higher-order moments of sums with heavy-tailed increments and application to the Haezendonck–Goovaerts risk measure," Statistics & Probability Letters, Elsevier, vol. 170(C).
    17. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    18. Cheung, Ka Chun & Lo, Ambrose, 2013. "General lower bounds on convex functionals of aggregate sums," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 884-896.
    19. Gao, Niushan & Munari, Cosimo & Xanthos, Foivos, 2020. "Stability properties of Haezendonck–Goovaerts premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 94-99.
    20. Gómez, Fabio & Tang, Qihe & Tong, Zhiwei, 2022. "The gradient allocation principle based on the higher moment risk measure," Journal of Banking & Finance, Elsevier, vol. 143(C).
    21. Fabio Bellini & Pablo Koch-Medina & Cosimo Munari & Gregor Svindland, 2018. "Law-invariant functionals on general spaces of random variables," Papers 1808.00821, arXiv.org, revised Jan 2021.
    22. Wang, Xing & Peng, Liang, 2016. "Inference for intermediate Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 231-240.
    23. Xun, Li & Jiang, Renqiao & Guo, Jianhua, 2021. "The conditional Haezendonck–Goovaerts risk measure," Statistics & Probability Letters, Elsevier, vol. 169(C).
    24. Asimit, Alexandru V. & Chi, Yichun & Hu, Junlei, 2015. "Optimal non-life reinsurance under Solvency II Regime," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 227-237.

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    More about this item

    Keywords

    IM30; IE13; Orlicz premiums; Haezendonck–Goovaerts risk measures; Conditional value at risk; Quantiles; Expectiles;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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