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Monetary Risk Measures

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  • Guangyan Jia
  • Jianming Xia
  • Rongjie Zhao

Abstract

In this paper, we study general monetary risk measures (without any convexity or weak convexity). A monetary (respectively, positively homogeneous) risk measure can be characterized as the lower envelope of a family of convex (respectively, coherent) risk measures. The proof does not depend on but easily leads to the classical representation theorems for convex and coherent risk measures. When the law-invariance and the SSD (second-order stochastic dominance)-consistency are involved, it is not the convexity (respectively, coherence) but the comonotonic convexity (respectively, comonotonic coherence) of risk measures that can be used for such kind of lower envelope characterizations in a unified form. The representation of a law-invariant risk measure in terms of VaR is provided.

Suggested Citation

  • Guangyan Jia & Jianming Xia & Rongjie Zhao, 2020. "Monetary Risk Measures," Papers 2012.06751, arXiv.org.
    • Handle: RePEc:arx:papers:2012.06751
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    References listed on IDEAS

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    1. Jianming Xia, 2020. "Decision Making under Uncertainty: A Game of Two Selves," Papers 2012.07509, arXiv.org.
    2. Carlier, G. & Lachapelle, A., 2011. "A numerical approach for a class of risk-sharing problems," Journal of Mathematical Economics, Elsevier, vol. 47(1), pages 1-13, January.
    3. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    4. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    5. Song, Yongsheng & Yan, Jia-An, 2009. "Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 459-465, December.
    6. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    7. repec:dau:papers:123456789/3821 is not listed on IDEAS
    8. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    9. repec:dau:papers:123456789/342 is not listed on IDEAS
    10. Steven Kou & Xianhua Peng & Chris C. Heyde, 2013. "External Risk Measures and Basel Accords," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 393-417, August.
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    Cited by:

    1. Moresco, Marlon Ruoso & Righi, Marcelo Brutti, 2022. "On the link between monetary and star-shaped risk measures," Statistics & Probability Letters, Elsevier, vol. 184(C).
    2. Marlon Moresco & Marcelo Brutti Righi, 2021. "On the link between monetary and star-shaped risk measures," Papers 2108.13500, arXiv.org.
    3. Jianming Xia, 2020. "Decision Making under Uncertainty: A Game of Two Selves," Papers 2012.07509, arXiv.org.
    4. Xia Han & Qiuqi Wang & Ruodu Wang & Jianming Xia, 2021. "Cash-subadditive risk measures without quasi-convexity," Papers 2110.12198, arXiv.org, revised Mar 2022.
    5. Dejian Tian & Xunlian Wang, 2023. "Dynamic star-shaped risk measures and $g$-expectations," Papers 2305.02481, arXiv.org.

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