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Upper bounds for strictly concave distortion risk measures on moment spaces

Author

Listed:
  • Cornilly, D.
  • Rüschendorf, L.
  • Vanduffel, S.

Abstract

The study of worst-case scenarios for risk measures (e.g., Value-at-Risk) when the underlying risk (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper, we tackle the open problem of deriving upper bounds for strictly concave distortion risk measures on moment spaces. Building on early results of Rustagi (1957, 1976), we show that in general this problem can be reduced to a parametric optimization problem. We completely specify the sharp upper bound (and corresponding maximizing distribution function) when the first moment and any other higher moment are fixed. Specifically, in the case of a fixed mean and variance, we generalize the Cantelli bound for (Tail) Value-at-Risk in that we express the sharp upper bound for a strictly concave distorted expectation as a weighted sum of the mean and standard deviation.

Suggested Citation

  • Cornilly, D. & Rüschendorf, L. & Vanduffel, S., 2018. "Upper bounds for strictly concave distortion risk measures on moment spaces," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 141-151.
    • Handle: RePEc:eee:insuma:v:82:y:2018:i:c:p:141-151
    DOI: 10.1016/j.insmatheco.2018.07.002
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    References listed on IDEAS

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    Cited by:

    1. Rüschendorf, L., 2019. "Analysis of risk bounds in partially specified additive factor models," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 115-121.
    2. Silvana Pesenti & Qiuqi Wang & Ruodu Wang, 2020. "Optimizing distortion riskmetrics with distributional uncertainty," Papers 2011.04889, arXiv.org, revised Feb 2022.
    3. Cornilly, Dries & Vanduffel, Steven, 2019. "Equivalent distortion risk measures on moment spaces," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 187-192.
    4. Wakker, Peter P. & Yang, Jingni, 2021. "Concave/convex weighting and utility functions for risk: A new light on classical theorems," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 429-435.
    5. Carole Bernard & Silvana M. Pesenti & Steven Vanduffel, 2022. "Robust Distortion Risk Measures," Papers 2205.08850, arXiv.org, revised Mar 2023.

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    More about this item

    Keywords

    Value-at-Risk (vaR); Coherent risk measure; Model uncertainty; Cantelli bound; Distortion function;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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