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Risk excess measures induced by hemi-metrics

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  • Faugeras, Olivier
  • Rüschendorf, Ludger

Abstract

The main aim of this paper is to introduce the notion of risk excess measure, to analyze its properties and to describe some basic construction methods. To compare the risk excess of one distribution Q w.r.t. a given risk distribution P, we propose to apply the concept of hemi-metric on the space of probability measures. This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures. Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings. Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the values rho(Q), rho(P) of a risk measure rho. We argue that the difference rho(Q)-rho(P) neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure. We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.

Suggested Citation

  • Faugeras, Olivier & Rüschendorf, Ludger, 2018. "Risk excess measures induced by hemi-metrics," TSE Working Papers 18-922, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:32655
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    References listed on IDEAS

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    1. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    2. Burgert, Christian & Ruschendorf, Ludger, 2006. "Consistent risk measures for portfolio vectors," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 289-297, April.
    3. 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.
    4. repec:dau:papers:123456789/353 is not listed on IDEAS
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    Cited by:

    1. Faugeras, Olivier P. & Pagès, Gilles, 2024. "Risk quantization by magnitude and propensity," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 134-147.
    2. Asier Estevan & Roberto Maura & Óscar Valero, 2023. "Quasi-Metrics for Possibility Results: Intergenerational Preferences and Continuity," Mathematics, MDPI, vol. 11(2), pages 1-19, January.

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    Keywords

    risk measure; mass transportation; hemi-metric; stochastic order;
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