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Asymptotic Properties of Generalized Shortfall Risk Measures for Heavy-tailed Risks

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  • Tiantian Mao
  • Gilles Stupfler
  • Fan Yang

Abstract

We study a general risk measure called the generalized shortfall risk measure, which was first introduced in Mao and Cai (2018). It is proposed under the rank-dependent expected utility framework, or equivalently induced from the cumulative prospect theory. This risk measure can be flexibly designed to capture the decision maker's behavior toward risks and wealth when measuring risk. In this paper, we derive the first- and second-order asymptotic expansions for the generalized shortfall risk measure. Our asymptotic results can be viewed as unifying theory for, among others, distortion risk measures and utility-based shortfall risk measures. They also provide a blueprint for the estimation of these measures at extreme levels, and we illustrate this principle by constructing and studying a quantile-based estimator in a special case. The accuracy of the asymptotic expansions and of the estimator is assessed on several numerical examples.

Suggested Citation

  • Tiantian Mao & Gilles Stupfler & Fan Yang, 2024. "Asymptotic Properties of Generalized Shortfall Risk Measures for Heavy-tailed Risks," Papers 2411.07212, arXiv.org.
    • Handle: RePEc:arx:papers:2411.07212
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    References listed on IDEAS

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    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)

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