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Extreme M-quantiles as risk measures: From L1 to Lp optimization

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  • Daouia, Abdelaati
  • Girard, Stéphane
  • Stupfler, Gilles

Abstract

The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of Lp optimization. These generalized Lp-quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for 1 p 2. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate Lp-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme Lp-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme Lp-quantiles and elaborate the choice of p through applications to some simulated and financial real data.

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  • Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2017. "Extreme M-quantiles as risk measures: From L1 to Lp optimization," TSE Working Papers 17-841, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:32050
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    Cited by:

    1. Laurent Gardes & Stéphane Girard & Gilles Stupfler, 2020. "Beyond tail median and conditional tail expectation: Extreme risk estimation using tail Lp‐optimization," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 922-949, September.
    2. Stéphane Girard & Gilles Claude Stupfler & Antoine Usseglio-Carleve, 2021. "Extreme Conditional Expectile Estimation in Heavy-Tailed Heteroscedastic Regression Models," Post-Print hal-03306230, HAL.

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    Keywords

    Asymptotic normality; Dependent observations; Expectiles; Extrapolation; Extreme values; Heavy tails; Lp optimization; Mixing; Quantiles; Tail risk;
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