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Estimation of tail risk based on extreme expectiles

Daouia, Abdelaati; Girard, Stéphane; Stupfler, Gilles

Authors

Abdelaati Daouia

Stéphane Girard

Gilles Stupfler



Abstract

We use tail expectiles to estimate alternative measures to the Value at Risk (VaR) and Marginal Expected Shortfall (MES), two instruments of risk protection of utmost importance in actuarial science and statistical _nance. The concept of expectiles is a least squares analogue of quantiles. Both are M-quantiles as the minimizers of an asymmetric convex loss function, but expectiles are the only M-quantiles that are coherent risk measures. Moreover, expectiles de_ne the only coherent risk measure that is also elicitable. The estimation of expectiles has not, however, received any attention yet from the perspective of extreme values. Two estimation methods are proposed here, either making use of quantiles or relying directly on least asymmetrically weighted squares. A main tool is to _rst estimate large values of expectile-based VaR and MES located within the range of the data, and then to extrapolate the obtained estimates to the very far tails. We establish the limit distributions of both of the resulting intermediate and extreme estimators. We show via a detailed simulation study the good performance of the procedures, and present concrete applications to medical insurance data and three large US investment banks.

Journal Article Type Article
Journal Journal of the Royal Statistical Society: Series B
Print ISSN 1369-7412
Electronic ISSN 1467-9868
Publisher Wiley
Peer Reviewed Peer Reviewed
Volume 80
Issue 2
APA6 Citation Daouia, A., Girard, S., & Stupfler, G. (in press). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80(2), doi:10.1111/rssb.12254
DOI https://doi.org/10.1111/rssb.12254
Keywords Asymmetric squared loss; Coherency; Expectiles; Extrapolation; Extreme values; Heavy tails; Marginal expected shortfall; Value at Risk
Publisher URL http://onlinelibrary.wiley.com/doi/10.1111/rssb.12254/full
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
Additional Information This is the peer reviewed version of the following article: Daouia, A., Girard, S. and Stupfler, G. (2017), Estimation of tail risk based on extreme expectiles. J. R. Stat. Soc. B. doi:10.1111/rssb.12254
which has been published in final form at http://onlinelibrary.wi...0.1111/rssb.12254/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

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