Estimation of tail risk based on extreme expectiles

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.

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), https://doi.org/10.1111/rssb.12254

• Journal Article Type: Article
• Acceptance Date: Aug 15, 2017
• Online Publication Date: Oct 10, 2017
• Deposit Date: Aug 17, 2017
• Publicly Available Date: Oct 11, 2018
• 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
• 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
• Public URL: https://nottingham-repository.worktribe.com/output/887315
• Publisher URL: http://onlinelibrary.wiley.com/doi/10.1111/rssb.12254/full
• 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.wiley.com/doi/10.1111/rssb.12254/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.