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

Daouia, Abdelaati; Girard, St�phane; 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.

Citation

Daouia, A., Girard, S., & Stupfler, G. (2019). Extreme M-quantiles as risk measures: from L1 to Lp optimization. Bernoulli, 25(1), 264-309

• Journal Article Type: Article
• Acceptance Date: Sep 9, 2017
• Online Publication Date: Dec 12, 2018
• Publication Date: Feb 1, 2019
• Deposit Date: Sep 11, 2017
• Publicly Available Date: Dec 12, 2018
• Journal: Bernoulli
• Print ISSN: 1350-7265
• Electronic ISSN: 1573-9759
• Publisher: Bernoulli Society for Mathematical Statistics and Probability
• Peer Reviewed: Peer Reviewed
• Volume: 25
• Issue: 1
• Pages: 264-309
• Keywords: Asymptotic normality; Dependent observations; Expectiles; Extrapolation;Extreme values; Heavy tails; Lp optimization; Mixing; Quantiles; Tail risk
• Public URL: https://nottingham-repository.worktribe.com/output/881544
• Publisher URL: https://projecteuclid.org/euclid.bj/1544605247
• Contract Date: Sep 11, 2017