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 finance. 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 define 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 first 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.

Estimation of conditional extreme risk measures from heavy-tailed elliptical random vectors

“…This assumption is widespread in literature of extreme quantiles (see, e.g, Daouia et al (2017a)). A first consequence is that Φ R , or equivalently F R1 is attracted to the maximum domain of Pareto-type distributions with tail index γ.…”

A Renewal Shot Noise Process with Subexponential Shot Marks

“…As a referee kindly points out, the asymptotic study has an immediate implication for risk management. There is a vast literature devoted to studies that focus on tail probabilities as an essential risk measurement tool; see, e.g., Asmussen et al (1999); Yang and Wang (2013); Kelly and Jiang (2014); Landsman et al (2016); Daouia et al (2018) and Tang et al (2019).…”

On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles

“…In short, the ordinary least squares regression methodology is a special case of asymmetric least squares regression with expectiles (Daouia et al 2018). A similar expression reports the sample mean:…”

“…The displacement of VaR by expected shortfall between Basel II and III is the iconic example. In turn, a future Basel accord may replace expected shortfall with the joint vector Θ(VaR, ES) , median shortfall (Embrechts et al 2014, p. 43;Kou et al 2013, p. 402), or expectile-based expected shortfall (Daouia et al 2018;Gschöpf et al 2015).…”

“…The literature already contains specifications of expectile-based VaR (Bellini and Bernardino 2017, p. 488;Kuan et al 2009, pp. 262-64) and expectile-based expected shortfall (Daouia et al 2018). …”