Estimation of tail risk based on extreme expectiles

We use tail expectiles to estimate alternative measures to the value at risk and marginal expected shortfall, which are 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 first to estimate large values of expectile-based value at risk and marginal expected shortfall within the range of the data, and then to extrapolate the estimates obtained 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.