Estimation of Tail Risk based on Extreme Expectiles

We use tail expectiles to estimate alternative measures to the Value at Risk (VaR), Expected Shortfall (ES) and Marginal Expected Shortfall (MES), three 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 expectiles and quantiles were embedded in the more general class of M-quantiles as the minimizers of an asymmetric convex loss function. It has been proved very recently that the only M-quantiles that are coherent risk measures are the expectiles. Moreover, expectiles define the only coherent risk measure that is also elicit able. The elicit ability corresponds to the existence of a natural backtesting methodology. The estimation of expectiles did not, however, receive yet any attention from the perspective of extreme values. The first estimation method that we propose enables the usage of advanced high quantile and tail index estimators. The second method joins together the least asymmetrically weighted squares estimation with the tail restrictions of extreme-value theory. A main tool is to first estimate the large expectile-based VaR, ES and MES when they are covered by the range of the data, and then extrapolate these estimates to the very far tails. We establish the limit distributions of the proposed estimators when they are located in the range of the data or near and even beyond the maximum observed loss. We show through a detailed simulation study the good performance of the procedures, and also present concrete applications to medical insurance data and three large US investment banks.