Assessing coherent Value-at-Risk and expected shortfall with extreme expectiles
Résumé
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. The flexibility and virtues of these least squares analogues of quantiles are now well established in actuarial science, econo-metrics and statistical finance. Both quantiles and expectiles were embedded in the more general class of M-quantiles as the minimizers of a generic asymmetric convex loss function. It has been proved very recently that the only M-quantiles that are coherent risk measures are the expectiles. Also, in contrast to the quantile-based expected shortfall, expectiles benefit from the important property of elicitability that corresponds to the existence of a natural backtesting methodology. Least asymmetrically weighted squares estimation of expectiles did not, however, receive yet as much attention as quantile-based risk measures from the perspective of extreme values. In this article, we develop new methods for estimating the Value-at-Risk and expected shortfall measures via high expectiles. We focus on the challenging domain of attraction of heavy-tailed distributions that better describe the tail structure and sparseness of most actuarial and financial data. We first estimate the intermediate large expec-tiles 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. Monte Carlo experiments and a concrete application are given to illustrate the utility of extremal expectiles as an efficient instrument of risk protection.
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