Risk management with expectiles

Expectiles (EVaR) are a one-parameter family of coherent risk measures that have been recently suggested as an alternative to quantiles (VaR) and to expected shortfall (ES). In this work we review their known properties, we discuss their financial meaning, we compare them with VaR and ES and we study their asymptotic behaviour, refining some of the results in Bellini et al. [(2014). Generalized Quantiles as Risk Measures. Insurance: Mathematics and Economics, 54:41-48]. Moreover, we present a real-data example for the computation of expectiles by means of simple Garch(1,1) models and we assess the accuracy of the forecasts by means of a consistent loss function as suggested by Gneiting [(2011). Making and Evaluating Point Forecast. Journal of the American Statistical Association, 106 (494): 746-762]. Theoretical and numerical results indicate that expectiles are perfectly reasonable alternatives to VaR and ES risk measures.