TERES: Tail Event Risk Expectile Shortfall

Abstract

We propose a generalized risk measure for expectile-based expected shortfall estimation. The generalization is designed with a mixture of Gaussian and Laplace densities. Our plug-in estimator is derived from an analytic relationship between expectiles and expected shortfall. We investigate the sensitivity and robustness of the expected shortfall to the underlying mixture parameter specification and the risk level. Empirical results from US, German and UK stock markets and for selected NASDAQ blue chip companies indicate that expected shortfall can be successfully estimated using the proposed method on a monthly, weekly, daily, and intra-daily basis using a one-year or one-day time horizon across different risk levels.

How Sensitive are Tail-Related Risk Measures in a Contamination Neighbourhood?

“…We remark that the Power-like contamination distribution is a typical example that the probability level ratio w α /α = 1. For a realistic degree of tail heaviness, the ratio w α /α is less than 1, and increases with the degree of tail heaviness, Mihoci et al (2017).…”

“…Finally, estimations of VaR and ES at level α = 0.5%, 1%, 5% are compared also in Table 2 via three methods including the historical simulation, written by q * α , ES * α ; Laplace approximations at level α = α/ by use of Theorem 2.1, denoted by q α (1), ES α (1); and approximations based directly on the normal-Laplace mixture model, written by q α ( ), ES α ( ). For all the estimations of ES, we keep the historical simulations of VaR, as in Mihoci et al (2017).…”

“…Cont et al (2010) pointed out that ES appears to lack robustness with respect to small changes in the underlying cdf. The recent contribution by Mihoci et al (2017) provides evidence on expected shortfall robustness through its link with expectile, which is given by minimizing the asymmetric weighted least square error, Newey and Powell (1987) e α = arg min…”

How Sensitive are Tail-related Risk Measures in a Contamination Neighbourhood?

“…Cont et al [2010] pointed out that ES appears to lack robustness with respect to small changes in the underlying cdf. The recent contribution by Mihoci et al [2021] provides evidence on expected shortfall robustness through its link with expectile, which is given as the minimizer of the expected value of an asymptotic piece-wise quadratic loss [Newey and Powell, 1987]:…”

“…Here LPM α = qα −∞ x dF (x) stands for the lower partial moment at α quantile. As a consequence, we get an alternative expression of ES as follows [Taylor, 2008, Mihoci et al, 2021.…”