Motivated by the Basel 3 regulations, recent studies have considered joint forecasts of Value-at-Risk and Expected … Shortfall. A large family of scoring functions can be used to evaluate forecast performance in this context. However, little …

results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. We also … present a particularly short proof of the subadditivity of expected shortfall. …

We provide an accurate closed-form expression for the expected shortfall of linear portfolios with elliptically … understatement of expected shortfall by a factor varying from at least four to more than 100 across different tail quantiles and … distributions. More generally, our findings point to the extra scrutiny required when deploying new methods for expected shortfall …

profit, variance and expected shortfall of the insurer’s risk. The aggregate claim amounts are assumed to be distributed as … the expected shortfall of the insurer’s risk. In the decision making process, we concentrate on multi-attribute decision …

estimator to estimate one-step-ahead daily expected shortfall conditional on previously observed daily, weekly, and monthly …

using historical data on European financial stocks that forecasts portfolio Value at Risk (VaR) and Expected Shortfall (ES). …

-called Range-Value-at-Risk (RVaR), as their preferences. The family of RVaR includes the Value-at-Risk (VaR) and the Expected … Shortfall (ES), the two popular and competing regulatory risk measures, as special cases. We first establish an inequality for …