Measuring risks in the extreme tail: The extreme VaR and its confidence interval

Contrary to the current regulatory trend concerning extreme risks, the purpose of this paper is to emphasize the necessity of considering the Value-at-Risk (VaR) with extreme confidence levels like 99.9%, as an alternative way to measure risks in the "extreme tail". Although the mathematical definition of the extreme VaR is trivial, its computation is challenging in practice, because the uncertainty of the extreme VaR may not be negligible for a finite amount of data. We begin to build confidence intervals around the unknown VaR. We build them using two different approaches, the first using Smirnov's result (Smirnov, 1949 [24]) and the second Zhu and Zhou's result (Zhu and Zhou, 2009 [25]), showing that this last one is robust when we use finite samples. We compare our approach with other methodologies which are based on bootstrapping techniques, Christoffersen et al. (2005) [7], focusing on the estimation of the extreme quantiles of a distribution. Finally, we apply these confidence intervals to perform a stress testing exercice with historical stock returns during financial crisis, for identifying potential violations of the VaR during turmoil periods on financial markets.