Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

Abstract

Abstract:We provide an accurate closed-form expression for the expected shortfall of linear portfolios with elliptically distributed risk factors. Our results aim to correct inaccuracies that originate in Kamdem (2005) and are present also in at least thirty other papers referencing it, including the recent survey by Nadarajah et al. (2014) on estimation methods for expected shortfall. In particular, we show that the correction we provide in the popular multivariate Student t setting eliminates understatement of expected shortfall by a factor varying from at least four to more than 100 across different tail quantiles and degrees of freedom. As such, the resulting economic impact in financial risk management applications could be significant. We further correct such errors encountered also in closely related results in Kamdem (2007 and) for mixtures of elliptical distributions. More generally, our findings point to the extra scrutiny required when deploying new methods for expected shortfall estimation in practice.

Quantifying Risk in Traditional Energy and Sustainable Investments

“…The papers of the Special Issue "Advances in Modelling Value at Risk and Expected Shortfall" of Journal of Risk and Financial Management present a recent state of the art in these market risk measures and its implications for stability of financial system. [32] proposes a novel method to estimate VaR and ES implied by financial options, whereas [33] provide a closed-form expression for ES of portfolios when risk factors are elliptically distributed. On the other hand, [34] develop a new VaR model based on financial markets overnight information.…”