Analytical VaR and Expected Shortfall for Quadratic Portfolios

Abstract

Expected shortfall is a coherent risk measure proposed to remedy the weakness of the widely used Value-at-Risk (VaR). However, calculation of expected shortfall is time consuming due to the lack of closed-form formulae. In this article, we employ the Fourier transform techniques to derive analytic expressions for VaR and expected shortfall for quadratic portfolios exposed to multivariate normally distributed risk factors. Our numerical results show that the proposed analytical expressions for the two risk measures are accurate and much faster than alternative Monte Carlo simulations. We thus argue that expected shortfall should be used in conjunction with VaR to provide useful information for aggregating and assessing portfolio risks. From this perspective, the derived analytic expressions provide an efficient way to calculate the coherent risk measure to fulfill the goal of integrated risk management.