On the evaluation of marginal expected shortfall

In the analysis of systemic risk, Marginal Expected Shortfall (MES) may be considered to evaluate the marginal impact of a single stock on the market Expected Shortfall (ES). These quantities are generally computed using log-returns, in particular when there is also a focus on returns conditional distribution. In this case, the market log-return is only approximately equal to the weighed sum of equities log-returns. We show that the approximation error is large during turbulent market phases, with a subsequent impact on MES. We then suggest how to improve the evaluation of MES by means of a second-order approximation.