Measuring the contribution of a bank or an insurance company to overall systemic risk is a key concern, particularly in the aftermath of the 2007--2009 financial crisis and the 2020 downturn. In this paper, we derive worst-case and best-case bounds for the marginal expected shortfall (MES) -- a key measure of systemic risk contribution -- under the assumption that individual firms' risk distributions are known but their dependence structure is not. We further derive tighter MES bounds when parti...