An emerging literature proposes using conditional value at risk (CoVaR) and marginal expected shortfall (MES) to measure financial institution systemic risk. We identify two weaknesses in this literature: (1) it lacks formal statistical hypothesis tests; and, (2) it confounds systemic and systematic risk. We address these weaknesses by introducing a null hypothesis that stock returns are normally distributed. This allows us to separate systemic from systematic risk and construct hypothesis tests for the presence of systemic risk. We calculate the sampling distribution of these new test statistics and apply our tests to daily stock returns data over the period 2006-2007. The null hypothesis is rejected in many instances, consistent with tail dependence and systemic risk but the CoVaR and MES tests often disagree about which firms are potentially “systemic.” The highly restrictive nature of the null hypothesis and the wide range of firms identified as systemic makes us reluctant to interpret rejections as clear evidence of systemic risk. The introduction of hypothesis testing is our primary contribution, and the results highlight the importance of generalizing the approach to less restrictive stock return processes and to other systemic risk measures derived from return data.