Stress tests with handpicked scenarios might misrepresent risks either because dangerous scenarios are not considered or because the scenarios considered are too implausible. To overcome these two pitfalls we propose a systematic search for the worst case within a relative entropy ball of sufficiently plausible scenarios. For this purpose we use mixed scenarios, which are risk factor distributions rather than realisations. A Maximum Loss theorem explicitly gives the worst case distribution. The method is illustrated in a number of example applications: linear and quadratic portfolios, stressed default probabilities, stressed correlations, macroeconomic stress tests.