This paper develops a method for selecting and analysing stress scenarios for financial risk assessment, with particular emphasis on identifying sensible combinations of stresses to multiple factors. We focus primarily on reverse stress testing - finding the most likely scenarios leading to losses exceeding a given threshold. We approach this problem using a nonparametric empirical likelihood estimator of the conditional mean of the underlying market factors given large losses. We then scale confidence regions for the conditional mean by a coefficient that depends on the tails of the market factors to estimate the most likely loss scenarios. We provide rigorous justification for the confidence regions and the scaling procedure when the joint distribution of the market factors and portfolio loss is elliptically contoured. We explicitly characterize the impact of the heaviness of the tails of the distribution, contrasting a broad spectrum of cases including exponential tails and regularly varying tails. The key to this analysis lies in the asymptotics of the conditional variances and covariances in extremes. These results also lead to asymptotics for marginal expected shortfall and the corresponding variance, conditional on a market stress; we combine these results with empirical likelihood significance tests of systemic risk rankings based on marginal expected shortfall in stress scenarios.