Given a macroeconomic stress scenario defined in terms of the magnitude of common shocks across balance sheet, we perform an asymptotic analysis of default contagion, using analytical methods, and derive an expression for the fraction of defaulted nodes in the limit where the number of nodes is large, in terms of the empirical distribution of the in and out-degrees and the proportion of weak links in the network. We show that the size of the default cascade generated by the macroeconomic shock may exhibit a phase transition when the macroeconomic shock affecting the financial institutions reaches a certain threshold, beyond which the fraction of defaults is close to one. This result is used obtain a criterion for the resilience of a large network to macro-economic shocks The asymptotic results are shown to be in good agreement with simulations for networks whose sizes are realistic, showing the relevance of the large network limit for macro-prudential regulation.