Today's distributed systems operate in complex environments that inevitably involve faults and even adversarial behaviors. Predicting their performance under such environments directly from formal designs remains a longstanding challenge. We present the first formal framework that systematically enables performance prediction of distributed systems across diverse faulty scenarios. Our framework features a fault injector together with a wide range of faults, reusable as a library, and model compositions that integrate the system and the fault injector into a unified model suitable for statistical analysis of performance properties such as throughput and latency. We formalize the framework in Maude and implement it as an automated tool, PERF. Applied to representative distributed systems, PERF accurately predicts system performance under varying fault settings, with estimations from formal designs consistent with evaluations on real deployments.