Reliable networks are those in which network elements have a positive probability of failing. Conventional performance measures for such networks concern themselves either with expected network performance or with the performance of the network when it is performing well. In reliable networks modeling critical functions, decision makers are often more concerned with network performance when the network is not performing well. In this paper, we study the single-source single-destination maximum flow problem through reliable networks and propose two risk measures to evaluate such downside performance. We propose an algorithm called COMPUTE-RISK to compute downside risk measures, and report our computational experience with the proposed algorithm.