Negative interest rates are a new (and controversial) monetary policy tool. This paper studies a novel signalling channel and asks whether negative rates can be 1) an effective and 2) an optimal policy tool. 1) We build a financial-friction new- Keynesian model in which monetary policy can set a negative reserve rate, but deposit rates are constrained by zero. All else equal, a negative rate contracts bank net worth and increases credit spreads (the costly ''interest margin" channel). However, it also signals lower future deposit rates, even with current deposit rates constrained, boosting aggregate demand and net worth. Quantitatively, we find the signalling channel dominates, but the effectiveness of negative rates depends crucially on three factors: i) degree of policy inertia, ii) level of reserves, iii) zero lower bound duration. 2) In a simplified model we prove two necessary conditions for the optimality of negative rates: i) time-consistent policy setting, ii) preference for policy smoothing.