We empirically examine whether access to deposits with inelastic rates (core deposits) permits a bank to make contractual agreements with borrowers that are infeasible if the bank must pay market rates for funds. Such access insulates a bank's costs of funds from exogenous shocks, allowing it to insulate its borrowers against exogenous credit shocks. We find that, controlling for loan market competition, banks funded more heavily with core deposits provide more loan rate smoothing in response to exogenous changes in aggregate credit risk. Thus we provide evidence for a novel channel linking bank liabilities to relationship lending. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.