We study the interaction between monetary policy and household debt dynamics. To this end, we develop a dynamic stochastic general equilibrium model where household debt is amortized gradually, and only new loans are constrained by the current value of collateral. Long-term debt implies that swings in leverage do not simply reflect shifts in borrowing, and brings modelimplied debt dynamics closer to their empirical counterparts. The model implies that contractive monetary policy has muted influence on household debt, increasing debt-to-GDP in the short run, while reducing it only in the medium run. If the interest rate is systematically raised whenever the debt-to-GDP ratio or the real debt level is high, equilibrium indeterminacy and greater volatility of debt itself follows. Responding to debt growth does not cast this destabilizing influence.