Climate models predict that tropical lower-stratospheric humidity will increase as the climate warms. We examine this trend in two state-of-the-art chemistry-climate models. Under high greenhouse gas emissions scenarios, the stratospheric entry value of water vapor increases by approx. 1 part per million by volume (ppmv) over this century in both models. We show with trajectory runs driven by model meteorological fields that the warming tropical tropopause layer (TTL) explains 50-80% of this increase. The remainder is a consequence of trends in evaporation of ice convectively lofted into the TTL and lower stratosphere. Our results further show that, within the models we examined, ice lofting is primarily important on long time scales - on interannual time scales, TTL temperature variations explain most of the variations in lower stratospheric humidity. Assessing the ability of models to realistically represent ice-lofting processes should be a high priority in the modeling community.