Magnetic tunnel junctions (MTJs) are leading candidates for memory cells in spin-transfer-torque magnetic random-access memory, and to that end larger perpendicular magnetic anisotropy and smaller Gilbert damping $\ensuremath{\alpha}$ are sought. The authors' calculations show that in Ag/MgO/Fe MTJs the decrease in $\ensuremath{\alpha}$ with increasing MgO thickness, previously observed in experiments, is due to MgO's suppression of spin pumping upon insertion at the Ag/Fe interface. Other interfacial effects are also investigated.