  In this paper we analyze different contributions to the magnetoresistance of
magnetic tunneling junctions at low voltages. A substantial fraction of the
resistance drop with voltage can be ascribed to variations of the density of
states and the barrier transmission with the bias. However, we found that the
anomaly observed at zero bias and the magnetoresistance behavior at very small
voltages, point to the contribution of inelastic magnon-assisted tunneling. The
latter is described by a transfer parameter $T^{J}$, which is one or two orders
of magnitude smaller than $T^{d}$, the direct transmission for elastic
currents. Our theory is in excellent agreement with experimental data, yielding
estimated values of $T^{J}$ which are of the order of $T^{d}$ / $T^{J}$ ~ 40.
