Recent experiments on c-axis tunneling into a vortex core of YBCO have yielded a sub-gap differential conductance which is not invariant under a reversal of the applied bias. We present a new result for the differential conductance G(E) of a normal-superconducting contact which predicts that whatever the strength of contact, in the absence of open channels in the superconductor, G(E) is necessarily an even function of E and that an asymmetric current-voltage characteristic implies the existence of propagating bound-states within the vortex core. In the tunneling limit, asymmetries arise from the difference between particle and hole densities of states and if the Fermi energy is close to a singularity, is predicted to be sensitive to doping.