The second harmonic of the magnetoinductance (MI) signal of a melt-extracted amorphous magnetic wire varies significantly with the strength of the DC axial field applied to the wire. Its amplitude at frequencies between 200 kHz and 1 MHz was measured with a spectrum analyzer. We found that it has a four-peak structure, and that the peak amplitude increases with the frequency of the driving current. When a tensile stress is applied to the wire, the central peaks diminish, while the outer peaks are more widely spaced on the field axis. A simple analytical model of the MI signal shows that a large second harmonic of this signal and a large asymmetry in the circumferential hysteresis loop of the wire occur together and are manifestations of the same material property.