Magneto-optical studies on a series of CdTe/Cd1-xMnxTe multiple-quantum- well structures with x0.08 have identified the 1S and 2S states of both the light- and heavy-hole n=1 excitons. This has allowed changes of the exciton binding energies to be studied as a function of the depth of the confining potentials, which were tuned through the sp3-d exchange interaction in the barrier layers by application of a magnetic field. Calculations of these binding energies by a variational technique are in general agreement with the observations. The exchange-induced splitting of the heavy-hole exciton is found to be consistent with between 0.35 and 0.45 of the band offset being in the valence band, which accounts for the absence of any evidence that the valence-band structure changes from a type-I structure to a type-II structure above a certain value of a magnetic field. However, this offset is found to be too large to account for the exchange-induced splitting of the light-hole exciton, which appears to be anomalously large. Calculations have shown that this anomaly cannot be explained in terms of the diffusion of Mn ions from the barrier regions into the wells. An alternative explanation is given in terms of field-dependent interface potentials wherein the sp3-d exchange interaction is considered to be different in the interface regions of the multiple quantum wells to that in the barrier regions.