  We investigate the electronic structure and the ferroelectric properties of
the recently discovered multiferroic ScFeO$_3$ by means of ab-initio
calculations. The $3d$ manifold of Fe in the half-filled configuration
naturally favors an antiferromagnetic ordering, with a theoretical estimate of
the antiferromagnetic N\'eel temperature in good agreement with the
experimental values. We find that the inversion symmetry-breaking is driven by
the off-centering of Sc atoms, which results in a large ferroelectric
polarization of $\sim$105\,$\mu$C/cm$^{2}$. Surprisingly the ferroelectric
polarization is sensitive to the local magnetization of the Fe atoms resulting
in a large negative magnetoelectric interaction. This behavior is unexpected in
type-I multiferroic materials because the magnetic and ferroelectric orders are
of different origins.
