  Using a joint approach of density functional theory and model calculations,
we demonstrate that a prototypical charge ordered half-doped manganite,
La$_{1/2}$Ca$_{1/2}$MnO$_3$ is multiferroic. The combination of a peculiar
charge-orbital ordering and a tendency to form spin dimers breaks inversion
symmetry, leads to a ferroelectric ground-state with a polarization up to
several $\mu C/cm^2$. The presence of improper ferroelectricity does not depend
on hotly debated structural details of this material: in the Zener-polaron
structure we find a similar dramatic ferroelectric response with a large
polarization of purely magnetic origin.
