  Density functional theory (DFT) calculations are used to study the magnetic
and ferroelectric properties of Sr$_{1-x}$Ba$_{x}$MnO$_3$, with focus on
$x=0.5$, under isotropic volume expansion/compression and biaxial strain. Our
results indicate that, unexpectedly, Ba substitution alters the electronic
structure in a way that, at fixed lattice parameter, notably enhances the
interatomic magnetic exchange interactions. However, increasing Ba-content also
causes a volume expansion which tends to weaken these interactions, leading to
a net effect of weakly suppressed magnetism, as observed in experiments. The
ferroelectric properties, on the other hand, are found to be less affected by
changes in the electronic structure and can largely be understood in terms of
the volume expansion caused by Ba-substitution. The calculated electric
polarization as a function of biaxial strain in Sr$_{1-x}$Ba$_{x}$MnO$_3$ for
$x=0$ and $x=0.5$, shows that the difference between the two is mainly due to
differences in the magnetic order at certain strain values, accompanied by
enormous magnetoelectric coupling.
