  By means of density functional theory (DFT) and the generalized gradient
approximation (GGA) we present a structural, electronic and magnetic study of
FePt, CoPt, FeAu and FePd based L1$_0$ ordered cuboctahedral nanoparticles,
with total numbers of atoms, N$_{tot}$ = 13, 55, 147. After a conjugate
gradient relaxation, the nanoparticles retain their L1$_0$ symmetry, but the
small displacements of the atomic positions tune the electronic and magnetic
properties. The value of the total magnetic moment stabilizes as the size
increases. We also show that the Magnetic Anisotropy Energy (MAE) depends on
the size as well as the position of the Fe-atomic planes in the clusters. We
address the influence on the MAE of the surface shape, finding a small in-plane
MAE for (Fe,Co)$_{24}$Pt$_{31}$ nanoparticles.
