We present ab initio density-functional calculations for the electronic structure of the dilute magnetic semiconductors Mn$_x$Ga$_{1-x}$As and Mn$_x$In$_{1-x}$As with a realistic $x = 0.063$. We find that the introduction of Mn perturbs the position of the nearest As atoms, but does not break the tetrahedral symmetry. Neither material is found to be strictly half metallic. However, in both materials the Mn content results in a majority-spin valence-band maximum that is ∼0.5 eV above the minority-spin valence-band maximum. This large valence-band split is primarily due to the hybridization of As 4$p$ and Mn 3$d$ orbitals. It results in a significant energy range where holes have a well-defined spin. The effective masses of holes in this range are found to be comparable to those of GaAs and InAs. Hence, in an ideal, disorder-free situation, spin-polarized transport may be explained by conventional transport in the context of a simple band picture. This leads to a theoretical limit of 100% spin injection from these materials. Attaining this limit in a sufficiently ordered material also requires a careful “engineering” of the Fermi-level position and a sufficiently low temperature.