The electronic structure of the heavy-fermion compound ${\mathrm{Yb}}_{4}{\mathrm{As}}_{3}$ is investigated using energy band calculations within the so-called local spin density $\mathrm{approximation}+U$ approach. We find as a generic feature a very rigid pinning of the energy of Yb $4f$ hole states close to the top of the pnictide $p$ band and the Fermi level pinned to those hole states, providing for a very low-carrier concentration. Calculated Fermi surface and optical properties compare well with experimental data. We also present electronic structure results for the related compounds ${\mathrm{Yb}}_{4}{\mathrm{P}}_{3},{\mathrm{Yb}}_{4}{\mathrm{Sb}}_{3},{\mathrm{Yb}}_{4}{\mathrm{Bi}}_{3},$ and ${\mathrm{Yb}}_{4}{(\mathrm{P}}_{0.5}{\mathrm{As}}_{0.5}{)}_{3}.$