We present a systematic assessment of the structural properties, the electronic density of states, the charge densities, and the phase stabilities of AgInSe2 and AuInSe2 using screened exchange hybrid density functional theory, and compare their properties to those of CuInSe2. For AgInSe2, hybrid density functional theory properly captures several experimentally measured properties, including the increase in the band gap and the change in the direction of the lattice distortion parameter u in comparison to CuInSe2. While the electronic properties of AuInSe2 have not yet been experimentally characterized, we predict it to be a small gap (≈ 0.15 eV) semiconductor. We also present the phase stability of AgInSe2 and AuInSe2 according to screened-exchange density functional theory, and compare the results to predictions from conventional density functional theory, results tabulated from several online materials data repositories, and experiment (when available). In comparison to conventional density functional theory, the hybrid functional predicts phase stabilities of AgInSe2 in better agreement with experiment: discrepancies in the calculated formation enthalpies are reduced by approximately a factor of three, from ≈ 0.20 eV/atom to ≈ 0.07 eV/atom, similar to the improvement observed for CuInSe2. We further predict that AuInSe2 is not a stable phase, and can only be present under non-equilibrium conditions.