  We investigate magnetic excitations in iridium insulators with pentavalent
Ir$^{5+}$ ($5d^4$) ions with strong spin-orbit coupling. We obtain a
microscopic model based on the local Ir$^{5+}$ multiplets involving $J=0$
(singlet), $J=1$ (triplet), and $J=2$ (quintet) spin-orbital states. We get
effective interactions between these multiplets on square and
face-centered-cubic (fcc) structures of magnetic ions in the
layered-perovskites and the double-perovkites, in particular Ba$_2$YIrO$_6$.
Further, we derive an effective spin-orbital Hamiltonian in terms of bond
bosons and explore possible instabilities towards magnetic and quadrupole
orderings. Additionally, we study charge excitations with help of the
variational cluster perturbation theory and calculate the electronic charge gap
as a function of hopping and Coulomb interactions. Based on both electronic and
magnetic phase diagrams, we verify the possibility of excitonic magnetism due
to condensation of spin-orbital excitons in Ir$^{5+}$ iridates.
