The electronic structure of $\mathrm{Ba}{\mathrm{Fe}}_{12}{\mathrm{O}}_{19}$ hexaferrite is calculated using the density functional theory and generalized gradient approximation (GGA). The $\mathrm{GGA}+\mathrm{U}$ method is used to improve the description of strongly correlated $3d$ electrons of Fe. The calculation is performed for a number of spin configurations. From differences of the total energies 13 independent exchange integrals are determined as functions of the parameter $U$. Their magnitude decreases with increasing $U$, pointing to the dominating role played by the antiferromagnetic superexchange. The Curie temperature ${T}_{C}$ is calculated using the molecular field and the random phase approximations. ${T}_{C}$ determined by the random phase approximation agrees with the experimental ${T}_{C}$ for $U\ensuremath{\approx}6--7\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$.