We use the first principles methods to study the electronic structure and optical properties of G-type anti-ferromagnetic hexagonal-YMnO$_3$. Ground state properties of this material were calculated within density functional theory (DFT) using the DFT + U formalism. We calculated the quasiparticle band structure of this material using many body perturbation theory within the GW approximation. In order to understand the optical response of this material, we solved the Bethe–Salpeter equation and calculated the absorption spectrum. Our calculated optical band gap of 1.45 eV agrees well with the experimental value of 1.55 eV. We find an exciton binding energy of 0.21 eV for this material.