Bismuth-based solar absorbers are of interest due to similarities in the chemical properties of bismuth halides and the exceptionally efficient lead halide hybrid perovskites. Whilst they both experience the same beneficial relativistic effects acting to increase the width of the conduction band, bismuth is non-toxic and non-bioaccumulating, meaning the impact of environmental contamination is greatly reduced. Here, we use hybrid density functional theory, with the addition of spin orbit coupling, to examine two candidate bismuth containing photovoltaic absorbers, BiSI and BiSeI, and show that they possess electronic structures suitable for photovoltaic applications. Furthermore, we calculate band alignments against commonly used hole transporting and buffer layers, which indicate band misalignments are likely to be the source of the poor efficiencies reported for devices containing these materials. Based on this we have suggested alternative device architectures expected to result in improved power conversion efficiencies.