  We theoretically investigate lead iodide perovskites of general formula
APbI$_3$ for a series of metallic cations (namely Cs$^+$, Rb$^+$, K$^+$, Na$^+$
and Li$^+$) by means of the density functional theory, GW method and
Bethe-Salpeter equation including spin-orbit coupling. We demonstrate that the
low-energy edges (up to 1.3 eV) of the absorption spectra are dominated by
weakly bound excitons, with binding energies $E_b$ $\sim$ 30-80~meV, and the
corresponding intensities grow as metallic %A$^+$ cations become lighter. The
middle parts of the spectra (1.8-2.4 eV), on the other hand, contain optical
dipole transitions comprising more strongly bound excitons ($E_b$ $\sim$
150-200~meV) located at PbI$_3$. These parts of the spectra correspond to the
optical-gain wavelengths which are experimentally achieved in optically-pumped
perovskite lasers. Finally, the higher energy parts, from about 2.8~eV
(LiPbI$_3$) to 4.3 eV (CsPbI$_3$), contain optical transitions with very
strongly bound excitons ($E_b$ $\sim$ 220-290~meV) located at the halide atoms
and the empty states of the metallic cations.
