We study the transition from type-I to type-II AlInAs/AlGaAs self-assembled quantum dots (QDs) which is induced by changing the Al composition of an AlGaAs matrix. We found theoretical evidence for type-II band alignment above the crossover Al concentration > = x x (0.43) c. For this purpose, we obtained the phase diagram for different QD radii, and we can identify the region where the electron states derived from the X-valley are lower than those derived from the Γ-point. The spatial distribution of electrons and holes is very sensitive to QD size variations. Furthermore, the effect of the QD type on the exciton binding energy is investigated. The ground-state exciton binding energy is always significantly smaller for type-II than for corresponding type-I systems, which originates from the redistribution of the electron and hole wave function.