An enhanced dynamic address translation facility product is created such that, in one embodiment, a virtual address to be translated and an initial origin address of a translation table of the hierarchy of translation tables are obtained. Dynamic address translation of the virtual address proceeds. In response to a translation interruption having occurred during dynamic address translation, bits are stored in a translation exception qualifier (TXQ) field to indicate that the exception was either a host DAT exception having occurred while running a host program or a host DAT exception having occurred while running a guest program. The TXQ is further capable of indicating that the exception was associated with a host virtual address derived from a guest page frame real address or a guest segment frame absolute address. The TXQ is further capable of indicating that a larger or smaller host frame size is preferred to back a guest frame.