Answering your own question

This figure of speech is called hypophora. If you visit the Wikipedia entry for figures of speech or this web page about rhetoric, you can find more information about this and other devices.

Apparently a number of terms apply to this tactic. Richard A. Lanham, A Handlist of Rhetorical Terms (1968) offers these:

anthypophora: asking questions and answering them.

dialogismus: speaking in another man's person.

hypophora: asking questions and answering them.

ratiocinatio: a question addressed by the speaker to himself.

sermocinatio: the speaker answers the remarks or questions of a pretended interlocutor.

The past master of sermocinatio was Donald Rumsfeld, who, as U.S. Secretary of Defense from 2001 to 2006, habitually rephrased or replaced reporters' questions at press conferences with the questions he preferred to answer.