Qatar is building itself up to be a knowledge center and their Financial Center is based on assets and insurance.  Oil is bountiful is this small country, but they are aware that hydrocarbons will run out eventually.

While banks are not central to their future plans, they recognize them as undervalued assets today.  Reeling from litigation, bank stocks have not returned to their pre-financial crisis highs.  Qatar sees them as a good long-term investment.

They are not being purchased on behalf of the country however.  The purchase vehicles benefit only the royals.