I'm not loaded! Where on earth did you get that idea from?!? The way the exchange worked with Oxford was that it was cost neutral - I paid my Oxford tuition fees and got a semester in Princeton in exchange. Bloody good deal for me!
That's how it would work if I went to study abroad.
If you were to actually go to Princeton (or any of the Ivys - Harvard
, Yale, Brown, etc), they would meet 100% of your need for aid. However actually trying to get in costs an arm and a leg.