As a nearly lifelong, immersed-in-evangelical-Christianity woman, I have heard the word redeemed over and over. It is in our hymns (There is a Redeemer, Redeemed by the blood of the lamb, and so many more) and our Bible stories (Ruth). But as a lifelong lover of words, it helps me to know what a word means.
Redeem:: to compensate for the faults or bad aspects of something, atone or make amends for.
Christ has done more than compensate for my faults or bad aspects. He has replaced my unrighteousness with his righteousness. He has atoned for my sin, all of them, knowing I would do them. Not after the fact, but before I was even born.
As one who has been redeemed, I have "gained something in exchange for payment," but I did not make that payment. Christ did. He paid and I was redeemed.
This is the hope of the resurrection, the proof of my redemption.