I had the exact same situation with my 13 year old, who I have 50/50 custody of (search for other posts of mine to read in detail). She would threaten to "live with Mom" every time she didn't get her way or when she'd get into trouble (which Mom would give in to her all the time). I finally got sick of the threats one day and told her that it was her choice to leave, and she moved out.
In the three months that followed, I spoke with her three times...all three were when SHE called ME. I told her each time that I loved her, and that I was here for her when and if she decided to come back. During that three months, I went on a week long N.C. vacation with my wife's family (including my son and step daughter), and got re-married (which she was supposed to be in the bridal party...already had her dress and shoes)...all without her. She was invited, but chose not to be a part of our lives and therefore missed out. When we got back from our honeymoon, she called sobbing, begging us to let her come back and saying how sorry she was, etc, etc, etc.
To this day (six months later), I have YET to hear one threat to leave, and her respect for our home, our rules and me and my new wife is tremendous. It was the hardest thing I've ever had to do (let her go), but three months away from me, and more importantly three months with her IDIOT Mother did WONDERS! It was the best thing I've ever done as a Father. My other kids saw it, and know that threats do not work with me and neither of them have ever talked to me like that.
Trust me, she WILL NOT be like that forever. Simply tell her, "I love you, I'm sorry that you feel that way, I hope you change your mind, but I will respect your decision and will not fight you. Give me a call sometime and let me know how you're doing". And then drop it. Don't call. Send a birthday card and gift if her birthday falls during the period that she's gone. But go on with your life. She will miss out on things, and she will miss YOU, I guarantee it.
Good luck to you.