If your child is a teen runaway, there is a problem. Before you jump to the conclusion that they are just a bad kid, determine what really pushed them in this direction. As a parent, there is nothing more fearful then knowing that your child is out there, facing whatever scum walks the earth and that you can not protect them. But, before you can get them to stay home with you, you need to work with them to determine why they left in the first place.
To do this, you need to do several things. First, stop accusing and leave the anger at the door. You need to have a frank but fair conversation with them. Determine why they left. Do not yell at them, punish them or demand answers. Calmly determine why they left, they must tell you here.
Tell them why being a runaway teen is dangerous. Explain why this is not something that is okay and that you will not allow it. Make sure they know that you are open to helping them to work through problems they have at school, with their peers or with you so they can avoid being a runaway teen.
Get them help. Physically, they need to be examined. Emotionally they need to work with a child therapist to determine what they need to resolve with you or their problems to keep them at home, safe. In fact, the biggest benefit they will need is working with you side by side to determine what the solution is.
A teen runaway is a call for help. In some form, they can no longer handle what is in their lives threatening them. They are looking for a way to run, to hide or to start over because something is pushing them to do that. You do not have to be a bad parent to face a runaway teen. In fact, most will deal with this. Yet, what you need to be is an open parent that will welcome your teen runaway back into your home and work on repairing problems so that they will stay at home with you.