firstly try and get your feelings of failure out of the way. You are a worried parent who needs to budget and can raise their voiuce when a large unexpected bill comes in.
Im am pleased to see your daughter has actually voiced her reasons for harming herself although it may be that there is more to it that just a fear of your shouting. Perhaps her peers or stress at school (I presume she is in her teenage years?)
Heres a link about self harming, perhaps you could both read it together and then sit and talk about what you both think. tell her why you get annoyed about bills, suggest some compromises so that the risk of you shouting is lowered. Ie how about a limit on her phone bill, many suppliers will allow you to limit the monthly usage.
Most importantly see if there is anyone your daughter can talk to, someone who she can call when she feels the need to harm herself who can talk things through with her.
http://www.mind.org.uk/Information/Booklets/Un derstanding/Understanding+self-harm.htm
good luck