What is the benefit to a lender to do a deed in lieu of a foreclosure instead of a foreclosure?

I am not licensed in AZ, nor have I practiced there; however, your question is more practical than state-specific so I offer the following. As you likely know, a Deed in Lieu of Foreclosure (DILOF) essentially allows you to turn over the property without incurring a deficiency judgement or demand for any further payment on your mortgage loan. Usually, the lender will consider accpeting a DILOF to avoid the costs that the lender incurs via the foreclosure process. However, the Lender may likely proceed with the costs and time associated with a foreclosure if the market price for the home is much less than the outstanding debt-as seems to be your case. Thus, the lender may not want to accept a 90K home in exchange for a 190K debt.

I would contact the loss mitigation department of your lender and inquire about a potential loan modification. Hopefully, with the appropriate documentation, you can strike an agreement which will modify the terms of your loan and allow you to stay in your home. The hope is that the real estate market will pick up and your equity will eventually surpass your debt.