If you really want to refinance I think you should be able to do it if you originally put down 35%. You may end up with a close to 100% financing mortgage since the value of the house has fallen but you should be able to get a lower rate than your current 6.25%. The way to look at refinancing is calculate how much the refinance is going to cost you and how long the payback will be. For example, you say that you can reduce your monthly cost by $700 for a $8500 refinance cost. That means you would get your money back in about a year which is pretty good. By the way - $8500 sounds like a LOT of money to refinance a mortgage. It sounds like they are getting you to "buy down" the rate. All that means is that you are paying the interest in advance to get a better rate and if you sell early you lose that money. But I have to warn you like JB mentiioned - if you get a principal reduction it may be taxable and could affect your credit. Before you do anything else why not talk to another bank or mortgage company and see what they offer if your mortgage company won't deal with you? It never hurts to ask. I don't know enough about all of the details to make a firm recommendation except to warn that if a deal sounds too good it probably is a rip off of some sort. Everything I have read recently said that you never give any money to a company upfront.
To recap, if you can save $700 per month through an honest refinance then go for it. Just make sure that it is worth it.
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