When you calculated savings of $3,800 after closing costs, did you compare the refi case to just paying the existing mortgage on time for 5 years, or did you compare it to dumping an amount equal to the closing costs into your existing mortgage and then paying it normally until it's paid in full?
If you didn't do the latter comparison, run that and see how the numbers look.
Wow, I've never even heard of a 5 year mortgage. I probably wouldn't because I'm lazy. But if it will really save you almost $4K, why not?
You have me wondering what mortgage rates are now for a 10 year mortgage. Bankrate says 1.75% but I can never actually get the lowest rates that people talk about. We refied into a 15 year at 2.75% last year. If we could get 10 years at 1.75%, that seems worth it. An online calculator says we'd save $20K even if we got 2.00%. The idea of paying off our house in 10 years is very attractive. But again, lazy....