YES, they definitely exist and are available from almost any lender.
Every rate comes with a cost (or credit) associated with that rate. The cost/credit changes daily depending on the bond market, and there are often intra-day changes.
There is no set formula for how the credit increases for each higher 1/8th of a point, but it is not uncommon to receive a 1pt credit (of the loan amount) for taking a rate 1/8th of a pt higher.
IN THIS EXAMPLE: On a $400,000 loan you would receive a credit of $4000.00 (1.00%) and have a payment that is $29 higher each month.($348 yr)
If you keep the loan long enough, it will cost you
much more than $4000.
After all fees are paid for, the excess funds can (usually) ONLY be used to cover certain things such as interest, insurance, property taxes and reserves required for an impound account. This can be a problem for those with low property taxes who do not want an impound account.
Some lenders *might* allow any excess to reduce principal balance but it is rare.