What type of mortgage loan is likely to be tied to a publicly available index that is mutually acceptable to the lender and the borrower?

A renegotiable rate mortgage does not rely on a publicly available index; instead, it allows borrowers to renegotiate the interest rate after a specified period. This type of mortgage is typically not tied directly to market indices, making it less predictable in terms of interest rate adjustments compared to an adjustable rate mortgage.

Graduated payment mortgages have a fixed interest rate that increases over time, allowing for lower initial payments that gradually rise. This type does not fluctuate with a public index and lacks the market-driven adjustments that characterize adjustable rate mortgages, making it fundamentally different from an ARM.

Freddie Mac is not a type of mortgage but rather a government-sponsored enterprise that buys and securitizes mortgages. While it plays a significant role in the mortgage market, it does not represent a specific mortgage product tied to an index, making it irrelevant in the context of the question.