Are you asking which banks will waive LMI up to 85% or which banks have their own underwritting authority to approve loans without the insurer even seeing it?
The answer to the first question tends to be subjective depending on what lenders are offering at the time and occassionally on a case by case basis. If this is what you're looking for, CBA & Westpac sometimes do this.
The second question also is a bit more complex. CBA, Westpac, ANZ, NAB, AMP, Suncorp & ING all have a Deffered Underwritting Authority (DUA) of some description with their insurers. I'm sure there's a few more that I can't recall. The parameters of the DUA do change from one lender to another and can be based on the loan amount, property location and any number of factors based on the 'risk rating' or 'credit score' of the application.
In most cases the lenders have structured their assessment policies and credit scoring to reflect the insurers criteria, so a quirky application may not pass the lenders criteria anyway.
If you're trying to figure out which lender is going to approve a loan for you when there's one or two problems in the past, the best way to proceed is to talk to a broker who's got some experience in these things.
