Mounting bills come on top of the £42bn the four have already paid out over the past five years.
The next nine largest lenders paid a further £6bn over the same period.
The pace is slowing slightly – the £19bn for the coming two years is compared with £22bn across 2013 and 2014. Payment protection insurance redress amounted to £26bn of the costs over the past five years.
Although the fallout from the mis-selling scandal is fading, it remains a substantial cost – the analysts predict it will cost just over £5bn over the next two years.
But other litigation could cost the banks around £10bn in 2015 and another £3bn in 2016.
However, banks have worked hard to become more resilient.
“We doubt that these ongoing charges will lead on their own to negative rating actions, primarily because we believe that the affected banks have sufficient capital buffers, according to our assessments,” said the note, written by analysts led by Nigel Greenwood.