Tasmanian-based lender Bank of us will pass on just 60 per cent of the Reserve Bank's cut to the cash rate.
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The central bank last week announced a 0.25 percentage point cut to the cash rate, taking it to a record low of 1.25 per cent.
Reserve Bank governor Philip Lowe later said the cut should be passed on in full to variable mortgage interest rates, arguing bank funding costs had recently reduced.
"Full pass-through would also mean that the economy receives the full benefit of today's policy decision," Dr Lowe said.
Federal Treasurer Josh Frydenberg had earlier called on banks to pass on any cut in full, also citing "sustained reductions" in banks' funding costs.
The Commonwealth Bank and the NAB passed on the full cut, while Westpac passed on 0.20 percentage points and the ANZ 0.18.
Bank of us on Wednesday announced it would pass on 0.15 percentage points.
Chief executive Paul Ranson said the bank's rates remained among the most competitive.
"We are committed to keeping home loan interest rates as low as possible for our borrowers," he said.
"As a small, customer owned bank, we must balance the interests of both borrowers and depositors, together with the increased cost of operating our business, in order to remain sustainable."
He said the bank earlier in 2019 made a "conservative" decision to only pass on a small increase to borrowers.
"Our aim is to remain competitive in the long term, not just the short term."
He said the rate reduction would be applied to all variable home loans for owner occupiers.
The new rates would apply from August 1 for existing loans and July 1 for new loans.