Its analysis found that debt-serviceability restrictions were likely to be the most effective tool that could be deployed by the Reserve Bank to support financial stability and house-price sustainability and would complement existing loan-to-value ratio restrictions.
It also showed that the restrictions would affect investors more while having a limited impact on first-home buyers.
The Reserve Bank said Robertson had agreed to add the restrictions to its toolkit on the condition that any implementation was designed to avoid impact, as much as possible, to first-home buyers.
"We will now work with the Treasury to update the wording for the MoU, which will need to be approved by the minister."
Reserve Bank Governor Adrian Orr said it did not have a remit to target house prices directly but its financial policy tools could help ensure prices did not deviate too far from sustainable levels.
"We believe that a 'sustainable house price' is the level that the price would be expected to move towards over several years, reflecting the underlying drivers of supply and demand for housing, including population growth, building costs, land supply and interest rates."
Orr said the Reserve Bank would discuss with the industry over the coming months the feasibility of implementing debt-to-income limits and other debt-servicing restrictions.
But any decision on implementing debt-serviceability restrictions would include a full public consultation first along with a regulatory impact statement.
Robertson said he retains the view that the development and design of any debt serviceability tool such as a debt-to-income ratio limit should apply only to investors.
"The Reserve Bank has clearly stated that there is no immediate plan to use DTIs and any decision to do so would only happen after a full public consultation. The Government has already put in place a number of measures to cool the housing market and it's important to give these initiatives time to assess their impact.
"I have accepted the Reserve Bank's advice on interest-only lending, which will not be included in the MoU. The Bank notes the use of interest-only lending has been trending downwards since 2016 for both investors and owner-occupiers. The Government's interest deductibility changes are likely to separately and significantly reduce demand for interest-only lending among investors.
"Further work by officials now needs to be done on the wording in the 2013 Memorandum of Understanding (MoU) on macro-prudential policy to ensure it is consistent with supporting the soundness of the financial system and the Government's housing policy objective of sustainable house prices, as set out in the section 68B direction."