Low-income tenants in the private rented sector face a “heat, eat or pay rent” problem because housing benefit rates have failed to keep up with the soaring cost of accommodation, a study has found.

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The four-year freeze on local housing allowance levels, which has been in place since April 2016, means some families must meet a shortfall of hundreds of pounds a month on their rent support, according to the Chartered Institute of Housing (CIH).

It said the ongoing housing benefit freeze meant even the lowest private rents were out of reach for many low-income families in most areas – making it more likely that tenants would be forced to choose between living necessities or paying the rent.

“Our research makes it clear just how far housing benefit for private renters has failed to keep pace with even the cheapest private rents. We fear this policy is putting thousands of private renters on low incomes at risk of poverty and homelessness,” said the CIH’s chief executive, Terrie Alafat.

The biggest “rent gaps” – the difference between rent and housing support levels in the cheapest third of rented properties in a local housing market area – were in London, where the median shortfall was £66 a week for a four-bedroom property and £43 for a two-bedroom, although this masked larger gaps of more than £140 a week in the inner city – adding up to more than £600 a month.

The problem was noticeable in locations near the capital that have been popular with commuters, such as Cambridge, Milton Keynes and Chelmsford, where the weekly shortfalls on a four-bed property were £68, £62 and £59 respectively.

There were striking shortfalls in areas such as Greater Glasgow (£82 a week on a four-bedroom home), Bristol (£71) and southern Greater Manchester (£53). The median national weekly rent gap on a four-bedroom home was £17. In a handful of areas, including Liverpool and Leeds, there was no rent gap for this size of property.

The weekly shortfall on two-bedroom private rented homes has followed a broadly similar pattern – highest in London, with significant gaps in places such as Bristol (£40), south-west Hertfordshire (£39) and Bath (£34). The median national weekly rent gap on this size of property was £8, according to the research.

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The CIH said that although ministers introduced a “targeted affordability fund” in 2014 to bridge the biggest gaps, the impact of this had been negligible because it had failed to meaningfully shrink the shortfall in the worst-hit areas.

The CIH called on the government to end the freeze on benefits. It estimated that the cost of realigning local housing allowance – which has lagged behind rent increases in many areas since 2013 – would be £1.2bn.

Matt Downie, the director of policy at the housing charity Crisis, said: “This report highlights just how much housing benefits for private renters are falling short of the levels needed, leaving many homeless people stuck in a desperate situation and putting yet more people at risk of homelessness.”

A government spokesperson said: “We spend £24bn a year on housing benefit each year. And since April we’ve provided additional, targeted housing support for low-income households by increasing more than 200 local housing allowance rates.

“Since 2011, we have provided a further £1bn in discretionary housing payment for local authorities to support vulnerable claimants with their housing costs.

“We have also delivered over 378,000 new affordable properties since 2010 and we are investing a further £9bn in affordable homes to buy and rent.”