Housing federation says 3.7m young people will be forced to live with parents as housebuilding and wages lag behind

House prices in England will soar by 42% by 2020 and rents will rise by even more, according to a report from the National Housing Federation which warns of the "colossal strain" facing the generation born in the 1990s.

Many will remain trapped in their parents' homes as property prices continue to outstrip earnings, warns the NHF. It forecasts that 3.7 million young people will be living with their parents by 2020, as the rate of housebuilding fails to keep up with the rising population.

"By 2020 the price of a first-time buyer's home will increase by 42% to £245,165. Although wages for 22- to 29-year-olds will increase by 36% by 2020, this poses a huge challenge for those wishing to be homeowners. Low-earning young people would have to spend 16 times their average wage just to buy a home," said the NHF report.

Rents will be driven even higher as young adults are priced out of the property market. "NHF research shows that private rents are likely to be broadly stable through 2013, but could increase sharply, by about 6% a year, between 2015 and 2020 as interest rates and house prices rise. In 2020, rents are expected to be 46% higher than today. But when the new flood of young adults born in the noughties starts university or a new job, they could push rents even higher in a country already chronically short of decent housing."

The NHF figures chime with research by estate agency Savills, which last week revised its forecast for house price growth in the UK over the long term. It predicted that house prices would rise by a further 18% by 2017, compared with its forecast for 11.5% growth just nine months ago, as the turnaround in the property market has gathered pace faster than previously expected.

A separate report from housing charity Shelter blames the situation on the chronic shortage of housebuilding in the UK – and said one solution should be a massive programme of garden cities and new towns across the south of England. "We should be building 250,000 homes per year just to meet newly arising need, let alone start tackling the backlog, but in England we're currently delivering less than half that amount each year," said Shelter's head of policy and research, Roger Harding.

Despite the Help to Buy scheme announced in the budget earlier this year, Shelter warns that the amount of cash set aside for affordable housing is shrinking. "The new housing settlement set out in the 2013 comprehensive spending review is nowhere near sufficient to deliver the homes required. Overall government spending on affordable housing per year has been reduced, from £1.1bn in the current spending round to £950m per year after 2015. This is on top of the 60% cut in 2010."

Shelter said short-term measures by government could help boost housebuilding by 50,000 units a year, but says that over the long term Britain has to establish more new towns and garden cities to provide another 43,000 homes a year.

The NHF, which represents housing associations, said current policy measures had the potential to make matters worse for young adults. "Making mortgage deposits cheaper to aspirant homeowners without increasing housing supply doesn't tackle the root of the problem. It simply risks creating another housing bubble where prices are pushed up. These sticking-plaster answers will not heal our critically injured housing market."

It has launched a website, yestohomes.co.uk, to help people lobby local decision-makers for the provision of more affordable homes in their area.