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Royal Bank of Scotland (LON:RBS) and Standard Chartered (LON:STAN) emerged as the worst performers in UK stress tests.
But shares in the sector rose as investors welcomed the fact that all the banks passed the tests carried out by the Bank of England (BoE).
The BoE found that partly state-owned RBS and Asia-focused Standard did not have sufficient capital to withstand potential shocks.
RBS did not meet the individual capital guidance from the Prudential Regulation Authority and Standard did not meet its tier one minimum capital requirement.
But the BoE did not demand any corrective action because the pair have already taken steps to improve their financial strength.
RBS said action taken so far in 2015 and its future plans to boost its capital meant it did not have to alter its capital plan as a result of the stress test.
Finance chief Ewen Stevenson said: "We're pleased with the progress we have made relative to the 2014 stress test, but recognise we still have much to do."
Standard said action that it had taken since the end of last year had strengthened the group's capital and leverage position.
"Standard believes the results demonstrate the group's resilience to a marked slowdown across the key markets in which it operates," it said.
Chief executive Bill Winters said: "We are operating at capital levels above current minimum regulatory requirements and have a number of additional levers at our disposal."
HSBC (LON:HSBA) was 8.6p up at 538.1p, Lloyds Banking Group (LON:LLOY) lifted 1.78p to 74.74p and Barclays (LON:BARC) rose 9.6p to 232.8p.
Royal Bank of Scotland (LON:RBS) and Standard Chartered (LON:STAN) increased 7.7p to 310.1p and 1.4p to 558.6p respectively.
David Lock at Deutsche Bank said the results showed that the banks were doing more to strengthen their finances than the sector as a whole.
"Overall, this reads more positively than we expected," he said.
The BoE did the stress test against a hypothetical background in which oil fell to US$38 a barrel and the global economy worsened.
It told all the banks and building societies, which included HSBC (LON:HSBA), Lloyds (LON:LLOY), Barclays (LON:BARC), Santander UK and Nationwide Building Society, that they would have to reserve capital to protect their UK exposures as part of a new measure imposed by the bank known as a counter-cyclical capital buffer.
Professor of finance economics at Warwick Business School, John Thanassoulis, noted that all the banks had passed the tests - albeit with RBS and Standard being given the benefit of the doubt on the basis that they had pledged to raise more funds.
“It was a test about tail risk, that is checking if the banks were robust to deal with, in particular, big problems in China and emerging markets," he said.
"This is why Standard Chartered, which does a lot of its business in those markets, was called up.
“The tests were brought in to try to give investors’ confidence in the height of the financial crisis – to show them that this was the lowest that it could go, a bottom below which banks would never fall.
“Now, moving out of the financial crisis there is a risk banks may stop listening. But I believe it is a useful exercise, it keeps the banks on their toes and gives the Bank of England a focus for their investigations with different banks.”