State-backed Lloyds Banking Group has floated a larger-than-expected 35% chunk of TSB on the stock market.
The offer at 260p per share values the business at £1.3 billion and will see approximately 30% of the stock allocated to around 60,000 ordinary retail investors.
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It raises £455 million and sees TSB return to the market for the first time since 1995 when it merged with Lloyds.
The bank, which has 631 branches, has been re-launched after Lloyds was forced to offload the sites under European rules on state aid. It must dispose of its holdings in the business by the end of next year.
Lloyds said that, due to "significant investor demand" for TSB shares, the offer had been set above the initial size of 25%.
Shares rose by as much as 13% as conditional trading began, adding more than £170 million to TSB's market value.
Increasing expectations of an interest rate rise in the wake of remarks by Bank of England governor Mark Carney are thought to have spurred the demand as it would mean better profit margins for TSB.
The valuation is just above the mid-point of the 220p-290p expected price range announced earlier this month, and makes the brand worth 82% of its book value on the Lloyds balance sheet.
Lloyds chief executive Antonio Horta-Osorio said: "The successful initial public offering of TSB is an important further step for Lloyds Banking Group as we act to meet our commitments to the European Commission.
"The significant investor demand for shares in TSB, which reflects investors' confidence in the prospects for the business, has meant that we have been able to set the offer size at 35%.
"TSB has a national network of branches, a strong capital base, robust liquidity and significant economic protection against legacy issues.
"It is already operating on the UK high street and is proving to be a strong and effective challenger, further enhancing competition in the UK banking sector."
Lloyds Banking Group, which also includes Halifax and Bank of Scotland, remains 25% owned by the Treasury after it was rescued by the taxpayer during the financial crisis.
Lloyds had previously been preparing to sell the branches to the Co-operative Bank before the discovery of a £1.5 billion hole in the mutual's balance sheet caused the deal to collapse in April 2013.
The TSB offer comes amid the disappointing performance of other recent stock market flotations, with over-50s insurance and holidays group Saga among those to have seen shares declined while fashion retailer FatFace shelved its float plans.
Retail investors in the bank have been offered the sweetener of one free share for every 20 acquired, up to the value of £2,000, and held for a year after the float.
TSB, which has 4.5 million customers, is the seventh largest retail banking group in the UK. It is aiming to become a larger player in the current account market, growing from 4.2% to 6% over the next four to five years.
In its prospectus for potential investors, TSB said it did not expect to pay a dividend until the 2017 financial year. It reported profits of £172 million for 2013 and a figure of £60 million for the first three months of 2014.
The bank this month announced that staff were to be handed free shares worth £100 as part of a John Lewis-style reward scheme.
TSB, famous for its 1980s slogan "The bank that likes to say yes", traces its history back more than 200 years. It was first floated in 1986 before being swallowed up in the Lloyds merger nine years later.
Shore Capital said the implied pre-tax loss to Lloyds relative to the book value on the sale was £113 million.
The analysts have described the shares as "enticing" and "priced to go" with potential to take current account market share and expand mortgage distribution.
There was also potential for profitability to improve as the loan book builds and as interest rates rise, while Lloyds has offered an indemnity against past "legacy" issues.