A string of banks and utilities have sent letters under other names - a practice that is criticised by some as potentially misleading.
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TSB says it plans to grow its balance sheet by 40%-50% over the next five years.
The bank is aiming to become a larger player in the current account market, growing from 4.2% to 6% during that time.
Chief executive Paul Pester says it has already seen four to five as many people opening accounts every week since the TSB brand was re-launched last September than it had before.
Growth plans will also see TSB mortgages becoming available through brokers again from the start of next year.
TSB chief executive Paul Pester says there is "strong appetite" from investors for the flotation of TSB, with optimism in the UK and overseas over the strength of Britain's economic recovery.
Further tranches of TSB will be floated later, with Lloyds obliged to dispose of its remaining interest in the business by the end of 2015.
Details of the pricing of next month's offer have yet to be announced but reports put the book value of TSB at about £1.5 billion.
TSB is protected against compensation claims by its parent company, the bank's chief executive Paul Pester says.
TSB boss says it has "indemnity" against any claims for misconduct (PPI etc) until it lists. Lloyds would deal with compensation.
Misconduct claims for the missale of payment protection insurance (PPI) have cost the industry billions of pounds, according to estimates from the Financial Conduct Authority.
Lloyds chief executive Antonio Horta-Osario says TSB has "a strong balance sheet" and is well protected from the issues that caused the banking crisis.
– Lloyds chief executive Antonio Horta-Osorio
TSB has a national network of branches, a strong balance sheet 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.
TSB will offer less than a quarter of its available shares to ordinary members of the public, its boss has said.
ITV News Business Editor Joel Hills reports:
TSB is to list next month. 25% stake in the bank will be sold. Boss says 15 - 20% of shares will go to small (retail) investors.
Paul Pester says TSB will not pay a dividend until 2017 at earliest as all profits will be channelled into growing bank.
Shares in TSB will be available to members of the public via intermediaries, as well as institutional investors.
Lloyds Banking Group said the retail offering will contain an incentive allowing each retail investor to receive one free share for every 20 shares acquired up to the value of £2,000 and kept for a year after the float.
A 25% stake in TSB is to be floated on the stock market next month, owner Lloyds Banking Group has announced.
Barclays looks set to axe hundreds more jobs, mainly at its investment banking arm due to a strategic review that would be published on May 8.
The review is likely to mean hundreds of job cuts in addition to plans announced earlier this year by the bank to slash 12,000 posts.
Chief executive Antony Jenkins wrote to all group staff this morning, saying the company would answer questions including how best to "simplify" its operations and which of its businesses it should "focus on and invest in".
Mr Jenkins warned staff that regulation and the wider economy were having a "significant effect on some parts of our business which we need to address proactively".
He added: "The future for Barclays will be as a strong, focused, international bank. And the investment bank will continue to be part of that mix".
The bank Virgin Money has backed a call from consumer group Which? for a full competition inquiry into retail banking after "woeful" switching rates.