The mutual also said that chief executive Niall Booker is to step aside from his duties as deputy boss of its parent, the Co-operative Group, after it set up a board committee to consider “third-party approaches” in relation to its plans to plug a £1.5 billion black hole in its balance sheet.
Earlier this week, a powerful group of investors urged the bank to convert more of its debt into shares, which could see majority ownership handed to its bondholders.
A spokesman said Bulmer, who has been with the Co-op for six years, was a “valued member of the business” but had decided to embark on a “new direction in his career”.
He added: “The business wanted Rod to stay as he’s been a key driver of the retail business, in particular its customer service.”
Co-op Bank, which last month admitted it would take several years to return to profit after slumping to a loss of £709.4 million in the first half of the year, is putting the finishing touches to plans that will see bondholders take heavy losses.
This move will generate about £1bn in capital, while a further £500m will be raised by selling its insurance businesses.
Bulmer, a former Santander UK executive, stepped up to the role of Co-op Bank’s acting chief in May when previous boss Barry Tootell resigned following a credit rating downgrade and the collapse of a £750m bid for 632 Lloyds branches.