The resignation will see Alan Schwartz, with whom Mr Spector had shared both roles, become the firm's sole president.
News of Mr Spector's departure follows a report from Standard & Poor's that asserted that the US bank's performance could be hit in the long-term as a result of the collapse of two funds managed by the firm, which had heavily invested in mortgage-backed securities.
Bear's chairman and chief executive James Cayne, who had been expected to be succeeded by Mr Spector, said: "In light of the recent events concerning BSAM's high-grade and enhanced-leverage funds, we have determined to make changes in our leadership structure."
Other appointments announced by Bear will see Samuel Molinaro become chief operating officer, while Jeffrey Mayer, co-head of the firm's fixed income unit, has been named to Bear's executive committee.
Recent figures show that Bear has lost 33 per cent of its stock market value this year.