David Willetts, the universities and science minister, announced the appointment today.
The government is pursuing plans to sell the SLC’s pre-2012 loans, with banks and pension funds among potential buyers.
Mr Brodie retired in July 2013 from the role of vice-chairman of UBS Investment Bank.
Besides his Sussex role, he is chairman of the Chelsea and Westminster Health Charity, an NHS charity supporting the Chelsea and Westminster Hospital.
Mr Willetts said Mr Brodie “brings with him a wealth of experience at the highest levels from across the public, private and higher education sectors. His appointment will provide support and leadership to the SLC as it embarks on a major programme of transformation, replacing its ageing core ICT systems and fully embracing digital technologies to improve radically the way it delivers services to customers.”
Mr Willetts said he also wanted to extend his “deepest appreciation” to Dame Glynis Breakwell, the University of Bath vice-chancellor, “for her work and commitment over the past three months as the interim chair of the SLC”.
Ed Smith, the University of Birmingham pro-chancellor, had stepped down as chair of the SLC in the autumn.
The announcement comes amid a heated political row over claims Downing Street has “politicised” quango appointments by pushing out non-Conservatives. It follows comments by Baroness Sally Morgan, whose term as Ofsted chair is not being renewed by the government.
The Department for Business Innovation and Skills said in a press release on Mr Brodie’s appointment that in accordance with the Nolan principles on standards in public life “there is a requirement for appointees’ political activity (if any declared) to be made public”. It said Mr Brodie “has not been involved in any political activity in the last five years”.
