He will replace Sir Ronald Cohen, who announced in March that he would stand down as founding chair of the £600m social lender at the end of 2013.
McGrath was chair of Prudential from 2008 to 2012 and chair of the alternative investment management business the Man Group between 2000 and 2007.
He is also a member of the advisory board of the social investor Bridges Ventures and is on the advisory council of the Centre for Social Justice, the think tank set up by Iain Duncan Smith, the work and pensions secretary.
McGrath was earlier this year given the Beacon Award for City Philanthropy, which recognises philanthropy that has inspired others in the financial, professional or business sectors, and is awarded by the Beacon Fellowship Charitable Trust.
McGrath, who will not be paid for his new role, said in a statement that he had been involved in social investment for the past 10 years. "I am excited to have this opportunity to contribute to Big Society Capital’s leadership in driving entrepreneurship and innovation in tackling social issues," he said.
He will join the board immediately before taking over as chair on 1 January.
Cohen said in a statement that McGrath had extensive philanthropic and investment experience. "Big Society Capital has developed into a powerful force driving social improvement in the UK, and I have been privileged to be its founding chair," said Cohen.
Big Society Capital, which was launched in April 2012, is expected to receive £600m over five years from dormant bank accounts and from the UK’s largest high-street banks to invest in social investment finance providers.