Stevenson worked as a market researcher for Conrad Jameson Associates in Lower Belgrave Street, Belgrave, London around 1968.
He is a businessman who was Chair of the SRU Group of Companies from 1972 to 1996. He became Chair of Halifax plc in 1999 and when they merged with Bank of Scotland he became Chair of the merged group, HBOS plc, in 2001. He was also Chair of Pearson PLC. He is Chair of the House of Lords Appointments Commission and Chancellor of the University of the Arts London.