Finch has been chief executive of the Centre for Cities since it was set up in 2005. Centre for Cities is an independent research and policy institute committed to helping Britain's cities improve their economic performance.
Previously, Finch was a policy adviser at HM Treasury for 11 years, advising both Labour and Conservative chancellors.
His responsibilities there included financial services as well as holding the post of First Secretary (economic affairs) during a three-year posting to the British Embassy in Washington.
Finch will move to Fishburn Hedges in August. He replaces Lucy Burns, who is taking a career break.
He said: 'This is a fascinating time to move into public affairs, with enormous policy challenges facing the new Government and difficult decisions facing private and non-profit players.'