Cohen was born in Egypt on 1 August 1945, the son of Michael Mourad Cohen, a banker, and Sonia Douek. His family are Sephardi Jews, though his mother, Sonia Douek, is British by nationality. In 1957, following the Suez Crisis, Cohen's family was forced to abandon all their assets and flee Egyptian President Nassar's persecution of Jews. The family (including younger brother Andre) moved to England. His father started his own import/export business, but having arrived with just £10, life was a struggle.