This is a commonly misunderstood topic. The class of medical depends upon the "privileges" being exercised, not the certificate held.
E.g., a person holding a FAA ATP certificate can fly privately, without compensation, with a third class medical. Or fly for compensation or hire, part 91 and/or part 135, with a second class medical. The employer (part 91 or 135), contractor or insurance may request a first class medical -- and a college degree -- but that is not required by the FAR.
Cf. 61.23 and
http://www.faa.gov/licenses_certificates/medical_certification/faq/.
It is different in other countries and aviation rules.