Federated Core Trust III N-CSR Certified annual shareholder report (management investment company)

N-CSR Item 12(b) - Exhibits: Certifications

SECTION 906 CERTIFICATION

Pursuant to 18 U.S.C.§ 1350, the undersigned officers of Federated Core Trust III on behalf of Federated Project and Trade Finance Core Fund (the “Registrant”), hereby certify, to the best of our knowledge, that the Registrant’s Report on Form N-CSR for the period ended March 31, 2017 (the “Report”) fully complies with the requirements of Section 13(a) or 15(d), as applicable, of the Securities and Exchange Act of 1934 and that the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Registrant.

Dated: May 30, 2017

/s/ J. Christopher Donahue

J. Christopher Donahue

Title: President, Principal Executive Officer

Dated: May 30, 2017

/s/ Lori A. Hensler

Lori A. Hensler

Title: Treasurer, Principal Financial Officer

This certification is being furnished solely pursuant to 18 U.S.C.§ 1350 and is not being filed as part of the Report or as a separate disclosure document.