Alterra Capital 10-Q 2005 Q2 Quarterly report

EXHIBIT 32.1

CERTIFICATION OF

CHIEF EXECUTIVE OFFICER

OF MAX RE CAPITAL LTD.

This certification is provided pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, 18 U.S.C. § 1350, and accompanies the quarterly report on Form 10-Q (the “Form 10-Q”) for the quarter ended June 30, 2005 of Max Re Capital Ltd. (the “Issuer”).

I, Robert J. Cooney, the Chief Executive Officer of Issuer certify that:

(i) the Form 10-Q fully complies with the requirements of Section 13(a) or Section 15(d) of the Securities Exchange Act of 1934 (15 U.S.C. 78m(a) or 78o(d)); and

(ii) the information contained in the Form 10-Q fairly presents, in all material respects, the financial condition and results of operations of the Issuer.

A signed original of this written statement required by Section 906, or other document authenticating, acknowledging, or otherwise adopting the signature that appears in typed form within the electronic version of this written statement required by Section 906, has been provided to the Issuer and will be retained by the Issuer and furnished to the Securities and Exchange Commission or its staff upon request.

Dated: August 1, 2005

/s/ Robert J. Cooney

Name: Robert J. Cooney