Alterra Capital 10-K 2008 FY Annual report

EXHIBIT 32.1

CERTIFICATION PURSUANT TO 18 U.S.C. SECTION 1350 OF THE

CHIEF EXECUTIVE OFFICER

OF MAX CAPITAL GROUP LTD.

This certification is provided pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, 18 U.S.C. § 1350, and accompanies the annual report on Form 10-K (the “Form 10-K”) for the year ended December 31, 2008 of Max Capital Group Ltd. (the “Issuer”).

I, W. Marston Becker, the Chief Executive Officer of the Issuer certify that:

(i) the Form 10-K 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-K 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 SEC or its staff upon request.

Date: February 19, 2009

/s/    W. MARSTON BECKER

Name:    W. Marston Becker