Exhibit 32.1

Certification Pursuant to Chapter 63, Title 18 United States Code §1350

As Adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002

In connection with the Quarterly Report of Arch Capital Group Ltd. (the “Company”) on Form 10-Q for the period ending September 30, 2008 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), Constantine Iordanou, as President and Chief Executive Officer of the Company, certifies pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

(1)the Report fully complies with the requirements of section 13(a) of the Securities Exchange Act of 1934; and

(2)the information contained in such report fairly presents, in all material respects, the financial condition and results of operations of the Company.

Date: November 10, 2008

/s/ Constantine Iordanou

Constantine Iordanou

President and Chief Executive Officer

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 Arch Capital Group Ltd. and will be retained by Arch Capital Group Ltd. and furnished to the Securities and Exchange Commission or its staff upon request.