**Exhibit 31.1****CERTIFICATION BY CHIEF EXECUTIVE OFFICER**I, Stephen L. Way, certify that:1. I have reviewed this annual report on Form 10-K of HCC Insurance Holdings, Inc.; 2. Based on my knowledge, this report does not contain any untrue statement of a materialfact or omit to state a material fact necessary in order to make the statements made, inlight of the circumstances under which such statements were made, not misleading withrespect to the period covered by this report; 3. Based on my knowledge, the financial statements, and other financial informationincluded in this report, fairly present in all material respects the financial condition,results of operations and cash flows of the registrant as of, and for, the periodspresented in this report; 4. The registrant’s other certifying officer and I are responsible for establishing andmaintaining disclosure controls and procedures (as defined in Exchange Act Rules 13a-15(e)and 15d-15(e)) and internal control over financial reporting (as defined in Exchange ActRules 13a-15(f) and 15d-15(f)) for the registrant and have: a. Designed such disclosure controls and procedures, or caused suchdisclosure controls and procedures to be designed under our supervision, to ensurethat material information relating to the registrant, including its consolidatedsubsidiaries, is made known to us by others within those entities, particularlyduring the period in which this report is being prepared;  b. Designed such internal control over financial reporting, or caused suchinternal control over financial reporting to be designed under our supervision, toprovide reasonable assurance regarding the reliability of financial reporting andthe preparation of financial statements for external purposes in accordance withgenerally accepted accounting principles;  c. Evaluated the effectiveness of the registrant’s disclosure controls andprocedures and presented in this report our conclusions about the effectiveness ofthe disclosure controls and procedures, as of the end of the period covered by thisreport based on such evaluation; and  d. Disclosed in this report any change in the registrant’s internal controlover financial reporting that occurred during the registrant’s most recent fiscalquarter (the registrant’s fourth fiscal quarter in the case of an annual report)that has materially affected, or is reasonably likely to materially affect, theregistrant’s internal control over financial reporting; and5. The registrant’s other certifying officer and I have disclosed, based on our mostrecent evaluation of internal control over financial reporting, to the registrant’sauditors and the audit committee of registrant’s board of directors (or persons performingequivalent functions): a. All significant deficiencies and material weaknesses in the design oroperation of internal control over financial reporting which are reasonably likelyto adversely affect the registrant’s ability to record, process, summarize andreport financial information; and  b. Any fraud, whether or not material, that involves management or otheremployees who have a significant role in the registrant’s internal control overfinancial reporting.     March 15, 2006   /s/ Stephen L. Way     (Date)   Stephen L. Way, Chairman of the Board,    Chief Executive Officer and President