EXHIBIT 10.4

AMENDMENT TO EMPLOYMENT AGREEMENT

Amendment (“Amendment”), dated July 25, 2012, to the Employment Agreement, dated as of June 5, 2009 (the “Agreement”), among, Arch Insurance Group Inc., a Delaware corporation (the “Company”) and David McElroy (the “Executive”). Capitalized terms used without definition herein have the meanings given to them in the Agreement.

NOW, THEREFORE, in consideration of the premises and mutual covenants contained herein, the parties have agreed to amend the Agreement as follows:

1. The first sentence of Section 3.01 shall be hereby amended and restated as follows:

“During the Employment Period, the Executive shall serve as Chairman and Chief Executive Officer of the Company and shall have such responsibilities, powers and duties as may from time to time be prescribed by the Board of Directors of the Company; provided that such responsibilities, powers and duties are substantially consistent with those customarily assigned to individuals serving in such position at comparable companies or as may be reasonably required by the conduct of the business of the Company.”

2. The first sentence of Section 4.01 shall be hereby amended and restated as follows:

“During the Employment Period, the Executive’s base salary will be $600,000 per annum (the “Base Salary”).”

3. SECTION 4.07 shall be hereby added to the Agreement as follows:

“SECTION 4.07. Retention Shared-Based Awards. Subject to approval (“Board Approval”) by the Board of Directors of Arch Capital Group Ltd. (“Parent”), the Parent shall grant to the Executive share appreciation rights with respect to 25,000 of the Company’s common shares at an exercise price equal to the closing market price on the date of such Board Approval (the “Grant Date”), which award will vest in full on the fifth anniversary of the Grant Date. Subject to Board Approval, on the Grant Date, the Company shall also grant to the Executive 25,000 restricted common share units of the Company, which will vest in full on the fifth anniversary of the Grant Date and the shares under such award would not be transferred to him until his separation from service, as provided in the applicable award agreement. The other terms of such share-based awards shall be as set forth in the applicable award agreements as approved by the Board of Directors of the Company.”

5. SECTION 5.01 shall be hereby amended and restated as follows:

“SECTION 5.01. Term. The Employment Period will terminate on third anniversary of the date hereof; provided that (a) the Employment Period shall terminate prior to such date upon the Executive’s death or Permanent Disability, (b) the Employment

Period may be terminated by the Companies for any reason prior to such date, and (c) the Employment Period may be terminated by the Executive at any time prior to such date, if such termination shall be for Good Reason. In addition, this Agreement will be automatically extended on the same terms and conditions for successive one year periods following the original term until either the Companies or the Executive, at least ninety (90) days prior to the expiration of the original term or any extended term, shall give written notice of their intention not to renew the Agreement.”

6. ARTICLE 9 shall be hereby amended as restated as follows:

“ARTICLE 9

NONCOMPETITION AND NONSOLICITATION

SECTION 9.01. Noncompetition. The Executive acknowledges that during his employment with the Company, he will become familiar with trade secrets and other Confidential Information concerning the Company and its Affiliates and their respective predecessors, and that his services will be of special, unique and extraordinary value to the Company. In addition, the Executive hereby agrees that at any time during the Employment Period, and for a period ending two (2) year after the termination of the Executive’s employment (the “Noncompetition Period”), he will not directly or indirectly own, manage, control, participate in, consult with, render services for or in any manner engage in any business competing with the businesses of the Company or its Affiliates as such businesses exist or are in process as of the date of termination, within any geographical area in which the Company or its Affiliates engage or plan to engage in such businesses. It shall not be considered a violation of this Section 9.01 for the Executive to be a passive owner of not more than 2% of the outstanding stock of any class of a corporation which is publicly traded, so long as the Executive has no active participation in the business of such corporation.

SECTION 9.02. Nonsolicitation. The Executive acknowledges that during his employment with the Company, he will become familiar with trade secrets and other Confidential Information concerning the Company, its Affiliates and their respective predecessors, and that his services will be of special, unique and extraordinary value to the Company. The Executive hereby agrees that (a) during the Employment Period and for a period of two (2) year after the date of termination of employment (the “Nonsolicitation Period”) the Executive will not, directly or indirectly, induce or attempt to induce any employee of the Company or its Affiliates to leave the employ of the Company or its Affiliates, or in any way interfere with the relationship between the Company or its Affiliates and any employee thereof or otherwise employ or receive the services of any individual who was an employee of the Company or its Affiliates at the Date of Termination or within the six-month period prior thereto, and (b) during the Nonsolicitation Period, the Executive will not induce or attempt to induce any customer, supplier, client, insured, reinsured, reinsurer, broker, licensee or other business relation of

the Company or its Affiliates to cease doing business with the Company or its Affiliates.

SECTION 9.03. Enforcement. If, at the enforcement of Sections 9.01 or 9.02, a court holds that the duration or scope stated therein are unreasonable under circumstances then existing, the parties agree that the maximum duration and scope reasonable under such circumstances will be substituted for the stated duration or scope and that the court will be permitted to revise the restrictions contained in this Article 9 to cover the maximum duration and scope permitted by law.”

7. All other provisions of the Agreement shall remain in full force and effect. This amendment shall be governed by and construed in accordance with the laws of New York, without giving effect to principles of conflict of laws, and may be executed in two or more counterparts, each of which shall constitute one and the same instrument.

IN WITNESS WHEREOF, the parties hereto have executed this Amendment as of the date and year first above written.

	ARCH INSURANCE GROUP INC.


	By:	/s/ Mark D. Lyons
	Printed Name:	Mark D. Lyons
	Title:	Chairman and Chief Executive Officer


	/s/ David McElroy
	David McElroy