Exhibit 10.4

ARCH CAPITAL GROUP LTD.
Share Appreciation Right Agreement

AGREEMENT, made and entered into this 1st day of July, 2012, by and between Arch Capital Group Ltd. (the “Company”), a Bermuda company, and Maamoun Rajeh (the “SAR Holder”).

WHEREAS, the SAR Holder has been granted the following award under the Company’s 2012 Long Term Incentive and Share Award Plan (the “Plan”);

NOW, THEREFORE, in consideration of the premises and mutual covenants contained herein, and for other good and valuable consideration, the Company and the SAR Holder agree as follows:

(a)Grant. Pursuant to the provisions of the Plan, the terms of which are incorporated herein by reference, the Company hereby grants to the SAR Holder a Share Appreciation Right (the “SAR”) with respect to 19,071 Shares. The SAR represents a right to be paid, upon exercise of the SAR, an amount measured by (x) the difference between the Fair Market Value per Share on the date of exercise and the exercise price per Share of the SAR, multiplied by (y) the number of Shares with respect to which the SAR is exercised, with such amount to be paid in the form of Shares valued at their Fair Market Value on the date of exercise. The SAR is granted as of July 1, 2012 (the “Date of Grant”), and such grant is subject to the terms and conditions herein and the terms and conditions of the Plan. In the event there is any conflict between the terms of the Plan and this Agreement, the terms of the Plan shall control. Capitalized terms used herein but not defined shall have the meanings given to them in the Plan.

(b)Exercise Price. The exercise price of the SAR shall be equal to $39.69 per Share.

(c)Status of Shares. Upon issue, the shares received upon exercise of the SAR shall rank equally in all respects with the other Shares.

(d)Term of SAR. The SAR may be exercised only during the period (the “SAR Period”) set forth in paragraph (f) below and shall remain exercisable until the tenth anniversary of the Date of Grant. Thereafter, the SAR Holder shall cease to have any rights in respect thereof. The right to exercise the SAR shall be subject to sooner termination as provided in paragraph (j) below.

(e)No Rights of Shareholder. The SAR Holder shall not, by virtue hereof, be entitled to any rights of a shareholder in the Company, either at law or in equity. The SAR Holder acknowledges that there is no legal obligation of Arch Capital Group Ltd. or any of its Subsidiaries to offer a participation in the Plan and that the offer, even if made again in this form or in similar form in the future, will not create any obligation of Arch Capital Group Ltd. or any of its Subsidiaries to make any such offer.

(f)Exercisability. Except as otherwise set forth in paragraph (j) below, the SAR shall become exercisable in three equal annual installments on the first, second and third anniversaries of the Date of Grant, in each case subject to paragraph (j) below. Subject to paragraph (j) below, the SAR may be exercised at any time or from time to time during the SAR Period in regard to all or any portion of the SAR which is then exercisable, as may be adjusted pursuant to paragraph (g) below.

(g)Anti-dilution Adjustment. For the avoidance of doubt, the terms of Section 4(c) of the Plan, relating to anti-dilution adjustments, will apply to the SAR.

(h)Nontransferability. The SAR, or any interest therein, may not be assigned or otherwise transferred, disposed of or encumbered by the SAR Holder, other than by will or by the laws of descent and distribution. During the lifetime of the SAR Holder, the SAR shall be exercisable only by the SAR Holder or by his or her guardian or legal representative. Notwithstanding the foregoing, the SAR may be transferred by the SAR Holder to members of his or her “immediate family” or to a trust or other entity established for the exclusive benefit of solely one or more members of the SAR Holder’s “immediate family.” Any SAR held by the transferee will continue to be subject to the same terms and conditions that were applicable to the SAR immediately prior to the transfer, except that the SAR will be transferable by the transferee only by will or the laws of descent and distribution. For purposes hereof, “immediate family” means the SAR Holder’s children, stepchildren,

grandchildren, parents, stepparents, grandparents, spouse, siblings (including half brother and sisters), in laws, and relationships arising because of legal adoption.

(i)Exercise of SAR. In order to exercise the SAR, the SAR Holder shall submit to the Company an instrument specifying the whole number of Shares in respect of which the SAR is being exercised. Shares will be issued accordingly by the Company within 30 days. The payment upon a SAR exercise shall be solely the number of whole Shares calculated in paragraph (a) above. Fractional Shares shall be rounded down to the nearest whole Share with no cash consideration being paid upon exercise. Anything to the contrary herein notwithstanding, the Company shall not be obligated to issue any Shares hereunder if the issuance of such Shares would violate the provision of any applicable law, in which event the Company shall, as soon as practicable, take whatever action it reasonably can so that such Shares may be issued without resulting in such violations of law.

(j)Termination of Service.

1.In the event the SAR Holder ceases to be an employee of the Company due to his death or Permanent Disability (as defined in the Company’s Incentive Compensation Plan on the date hereof), the SAR, to the extent not already exercisable in full, shall become immediately exercisable in full and shall continue to be exercisable by the SAR Holder (or his Beneficiary or estate in the event of his death) for a period of three years following such termination of employment (but not beyond the SAR Period).

2.In the event of termination of employment (other than by the Company for Cause, as such term is defined in the Company’s Incentive Compensation Plan on the date hereof) after the attainment of Retirement Age (as defined in the Company’s Incentive Compensation Plan on the date hereof), the SAR shall continue to become exercisable on the schedule set forth in paragraph (f) above so long as the SAR Holder does not engage in any activity in competition with any activity of the Company or any of its Subsidiaries other than serving on the board of directors (or similar governing body) of another company or as a consultant for no more than 26 weeks per calendar year (“Competitive Activity”) and shall continue to be exercisable by the SAR Holder (or his Beneficiary or estate in the event of his death) for the remainder of the SAR Period. In the event the SAR Holder engages in a Competitive Activity, (A) the SAR, to the extent then exercisable, may be exercised for 30 days following the date on which the SAR Holder engages in such Competitive Activity (but not beyond the SAR Period) and (B) the SAR, to the extent then not exercisable, shall be immediately forfeited.

3.In the event the SAR Holder ceases to be an employee of the Company after a Change in Control (as defined below) due to termination (A) by the Company not for Cause or (B) by the SAR Holder for Good Reason (as defined below), in either case, on or before the second anniversary of the occurrence of the Change in Control, the SAR, to the extent not already exercisable in full, shall become immediately exercisable in full and shall continue to be exercisable by the SAR Holder for a period of 90 days following such termination of employment (but not beyond the SAR Period). “Good Reason” means, without the SAR Holder ’s written consent, (a) the material diminution of any material duties or responsibilities of the SAR Holder without the same being corrected within thirty (30) days after being given written notice thereof; or (b) a material reduction in the SAR Holder’s Base Salary without the same being corrected within thirty (30) days after being given written notice thereof.

4.In the event that the SAR Holder ceases to be an employee of the Company for any other reason, except due to a termination of the SAR Holder’s employment by the Company for Cause, (A) the SAR, to the extent then exercisable, may be exercised for 90 days following termination of employment (but not beyond the SAR Period) and (B) the SAR, to the extent then not exercisable, shall be immediately forfeited; provided that, in the event of a Redundancy (as defined below), the Committee, in its sole discretion, may, in accordance with its authority under the Plan, determine that the SAR, to the extent not exercisable, shall become exercisable and shall continue to be exercisable by the SAR Holder for a period of 90 days following such termination of employment (but not beyond the SAR Period).

5.In the event of a termination of the SAR Holder’s employment for Cause, the SAR shall immediately cease to be exercisable and shall be immediately forfeited.

6.For purposes of this SAR, service with any of the Company’s Subsidiaries (as defined in the Plan) shall be considered to be service with the Company.

7.    “Change in Control” shall mean:

(A) any person (within the meaning of the Securities Exchange Act of 1934, as amended (the “Exchange Act”), other than a Permitted Person, is or becomes the “beneficial owner” (as defined in Rule 13d-3

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under the Exchange Act), directly or indirectly, of Voting Securities representing 50% or more of the total voting power or value of all the then outstanding Voting Securities; or

(B) the individuals who, as of the date hereof, constitute the Board of Directors of the Company (the “Board”) together with those who become directors subsequent to such date and whose recommendation, election or nomination for election to the Board was approved by a vote of at least a majority of the directors then still in office who either were directors as of such date or whose recommendation, election or nomination for election was previously so approved, cease for any reason to constitute a majority of the members of the Board; or

(C) the consummation of a merger, consolidation, recapitalization, liquidation, sale or disposition by the Company of all or substantially all of the Company's assets, or reorganization of the Company, other than any such transaction which would (x) result in more than 50% of the total voting power and value represented by the voting securities of the surviving entity outstanding immediately after such transaction being beneficially owned by the former shareholders of the Company and (y) not otherwise be deemed a Change in Control under subparagraphs (A) or (B) of this paragraph.

“Permitted Persons” means (A) the Company; (B) any Related Party; or (C) any group (as defined in Rule 13b-3 under the Exchange Act) comprised of any or all of the foregoing.

“Related Party” means (A) a majority-owned subsidiary of the Company; (B) a trustee or other fiduciary holding securities under an employee benefit plan of the Company or any majority-owned subsidiary of the Company; or (C) any entity, 50% or more of the voting power of which is owned directly or indirectly by the shareholders of the Company in substantially the same proportion as their ownership of Voting Securities immediately prior to the transaction.

“Voting Security” means any security of the Company which carries the right to vote generally in the election of directors.

8.    “Redundancy” shall mean termination of employment by the Company due to its need to reduce the size of its workforce, including due to closure of a business or a particular workplace or change in business process. Whether a termination of employment is due to a "redundancy" shall be determined by the Committee in its sole and absolute discretion, such determination being final and binding on all parties hereto and all persons claiming through, in the name of or on behalf of such parties.

(k)     Obligations as to Capital. The Company agrees that it will at all times maintain authorized and unissued share capital sufficient to fulfill all of its obligations under the SAR.

(l)     Transfer of Shares. The SAR, the Shares issued hereunder, or any interest in either, may be sold, assigned, pledged, hypothecated, encumbered, or transferred or disposed of in any other manner, in whole or in part, only in compliance with the terms, conditions and restrictions as set forth in the governing instruments of the Company, applicable United States federal and state securities laws and the terms and conditions hereof.

(m)    Expenses of Issuance of Shares. The issuance of stock certificates upon the exercise of the SAR in whole or in part, shall be without charge to the SAR Holder. The Company shall pay any issuance, stamp or documentary taxes (other than transfer taxes) or charges imposed by any governmental body, agency or official (other than income taxes) by reason of the exercise of the SAR in whole or in part or the resulting issuance of Shares hereunder.

(n)    Withholding. No later than the date of exercise of the SAR granted hereunder, the SAR Holder shall pay to the Company or make arrangements satisfactory to the Committee regarding payment of any federal, state or local taxes of any kind required by law to be withheld upon the exercise of such SAR and the Company shall, to the extent permitted or required by law, have the right to deduct from any payment of any kind otherwise due to the SAR Holder, federal, state and local taxes of any kind required by law to be withheld upon the exercise of such SAR.

(o)    References. References herein to rights and obligations of the SAR Holder shall apply, where appropriate, to the SAR Holder’s legal representative or estate without regard to whether specific reference to such legal representative or estate is contained in a particular provision of this SAR.

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(p)    Notices. Any notice required or permitted to be given under this agreement shall be in writing and shall be deemed to have been given when delivered personally or by courier, or sent by certified or registered mail, postage prepaid, return receipt requested, duly addressed to the party concerned at the address indicated below or to such changed address as such party may subsequently by similar process give notice of:

If to the Company:

Arch Capital Group Ltd.:

Wessex House
45 Reid Street
Hamilton HM 12 Bermuda
Attn: Secretary

If to the SAR Holder:

The last address delivered to the Company by the SAR Holder in the manner set forth herein.

(q)    Governing Law. This agreement shall be governed by and construed in accordance with the laws of New York, without giving effect to principles of conflict of laws thereof.

(r)    Entire Agreement. This agreement and the Plan constitute the entire agreement among the parties relating to the subject matter hereof, and any previous agreement or understanding among the parties with respect thereto is superseded by this agreement and the Plan.

(s)    Counterparts. This agreement may be executed in two counterparts, each of which shall constitute one and the same instrument.

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IN WITNESS WHEREOF, the undersigned have executed this agreement as of the Date of Grant.

ARCH CAPITAL GROUP LTD.
By:	/s/ Dawna Ferguson
	Name:	Dawna Ferguson
	Title:	Secretary
	/s/ Maamoun Rajeh
	Maamoun Rajeh

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