Exhibit 10.4 EXECUTION COPY ARCH CAPITAL GROUP LTD. 20 Horseneck Lane Greenwich, CT 06830 November 20, 2001 Insurance Private Equity Investors, Warburg Pincus Private Equity VIII, L.P. L.L.C. (the "GE Private Equity Purchaser") Warburg Pincus International Partners, L.P. c/o GE Asset Management Warburg Pincus Netherlands International Incorporated Partners I, C.V. 3003 Summer Street Stamford, CT 06905 Warburg Pincus Netherlands International Partners II, C.V. (collectively, "Warburg") 466 Lexington Avenue New York, NY 10017 HFCP IV (Bermuda), L.P. ("H&F") c/o Hellman & Friedman LLC One Maritime Plaza Suite 1200 San Francisco, CA 94111 Ladies and Gentlemen: This letter agreement (this "Agreement") confirms the agreement reached today among each of the parties signatories hereto regarding the participation of the GE Private Equity Purchaser in the purchase of a portion of the Securities, as contemplated by, and on the terms set forth in, this Agreement and the Subscription Agreement dated as of October 24, 2001, as amended (the "Subscription Agreement") by and among Arch Capital Group Ltd. ("Arch"), Warburg and H&F (the "Original Signatories"), and certain other matters in connection therewith. Capitalized terms used herein without definition shall have the meanings ascribed to such terms in the Subscription Agreement. 1. Investment by GE Purchaser. Warburg hereby assigns, without recourse or warranty by it, to the GE Private Equity Purchaser (as set forth in Schedule 1 hereto) -2- the right, and obligation, to purchase an aggregate of $50,000,000 of the Securities on the terms and conditions set forth in the Subscription Agreement (except as explicitly modified hereby), as a Purchaser (as defined in the Subscription Agreement). The GE Private Equity Purchaser acknowledges that its investment will be required to be made on the Closing Date, simultaneously with the investments being made by the Original Signatories. The GE Private Equity Purchaser shall become a "Purchaser" under the Subscription Agreement and an "Investor" under the Shareholders Agreement (solely for purposes of Sections 3.4, 5.1, 5.2 and 5.3 and Articles II, IV and VIII thereof, and the provisions implementing the provisions described in paragraph 2 below); provided that: (a) Warburg and H&F shall jointly have the sole right (on behalf of themselves and all other Purchasers) to make any and all determinations with respect to, or to take any and all actions necessary to effectuate the provisions of, Section B of the Subscription Agreement (including the right to approve any amendment or acceleration of, or to waive compliance by Arch with, any of the terms thereof), provided that the consequences of such determinations and actions by Warburg and H&F do not apply differently to the GE Private Equity Purchaser than to Warburg and H&F (or, if they apply differently, it is because of differences in the treatment of Warburg and H&F as opposed to other Purchasers existing in the Subscription Agreement (as modified by this Agreement) and such differences are not made more adverse to the GE Private Equity Purchaser or more favorable to Warburg and H&F as a result of such determination or action); (b) Warburg and H&F shall have the sole right to determine whether each condition for the Purchasers contained in Section C of the Subscription Agreement is satisfied; (c) the failure of the conditions set forth in Section C.2 of the Subscription Agreement due to any breach by the GE Private Equity Purchaser of any representation, warranty or covenant shall not affect the obligation of the Company to sell the Securities on the Closing Date to either Warburg or H&F (d) the GE Private Equity Purchaser shall be subject to Section D.1 and D.2 of the Subscription Agreement, including the covenants thereunder; (e) the GE Private Equity Purchaser shall have no rights (including no right to consent to any action proposed to be taken by Arch under, or any right to waive compliance by Arch with, any covenant or agreement) as a "Purchaser" under Section D.4 of the Subscription Agreement, it being acknowledged that each GE Private Equity Purchaser shall, however, have the obligations of a "Purchaser" under Sections D.4(d), (g) and (i) thereof; provided that any information provided to the Company -3- pursuant to Section D.4(g) shall be held confidentially and not used for any purpose other than as set forth in Section D.4(g); (f) the GE Private Equity Purchaser shall not be considered an "original signatory" to the Subscription Agreement for purposes of Section E.6 thereof, provided that no amendment, modification or waiver of Section E of the Subscription Agreement shall affect the GE Private Equity Purchaser differently than Warburg and H&F (or, if they affect them differently, it is because of differences in the treatment of Warburg and H&F as opposed to other Purchasers existing in the Subscription Agreement (as modified by this Agreement) and such differences are not made more adverse to the GE Private Equity Purchaser or more favorable to Warburg and H&F as a result of such determination or action); (g) no consent of the GE Private Equity Purchaser shall be required to effect any modification or amendment to the Subscription Agreement (including, without limitation, Schedules A and B, and Exhibits I, II and III thereto), unless such amendment or modification affects the GE Private Equity Purchaser differently than Warburg and H&F (or, if they affect them differently, it is because of differences in the treatment of Warburg and H&F as opposed to other Purchasers existing in the Subscription Agreement (as modified by this Agreement) and such differences are not made more adverse to the GE Private Equity Purchaser or more favorable to Warburg and H&F as a result of such determination or action); (h) the GE Private Equity Purchaser shall have no rights under Section F.2 (except the right to be reimbursed by the Company, together with the other GE Private Equity Purchaser, for up to an aggregate of up to $50,000 in counsel fees), and no right to assign under Section F.4, of the Subscription Agreement (except that the GE Private Equity Purchaser may assign its rights and obligations under the Subscription Agreement in connection with a transfer of Securities to (i) any person or entity that directly or indirectly through one or more intermediaries controls, or is controlled by or under common control with, such GE Private Equity Purchaser, (ii) an entity over which such GE Private Equity Purchaser has management rights, or (iii) if such GE Private Equity Purchaser is affiliated with a trustee of a pension trust, to a successor trustee or trust, in each case so long as such transferee becomes a party to, and bound by, this Agreement, Amendment No. 1 to the Subscription Agreement and the Shareholders Agreement (a "Permitted Transferee"); (i) Amendment No. 1 to the Subscription Agreement will contain a covenant by the Company to conduct its business in compliance with law including, without limitation, the Foreign Corrupt Practices Act; -4- (j) so long as the GE Private Equity Purchaser owns any equity interest in the Company, the Company agrees to furnish to the GE Private Equity Purchaser a copy of its annual and quarterly reports filed under the Securities and Exchange Act of 1934; and (k) for the avoidance of doubt, the GE Private Equity Purchaser shall become parties to the Shareholders Agreement as an "Investor" solely for purposes of Sections 3.4, 5.1, 5.2 and 5.3 and Articles II, IV and VIII thereof and the provisions thereof implementing the provisions of paragraph 2 below; it being further understood that Warburg and H&F can consent on behalf of all other Investors to (A) any amendment or modification whatsoever of the Sections of the Shareholders Agreement that do not apply to the GE Private Equity Purchaser and (B) any amendment or modification of the Sections of the Shareholders Agreement that do apply to the GE Private Equity Purchaser, so long as in the case of clause (B), such amendment or modification does not affect the GE Private Equity Purchaser differently than Warburg and H&F (or, if it affects them differently, it is because of differences in the treatment of Warburg and H&F as opposed to other Purchasers existing in the Subscription Agreement or the Shareholders Agreement (as modified by this Agreement) and such differences are not made more adverse to the GE Private Equity Purchaser or more favorable to Warburg and H&F as a result of such determination or action). 2. Registration; Tag-Along. Arch, the Original Signatories and the GE Private Equity Purchaser agree that the Shareholders Agreement will be amended and restated to provide that: (a) if the GE Private Equity Purchaser exercises its right under Section 4.3 thereof, any cutback pursuant to Section 4.4 thereof will treat the GE Private Equity Purchaser at least as favorably as Warburg and H&F (i.e., the GE Private Equity Purchaser will have priority under clause (b), and not under clause (c), thereof); (b) Warburg and H&F agree that in the case of a Warburg Demand or an H&F Demand involving an underwritten public offering, when selecting an underwriter, the consent of the General Electric Pension Trust will be required for the selection of any underwriter in which the General Electric Company has a direct or indirect interest of 5% or more if the GE Private Equity Purchaser will be a participating seller in the offering; (c) the GE Private Equity Purchasers will have the obligations of a Selling Investor (considered together for the purpose of determining whether the Selling Investor has met the $50 million threshold), and the rights of a Tag-Along Investor, under Section 5.1 thereof; provided that (1) each Investor participating in a transaction -5- under Section 5.1 shall only be responsible for its pro rata portion of any indemnification (except in respect of representations specifically relating to such Investor) and (2) in the event that H&F is a Selling Investor, the GE Private Equity Purchasers shall only be permitted to elect to participate as a Tag-Along Investor if Warburg so elects; (d) the GE Private Equity Purchaser shall be subject to the restrictions of Section 5.2 of the Shareholders Agreement with respect to the Securities acquired by it under the Subscription Agreement and any securities acquired in respect thereof, to the same extent that Warburg and H&F are restricted with respect to the Securities acquired by them under the Subscription Agreement and any securities acquired in respect thereof, provided that the GE Private Equity Purchaser will be permitted to transfer Securities to a Permitted Transferee. "Tag-Along Investor," "Third Party Sale" and "Selling Investor" have the meanings given to them in the Shareholders Agreement and to the extent necessary the term "Tag-Along Investor" shall be deemed to include more than one party. 3. Further Assurances. Subject to the terms and conditions of this Agreement, each of the parties hereto agrees to use its reasonable best efforts to take, or cause to be taken, all action, and to do, or cause to be done, all things necessary or desirable under applicable legal requirements, to consummate and make effective the transactions contemplated by this Agreement. If at any time after the Closing Date, any further action is necessary or desirable to carry out the purposes of this Agreement, the parties hereto shall use their reasonable best efforts to take or cause to be taken all such necessary or desirable action and execute, and deliver and file, or cause to be executed, delivered and filed, all necessary or desirable documentation. The GE Private Equity Purchaser agrees (to the full extent of their current or future ownership of securities of Arch) to vote in favor of all matters to be submitted to shareholders of Arch in connection with the foregoing or the transactions contemplated by the Subscription Agreement (and the grants of any shares or options contemplated thereby or in connection therewith). Each of the parties will consult with each other with respect to the issuance of any press release or public announcement with respect to the foregoing. 4. Notices. All notices or other communications given or made hereunder shall be validly given or made if in writing and delivered by facsimile transmission or in person at, or mailed by registered or certified mail, return receipt requested, postage prepaid, to, the addresses (and shall be deemed effective at the time of receipt thereof): -6- (a) If to Arch: Arch Capital Group Ltd. 20 Horseneck Lane Greenwich, CT 06830 Attention: Peter Appel, President and Chief Executive Officer Facsimile: (203) 861-7240 (b) If to the GE Private Equity Purchaser: c/o GE Asset Management Incorporated 3003 Summer Street Stamford, CT 06905 Attention: Michael M. Pastore, Esq. or to such other address as the party to whom notice is to be given may have previously furnished notice in writing to the other in the manner set forth above. A notice hereunder shall not be deemed given until copies thereof are given as contemplated above. Notices to all other parties hereto shall be given in accordance with the Subscription Agreement. 5. Entire Agreement; Amendment. This Agreement, together with the Subscription Agreement, Amendment No. 1 to the Subscription Agreement (including the exhibits thereto) the Shareholders Agreement contains all of the terms agreed upon by the parties with respect to the subject matter hereof. This Agreement may be amended or the provisions thereof waived only by a written instrument signed by the party against whom enforcement of any waiver, change, modification, extension or discharge is sought. 6. Headings. The headings of the sections of this Agreement are inserted for convenience only and shall not constitute a part hereof. 7. Assignment. This Agreement and all of the provisions hereof shall be binding upon and inure to the benefit of the parties hereto and their respective permitted assigns, but neither this Agreement nor any of the rights, interests or obligations hereunder shall be assigned by any party without the prior written consent of the other party; provided that this Agreement may be assigned by a Purchaser consistent with an assignment in accordance with Section F.4 of the Subscription Agreement and Section 1(h) of this Agreement, so long as the assignee executes an agreement in the form of this Agreement. 8. Severability. In the event that any provision or any part of this Agreement is held to be illegal, invalid or unenforceable, such illegality, invalidity or unenforceability shall not effect the validity or enforceability of any other provision or part thereof. -7- 9. Governing Law. This Agreement shall be governed in all respects, including validity, interpretation and effect, by the substantive laws of the State of New York, without giving effect to principles of conflicts of laws. 10. Counterparts. This Agreement and any instrument delivered in connection herewith may be executed in any number of counterparts with the same effect as if the signatures on all counterparts are upon the same instrument. [Signature pages follow] Please confirm that the foregoing is in accordance with your understanding by signing and returning to us the duplicate enclosed copy of this Agreement. Very truly yours, ARCH CAPITAL GROUP LTD. By: /s/ Louis T. Petrillo ------------------------------------- Name: Louis T. Petrillo Title: Senior Vice President and General Counsel Agreed to and Accepted As of the Date First Above Written: INSURANCE PRIVATE EQUITY INVESTORS, L.L.C. By: GE Asset Management Incorporated, its Manager, By: /s/ Patrick McNeela -------------------------------------------- Name: Patrick McNeela Title: Vice President Notice Information for Insurance Private Equity Investors, L.L.C.: c/o GE Asset Management Incorporated 3003 Summer Street Stamford, CT 06905 Attention: Michael M. Pastore, Esq. HFCP IV (BERMUDA), L.P. By: H&F Investors (Bermuda) IV, L.P., its General Partner By: H&F Corporate Investors IV (Bermuda), Ltd. its General Partner By: /s/ David R. Tunnell ---------------------------------------------- Name: David R. Tunnell Title: Authorized Signatory WARBURG PINCUS PRIVATE EQUITY VIII, L.P. WARBURG PINCUS INTERNATIONAL PARTNERS, L.P. WARBURG PINCUS NETHERLANDS INTERNATIONAL PARTNERS I, C.V. WARBURG PINCUS NETHERLANDS INTERNATIONAL PARTNERS II, C.V. By: Warburg, Pincus & Co., its General Partner By: /s/ Kewsong Lee ----------------------------------- Name: Kewsong Lee Title: Partner SCHEDULE 1 Amount Warburg Purchaser Assigned ----------------- -------- Warburg Pincus Private Equity VIII, L.P. $25,000,000 Warburg Pincus International Partners, L.P. $24,000,000 Warburg Pincus Netherlands International Partners I, C.V. $600,000 Warburg Pincus Netherlands International Partners II, C.V. $400,000 ================= Total $50,000,000