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)


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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


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                                      -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;


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                                      -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


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                                      -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):


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      (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.


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     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]




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     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.




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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


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                                   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