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                                                                    EXHIBIT 10.1

                                 AMENDMENT NO. 4

          Reference is made to the Shareholders Agreement dated as of November
20, 2001, as amended (the "SHAREHOLDERS AGREEMENT"), by and among Arch Capital
Group Ltd., a company organized under the laws of Bermuda (the "COMPANY"), the
parties listed on the signature pages hereto and certain other persons. This
amendment ("AMENDMENT") to the Shareholders Agreement is made as of September
16, 2002.

          1.   Section 1.1 of the Shareholders Agreement is amended to add the
following definitions:

          "H&F REPRESENTATIVE" shall mean the individual designated by the H&F
     Purchasers to act on behalf of H&F, as shareholders of the Company, under
     Section 3.3, which individual is identified to the Company in writing from
     time to time; PROVIDED that the H&F Representative shall not be an H&F
     Director.

          "WARBURG REPRESENTATIVE" shall mean the individual designated by the
     Warburg Purchasers to act on behalf of Warburg, as shareholders of the
     Company, under Section 3.3, which individual is identified to the Company
     in writing from time to time; PROVIDED that the Warburg Representative
     shall not be a Warburg Director.

          2.   The first paragraph of Section 3.3 (preceding clauses (1) - (17))
of the Shareholders Agreement is amended and restated in its entirety as
follows:

          Section 3.3. INVESTOR PROTECTION MATTERS. Except as specifically set
     forth herein, in accordance with the Company's Bye-laws, the Board shall
     act by the vote of a majority of the Directors present at a meeting, and
     the required quorum for a meeting of the Board shall be a majority of the
     whole Board. Notwithstanding the foregoing, and except as specifically set
     forth in the Subscription Agreement, (a) prior to the Approval Date, unless
     also approved by the Warburg Representative and the H&F Representative, and
     (b) following the Approval Date, unless also approved by (i) the Warburg
     Representative, if at such time Warburg's Retained Percentage equals or
     exceeds 25%, and (ii) the H&F Representative, if at such time H&F's
     Retained Percentage equals or exceeds 50%, the Company shall not (and shall
     not permit any of its Subsidiaries to):

          3.   The validity and effects of this Amendment shall be governed by
and construed and enforced in accordance with the laws of the State of New York.

          4.   This Amendment may be executed in any number of counterparts,
each of which shall be considered an original and all of which together shall be
deemed to be one and the same instrument.

                            [Signature Pages Follow]

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          IN WITNESS WHEREOF, each party has executed this Amendment as of the
date first above-written.

                                    ARCH CAPITAL GROUP LTD.

                                    By:  /s/ John D. Vollaro
                                         ---------------------------------------
                                         Name: John D. Vollaro
                                         Title: Executive Vice President & Chief
                                                Financial Officer

                                    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

                                    WARBURG PINCUS (BERMUDA) PRIVATE
                                      EQUITY VIII, L.P.

                                    By:  Warburg Pincus (Bermuda)
                                         Private Equity Ltd.,
                                         its General Partner

                                           By: /s/ Kewsong Lee
                                               ---------------------------------
                                               Name: Kewsong Lee
                                               Title: Partner

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                                    WARBURG PINCUS (BERMUDA)
                                      INTERNATIONAL PARTNERS, L.P.

                                    By:  Warburg Pincus (Bermuda)
                                         International Ltd.,
                                         its General Partner

                                           By: /s/ Kewsong Lee
                                               ---------------------------------
                                               Name: Kewsong Lee
                                               Title: Partner

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                                 HFCP IV (BERMUDA), L.P.,

                                 By:  H&F Investors IV (Bermuda), L.P.

                                      By:   H&F Corporate Investors IV (Bermuda)
                                            Ltd.,
                                            its General Partner

                                            By:  /s/ Georgia Lee
                                                 -------------------------------
                                                 Name: Georgia Lee
                                                 Title: Vice President

                                 H&F INTERNATIONAL PARTNERS
                                   IV-A (BERMUDA), L.P.

                                 By:  H&F Investors IV (Bermuda), L.P.,
                                      its General Partner

                                      By:   H&F Corporate Investors IV
                                            (Bermuda), Ltd.,
                                            its General Partner,

                                            By:  /s/ Georgia Lee
                                                 -------------------------------
                                                 Name: Georgia Lee
                                                 Title: Vice President
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                                 HFCP IV (BERMUDA), L.P.,

                                 By:  H&F Investors IV (Bermuda), L.P.

                                      By:   H&F Corporate Investors IV (Bermuda)
                                            Ltd.,
                                            its General Partner

                                            By:  /s/ Georgia Lee
                                                 -------------------------------
                                                 Name: Georgia Lee
                                                 Title: Vice President

                                 H&F INTERNATIONAL PARTNERS
                                   IV-A (BERMUDA), L.P.

                                 By:  H&F Investors IV (Bermuda), L.P.,
                                      its General Partner

                                      By:   H&F Corporate Investors IV
                                            (Bermuda), Ltd.,
                                            its General Partner,

                                            By:  /s/ Georgia Lee
                                                 -------------------------------
                                                 Name: Georgia Lee
                                                 Title: Vice President<Page>
                                                                    Exhibit 10.2

                             ARCH CAPITAL GROUP LTD.

                                    AGREEMENT

           THIS AGREEMENT, dated November 12, 2002, between Arch Capital Group
Ltd. (the "COMPANY"), a Bermuda company, and Robert Clements (the "EXECUTIVE");

           NOW, THEREFORE, in consideration of the premises and mutual covenants
contained herein, the parties agree as follows:

         1. To the extent that (a) the Executive seeks capital gains treatment
for U.S. federal income tax purposes (or any corresponding provisions of state
or local tax law) on any gains on the sales (the "REPURCHASES") to the Company
of common shares of the Company pursuant to the Share Repurchase Agreement,
dated November 6, 2002, and (b) the proceeds of such Repurchases are
characterized as ordinary dividend income for U.S. federal income tax purposes
(or any corresponding provisions of state or local tax law), then the Company
shall indemnify and hold the Executive harmless, on an after-tax basis, for any
such ordinary income taxes payable in excess of the capital gains taxes (long-
or short-term, depending on the applicable holding periods) relating to such
Repurchases.

         2. The Executive agrees to (a) file on a timely basis the income tax
returns relating to such Repurchases and (b) provide the Company with any
information reasonably requested by the Company relating to the reporting
position to be taken by the Executive for such Repurchases.

           3. This agreement shall be governed by and construed in accordance
with the laws of Connecticut, without giving effect to principles of conflict of
laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement on
the date first above written.

                                      ARCH CAPITAL GROUP LTD.

                                      By: /s/ John D. Vollaro
                                          --------------------------------------
                                         Name: John D. Vollaro
                                         Title: Executive Vice President & Chief
                                               Financial Officer

                                      /s/ Robert Clements
                                      ------------------------------------------
                                      Robert Clements

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