Exhibit 10.2
                                                                  EXECUTION COPY


                                 AMENDMENT NO. 2


     Reference is made to the Subscription Agreement dated as of October 24,
2001, as amended November 20, 2001 (the "Subscription 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. Capitalized terms used without definition herein have the meanings
given to them in the Subscription Agreement.

     This amendment ("Amendment") to the Subscription Agreement, the
Shareholders Agreement, the Certificate and the form of Bye-Law 75 is made as of
January 3, 2002.

     For good and valid consideration, the receipt of which is hereby
acknowledged, the parties hereto agree as follows:

A.   AMENDMENTS TO SUBSCRIPTION AGREEMENT

     1. Section B.2 of the Subscription Agreement is amended and restated in its
entirety as follows:

          "2. Adjustment for Triggering Event. In the event that a Triggering
     Event occurs, then, within five business days of the occurrence of such
     Triggering Event or the Applicable Date (whichever is later), the Company
     shall issue and deliver to each Purchaser one or more certificates
     registered in the name of such Purchaser (or its designee) representing
     that number of Preference Shares equal to the difference between (i) such
     Purchaser's Total Purchase Price divided by an amount equal to the
     difference between the Per Share Price and $1.50 and (ii) such Purchaser's
     Total Purchase Price divided by the Per Share Price."

     2. Section B.3.f of the Subscription Agreement is amended and restated in
its entirety as follows:

          "(f) In addition to paragraph (e), if the Adjustment Basket is less
     than zero and in the event that a Triggering Event occurs, then, on the
     Second Applicable Date, the Company shall also issue and deliver to each
     Purchaser a number of Preference Shares equal to the difference between (A)
     such Purchaser's Total Purchase Price divided by an amount equal to [P -
     $1.50 - B/12.86 million] and (B) the Purchaser's Total Purchase Price
     divided by an amount equal to (P - $1.50)."

     3. Section D.4.b of the Subscription Agreement is amended to restate clause
(ii) in its entirety as follows: "(ii) the proposal to obtain the Requisite
Nasdaq Approval".


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                                      -2-


     4. Section E.2 of the Subscription Agreement is amended by replacing the
last sentence thereof with the following:

     "Until the latest of (a) the receipt of Requisite Shareholder Approval, (b)
     the receipt of Requisite Regulatory Approval and (c) ninety days following
     the Fourth Anniversary Adjustment Date, to the extent that the number and
     kind of outstanding capital shares of the Company change from time to time,
     a corresponding adjustment shall be made by the Company in the number and
     kind of outstanding capital shares of Newco."

     5. Section E.3 of the Subscription Agreement is amended by replacing the
last sentence thereof with the following:

     "An "Exchange Trigger Event" shall mean any one or more of the following:
     (a) failure to obtain the Requisite Shareholder Approval (unless such
     failure was due to a breach by the Purchaser of a covenant hereunder)
     within five months of the Closing Date, (b) failure to obtain the Requisite
     Regulatory Approval (unless such failure was due to a breach by the
     Purchaser of a covenant hereunder) within six months of the Closing Date or
     (c) if the Adjustment Basket (as determined pursuant to paragraphs (a)
     through (g) of Section B.3) is less than zero and its absolute value at any
     time exceeds $250 million."

     6. Section E.4 of the Subscription Agreement is amended by replacing
"ninety days following the consummation of the Final Adjustment contemplated by
Section B.3. of this Agreement" with "ninety days following the Fourth
Anniversary Adjustment Date".

     7. Section F.5 of the Subscription Agreement is amended by replacing
"Second Applicable Date" with "Fourth Anniversary Adjustment Date".

     8. Schedule A of the Subscription Agreement is amended to add the following
definitions:

     "Exchange Act" means the United States Securities Exchange Act of 1934, as
amended.

     "Fourth Anniversary Adjustment Date" means (i) the last date on which an
adjustment could be required to be determined under Section B.3.g. hereof or
(ii) if such an adjustment is required to be determined, the date of completion
of such adjustment.

     "Requisite Nasdaq Approval" has the meaning given to such term in the
Certificate.


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                                      -3-


     "Triggering Event" means either:

     (i)  the closing price of the Common Shares being at or above $30 per share
          (as adjusted, for any event which would subject the exercise price of
          the Warrants to an adjustment, by the same percentage as the
          percentage adjustment of such exercise price) for 20 out of 30
          consecutive trading days at any time following the Closing Date; or

     (ii) the acquisition by any person, entity or "group" (within the meaning
          of Section 13(d)(3) or 14(d)(2) of the Exchange Act) of beneficial
          ownership (within the meaning of Rule 13d-3 promulgated under the
          Exchange Act) of 40% or more, in the aggregate, of either the voting
          power of the then outstanding Common Shares or the combined voting
          power of the Company's then outstanding voting securities entitled to
          vote generally in election of directors; provided, however, that if
          such acquisition results in whole or in part from -------- ------- a
          transfer of any Common Shares or other voting securities by Marsh &
          McLennan Companies, Inc. or any of its subsidiaries (or by any parent
          of Marsh & McLennan Companies, Inc. or any of its subsidiaries), such
          acquisition shall not constitute a Triggering Event unless such
          transfer is effected pursuant to an offer by such acquiror to purchase
          all of the Company's outstanding Common Shares.

     9. Schedule B of the Subscription Agreement is amended to add the following
at the end of the definition of "C":

     "For the avoidance of doubt, the foregoing shall not include, and in no
     event shall the Per Share Price be reduced by, any payments made in
     exchange for cancellation of the Company's class B warrants outstanding on
     the date of this Agreement."

B.   AMENDMENTS TO CERTIFICATE OF DESIGNATIONS

     1. Paragraph (f)(3)(B) shall be amended by replacing "Requisite Shareholder
Approval" with "Requisite Nasdaq Approval".

     2. Paragraph (g)(2) shall be amended and restated in its entirety as
follows:

          "(2) Mandatory Conversion. Following the later of (a) receipt of the
     Requisite Shareholder Approval and the Requisite Regulatory Approval, and
     (b) 90 days after the Fourth Anniversary Adjustment Date, the Preference
     Shares shall automatically convert into Common Shares. The number of Common
     Shares deliverable upon conversion of a Preference Share shall be as set
     forth in paragraphs (g)(1) and (h)."


<PAGE>
                                      -4-


     3. Paragraph (h) shall be amended by replacing the heading "Conversion
Price Adjustments" with "Adjustments".

     4. Paragraph (i) shall be amended by adding the following definitions:

          "Fourth Anniversary Adjustment Date" means (i) the last date on which
     an adjustment could be required to be determined under Section B.3.g. of
     the Subscription Agreement or (ii) if such an adjustment is required to be
     determined, the date of completion of such adjustment.

          "Requisite Nasdaq Approval" means the approval by the holders of
     Common Shares and Preference Shares of the issuance of Common Shares
     issuable upon conversion of all Preference Shares issued under the
     Subscription Agreement and the Management Subscription Agreement and the
     issuance of Common Shares issuable upon exercise of all Class A Warrants
     issued under the Subscription Agreement and the Management Subscription
     Agreement (including by operation of the anti-dilution adjustments in the
     Class A Warrants), to the extent that the number or voting power of all
     such Common Shares and the 140,380 Common Shares issued on November 20,
     2001 would exceed 19.9% of the total number or the total voting power of
     the Common Shares issued and outstanding on November 19, 2001.

          "Shareholders Agreement" means the Shareholders Agreement, dated as of
     November 20, 2001, by and between the Company and the shareholders named
     therein, as amended from time to time in accordance with its terms.

     5. Paragraph (i) shall be amended by replacing the definition of "Requisite
Regulatory Approval" in its entirety with the following:

          "Requisite Regulatory Approval" means approval by the insurance
     authorities in the States of Florida, Missouri, Nebraska and Wisconsin of
     the acquisition of greater than 9.9% of the total voting power of all
     shares of the Company entitled to vote generally in the election of
     directors by the "Warburg Purchasers" and the "H&F Purchasers" (as defined
     in the Shareholders Agreement) who are original signatories to the
     Shareholders Agreement.

     6. Paragraph (i) shall be amended by replacing the definition of "Requisite
Shareholder Approval" in its entirety with the following:

          "Requisite Shareholder Approval" means (a) the approval by the holders
     of Common Shares and Preference Shares of an amendment to bye-law 45 in the
     form included in Exhibit III to the Subscription Agreement and (b) the
     Requisite Nasdaq Approval.


<PAGE>
                                      -5-


     7. Paragraph (i) shall be amended by replacing the definition of
"Subscription Agreement" in its entirety with the following:

          "Subscription Agreement" means the Subscription Agreement, dated as of
     October 24, 2001, by and between the Company and each of the Purchasers
     named therein, as amended from time to time in accordance with its terms.

     8. Paragraph (i) shall be amended by deleting the definition of "Conversion
Price" in its entirety.

C.   AMENDMENTS TO FORM OF BYE-LAW 75

     Exhibit III to the Subscription Agreement is amended and restated in the
form annexed hereto.

D.   AMENDMENTS TO SHAREHOLDERS AGREEMENT

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

     "Requisite Nasdaq Approval" shall have the meaning assigned in the
Certificate of Designations for the Preference Shares.

     2. Section 3.1(c)-(e) of the Shareholders Agreement is amended and restated
in its entirety as follows:

          "(c) Effective as of 12:00 a.m. on the date immediately following the
     Approval Date, the size of the Board shall be increased such that the Board
     shall then and thereafter consist of 17 Directors (such number not to be
     increased without the consent of Warburg and H&F) and (i) five individuals
     designated by Warburg (together with the Initial Warburg Director, and any
     other replacements or substitutions therefor, the "Warburg Directors"), and
     (ii) two individual designated by H&F (together with the Initial H&F
     Director, and any other replacements or substitutions therefor, the "H&F
     Directors") shall each be appointed by the Board as a Director to serve in
     such classes of Directors as may be necessary to assure that each class in
     Directors is as near in equal in number as possible and that the Warburg
     Directors and the H&F Directors, respectively, are distributed among
     different classes.

          (d) Following the Approval Date, for so long as Warburg's Retained
     Percentage is "x" as set forth in the table below, the slate of nominees
     recommended by the Board to shareholders for election as directors of the
     Company at each annual meeting of shareholders shall include such number of
     individuals designated by War-


<PAGE>
                                      -6-


     burg, which together with the number of Warburg Directors whose term is not
     scheduled to expire, is equal to the number set forth opposite such
     Warburg's Retained Percentage in the table below:

                   Warburg's Retained               Number of Nominees
                       Percentage

                         x > 75%                            6
                           -
                      65% < x < 75%                         5
                          -
                      55% < x < 65%                         4
                          -
                      40% < x < 55%                         3
                          -
                      25% < x < 40%                         2
                          -
                      10% < x < 25%                         1;
                                            -

     provided that, if the Approval Date has not occurred, for so long as
     Warburg's Retained Percentage is equal to or exceeds 10%, at least one
     Warburg Director shall be included in the slate of nominees recommended by
     the Board to shareholders for election as directors of the Company at each
     annual general meeting of shareholders at which a Warburg Director's term
     is scheduled to expire. For so long as Warburg has the power to have at
     least two Directors included in the slate of nominees recommended by the
     Board, power with respect to one such Director shall be exercised by
     Warburg Pincus (Bermuda) Private Equity VIII, L.P and power with respect to
     one such Director shall be exercised by Warburg Pincus (Bermuda)
     International Partners, L.P.

          (e) Following the Approval Date, for so long as H&F's Retained
     Percentage is "x" as set forth in the table below, the slate of nominees
     recommended by the Board to shareholders for election as directors of the
     Company at each annual meeting of shareholders shall include such number of
     individuals designated by H&F, which together with the number of H&F
     Directors whose term is not scheduled to expire, is equal to the number set
     forth opposite such H&F's Retained Percentage in the table below:

                 H&F's Retained                 Number of Nominees
                   Percentage
                     x > 60%                            3
                       -
                  35% < x < 60%                         2
                      -
                  20% < x < 35%                         1;
                      -

     provided that, if the Approval Date has not occurred, for so long as H&F's
     Retained Percentage is equal to or exceeds 20%, at least one H&F Director
     shall be included in


<PAGE>
                                      -7-


     the slate of nominees recommended by the Board to shareholders for election
     as directors of the Company at each annual general meeting of shareholders
     at which an H&F Director's term is scheduled to expire. For so long as H&F
     has the power to have at least one Director included in the slate of
     nominees recommended by the Board, such power shall be exercised by HFCP IV
     (Bermuda), L.P."

     3. Section 3.1 of the Shareholders Agreement is amended to add the
following as clause (g):

          "(g) The Company shall use its best efforts (i) to cause a special
     meeting of the Board to be called upon the request of at least three
     Directors and (ii) to cause to be submitted, at the 2002 annual general
     meeting of the Company's shareholders, a proposal to amend Bye-Law 20 of
     the Company to replace "by a majority of the total number of Directors"
     with "by three Directors or a majority of the total number of Directors
     (whichever is fewer)"."

     4. Section 3.3 of the Shareholders Agreement is amended to add the
following after the semicolon in clause (13):

     "provided, however, that the foregoing shall not apply to any merger or
     consolidation solely between or among wholly owned Subsidiaries of the
     Company, other than any such transaction between Subsidiaries which are
     considered "Core Insurance Operations" under Section E, and Subsidiaries
     which are not considered "Core Insurance Operations";".

     5. Section 3.3 of the Shareholders Agreement is amended to delete the
period at the end of clause (17) and adding the following before the last
paragraph:

     "; provided, however, transactions solely between or among the Company
     and/or one or more of its wholly owned Subsidiaries shall be excluded from
     clauses (6), (8), (9), (11) and (14) (and from clause (17) to the extent
     relating to an agreement with respect to a transaction excluded by this
     proviso), other than any such transaction between Subsidiaries which are
     considered "Core Insurance Operations" under Section E, and Subsidiaries
     which are not considered "Core Insurance Operations". In addition, prior to
     the Approval Date, and after the Approval Date for so long as the Warburg
     Directors and the H&F Directors together constitute a majority of the
     Board, (i) the notice for each meeting of the Board called by the Chairman
     of the Board, the President of the Company, the Warburg Directors or the
     H&F Directors shall include a list of topics to be discussed at the meeting
     (the "Agenda") and (ii) the Board shall not act on any matter that is not
     within the Agenda without the consent of at least one Warburg Director and
     at least one H&F Director."


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                                      -8-


     6. Section 5.3 of the Shareholders Agreement is amended to replace the
first two sentences only with the following:

          "Section 5.3. Restrictions on Conversion. Prior to the receipt of the
     Requisite Nasdaq Approval, no Investor shall convert any Preference Share
     or exercise any Class A Warrant, if the number of Common Shares to be
     issued to such Investor upon such conversion or exercise, together with all
     Common Shares issued upon prior conversions or exercise by such holder,
     would exceed such Investor's Permissible Conversion Amount. An Investor's
     "Permissible Conversion Amount" shall be a number of Common Shares equal to
     the product of (a) the total number of Common Shares issuable to such
     Investor upon conversion or exercise of all such Investor's Initial Shares,
     and (b) a fraction the numerator of which is (i) (A) the lesser of (x) the
     product of .199 times the total number of Common Shares issued and
     outstanding on November 19, 2001 and (y) the product of .199 times the
     total voting power of the Common Shares issued and outstanding on November
     19, 2001, minus (B) the 140,380 Common Shares issued on November 20, 2001,
     and the denominator of which is (ii) the total number of Common Shares
     issuable upon conversion or exercise of all Initial Shares."

E.   MISCELLANEOUS

     1. This Amendment shall be governed by, and construed and enforced in
accordance with, the laws of the State of New York.

     2. 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]


<PAGE>



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


                     ARCH CAPITAL GROUP LTD.


                     By:      /s/ Louis T. Petrillo
                              ------------------------------------------------
                              Name:  Louis T. Petrillo
                              Title:  Senior Vice President, General Counsel &
                                      Secretary


                     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


<PAGE>


                     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




<PAGE>


                     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>


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

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

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


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


                     H&F EXECUTIVE FUND IV
                          (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>


                                                        Annex to Amendment No. 2

                                   Bye-Law 75

75.      Certain Subsidiaries

     With respect to any company incorporated under the laws of Bermuda,
Barbados or the Cayman Islands all of the voting shares of which are owned
(directly or indirectly through subsidiaries) by the Company, and any other
subsidiary of the Company designated by the Board of the Company (together, the
"Designated Companies"), the board of directors of each such Designated Company
shall consist of the persons who have been elected by the Members as Designated
Company Directors. Notwithstanding the general authority set out in Bye-law
2(1), the Board shall vote all shares owned by the Company in each Designated
Company to ensure the constitutional documents of such Designated Company
require such Designated Company Directors to be elected as the directors of such
Designated Company, and to elect such Designated Company Directors as the
directors of such Designated Company. The Company shall enter into agreements
with each such Designated Company to effectuate or implement this Bye-law.