Document:

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                                                                     EXHIBIT 4.1

COMMON SHARES                                                      COMMON SHARES

NUMBER

AXS                                 [AXIS LOGO]                 SHARES

[LOGO]

ORGANIZED UNDER THE LAWS
       OF BERMUDA                            CUSIP G0692U 10 9
                                             SEE REVERSE FOR CERTAIN DEFINITIONS

                          AXIS CAPITAL HOLDINGS LIMITED

THIS IS TO CERTIFY THAT

is the registered holder of

                              CERTIFICATE OF STOCK

  FULLY PAID AND NON-ASSESSABLE COMMON SHARES, PAR VALUE US $0.10 PER SHARE, OF

AXIS Capital Holdings Limited transferable on the books of the Company by the
holder hereof in person or by duly authorized attorney upon surrender of this
certificate properly endorsed. This certificate and the shares represented
hereby are issued and shall be held subject to all the provisions of the
Memorandum of Association and Bye-Laws of the Company, copies of which are on
file with the Transfer Agent, to all of which the holder by acceptance hereof
assents.

     This certificate is not valid unless countersigned and registered by the
Transfer Agent and Registrar.

     Witness the seal of the Company and the signatures of its duly authorized
officers.

Dated

COUNTERSIGNED AND REGISTERED:
        THE BANK OF NEW YORK
                     TRANSFER AGENT
                      AND REGISTRAR
BY  /s/ [ILLEGIBLE]                    /s/ [ILLEGIBLE]       /s/ [ILLEGIBLE]
                                                              PRESIDENT AND
               AUTHORIZED SIGNATURE          SECRETARY   CHIEF EXECUTIVE OFFICER

 AMERICAN BANK NOTE COMPANY    PRODUCTION COORDINATOR: SUE McNAMEE: 931-490-1722
    711 ARMSTRONG LANE                       PROOF OF MAY 30, 2003
COLUMBIA, TENNESSEE 38401                 AXIS CAPITAL HOLDINGS LIMITED
       (931) 388-3003                         TSB 12043 MODEL FACE

SALES: R.JOHNS: 212-269-0339                    OPERATOR: KOSHY

ETHER 13 / LIVE JOBS / A /
   Axis 12043 model face                              NEW

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                          AXIS CAPITAL HOLDINGS LIMITED

     The following abbreviations, when used in the inscription on the face of
this certificate, shall be construed as though they were written out in full
according to applicable laws or regulations:

<Table>
     <S>                                        <C>
     TEN COM - as tenants in common             UNIF GIFT MIN ACT-_________Custodian____________
     TEN ENT - as tenants by the entireties                        (Cust)              (Minor)
     JT TEN  - as joint tenants with right of                   under Uniform Gifts to Minors
               survivorship and not as tenants                  Act_____________
               in common                                              (State)
</Table>

     Additional abbreviations may also be used though not in the above list.

          For value received, ___________________________ hereby sell,
          assign and transfer unto
          PLEASE INSERT SOCIAL SECURITY OR OTHER
              IDENTIFYING NUMBER OF ASSIGNEE

          ____________________________________________________________
                (PLEASE PRINT OR TYPEWRITE NAME AND ADDRESS,
                      INCLUDING ZIP CODE, OF ASSIGNEE)

          ____________________________________________________________

          ____________________________________________________________

          _____________________________________________________ shares
          represented by the within certificate, and do hereby
          irrevocably constitute and appoint

          __________________________________________________ Attorney
          to transfer the said shares on the books of the within named
          Company with full power of substitution in the premises.
          Dated________________________

                       _______________________________________________
               NOTICE: THE SIGNATURE TO THIS ASSIGNMENT MUST
                       CORRESPOND WITH THE NAME AS WRITTEN UPON THE
                       FACE OF THE CERTIFICATE IN EVERY PARTICULAR,
                       WITHOUT ALTERATION OR ENLARGEMENT OR ANY CHANGE
                       WHATEVER.

          Signature(s) Guaranteed:

          ____________________________________________________
          THE SIGNATURE(S) MUST BE GUARANTEED BY AN ELIGIBLE
          GUARANTOR INSTITUTION (BANKS, STOCKBROKERS,
          SAVINGS ASSOCIATIONS AND CREDIT UNIONS WITH
          MEMBERSHIP IN AN APPROVED SIGNATURE GUARANTEE
          MEDALLION PROGRAM), PURSUANT TO S.E.C. RULE 17Ad-15.

AMERICAN BANK NOTE COMPANY     PRODUCTION COORDINATOR: SUE MCNAMEE: 931-490-1722
    711 ARMSTRONG LANE                     PROOF OF APRIL 21, 2003
COLUMBIA, TENNESSEE 38401                 AXIS CAPITAL HOLDINGS LIMITED
       (931) 388-3003                            TSB 11522 BK

SALES: R. JOHNS: 212-269-0339              OPERATOR:          KOSHY

/ ETHER 13 / LIVE JOBS /
    A/Axis 11522 bk                                   NEW<Page>

                                                                     EXHIBIT 4.2

                            FORM OF SERIES A WARRANT

                          AXIS CAPITAL HOLDINGS LIMITED

                SERIES A WARRANT FOR THE PURCHASE OF COMMON STOCK
                        OF AXIS CAPITAL HOLDINGS LIMITED

No. A-1
                                                    SERIES A WARRANT TO PURCHASE
                                         SHARES EQUAL TO HOLDER'S WARRANT AMOUNT
                                                              (AS DEFINED BELOW)

     THIS SECURITY HAS NOT BEEN REGISTERED UNDER THE UNITED STATES SECURITIES
ACT OF 1933, AS AMENDED. THIS SECURITY CANNOT BE OFFERED, TRANSFERRED OR SOLD
UNLESS (I) A REGISTRATION STATEMENT UNDER SUCH ACT IS IN EFFECT WITH RESPECT TO
SUCH SECURITIES OR A WRITTEN OPINION FROM COUNSEL ACCEPTABLE TO THE COMPANY IS
OBTAINED TO THE EFFECT THAT NO SUCH REGISTRATION IS REQUIRED AND (II) THE
TRANSFEREE IS APPROVED BY APPLICABLE REGULATORY AUTHORITIES, IF SUCH APPROVAL IS
REQUIRED. TRANSFERS OF THIS SECURITY ARE SUBJECT TO THE APPROVAL OF THE BOARD OF
DIRECTORS OF THE COMPANY, IN ITS SOLE DISCRETION. THIS SECURITY IS ALSO SUBJECT
TO ADDITIONAL RESTRICTIONS ON TRANSFER, VOTING AND OTHER MATTERS AS SET FORTH IN
THE COMPANY'S BYE-LAWS.

     This Warrant is being issued in consideration for the surrender by the
Holder (defined below) of that certain Series A Warrant for the purchase of
common stock of AXIS Specialty Limited ("AXIS Specialty"), issued by AXIS
Specialty in favor of the Holder and dated as of November 20, 2001 (the "Old
Warrant"), which exchange the parties intend will qualify as a reorganization
within the meaning of Section 368 of the United States Internal Revenue Code of
1986, as amended. The Old Warrant is being surrendered in exchange for this
Warrant in connection with the transactions contemplated by that certain
Offering Memorandum, Proxy Statement and Information Circular, dated as of
December 2, 2002, issued by AXIS Specialty, AXIS Mergeco Limited and the
Company.

     FOR VALUE RECEIVED, AXIS Capital Holdings Limited, a Bermuda corporation
(the "Company"), hereby certifies that [ ], its successor or permitted assigns
(the "Holder") is entitled, subject to the terms herein, to purchase a number of
fully paid and non-assessable shares of voting common stock of the Company, par
value $0.10 per share (the "Warrant Shares"), equal to the Holder's Warrant
Amount (as defined herein), at a purchase price per share equal to the Exercise
Price (as defined herein). The number of shares of Warrant Shares to be received
upon the exercise of this Warrant and the price to be paid for such shares are
subject to adjustment from time to time as hereinafter set forth.

     1.   DEFINITIONS. The following terms, as used herein, have the following
meanings:

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          "Acquisition Event" means any amalgamation, scheme of arrangement or
     consolidation as a result of which the Members of the Company immediately
     prior to such transaction shall, in the aggregate, beneficially own (as
     such term is defined in Rule 13D promulgated under the United States
     Securities Exchange Act of 1934, as amended) immediately following the
     consummation thereof (x) less than 50% of the Common Stock of the Company,
     if the Company is the entity surviving such amalgamation, scheme of
     arrangement or consolidation, or (y) less than 50% of common equity of the
     entity surviving such amalgamation or formed by such scheme of arrangement
     or consolidation if the Company is not the entity surviving such
     amalgamation, scheme of arrangement or consolidation.

          "Affiliate" shall have the meaning given to such term in Rule 12b-2
     promulgated under the United States Securities and Exchange Act of 1934, as
     amended.

          "Bye-Laws" means the Bye-Laws of the Company, as amended from time to
     time.

          "Common Stock" means the voting common shares, par value $0.10 per
     share, of the Company.

          "Exercised Percentage" with respect to any exercise of a portion of
     this Warrant as of any particular date, means a percentage equal to the
     product of (i) 100% and (ii) a fraction, the numerator of which shall be
     the number of shares of Common Stock issued upon the exercise of such
     portion of this Warrant as of such date of partial exercise, and the
     denominator of which shall be the number of shares of Common Stock
     outstanding as of such date of partial exercise (calculated on a fully
     diluted basis and assuming that all options, warrants, including this
     Warrant, and any other rights to purchase shares of Common Stock
     outstanding on such date, shall have been exercised on such date).

          "Exercise Price" means $100 per Warrant Share, subject to adjustment
     as provided herein.

          "Expiration Date" means November 20, 2011 at 5:00 p.m. New York City
     time.

          "Initial Public Offering" means the first registered public offering
     of the Common Stock under the United States securities laws after the date
     hereof or any amalgamation, scheme of arrangement or consolidation as a
     result of which the Members of the Company receive, as the consideration in
     such amalgamation, scheme of arrangement or consolidation, equity
     securities of a class that (i) has been registered as part of a public
     offering under the United States securities laws and (ii) is publicly
     traded on a securities exchange in or outside the United States.

          "Warrant Amount" as of any exercise date, means a number of Warrant
     Shares equal to:

          (i)     a fraction, the numerator of which is the Warrant Percentage
          for this Warrant as of such exercise date and the denominator of which
          is the total of the

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          Warrant Percentages for all of the Series A Warrants of the Company
          (including this Warrant) as of such exercise date; MULTIPLIED BY

          (ii)    the difference of:

                  (A) (1) the aggregate number of shares of Common Stock which
                  would be outstanding on such date on a fully diluted basis and
                  assuming that all options, warrants (but excluding this
                  Warrant and each other Series A Warrant of the Company) and
                  any other rights to purchase shares of Common Stock
                  outstanding on such date shall have been exercised on such
                  date DIVIDED BY (2) (x) one hundred percent MINUS (y) the
                  total of the Warrant Percentages for all of the Series A
                  Warrants of the Company (including this Warrant) as of such
                  exercise date, MINUS

                  (B) the aggregate number of shares of Common Stock which would
                  be outstanding on such date on a fully diluted basis and
                  assuming that all options, warrants (but excluding this
                  Warrant and each other Series A Warrant of the Company) and
                  any other rights to purchase shares of Common Stock
                  outstanding on such date shall have been exercised on such
                  date;

     PROVIDED, HOWEVER, that the Warrant Amount shall be fixed in accordance
     with the foregoing formula (subject to changes pursuant to Section 6 or 7
     hereof) at the close of business on the day immediately preceding the date
     of consummation of an Initial Public Offering, if any, or the close of
     business on the day immediately preceding the date of consummation of an
     Acquisition Event, if any, whichever first occurs.

          "Warrant Percentage" as of any exercise date, means 10.367% less (i)
     in any case when a portion of the Warrant shall have been previously
     exercised on one specific occasion, the Exercised Percentage, or (ii) in
     the event that a portion of the Warrant shall have been previously
     exercised on more than one occasion, the sum of the Exercised Percentages
     calculated with respect to each such prior exercise. In the event that, as
     of any exercise date, a portion of the Warrant shall have been previously
     exercised on more than one occasion, the calculation of the Exercised
     Percentage with respect to each such prior exercise shall be made
     successively, commencing with the earliest such prior exercise.

     2.   EXERCISE OF WARRANT. (a) The Holder is entitled to exercise this
Warrant in whole or in part at any time, or from time to time, until the
Expiration Date, for the Holder's Warrant Amount on such exercise date. To
exercise this Warrant, the Holder shall execute and deliver to the Company a
Warrant Exercise Notice substantially in the form attached hereto. Subject to
Section 2(d) below, no earlier than ten days after delivery of the Warrant
Exercise Notice, the Holder shall deliver to the Company this Warrant
Certificate, including the Warrant Exercise Subscription Form attached hereto
duly executed by the Holder, together with payment of the applicable Exercise
Price. Upon such delivery and payment, the Holder shall be deemed to be the
holder of record of the Warrant Shares subject to such exercise, notwithstanding
that

                                        3
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the register of members of the Company shall then be closed or that certificates
representing such Warrant Shares shall not then be actually delivered to the
Holder.

     (b)  Subject to Section 2(d) below, the Exercise Price may be paid in cash
or by certified or official bank check or bank cashier's check payable to the
order of the Company or by any combination of such cash or check. The Company
shall pay any and all documentary, stamp or similar issue or transfer taxes
payable in respect of the issue or delivery of the Warrant Shares.

     (c)  Upon surrender of this Warrant Certificate in conformity with the
provisions hereunder, the Company shall transfer to the Holder appropriate
evidence of ownership of the Warrant Shares or any other securities or property
(including any money) to which the Holder is entitled, registered or otherwise
placed in, or payable to the order of, the name or names of the Holder or such
transferee as may be directed in writing by the Holder, and shall deliver such
evidence of ownership and any other securities or property (including any money)
to the Person or Persons entitled to receive the same, together with an amount
in cash in lieu of any fraction of a share as provided in Section 3 below.

     (d)  In lieu of making the cash payment required to exercise the Warrant
pursuant to this Section 2, the Holder may elect to (i) deliver as payment, in
whole or in part of the aggregate Exercise Price, shares of Common Stock having
a value calculated by reference to the aggregate Daily Price (as defined below)
on the day immediately preceding the date on which the Holder delivers written
notice to the Company pursuant to Section 2(a) equal to or in excess of the
applicable portion of the aggregate Exercise Price for the Warrant Shares or
(ii) exchange this Warrant for shares of Common Stock, in which event the
Company will issue to the Holder the number of shares of Common Stock equal to
the result obtained by (A) subtracting the Exercise Price from the Daily Price,
(B) multiplying the difference by the number of Shares for which the Warrant is
being exercised, and (C) dividing the product by the Daily Price as set forth in
the following equation:

          X = (A - B) X C where:
              -----------
                   A

          X = the number of shares of Common Stock issuable upon exercise
                  pursuant to this Section 2(d).

          A = the Daily Price (as defined below) on the day immediately
              preceding the date on which the Holder delivers written notice to
              the Company pursuant to Section 2(a).

          B = the Exercise Price

          C = the number of Shares for which the Warrant is being exercised

          If the foregoing calculation results in a negative number, then no
shares of Common Stock shall be issued upon exercise pursuant to this Section
2(d)(ii).

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     3.   NO FRACTIONAL SHARES. The Company shall not be required to issue a
fractional share of common stock upon the exercise of this Warrant. If any
fractional interest in a share of common stock would be deliverable upon the
exercise of this Warrant in whole or in part, the Company, in lieu of delivering
any such fractional share, shall pay an amount in cash equal to the book value
per share at the end of the most recent fiscal quarter multiplied by the
fraction of the fractional share which would otherwise have been issued
hereunder. The Company agrees that it will not change the par value of the
Common Stock from par value $0.10 per share to any higher par value which
exceeds the Exercise Price then in effect, and will reduce the par value of the
Common Stock upon any event that would, but for this provision, reduce the
Exercise Price below the par value of the Common Stock.

     4.   RESERVATION OF SHARES. The Company agrees that it will at all times
reserve for issuance and delivery upon exercise of this Warrant such number of
its authorized but unissued shares of Common Stock sufficient to permit the
exercise in full of this Warrant. All such shares shall be duly authorized and,
when issued upon such exercise, shall be validly issued, fully paid and
non-assessable, free and clear of all liens, security interests, charges and
other encumbrances or restrictions on sale, except to the extent set forth in
the Bye-Laws.

     5.   TRANSFER RESTRICTIONS. (a) This Warrant may not be assigned or
otherwise transferred, disposed of or encumbered by the Holder in whole or in
part except pursuant to an assignment approved by the Board of Directors of the
Company in accordance with the following rules:

          (i)     An assignment, as to all or any of the Warrant Shares, shall
          be in writing in a form acceptable to the Board of Directors, shall be
          signed by or on behalf of the assignor and shall be accompanied by
          satisfactory evidence of the authority of any person signing on behalf
          of the assignor.

          (ii)    The Board of Directors shall have the same obligations,
          powers, and discretions in regard to the assignment of the Warrant as
          shall be vested in them by the Bye-Laws of the Company in regard to
          the transfer of shares of Common Stock; PROVIDED, HOWEVER, that the
          provisions of the Bye-Laws referring to share certificates shall not
          apply in regard to the Warrant. Additionally, the Board of Directors
          may require, before approving an assignment, that (A) the assignor
          surrender the original of this Warrant Certificate (or, in the case of
          a subsequent Holder, the original Warrant Certificate issued by the
          Company to him) to be endorsed with a notation of the assignment and,
          in the event of a partial assignment, returned to the assignor and/or
          (B) that the assignee enter into an agreement with the Company
          acknowledging the terms of the Warrant and (C) that the assignee
          confirms that it is an "accredited investor", as such term is defined
          in the United States Securities Act of 1933, as amended.

          (iii)   If the Board of Directors declines to approve an assignment
          they shall notify the assignee within one month of such refusal. If
          the Board of Directors approves an assignment they shall within the
          same period issue to the assignee a Warrant Certificate on
          substantially the terms of this Warrant and, as regards the

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          assigned Warrant, the Company shall recognize the assignee as the
          Holder for all purposes and the assignor shall cease to have any
          interest herein.

     (b)  Notwithstanding the foregoing, the Holder shall be entitled, without
obtaining the consent of the Company, to make an assignment of its interest in
this Warrant in whole or in part to any Affiliate of the Holder; PROVIDED that
concurrently with any such assignment of this Warrant or any part hereof, and as
a condition precedent to such valid assignment to such assignee, the assignee
shall execute and deliver to the Company an agreement to comply with all
restrictions and conditions relating to the ownership of this Warrant as are
herein set forth.

     (c)  Except as provided in paragraph (b), neither the Warrant nor any of
the Warrant Shares, nor any interest in either, may be sold, assigned, pledged,
hypothecated, encumbered or in any other manner transferred or disposed of, in
whole or in part, except in compliance with the terms, conditions and
restrictions as set forth in the Bye-Laws of the Company, applicable United
States federal and state securities laws and the terms and conditions hereof.
Each certificate for Warrant Shares issued upon exercise of the Warrant, unless
at the time of exercise such Warrant Shares are registered under the United
States Securities Act of 1933, as amended, shall bear the following legend:

          "These securities have not been registered under the United States
     Securities Act of 1933, as amended. These securities cannot be offered,
     transferred or sold unless (i) a registration statement under such Act is
     in effect with respect to such securities or a written opinion from counsel
     acceptable to the Company is obtained to the effect that no such
     registration is required and (ii) the transferee is approved by applicable
     regulatory authorities, if such approval is required. The Company reserves
     the right to refuse the transfer of such securities until such conditions
     have been fulfilled. In all events transfers of these securities are
     subject to the approval of the Board of Directors of the Company, in its
     sole discretion. The Bye-Laws of the Company contain other significant
     restrictions on transfers of shares of the Company."

     (d)  Notwithstanding the foregoing, in the event that the laws of any
jurisdiction to which the Holder, or any Affiliate of the Holder, is subject,
make it illegal for such entity to hold the Warrant or the Warrant Shares, the
Company shall use its best efforts to facilitate, and shall not unreasonably
withhold its permission to allow for, the transfer, sale or assignment of the
Warrant and/or the Warrant Shares.

     6.   ANTI-DILUTION PROVISIONS. (a) In case the Company shall at any time
after the date hereof (i) declare a dividend or make a distribution on Common
Stock payable in Common Stock, (ii) subdivide or split the outstanding Common
Stock, (iii) combine or reclassify the outstanding Common Stock into a smaller
number of shares, or (iv) issue any shares of its capital stock in a
reclassification of Common Stock (including any such reclassification in
connection with a consolidation, scheme of arrangement or amalgamation in which
the Company is the surviving entity), the Exercise Price in effect at the time
of the record date for such dividend or distribution or of the effective date of
such subdivision, split, combination or reclassification shall be
proportionately adjusted so that, giving effect to Section 6(i), the exercise of
this Warrant after such time shall entitle the holder to receive the aggregate
number of shares of Common Stock or other securities of the Company (or shares
of any security into which such

                                        6
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shares of Common Stock have been reclassified) which, if this Warrant had been
exercised immediately prior to such time, such holder would have owned upon such
exercise and been entitled to receive by virtue of such dividend, distribution,
subdivision, split, combination, amalgamation, scheme of arrangement or
reclassification. Such adjustment shall be made successively whenever any event
listed above shall occur.

     (b)  In case the Company shall issue or sell any Common Stock (other than
Common Stock issued (i) upon exercise of the Warrants, (ii) pursuant to the
Company's stock option plan or pursuant to any similar Common Stock related
employee compensation plan of the Company approved by the Company's Board of
Directors or (iii) upon exercise or conversion of any security the issuance of
which caused an adjustment under paragraphs (c) or (d) hereof) without
consideration or for a consideration per share less than the Current Market
Price Per Common Share (as defined in paragraph (f) immediately preceding such
issuance or sale or immediately preceding the announcement thereof, if earlier),
the Exercise Price to be in effect after such issuance or sale shall be
determined by multiplying the Exercise Price in effect immediately prior to such
issuance or sale or immediately preceding the announcement thereof, if earlier,
as the case may be, by a fraction, the numerator of which shall be the sum of
(x) the number of shares of Common Stock outstanding immediately prior to the
time of such issuance or sale multiplied by the Current Market Price Per Common
Share immediately prior to such issuance or sale or immediately preceding the
announcement thereof, as the case may be, and (y) the aggregate consideration,
if any, to be received by the Company upon such issuance or sale, and the
denominator of which shall be the product of the aggregate number of shares of
Common Stock outstanding immediately after such issuance or sale and the Current
Market Price Per Common Share immediately prior to such issuance or sale or
immediately preceding the announcement thereof, as the case may be. In case any
portion of the consideration to be received by the Company shall be in a form
other than cash, the fair market value of such noncash consideration shall be
utilized in the foregoing computation. Such fair market value shall be
determined by the Board of Directors of the Company; PROVIDED that if the Holder
shall object to any such determination, the Board of Directors shall retain an
independent appraiser reasonably satisfactory to the Holder to determine such
fair market value. The Holder shall be notified promptly of any consideration
other than cash to be received by the Company and furnished with a description
of the consideration and the fair market value thereof as determined by the
Board of Directors.

     (c)  In case the Company shall fix a record date for the issuance of
rights, options or warrants to the holders of its Common Stock or other
securities entitling such holders to subscribe for or purchase for a period
expiring within 60 days of such record date shares of Common Stock (or
securities convertible into share of Common Stock) at a price per share of
Common Stock (or having a conversion price per share of Common Stock, if a
security convertible into shares of Common Stock) less than the Current Market
Price Per Common Share on such record date or immediately preceding the
announcement thereof if earlier, the maximum number of shares of Common Stock
issuable upon exercise of such rights, options or warrants (or conversion of
such convertible securities) shall be deemed to have been issued and outstanding
as of such record date and the Exercise Price shall be adjusted pursuant to
paragraph (b) hereof as though such maximum number of shares of Common Stock had
been so issued for an aggregate consideration payable by the holders of such
rights, options, warrants or convertible securities prior to their receipt of
such shares of Common Stock. In case any portion of such

                                        7
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consideration shall be in a form other than cash, the fair market value of such
noncash consideration shall be determined as set forth in paragraph (b) hereof.
Such adjustment shall be made successively whenever such record date is fixed;
and in the event that such rights, options or warrants are not so issued or
expire unexercised, or in the event of a change in the number of shares of
Common Stock to which the holders of such rights, options or warrants are
entitled (other than pursuant to adjustment provisions therein comparable to
those contained in this Section 6), the Exercise Price shall again be adjusted
to be the Exercise Price which would then be in effect if such record date had
not been fixed, in the former event, or the Exercise Price which would then be
in effect if such holder had initially been entitled to such changed number of
shares of Common Stock, in the latter event.

     (d)  In case the Company shall issue rights, options (other than options
issued pursuant to a plan described in clause (b)(i) above) or warrants
entitling the holders thereof to subscribe for or purchase Common Stock (or
securities convertible into shares of Common Stock) or shall issue convertible
securities, and the price per share of Common Stock of such rights, options,
warrants or convertible securities (including, in the case of rights, options or
warrants, the price at which they may be exercised) is less than the Current
Market Price Per Common Share immediately preceding such issuance of rights,
options or warrants or immediately preceding the announcement thereof, if
earlier, the maximum number of shares of Common Stock issuable upon exercise of
such rights, options or warrants or upon conversion of such convertible
securities shall be deemed to have been issued and outstanding as of the date of
such sale or issuance, and the Exercise Price shall be adjusted pursuant to
paragraph (b) hereof as though such maximum number of shares of Common Stock had
been so issued for an aggregate consideration equal to the aggregate
consideration paid for such rights, options, warrants or convertible securities
and the aggregate consideration payable by the holders of such rights, options,
warrants or convertible securities prior to their receipt of such shares of
Common Stock. In case any portion of such consideration shall be in a form other
than cash, the fair market value of such noncash consideration shall be
determined as set forth in paragraph (b) hereof. Such adjustment shall be made
successively whenever such rights, options, warrants or convertible securities
are issued; and in the event that such rights, options or warrants expire
unexercised, or in the event of a change in the number of shares of Common Stock
to which the holders of such rights, options, warrants or convertible securities
are entitled (other than pursuant to adjustment provisions therein comparable to
those contained in this Section 6), the Exercise Price shall again be adjusted
to be the Exercise Price which would then be in effect if such rights, options,
warrants or convertible securities had not been issued, in the former event, or
the Exercise Price which would then be in effect if such holders had initially
been entitled to such changed number of shares of Common Stock, in the latter
event. No adjustment of the Exercise Price shall be made pursuant to this
paragraph (d) to the extent that the Exercise Price shall have been adjusted
pursuant to paragraph (c) upon the setting of any record date relating to such
rights, options, warrants or convertible securities and such adjustment fully
reflects the number of shares of Common Stock to which the holders of such
rights, options, warrants or convertible securities are entitled and the price
payable therefor.

     (e)  In case the Company shall fix a record date for the making of a
distribution to holders of Common Stock (including any such distribution made in
connection with a consolidation, scheme of arrangement or amalgamation in which
the Company is the surviving entity) of evidences of indebtedness, assets or
other property (other than dividends payable in

                                        8
<Page>

Common Stock or rights, options or warrants referred to in, and for which an
adjustment is made pursuant to, paragraph (c) hereof), the Exercise Price to be
in effect after such record date shall be determined by multiplying the Exercise
Price in effect immediately prior to such record date or immediately preceding
the announcement thereof, by a fraction, the numerator of which shall be the
Current Market Price Per Common Share on such record date, less the fair market
value (determined as set forth in paragraph (b) hereof) of the portion of the
assets, other property or evidence of indebtedness so to be distributed which is
applicable to one share of Common Stock, and the denominator of which shall be
such Current Market Price Per Common Share. Such adjustments shall be made
successively whenever such a record date is fixed; and in the event that such
distribution is not so made, the Exercise Price shall again be adjusted to be
the Exercise Price which would then be in effect if such record date had not
been fixed.

     (f)  For the purpose of any computation under this Section 6, on any
determination date (i) on or prior to the initial public offering of the
Company's common stock registered under the United States Securities Act of
1933, as amended, the Current Market Price Per Common Share shall, subject to
the penultimate sentence of this paragraph (f), be the fair market value per
share of the applicable class of Common Stock as reasonably determined by the
Board of Directors of the Company, and (ii) after such registered public
offering, the Current Market Price Per Common Share shall be deemed to be the
average (weighted by daily trading volume) of the Daily Prices (as defined
below) per share of the applicable class of Common Stock for the 20 consecutive
trading days immediately prior to such date. "DAILY PRICE" means (A) if the
shares of such class of Common Stock are then listed and traded on the New York
Stock Exchange, Inc. ("NYSE"), the closing price on the NYSE on such day as
reported on the NYSE Composite Transactions Tape; (B) if the shares of such
class of Common Stock are then not listed and traded on the NYSE, the closing
price on such day as reported by the principal national securities exchange on
which the shares are listed and traded; (C) if the shares of such class of
Common Stock are then not listed and traded on the NYSE or any such national
securities exchange, the last reported sale price on such day on the National
Market of the National Association of Securities Dealers, Inc. Automated
Quotation System ("NASDAQ"); or (D) if the shares of such class of Common Stock
are then not listed and traded on the NYSE or any national securities exchange
or traded on the NASDAQ National Market, the average of the highest reported bid
and lowest reported asked price on such day as reported by NASDAQ. If on any
determination date the shares of such class of Common Stock are not quoted by
any such organization, the Current Market Price Per Common Share shall be the
fair market value of such shares on such determination date as determined by the
Board of Directors. If the Holder shall object to any determination by the Board
of Directors of the Current Market Price Per Common Share, the Current Market
Price Per Common Share shall be the fair market value per share of the
applicable class of Common Stock as determined by an independent appraiser
retained by the Company at its expense and reasonably acceptable to the Holder.
For purposes of any computation under this Section 6, the number of shares of
Common Stock outstanding at any given time shall not include shares owned or
held by or for the account of the Company.

     (g)  No adjustment in the Exercise Price shall be required unless such
adjustment would require an increase or decrease of at least one percent in such
price; PROVIDED that any adjustments which by reason of this paragraph are not
required to be made shall be carried forward and taken into account in any
subsequent adjustment. All calculations under this

                                        9
<Page>

Section 6 shall be made to the nearest one tenth of a cent or to the nearest
hundredth of a share, as the case may be.

     (h)  In the event that, at any time as a result of the provisions of this
Section 6, the holder of this Warrant upon subsequent exercise shall become
entitled to receive any shares of capital stock of the Company other than Common
Stock, the number of such other shares so receivable upon exercise of this
Warrant shall thereafter be subject to adjustment from time to time in a manner
and on terms as nearly equivalent as practicable to the provisions contained
herein.

     (i)  Upon each adjustment of the Exercise Price as a result of the
calculations made in this Section 6, the number of shares for which this Warrant
is exercisable immediately prior to the making of such adjustment shall
thereafter evidence the right to purchase, at the adjusted Exercise Price, that
number of shares of Common Stock obtained by (i) multiplying the number of
shares covered by this Warrant immediately prior to this adjustment of the
number of shares by the Exercise Price in effect immediately prior to such
adjustment of the Exercise Price and (ii) dividing the product so obtained by
the Exercise Price in effect immediately after such adjustment of the Exercise
Price; PROVIDED that no adjustment shall be necessary pursuant to this paragraph
(i) to the extent (but only to the extent) that the number of shares of Common
Stock for which this Warrant is exercisable has already increased (or would
increase, if such Warrant were actually exercised) as a result of the
transaction in question by operation of the definition of "Warrant Percentage."

     (j)  Not less than 10 nor more than 30 days prior to the record date of any
action which requires an adjustment pursuant to this Section 6, the Company
shall file in the custody of the secretary of the Company at its principal
executive office an officer's certificate signed by the chairman, president or
chief financial officer of the Company showing the adjusted Exercise Price
determined as herein provided, setting forth in reasonable detail the facts
requiring such adjustment and the computation of such adjustment. Each such
officer's certificate shall be made available at all reasonable times for
inspection by the Holder and the Company shall, promptly after such adjustment,
mail a copy, by first-class mail, of such certificate to the Holder.

     (k)  The Holder shall, at its option, be entitled to receive, in lieu of
the adjustment pursuant to this Section 6 otherwise required, on the date of
exercise of the Warrant, the evidences of indebtedness, assets or other property
which such Holder would have been entitled to receive if it had exercised its
Warrant for shares of Common Stock immediately prior to the record date with
respect to such distribution. The Holder may exercise its option under this
paragraph (k) by written notice to the Company within fourteen days of its
receipt of the certificate of adjustment required pursuant to paragraph (j)
above to be delivered by the Company in connection with such distribution.

     7.   CONSOLIDATION, SCHEME OF ARRANGEMENT, AMALGAMATION, OR SALE OF ASSETS.
In case of any consolidation of the Company with, or amalgamation of the Company
into, any other Person, any amalgamation of another Person into the Company
(other than an amalgamation which does not result in any reclassification,
conversion, exchange or cancellation of outstanding shares of Common Stock), any
scheme of arrangement, or any sale or transfer of all or substantially all of
the assets of the Company or of the Person formed by such consolidation or

                                       10
<Page>

resulting from such amalgamation or which acquires such assets, as the case may
be, the Holder shall have the right thereafter to exercise this Warrant for the
kind and amount of securities, cash and other property receivable upon such
consolidation, scheme of arrangement, amalgamation, sale or transfer by a holder
of the number of shares of Common Stock for which this Warrant may have been
exercised immediately prior to such consolidation, scheme of arrangement,
amalgamation, sale or transfer, assuming (i) such holder of Common Stock is not
a Person with which the Company consolidated or into which the Company
amalgamated or which amalgamated into the Company or to which such sale or
transfer was made, as the case may be ("CONSTITUENT PERSON"), or an Affiliate of
a constituent Person and (ii) in the case of a consolidation, amalgamation, sale
or transfer which includes an election as to the consideration to be received by
the holders, such holder of Common Stock failed to exercise its rights of
election, as to the kind or amount of securities, cash and other property
receivable upon such consolidation, amalgamation, sale or transfer (provided
that if the kind or amount of securities, cash and other property receivable
upon such consolidation, amalgamation, sale or transfer is not the same for each
share of Common Stock held immediately prior to such consolidation,
amalgamation, sale or transfer by other than a constituent Person or an
Affiliate thereof and in respect of which such rights of election shall not have
been exercised ("NON-ELECTING SHARE"), then for the purpose of this Section the
kind and amount of securities, cash and other property receivable upon such
consolidation, amalgamation, sale or transfer by each non-electing share shall
be deemed to be the kind and amount so receivable per share by a plurality of
the non-electing shares). Adjustments for events subsequent to the effective
date of such a consolidation, amalgamation and sale of assets shall be as nearly
equivalent as may be practicable to the adjustments provided for in this
Warrant. In any such event, effective provisions shall be made in the
certificate or articles of incorporation of the resulting or surviving
corporation, in any contract of sale, conveyance, lease or transfer, or
otherwise so that the provisions set forth herein for the protection of the
rights of the Holder shall thereafter continue to be applicable; and the entity
whose securities, cash or other property for which this Warrant shall have
become exercisable shall expressly assume the obligation to deliver, upon
exercise, such securities, cash or other property. The provisions of this
Section 7 shall similarly apply to successive consolidations, amalgamations,
sales, leases or transfers.

     8.   LOSS OR DESTRUCTION OF WARRANT. Upon receipt by the Company of
evidence satisfactory to it (in the exercise of its reasonable discretion) of
the loss, theft, destruction or mutilation of this Warrant Certificate, and (in
the case of loss, theft or destruction) of reasonably satisfactory
indemnification, and upon surrender and cancellation of this Warrant
Certificate, if mutilated, the Company shall execute and deliver a new Warrant
Certificate of like tenor and date.

     9.   NOTICES. Any notice, demand or delivery authorized by this Warrant
Certificate shall be in writing and shall be given to the Holder or the Company,
as the case may be, at its address (or facsimile number) set forth below, or
such other address (or facsimile number) as shall have been furnished to the
party giving or making such notice, demand or delivery:

                                       11
<Page>

     If to the Company:

          AXIS Capital Holdings Limited
          c/o Conyers Dill & Pearman
          2 Church Street
          Clarendon House
          P.O. Box HM 666
          Hamilton HM CX, Bermuda
          Attention: Graham B.R. Collis
          Fax: (441) 299-4965

     With a copy to:

          [   ]

     If to the Holder:

          [   ]

     With a copy to:

          [   ]

Each such notice, demand or delivery shall be effective (i) if given by
facsimile, when receipt acknowledged or (ii) if given by any other means, when
received at the address specified herein.

     10.  RIGHTS OF THE HOLDER. Prior to the exercise of any Warrant, the Holder
shall not, by virtue hereof, be entitled to any rights of a shareholder of the
Company, including, without limitation, the right to vote, to receive dividends
or other distributions, to exercise any preemptive right or to receive any
notice of meetings of shareholders or any notice of any proceedings of the
Company except as may be specifically provided for herein.

     11.  GOVERNING LAW. THIS WARRANT CERTIFICATE AND ALL RIGHTS ARISING
HEREUNDER SHALL BE CONSTRUED AND DETERMINED IN ACCORDANCE WITH THE LAWS OF
BERMUDA, AND THE PERFORMANCE THEREOF SHALL BE GOVERNED AND ENFORCED IN
ACCORDANCE WITH SUCH LAWS.

     12.  AMENDMENTS; WAIVERS. Any provision of this Warrant may be amended or
waived if, and only if, such amendment or waiver is in writing and signed, in
the case of an amendment, by the Holder and the Company, or in the case of a
waiver, by the party against whom the waiver is to be effective. No failure or
delay by either party in exercising any right, power or privilege hereunder
shall operate as a waiver thereof nor shall any single or partial exercise
thereof preclude any other or further exercise thereof or the exercise of any
other right, power or privilege. The rights and remedies herein provided shall
be cumulative and not exclusive of any rights or remedies provided by law.

                                       12
<Page>

     13.  ENTIRE AGREEMENT; SUPERSESSION OF PRIOR WARRANT. The Company and the
Holder hereby acknowledge and agree that this warrant constitutes the entire
agreement between them with respect to the subject matter of this Warrant, and
this Warrant shall supersede all contemporaneous oral and all prior oral and
written agreements and understandings with respect to the subject matter hereof.
For avoidance of doubt, the Company, AXIS Specialty and the Holder hereby
acknowledge and agree that this Warrant supersedes and replaces in its entirety
the Old Warrant, and the Old Warrant is hereby cancelled and of no further force
or effect. Simultaneously with the execution of this Warrant, the Holder has
returned to AXIS Specialty the original Old Warrant.

                                       13
<Page>

     IN WITNESS WHEREOF, the Company has duly caused this Warrant to be signed
and attested by its duly authorized officers and to be dated as of December 31,
2002.

                                         AXIS Capital Holdings Limited

                                         By:
                                            ------------------------------------
                                            Name:
                                            Title:

Attest:
       -----------------------

ACKNOWLEDGED AND AGREED:

------------------------------
[   ]

For Purposes of Section 13 only:

AXIS Specialty Limited

By:
     -------------------------
     Name:
     Title:

<Page>

                       WARRANT EXERCISE OR EXCHANGE NOTICE

                (To be delivered prior to exercise of the Warrant
             by execution of the Warrant Exercise Subscription Form)

To:  AXIS Capital Holdings Limited

     The undersigned hereby notifies you of its intention to exercise or
exchange the Warrant to purchase shares of common stock, par value $0.10 per
share, of AXIS Capital Holdings Limited.

     The undersigned intends to exercise the Warrant to purchase ________ shares
of Common Stock (the "SHARES") at $ __________per Share (the Exercise Price
currently in effect pursuant to the Warrant). The undersigned intends to pay the
aggregate Exercise Price for the Shares in cash, certified or official bank or
bank cashier's check (or a combination of cash and check) as indicated below.

                                      -OR-

     The undersigned intends to exercise the Warrant to purchase shares of
Common Stock (the "SHARES") and wishes, in lieu of paying the Exercise Price of
$___________ per Share currently in effect pursuant to the Warrant, to receive
that number of shares reduced by a number of shares of Common Stock having a
value calculated by reference to the aggregate Daily Price (as defined in the
Warrant) equal to the aggregate Exercise Price for the Shares.

                                      -OR-

     The undersigned intends pursuant to Section 2(d)(ii) of the Warrant to
exchange the Warrant (or a portion thereof), insofar as it relates _________ to
shares of Common Stock, for the number of shares of Common stock determined
pursuant to such Section 2(d)(ii).

                                      -OR-

     The undersigned intends to exercise the Warrant to purchase shares of
Common Stock (the "SHARES") at the Exercise Price of $_________ per share
currently in effect pursuant to the Warrant, and intends to pay $__________ of
the aggregate Exercise Price for the Shares in cash, certified or official bank
or bank cashier's check (or a combination of cash and check) as indicated below,
and to deliver as payment of $__________ of the aggregate Exercise Price that
number of shares of Common Stock having a value calculated by reference to the
aggregate Daily Price (as defined in the Warrant) equal to or in excess of such
portion of the aggregate Exercise Price for the Shares.

                                       A-1
<Page>

Date:_____________

                                                     ---------------------------
                                                         (Signature of Owner)

                                                     ---------------------------
                                                              (Address)

                                                     ---------------------------
                                                              (Address)

Payment:  $_______ cash
          $_______ check
          ________ shares of Common Stock having a Daily Price of $_____________

                                       A-2
<Page>

                       WARRANT EXERCISE SUBSCRIPTION FORM

                (To be executed only upon exercise of the Warrant
                   after delivery of Warrant Exercise Notice)

To:  AXIS Capital Holdings Limited

     The undersigned irrevocably exercises the Warrant for the purchase of
shares of common stock (the "SHARES"), par value $0.10 per share, of AXIS
Capital Holdings Limited (the "COMPANY") at $_________ per Share (the Exercise
Price currently in effect pursuant to the Warrant) and herewith makes payment of
$_________ (such payment being made in cash or by certified or official bank or
bank cashier's check payable to the order of the Company or by any permitted
combination of such cash or check), all on the terms and conditions specified in
the within Warrant Certificate, surrenders this Warrant Certificate and all
right, title and interest therein to the Company and directs that the Shares
deliverable upon the exercise of this Warrant be registered or placed in the
name and at the address specified below and delivered thereto.

                                      -OR-

     The undersigned irrevocably exercises the Warrant for the purchase of
shares of Common Stock (the "SHARES"), par value $0.10 per share, of AXIS
Capital Holdings Limited (the "COMPANY") at $_________ per Share (the Exercise
Price currently in effect pursuant to the Warrant) (provided that in lieu of
payment of $_________, the undersigned will receive a number of Shares reduced
by a number of shares of Common Stock having a value calculated by reference to
the aggregate Daily Price (as defined in the Warrant) equal to the aggregate
Exercise Price for the Shares), all on the terms and conditions specified in the
Warrant Certificate, surrenders this Warrant Certificate and all right, title
and interest therein to the Company and directs that the Shares deliverable upon
the exercise of this Warrant be registered or placed in the name and at the
address specified below and delivered thereto.

                                      -OR-

     The undersigned irrevocably exercises the Warrant for the purchase of
shares of Common Stock (the "SHARES"), par value $0.10 per share, of AXIS
Capital Holdings Limited (the "COMPANY") at $_________ per share (the Exercise
Price currently in effect pursuant to the Warrant) (such payment being made by
delivering that number of shares of Common Stock having a value calculated by
reference to the aggregate Daily Price (as defined in the Warrant) equal to or
in excess of the aggregate Exercise Price for the Shares), all on the terms and
conditions specified in the Warrant Certificate, surrenders this Warrant
Certificate and all right, title and interest therein to the Company and directs
that the Shares deliverable upon the exercise of this Warrant be registered or
placed in the name and at the address specified below and delivered thereto.

                                      -OR-

     The undersigned irrevocably exercises the Warrant for the purchase shares
of Common Stock (the "Shares"); par value $0.10 per share, of AXIS Capital
Holdings Limited (the

                                       A-3
<Page>

"Company") at $_________ per Share (the Exercise Price currently in effect
pursuant to the Warrant), and herewith makes payment of $_________ of the
aggregate Exercise Price for the Shares in cash, certified or official bank or
bank cashier's check (or a combination of cash and check), and herewith delivers
as payment of $_________ of the aggregate Exercise Price that number of shares
of Common Stock having a value calculated by reference to the aggregate Daily
Price (as defined in the Warrant) equal to or in excess of such portion of the
aggregate Exercise Price for the Shares, all on the terms and conditions
specified in the Warrant Certificate, surrenders this Warrant Certificate and
all right, title and interest therein to the Company and directs that the Shares
deliverable upon the exercise of this Warrant be registered or placed in the
name and at the address specified below and delivered thereto.
Date:

Date:_____________

                                                     ---------------------------
                                                            (Signature of Owner)

                                                     ---------------------------
                                                            (Address)

                                                     ---------------------------
                                                            (Address)

                                       A-4
<Page>

Securities and/or check to be issued to:________________________________________

Please insert social security or identifying number:____________________________

Name:___________________________________________________________________________

Street Address:_________________________________________________________________

City, State and Zip Code:_______________________________________________________

Any unexercised portion of the Warrant evidenced by the within Warrant
Certificate to be issued to:

Please insert social security or identifying number:____________________________

Name:___________________________________________________________________________

Street Address:_________________________________________________________________

City, State and Zip Code:_______________________________________________________

                                       A-5
<Page>

                             WARRANT ASSIGNMENT FORM

                                                              Date:_____________

FOR VALUE RECEIVED,________________________________________________hereby sells,

assigns and transfers unto___________________________________(the "ASSIGNEE"),
                        (please type or print in block letters)

________________________________________________________________________________
                                (insert address)

its right to purchase up to shares of common stock, par value $0.10 per share,
of AXIS Capital Holdings Limited (the "Company") represented by this Warrant and
does hereby irrevocably constitute and appoint Attorney, to transfer the same on
the books of the Company, with full power of substitution in the premises.

                                              Signature:
                                                        ------------------------

                                       A-6

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