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                                                                  EXHIBIT 10.1.3

                           [Form of Class A Warrants]

                 [    ] CLASS A WARRANTS, EACH WARRANT ENTITLING
                     THE HOLDER TO PURCHASE ONE COMMON SHARE
                           OF ARCH CAPITAL GROUP LTD.

                             ARCH CAPITAL GROUP LTD.

THIS WARRANT (THE "WARRANT") AND THE UNDERLYING COMMON SHARES MAY NOT BE
TRANSFERRED EXCEPT (I) IN COMPLIANCE WITH THE PROVISIONS OF THE AGREEMENTS
PURSUANT TO WHICH THE CLASS A WARRANTS WERE ORIGINALLY ISSUED AND ANY APPLICABLE
STATE SECURITIES OR "BLUE SKY" LAWS AND (II) (A) PURSUANT TO AN EFFECTIVE
REGISTRATION UNDER THE SECURITIES ACT OF 1933, AS AMENDED (THE "ACT"), (B) IN
COMPLIANCE WITH RULE 144 UNDER THE ACT, (C) INSIDE THE UNITED STATES TO A
"QUALIFIED INSTITUTIONAL BUYER" AS DEFINED IN AND IN COMPLIANCE WITH RULE 144A
UNDER THE ACT, (D) OUTSIDE THE UNITED STATES IN COMPLIANCE WITH RULE 904 OF
REGULATION S UNDER THE ACT OR (E) INSIDE THE UNITED STATES TO AN INSTITUTIONAL
"ACCREDITED INVESTOR", AS DEFINED IN RULE 501(A) (1), (2), (3) OR (7) UNDER THE
ACT IN A TRANSACTION WHICH, IN THE OPINION OF COUNSEL REASONABLY SATISFACTORY TO
THE COMPANY, QUALIFIES AS AN EXEMPT TRANSACTION UNDER THE ACT AND THE RULES AND
REGULATIONS PROMULGATED THEREUNDER.

      ARCH CAPITAL GROUP LTD., a Bermuda exempted limited company (the
"COMPANY"), hereby certifies that, for value received, [     ] (the "HOLDER"),
or its assigns, is entitled, subject to the terms set forth below, to purchase
from the Company, at any time and from time to time in whole or in part, an
aggregate of [    ] fully paid and nonassessable common shares, par value $.01
per share ("COMMON SHARES"), of the Company during the period beginning on the
date of issuance hereof (the "CLOSING DATE") and ending on September 15, 2002
(the "EXERCISE PERIOD").

      1. PURCHASE PRICE. Such Common Shares shall be purchased at a purchase
price per share, subject to the provisions of Paragraph 3 hereof, equal to
$20.00 (as adjusted in accordance with the terms hereof, the "PURCHASE PRICE").
The number and character of such Common Shares are subject to adjustment as
provided below, and the term "Common Shares" shall mean, unless the context
otherwise requires, the Common Shares or other securities or property at the
time deliverable upon the exercise of this Warrant.
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      2. EXERCISE OF WARRANT. The purchase rights evidenced by this Warrant
shall be exercised by the Holder surrendering this Warrant, with the form of
subscription at the end hereof duly executed by such Holder (the "EXERCISE
NOTICE"), to the Company at its offices, 20 Horseneck Lane, Greenwich,
Connecticut 06830, accompanied by payment as specified below of the aggregate
Purchase Price determined as of the Determination Date (as defined below) of the
Common Shares being purchased pursuant to such exercise. Payment of the
aggregate Purchase Price may be made, at the option of the Holder, (i) in cash,
(ii) by certified check or bank cashier's check payable to the order of the
Company in the amount of such Purchase Price, (iii) by delivering Common Shares
with an aggregate Market Price (as hereinafter defined) as of the day prior to
the Company's receipt of the Exercise Notice (the "DETERMINATION DATE") equal to
the product of the Purchase Price and the number of Common Shares being
purchased, (iv) by the Company reducing, at the request of the Holder, the
number of Common Shares for which this Warrant is exercisable by a number of
Common Shares (the "SURRENDERED STOCK") such that the product of (a) the Market
Price per Common Share as of the Determination Date less the Purchase Price in
effect on the Determination Date multiplied by (b) the Surrendered Stock equals
or exceeds the product of (x) the Purchase Price in effect on the Determination
Date and (y) the number of Common Shares being purchased, or any combination of
the methods of payment described in clauses (i) through (iv) above.

      2.1. PARTIAL EXERCISE. This Warrant may be exercised for less than the
full number of Common Shares at the times called for hereby, in which case the
number of Common Shares receivable upon the exercise of this Warrant as a whole,
the sum payable upon the exercise of this Warrant as a whole, shall be
proportionately reduced. Upon any such partial exercise, the Company at its
expense will forthwith issue to the Holder hereof a new Warrant or Warrants of
like tenor for the number of Common Shares as to which rights have not been
exercised, such Warrant or Warrants to be issued in the name of the Holder
hereof or his nominee (upon payment by such Holder of any applicable transfer
taxes).

      2.2. DELIVERY OF CERTIFICATES FOR COMMON SHARES ON CONVERSION. As soon as
practicable after the exercise of this Warrant and payment of the Purchase
Price, and in any event within ten (10) days thereafter, the Company, at its
expense, will cause to be issued in the name of and delivered to the Holder
hereof a certificate or certificates for the number of fully paid and
nonassessable Common Shares or other securities or property to which such Holder
shall be entitled upon such exercise, plus, in lieu of any fractional share to
which such Holder would otherwise be entitled, cash in an amount determined in
accordance with Paragraph 3.9 hereof. The Company agrees that the Common Shares
so received shall be deemed to be issued to the Holder as the record owner of
such Common Shares as of the close of business on the date on which this Warrant
shall have been surrendered and payment for such Common Shares made as
aforesaid.
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      3. ANTI-DILUTION PROVISIONS AND OTHER ADJUSTMENTS. In order to prevent
dilution of the right granted hereunder, the Purchase Price shall be subject to
adjustment from time to time in accordance with this Paragraph 3. Upon each
adjustment of the Purchase Price pursuant to this Paragraph 3, the registered
holder hereof shall thereafter be entitled to acquire upon exercise of this
Warrant, at the Purchase Price resulting from such adjustment, the number of
shares of the Company's Common Shares obtainable by multiplying the Purchase
Price in effect immediately prior to such adjustment by the number of shares of
the Company's Common Shares acquirable upon conversion thereof immediately prior
to such adjustment and dividing the product thereof by the Purchase Price
resulting from such adjustment.

      3.1. ADJUSTMENT FOR ISSUE OR SALE OF COMMON SHARES. Except as provided in
Paragraph 3.5 below, if and whenever on or after the date of issuance hereof the
Company shall issue or sell, or shall in accordance with subparagraphs (1) to
(8) of this Paragraph 3.1 inclusive, be deemed to have issued or sold any Common
Shares to any person for a consideration per share less than the Market Price
(or, in the case of an issuance or sale in connection with an underwritten
public offering, 99% of the Market Price less underwriting discounts and
commissions) in effect immediately prior to the time of such issue or sale, then
forthwith upon such issue or sale (each, a "TRIGGERING TRANSACTION"), the
Purchase Price shall, subject to subparagraphs (1) to (8) of this Paragraph 3.1,
be reduced to the Purchase Price (calculated to the nearest tenth of a cent)
determined by multiplying the Purchase Price in effect immediately prior to the
time of such Triggering Transaction by a fraction, the numerator of which shall
be the sum of (x) the product of the Number of Common Shares Deemed Outstanding
(as defined below) immediately prior to such Triggering Transaction multiplied
by the Market Price immediately prior to such Triggering Transaction plus
(y) the total amount, if any, received or receivable at any time by the Company
as consideration for the issuance or sale of such Common Shares, and the
denominator of which shall be the product of (x) the Number of Common Shares
Deemed Outstanding immediately after such Triggering Transaction, multiplied by
(y) the Market Price immediately prior to such Triggering Transaction; PROVIDED
that in the case of a sale of Common Shares pursuant to a purchase, underwriting
or similar agreement, the Market Price shall mean the Market Price in effect
upon the date such agreement is executed by the Company.

      For purposes of this Paragraph 3, the term "NUMBER OF COMMON SHARES DEEMED
OUTSTANDING" at any given time shall mean the sum of (i) the number of Common
Shares outstanding at such time, and (ii) the number of Common Shares deemed to
be outstanding under subparagraphs (1) to (8) of this Paragraph 3, inclusive, at
such time. For purposes of this Paragraph 3, inclusive, at such time. For
purposes of this Paragraph 3, the term "MARKET PRICE" shall mean, as of any
date, (a) for any period during which a security shall be listed for trading on
a national securities exchange or on the Nasdaq Stock Market ("NASDAQ"), or (in
the case of non-United States securities) similar securities exchange, the
closing price per
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                                      -4-

share of such security as of such day, or, in case no reported sale occurs on
such trading day, the mean of the reported closing bid and asked prices per
share for the prior five trading days ending on such day, (b) for any period
during which such security shall not be so listed, but when prices for such
security shall be reported by Nasdaq or similar system in the case of non-U.S.
securities, the mean of the most recent average bid and asked prices per share
as quoted by Nasdaq or such other system or (c) the market price per share of
such security as determined by the Board of Directors of the Company (the "BOARD
OF DIRECTORS") as of the next preceding day, in the event neither (a) nor (b)
above shall be applicable, or in the event the Board of Directors shall be in
good faith determine that application of (a) or (b) would not result in a fair
determination of the Market Price.

      For purposes of determining the adjusted Purchase Price under this
Paragraph 3.1, the following subparagraphs (1) to (8), inclusive, shall be
applicable:

            (1) In case the Company at any time shall in any manner issue or
      sell (whether directly or by assumption in a merger or otherwise) any
      rights to subscribe for or to purchase, or any options for the purchase
      of, Common Shares or any other securities convertible into or exchangeable
      or exercisable for Common Shares (such rights or options being herein
      called "OPTIONS" and such convertible, exchangeable or exercisable shares
      or securities being herein called "CONVERTIBLE SECURITIES"), whether or
      not such Options or the right to convert, exercise or exchange any such
      Convertible Securities are immediately exercisable, and the price per
      share for which Common Shares are issuable upon exercise, conversion or
      exchange (determined by dividing (x) the total amount, if any, received or
      receivable by the Company as consideration for the issuing of such
      Options, plus the minimum aggregate amount of additional consideration
      payable to the Company upon the exercise of all such Options, by (y) the
      total maximum number of Common Shares issuable upon the exercise of such
      Options or the conversion or exchange of such Convertible Securities)
      shall be less than the Market Price (or, in the case of an issuance or
      sale in connection with an underwritten public offering, less than 99% of
      the Market Price less underwriting discounts and commissions), determined
      as of the date of such issuance or sale (or, in the case of an issuance or
      sale pursuant to a purchase, underwriting or similar agreement, the Market
      Price determined as of the date such agreement is executed by the
      Company), then the total maximum number of Common Shares issuable upon
      exercise of such Options or conversion or exchange of all such Convertible
      Securities shall (as of the date of the grant of such Option) be deemed to
      be outstanding and to have been issued and sold by the Company for such
      price per share. No adjustment of the Purchase Price shall be made upon
      exercise of such Options or conversion or exchange of such Convertible
      Securities, except as otherwise provided in subparagraph (3) below.
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            (2) In case the Company at any time shall in any manner issue or
      sell (whether directly or by assumption in a merger or otherwise) any
      Convertible Securities, whether or not the rights to exchange, exercise or
      convert thereunder are immediately exercisable, and the price per share
      for which Common Shares are issuable upon such conversion, exercise or
      exchange (determined by dividing (x) the total amount received or
      receivable by the Company as consideration for the issue or sale of such
      Convertible Securities, plus the minimum aggregate amount of additional
      consideration, if any, payable to the Company upon the conversion or
      exchange thereof, by (y) the total maximum number of Common Shares
      issuable upon the conversion or exchange of all such Convertible
      Securities) shall be less than the Market Price (or, in the case of an
      issuance or sale in connection with an underwritten public offering, less
      than 99% of the Market Price less underwriting discounts and commissions),
      determined as of the date of such issuance or sale (or, in the case of an
      issuance or sale pursuant to a purchase, underwriting or similar
      agreement, the Market Price determined as of the date such agreement is
      executed by the Company), then the total maximum number of Common Shares
      issuable upon conversion or exchange of all such Convertible Securities
      shall (as of the date of the issue or sale of such Convertible Securities)
      be deemed to be outstanding and to have been issue and sold by the Company
      for such price per share. No adjustment of the Purchase Price shall be
      made upon the actual issue of such Common Shares upon exercise of the
      rights to exchange or convert under such Convertible Securities, except as
      otherwise provided in subparagraph (3) below.

            (3) If the exercise price provided for in any Options referred to in
      subparagraph (1), the additional consideration, if any, payable upon the
      conversion or exchange of any Convertible Securities referred to in
      subparagraph (1) or (2), or the rate at which any Convertible Securities
      referred to in subparagraph (1) or (2) are convertible into or
      exchangeable for Common Shares, shall change at any time (other than under
      or by reason of provisions designed to protect against dilution of the
      type set forth in Paragraph 3.1 or 3.3), the Purchase Price in effect at
      the time of such change shall forthwith be readjusted to the Purchase
      Price which would have been in effect at the time had such Options or
      Convertible Securities still outstanding provided for such changed
      purchase price, additional consideration or conversion or exchange rate,
      as the case may be, at the time initially granted, issued or sold. If the
      exercise price provided for in any Option referred to in subparagraph (1),
      the additional consideration, if any, payable upon the conversion or
      exchange of any Convertible Securities referred to in subparagraph (1) or
      (2), or the rate at which any Convertible Securities referred to in
      subparagraph (1) or (2) are convertible into or exchangeable for Common
      Shares, shall be reduced at any time under or by reason of provisions with
      respect thereto designed to protect against dilution, then the Purchase
      Price then in effect hereunder shall forthwith be adjusted to such
      respective amount as would have been obtained had such
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      Option or Convertible Security never been issued as to such Common Shares
      and had adjustments been made upon the issuance of Common Shares delivered
      as aforesaid, but only if as a result of such adjustment the Purchase
      Price then in effect hereunder is hereby reduced and provided that there
      shall be no duplication of any adjustments otherwise made in accordance
      with the terms hereof.

            (4) On the expiration of any Option or the termination of any right
      to convert or exchange any Convertible Securities, the Purchase Price then
      in effect hereunder shall forthwith be increased to the Purchase Price
      which would have been in effect at the time of such expiration or
      termination had such Option or Convertible Securities, to the extent
      outstanding immediately prior to such expiration or termination, never
      been issued.

            (5) In case any Options shall be issued in connection with the issue
      or sale of other securities of the Company, together comprising one
      integral transaction in which no specific consideration is allocated to
      such Options by the parties thereto, such Options shall be deemed to have
      been issued without consideration.

            (6) In case any Common Shares, Options or Convertible Securities
      shall be issued or sold or deemed to have been issued or sold for cash,
      the consideration received therefor shall be deemed to be the amount
      received by the Company therefor. In case any Common Shares, Options or
      Convertible Securities shall be issued or sold for a consideration other
      than cash, the amount of the consideration other than cash received by the
      Company shall be the fair value of such consideration as determined in
      good faith by the Board of Directors.

            (7) The number of Common Shares outstanding at any given time shall
      not include shares owned or held by or for the account of the Company, but
      the disposition of any shares so owned or held shall be considered an
      issue or sale of Common Shares for the purpose of this Paragraph 3.1.

            (8) In case the Company shall declare a dividend or make any other
      distribution upon the stock of the Company payable in Common Shares,
      Options or Convertible Securities, then in such case any Common Shares,
      Options or Convertible Securities, as the case may be, issuable in payment
      of such dividend or distribution shall be deemed to have been issued or
      sold without consideration.

      3.2. DETERMINATION OF DATE OF ISSUANCE OR SALE OF COMMON SHARES. For
purposes of Paragraph 3.1, in case the Company shall take a record of the
holders of its Common Shares for the purpose of determining holders entitled
(x) to receive a dividend or other distribution payable in Common Shares,
Options or Convertible Securities, or (y) to subscribe for or purchase Common
Shares, Options or Convertible Securities, then such record date shall be
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                                      -7-

deemed to have been issued or sold upon the declaration of such dividend or the
making of such other distribution or the date of the granting of such right or
subscription or purchase, as the case may be.

      3.3. SUBDIVISIONS AND COMBINATIONS. In case the Company shall at any time
subdivide (other than by means of a dividend payable in Common Shares covered by
subparagraph 3.1(8)) its outstanding Common Shares into a greater number of
shares, the Purchase Price in effect immediately prior to such subdivision shall
be appropriately reduced, and, conversely, in case the outstanding Common Shares
of the Company shall be combined into a smaller number of shares, the Purchase
Price in effect immediately prior to such combination shall be proportionately
increased.

      3.4. REORGANIZATION, RECLASSIFICATION, CONSOLIDATION, MERGER OR SALE OF
ASSETS. If any capital reorganization or reclassification of the shares of the
Company, or consolidation or merger of the Company with another corporation, or
the sale of all or substantially all of its assets to another corporation shall
be effected in such a way that holders of Common Shares shall be entitled to
receive stock, securities, cash or other property with respect to or in exchange
for Common Shares, then, as a condition of such reorganization,
reclassification, consolidation, merger or sale, lawful and adequate provision
shall be made whereby the Holder of this Warrant shall have the right to acquire
and receive upon exercise hereof such common stock, securities, cash or other
property issuable or payable (as part of the reorganization, reclassification,
consolidation, merger or sale) with respect to or in exchange for such number of
outstanding Common Shares of the Company as would have been received upon
exercise of this Warrant at the Purchase Price then in effect. The Company will
not effect any such consolidation, merger or sale unless prior to the
consummation thereof the successor corporation (if other than the Company)
resulting from such consolidation or merger or the corporation purchasing such
assets shall assume by written instrument, mailed or delivered to the Holder of
this Warrant at the last address of such Holder appearing on the books of the
Company, the obligation to deliver to such holder such common stock, securities
or assets as, in accordance with the foregoing provisions, such Holder may be
entitled to purchase. In the event that pursuant to the terms of this Paragraph
3.4, this Warrant becomes exercisable for securities other than Common Shares,
then the provisions of Section 3 shall apply to such securities as though such
securities were Common Shares. If a purchase, tender or exchange offer is made
to and accepted by the holders of more than 50% of the outstanding Common Shares
of the Company, the Company shall not effect any consolidation, merger or sale
with the person having made such offer or with any Affiliate of such person
unless prior to the consummation of such consolidation, merger or sale the
holder of this Warrant shall have been given a reasonable opportunity to then
elect to receive upon the exercise of this Warrant either the common shares,
securities or assets then issuable with respect to the Common Shares of the
Company or the common shares, securities or assets, or the equivalent, issued to
previous holders of the Common Shares in accordance with such offer. For
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purposes hereof the term "AFFILIATE" with respect to any given person shall mean
any person controlling, controlled by or under common control with the given
person.

      3.5. NO ADJUSTMENT FOR EXERCISE OF CERTAIN OPTIONS, WARRANTS, ETC. The
provisions of this Section 3 shall not apply to any Common Shares issued,
issuable or deemed outstanding under subparagraphs (1) to (8) of paragraph 3.1,
inclusive: (i) to any person pursuant to any stock option, stock purchase or
similar plan or arrangement for the benefit of employees or directors of the
Company or its subsidiaries; PROVIDED that any Common Shares issued pursuant to
any such plan or arrangement (in excess of 1,700,000 shares) shall be approved
by the Board of Directors or a duly organized committee thereof, it being
further provided that, with respect to Class A Warrants exercised by the Initial
Investors or their Affiliates (as defined in the Registration Statement filed by
the Company in connection with the Company's public offering of its Common
Shares), this Section 3 shall apply with respect to any Common Shares issued
pursuant to any such plan or arrangement in excess of 1,700,000 shares
(ii) pursuant to options, warrants and conversion rights in existence on the
date of issuance hereof or (iii) pursuant to the exercise of any Class A
Warrants or Class B Warrants described in the Company's prospectus relating to
the initial public offering of its Common Shares.

      3.6. NOTICES OF RECORD DATE, ETC. In the event that:

            (1) the Company shall declare any cash dividend upon its Common
      Shares, or

            (2) the Company shall declare any dividend upon its Common Shares
      payable in common shares or make any special dividend or other
      distribution to the holders of its Common Shares, or

            (3) the Company shall offer for subscription PRO RATA to the holders
      of its Common Shares any additional common shares of any class or other
      rights, or

            (4) there shall be any capital reorganization or reclassification of
      the shares of the Company, including any subdivision or combination of its
      outstanding Common Shares, or consolidation or merger of the Company with,
      or sale of all or substantially all of its assets to, another corporation,
      or

            (5) there shall be a voluntary or involuntary dissolution,
      liquidation or winding up of the Company;

then, in connection with such event, the Company shall give to the Holder of
this Warrant:
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            (i)   at least ten (10) days' prior written notice of the date on
                  which the books of the Company shall close or a record shall
                  be taken for such dividend distribution or subscription rights
                  or for determining rights to vote in respect of any such
                  reorganization, reclassification, consolidation, merger, sale,
                  dissolution, liquidation or winding up; and

            (ii)  in the case of any such reorganization, reclassification,
                  consolidation, merger, sale, dissolution, liquidation or
                  winding up, at least ten (10) days prior written notice of the
                  date when the same shall take place.

Such notice in accordance with the foregoing clause (i) shall also specify, in
the case of any such dividend, distribution or subscription rights, the date on
which the holders of Common Shares shall be entitled thereto, and such notice in
accordance with the foregoing clause (ii) shall also specify the date on which
the holders of Common Shares shall be entitled to exchange their Common Shares
for securities or other property deliverable upon such reorganization,
reclassification, consolidation, merger, sale, dissolution, liquidation or
winding up, as the case may be. Each such written notice shall be given by first
class mail, postage prepaid, addressed to the Holder of this Warrant at the
address of such Holder as shown on the books of the Company.

      3.7. ADJUSTMENTS FOR OTHER DISTRIBUTIONS. If at any time or from time to
time on or after the date of issuance hereof, the Company shall distribute,
grant or issue to all holders of its Common Shares any of its assets or debt
securities or any rights or warrants to purchase assets or securities of the
Company (including securities or cash, but excluding (x) distributions covered
by Section 3.1 or (y) cash dividends or other cash distributions that are paid
out of consolidation current net earnings or earnings retained in the business
as shown on the books of the Company unless such cash distributions in any
twelve month period exceed 3.00% of the average Market Price of the Common
Shares for the fifteen (15) trading days ending on the first day of such twelve
month period) (collectively, the "PURCHASE RIGHTS"), then the Holder of this
Warrant shall be entitled to, at its option:

            (i)   have the Purchase Price reduced to the Purchase Price
                  (calculated to the nearest tenth of a cent) determined by
                  multiplying the Purchase Price in effect immediately prior to
                  the time of distribution of such Purchase Rights by a
                  fraction, the numerator of which shall be the difference
                  between (x) the Market Price immediately prior to such
                  distribution and (y) the fair market value of the assets,
                  securities, rights or warrants applicable to one Common Share
                  (which fair market value shall be determined by the Company in
                  accordance with generally accepted accounting principals) and
                  the denominator of which shall be the Market Price immediately
                  prior to such distribution; PROVIDED that in the case of a
                  distribution, grant or issuance pursuant to a purchase,
                  underwriting
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                                      -10-

                  or similar agreement, the Market Price shall mean the Market
                  Price in effect upon the date such agreement is executed by
                  the Company; or

            (ii)  acquire (within thirty (30) days after the later to occur of
                  the initial exercise date of such Purchase Rights or receipt
                  by such Holder of the notice concerning Purchase Rights to
                  which such Holder shall be entitled under Paragraph 3.6) and
                  upon the terms applicable to such Purchase Rights either:
                  (x) the aggregate Purchase Rights which such Holder could have
                  acquired if it had held the number of Common Shares acquirable
                  upon exercise of this Warrant immediately before the grant,
                  issuance or sale of such Purchase Rights; provided that if any
                  Purchase Rights were distributed to holders of Common Shares
                  without the payment of additional consideration by such
                  holders, corresponding Purchase Rights shall be distributed to
                  the exercising holder of this Warrant as soon as possible
                  after such exercise and it shall not be necessary for the
                  exercising holder of this Warrant specifically to request
                  delivery of such rights; or (y) in the event that any such
                  Purchase Rights shall have expired or shall expire prior to
                  the end of said thirty (30) day period, the number of Common
                  Shares or the amount of property which such Holder could have
                  acquired upon such exercise at the same time or times at which
                  the Company granted, issued or sold such expired Purchase
                  Rights.

      3.8. ADJUSTMENT BY BOARD OF DIRECTORS. If any event occurs as to which the
provisions of this Section 3 are not strictly applicable or if strictly
applicable would not fairly protect the rights of the Holder of this Warrant in
accordance with the essential intent and principles of such provisions, then the
Board of Directors shall make an adjustment in the application of such
provisions, in accordance with such essential intent and principles, so as to
protect such rights as aforesaid, but in no event shall any adjustment have the
effect of increasing the Purchase Price as otherwise determined pursuant to any
of the provisions of this Section 3 except in the case of a combination of
shares of a type contemplated in Paragraph 3.3 and then in no event to an amount
larger than the Purchase Price as adjusted pursuant to Paragraph 3.3.

      3.9. FRACTIONAL SHARES. The company shall not issue fractions of Common
Shares upon exercise of this Warrant or scrip in lieu thereof. If any fraction
of a Common Share would, except for the provisions of this Paragraph 3.9, be
issuable upon exercise of this Warrant, the company shall in lieu thereof pay to
the person entitled thereto an amount in cash equal to the product of such
fraction, calculated to the nearest one-hundredth (1/100), and the Market Price
of a Common Share as of such date of exercise.
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      3.10. OFFICERS' STATEMENT AS TO ADJUSTMENTS. Whenever the Purchase Price
shall be adjusted as provided in this Section 3, the Company shall forthwith
file at the office designated for the exercise of this Warrant a statement,
signed by the chairman of the Board, the President, any Vice President or the
Treasurer of the Company, showing in reasonable detail the facts requiring such
adjustment and the Purchase Price that will be effective after such adjustment.
The Company shall also cause a notice setting forth any such adjustments to be
sent by mail, first class, postage prepaid, to the record holder of this warrant
at his or its address appearing on the Warrant register of the Company. If such
notice relates to an adjustment resulting from an event referred to in Paragraph
3.6, such notice shall be included as part of the notice required to be mailed
and published under the provisions of Paragraph 3.6.

      4. NO DILUTION OR IMPAIRMENT. The Company will not, by amendment of its
charter or through reorganization, reclassification, consolidation,
amalgamation, merger, dissolution, sale of assets or any other voluntary action,
avoid or seek to avoid the observance or performance of any of the terms of this
Warrant, but will at all times in good faith assist in the carrying out of all
such terms and in the taking of all such action as may be necessary or
appropriate in order to protect the rights of the Holder hereof against dilution
or other impairment. Without limiting the generality of the foregoing, the
Company will not increase the par value of any Common Share receivable upon the
exercise of this Warrant above the amount payable therefor upon such exercise,
and at all times will take all such action as may be necessary or appropriate in
order that the Company may validly and legally issue fully paid and
nonassessable Common Shares upon the exercise of this Warrant.

      5. RESERVATION OF COMMON SHARES, ETC., ISSUABLE UPON EXERCISE OF WARRANTS.
The Company shall at all times reserve and keep available out of its authorized
but unissued Common Shares, solely for issuance and delivery upon the exercise
of this Warrant and other similar warrants, such number of its duly authorized
Common Shares as from time to time shall be issuable upon the exercise of this
Warrant and all other similar warrants at the time outstanding.

      6. REPLACEMENT OF WARRANT. Upon receipt of evidence reasonably
satisfactory to the Company of the loss, theft, destruction or mutilation of
this Warrant and (in the case of loss, theft or destruction) upon delivery of an
indemnity agreement (with surety if reasonably required) in an amount reasonably
satisfactory to it, or (in the case of mutilation) upon surrender and
cancellation thereof, the Company will issue, in lieu thereof, a new Warrant of
like tenor.

      7. REMEDIES. The Company stipulates that the remedies at law of the Holder
of this Warrant in the event of any default by the Company in the performance of
or compliance with any of the terms of this Warrant are not and will not be
adequate, and that the same may be specifically enforced.
<Page>

                                      -12-

      8. NEGOTIABILITY, ETC. This Warrant is issued upon the following terms, to
all of which each taker or owner hereof consents and agrees:

      (a)   Neither this Warrant, the Common Shares underlying this Warrant (the
            "UNDERLYING STOCK") nor the rights of the Holder hereunder may be
            transferred except in compliance with the provisions of a certain
            Subscription Agreement executed in connection with the issuance of
            this Warrant, copies of which are on file at the principal office of
            the Company.

            The provisions of this Section 8 shall be binding upon any
            transferee of this Warrant and upon each holder of Underlying Stock.

      (b)   Subject to the limitation described in this Section 8, title to this
            Warrant may be transferred by endorsement (by the Holder hereof
            executing the form of assignment at the end hereof including
            guaranty of signature) and delivery in the same manner as in the
            case of a negotiable instrument transferable by endorsement and
            delivery.

      (c)   Any person in possession of this Warrant properly endorsed and, if
            not the original holder hereof, to whom possession was transferred
            in accordance with the provisions was transferred in accordance with
            the provisions of clauses (a) and (b) of this Section 8 is
            authorized to represent himself as absolute owner hereof and is
            granted power to transfer absolute title hereto by endorsement and
            delivery hereof to a bona fide purchaser hereof for value; each
            prior taker or owner waives and renounces all of his equities or
            rights in this Warrant in favor of every such bona fide purchaser,
            and every such bona fide purchaser shall acquire title hereto and to
            all rights represented hereby.

      (d)   Until this Warrant is transferred on the books of the Company, the
            Company may treat the registered Holder of this Warrant as the
            absolute owner hereof for all purposes without being affected by any
            notice to the contrary.

      (e)   The Company shall not be required to pay any U.S. federal or state
            transfer tax or charge that may be payable in respect of any
            transfer involved in the transfer or delivery of this Warrant or the
            issuance or conversion or delivery of certificates for Common Shares
            in a name other than that of the registered Holder of this warrant
            or to issue or deliver any certificates for Common Shares upon the
            exercise of this Warrant until any and all such taxes and charges
            shall have been paid by the Holder of this Warrant or until it has
            been established to the Company's satisfaction that no such tax or
            charge is due.
<Page>

                                      -13-

      9. NO RIGHTS TO VOTE OR RECEIVE DIVIDENDS OR OTHER DISTRIBUTIONS. Prior to
the exercise of this Warrant, the Holder hereof shall not be entitled to any
rights of a shareholder of the Company with respect to Common Shares for which
this Warrant shall be exercisable, including, without limitation, the right to
vote, to receive dividends or other distributions or to exercise any preemptive
rights, and shall not be entitled to receive any notice of any proceedings of
the Company, except as provided herein.

      10. MAILING OF NOTICES, ETC. All notices and other communications from the
Company to the Holder of this Warrant shall be mailed by first-class certified
mail, postage prepaid, to the address furnished to the Company in writing by the
last Holder of this Warrant who shall have furnished an address to the Company
in writing.

      11. CHANGE, WAIVER, ETC. The terms of this Warrant may not be changed,
waived, discharged or terminated orally but only by an instrument in writing
signed by the Holder of this Warrant against which enforcement of the change,
waiver, discharge or termination is sought.
<Page>

                                       S-1

                                           ARCH CAPITAL GROUP LTD.

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

Attest:

_____________________
<Page>

                  [To be signed only upon exercise of Warrant]

To Arch Capital Group Ltd.

            The undersigned, the Holder of the within Warrant, hereby
irrevocably elects to exercise the purchase right represented by such Warrant
for, and to purchase thereunder, ________________ Common Shares of Arch Capital
Group Ltd. and herewith makes payment of $_________ therefor and/or requests
that the number of Common Shares for which the within Warrant is exercisable to
reduced by __________ Common Shares (in addition to the Common Shares being
purchased) and/or delivers _________ Common Shares, the aggregate of such
payment being equal to the aggregate purchase price for the Common Shares being
purchased, and requests that the certificates for the Common Shares being
purchased be issued in the name of, and be delivered to, _____________, whose
address is ________________.

Dated:

------------------

                                        ---------------------------------------
                                        (Signature must conform in all respects
                                        to name of Holder as specified on the
                                        face of the Warrant)

                                        ---------------------------------------
                                                       Address
<Page>
                  [To be signed only upon transfer of Warrant]

            FOR VALUE RECEIVED, the undersigned hereby sells, assigns and
transfers unto ______________ the within Warrant and appoints _______________
attorney to transfer said right on the books of Arch Capital Group Ltd. with
full power of substitution in the premises.

Dated:

------------------

                                        ---------------------------------------
                                        (Signature must conform in all respects
                                        to name of Holder as specified on the
                                        face of the Warrant)

                                        ---------------------------------------
                                                       Address

In the presence of

------------------<Page>
                                                                  Exhibit 10.1.4

                           [FORM OF BYE-LAW AMENDMENT]

                              AMENDED BYE-LAW 45(a)

45. LIMITATION ON VOTING RIGHTS OF CONTROLLED SHARES

      (1) Except as provided in the other paragraphs of this Bye-law 45, every
Member of record owning shares conferring the right to vote present in person or
by proxy shall have one vote, or such other number of votes as may be specified
in the terms of the issue and rights and privileges attaching to such shares or
in these Bye-laws, for each such share registered in such Member's name.

      (2) If, as a result of giving effect to the foregoing provisions of this
Bye-law 45 or otherwise, the votes conferred by the Controlled Shares, directly
or indirectly or by attribution, to any U.S. Person would otherwise represent
more than 9.9% of the voting power of all shares entitled to vote generally at
an election of Directors, the votes conferred by the Controlled Shares on such
U.S. Person shall be reduced by whatever amount is necessary so that after any
such reduction the votes conferred by the Controlled Shares to such U.S. Person
shall constitute 9.9% of the total voting power of all shares of the Company
entitled to vote generally at any election of Directors (provided, however, that
(a) votes shall be reduced only in shares of the Company, if any, held by such
U.S. Person within the meaning of Section 958(a) of the Internal Revenue Code of
1986, as amended (the "Code"), (b) votes shall be reduced only in shares of the
Company, if any, held by such U.S. Person directly or held by such U.S. Person
indirectly solely through one or more foreign entities, (c) votes shall be
reduced in shares of the Company held by the H&F Funds and the WP Funds only if
and to the extent that reductions in the vote of other shares do not result in
satisfaction of the 9.9% threshold set forth in this paragraph (2), (d) such
reduction (or portion thereof) in votes conferred by shares held by an H&F Fund
shall not be effective on or after such date, if any, as such H&F Fund objects
in writing to such reduction (or portion thereof) after reasonable notice given
to such H&F Fund in advance of any meeting of shareholders (which notice the
Company shall provide in writing) and (e) such reduction (or portion thereof) in
votes conferred by shares held by a WP Fund shall not be effective on or after
such date, if any, as such WP Fund objects in writing to such reduction (or
portion thereof) after reasonable notice given to such WP Fund in advance of any
meeting of shareholders (which notice the Company shall provide in writing)).

(3)   (a)   "CONTROLLED SHARES" in reference to any person means all shares of
            the Company directly, indirectly or constructively owned by such
            person within the meaning of Section 958 of the Code;

      (b)   "U.S. PERSON" means a United States person as defined in Section
            7701(a)(30) of the Code;

      (c)   "H&F FUND" means [      ]; and

      (d)   "WP FUND" means [      ].

      (4) Upon written notification by a Member of the Board, the number of
votes conferred by the total number of shares held by such Member shall be
reduced to that percentage of the total voting power of the Company, as so
designated by such Member (subject to acceptance of such reduction by the Board
in its sole discretion), so that (and to the extent that) such Member may meet
any applicable insurance or other regulatory requirement or voting threshold or
limi-

----------
1     Between the date hereof and closing, the parties agree that to the extent
      that the form of amendment to bye-law 45 leaves open the possibility that
      the cutback rules fail to protect shareholders who are unaffiliated with
      the Purchasers from adverse tax consequences resulting from "controlled
      foreign corporation" status of the Company, the parties shall cooperate to
      develop alternative language for such bye-law (it being understood,
      however, that the Purchasers shall not suffer a cutback except as the
      Purchasers are or may be cut back as set forth herein).

                                      A-1
<Page>

tation that may be applicable to such Member or to evidence that such person's
voting power is no greater than such threshold.

      (5) Notwithstanding the foregoing provisions of this Bye-law 45, after
having applied such provisions as best as they consider reasonably practicable,
the Board may make such final adjustments to the aggregate number of votes
conferred, directly or indirectly or by attribution, by the Controlled Shares on
any U.S. Person that they consider fair and reasonable in all the circumstances
to ensure that such votes represent 9.9% (or the percentage designated by a
Member pursuant to paragraph (4) of this Bye-law 45) of the aggregate voting
power of the votes conferred by all the shares of the Company entitled to vote
generally at any election of Directors. Such adjustments intended to implement
the 9.9% limitation set forth in paragraph (2) of this Bye-law 45 shall be
subject to the proviso contained in such paragraph (2), but adjustments intended
to implement the limitation set forth in a notification pursuant to paragraph
(4) of this Bye-law 45 shall not be subject to the proviso contained in
paragraph (2).

      (6) Each Member shall provide the Company with such information as the
Company may reasonably request so that the Company and the Board may make
determinations as to the ownership (direct or indirect or by attribution) of
Controlled Shares to such Member or to any person to which Shares may be
attributed as a result of the ownership of Shares by such Member.

                                      A-2
<Page>

                               AMENDED 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 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 Alternate Directors of the Company in accordance with Bye-law
14 or 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 Alternate Directors of the Company or Designated
Company Directors to be elected as the directors of such Designated Company, and
to elect such Alternate Directors or Designated Company Directors, as the case
may be, 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.

                                      A-3

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