Exhibit 4.1

CERTIFICATE OF DESIGNATION

OF

8.00% NON-CUMULATIVE PREFERRED SHARES, SERIES A

OF

ARCH CAPITAL GROUP LTD.

Arch Capital Group Ltd., an exempted company with limited liability registered under the laws of Bermuda (the “Company”), CERTIFIES that pursuant to resolutions of a duly authorized committee the board of directors (the “Board of Directors”) of the Company adopted on January 20, 2006, the creation of the series of 8.00% Non-Cumulative Preferred Shares, Series A, US$0.01 par value per share, US$25.00 liquidation preference per share (the “Series A Non-Cumulative Preferred Shares”), were authorized and the designation, preferences and privileges, voting rights, relative, participating, optional and other special rights, and qualifications, limitations and restrictions of the Series A Non-Cumulative Preferred Shares, in addition to those set forth in the Bye-Laws (as amended, restated, supplemented, altered or modified from time to time, the “Bye-Laws”) of the Company, were fixed as follows:

Section 1.               Designation.  The distinctive serial designation of the Series A Non-Cumulative Preferred Shares is “8.00% Non-Cumulative Preferred Shares, Series A.” Each share of the Series A Non-Cumulative Preferred Shares shall be identical in all respects to every other share of Series A Non-Cumulative Preferred Shares, except as to the respective dates from which dividends thereon shall accrue, to the extent such dates may differ as permitted pursuant to Section 4(a) below.

Section 2.               Number of Shares.  The authorized number of shares of Series A Non-Cumulative Preferred Shares shall be 8,000,000.  Shares of Series A Non-Cumulative Preferred Shares that are redeemed, purchased or otherwise acquired by the Company shall be cancelled.

Section 3.               Definitions.  As used herein with respect to Series A Non-Cumulative Preferred Shares:

“Business Day” means a day that is a Monday, Tuesday, Wednesday, Thursday or Friday and is not a day on which banking institutions in New York City generally are authorized or obligated by law or executive order to close.

“Certificate of Designation” means this Certificate of Designation relating to the Series A Non-Cumulative Preferred Shares, as it may be amended, restated, supplemented, altered or modified from time to time.

“Commission” means the U.S.  Securities and Exchange Commission.

“Common Shares” means the Common Shares, par value $0.01 per share, of the Company.

“Companies Act” means the Companies Act 1981 of Bermuda, as amended from time to time.

“Dividend Payment Date” has the meaning specified in Section 4(a).

“Dividend Period” has the meaning specified in Section 4(a).

“Dividend Record Date” has the meaning specified in Section 4(a).

“Junior Shares” means the Common Shares, and any other class or series of shares of the Company that ranks junior to the Series A Non-Cumulative Preferred Shares either as to the payment of dividends (whether such dividends are cumulative or non-cumulative) or as to the distribution of assets upon any liquidation, dissolution or winding-up of the Company.

“Liquidation Preference” has the meaning specified in Section 6(b).

“Parity Shares” means the Series A Non-Cumulative Preferred Shares and any other class or series of shares of the Company that ranks equally with the Series A Non-Cumulative Preferred Shares with respect to both (a) the payment of dividends (whether such dividends are cumulative or non-cumulative) and (b) the distribution of assets upon a liquidation, dissolution or winding-up of the Company.

“Preferred Shares” means any and all series of preference shares of the Company, including the Series A Non-Cumulative Preferred Shares.

“Preferred Shares Directors” has the meaning specified in Section 8(b).

“Taxing Jurisdiction” has the meaning specified in Section 5(a).

“Voting Preferred Shares” means, with regard to any election or removal of a Preferred Shares Director or any other matter as to which the holders of Series A Non-Cumulative Preferred Shares are entitled to vote as specified in Section 8 of this Certificate of Designation, any and all series of Parity Shares upon which like voting rights have been conferred and are exercisable with respect to such matter.

Section 4.               Dividends.

(a)           Rate.  Dividends on the Series A Non-Cumulative Preferred Shares will not be mandatory.  Subject to applicable Bermuda law and regulation, holders of Series A Non-Cumulative Preferred Shares shall be entitled to receive, only when, as and if declared by the Board of Directors or a duly authorized committee of the Board of Directors, out of lawfully available funds for the payment of dividends under Bermuda law and regulation, non-cumulative cash dividends at the annual rate of 8.00% applied to the liquidation preference amount of US$25.00 per share of Series A Non-Cumulative Preferred Shares.  Such dividends shall be payable

2

quarterly in arrears (as provided below in this Section 4(a)), but only when, as and if declared by the Board of Directors or a duly authorized committee of the Board of Directors, on the 15th day of February, May, August and November of each year (each, a “Dividend Payment Date”), commencing on May 15, 2006; provided that if any such Dividend Payment Date would otherwise occur on a day that is not a Business Day, such Dividend Payment Date shall instead be (and any dividend payable on the Series A Non-Cumulative Preferred Shares on such Dividend Payment Date shall instead be payable on) the immediately succeeding Business Day.

Dividends, if so declared, that are payable on Series A Non-Cumulative Preferred Shares on any Dividend Payment Date will be payable to holders of record of Series A Non-Cumulative Preferred Shares as they appear on the share register of the Company on the applicable record date, which shall be the 15th calendar day of the month immediately preceding the applicable Dividend Payment Date or such other record date fixed by the Board of Directors or a duly authorized committee of the Board of Directors that is not more than 60 nor less than 10 days prior to such Dividend Payment Date (each, a “Dividend Record Date”).  Any such day that is a Dividend Record Date shall be a Dividend Record Date whether or not such day is a Business Day.

Each dividend period (a “Dividend Period”) shall commence on and include a Dividend Payment Date (other than the initial Dividend Period, which shall commence on and include the date of original issue of the Series A Non-Cumulative Preferred Shares, provided that, for any share of Series A Non-Cumulative Preferred Shares issued after such original issue date, the initial Dividend Period for such shares may commence on and include such other date as the Board of Directors or a duly authorized committee of the Board of Directors shall determine and publicly disclose at the time such additional shares are issued) and shall end on and include the calendar day preceding the next Dividend Payment Date.  Dividends payable on the Series A Non-Cumulative Preferred Shares in respect of any Dividend Period shall be computed on the basis of a 360-day year consisting of twelve 30-day months, except that dividends for the initial Dividend Period will be calculated based upon the actual number of calendar days from the original issue date to the calendar day preceding the first Dividend Payment Date, divided by a 360 day year.  Dividends payable in respect of a Dividend Period shall be payable in arrears (i.e., on the first Dividend Payment Date after such Dividend Period).

Dividends on the Series A Non-Cumulative Preferred Shares shall be non-cumulative.  Accordingly, if the Board of Directors or a duly authorized committee of the Board of Directors does not declare a dividend on the Series A Non-Cumulative Preferred Shares payable in respect of any Dividend Period before the related Dividend Payment Date, in full or otherwise, then such undeclared dividends shall not cumulate and will not accrue and will not be payable and the Company shall have no obligation to pay such undeclared dividends for the applicable Dividend Period on the related Dividend Payment Date or at any future time or to pay interest with respect to such dividends, whether or not dividends are declared on Series A Non-Cumulative Preferred Shares or any other preference shares the Company may issue in the future.

Holders of Series A Non-Cumulative Preferred Shares shall not be entitled to any dividends or other distributions, whether payable in cash, securities or other property, other than

3

dividends (if any) declared and payable on the Series A Non-Cumulative Preferred Shares as specified in this Section 4 (subject to the other provisions of this Certificate of Designation).

(b)           Priority of Dividends.  So long as any Series A Non-Cumulative Preferred Shares remain outstanding for any Dividend Period, unless the full dividends for the latest completed Dividend Period on all outstanding Series A Non-Cumulative Preferred Shares and any Parity Shares have been declared and paid (or declared and a sum sufficient for the payment thereof has been set aside), (1) no dividend shall be declared or paid on the Common Shares or any other Junior Shares (other than a dividend payable solely in Junior Shares), and (2) no Common Shares or other Junior Shares shall be purchased, redeemed or otherwise acquired for consideration by the Company, directly or indirectly (other than (i) as a result of a reclassification of Junior Shares for or into other Junior Shares, or the exchange or conversion of one Junior Share for or into another Junior Share, (ii) through the use of the proceeds of a substantially contemporaneous sale of Junior Shares and (iii) as permitted by the Bye-Laws in effect as of the date of this Certificate of Designation).

When dividends are not paid (or declared and a sum sufficient for payment thereof set aside) in full on any Dividend Payment Date (or, in the case of Parity Shares having dividend payment dates different from the Dividend Payment Dates, on a dividend payment date falling within the Dividend Period related to such Dividend Payment Date) upon the Series A Non-Cumulative Preferred Shares and any Parity Shares, all dividends declared by the Board of Directors or a duly authorized committee thereof on the Series A Non-Cumulative Preferred Shares and all such Parity Shares and payable on such Dividend Payment Date (or, in the case of Parity Shares having dividend payment dates different from the Dividend Payment Dates, on a dividend payment date falling within the Dividend Period related to such Dividend Payment Date) shall be declared by the Board of Directors or such committee of the Board of Directors pro rata so that the respective amounts of such dividends shall bear the same ratio to each other as all declared but unpaid dividends per share on the Series A Non-Cumulative Preferred Shares and all Parity Shares payable on such Dividend Payment Date (or, in the case of Parity Shares having dividend payment dates different from the Dividend Payment Dates, on a dividend payment date falling within the Dividend Period related to such Dividend Payment Date) bear to each other.

(c)           Restrictions on Payment of Dividends.  Pursuant to and subject to the Companies Act, the Company may not lawfully declare or pay a dividend if the Company has reasonable grounds for believing that the Company is, and would after payment of the dividend be, unable to pay its liabilities as they become due, or that the realizable value of the Company’s assets would, after payment of the dividend, be less than the aggregate value of the Company’s liabilities, issued share capital and share premium accounts.

Section 5.               Payment of Additional Amounts.

(a)           The Company will make all payments on the Series A Non-Cumulative Preferred Shares free and clear of and without withholding or deduction at source for, or on account of, any present or future taxes, fees, duties, assessments or governmental charges of whatever nature imposed or levied by or on behalf of Bermuda or any other jurisdiction in which the Company is organized (a “Taxing Jurisdiction”) or any political subdivision or taxing authority

4

thereof or therein, unless such taxes, fees, duties, assessments or governmental charges are required to be withheld or deducted by (x) the laws (or any regulations or rulings promulgated thereunder) of a Taxing Jurisdiction or any political subdivision or taxing authority thereof or therein or (y) an official position regarding the application, administration, interpretation or enforcement of any such laws, regulations or rulings (including, without limitation, a holding by a court of competent jurisdiction or by a taxing authority in a Taxing Jurisdiction or any political subdivision thereof).  If a withholding or deduction at source is required, the Company will, subject to certain limitations and exceptions described below, pay to the holders of the Series A Non-Cumulative Preferred Shares such additional amounts as dividends as may be necessary so that every net payment made to such holders, after the withholding or deduction, will not be less than the amount provided for in Section 4(a) to be then due and payable.

(b)           The Company will not be required to pay any additional amounts for or on account of:

(1)            any tax, fee, duty, assessment or governmental charge of whatever nature that would not have been imposed but for the fact that such holder was a resident, citizen, domiciliary or national of, or engaged in business or maintained a permanent establishment or was physically present in, the relevant Taxing Jurisdiction or any political subdivision thereof or otherwise had some connection with the relevant Taxing Jurisdiction other than by reason of the mere ownership of, or receipt of payment under, such Series A Non-Cumulative Preferred Shares or any Series A Non-Cumulative Preferred Shares presented for payment more than 30 days after the Relevant Date.  The “Relevant Date” means, in respect of any payment, the date on which such payment first becomes due and payable, but if the full amount of the moneys payable has not been received by the dividend disbursing agent on or prior to such due date, it means the first date on which, the full amount of such moneys having been so received and being available for payment to holders, notice to that effect shall have been duly given to the holders of the Series A Non-Cumulative Preferred Shares;

(2)            any estate, inheritance, gift, sale, transfer, personal property or similar tax, assessment or other governmental charge or any tax, assessment or other governmental charge that is payable otherwise than by withholding or deduction from payment of the liquidation preference;

(3)            any tax, fee, assessment or other governmental charge that is payable otherwise than by withholding or deduction from payment of the liquidation preference of or any dividends on the Series A Non-Cumulative Preferred Shares;

(4)            any tax, fee, duty, assessment or other governmental charge that is imposed or withheld by reason of the failure by the holder of such Series A Non-Cumulative Preferred Shares to comply with any reasonable request by the Company addressed to the holder within 90 days of such request (a) to provide information concerning the nationality, citizenship, residence or identity of the holder or (b) to make any declaration or other similar claim or satisfy any information or reporting requirement, which is required or imposed by statute, treaty, regulation or administrative practice of the relevant

5

Taxing Jurisdiction or any political subdivision thereof as a precondition to exemption from all or part of such tax, fee, duty, assessment or other governmental charge;

(5)            any withholding or deduction required to be made pursuant to any EU Directive on the taxation of savings implementing the conclusions of the ECOFIN Council meetings of 26-27 November 2000, 3 June 2003 or any law implementing or complying with, or introduced in order to conform to, such EU Directive; or

(6)            any combination of items (1), (2), (3), (4) and (5).

(c)           In addition, the Company will not pay additional amounts with respect to any payment on any such Series A Non-Cumulative Preferred Shares to any holder who is a fiduciary, partnership, limited liability company or other pass-thru entity other than the sole beneficial owner of such Series A Preferred Shares if such payment would be required by the laws of the relevant taxing jurisdiction (or any political subdivision or relevant taxing authority thereof or therein) to be included in the income for tax purposes of a beneficiary or partner or settlor with respect to such fiduciary or a member of such partnership, limited liability company or other pass-thru entity or a beneficial owner to the extent such beneficiary, partner or settlor would not have been entitled to such additional amounts had it been the holder of the Series A Non-Cumulative Preferred Shares.

Section 6.               Liquidation Rights.

(a)           Voluntary or Involuntary Liquidation.  Upon any voluntary or involuntary liquidation, dissolution or winding-up of the affairs of the Company, holders of Series A Non-Cumulative Preferred Shares and any Parity Shares shall be entitled to receive, out of the assets of the Company or proceeds thereof (whether capital or surplus) available for distribution to shareholders of the Company, after satisfaction of all liabilities and obligations to creditors of the Company, if any, but before any distribution of such assets or proceeds is made to or set aside for the holders of Common Shares and any other Junior Shares as to such a distribution, in full a liquidating distribution in an amount equal to US$25.00 per Series A Non-Cumulative Preferred Share, plus any declared and unpaid dividends.

(b)           Partial Payment.  If, in any distribution described in Section 6(a) above, the assets of the Company or proceeds thereof are not sufficient to pay the Liquidation Preferences in full to all holders of Series A Non-Cumulative Preferred Shares and all holders of any Parity Shares, the amounts paid to the holders of Series A Non-Cumulative Preferred Shares and to the holders of all such other Parity Shares shall be paid pro rata in accordance with the respective aggregate Liquidation Preferences of the holders of Series A Non-Cumulative Preferred Shares and the holders of all such other Parity Shares but only to the extent the Company has assets or proceeds thereof available after satisfaction of all liabilities to creditors.  In any such distribution, the “Liquidation Preference” of any holder of Preferred Shares of the Company shall mean the amount otherwise payable to such holder in such distribution (assuming no limitation on the assets of the Company available for such distribution), including any declared and unpaid dividends (and, in the case of any holder of shares other than Series A Non-Cumulative Preferred Shares and on which dividends accrue on a cumulative basis, an amount equal to any unpaid, accrued cumulative dividends, whether or not declared, as applicable).  Holders of Series A

6

Non-Cumulative Preferred Shares will not be entitled to any other amounts from the Company after they have received their full Liquidation Preference.

(c)           Residual Distributions.  If the Liquidation Preference has been paid in full to all holders of Series A Non-Cumulative Preferred Shares and any holders of Parity Shares, the holders of other shares of the Company shall be entitled to receive all remaining assets of the Company (or proceeds thereof) according to their respective rights and preferences.

(d)           Merger, Consolidation and Sale of Assets not Liquidation.  For purposes of this Section 6, the consolidation, amalgamation, merger, arrangement, reincorporation, de-registration or reconstruction involving the Company or the sale or transfer of all or substantially all of the shares or the property or business of the Company shall not constitute a liquidation, dissolution or winding-up.

Section 7.               Redemption.

(a)           Optional Redemption.

(1)           The Series A Non-Cumulative Preferred Shares may not be redeemed by the Company prior to February 1, 2011, subject to the exceptions set forth in Section 7(a)(2) and Section 7(f).  On or after February 1, 2011, the Company, at its option, may redeem, in whole at any time or in part from time to time, the shares of Series A Non-Cumulative Preferred Shares at the time outstanding, upon notice given as provided in Section 7(c) below, at a redemption price equal to US$25.00 per share, together (except as otherwise provided herein below) with an amount equal to any dividends that have been declared but not paid prior to the redemption date (but with no amount in respect of any dividends that have not been declared prior to such date).  The redemption price for any shares of Series A Non-Cumulative Preferred Shares redeemed pursuant to this Section 7 shall be payable on the redemption date to the holder of such shares against book entry transfer or surrender of the certificate(s) evidencing such shares to the Company or its agent.  Any declared but unpaid dividends payable on a redemption date that occurs subsequent to the Dividend Record Date for a Dividend Period shall not be paid to the holder entitled to receive the redemption price on the redemption date, but rather shall be paid to the holder of record of the redeemed shares on such Dividend Record Date relating to the Dividend Payment Date as provided in Section 4 above.  Holders of Series A Non-Cumulative Preferred Shares shall have no right to require the redemption of Series A Non-Cumulative Preferred Shares.

Prior to delivering notice of redemption as provided below, the Company will file with its corporate records a certificate signed by one of the Company’s officers affirming the Company’s compliance with the redemption provisions under the Companies Act relating to the Series A Non-Cumulative Preferred Shares, and stating that there are reasonable grounds for believing that the Company is, and after the redemption will be, able to pay its liabilities as they become due and that the redemption will not render the Company insolvent or cause it to breach any provision of applicable Bermuda law or regulation.  The Company will mail a copy of this certificate with the notice of any redemption.

7

(2)           At any time prior to February 1, 2011, if the Company shall have submitted to the holders of Common Shares a proposal for an amalgamation, consolidation, merger, arrangement, reconstruction, reincorporation, deregistration or any other similar transaction involving the Company that requires, or the Company shall have submitted any proposal for any other matter that, as a result of any change in Bermuda law or regulation after such Series A Non-Cumulative Preferred Shares are issued (whether by enactment or official interpretation), that requires, in either case, a vote of the holders of the Series A Non-Cumulative Preferred Shares at the time outstanding, voting separately as a single class (alone or with one or more other classes or series of preference shares), the Company shall have the option (but not obligation), by not less than 30 days nor more than 60 days prior written notice, to the relevant holders, in such form and given in such manner as to be in accordance with paragraph (c) below, to redeem all of the outstanding Series A Non-Cumulative Preferred Shares pursuant to this clause for cash at a redemption price of US$26.00 per share being redeemed, plus all declared and unpaid dividends, if any, to the date of redemption, without interest on such unpaid dividends.

(b)           No Sinking Fund.  The Series A Non-Cumulative Preferred Shares will not be subject to any mandatory redemption, sinking fund, retirement fund or purchase fund or other similar provisions.  Holders of Series A Non-Cumulative Preferred Shares will have no right to require redemption, repurchase or retirement of any shares of Series A Non-Cumulative Preferred Shares.

(c)           Notice of Redemption.  Notice of every redemption of Series A Non-Cumulative Preferred Shares shall be given by first class mail, addressed to the holders of record of the shares to be redeemed at their respective last addresses appearing on the share register of the Company, mailed at least 30 days and not more than 60 days before the date fixed for redemption.  Any notice mailed as provided in this subsection shall be conclusively presumed to have been duly given, whether or not the holder receives such notice, but failure duly to give such notice by mail, or any defect in such notice or in the mailing thereof, to any holder of Series A Non-Cumulative Preferred Shares designated for redemption shall not affect the validity of the proceedings for the redemption of any other Series A Non-Cumulative Preferred Shares.  Notwithstanding the foregoing, if the Series A Non-Cumulative Preferred Shares or any depositary shares representing interests in the Series A Non-Cumulative Preferred Shares are issued in book-entry form through The Depository Trust Company or any other similar facility, notice of redemption may be given to the holders of Series A Non-Cumulative Preferred Shares at such time and in any manner permitted by such facility.  Each such notice given to a holder shall state: (1) the redemption date; (2) the number of Series A Non-Cumulative Preferred Shares to be redeemed and, if less than all the Series A Non-Cumulative Preferred Shares held by such holder are to be redeemed, the number of such Series A Non-Cumulative Preferred Shares to be redeemed from such holder; (3) the redemption price; and (4) that the Series A Non-Cumulative Preferred Shares should be delivered via book entry transfer or the place or places where certificates for such Series A Non-Cumulative Preferred Shares are to be surrendered for payment of the redemption price.

(d)           Partial Redemption.  In case of any redemption of only part of the Series A Non-Cumulative Preferred Shares at the time outstanding, the shares to be redeemed shall be selected either pro rata or in such other manner as the Company may determine to be fair and

8

equitable.  Subject to the provisions hereof, the Company shall have full power and authority to prescribe the terms and conditions upon which Series A Non-Cumulative Preferred Shares shall be redeemed from time to time.  If fewer than all the shares represented by any certificate are redeemed, a new certificate shall be issued representing the unredeemed shares without charge to the holder thereof.

(e)           Effectiveness of Redemption.  If notice of redemption has been duly given and if on or before the redemption date specified in the notice all funds necessary for the redemption have been set aside by the Company for the benefit of the holders of the Series A Non-Cumulative Preferred Shares called for redemption, so as to be and continue to be available therefor, then, notwithstanding that any certificate for any share so called for redemption has not been surrendered for cancellation or transferred via book entry, on and after the redemption date, no further dividends will be declared on all shares so called for redemption, all shares so called for redemption shall no longer be deemed outstanding and all rights with respect to such shares shall forthwith on such redemption date cease and terminate, except only the right of the holders thereof to receive the amount payable on such redemption, without interest.

(f)            Tax Redemption.

(1)           If, as a result of a “change in tax law,” there is a substantial probability that the Company or any successor company would be required pay additional amounts with respect to the Series A Non-Cumulative Preferred Shares with respect to the Series A Non-Cumulative Preferred Shares, the Company shall be entitled (but not obligated) at any time thereafter, by not less than 30 days nor more than 60 days prior written notice to the relevant holders of the Series A Non-Cumulative Preferred Shares, to redeem any or all Series A Non-Cumulative Preferred Shares pursuant to this clause for cash at a redemption price of US$25.00 per share being redeemed, plus all declared and unpaid dividends, if any, to the date of redemption, without interest on such unpaid dividends.  For the purposes of this provision, a “change in tax law” shall be (a) a change in or amendment to laws, regulations or rulings of any jurisdiction, political subdivision or taxing authority described in the next sentence, (b) a change in the official application or interpretation of those laws, regulations or rulings, (c) any execution of or amendment to any treaty affecting taxation to which any jurisdiction, political subdivision or taxing authority described in the next sentence is party after the date of original issuance of the Series A Non-Cumulative Preferred Shares, or (d) a decision rendered by a court of competent jurisdiction in Bermuda or any taxing jurisdiction or any political subdivision, whether or not such decision was rendered with respect to the Company.  The jurisdictions, political subdivisions and taxing authorities referred to in the previous sentence are (a) Bermuda or any political subdivision or governmental authority of or in Bermuda with the power to tax, (b) any jurisdiction from or through which the Company or its dividend disbursing agent is making payments on the Series A Non-Cumulative Preferred Shares or any political subdivision or governmental authority of or in that jurisdiction with the power to tax, or (c) any other jurisdiction in which the Company or its successor company is organized or generally subject to taxation or any political subdivision or governmental authority of or in that jurisdiction with the power to tax.

(2)           If the entity formed by a consolidation, merger or amalgamation involving the Company or the entity to which the Company conveys, transfers or leases substantially all of

9

its properties and assets is required to pay additional amounts in respect of any tax, assessment or governmental charge imposed on any holder of Series A Non-Cumulative Preferred Shares as a result of a change in tax law that occurred after the date of the consolidation, merger, amalgamation, conveyance, transfer or lease, the Company shall be entitled (but not obligated) at any time thereafter by not less than 30 days nor more than 60 days prior written notice to the relevant holders of Series A Non-Cumulative Preferred Shares, in such form and given in such manner as in accordance with paragraph (c) above, to redeem any or all Series A Non-Cumulative Preferred Shares pursuant to this clause for cash at a redemption price of US$25.00 per share being redeemed, plus all declared and unpaid dividends, if any, to the date of redemption, without interest on such unpaid dividends.

Section 8.               Voting Rights.

(a)           General.  The holders of Series A Non-Cumulative Preferred Shares shall not have any voting rights except as set forth below or as otherwise from time to time required by law.

(b)           Right To Elect Two Directors Upon Nonpayment Events.  Whenever dividends on any Series A Non-Cumulative Preferred Shares shall not have been declared and paid for the equivalent of six or more Dividend Periods, whether or not consecutive (a “Nonpayment Event”), the holders of Series A Non-Cumulative Preferred Shares, together with the holders of any outstanding shares of Voting Preferred Shares, voting together as a single class, shall be entitled to elect two additional directors to the board of directors of the Company (the “Preferred Shares Directors”), provided that it shall be a qualification for election for any such Preferred Shares Director that the election of such director shall not cause the Company to violate the corporate governance requirements of any securities exchange or other trading facility on which securities of the Company may then be listed or quoted that listed or quoted companies must have a majority of independent directors.  Each Preferred Shares Director will be added to an already existing class of directors.  The number of Preferred Shares Directors on the Board of Directors shall never be more than two at any one time.

In the event that the holders of the Series A Non-Cumulative Preferred Shares, and any such other holders of Voting Preferred Shares, shall be entitled to vote for the election of the Preferred Shares Directors following a Nonpayment Event, such directors shall be initially elected following such Nonpayment Event only at a special meeting called at the request of the holders of record of at least 20% of the aggregate voting power of the Series A Non-Cumulative Preferred Shares or of any other such series of Voting Preferred Shares then outstanding (unless such request for a special meeting is received less than 90 days before the date fixed for the next annual or special meeting of the shareholders of the Company, in which event such election shall be held only at such next annual or special general meeting of shareholders), and at each subsequent annual general meeting of shareholders of the Company, so long as the rights related to a Nonpayment Event remain in effect.  Such request to call a special general meeting for the initial election of the Preferred Shares Directors after a Nonpayment Event shall be made by written notice, signed by the requisite holders of Series A Non-Cumulative Preferred Shares or Voting Preferred Shares, and delivered to the Secretary of the Company in such manner as provided for in Section 11 below, or as may otherwise be required by Bermuda law and regulation.

10

If and when dividends have been paid (or declared and a sum sufficient for payment thereof set aside) in full on the Series A Non-Cumulative Preferred Shares for at least four Dividend Periods (whether or not consecutive) after a Nonpayment Event, then the right of the holders of Series A Non-Cumulative Preferred Shares to elect the Preferred Shares Directors shall cease (but subject always to revesting of such voting rights in the case of any future Nonpayment Event pursuant to this Section 8) and the number of Dividend Periods in which dividends have not been declared and paid shall be reset to zero, and, if and when any rights of holders of Series A Non-Cumulative Preferred Shares and Voting Preferred Shares to elect the Preferred Shares Directors shall have ceased, the terms of office of all the Preferred Shares Directors shall forthwith terminate and the number of directors constituting the Board of Directors shall automatically decrease by two.  In determining whether dividends have been paid for four Dividend Periods following a Nonpayment Event, the Company may take account of any dividends it elects to pay for such a Dividend Period after the regular Dividend Payment Date for that period has passed.

Any Preferred Shares Director may be removed at any time without cause by the holders of record of a majority of the aggregate voting power, as determined by the Bye-laws of the Company, of Series A Non-Cumulative Preferred Shares and Voting Preferred Shares then outstanding (voting together as a single class), when they have the voting rights described above.  Until the right of the holders of Series A Non-Cumulative Preferred Shares and any Voting Preferred Shares to elect the Preferred Shares Directors shall cease, any vacancy in the office of a Preferred Shares Director (other than prior to the initial election of Preferred Shares Directors after a Nonpayment Event) may be filled by the written consent of the Preferred Shares Director remaining in office, or if none remains in office, by a vote of the holders of record of a majority of the outstanding shares of the Series A Non-Cumulative Preferred Shares and any Voting Preferred Shares (voting together as a single class), when they have the voting rights described above.  Any such vote of shareholders to remove, or to fill a vacancy in the office of, a Preferred Shares Director may be taken only at a special meeting of such shareholders, called as provided above for an initial election of Preferred Shares Directors after a Nonpayment Event (unless such request is received less than 90 days before the date fixed for the next annual or special general meeting of the shareholders of the Company, in which event such election shall be held at such next annual or special general meeting of shareholders).  The Preferred Shares Directors shall each be entitled to one vote per director on any matter that shall come before the Board of Directors for a vote, unless otherwise adjusted pursuant to the Bye-Laws.  Each Preferred Shares Director elected at any special general meeting of shareholders of the Company or by written consent of the other Preferred Shares Director shall hold office until the next annual general meeting of the shareholders of the Company if such office shall not have previously terminated as above provided.

(c)           Variation of Rights.  Subject to the terms of the Bye-Laws and the Companies Act, any or all of the special rights attached to the Series A Non-Cumulative Preferred Shares may, whether or not the Company is being wound-up, be varied with the consent in writing of the holders a majority of the voting power represented by the issued Series A Non-Cumulative Preferred Shares or with the sanction of a resolution passed by a majority of the voting power represented by the votes cast at a separate general meeting of the holders of the Series A Non-Cumulative Preferred Shares in accordance with the Companies Act.  Subject to the

11

terms of the Bye-Laws and the Companies Act, rights conferred upon the holders of Series A Non-Cumulative Preferred Shares shall not be deemed to be varied by the creation or issue of further shares ranking pari passu therewith.

(d)           Changes for Clarification.  Subject to applicable Bermuda law and regulation, without the consent of the holders of the Series A Non-Cumulative Preferred Shares, so long as such action does not affect the special rights, preferences, privileges and voting powers, and limitations and restrictions, of the Series A Non-Cumulative Preferred Shares taken as a whole, the Company may amend, restate, alter, supplement, modify or repeal any terms of the Series A Non-Cumulative Preferred Shares:

(i)            to cure any ambiguity, or to cure, correct or supplement any provision contained in this Certificate of Designation that may be defective or inconsistent; or

(ii)           to make any provision with respect to matters or questions arising with respect to the Series A Non-Cumulative Preferred Shares that is not inconsistent with the provisions of this Certificate of Designation.

(e)           Changes After Provision for Redemption.  No vote or consent of the holders of Series A Non-Cumulative Preferred Shares shall be required pursuant to Section 8(b), (c) or (d) above if, at or prior to the time when any such vote or consent would otherwise be required pursuant to such Section, all outstanding Series A Non-Cumulative Preferred Shares shall have been redeemed, or shall have been called for redemption upon proper notice and sufficient funds shall have been set aside for such redemption, in each case pursuant to Section 7 above.

(f)            Procedures for Voting and Consents.  The rules and procedures for calling and conducting any meeting of the holders of Series A Non-Cumulative Preferred Shares (including, without limitation, the fixing of a record date in connection therewith), the solicitation and use of proxies at such a meeting, the obtaining of written consents and any other aspect or matter with regard to such a meeting or such consents shall be governed by any rules the Board of Directors or a duly authorized committee of the Board of Directors, in its discretion, may adopt from time to time, which rules and procedures shall conform to the requirements of the Bye-Laws, applicable law and any national securities exchange or other trading facility on which the Series A Non-Cumulative Preferred Shares is listed or traded at the time.  Whether the vote or consent of the holders of a plurality, majority or other portion of the shares of Series A Non-Cumulative Preferred Shares and any Voting Preferred Shares has been cast or given on any matter on which the holders of shares of Series A Non-Cumulative Preferred Shares are entitled to vote shall be determined by the Company by reference to the aggregate voting power, as determined by the Bye-Laws, of the shares voted or covered by the consent.

(g)           Bye-Laws. For the avoidance of doubt, the provisions of this Section 8 shall be subject to Bye-law 45 (as may be amended, restated, supplemented, altered or modified from time to time) of the Bye-Laws of the Company.

Section 9.               Ranking.  The Series A Non-Cumulative Preferred Shares will, with respect to the payment of dividends and distributions of assets upon liquidation, dissolution and winding-up, rank senior to Junior Shares and pari passu with any Parity Shares of the Company,

12

including other series of Preferred Shares of the Company that the Company may issue from time to time in the future.

Section 10.             Record Holders.  To the fullest extent permitted by applicable law, the Company and the transfer agent for the Series A Non-Cumulative Preferred Shares may deem and treat the record holder of any share of Series A Non-Cumulative Preferred Shares as the true and lawful owner thereof for all purposes, and neither the Company nor such transfer agent shall be affected by any notice to the contrary.

Section 11.             Notices.  All notices or communications in respect of Series A Non-Cumulative Preferred Shares shall be sufficiently given if given in writing and delivered in person or by first class mail, postage prepaid, or if given in such other manner as may be permitted in this Certificate of Designation, the Bye-Laws or by applicable law.

Section 12.             No Preemptive Rights.  No share of Series A Non-Cumulative Preferred Shares shall have any rights of preemption whatsoever as to any securities of the Company, or any warrants, rights or options issued or granted with respect thereto, regardless of how such securities, or such warrants, rights or options, may be designated, issued or granted.

Section 13.             Other Rights.  The shares of Series A Non-Cumulative Preferred Shares shall not have any voting powers, preferences or relative, participating, optional or other special rights, or qualifications, limitations or restrictions thereof, other than as set forth herein or in the Bye-laws or as provided by applicable law.  The Series A Non-Cumulative Preferred Shares are not convertible into or exchangeable for any other securities or property of the Company.

13

IN WITNESS WHEREOF, ARCH CAPITAL GROUP LTD. has caused this certificate to be signed by John D. Vollaro, its Executive Vice President and Chief Financial Officer, this 1^st day of February, 2006.

14