Exhibit 3.1

SECOND AMENDMENT

TO AMENDED AND RESTATED

CERTIFICATE OF INCORPORATION

OF

RADIAN GROUP INC.

Radian Group Inc., a corporation organized and existing under the laws of the State of Delaware (the “Corporation”), does hereby certify that:

1. This Second Amendment to the Amended and Restated Certificate of Incorporation, as amended (the “Certificate of Incorporation”), of the Corporation has been duly adopted in accordance with the provisions of Section 242 of the General Corporation Law of the State of Delaware.

2. This Second Amendment to the Certificate of Incorporation amends Article FOURTH of the Certificate of Incorporation by adding new Section 4.4 of Article FOURTH to read in its entirety as follows:

Section 4.4 Transfer Restrictions.

Section 4.4.1 Certain Definitions.

As used in Section 4.4 of this Article FOURTH:

“Acquire” or “Acquisition” and similar terms mean the direct or indirect acquisition of record, legal, beneficial or any other ownership of Corporation Securities by any means, including (a) the exercise of any rights under any option, warrant, convertible security, pledge or other security interest or similar right to acquire shares or (b) the entering into of any swap, hedge or other arrangement that results in the acquisition of any of the economic consequences of ownership of Corporation Securities if, as a result of such direct or indirect acquisition, the acquirer would be considered an owner of Corporation Securities under the direct, indirect or constructive ownership rules of Section 382 of the Code.

“Agent” shall have the meaning set forth in Section 4.4.3(b) of this Article FOURTH.

“Business Day” means any day, other than a Saturday, Sunday or day on which banks located in New York City, New York or Philadelphia, Pennsylvania, are authorized or required by law to close.

“Code” means the Internal Revenue Code of 1986, as amended, and the rules and regulations promulgated thereunder.

“Controlled Person” shall have the meaning set forth in Section 4.4.3(f) of this Article FOURTH.

“Corporation Securities” means (a) shares of Common Stock, (b) shares of Preferred Stock of any class or series of Preferred Stock, (c) warrants, rights or options (including within the meaning of Treasury Regulation Section 1.382–2T(h)(4)(v) (or any successor provision)) to purchase other Corporation Securities of the Corporation, and (d) any other interests that would be treated as “stock” of the Corporation pursuant to Treasury Regulation Section 1.382-2T(f)(18) (or any successor provision).

“Effective Date” means the date of filing of the Second Amendment to the Amended and Restated Certificate of Incorporation first containing this provision.

“Entity” means an entity within the meaning of Treasury Regulation Section 1.382–3(a)(1) (or any successor provision).

“Excess Securities” shall have the meaning set forth in Section 4.4.3(a) of this Article FOURTH.

“Exchange Act” means the Securities Exchange Act of 1934, as amended, and the rules and regulations promulgated thereunder.

“Exempt Person” means any Existing Holder, unless and until such time as such Existing Holder shall (i) have a Percentage Stock Ownership that is more than the Existing Holder Ownership Cap of such Existing Holder or (ii) no longer be a “five-percent shareholder” of the Corporation Securities pursuant to Treasury Regulation Section 1.382–2T(g)(1) (or any successor provision). Notwithstanding the foregoing, no Exempt Person shall cease to be an Exempt Person solely as the result of an Acquisition of Corporation Securities by the Corporation which, by reducing the number of Corporation Securities outstanding, increases the Percentage Stock Ownership of such Person.

“Existing Holder” means any Person who, immediately before the Effective Date, is a “five-percent shareholder” of the Corporation Securities pursuant to Treasury Regulation Section 1.382–2T(g)(1) (or any successor provision).

“Existing Holder Initial Ownership” means, with respect to any Existing Holder, the aggregate Percentage Stock Ownership of such Existing Holder immediately before the Effective Date (as reflected in the most recent Schedule 13D, Schedule 13F or Schedule 13G filed by such Existing Holder before the Effective Date).

“Existing Holder Ownership Cap” means, as determined from time to time with respect to any Existing Holder, the sum of (a) the difference of (i) the Existing Holder Initial Ownership of such Existing Holder minus (ii) the total shares of Stock that such Existing Holder has disposed of on or after the Effective Date plus (b) the difference (which difference shall in no event be less than zero) of (i) 150,000 shares of Common Stock (subject to adjustment for any stock split, reverse stock split, recapitalization or similar transaction) minus (ii) the total shares of Stock that such Existing Holder has Acquired on or after the Effective Date; provided, however that in no event shall the Existing Holder Ownership Cap of such Existing Holder ever exceed the Existing Holder Initial Ownership of such Existing Holder. For purposes of clause (a)(ii) of this definition, “disposed” means any direct or indirect sale, transfer, assignment, conveyance, pledge or other disposition or other action in any manner whatsoever, whether voluntary or involuntary, by operation of law or otherwise, that reduces the Percentage Stock Ownership of the Existing Holder.

“Five Percent Shareholder” means a Person that is identified as a “five-percent shareholder” of the Corporation Securities pursuant to Treasury Regulation Section 1.382–2T(g)(1) (or any successor provision), but excluding (a) any “direct public group” with respect to the Corporation, as that term is defined in Treasury Regulation Section 1.382–2T(j)(2)(ii) (or any successor provision) or (b) any Exempt Person.

“Percentage Stock Ownership” and similar terms means the direct and indirect percentage stock ownership of any Person for purposes of Section 382 of the Code as determined in accordance with Treasury Regulation Section 1.382–2T(g), (h), (j) and (k) (or any successor provisions) including any ownership by application of constructive ownership rules.

“Person” means an individual, corporation, estate, trust, association, limited liability company, partnership, joint venture or similar organization, and also includes a group of Persons that is an “entity” within the meaning of Treasury Regulation Section 1.382–3(a)(1) (or any successor provision).

“Prohibited Distributions” shall have the meaning set forth in Section 4.4.3(b) of this Article FOURTH.

“Prohibited Transfer” shall have the meaning set forth in Section 4.4.2(a) of this Article FOURTH.

“Purported Transferee” shall have the meaning set forth in Section 4.4.3(a) of this Article FOURTH.

“Request” shall have the meaning set forth in Section 4.4.2(b) of this Article FOURTH.

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“Restriction Release Date” means such date, after the Effective Date, that (i) the repeal of Section 382 or any successor statute occurs if the Board determines that Section 4.4 of this Article FOURTH is no longer necessary for the preservation of Tax Benefits, (ii) is the beginning of a taxable year of the Company to which the Board determines that no Tax Benefits may be carried forward, or (iii) is such date as the Board determines that a limitation on the use of the Tax Benefits under Section 382 would no longer be material to the Company. In addition, the Restriction Release Date shall occur on the close of business on the second Business Day after the final adjournment of the third consecutive annual meeting of the stockholders of the Company held after this Section 4.4 was most recently approved by the stockholders of the Company unless this Section 4.4 is re-approved by a majority of the stockholders voting at such meeting.

“Restricted Holder” means a Person that (a) is a Five Percent Shareholder and Acquires or proposes to Acquire additional Corporation Securities, or (b) is proposing to Acquire Corporation Securities, and after such proposed Acquisition of Corporation Securities, would be a Five Percent Shareholder.

“Securities Act” means the Securities Act of 1933, as amended, and the rules and regulations promulgated thereunder.

“Security” or “Securities” shall have the meaning set forth in Section 4.4.3(d) of this Article FOURTH.

“Stock” means any interest that would be treated as “stock” of the Corporation pursuant to Treasury Regulation Section 1.382–2T(f)(18) (or any successor provision).

“Tax Benefits” means all net operating loss carryovers, capital loss carryovers, general business carryovers, alternative minimum tax credit carryovers and foreign tax credit carryovers, if any, as well as any loss or deduction attributable to a “net unrealized built-in loss” within the meaning of Section 382 of the Code and the Treasury Regulations promulgated thereunder, of the Corporation or any of its subsidiaries.

“Transfer” means any direct or indirect Acquisition, sale, transfer, assignment, conveyance, pledge or other disposition or other action in any manner whatsoever, whether voluntary or involuntary, by operation of law or otherwise, by any Person that alters the Percentage Stock Ownership of any Person, or any attempt to do any of the foregoing. A Transfer shall also include the creation or grant of an option (including within the meaning of Treasury Regulation Section 1.382–2T(h)(4)(v) (or any successor provision)). A Transfer shall include a repurchase of Corporation Securities by the Corporation but shall not include an issuance or grant of Corporation Securities by the Corporation.

“Treasury Regulation” means a Treasury Regulation promulgated under the Code.

Section 4.4.2 Transfer Restrictions.

(a) In order to preserve the Tax Benefits, from and after the Effective Date and before the Restriction Release Date, no Transfer other than to the Corporation shall be permitted, and any such purported Transfer shall be null and void ab initio, as to the amount of any such purported Transfer of Corporation Securities that causes, after giving effect to such purported Transfer (or any series of Transfers of which such Transfer is a part), (i) any Person to become a Five Percent Shareholder or (ii) the Percentage Stock Ownership interest in the Corporation of any Five Percent Shareholder to increase (a “Prohibited Transfer”). The prior sentence is not intended to prevent the Corporation Securities from being DTC-eligible and shall not preclude the settlement of any transactions in the Corporation Securities entered into through the facilities of a national securities exchange or any national securities quotation system, provided, that if the settlement of the transaction would result in a Prohibited Transfer, such Transfer shall nonetheless be a Prohibited Transfer.

(b) The restrictions contained in this Article FOURTH are for the purposes of reducing the risk that any “ownership change” (as defined in the Code) of the Corporation Securities may limit the Corporation’s ability to utilize its Tax Benefits. In connection therewith, and to provide for effective policing of these provisions, a Restricted Holder who proposes to Acquire Corporation Securities shall, before the date of such proposed Acquisition, request in writing (a “Request”) that the Board of Directors of the Corporation review such proposed Acquisition and authorize or not authorize such proposed Acquisition in accordance with this Section 4.4.2(b) of

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Article FOURTH. A Request shall be made in accordance with this Section 4.4.2(b) of Article FOURTH and shall be delivered by fax and by registered mail, return receipt requested, to the Secretary of the Corporation at the principal executive offices of the Corporation. Such Request shall be deemed to have been received by the Corporation only when actually received by the Corporation. To be made in accordance with this Section 4.4.2(b) of Article FOURTH, a Request shall include (i) the name, address and telephone number of the Restricted Holder, (ii) a description of the Restricted Holder’s existing direct and indirect ownership of Corporation Securities, together with such ownership of all affiliates and associates of the Restricted Holder, (iii) a description of the Corporation Securities that the Restricted Holder proposes to Acquire, (iv) the date on which such proposed Acquisition is expected to take place (or, if such Acquisition is proposed to be made in a transaction on a national securities exchange or any national securities quotation system, a statement to that effect), (v) the name, address and telephone number of the proposed transferor of the Corporation Securities that the Restricted Holder proposes to Acquire (or, if such Acquisition is proposed to be made in a transaction on a national securities exchange or any national securities quotation system, a statement to that effect), (vi) a reasonably detailed description of the Acquisition, and (vii) a request that the Board of Directors authorize, if appropriate, such Acquisition pursuant to this Section 4.4.2(b) of Article FOURTH. The Board of Directors may authorize an Acquisition by a Restricted Holder, if it determines in its sole discretion, that, such Acquisition will not be likely to directly or indirectly limit the availability to the Corporation of the Tax Benefits or is otherwise in the best interests of the Corporation and, in such case, the restrictions set forth in Section 4.4.2(a) of this Article FOURTH shall not apply to such Acquisition. If the Board of Directors authorizes an Acquisition by a Restricted Holder, it may, in its sole discretion, deem such Restricted Holder to be an Existing Holder (and to determine the deemed Existing Holder Initial Ownership) under this Article FOURTH. Any determination by the Board of Directors not to authorize a proposed Acquisition by a Restricted Holder shall cause such proposed Acquisition to be deemed a Prohibited Transfer. Any determination to authorize a proposed Acquisition by a Restricted Holder granted hereunder may be granted in whole or in part, and may be subject to any limitations or conditions (including restrictions on the ability of the Restricted Holder to subsequently transfer Corporation Securities acquired through such authorized Acquisition), in each case as and to the extent the Board shall determine in its sole discretion. In addition, the Board of Directors may, in its sole discretion, require representations from the Restricted Holder or an opinion of counsel to be rendered by counsel selected by the Board of Directors, that the Transfer will not result in the application of any Section 382 limitation on the use of the Tax Benefits or other matters that the Board of Directors may determine. Any Restricted Holder who makes a Request to the Board of Directors shall reimburse the Corporation, on demand, for all costs and expenses incurred by the Corporation with respect to any proposed Acquisition of Corporation Securities, including the Corporation’s costs and expenses incurred in determining whether to authorize the proposed Acquisition, which costs may include any expenses of counsel and/or tax advisors engaged by the Board of Directors to advise the Board of Directors or deliver an opinion thereto.

Section 4.4.3 Treatment of Excess Securities.

(a) No employee or agent of the Corporation shall record any Prohibited Transfer, and the purported transferee of a Prohibited Transfer (the “Purported Transferee”) shall not be recognized as a stockholder of the Corporation for any purpose whatsoever in respect of the Corporation Securities that are the subject of the Prohibited Transfer (the “Excess Securities”). The Purported Transferee shall not be entitled with respect to such Excess Securities to any rights of a stockholder of the Corporation, including the right to vote such Excess Securities and to receive dividends or distributions, whether liquidating or otherwise, in respect thereof. Once the Excess Securities have been acquired in a Transfer that is not a Prohibited Transfer, such Corporation Securities shall cease to be Excess Securities.

(b) If the Board of Directors determines that a Prohibited Transfer has been recorded by an agent or employee of the Corporation notwithstanding the prohibition in Section 4.4.3(a) of this Article FOURTH, such recording and the Prohibited Transfer shall be null and void ab initio and have no legal effect and, upon written demand by the Corporation, the Purported Transferee shall transfer or cause to be transferred any certificate or other evidence of ownership of the Excess Securities within the Purported Transferee’s possession or control, together with any dividends or other distributions that were received by the Purported Transferee from the Corporation with respect to the Excess Securities (the “Prohibited Distributions”), to an agent designated by the Board of Directors (the “Agent”). In the event of an attempted Prohibited Transfer involving the purchase or Acquisition of Corporation Securities in violation of this Article FOURTH by a Restricted Holder, the Agent shall thereupon sell to a buyer or buyers, which may include the Corporation or the purported transferor, the Excess Securities transferred

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to it in one or more arm’s-length transactions (including over a national securities exchange or national securities quotation system on which the Corporation Securities may be traded); provided, however, that the Agent, in its sole discretion, shall effect such sale or sales in an orderly fashion and shall not be required to effect any such sale within any specific time frame if the Agent determines such sale or sales could disrupt the market for the Corporation Securities, could adversely affect the value of the Corporation Securities or may be in violation of applicable securities laws. If the Purported Transferee has resold the Excess Securities before receiving the Corporation’s demand to surrender the Excess Securities to the Agent, the Purported Transferee shall be deemed to have sold the Excess Securities for the Agent, and shall be required to transfer to the Agent any Prohibited Distributions and proceeds of such sale, unless the Corporation grants written permission to the Purported Transferee to retain a portion of such sales proceeds not exceeding the amount that the Purported Transferee would have received from the Agent pursuant to Section 4.4.3(c) of this Article FOURTH if the Agent, rather than the Purported Transferee, had resold the Excess Securities.

(c) The Agent shall apply any proceeds of a sale by it of Excess Securities and, if the Purported Transferee had previously resold the Excess Securities, any amounts received by it from a Purported Transferee, as follows: (i) first, to reimburse itself to the extent necessary to cover its costs and expenses incurred in accordance with its duties hereunder; (ii) second, to reimburse the Purported Transferee for the amounts paid by the Purported Transferee for the Excess Securities (or in the case of any Prohibited Transfer by gift, devise or inheritance or any other Prohibited Transfer without consideration, the fair market value, calculated on the basis of the closing market price for the Corporation Securities on the day before the Prohibited Transfer), and (iii) third, the remainder, if any, to the original transferor, or, if the original transferor cannot be readily identified, to an entity designated by the Corporation’s Board of Directors that is described in Section 501(c) of the Code, contributions to which must be eligible for deduction under each of Sections 170(b)(1)(A), 2055 and 2522 of the Code. The recourse of any Purported Transferee with respect to any Prohibited Transfer shall be limited to the amount payable to the Purported Transferee pursuant to clause (ii) of this Section 4.4.3(c) of this Article FOURTH. Except as may be required by law, in no event shall the proceeds of any sale of Excess Securities pursuant to this Article FOURTH inure to the benefit of the Corporation or the Agent, except to the extent used to cover expenses incurred by the Agent in performing its duties hereunder.

(d) In the event of any Transfer to the Corporation, or any Transfer that does not involve a transfer of securities of the Corporation within the meaning of Delaware law (“Securities,” and individually, a “Security”), that would in either case cause (i) any Person to become a Five Percent Shareholder or (ii) the Percentage Stock Ownership interest of any Five Percent Shareholder to increase, the application of Section 4.4.3(b) and Section 4.4.3(c) shall be modified as described in this Section 4.4.3(d). In such case, no such Five Percent Shareholder shall be required to dispose of any interest that is not a Security, but such Five Percent Shareholder and/or any Person whose ownership of Securities is attributed to such Five Percent Shareholder shall be deemed to have disposed of and shall be required to dispose of sufficient Securities (which Securities shall be disposed of in the inverse order in which they were acquired) to cause such Five Percent Shareholder, after such disposition, not to be in violation of this Article FOURTH. Such disposition shall be deemed to occur simultaneously with the Transfer giving rise to the application of this provision, and such number of Securities that are deemed to be disposed of shall be considered Excess Securities and shall be disposed of through the Agent as provided in Section 4.4.3(b) and Section 4.4.3(c), except that the maximum aggregate amount payable either to such Five Percent Shareholder, or to such other Person that was the direct holder of such Excess Securities, in connection with such sale shall be the fair market value of such Excess Securities at the time of the purported Transfer. All expenses incurred by the Agent in disposing of such Excess Securities shall be paid out of any amounts due such Five Percent Shareholder or such other Person. The purpose of this Section 4.4.3(d) is to extend the restrictions in Section 4.4.2(a) and Section 4.4.3(a) to situations in which there is a Five Percent Shareholder without a direct Transfer of Securities, and this Section 4.4.3(d), along with the other provisions of this Article FOURTH, shall be interpreted to produce the same results, with differences as the context requires, as a direct Transfer of Corporation Securities.

(e) If the Purported Transferee fails to surrender the Excess Securities or the proceeds of a sale thereof to the Agent within thirty (30) days from the date on which the Corporation makes a demand pursuant to Section 4.4.3(b) of this Article FOURTH or any written demand with respect to a deemed disposition pursuant to Section 4.4.3(d) of this Article FOURTH, then the Corporation may take any actions it deems necessary to enforce the provisions hereof, including the institution of legal proceedings to compel such surrender.

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(f) If any Person shall knowingly violate, or knowingly cause any other Person under control of such Person (a “Controlled Person”) to violate this Article FOURTH (including failure to surrender the Excess Securities or the proceeds of a sale thereof as demanded by the Corporation pursuant to Section 4.4.3(e) of this Article FOURTH), then that Person and any Controlled Person shall be jointly and severally liable to the Corporation for, and shall indemnify and hold the Corporation harmless against, any and all losses and damages suffered as a result of such violation, including any attorneys’ and auditors’ fees incurred in connection with such violation.

Section 4.4.4 Legends; Compliance.

(a) All certificates reflecting Corporation Securities on or after the Effective Date shall, until the Restriction Release Date, bear a conspicuous legend in substantially the following form:

THE TRANSFER OF SECURITIES REPRESENTED HEREBY IS SUBJECT TO RESTRICTION PURSUANT TO ARTICLE FOURTH OF THE AMENDED AND RESTATED CERTIFICATE OF INCORPORATION OF RADIAN GROUP INC., AS AMENDED, A COPY OF WHICH MAY BE OBTAINED UPON WRITTEN REQUEST TO THE CORPORATION AT ITS PRINCIPAL PLACE OF BUSINESS. IF THE TRANSFER RESTRICTIONS ARE VIOLATED, THEN THE TRANSFER WILL BE VOID AND THE PURPORTED TRANSFEREE OF THE STOCK WILL BE REQUIRED TO TRANSFER THE EXCESS SECURITIES (AS DEFINED IN THE CERTIFICATE OF INCORPORATION) TO THE CORPORATION’S AGENT.

(b) The Corporation shall have the power to make appropriate notations upon its stock transfer records and to instruct any transfer agent, registrar, securities intermediary or depository with respect to the requirements of this Article FOURTH for any uncertificated Corporation Securities or Corporation Securities held in an indirect holding system. As a condition to the registration of the Transfer of any Corporation Securities, any Person who is a beneficial, legal or record holder of Corporation Securities, and any proposed transferee of such Corporation Securities and any Person controlling, controlled by or under common control with the proposed transferee of such Corporation Securities, shall provide such information as the Corporation may request from time to time in order to determine compliance with this Article FOURTH or the status of the Tax Benefits of the Corporation.

(c) Nothing contained in this Article FOURTH shall limit the authority of the Board of Directors of the Corporation to take such other action to the extent permitted by law as it deems necessary or advisable to preserve the Corporation’s Tax Benefits. The Board of Directors of the Corporation shall have the power to determine all matters necessary for determining compliance with this Article FOURTH, including determining (i) the identification of Five Percent Shareholders, Exempt Persons and Restricted Holders, (ii) whether a Transfer or proposed Transfer is a Prohibited Transfer, (iii) the Percentage Stock Ownership in the Corporation of any Five Percent Shareholders, Exempt Persons and Restricted Holders, (iv) whether an instrument constitutes a Corporation Security, (v) the amount (or fair market value) due to a Purported Transferee, (vi) whether compliance with any restriction or limitation on stock ownership and Transfers are no longer required for preservation of Tax Benefits, (vii) the interpretation of the provisions of this Article FOURTH and the applicability to stockholders of the Corporation of the restrictions on Transfer set forth herein, including the establishment of presumptions and procedures related thereto, and the correction or clarification of any errors or ambiguities therein, and (viii) any other matters which the Board of Directors deems relevant. Without limiting the generality of the foregoing, for the purposes of determining the existence and identity of, and the amount of Corporation Securities owned by, any Person, the Corporation and the Board of Directors are entitled to rely on (a) the existence and absence of filings of Schedule 13D, Schedule 13F, or Schedule 13G under the Exchange Act (or any similar schedules) as of any date, and (b) its actual knowledge of the ownership of the Corporation Securities. In the case of an ambiguity in the application of any of the provisions of this Article FOURTH, including any definition used herein, the Board of Directors shall have the power to determine the application of such provisions with respect to any situation based on its reasonable belief, understanding or knowledge of the circumstances. In the event that this Article FOURTH requires an action by the Board of Directors but fails to provide specific guidance with respect to such action, the

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Board of Directors shall have the power to determine the action to be taken so long as such action is not contrary to the express provisions of this Article FOURTH. All such actions, calculations, interpretations and determinations that are done or made by the Board of Directors in good faith shall be final, conclusive and binding on the Corporation, the Agent, and all other parties to a Transfer. The Board of Directors may delegate all or any portion of its duties and powers under this Article FOURTH to a committee of the Board of Directors as it deems advisable or necessary.

(d) Nothing contained in this Article FOURTH shall be construed to give any Person other than the Corporation or the Agent any legal or equitable right, remedy or claim under this Article FOURTH. This Article FOURTH shall be for the sole and exclusive benefit of the Corporation and the Agent.

(e) With regard to any power, remedy or right provided herein or otherwise available to the Corporation or the Agent provided under this Article FOURTH, (i) no waiver will be effective unless expressly contained in a writing signed by the waiving party, and (ii) no alteration, modification, or impairment will be implied by reason of any previous waiver, extension of time, delay or omission in exercise, or other indulgence.

(f) If any provision of this Article FOURTH or the application of any such provision to any Person or under any circumstances shall be held invalid, illegal or unenforceable in any respect by a court of competent jurisdiction, such invalidity, illegality or unenforceability shall not affect any other provision of this Article FOURTH.

(g) The Board shall at least annually consider whether to make the determination provided by clause (iii) of the definition of Restriction Release Date in Section 4.4.1 of this Article FOURTH in light of all relevant factors, including the amount and anticipated utilization of the Corporation’s Tax Benefits and the Corporation’s market capitalization.

[SIGNATURE PAGE FOLLOWS]

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IN WITNESS WHEREOF, this Second Amendment to the Corporation’s Amended and Restated Certificate of Incorporation, as amended, has been executed by a duly authorized officer of the Corporation on this 12 day of May, 2010.