FIRST NBC BANK HOLDING COMPANY STOCK INCENTIVE PLAN 2012 AMENDMENT

Exhibit 10.4

FIRST NBC BANK HOLDING COMPANY

STOCK INCENTIVE PLAN

2012 AMENDMENT

Whereas, First NBC Bank Holding Company, a corporation organized and existing under the laws of the State of Louisiana (the “Company”), maintains a Stock Incentive Plan, which plan provides for the grant and award of incentive compensation related to the Company’s $1.00 per share par value common stock (“Common Stock”) and was first effective as of August 2, 2005, as the same has been amended from time to time (the “Plan”);

Whereas, the Company has now determined that the Plan should be amended and Section 4.8 of the Plan permits such amendment;

Now, Therefore, the Plan shall be amended as follows, such amendment to be effective upon its approval by a majority of the Company’s shareholders.

1. The following paragraph shall be added to Section 1 of the Plan, entitled “Shares Subject to the Plan” to read in its entirety as follows:

“Notwithstanding the preceding paragraph, an additional 418,000 shares of Common Stock shall be added to the Plan and available for grants and awards thereunder. The number of shares available for grant, award, transfer, issuance or other payment under the Plan shall be adjusted as follows: (a) reduced by the number of shares actually granted, awarded, transferred or issued hereunder; (b) increased by the number of shares covered by grants and awards that are not earned or that are cancelled, forfeited, terminated, expired or otherwise lapse for any reason; and (c) increased by the number of shares of Common Stock tendered in consideration of the payment of the exercise price of an Option or applicable taxes as provided herein.”

2. The following provisions shall be added to Section 2.6 of the Plan, concerning incentive stock options, to read in their entirety to read as follows:

3. Section 4.6 of the Plan, entitled “Withholding,” shall be amended and restated in its entirety as follows:

“4.6 Withholding. The Company shall have the right to withhold from any payment or distribution made hereunder, or to collect as a condition of any such payment or distribution, any taxes required by law to be withheld. Unless otherwise provided in an Incentive Agreement, this obligation may be satisfied by a common law employee, in

whole or in part, by directing the Company to withhold from any payment or distribution shares of Common Stock having a Fair Market Value equal to the amount required to be withheld, determined as income at a rate not in excess of the aggregate of the Federal income tax rate then applicable to supplemental wage payments, the then applicable employment tax rates, and the maximum marginal state income tax rate.”

4. The text of Section 4.1 of the Plan, entitled “Duration,” shall be removed and that section marked “Reserved”; Section 4.8 of the Plan, entitled “Amendment of the Plan,” shall be amended and restated in its entirety as follows:

“4.8 Duration, Amendment and Termination. This Plan commenced on August 2, 2005, its ‘Initial Effective Date’ and shall remain in effect until: (a) all incentives granted or awarded hereunder have been satisfied by the issuance of shares of Common Stock or cash payments or a combination thereof, or such incentives have expired, otherwise terminated or forfeited; or (b) restrictions or other performance objectives imposed on shares of Common Stock have been satisfied or lapsed. No incentive shall be granted or awarded hereunder ten years after the Initial Effective Date.

The Board of Directors may amend or terminate this Plan at any time; any such action may be taken without the approval of the Company’s shareholders, but only to the extent that: (a) shareholder approval is not required under applicable Federal or state law, regulation or stock exchange rules; and (b) such amendment does not increase the number of Shares of Common Stock available for grant, award or issuance hereunder.

The Board or a designated committee of the Board, as the case may be, shall possess the authority to amend the terms of any individual incentive granted or awarded hereunder; provided, however, that no such amendment shall materially impair the terms of any such incentive without the consent of the holder thereof and no such amendment shall: (a) reduce the exercise price of an Option; or (b) cancel or exchange an outstanding Option for cash, another incentive or for other Options with a lesser exercise price, except to the extent such action is contemplated under Section 4.4 hereof in connection with a corporate transaction involving the Company.”

5. Section 4.10 of the Plan, entitled “Definition of Fair Market Value,” shall be amended and restated in its entirety as follows:

“Section 4.10 Definition of Fair Market Value. As used in this Plan the term ‘Fair Market

Value’ means the closing sales price of a share of Common Stock as quoted on a national securities exchange or other recognized system of reporting as of the date specified herein; if no Common Stock is traded on such date, then Fair Market Value shall be determined as of the immediately preceding date on which Common Stock last traded on such national securities exchange or other recognized system of reporting. If Common Stock is not traded on any such national securities exchange or system of reporting, then Fair Market Value shall be determined by the Board or its designee in accordance with the provisions of Code Section 409A.”

6. The text of Section 4.11, entitled “Loans,” shall be removed, and that section marked “Reserved.”

7. Section 4.12 shall be added to the Plan to read in its entirety as follows:

“Section 4.12 Choice of Law. This Plan, including any agreement or form related thereto, shall be interpreted, construed and governed by the internal laws of the State of Louisiana, without regard to the conflicts of law provisions thereof.”

8. Section 4.13 shall be added to the Plan to read in its entirety as follows:

“Section 4.13 Participation. Common law employees of the Company and its affiliates shall be eligible to participate in this Plan, when designated by the Board of Directors or a committee thereof. Non-employee members of the Company’s Board of Directors shall also be eligible to participate in this Plan, subject to the terms and conditions set forth in that certain Addendum to the Plan approved by the Board of Directors on February 16, 2012. For purposes of this Section 4.13, the term ‘affiliate’ shall mean First NBC Bank and any other corporation or other form of entity of which the Company owns, from time to time, directly or indirectly, at least 50% of the total combined voting power of all classes of stock or other equity interests.”

This 2012 Amendment was adopted by the Board of Directors of the Company on May 3, 2012, to be effective upon its approval by the shareholders of the Company as provided herein.