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Exhibit 10.4

FIRST AMENDMENT TO ARCH CAPITAL GROUP LTD.

2002 LONG TERM INCENTIVE AND SHARE AWARD PLAN

        The Arch Capital Group Ltd. 2002 Long Term Incentive and Share Award Plan is amended as follows, effective September 17, 2003.

1.

Sections 7(a) of the Plan is amended and restated to read as follows:

"Annual Grant. On January 1 of each year, each Director in office on such date shall automatically be granted a NQSO to purchase 1500 Shares with an exercise price per Share equal to 100% of the Market Value of one Share on the date of grant; provided, however, that such price shall be at least equal to the par value of a Share. Each Option granted to a Director under this paragraph (a) shall become fully exercisable on the first anniversary of the date the Option is granted, and shall expire (unless terminated earlier under paragraph (d) below) on the tenth anniversary of the date of grant. Notwithstanding the foregoing, effective September 17, 2003, no additional Director Options shall be issued under this Section 7(a)."

2.

Section 7(b) of the Plan is amended and restated to read as follows:

"Initial Grants. Each Director will automatically be granted a NQSO on the date he or she is first elected to the Board to purchase 300 Shares with an exercise price per Share equal to 100% of the Market Value of one Share on the date of grant; provided, however, that such price shall be at least equal to the par value of a Share. Each Option granted to a Director under this paragraph (b) shall become exercisable in three equal installments, commencing on the date of grant and annually thereafter. Each Option granted under this paragraph (b) shall expire (unless terminated earlier under paragraph (d) below) on the tenth anniversary of the date of grant. Notwithstanding the foregoing, effective September 17, 2003, no additional Director Options shall be issued under this Section 7(b)."

3.

The first sentence of Section 8 of the Plan is amended and restated to read as follows:

"Director's Shares. Each Director may make an election in writing to receive the Director's annual retainer fees payable annually in the form of Shares instead of cash at least ten (10) days prior to the payment of such annual retainer fee."

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Exhibit 10.4

FIRST AMENDMENT TO ARCH CAPITAL GROUP LTD. 2002 LONG TERM INCENTIVE AND SHARE AWARD PLAN