Activist investors Carl Icahn and John Paulson, who secured seats on the board previous year, have also piled pressure on AIG's executives.
AIG said Hancock will remain with the company until a new CEO is installed.
Carl Icahn, who called for a breakup of the company two years ago to reduce regulatory burdens, welcomed the move in a tweet, saying "we fully support the actions taken today by the board of $AIG".
"I believe this is the right decision to make for the company and all its stakeholders", he said in a statement.
Mr. Hancock's resignation, which came less than a month after AIG posted a $3.04 billion loss for the fourth-quarter of 2016, was also well-received by analysts.
"Without wholehearted shareholder support for my continued leadership, a protracted period of uncertainty could undermine the progress we have made and damage the interests of our policyholders, employees, regulators, debtholders and shareholders", Hancock said in the statement.
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Shares (Berlin: DI6.BE - news) of AIG rose 1.9 percent to $64.63 in pre-market trading. "Of course, there aren't too many candidates with the skills needed to turn around this troubled global company, but several successful turnarounds" have occurred in the industry.
Doug Steenland, the insurer's chairman, thanked Hancock for his work in helping repay a bailout that swelled to $182.3 billion.
Hancock is AIG's fifth chief executive since Maurice "Hank" Greenberg was ousted in 2005.
Mr. Icahn quieted his criticism after he won representation on AIG's board, the insurer slashed its management ranks and proposed a turnaround plan that included shrinking its USA casualty book and selling certain units - but falling short of a full breakup. Earlier this year, the New York-based insurer agreed to pay $9.8 billion for reinsurance coverage from Warren Buffett's Berkshire Hathaway (BRK.A) .
Bowing to pressure from Icahn and hedge fund manager John Paulson, AIG unveiled its turnaround plan a year ago, but those efforts fell short.