The American International Group said Thursday that it had completed a plan to repay United States taxpayers for bailing out the insurance giant during the height of the financial crisis.

Under the plan, the Federal Reserve Bank of New York would be repaid the nearly $20 billion that it is owed and the Treasury Department would convert the $49.1 billion in preferred stock that it holds into 1.66 billion common shares. Over all, A.I.G. received a bailout package of nearly $180 billion.

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