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“European banks are having trouble borrowing in general, including in dollars,” said Joseph Gagnon, a former Fed official and a senior fellow at the Peterson Institute for International Economics. “The Fed did the Europeans a favor.”

Foreign central banks are reducing by half a percentage point, to about 0.6 percent, the rate they charge commercial banks for dollar loans. Commercial banks need dollars because it is the No. 1 currency for international trade. The lower rate is designed to get credit flowing again.

To get the dollars to lend, central banks go to the Fed and exchange their currency for dollars under a special swap program. Foreign central banks pay the Fed whatever interest they earn from commercial banks.

The Fed had offered dollar swaps from December 2007, when world financial markets were weakening because of fear about subprime mortgages, until February 2010. It reopened the program in May 2010, as European debt concerns grew, and planned to end it Aug. 1, 2012. On Wednesday, the Fed extended the program to Feb. 1, 2013.

If it all works, the market rates on dollar loans will drop, and stock and bond markets will calm down.

“It shows that policymakers are on the case,” said Roberto Perli, managing director at the International Strategy & Investment Group, an investment firm. He said it has symbolic value even if it does not have a big impact on credit markets.

The decision to cut the interest charged on the dollar swaps was taken by the Federal Reserve following a videoconference held by Fed officials on Monday morning. The Fed’s policy-setting panel approved it 9-1. The president of the Fed’s regional bank in Richmond, Va., voted no.

In New York, the stock market jumped at the opening bell and added to its gains throughout the day. It finished up 490.05 points, its seventh-largest one-day gain and its best since March 23, 2009, two weeks after the stock market’s post-meltdown low.

Wednesday’s advance also swung the Dow from a loss for the year to a gain. It closed at 12,045.68, its first close above 12,000 since Nov. 15.

Stocks closed 5 percent higher in Germany, 4.2 percent in France and 3.2 percent in Britain. European stocks had posted big gains earlier this week because investors saw hope that countries would settle on an attempted fix for the European debt crisis.

Stock markets in Asia, which closed before the central banks announced their move, finished lower for the day. The statement came out at 8 a.m., in the middle of the European trading day and an hour and a half before the market opened in New York.

Borrowing costs for countries across Europe fell, an encouraging sign. The yield on benchmark 10-year national bonds dropped 0.32 percentage points in Belgium. It also fell in Spain, France and Germany.

The yield on 10-year Italian bonds fell 0.08 points to 7.01 percent. The 7 percent level is significant because it is considered the point at which a country’s borrowing costs become unsustainable. Yields above that level forced Ireland, Portugal and Greece to seek bailouts.

In the U.S., the yield on the 10-year Treasury rose to 2.08 percent from 2 percent late Tuesday. That showed investors were willing to take money out of assets considered super-safe, such as U.S. government debt, and invest it in riskier assets like stocks.

It is also a sign of increased confidence in the U.S. economy, which is beginning to pick up after it faltered in the spring and summer. It grew at an annual rate of 2 percent in July, August and September, the strongest since late last year.

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