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Stocks surged Tuesday, as investors cheered the sixth interest rate cut by the Federal Reserve since September, and two Wall Street powerhouses, Lehman Brothers and Goldman Sachs, posted better-than-expected quarterly results.

The Dow Jones industrial average jumped 420.41 points, or 3.5 percent, to 12,392.66, topping the 416-point rally one week earlier.

All 30 of the Dow industrials advanced, led by American International Group and Bank of America. Despite a drumbeat of bad news about the economy and Wall Street in the past two months, the Dow has traded in a narrow range, between 12,000 and 12,500.

Lehman Brothers soared $14.74, or 46 percent, to $46.49. Goldman Sachs rose $24.57, or 16 percent, to $175.59.

The broader Standard & Poor’s 500 index posted a bigger percentage gain, closing up 54.14 points, or 4.2 percent, to 1330.74. The Nasdaq composite index advanced 91.25, or 4.2 percent, to 2268.26. The Russell 2000 index of small-company stocks rose 31.45, or 4.8 percent, to 681.93.

Advancing stocks outnumbered losers by more than an 8-1 ratio among New York Stock Exchange stocks. NYSE trading volume totaled 1.94 billion shares, down from 1.99 billion shares Monday.

All 10 major industry sectors in the S&P 500 closed higher, led by financial-service stocks. Mortgage finance agency Fannie Mae gained $6.01, or 27 percent, to $28.22. Freddie Mac rose $5.40, or 26 percent, to $26.02.

Treasury securities lost ground as stocks advanced and fears of fresh crises in credit markets eased.

The dollar staged its biggest rally in nine years against the yen, as traders cheered the Fed move.

Crude oil prices jumped in futures trading, reflecting fresh optimism about economic growth.

Crude oil for April delivery rose $3.74 a barrel, to $109.42.