NEW YORK (Reuters) - A volatile trading session ended with U.S. stocks mostly higher on Wednesday, even as Apple, the most valuable company in the United States, suffered its worst day of losses in almost four years.In a strange occurrence, Apple accounted for the entirety of the Nasdaq 100's fall of 1.1 percent, while the Dow industrials - which do not include Apple as a component - enjoyed the best day since November 23.With the drop, Apple shed nearly $35 billion in market capitalization, its biggest one-day market-cap loss ever. The company's market value, or market capitalization, now stands at $506.85 billion."Today's move is because of index weightings, with the Nasdaq down because of Apple's decline," said Rex Macey, chief investment officer of Wilmington Trust in Atlanta. "The S&P is up because Apple isn't as big a weight in that index, and the Dow is up even more because it isn't there at all."The broad market seesawed, with the S&P 500 dropping into negative territory before it rebounded off the 1,400 level, seen as a key support point over the past two weeks. Investors cited comments from President Barack Obama suggesting a potential near-term resolution to the "fiscal cliff" wrangling in Washington as a catalyst for the rebound.Shares of The Travelers Cos Inc (TRV.N) rose 4.9 percent to $74. The stock ranked as the Dow's top percentage gainer after the insurance company said it intended to resume stock buybacks it had temporarily suspended while it assessed its exposure to Superstorm Sandy. The company also said a preliminary estimate of net losses from Sandy was about $650 million after tax.The Dow Jones industrial average rose 82.71 points, or 0.64 percent, to 13,034.49 at the close. The Standard & Poor's 500 Index gained 2.23 points, or 0.16 percent, to 1,409.28. But the Nasdaq Composite Index fell 22.99 points, or 0.77 percent, to end at 2,973.70.Apple, the largest U.S. company by market capitalization and a big weight in both the S&P 500 and the Nasdaq, fell 6.4 percent to $538.79. Apple is down more than 20 percent from an all-time high reached in late September, putting the stock into bear market territory.Banking shares were led higher by a 6.3 percent jump in Citigroup (C.N) to $36.46 after the company said it would cut 4 percent of its workforce. The S&P financial sector index climbed 1.3 percent, and Bank of America hit a 52-week high of $10.55 before pulling back slightly. The stock, a Dow component, ended at $10.46, up 5.7 percent for the day.Still, Apple struggled throughout the session. Market participants cited a host of reasons for the drop in the iPad maker's stock, including a consultant's report about the company losing share in the tablet market and reports that margin requirements had been raised by at least one clearing firm, as well as year-end tax selling ahead of a possible rise in capital-gains tax rates next year.On the Washington front, Obama told the Business Roundtable, a group of chief executives, on Wednesday that a fiscal cliff deal was possible "in about a week" if Republicans acknowledged the need to raise taxes on the wealthiest Americans.Equities have struggled to gain ground recently because of concerns over the fiscal cliff - a series of mandatory spending cuts and tax increases effective in early January that could push the U.S. economy into recession next year. Recently equities have moved on any whiffs of sentiment from Washington in headlines about negotiations."Obama's comments generated a lot of optimism, but to the extent the market believes them, that's how much we're setting ourselves up for a decline if that deadline passes with no progress," said Macey, who helps oversee about $20 billion in assets.In an interview on CNBC after the market closed, U.S. Treasury Secretary Tim Geithner said that uncertainty over the fiscal cliff was standing in the way of stronger economic growth, and that there was no prospect for an agreement if tax rates didn't rise on the wealthiest taxpayers.The stock of Freeport-McMoRan Copper & Gold Inc (FCX.N) fell 16 percent to $32.17 and ranked as the S&P 500's biggest percentage decliner. The company said it was acquiring Plains Exploration & Production Co (PXP.N) and McMoRan Exploration Co (MMR.N) in two separate deals for $9 billion in cash and stock in a major expansion into energy.McMoRan Exploration soared 87 percent to $15.82 and Plains surged 23.4 percent to $44.50.About half of the stocks traded on the New York Stock Exchange closed in positive territory, while about 54 percent of Nasdaq-listed shares ended lower.Volume was higher than it has been in recent sessions, with about 6.93 billion shares changing hands on the New York Stock Exchange, the Nasdaq and NYSE MKT, above the daily average so far this year of about 6.48 billion shares.(Editing by Jan Paschal)