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Much of the trading was unwinding panic moves from last week when fears about the worst financial crisis in nearly 80 years swept across the world.
Wall Street looked set to follow Europe and Asia higher. European shares were up more than 5 percent. Japan's Nikkei, which was not traded on Monday because of a holiday, gained more than 14 percent.
MSCI's main world stock index was up 3.8 percent, gaining more than 12 percent for the week to date after plunging 20 percent last week.
Until this week's rally, the index had lost 47 percent of its value -- more than $20 trillion -- from an all-time high roughly a year ago.
Its emerging market stock counterpart was up 5.4 percent on the day, adding to Monday's 7.4 percent gain.
Investments seen recently as relatively safe compared with stocks -- the Japanese yen and government bonds -- fell.
‘The most important thing is to get money markets working again,’ said Neil Parker, market strategist at Royal Bank of Scotland.
‘If we can do that, with a combination of the measures that have already been put in place and further interest rate cuts from the central banks, then we could be reaching a turning point.’
The United States will announce plans later in the day to inject $250 billion into its banks, following similar moves by Britain, France, Germany and others on Monday.
Japan also joined the global push, saying it could inject public funds into regional banks to make sure small firms can get cash.
The pan-European FTSEurofirst 300 index was up 5.3 percent points after its strongest one-day percentage rise on record -- 10.1 percent -- on Monday.
‘Market players are hoping that the crisis has reached a turnaround point thanks to the extensive aid programmes,’ German bank Helaba said in a note.
Earlier, Japan's benchmark Nikkei surged 14.2 percent or 1,171.14 points to 9,447.57. The broader Topix gained 13.7 percent to 956.30.
ECONOMY WORRIES
Some analysts warned, however, that once the relief about the financial system had played through, the declining state of the world economy remained.
‘We will inevitably enter a phase of thinking about how the steps will actually impact the global economy,’ said Daiwa SB Investments' Ogawa.
This was underlined on Tuesday by the monthly ZEW German investor sentiment survey.
It showed the outlook for Europe's largest economy deteriorated by more than expected this month, with the index falling to -63.0 from -41.1 in September.
The yen fell against both the euro and dollar.
‘We've seen a very strong relief rally after severe falls on global stock markets, but to say that everything is over and it's going to be hunky-dory (okay) from now on is premature,’ Nordea FX strategist Niels Christensen said.
‘Governments and central banks have done what they can do, but that said, still, we don't have the money market functioning as it used to,’ he added.
The euro rose 2 percent against the yen to 141.43 yen, having rebounded off a three-year low of 132.15 yen hit on trading platform EBS on Friday.
The dollar was up 0.39 percent at 102.88 yen. The euro also gained 1.1 percent against the dollar to $1.3747.
On government bond markets, short-term euro zone debt prices fell as investors sold.
Two-year Schatz yields climbed 9 basis points to 3.263 percent, while 10-year Bund yields were up slightly at 4.097 percent.
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