US stocks ended the worst January since 2009 with the best one-day gains in more than four months Friday, after earnings from Microsoft Corp. exceeded expectations and the Bank of Japan stepped up monetary stimulus. The equity rally accelerated in the final hour of trading, with the strong finish a fitting end to a weak month that featured sharp reversals on an almost daily basis. Microsoft led the surge Friday with its biggest gain in three months. Nine of the S&P 500’s 10 main groups rose at least 1.7 percent. Amazon.com Inc. was a blemish, tumbling 7.6 percent as earnings for the holiday quarter missed estimates. “Part of the strength in the markets today is central banks in the developed world being accommodative, and the other is a surprisingly strong Chicago manufacturing number that was really a blowout,’’ said Phil Orlando, who helps oversee $360 billion as chief equity-market strategist at Federated Investors Inc. in New York. “Earnings have been better than expected so far.’’ Stocks swung between gains and losses last week as ­investors assessed corporate earnings and the degree to which central banks will intervene to help stem increasing volatility and a dimming outlook for global growth.