(Rewrites throughout to update prices, market details) By Susan Mathew Jan 31 (Reuters) - Latin American stocks and currencies rallied on Thursday, powered by the U.S. central bank's promise to be patient in hiking interest rates, helping stocks post their best January in 13 years. Markets globally cheered the U.S. Federal Reserve's stance, especially emerging markets that saw massive capital outflows last year thanks to a hawkish Fed raising the benchmark borrowing rate four times. "Equity markets liked what they heard – there is a sense that the Fed is very eager to prevent market volatility," said Paul Donovan, chief economist at UBS Global Wealth Management. Regionally, Mexican and Argentine stocks rose 0.8 percent each, while Brazil's Bovespa stock index climbed 0.4 percent to an all time high of 97,393.74. Over the month, the MSCI index of Latin American shares tacked up 14.9 percent in their best January since 2006. Brazil shares rose 11 percent and Argentine shares were up 20 percent over the month, its best January since 2013. Among currencies, MSCI's Latam currency index clocked its best month since June 2016. On the day, Brazil's real climbed more than 1 percent to a three month high, while the Argentine peso firmed 0.8 percent. "We now think it is most likely the Fed will hike one last time in September," said analysts at TD Securities. But, considering how much emerging markets have rallied over the past few months, a dovish Fed may not serve as a catalyst for a long terms rally unless it starts indicating a rate cut, Morgan Stanley analysts said. The Mexican peso, a weather vane for trade sentiment, built some gains after China's leader Xi Jinping told U.S. President Donald Trump in a letter that he hopes both sides will be able to meet each other halfway to reach a trade agreement before a March 1 deadline, Trump said on Thursday. Other Latin American currencies had closed trading for the day before Trump revealed the letter. The Mexican peso recovered from losses logged last session after rating agency Fitch downgraded debt-laden state oil firm Petróleos Mexicanos, or Pemex. But analysts warn the rally and monthly gains may be over done as political risk remain high. The Chilean peso hit its highest since early October, tracking copper prices that surged to multi week highs. The country's central bank raised the benchmark interest rate by 25 basis points to 3 percent after market close on Wednesday. The Colombia peso gained the most in the region, up 1.9 percent and posted its best day since August last year. The Colombian central bank held its key interest rates for a ninth consecutive month on Thursday. "The central bank will likely stay on hold until the second half of the year," analysts at Morgan Stanley said in a note. Latin American stock indexes and currencies at 2139 GMT: Stock indexes Latest Daily % change MSCI Emerging Markets 1049.93 1.28 MSCI LatAm 2947.57 2.35 Brazil Bovespa 97393.74 0.41 Mexico IPC 43987.94 0.82 Chile IPSA 5405.61 0.32 Argentina MerVal 36326.92 0.8 Colombia IGBC 11828.32 -0.61 Currencies Latest Daily % change Brazil real 3.6435 0.37 Mexico peso 19.1085 0.08 Chile peso 655.4 1.24 Colombia peso 3103.25 1.88 Peru sol 3.327 0.72 Argentina peso 37.2500 0.75 (interbank) (Reporting by Susan Mathew in Bengaluru; editing by Grant McCool)