* US stocks higher

* MSCI all country stock index hits record high

* Treasury bond yields fall

(Updates to 11:15 a.m. ET)

NEW YORK/LONDON, June 28 (Reuters) - The MSCI world stock index hit a record high while the U.S. dollar index inched lower on Friday after data showed that U.S. inflation was flat in May, fueling investor optimism the Federal Reserve could begin cutting interest rates in September.

U.S. stocks were moderately higher after the S&P 500 and Nasdaq hit record highs in early New York trading.

The Fed's preferred inflation measure, the personal consumption expenditures (PCE) index, showed that annual growth in prices was 2.6% in May, as economists had expected, down from 2.7% in April.

"When you compare what we got today with expectations, it is very much in-line and so the Fed will likely have enough comfort by the time of the September 18th meeting to cut rates for the first time," said Art Hogan, chief market strategists at B Riley Wealth in New York.

The chance of a rate cut in September inched up to 68% from 61% before the data, as per LSEG FedWatch data.

Investors were still digesting the U.S. presidential debate from late Thursday between Democratic President Joe Biden and Republican rival Donald Trump ahead of the November election.

Trump Media & Technology Group shares rose sharply early but were last down about 3%.

The Dow Jones Industrial Average rose 96.18 points, or 0.26%, to 39,264.37, the S&P 500 gained 14.49 points, or 0.26%, to 5,497.36 and the Nasdaq Composite gained 39.87 points, or 0.23%, to 17,899.56.

MSCI's gauge of stocks across the globe was up 1.60 points, or 0.20%, at 805.35 after hitting a record at 808.37 earlier. The STOXX 600 index fell 0.27%.

The dollar index, which measures the greenback against a basket of currencies, was down slightly at 105.85.

The U.S. dollar briefly fell against the Japanese yen after the PCE data. Against the yen, the dollar last was up 0.09% at 160.88.

The yen's slide to a 38-year low has fueled expectations of intervention by the Japanese authorities to stem the currency's weakness.

The euro was up 0.06% at $1.0708.

Worries about the outcome of the two-stage French parliamentary elections that start on Sunday pushed the risk premium on French government bonds over German bonds to its widest since the euro zone debt crisis in 2012.

In Treasuries, the yield on benchmark U.S. 10-year notes rose 4.3 basis points to 4.331%, from 4.288% late on Thursday.

U.S. crude lost 0.48% to $81.35 a barrel while Brent was flat at $86.39 per barrel.

(Reporting by Huw Jones in London and Caroline Valetkevitch in New York; additional reporting by Johann M Cherian, Rae Wee and Stella Qiu Editing by Kim Coghill, Jacqueline Wong, David Goodman and Christina Fincher)