She made the announcement a day after she was questioned for a fourth time in the investigation of her role in 2008 during an arbitration proceeding between the government and Bernard Tapie, a onetime cabinet minister and the former owner of the Adidas sportswear empire. Prosecutors had assigned Ms. Lagarde the status of “assisted witness” in the case. Placing her under formal investigation signals that prosecutors now believe they have evidence of wrongdoing, but the charge of negligence hardly suggests high crimes.

Image Stéphane Richard, Ms. Lagarde’s former chief of staff and currently the chief executive of Orange, has been placed under investigation on “suspicion of organized fraud.” Credit... Christophe Ena/Associated Press

Negligence by a government official that makes possible the misappropriation or embezzlement of public funds is punishable with a maximum fine of 15,000 euros, or $19,800, and up to one year in prison.

“After three years of investigation and dozens of hours of interrogation, the committee concluded that I had not been guilty of any infraction,” Ms. Lagarde said in the statement, “so it was reduced to claiming that I had been insufficiently vigilant during the arbitration.”

In the French legal system, a formal investigation suggests prosecutors believe they have enough of a case that they may ultimately bring criminal charges and trial. The reality is that many such investigations drag on for years with no charges being filed before being dropped.

Since 1944, when the I.M.F. was conceived, the fund’s top official has been a European, with the job alternating among Western European nations. Ms. Lagarde took over as managing director of the fund in 2011 after her French predecessor, Dominique Strauss-Kahn, left in disgrace amid charges — later dismissed — that he had sexually assaulted a woman in a New York hotel room.