Emap said that Savage's new role will see him continue to be "a guiding force in its future development" and added that the search for his replacement is under way.
Malcolm Gough, group managing director of Emap Business International, said: "This is an exciting time for Cannes Lions and we are looking forward to appointing a strong leader to take the festival into its next stage of growth."
Savage retained his position as chief executive of the festival when Emap acquired the business last summer in a deal worth £52.5m.
The media company's bid for the event followed the decision to sell by its 71-year-old owner and chairman, Roger Hatchuel.
At the time of the purchase, it was reported that Emap could expect to make an income of around €20m a year from the sale of delegate passes and entries for the event.
Hatchuel, who has run the festival for 17 years and was behind its relocation from Venice to Cannes, decided to sell after it became clear the business would not remain a family concern.
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