The stock fell just 6 cents, or about 0.2%, to $32.90 in trading on Wall Street after it was revealed that James Murdoch is expected to become the company's chief executive, replacing his father, Rupert Murdoch.
The shares had dropped about 1% earlier in the day as the broader markets rose.
Rupert Murdoch is expected to relinquish his CEO role at Fox but remain actively involved in the company. Lachlan Murdoch, James' older brother, is taking a larger role as executive chairman. Meanwhile, Chase Carey, is expected to step down as chief operating officer.
Timing of the transition was not immediately clear.
Carey is largely respected by Wall Street and investors might be nervous about the impending concentration of power in the Murdoch clan.
The appearance that the new CEO may be inheriting the job because of lineage rather than qualifications could be a concern for investors, wrote Cowen & Co. analyst Doug Creutz in a note to clients.
"Investors may be concerned that company performance would be worse than it could have been under an alternate scenario where the board sought out the best professional manager available," Creutz wrote.
James Murdoch's appointment would also come with baggage left over from his oversight of News International during the phone hacking scandal several years ago. The company had blamed the hacking on a single rogue reporter, but that defense crumbled.
Creutz said he was willing to give James Murdoch the benefit of the doubt that he can be a successful CEO but remained cautious.
"The fact that the company remains a family-run business remains a significant obstacle to our willingness to become more constructive on shares," Creutz said.
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