My dad and Warren Buffett wrong on airlines, the younger portfolio manager Bill Miller argues

Value investor Bill Miller III and his son Bill Miller IV — who work together at the investment firm the elder founded — told CNBC on Tuesday they disagree on the airline sector, a day after it was revealed that Warren Buffett just jumped into airline stocks.

The Millers, appearing together on "Squawk Box," said they're referred to as Bill III and Bill IV around the offices of Baltimore-based LMM.

Bill IV, a portfolio manager at LMM, said he'd like to see his dad take a "victory lap," and take the money he's made on airlines and put it into other investments.

But Bill III countered: "I got a vote in my camp yesterday. He's [my son] feeling a little less confident," referring to Buffett's new positions in four of the nation's major airlines.

In Securities and Exchange Commission filings, Buffett-run Berkshire Hathaway said it's taken a stake in American, United, and Delta. But CNBC reports Berkshire has also bought a stake Southwest Airlines.

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Bill III invested in airlines in 2011, and said he tried to convince Buffett at the time to do the same. Miller said he's made five times his money, but feels there's some room left for Buffett. "I think there's a fair amount left," he said.

Going up against Buffett and his dad, Bill IV said: "[My dad] has made a lot of money. He should take a victory lap and move on to other things that could be more profitable."

"His fund," the elder Miller joked.

"One of the things we really agree on is alternative asset managers; things like KKR, Apollo, Fortress Investment Group. Those things look incredibly cheap to us. That's a space we really like," Bill IV said.