Carl Icahn is keeping his family close.

Mr. Icahn's son, Brett Icahn, has scuttled plans to strike out on his own with a new hedge-fund firm, people familiar with the matter said.

The younger Mr. Icahn will instead stay at his father's public company, Icahn Enterprises LP, where he co-manages a portfolio of more than $6 billion of stocks with his longtime colleague, David Schechter, the people said.

Thanks to investment returns on some of Mr. Icahn's best-performing positions, the duo now manage $1 billion more than they did a year ago.

The Wall Street Journal first reported in May that Brett Icahn and Mr. Schechter were set to start their own Miami-based hedge-fund firm to specialize in shareholder activism, similar to the elder Mr. Icahn.

Icahn Enterprises was in negotiations to fund the new enterprise with $1 billion in startup cash in exchange for a partial ownership stake in the firm.

Before the deal was complete, however, talks fell apart.

Carl Icahn, 78 years old, hasn't laid out succession plans publicly and has no interest in stepping away from the firm he founded, according to people familiar with the matter. Icahn Enterprises manages both Mr. Icahn's vast fortune and runs several companies he has acquired.

The elder Mr. Icahn has credited Brett Icahn, 35, and Mr. Schechter, 39, with his 2012 investment in Netflix Inc., a pick that has netted Icahn Enterprises a profit of about $2 billion so far. Mr. Icahn has sold a significant part of the Netflix stake over the past year.

The duo also suggested to Mr. Icahn an investment in Apple Inc., which he made last summer.

Brett Icahn and Mr. Schechter both have spent more than a decade working at Icahn Enterprises and are responsible for what is known as the Sargon Portfolio.

Long one of the more outspoken voices in the hedge-fund industry, Mr. Icahn's company returned money managed for outside investors in 2011. He remains a forceful public presence through his firm's activist positions in companies like Hertz Global Holdings Inc. Earlier this week, Hertz announced its chairman and chief executive had resigned, weeks after Mr. Icahn criticized management. On Thursday, Hertz said it agreed to add three Icahn representatives to its board.

Icahn Enterprises managed more than $31 billion as of midyear, including roughly $11 billion in its investment division.

Write to Rob Copeland at rob.copeland@wsj.com and David Benoit at david.benoit@wsj.com

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