These instruments involved baskets of stocks put together by a bank. But Medallion didn’t buy the actual basket of stocks; it instead bought an option on that basket and sometimes gave the banks instructions on how to trade those stocks. Basket options have been criticized for having allowed hedge funds to borrow money more easily and allowing them to make bigger and potentially riskier trades.

The I.R.S. argued that the basket option trades should have been taxed at the higher rate because they were mainly the result of short-term trading.

The disagreement drew the attention of Congress, and led to rule changes. Following a report from the Senate Permanent Committee on Investigations, the I.R.S. issued new guidance in 2015 that sought to clamp down on this type of trading by making it more difficult and costly for hedge funds to buy basket options. Such investment vehicles had to be declared on the tax returns of any investor who used them, the agency said.

The I.R.S. had said its guidance on basket options would be retroactive, and applied to all transactions as far back as Jan. 1, 2011.

Still, some senators were critical of the I.R.S. for taking so long to change its rules and start investigating the trading practice, including at Renaissance.

Senator Carl Levin, the Michigan Democrat who headed the Senate committee in 2014 and died in July, said the I.R.S. guidance would stop banks and hedge funds from using “dubious structured financial products” that had cost taxpayers billions.

Elise Bean, a former aide to Mr. Levin, said she wished her former boss had lived to see the settlement. “It’s good to see that, despite a yearslong, knock-down, bare-knuckles battle, the I.R.S. prevailed in compelling at least one set of billionaires to pay the taxes they owe,” she said.

Jesse Drucker contributed reporting.

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