In late 2012, Arkansas hired Bret Bielema as its football coach, paying him a salary of $3.2 million a year, plus bonuses, making him one of the best-compensated coaches in his industry and his state’s highest-paid employee.

During Bielema’s first season, the Razorbacks won their first three games before losing their last nine, prompting some fans to wonder whether Arkansas had overpaid him. One fan took to Facebook to sarcastically thank the coach for the three wins: “Good job, Bielema, here’s $3 million dollars for that.” Another posted on Twitter that the coach should share his paycheck with his players who “get paid nothing but bring $ to the university.”

But, according to a new study by researchers at Vanderbilt, coaches like Bielema who command what are widely seen as robust salaries are worth the money because of the value they bring to their universities. The Vanderbilt study, which included 947 contracts from 2005 to 2013, benchmarked coaching salaries against those of chief executive officers at public companies — another group that is often accused of being paid too much.

“Coaches are running large programs that have tremendous value,” said Randall S. Thomas, a law and business professor and one of the authors of the study. “They are creating great value, and they are being paid for creating that value.”