Here’s a novel idea for companies that want to boost the value of their stocks: Pay the CEO less.

That seems to be one of the counter-intuitive lessons from a new study of 429 large-cap U.S. companies: Shareholder returns at the 20% with the lowest CEO compensation outperformed the 20% with the highest pay by as much as 39% over a 10 year period ending in 2015, according to MSCI, a research firm for institutional investors.

“Long-term institutional investors typically bear the cost of this misalignment, yet they have routinely approved CEO pay packages,” MSCI Corporate Governance Research Executive Director Ric Marshall says in a blog post. “Closer scrutiny of the relationship between CEO pay and performance over longer time periods could lead to different conclusions.”

That might surprise the boards of Big Media companies, which famously lavish moguls with eyepopping compensation packages. For example, CBS’ Les Moonves, Viacom’s Philippe Dauman, and Disney’s Bob Iger were among the USA’s 10 highest paid CEOs in 2015.

But MSCI says that its study turned up “very little statistical evidence to support” the proposition that higher pay serves shareholders. Indeed, “we found a small but consistently negative relationship, a possible indicator that superior performance may have been linked to lower rather than higher pay awards.”

Researchers say that company decisions may be distorted by the emphasis on annual reviews and CEO pay reports, which puts too much focus on short term results.

The report calls for additional disclosures in the annual proxy reports that describe top executives’ pay. Among other things, MSCI wants an assessment of executives’ realized pay over their entire tenures, with comparisons to the company performance in the same period. It wants annual updates on realized pay from executives’ stock awards, not just estimates of potential returns.

The firm also wants more details about pay incentives used to hire new CEOs, how that affected other executives’ pay, and one-time equity incentives.