Last October, Bank of America's board gave CEO Brian Moynihan another hat to wear, naming him chairman of the company's board.

But this was no ordinary crowning of a CEO as the board's leader. Back in 2009, as the financial crisis raged, a majority of Bank of America shareholders voted to separate the company's top two leadership roles, stripping then-CEO Ken Lewis of his chairman's title, a rare feat for investors. A few years and a CEO later, the board decided to delete that requirement for an independent chairman from its bylaws, and named Moynihan to the role.

Unsurprisingly, some shareholders and their advisers are none too pleased. Both CalPERS and CalSTRS, two of the country's largest pension funds, have said they are voting against the change, noting the company's underperformance and the need for more oversight. Proxy advisory firms Glass Lewis and Institutional Shareholder Services (ISS), which advise investors on how to vote their shares, have also recommended a vote against Moynihan holding both roles.

"Instead of trusting shareholders to approve its change, the board acted alone and by its own admission did not discuss the change with shareholders in advance of taking action," ISS wrote in a report. "The old adage – it is easier to ask forgiveness than it is to seek permission – is not a principle of good governance."

The decision on whether or not Moynihan will keep both titles will be made next Tuesday, when shareholders will have a chance to vote on the contentious issue.

The flap at Bank of America may be more focused on how the bank made the change, but it's also helping to reignite the long-simmering debate over whether public company CEOs should really chair their own boards. Many advocates of good corporate governance and members of the activist investor community think it's better to separate the roles, questioning how the CEO can preside over board meetings, set the agenda and help the board come to a consensus over tricky issues without inherent conflicts of interest.

"How in the world can the person being monitored be the one doing the monitoring?" asks Charles Elson, the director of the John L. Weinberg Center for Corporate Governance at the University of Delaware. "It makes no sense in theory."

Bank of America spokesman Lawrence Grayson pointed to the company's results in the second quarter, its turnaround strategy of a more simplified business model, and prior statements from the board in response to questions about the timing and process of the board's decision. "The board seeks the same flexibility on corporate governance as 97 percent of the S&P 500," he said. "We respectfully recognize that stockholders have varying views, which is why the board committed to holding the vote."

Some management thinkers do hold that splitting the roles has the potential to slow down decision-making or make CEOs more risk-averse. Still others think such theories of good governance have become too 'one size fits all' and should be decided on a case-by-case basis. Former representative Barney Frank, co-author of the Dodd-Frank financial reform bill, praised Moynihan recently, saying in an interview with Politico that he "never understood the argument that in principle it’s bad to have the CEO also be the chair."

Former FDIC head Sheila Bair also commented on the issue this month, telling Yahoo Finance that "too much is made of separating these roles." She also noted that "it's really more about the people and whether they are competent and setting the right tone and culture." She did not comment specifically on Bank of America.

Whether right or wrong, a growing number of companies have been giving the job to two different people. According to the executive search firm Spencer Stuart, 74 percent of companies in the S&P 500 had the same person in the chairman and CEO roles in 2004. Five years later, that number had dropped to 63 percent. It fell to 53 percent last year.

In tandem, the number of independent chairmen has risen significantly over the past decade, growing from just 9 percent in 2004 to 28 percent last year.

The concept of splitting the jobs—a way of putting less power into one person's hands—gained currency following the corporate scandals early in the last decade, said Matthew Semadeni, a professor at Arizona State University who studies corporate governance. "It has face validity," he said. "It became such an article of faith after Sarbanes-Oxley that you should split the two roles," referring to the 2002 legislation.

But that conventional wisdom isn't clear-cut, Semadeni said. So many academics have tried to find a correlation between the role split and company performance that Semadeni calls it "the El Dorado search" for many of them, a sort of unending quest for legendary gold. Study after study, he said, has had trouble finding an unambiguous link.  

His own research shows that splitting the jobs may have some effect on performance, but only in certain situations. In a 2012 paper, Semadeni found that if a company is performing poorly and chooses to split the jobs by giving the CEO a demotion, performance tended to reverse, improving after investors got the signal that the firm knew things needed fixing. But if the company was already performing well, performance actually fell if the CEO was demoted.

"If it ain't broke, don't fix it," Semadeni said.

Other governance experts, meanwhile, said that because companies only recently started turning to non-executive chairmen in any significant numbers, the jury is still out. "At this point I don’t have great confidence that you can make a conclusive judgement," says the University of Delaware's Elson. In the past, he said, boards were more advisory in nature, and it made sense to have the CEO in the chairman's job. But as they've gained more power as monitors in the wake of corporate accounting scandals and the financial crisis, "it just doesn't make any sense anymore." Elson owns stock in Bank of America and said he had voted against the bylaw change.

While many companies nominate a lead independent director to serve as a counterweight to a combined CEO-chairman—indeed, Bank of America named one when Moynihan got the nod—that approach isn't as strong as a true independent chair, according to Paul Hodgson, a partner at compensation research firm BHJ Partners who writes about corporate governance.

An independent chair "just has a lot more authority and can stand up to the CEO in a more authoritative way" than lead independent directors can, he said. "That balance of power that’s instituted keeps management in check."

Hodgson found in a 2012 study that companies tend to spend more on compensation when the same person is both CEO and chairman. (Yes, that's more, not less: CEOs who were also chairmen made a median $16.1 million in his study of 180 large companies, while companies spent a median $9.3 million when there was a separate independent chair, even though the company was paying two people.)

As for stock performance, Hodgson's study found that over the short term, CEOs who held both gigs actually outperformed those where the role was split. But over a five-year period, companies with a split CEO and chairman outperformed.

In other words, the best answer over whether to give the jobs to two people, or let one person hold both, may be that it depends—on how the company is performing, on how the change is made, and on what yardsticks are being used to measure performance. What Bank of America's shareholders think is the right answer, of course, will become very clear next week.

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