Time and again big banks have wrestled with the issue of whether they should separate the roles of chairman and chief executive, and they have often fought off shareholder efforts to force a split or have given in and undone it later.

But ever since Wells Fargo was fined $185 million this month for setting up thousands of unauthorized customer accounts, corporate governance has once again become a hot topic. Analysts and activist investors have renewed calls for the bank to establish an independent chairman of the board rather than letting its chief executive, John Stumpf, fill that role.

Independent chairmen, some say, could potentially uncover or minimize the damage caused by fraud while also meting out punishment more impartially after the fact. Perhaps an outsider might have pressed harder for Stumpf to clean up Wells' problems, or claw back the pay of Carrie Tolstedt, who as Wells Fargo's head of community banking failed to stop some 5,300 employees from creating roughly 2 million fake deposit and credit card accounts for customers — inflating the bank's numbers in order to earn higher fees. Tolstedt is set to retire at the end of the year with a package worth tens of millions of dollars.

Whether Stumpf is forced to resign, as some lawmakers and others have called for him to do, the larger issue is whether this fiasco lends credence to the argument that the CEO and chairman positions at big banks should be split up.

Robert McCormick, chief policy officer at the proxy advisory firm Glass Lewis, said it is "a pretty insurmountable conflict of interest" for one individual to hold both jobs. Not only is it difficult, as the CEO, to "oversee yourself" on behalf of shareholders, but in the event of scandal or malfeasance you may be disinclined to hold yourself accountable. "How," he asks, "do you impartially evaluate your own performance?"

Gerald Armstrong is an activist investor who has long advocated keeping the CEO and chairman positions separate. "John Stumpf was accountable only to himself as chairman, and until last November he was president, CEO and chairman," Armstrong points out. He intends to submit a new proposal to split up the roles, which would be voted on at the next Wells Fargo shareholders meeting in April 2017.

Yet however much the proposal might seem a no-brainer of governance to some, it is common for bank CEOs to lead their company's board — an arrangement which shortens the chain of command and gives the top executive greater discretion to steer the bank as he sees fit.

Of the 10 largest U.S. banks by total assets, only Citigroup has an independent chairman of the board; the others have all vouchsafed to their CEOs the dual role of chairman. That is no coincidence, said Stephan Biggar, an analyst at Argus Research. Citigroup has been more troubled than its peers in the years since the financial crisis.

Bank of America, following its acquisition of Merrill Lynch in 2009, was forced to split up the CEO and chairman jobs after 51% of its shareholders voted in favor of such a move. It, too, had its share of problems at the time, including a high number of writeoffs in its mortgage business. "So you can see the pattern here: Banks that have difficulties have a greater chance of having the roles separated," Biggar said.

Bank of America has since recombined the roles. The move, initially a unilateral decision by the bank's board to remove the mandate for an independent chairman from its bylaws, unsurprisingly drew shareholder ire for undoing the majority vote of a few years before. A spokesman for Bank of America defended the decision by saying its board sought "the same flexibility on corporate governance as 97% of the S&P 500."

Ultimately shareholders were given a say, and 63% of them voted in favor of letting B of A's Brian Moynihan keep both jobs at a special meeting a year ago.

While Biggar himself favors splitting up the roles, he understands why some analysts and investors are lackadaisical on the issue. "It's up to shareholders at the end of the day," he said. "And if someone is doing well in both roles, then why shake things up?"

While it is still not standard to separate the roles, said McCormick, there is a slow trend in that direction.

Watching the watchmen is generally good governance. But it provides no guarantee that scandals will be averted. Biggar points to the "London whale" incident of 2012, in which a London office of JPMorgan placed large bets that resulted in more than $6 billion in losses — and about $1 billion in fines — for the nation's largest bank.

"Would it have been different if [CEO Jamie Dimon] hadn't been in both roles? No, I think it would have still happened," Biggar said.

Following the incident, shareholder proposals to take away Dimon's chairmanship failed to pass, receiving 40% of the vote in 2012 and only 32% in 2013. Shareholders voted again in 2015 on splitting the chairman and CEO jobs at JPMorgan, and 36% voted in favor in what many saw as a vote of confidence in Dimon.

Armstrong thinks otherwise — at least as far as Wells Fargo's new fiasco is concerned. "I do believe that had we had an independent chairman at Wells Fargo, the $185 million fine would not have been imposed," he said. "I think it would have been a miniscule fine, if any, and probably the problem could have been avoided or corrected quickly."

Allowing a bank's CEO to lead the board is like "putting Count Dracula in charge of the blood bank," he said.

The matter of whom to hold responsible, and how, came to a head on Tuesday when Stumpf appeared before the Senate Banking Committee to be grilled on the unfolding scandal. Sen. Elizabeth Warren, a Democrat from Massachusetts, hammered Stumpf on the question of whether he would give back any of the millions of dollars he had earned in compensation. Wells Fargo's CEO insisted that the decision was up to the bank's 15-member board, which, he said, "has the tools to hold senior management accountable, including me."

Stumpf does not sit on the board's compensation committee, which is currently considering whether to claw back any of his or Tolstedt's earnings. It remains to be seen how Wells Fargo will try to reassure shareholders, but it is unlikely the board will strip Stumpf of his chairmanship. "Removing the title of chairman from a CEO would be in essence a vote of no confidence in that CEO," McCormick said.

In any case, Armstrong, who first bought Wells Fargo stock in 1969 and said he now holds more than 23,000 shares in his own name and more in a retirement trust, is not out for blood. And he said he would favor splitting up the CEO and chairman positions even if it did not boost Wells Fargo's stock price at all. He just wants good governance.

"I want to see the corporation improve," he said, "and then I will be happy."

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