Brian T. Moynihan, who runs Bank of America, the second-largest U.S. lender by assets, was among a group of underpaid chief executive officers last year, according to a study by pay expert Graef Crystal.

At least 99 companies in the Standard & Poor's 500 Index paid their CEOs at or below what Crystal calls their "going rate" last year, according to his study, which tracked 177 firms in the index. Moynihan, 54, was paid $13.1 million in 2013, 30 percent below what Crystal calculated as his going rate.

Chief executives typically receive boosted pay when their company's performance improves, Crystal said. When performance stalls, compensation doesn't always follow suit. Some U.S. companies have been able to tie pay to performance better than others ? and improvement at those firms is happening at a "glacial pace," Crystal said.

"If you go down the aisle at the executive-pay supermarket, the quality relationship to price is not as robust as in a regular supermarket," Crystal, 79, said in a phone interview from his home in Las Vegas. "You can pick up someone of very good performance for less than someone with very poor performance."

For the study, Crystal developed his so-called going rate, or what each of 177 chief executives would be paid on average based on his or her company's sales, total return versus the S&P 500 and number of years they have served. He chose S&P 500 companies with a fiscal year ended Dec. 31, 2013, that have filed 2013 proxy statements as of March 28, and whose CEO has served for at least one year.

He then compared each going rate to the company-reported CEO pay in their summary compensation tables and compiled the deviations between those numbers.

The study used information from the U.S. Securities and Exchange Commission-mandated summary compensation table, which reports some awards in the year they're granted rather than for the year they're earned. Some awards are restricted, vesting and paying out over a set time frame, and the receipt of others may depend on future performance goals. The summary compensation table also counts changes in pension and the value of perks.

Moynihan, who has a four-year tenure and a 2013 return that's 2 points over the S&P 500's advance, would have a going rate of $18.8 million in 2013 ? 30 percent below what he was paid that year, according to Crystal's model.

On an awarded basis, or what the board of directors deems appropriate to compensate an executive for a particular year's performance, Moynihan was paid $14 million for 2013. That includes $12.5 million in stock grants and a $1.5 million salary. His awarded compensation, which was 17 percent higher than a year earlier after profit more than doubled, didn't include a cash bonus.

As much as 80 percent of some executives' pay comes from cashing out awards that were granted for performance in prior years, Brent Longnecker, CEO of executive compensation consulting firm Longnecker & Associates, said by phone from the company's Houston office.

"If you have four- and five-year plans with long-term incentives, you most likely still have stock that has some value," Longnecker said of companies underperforming in the last year. "If you decide to cash out the last seven years, that would look high, but that's for all those years that shareholders make money."

Half of Moynihan's incentive compensation for 2013 was granted in restricted stock units that are earned by achieving specific performance goals over a three year period, Bank of America said last month in a regulatory filing. Lawrence Grayson, a spokesman for the Charlotte, North Carolina-based lender, declined to comment.

Crystal, a former Bloomberg News columnist, worked as a pay consultant for about 20 years, advising companies including American Express Co. and Bank of America, as well as the SEC and the Financial Accounting Standards Board.

Some companies paid their executives more than what Crystal calculated as their going rate. AT&T Inc., the second-largest U.S. wireless carrier, paid CEO Randall Stephenson $23.2 million in 2013, according to the company's summary compensation table. That's 19 percent higher than his going rate, even as AT&T's shares lagged the S&P 500's advance by 23 points last year.

"AT&T is committed to paying for performance, and Mr. Stephenson's compensation reflect this," Brad Burns, a spokesman for the Dallas-based company said in an e-mail. Last year, 92 percent of Stephenson's target compensation was tied to performance, he said.

"If you've been there for a long time and we think you're great, and we had a down year, we're probably going to give you the benefit of the doubt," Alan Johnson, a managing director at compensation consultant Johnson Associates, said in a phone interview from the company's office in New York. "Sometimes it's appropriate, sometimes it's nostalgia."

International Business Machines Corp., run by CEO Virginia Rometty, was among companies that more closely tied pay to performance last year, according to the study. Rometty's going rate was $15.4 million, and she received pay 9.5 percent below that, Crystal's study and company filings show. That occurred as IBM's total return fell behind the S&P 500 by 33 points, according to the study.

Rometty, 56, forwent annual bonuses for herself and top executives for 2013 after the Armonk, New York-based company reported a seventh straight quarterly sales decline in January.

"The cash flow is less important than the message; these people are not going to be missing any meals," Johnson said. "But we can send a message to you through bonus cuts."

A spokeswoman for IBM said the company couldn't immediately comment.

The model was capable of explaining 48 percent of the variation in CEO pay, Crystal said. The probability of a chance result in each of the three factors ? sales, total return versus the S&P 500 and tenure ? was less than 1 percent.

After controlling for the variables measured in the study, Crystal tested whether being a male or female further explained the variations and said it didn't.

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