“The frustration is with the stridency of the point of view on one side,” Rothenberg said. “We have a body of data that suggests something different.”

American Funds, the third-biggest mutual-fund family in the U.S., oversees $993 billion, excluding money-market funds, Morningstar data show. Vanguard’s mutual funds and ETFs total $2.1 trillion, with the majority in index-based products. Boston-based Fidelity Investments manages $1.1 trillion in mutual funds.

Capital Group’s largest offering, the $123 billion Growth Fund of America, has seen its assets drop 31 percent in the five years ended Aug. 31. During that time the fund returned an annual average of 6.4 percent, compared with 7.3 percent for the S&P 500.

Since July 2011, three funds have surpassed Growth Fund of America in size. All are index funds run by Vanguard.

‘Hit a Wall’

“After decades of really great success, they hit a wall and suffered subpar performance in the 2008-2009 period,” Burton Greenwald, a mutual-fund consultant in Philadelphia, said in an interview. “They’ve got a lot of mending to do in terms of their reputation.”

The Capital Group study examined 17 of the company’s mutual funds that invest in equities or both equities and bonds. It measured their performance over every one-, three-, five-, 10-, 20- and 30-year period, on a rolling monthly basis, from Dec. 31, 1933, through Dec. 31, 2012.

Rothenberg, 67, said the company has put in place a succession plan for his retirement, though no date has been set yet.

Timothy D. Armour, president of Capital Research & Management, was the “logical” person to take over as chairman when he steps down, Rothenberg said, and a core of younger executives had already taken over most decision-making on the firm’s management committee, the senior-most executive body.

That group, he said, includes Armour; Rob Lovelace, grandson of founder Jonathan Bell Lovelace; Shaw Wagener and Darcy Kopcho.