Comparable stocks, notably State Street and similar businesses within JPMorgan Chase, have outperformed BNY substantially, Marcato noted.
Though Tuesday's demands mark the most recent public activist attack on BNY, which was founded by the U.S.'s first treasury secretary, Alexander Hamilton, in 1784 and now acts as the custodian of some $28.5 trillion in assets, the company has been under pressure from other quarters, too.
During the course of the past year, some analysts have cracked down on BNY for failing to spin off certain assets or to reap greater benefits from its merger with the smaller player Mellon Financial in 2007. And last June, the hedge fund Trian Fund Management, whose principles are known for their sharp rebukes of corporate management teams, disclosed a $1 billion stake in BNY (with about 2.7 percent of the shares outstanding, Trian, which has added slightly to its position since then, is now the seventh-largest shareholder).
But unlike Marcato on Tuesday, Trian has kept its concerns about BNY largely in the private sphere so far, reportedly suggesting cost-cutting and other measures to the custodial bank in closed conversations. Even so, BNY in December named Trian co-founder Ed Garden to its board, employing what is widely regarded as a classic move to assuage a frustrated activist shareholder.
"When Trian goes on a board, our goal is always to work collaboratively with management and we have a strong track record of results," Garden said in a statement to CNBC. "I have been on the BNY Mellon Board for about three months. I am encouraged by the board's reaction to our ideas and management's sense of urgency in achieving significant operating improvement."
Marcato, the San Francisco hedge fund founded in 2010 by Pershing Square alumnus McGuire, has been a low-profile but successful investor in recent years. Managing some $3 billion, including a cornerstone investment from the hedge fund unit within the Blackstone Group, Marcato has generated double-digit returns every year since its inception except for 2014, when it returned 5.3 percent.
For February alone, Marcato returned nearly 9 percent—offsetting losses in January to be up about 2.4 percent year to date.
The hedge fund's holdings, which consist mostly of long-term equity positions, include just 13 different names, of which BNY is the largest. Other top positions include the auto-supply company Lear and InterContinental Hotels Group, both of which Marcato has been active with; the auction house Sotheby's, its eigth-biggest position, has also been a recent target.
Marcato's lengthy presentation on BNY doesn't spare any executive feelings in its analysis.
Sections entitled "Poor Management through a Changing Cycle" and "Gerald Hassell's 2011 Investor Day Targets Have Failed to Materialize," detail the ways in which the activist fund believes the company has proved a disappointing investment. They include a cumbersome cost structure thanks to too many mergers and acquisitions, including, most notably, the 2007 Mellon deal; a ballooning headcount; deteriorating return on equity figures despite a stated goal of 10 percent by the year 2014; flat earnings per share numbers in recent years; and weakening profit margins.
And Marcato's BNY presentation is notable for its scope as well as its tough tone.
In addition to scouring the public record for flaws and inconsistencies in the bank's financials and oversight, the hedge fund canvassed the financial industry for potential replacements for Hassell and even consulted with a brand manager on ideas for reshaping BNY's logo and advertising efforts. The result: a spiffed-up, multicolored version of BNY's iconic arrow-shaped logo that could be used in outdoor campaigns as a sculpture as well as on paper alongside the slogan "We're Here."
—CNBC's Scott Wapner contributed to this report.