The boards are fond of personally vetting investment firms — something experts in model boardrooms say they should not be doing. Politics can often intrude. The teachers’ union, for example, keeps a list of investment firms it sees as unacceptable because of their connections to groups that, say, favor charter schools.

Ranji Nagaswami, who served as Mr. Bloomberg’s first chief investment adviser, said that changes in the governing structure — consolidating, professionalizing and depoliticizing the pension boards — could result in “vastly improving outcomes.”

In the existing environment, important questions about cost and sustainability can be broached only with great diplomacy. In 2010, Blackstone Advisory Partners, a private equity firm, found out what can happen otherwise. On a conference call with investors, a company official answered a fiscal question by saying retirement benefits for public workers across the country were excessive. When New York City’s trustees got wind of the comment, they called for Blackstone’s chairman to apologize in person. A few months later, he did, and when that proved insufficient, Blackstone issued a statement saying it opposed “scapegoating public employees.”

When the dust finally settled, Blackstone survived as one of the system’s biggest investors.

New York’s pension system is also the only major governmental system in the country to outsource virtually all of its investment decisions to outside money managers, pension experts said. That inevitably leads to higher investment fees. In 1997, the city’s biggest fund, the New York City Employees’ Retirement System, known as Nycers, spent $17.3 million in investment fees for a $31.7 billion portfolio. By 2010, it was spending $175 million for a $35.4 billion portfolio.

Some have argued the pension system would perform better if it hired its own professionals to manage the money in-house. Fees would drop, and the overall strategy would be more coherent, they contend.

For years, pension officials saw little reason to alter investment strategies or governing, in part because the stock boom of the 1990s made it seem as if they had a winning strategy. Even after the tech crash beginning in 2000, the city’s pension reports relied on an unusual calculation that made the system appear 99 percent funded. When that calculation was disallowed in 2006, Nycers’s reported funding level tumbled to 64 percent. Even then, some economists said the city was still underestimating its total obligation.

As it turns out, Robert C. North Jr., the system’s actuary, had been preparing his own stark projections, buried in annual reports. They are based on fair-market values and reflect what an insurer would charge for annuities designed exactly like the pensions. He estimated last year that Nycers was only 40 percent funded, a figure normally associated with funds in severe distress.