In 2012, the pension trustees asked the State Legislature to lower the system's long-term investment expectation to 7 percent, at the prodding of Mr. North. Many experts say 7 percent is still too high. But cutting the assumption to 7 percent from 8, along with some other adjustments, was going to cost the city an additional $2.8 billion in contributions in the first year. Mr. North eventually lowered the contributions total to $600 million by spreading the cost over 22 years.
The additional cash will certainly help the pension system, but it will still take years and luck in the markets for the city to close the gap from the years it should have been contributing more.
It is the pursuit of higher returns that has led the trustees, and lawmakers in Albany, to authorize more aggressive, alternative strategies, mirroring a national trend among public pension systems. The approach carries the possibility of a greater upside but also brings greater risks and costlier fees.
"There's nothing wrong with taking the risk," Mr. North said. "The risk, however, should be recognized and understood as it is mostly borne by future generations," people who were not consulted on these decisions.
In New York, private equities — stocks that do not trade on any exchange or have a published price — have become a favored asset class. Private equity investments are typically done through partnerships with specialized firms, which last for several years. Until they run their course, the returns cannot be calculated accurately. The fees are high, and it is not yet clear that the partnerships are delivering consistently higher returns.
In one example, John Murphy, a former executive director of Nycers, the fund for general workers, said he noticed in 2011 that the Allegra Capital Partners IV fund had just come to an end, making a final accounting possible. After long delays, he received data showing that Allegra had lost 8.24 percent per year, on average.
"Nycers invested $24 million and got back $11.66 million," Mr. Murphy said. "This is clearly an imprudent strategy for a large pension fund."
Mr. Murphy said that Nycers had lost money in just five years out of the last 30 — all in the 2000s, after the system adopted its private equities program. Out of curiosity, he calculated what the returns might have been if Nycers had continued its strategy of investing solely in publicly traded stocks and high-rated bonds. The answer was: billions of dollars ahead of where it is today.
The city's decades-old structure for its pension system almost changed in 2011. Mr. Bloomberg and the comptroller at the time, John C. Liu, bitter rivals, unveiled an ambitious plan to consolidate the five plans.
The aims included hiring professional in-house investors and breaking the link between the pension system and the political calendar. Most of the ideas died, however. Some labor leaders felt that they had not been consulted, and Mr. Liu became hobbled by a federal investigation into his campaign finances.
Now, of course, there is a new mayor and a new comptroller, both of whom have been staunch labor allies. A new chief investment officer started last month, and Mr. North, the system's actuary for nearly 25 years, is expected to retire this year.
It is unclear whether pensions will be a top priority.
Mr. de Blasio, notably, did not mention the word "pension" during his hourlong budget presentation in May. Mr. Bloomberg, by contrast, raised the issue often and made his final formal speech a stemwinder on pension costs.
In May, Comptroller Scott M. Stringer announced he would try to commit $1 billion to smaller investment firms led by minorities and women, despite research showing that initiatives geared toward emerging firms make it harder to achieve top investment returns.
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Mr. North said that Mr. de Blasio's recent deal with the teachers' union — and two subsequent deals with health care workers and nurses — would necessitate bigger pension contributions from the city. But precisely how much bigger remains unknown because the contracts are complex.
After Mr. North leaves, it would be easy for New York to tweak key assumptions and lowball its contributions. That would save the city money, but it could wreak havoc on the future. He has urged the trustees to be mindful of the city's not-so-distant past.
"It's less than 40 years since we were near bankruptcy," he said.
—By David W. Chen and Mary Williams Walsh, The New York Times