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Reviving the market for bonds sold by state and local governments is shaping up as one of the stiffest tests in the Federal Reserve’s campaign to restore financial normalcy.

The Fed has committed trillions of dollars to keep money flowing through markets vital to economic growth, including huge purchases of government and mortgage securities and new programs to backstop money-market funds and corporate debt markets.

Those efforts have helped to fuel the markets’ partial recovery, say investors and portfolio managers, with the Dow Jones Industrial Average up 26% from its March 23 low.

But the central bank is limited in its efforts to revive the $4 trillion market for municipal securities, which back everything from school facilities to stadiums and highways. The Fed has so far intervened in only a few corners of the market, which is fraught with idiosyncrasies that make it difficult to categorize debt as investment-grade or risky, the line in the sand drawn by the Fed to ascertain what it backstops during a crisis.

The constraints stem in part from the coronavirus’s decimation of state and local finances, which could make the risks even harder to judge, and the Fed’s traditional deference to Congress in handling local government financing decisions.

“The Fed doesn’t want to be in a position to say you have to raise taxes or cut pay to policemen or firemen” to secure or repay a loan from the central bank, said Scott Alvarez, who was the Fed’s general counsel from 2004 to 2017.

Treasury Secretary Steven Mnuchin told Democratic lawmakers Wednesday that Fed and Treasury officials would soon unveil a program to backstop financing for states. The devil will be in the details, and those designs—along with other announced plans for lending to small and midsize businesses—could be unveiled as soon as Thursday, according to people familiar with the matter.

While the Fed has the authority to purchase municipal debt with maturities of six months or less, it hasn’t exercised that authority. A more likely route would be to establish an emergency lending program to backstop longer-dated muni debt.

The $2 trillion rescue package that Washington approved last month includes $454 billion that the Treasury can use to absorb losses on any Fed lending facilities. That bill provided $200 billion in direct funding for states and cities, but they are likely to need another $300 billion to $600 billion, said Tom Kozlik, head of municipal credit at Hilltop Securities.

The aid to cities and states in the recent rescue package “will not be enough to offset the cost many states and municipalities are encountering,” said Boston Fed President Eric Rosengren. The Fed can help with financial markets, but those efforts will be less effective without more direct aid, he said.

Officials are trying to avoid a rerun of state and local government layoffs after the 2007-09 recession, which contributed to an underwhelming economic recovery despite unprecedented Fed stimulus.

The Fed typically seeks to steer clear of concerns about the potential loss of taxpayer funds by focusing on purchases of assets such as highly-rated bonds whose default is widely judged to be minimal. Such judgments are harder to come by in the market for municipal bonds, where even the strongest borrowers have been hammered by the challenges arising from an unprecedented shutdown of business and commerce around the country.

States face not just the burden of boosting spending on public-health responses, but also a drop in revenue from sharp declines in sales-tax collections.

“In almost every way, states are at the front lines of fighting this,” said Joe Torsella, Pennsylvania’s state treasurer.

Fears that state and local finances will be permanently damaged are evident in the investor flight from this market, which until recently has ranked among the most resilient.

In March, investors pulled $32.8 billion from municipal-bond mutual and exchange-traded funds, according to Refinitiv, the largest monthly outflows since data collection began in 1992. State and local governments canceled billions of dollars of planned borrowing. The S&P Municipal Bond Index gave up more than a year’s worth of gains.

The Fed has long resisted lending to states and companies, having spurned requests from lawmakers in 2008 to aid ailing U.S. automakers and ruled out a muni-debt backstop.

The central bank has already broken some taboos during the current crisis. It is in the process of unveiling lending facilities for large and midsize companies, and it has dipped a toe into muni-debt markets by expanding a money-market lending backstop to include certain types of municipal debt—and by purchasing some highly-rated municipal debt in a facility backing the market for very-short-term commercial debt.

Analysts and state officials said the Fed could provide support by buying a broad-based muni index, avoiding the prospect of picking winners and losers outright.

Among the issues the Fed must weigh is who ultimately benefits. The yields on bonds issued by Montgomery County, Md., an affluent suburb of Washington, D.C., and Cook County, Ill., home to Chicago and where more than 700,000 people live in poverty, both jumped more than 2 percentage points over a week in March, indicating lower prices.

Yields on the Montgomery County bonds have since declined more than those on the Cook County bonds—indicating that while the market views the Montgomery County bonds as a better risk, the Cook County securities are potentially the ones more in need of support. Those sorts of regional and distributional issues carry significant risk for the Fed, investors said.

“It would be very problematic for the institution and its credibility to decide between New York and Montana,” said Mark Spindel, a Washington-based investment manager who co-wrote a history of the Fed.

The prospect of increased lending to businesses and local governments, often in consultation with the Treasury Department, could reshape the Fed’s longstanding autonomy from the executive branch.

During and after World War II, the central bank pegged Treasury yields to finance war spending and the recovery. A bruising fight with the Truman administration, which resulted in the resignation of the Fed chairman, ultimately led to a formal agreement in 1951 to end the Fed’s policy of fixing Treasury yields.

“I think it is possible that we will have a central bank when this is all over that has sacrificed a piece of its independence,” said Jeremy Stein, a former Fed governor who now teaches at Harvard.

Fears about the loss of central-bank independence are overstated given the gravity of the current crisis, said Mr. Torsella.

While political and constitutional tensions loom, “smart, well-intentioned people can figure out how to do this in a way” that “simply restores the functioning of this market,” said Mr. Torsella. “I want to make sure we have a fighting chance of getting back to those more normal times.”