A major impediment to the reform of government-run mortgage businesses Fannie Mae and Freddie Mac is a running disagreement about how they should be accounted for on the government’s budget.

Fannie and Freddie ultimately received $187 billion in taxpayer funds following their 2008 bailout, but they were never fully nationalized, partly to avoid having the trillions in loans they guarantee show up on the federal government’s ledger. Fannie and Freddie don’t issue home loans, but buy them from lenders and package them into guaranteed securities to sell to investors.

Instead, they were brought into the legal status of conservatorship, under the management of the Federal Housing Finance Agency.

The consequences of that decision are now playing out, as Congress wrestles with what to do with them. Both parties have said the two companies must be dissolved, and the Senate Banking Committee chairman earlier this month announced a bill that would do just that, after the House Financial Services Committee passed a more conservative approach to the same goal last summer.

Even more importantly, however, the two companies have returned to positive cash flow, in a big way.

In 2013 alone, Fannie and Freddie paid out $130 billion in dividends to the Treasury. Through the first quarter of 2014, they will have produced $203 billion in total for the government, according to the FHFA.

That cash flow has complicated the issue of resolving the two government-sponsored enterprises.

For one thing, activist hedge funds want a piece of it.

Led by Fairholme Capital, owners of Fannie and Freddie’s shares sued the government for taking, starting in 2012, all of Fannie and Freddie’s income above what was needed to keep them afloat.

Fairholme and other hedge funds also have proposed that the government sell them Fannie and Freddie’s mortgage insurance business.

It’s clear why they’re interested: Over the next 10 years, the Office of Management and Budget projects the two GSEs to bring in another $164 billion even as their business slows, if they’re not shuttered before then.

But two new analyses undercut the administration’s claims about Fannie and Freddie’s profitability.

According to reports from the conservative Heritage Foundation and the libertarian Cato Institute, the loan guarantees made by the GSEs should be considered liabilities alongside the incoming revenues.

When the possibility of loan losses is taken into account, Fannie and Freddie might actually be net liabilities to the government. The Congressional Budget Office, Congress' nonpartisan official scorekeeper, estimated in its most recent budget outlook that the GSEs' operations would cost the government roughly $19 billion over the next decade.

The CBO considers Fannie and Freddie, and their liabilities related to mortgage guarantees, effectively part of the government. The administration, on the other hand, “treats them as if they were a third-party" entity, noted Massachusetts Institute of Technology professor and former CBO analyst Deborah Lucas in an event at the Bipartisan Policy Center on Wednesday.

Not only does the CBO analysis yield a drastically different picture of Fannie and Freddie’s current budgetary status, but it also has implications for the proposals circulating in Congress.

For instance, the House GOP bill, which would eliminate government backstops in the secondary mortgage market, would have to yield more than $164 billion in deficit savings to break even for the federal government, by the OMB analysis. But CBO scored the bill as saving $5.7 billion.

The Senate bill, which would retain a government role in guaranteeing mortgage-backed securities, is harder to assess, because it hasn’t received a CBO score. But it would involve bringing the mortgage-insurance business of Fannie and Freddie onto the government’s books and explicitly guaranteeing the $5.2 billion of outstanding debt, according to Cato.

The study’s authors, Mark Calabria and Ike Brannon, write that while a “booming economy” would mean few risks for taxpayers, “if the housing market were to have another swoon, the government would undoubtedly find itself having to cover some portion of this debt.”

Heritage recommends putting the GSEs on budget, right now, to clarify the situation. “Improper accounting in the budget for the downside risks that the GSEs pose for American taxpayers is creating the illusion that the GSEs are a free lunch for Washington,” wrote Romina Boccia.

Other analysts disagree, pointing to the hedge funds’ interest in taking over the mortgage businesses as evidence of their value.

Just one more thorny issue to untangle in the long process of determining Fannie and Freddie’s fate.