Even after gobbling up $98 billion in government relief since last year, mortgage giants Fannie Mae and Freddie Mac remain deeply insolvent.

That conclusion, issued Monday by one private-sector investment firm, underscores the difficult choices facing Congress as it tries to extricate the US from a housing-market bust.

America is relying on these two entities to bail out the troubled housing market, yet Fannie and Freddie have needed bailouts of their own.

Analysts at Keefe, Bruyette & Woods in New York outlined what they see as a likely outcome for current investors in these government-sponsored enterprises (GSEs): share prices falling to zero.

This doesn’t mean Fannie Mae or Freddie Mac will go out of business. The government can’t afford to let that happen, financial experts say, because mortgage markets would break down. These two firms end up owning or providing a guarantee on most US home loans. But as the losses on bad GSE-backed loans mount, Congress is being forced to weigh what to do.

Policymakers are considering how to navigate out of the current mess of defaults, foreclosures, and falling home prices. But equally important, housing experts say, the system must be reset so the goal of stable mortgage markets can be achieved without placing huge risks on US taxpayers.

“I fear that the longer we wait, the more it’s going to cost the American taxpayer,” Sen. Richard Shelby (R) of Alabama said at a hearing on the issue earlier this month. “The question of what to do with GSEs is very difficult and complex. Yet it is a question that we ignore at our peril.”

Last year, as US mortgage problems deepened, regulators put Fannie and Freddie into government conservatorship.

Since then, the corporations have continued to back new mortgages across the United States, helping to prevent a deeper meltdown in the housing market. Private shares in the firms have retained a small value, as investors hope for an outcome in which the firms recover and exit conservatorship.

But according to estimates of the analysts at Keefe, Bruyette & Woods, the firms can’t pay their debts to the government, even over a 10-year period, and need new capital. They argue that the solution will be for private banks - the key customers of Fannie and Freddie - to invest that needed capital. In effect, the firms would become cooperatives owned by the mortgage banking industry.

“There have been many recommendations made about potential structures for the GSEs,” but most don’t address this vital question of where the needed capital will come from, the analysts’ report said.

A separate report this month, by the Government Accountability Office, outlined other options for what comes after conservatorship for Fannie and Freddie:

• Nationalize them and operate as a government agency.
• Reconstitute them as they were before the crisis, with better regulation.
• Privatize or terminate them.

Many conservatives argue that private markets can do Fannie Mae and Freddie Mac’s job. Backers of GSEs cite the Depression and the recent credit bust as cases when private firms were unable to keep credit flowing without government help.

For years, Fannie and Freddie existed as awkward hybrid institutions. They were chartered by Congress to help credit flow, but lacked explicit backstopping from the government. They gathered huge lobbying clout, lost their way amid accounting scandals, and parlayed implicit government support into rewards for their private shareholders.

Even on the left, though, many experts say the poor design and regulation of the GSEs contributed to the financial crisis.

“Please do not re-create Fannie Mae and Freddie Mac” as public-private hybrids, former Federal Reserve Chairman Paul Volcker, an adviser to President Obama, urged lawmakers in September.