But doing both seems virtually impossible.

Earlier this month, Fannie Mae (FNMA) asked for $7.8 billion in aid to cover its losses during the latest quarter.  It didn’t make big news — the mortgage giant and its cousin Freddie Mac (FMCC) have made so many such requests, most taxpayers have grown numb to it. Freddie asked for $6 billion — it’s largest quarterly request since April 2010.

The government took over the mortgage giants in 2008 to shoulder soured loans following the collapse of the housing market. Since then, taxpayers have spent about $169 billion to rescue the two companies – the costliest bailout of the financial crisis. The government estimates it could cost up to $220 billion to keep both companies afloat through 2014 after dividend payments.

Freddie and Fannie, which own or guarantee about half of all mortgages in the U.S., continue to suffer from bad lending practices during the housing bubble. And a deteriorating housing market has made it difficult for Freddie and Fannie to turn a profit. What’s more, at least according to some experts, the companies could see bigger losses from a newly expanded government program that helps homeowners lower their mortgage payments through refinancing.

Fannie’s latest net loss of $5.1 billion for the three months ending September 30 was not only due to defaults on loans it guaranteed, but also to homeowners paying less interest after refinancing at historically low mortgage rates. Given weak home prices and high unemployment, it’s not terribly surprising to find borrowers defaulting. Less expected, and arguably ironic, are Fannie’s losses from lower interest rates.

After all, the very purpose of the government’s continued efforts to keep interest rates low is to give the tepid economy a boost. The Federal Housing Finance Agency last month announced it would expand its mortgage refinancing program so that many more borrowers could take advantage of today’s record low mortgage rates, which averaged 4.07% in October for 30-year fixed-rate home loans. The FHFA’s program started in 2009 and is meant to help those borrowers typically unable to qualify for traditional loans because their homes have lost so much value. Under the program, Freddie and Fannie have financed new loans for nearly one million borrowers. And with the program’s expansion, officials hope another one million could gain.

To the individual homeowner, this might sound like a win. Officials have estimated that refinancing could save the average borrower about $2,500 a year. But such savings would effectively translate to a loss for Freddie and Fannie, and by extension, taxpayers. If the program works out the way FHFA officials hope with one million more borrowers able to refinance, this could create an additional $2.5 billion in losses for Freddie and Fannie, says Radhakrishnan Gopalan, finance professor at Washington University in St. Louis.

“The only mitigating factor would be if the refinancing actually avoids some default,” Gopalan says. “But given that the new program is targeting borrowers who are current on their mortgages, I think the gain on that front to Fannie and Freddie will be marginal at best.”

In a way, the FHFA’s program’s intention is more to help jumpstart the broader economy than stem foreclosures. It’s true that allowing borrowers to refinance would lower their monthly mortgage payments and thereby arm households with extra cash to spend elsewhere. But many economists say the program is still only a piecemeal solution and too small to spur much more spending. Given the uncertain economy, they’re more likely to save than spend their money. And while the government aims to provide refinancing to one million more borrowers, that’s still a relatively small share of the 10 million households with mortgages worth more than the value of their homes.

So it’s hard not to wonder what the government is really trying to save here.