You might want to check with a banker. Apparently nearly EVERY conventional loan is backed by FHA, Fannie Mae, Freddie Mac, or one of the other Government backed underwriters. That is why FHA, the FMs and other Government underwriters are losing their AAA ratings for the first time EVER.
While FHA and VA "only" carry about 7.5Million mortgages on the books Fannie Mae, Freddie Mac and others now guarantee the mass of Conventional/ARM/Jumbo/AltA and SubPrime loans.
In the interest of your education, you have muddied the waters with your imprecise use of the phrases "backed by" and "guaranteed."
The loans approved through FHA and VA are insured
against loss by those government agencies because of the higher purpose of making loans (therefore homeownership) available to those unable to come up with 20% down payment. That allows the borrower in most cases to make as little as 3% down payment. The loans are still originated by common lenders, but the underwriting criteria are particular to FHA or VA.
Freddie Mac and Fannie Mae exist to create a secondary market
for such loans. Once the lender makes the loan, underwritten to Fred or Mae's guidelines, they will fund the loan in the market and send the money back to the lender so he can make a new loan. They make their money on the mark-up from the bonds they sell. Freddie Mac works primarily with conventional loans, Fannie works with both gov't insured loans and conventional.
Conventional loans are insured by private mortgage insurors, they are not "guaranteed" by the government; by definition they are conventional and require a 20% down payment. I believe that part of the subprime problem is that many borrowers have gotten around the 20% problem (better described as equity) by taking on a second mortgage for that amount. Those 2nds have a much higher rate and often a balloon payment ,a formula for short term disaster.