Deanna White told a contractor she couldn’t afford the $42,200 loan he recommended for improvements to her house in Inglewood, Calif. The contractor, she recalled, said she wouldn’t be on the hook because the loan was part of a “government program.” She applied and was approved.

Two years later, Ms. White is struggling to make payments on the loan, which was packaged with more than 10,000 similar loans into bonds and sold to investors. Under its terms, Ms. White’s five-bedroom house could be foreclosed on if she defaults.

...