A new and extensive analysis of 2.4 million loans insured by the Federal Housing Administration in recent years shows a pattern of risky lending that could generate $20 billion in losses and harm thousands of the nation’s most vulnerable borrowers. By ignoring risks in loans it insured in 2009 and 2010, the study concludes, the F.H.A. is imperiling both borrowers and taxpayers who stand behind the agency. The analysis emerged less than a month after the F.H.A.’s auditor submitted a troubling report on the financial soundness of its insurance fund. In mid-November, the auditor estimated that the fund, which backs...