WASHINGTON  The Treasury Department said on Sunday that its scaled-down program to help banks unload their troubled mortgages and mortgage securities would begin operating at full strength by the end of this month, more than a year after Congress authorized $700 billion for that purpose.

Treasury officials said that five out of the nine money-management firms it selected to buy up unwanted mortgage-backed securities had raised the minimum amount of money from private investors  $500 million each  to qualify for matching investments and loans from the federal government.

Administration officials said they expected the remaining four firms to complete their financing by the end of this month.

Three of the biggest investment firms  BlackRock, a group led by the Wellington Management Company and a group led by AllianceBernstein  closed deals for private financing totaling about $1.9 billion. Two other firms, Invesco and the TCW Group, lined up their private investors last week.