WASHINGTON — The Federal Housing Administration, a government agency that insures mortgages, said it was taking steps to shore up its books and avoid a taxpayer bailout.

An independent audit released on Friday projects the agency’s losses will swamp its revenue, with a shortfall of $16.3 billion in its portfolio of insured home mortgages. That has raised the question of whether it will need an infusion of cash from taxpayers for the first time in its eight-decade history.