Six U.S. mortgage insurers, including units of Radian Group Inc. and Genworth Financial Inc., formed a lobbying association to push for a broader role for the industry in a revamped housing-finance system.

The organization, U.S. Mortgage Insurers, will work to ensure Congress considers the industry’s interests in federal legislation to shrink taxpayer risk in the housing market, Rohit Gupta, chief executive officer of Genworth Mortgage Insurance, said on Monday.

“We want to build on the momentum in the industry,” Gupta said on a call with reporters. “We have attracted new capital and new companies have entered the market.”

Private mortgage insurers backed about 11.3 percent of the $1.9 trillion in mortgage originations last year, up from 4.3 percent in 2010, according to USMI. The industry backs about a third of insured loans, with the federal government backing the rest through the Federal Housing Administration.

The group’s other members include Arch Mortgage Insurance Ltd., MGIC Investment Corp., Essent Group Ltd. and National Mortgage Insurance Corp.

Mortgage insurers cover losses when homeowners default and foreclosures fail to recoup costs. The firms are rebounding after posting years of losses when home prices tumbled in the financial crisis.

Shares of Philadelphia-based Radian and Richmond, Virginia-based Genworth more than doubled last year as investors bet on a housing recovery. Milwaukee-based MGIC Investment Corp. more than tripled in 2013.

USMI replaces Mortgage Insurance Cos. of America, which closed earlier this year.

“We just felt this was the time to form a new industry trade association,” Teresa Bryce Bazemore, president of the Radian mortgage unit, said on the call.