Bermuda-based Arch Capital Group has acquired American International Group’s mortgage-insurance unit in a $3.4 billion deal. Arch will pay $2.2 billion in cash plus stock for the business, United Guaranty. AIG will also retain some portions of the business through 2016. The mortgage-insurance business makes money off of homebuyers who put down less than 20 percent of the purchase price, and are required to obtain mortgage insurance to protect banks from losses. Arch, which has a market value of $9 billion, has been expanding its mortgage-insurance business, which currently has 3 percent of the U.S. market. The new deal with AIG will make Arch the nation’s largest mortgage insurer.