But raising the credit bar could have a dangerous side effect. In many of the nation’s hardest-hit housing markets, the FHA backs around half of all new home loans. If the agency pulls back too quickly, the nascent housing recovery could fizzle, endangering the economy. The dilemma puts the 52-year-old former mortgage banker squarely in the middle of the debate over how much the government should do to prop up the housing market, and how much risk taxpayers should take on to do it. “How big a role do we need to play to keep the housing system functioning?” says Mr. Stevens, referring to the FHA. “Overcorrecting in either direction would be a terrible thing to do right now.”