Bid credits are the amount saved when a contractor removes the cost of insurance from its bid. 

"If you look at the number of CCIPs fifteen years ago, they might have been five to eight percent" of the wrap-up total,  Resnick said. "Now it's probably closer to 50-50 when it comes from OCIPs vs. CCIPS."

Contractors with rolling CCIP programs and numerous staff members devoted to risk and insurance may be favored by insurers with more attractive rates and "can do a better deal" than an owner, said Anthony Rastall, partner in JLT Specialty Ltd., a London-based broker.

That may be the case especially if the wrap-up is a "one-off" for the owner.

Other reasons why contractor control may be better is that the contractor will do a better job at safety and claims management and the owner is a legal and contractual step removed from the subcontractors performing the work.

Yet owners sometimes prefer to control the wrap-up program—a complicated job with lots of paperwork—because their lenders prefer that arrangement. Under a wrap-up, the owner will also have control of payments for claims—something that may make a contractor queasy.

And a small number of big contractors will insist that they control the risk and insurance or will walk away from the projects, said Rastall.