In general, it seems to me that any insurance which serves a public interest shouldn't be run by market forces, for the simple reason that the profit-motive is incompatible with achieving other socially valuable goods. For example, if the aim is to make sure that people hurt in car accidents are compensated fairly for their injuries, then it seems the private sector should have nothing to do with it. (After all, private insurers will be trying to make as much money as possible, which is not compatible with the concept of fair compensation.) Same holds for health insurance, insurance against work injuries, disability insurance generally, and probably also for life insurance. It won't hold for property insurance, though, because protecting anything other than minimal personal property (what you need to live and work) just doesn't seem like a social good.