Most of the banks I have seen have a clause allowing them to buy insurance that covers what you owe -and usually costs more than what you would pay for full pri

Most of the banks I have seen have a clause allowing them to buy insurance that covers what you owe -and usually costs more than what you would pay for full private insureance- and bill it back to you... if you don't make the full payments, then the bank will foreclose. So there is usually that extra step of them taking out a policy.