Mortgages: Private profit; Public risk

Why doesn’t the government issue mortgages? Why do we have this system where private banks collect mortgage interest, while the significant risks are shouldered by the government? If we the public are holding the risks, shouldn’t we also collect the profits? Excerpt from Globe and Mail article:

The bankers’ effort is all the more notable given the unique structure of the Canadian mortgage business. Banks get the profits from mortgages with their decades of interest payments, but have little risk of direct loss because of mortgage insurance.

Consumers cover the premiums and, because most mortgage insurance is underwritten by CMHC, the federal government ultimately takes the risk.

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Mortgages: Private profit; Public risk