When you say you get equity by owning, that’s the exact same thing as saying “you own the home when the mortgage payments end.”
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Not the same thing.
Because you don’t own your home even when the mortgage payments end, you just own a deed or title. As I said.
What is more, you can have equity in your home even when you are not done making mortgage payments. So yes, you own some equity in the deed, as long as you can resell the title /deed that you own, but the property belongs to the government. The title (and it’s equity) will be yours as long as you keep making the tax payments.
You must be great fun at parties.