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What you're saying
Say my mortgage is 1k a month.
I can't afford 1k, so I let you move in and pay me "rent"
You pay me 1k each month.
After a year, I kick you out and move back in.
Where'd your 12k go?
Bank going to give 12k in cash to you and increase my mortgage by 12k?
So let's say your mortgage is 1k/mo. We're not talking PITI, just mortgage.
Let's pretend the house is worth $30,000 and the landlord has $7200 in equity at the beginning of the lease (20%.)
Let's also say of that mortgage 700 is interest and 300 is principal. So a renter pays you 1k/mo rent for 12 months. They get 300/mo principal totaling $3,600 after a year.
At this point there are a few different ways of handling it.
Option 1: The tenant is buying debt from the bank with the principal payments over the course of the year. So $3600 of the debt they now own, when you pay the bank the payments go proportionally to the tenant which actively reduces their equity, but they collect all the interest income for the share they own.
Option 2: You could have shares of the home. As money is paid in against principal the tenant gains share of the overall home - so in this case a bit over 10% over the course of the term. Since this is shares as the valuation of the home increases so do the share prices and the typical compound interest applies. This can be very messy because when you want to cash out the principal owner either need to buy you out by taking another mortgage or paying you, but it could be done.
When you factor in things like the appreciation of the home the first option gives the appreciation to the landlord without reducing their mortgage. They would gain in net worth because the home is worth more money, even though they still have a mortgage that hasn't decreased in price.
The second option takes the equity gain away from the landlord proportional to shares owned. Although harder to track it's still possible.
This is ignoring the real world scenario where the mortgage is really only $500/mo for a lot of long time landlords who bought 20+ years ago or refinanced in 2020 and the rent is $2000-3000+++ There would have to be something that takes into account excess paid beyond the mortgage/taxes. This also doesn't make any sense at all if the house is paid off either since there would be only taxes, no mortgage. Obviously there would need to be some nuance in implementing a solution but it's not impossible.