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I can’t think why it’d make sense for her to continue living here and having cash savings while I have a mortgage.
It's a lot of risk to add for not very much benefit. Having cash savings depreciate won't kill you. Having a mortgage won't kill you - in fact it enables you to wisely invest spare cash on something that earns you more than the 2% you should be paying on it. But having a relationship fail when you both have a share in the same home can cause a lot of grief.
The cash savings problem can be dealt with easily - invest the cash somewhere. This could be a property whilst exploiting the FTBers schemes or it could be investment ISAs or even Pension contributions.
The mortgage is obviously harder to close off if you don't have the cash to pay it off - but most people don't want to pay their mortgage off because they are happy to pay the %2 or whatever so they can spend the money on other things or invest it elsewhere. It's so cheap it's almost free.
As above, some kind of alternate legal agreement might be the better way to go.
It would preclude her from first-time buyer schemes, yeah. But hopefully we’ll stay together and live in this place. Upside is emotional but also financial. I can’t think why it’d make sense for her to continue living here and having cash savings while I have a mortgage.