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Or pay off a mortgage that's costing you more than the banks paying you interest?

The point, though, is that by adding a little extra to your mortgage payment each month, you are not "paying off a mortgage." Until your mortgage is completely paid off, you will still owe that mortgage payment to the bank every single month until it is paid off completely. So if you hit a rough patch such as being unemployed, it will not matter one iota to the bank that you have been prepaying your principal every month. If you do not make your expected monthly mortgage payment, you are in default and at risk of foreclosure.

If, instead, you were to save up that money in a separate account or even invest it and you hit that period of unemployment, you can continue to pay your mortgage and are not in danger of defaulting and foreclosure. In addition, when you have finally saved enough, you can then pay off your mortgage if you feel that is your best move.

But pouring more money each month into an illiquid asset is actually riskier than putting that money aside as long as there is still a balance on the mortgage.

As someone who has dealt with long-term unemployment, I can tell you that not having to worry about having money to pay the mortgage was one less stressful item with which I had to deal, and that was because I had opted not to prepay and to set the money aside instead.
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