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Everyone has to make their own call. For me I wanted to be clear of debt as fast as possible because who knows when the rates were going to go up. I'd heard of horror rates in the 80s. The more you can overpay early the quicker your debt disappears so if rates do go up and you can't afford to overpay you've already removed as much debt as you can. Also, in my mind, clearing debt is a certainty whereas making money on investments is not (I don't have time to pick a scheme let alone pick shares, etc).
I'm not sure what you mean re: remortgaging. I remortgaged (with the same lender) for a better rate and I keep all my overpayment 'surplus'. Not that I'd even considered taking it out again but it's possible. Maybe it's an issue going to another lender?
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hippy
Well that's where it gets complicated.
For instance a rate of around 2.25% going up to 6% after 7 years roughly matches up with savings at 4.5% per annum.
That 6.5 years of being mortgage free is matched by the decrease in your cash (i.e. you've paid off at the end of 11.5 years but with no savings or you've got ~ £95k in the bank at that point and a mortgage of ~£85k).
It's obviously not set in stone but it is worth considering whilst you have low interest rates and can get decent returns. There are further considerations in terms of overpayment restrictions, early repayment fees, etc.
At the moment the ability to access the cash is the key consideration for me. Although you can get cash out if you overpay that is limited to that mortgage deal, you won't be able to get that cash after you've remortgaged (unless you do it at the start of the mortgage with a higher principal).