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Then the optimum arrangement for the rainy day fund would be an offset mortgage. That way the rainy day fund is still doing its work reducing your interest payments, but is always available.
If I could port my current mortgage (which is interest-only, and we manage the repayment portion separately) to a fixed rate interest-only offset mortgage it'd be perfect, but my current mortgage provider (Santander) doesn't offer one of these, and the chances of getting a new interest-only mortgage are very slim.
It's a bit flawed.
If you've got the money and you're earmarking it to pay a lump sum later then you may as well overpay as you go. That way you're reducing the interest throughout the term rather than just in chunks every 2/3/5 years.
This assumes you're allowed some overpayments. Most do allow some.
If the mortgage is 1.8% then any way of saving that lump sum and earning more than 1.8% (riskier certainly, but historically not too tough to find) would make you more money than you're spending on the extra interest, so even though you'd spend lots on long term interest, you'd earn that plus some more on the investments.