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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.
[removed rubbish advice]