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How does the importance of a cash emergency fund change when you live in a country with nationalized healthcare?
The issue is how likely you will have zero income for six months, and what are your monthly expenses. If you know the maximum medical bill you face that may allow you to save a smaller amount. But you still have to protect for that loss of income. The interuption could be because of job loss, medical emergency, or other family crisis.  If I told you that  the chances you would face a crisis dropped by 50%, would you decide that the need for an emergency fund went away? Or would you still create a fund? I think the need still exists just to avoid the downside if you aren't prepared.