re: Have some extra cash saved up thinking about paying off car

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I think insurance is irrelevant

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If the asset is insured for the full value due on the note, then it doesn't matter. But that isn't the case for cars.

The fact that the loan was used to buy the car (the depreciating asset) isn't what's relevant here - what matters is that the note is secured by the car until it is paid off. This means that if you lose the car you have to pay the difference between what's left on the note minus the insurance check.

But look at a different example. Suppose I borrow against my home equity in order to finance the car (I am not saying this is a good idea). This timethe car is not secured by the loan at all and the HE loan should be considered completely independently of the asset it financed.