Yes, normally you will need full coverage on a vehicle which you are still paying a lien holder for the loan you have out on it.
While your state will require you have at least your state's minimum liability insurance on your vehicle if you have a lien holder on the vehicle they will require you to carry not only Liability coverages but also Physical Damage coverages of Collision and Comprehensive (which are commonly referred to as "full coverage").
It is not a state requirement to have full coverage (meaning at least the minimum state auto insurance limits plus Physical Damage coverages) but normally a requirement listed in your paperwork from your finance company or lender. The lender is your lien holder and thus the car is their asset until you pay it off. This is why they have a say about what insurance coverages you obtain and maintain on the financed vehicle.
If you drop the required auto insurance coverages from a financed vehicle it is a violation of your finance contract and may put your loan in jeopardy. Also the lender could place single interest coverage (force placed insurance) on the vehicle and add the premium to the loan. This type of coverage is expensive and does not provide any coverage for you, just the lender.
To find out if you must have full coverage on a specific financed vehicle you would need to speak to the lien holder and/or read through your lease or loan paperwork. Most all lien holders require full coverage since the car is their asset and if it is damaged or totaled out and you do not have insurance to cover it than they will have to come after you personally to repair it or pay off a totaled car which is much harder than your insurance company taking care of it for you.
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