AXA is advising the UK government on how claims will work in future, when computers are driving us around.
The company has deliberately crashed a number of cars to recreate various "autonomous" scenarios. They mocked-up three different accidents.
Insurers regularly crash their own cars, using the results to work out how much to charge us in premiums.
The head of Accident Research for AXA, Bettina Zahnd, says these new, driverless tests are vital: "We want to demonstrate that we cannot have the same accident investigation methods as we had before, for autonomous cars, we need to know whether the driver or the car was in charge."
Driverless cars won't be with us overnight.
They'll come along in stages. You may already have a vehicle that parks itself or brakes automatically if you're about to hit the person in front.
That last system has already reduced low speed rear end accidents.
But in the end, computers will take over more and more of the driving and it could make driving cheaper.
David Williams from AXA UK is advising the government on its driverless car policies: "The good news is that people can expect their premiums to reduce in the long run.
"Insurance premiums are based on the cost of claims and we are going to see much safer roads and fewer accidents so that will feed through into producing lower premiums."
These tests will also begin answering another vital question. If you crash in a computer driven car, is it your fault?
David Williams says drivers needn't be concerned: "I don't think people need to worry whether it's their fault or the computer's fault.
"The UK government has decided that an insurance policy will be required and that will deal with the claim in the first instance.
"So people will be compensated and then the insurance companies and motor manufacturers can argue among themselves in the background."
The UK wants to be a world leader in developing driverless cars. The industry could be worth billions of pounds.
Sorting out how the insurance will work is vital to smooth the way for the vehicle makers to test on our roads.
One of the world's biggest insurers has begun testing the impact of driverless cars on our premiums.