Looks like they'll survive just fine.
Self-driving cars are poised to put a lot of people out of business. In fact, tallying up the number of industries that are poised to take a hit from the robot takeover shows us not only how much we rely on cars and drivers to get around, but how much we rely on automobiles and their human counterparts to make the economy tick. Aside from the obvious toll that the change will have on the bus, truck, and taxi industry, the automobile insurance industry also stands to suffer.
A recent study found that by 2040, the auto insurance industry will shrink by 60%. However, a newer report conducted by Moody's shows that before the industry crash, auto insurance companies stand to win big during the transition. Before cars go fully autonomous, there will be an intermediary period where insurance premiums stay at their current prices because auto insurance companies will wait to see how much insuring self-driving cars costs them. During that point in time, insurance companies are likely to reap all of the profits of having the same amount of money coming in and less cash going out to cover crashes. This may just buy the insurance industry time to figure out how to move forward into a future where it's services are less needed.