Most moving companies are going to require you to have some sort of moving insurance for your move. Not only does this protect your stuff, but it protects them in case something goes wrong and liability becomes an even bigger concern for them and for you. The company has its own insurance (both business and liability insurance) to protect them in case someone were to sue them.
If you are concerned about the cost of your move, you may want to talk to your home insurance company about your options. There are some home insurance companies that also offer moving insurance at a lower rate if you an existing policy set up with them. Your mover may have discounted insurance if you buy directly from them, or you can compare prices and find what is most affordable.