In the U.S. there is a lot of competition which drives the price down. Technically, more claims mean higher preimum, but there is always a company willing to write a policy for less. The insurers that have been in avation for a long time are waiting for the new kids on the block to finally have so many losses that their low premiums can't make up for it and they're forced out of the sector, but they're sticking around. The thing is these companies(like AIG) are huge and write many different lines of business. They can shift funds between these different segments, and if the overall company loss ratio is positive, they don't worry too much about aviation, which is a very small part of their overall business portfolio.
So to sum it up, your premiums stay the same, because they know the competition will write it for less.