Save you money? Actually, It depends
One of my former employers provided the software algorithms for a few of these pay as you drive schemes.
They were broken, we knew they were broken, we knew what the problem was. But due to the age of the system, we were unable to do anything about it.
The problem is that when you take out insurance, you do so based on the price of the policy on that day. Insurers however, keep updating the same policies they have to offer all the time, in line with the latest rates.
This means that on a month by month basis, the BASE rate that the premium is calculated from, goes up.
If you get an insurance quote, agree an instalment scheme (these are pretty extortionate, credit cards are often cheaper!) or pay it off all in one go, then for you, this isn't a problem.
If, like these pay as you drive customers, your mileage adjustment is calculated each month. The quoting system we used could not store more than TWO copies (this month's and NEXT month's) of the rates used to work out your premium, which means that the adjustment you're paying is NOT just for your mileage, but for the increase in the price of insurance on top.
This was pro-rata adjusted appropriately, but still, in once case i tracked an 80% increase in price over the 12 months of one customers policy, every month the mileage increased and the 'driving' improved, but the premiums still went up, every month an extra charge was added because of the increase in market price at the time.