There is a good technical reason why you should invest in an annual service contract too. Your alarm probably has two back up battery systems, both of which need to be working. One battery is in your control panel and keeps the whole system working during any power outage. The other battery is outside in your bell box. The sole purpose of this battery is to keep your alarm sounding if an intruder vandalises your system.
Sometimes you hear alarm systems going off during a power cut. This is because the internal battery has failed but the external battery is still working. It is actually a fault.