"I have changed your billing cycle from monthly to annually billing cycle for our services"
I have changed your billing cycle for services rendered from monthly, to yearly.
However, as a communication from a company, it is better phrased as:
Your billing cycle for services rendered has been changed from monthly, to yearly.
annual = calculated over a period of a year
This refers, for example, to how simple interest might be calculated on a savings deposit. But here, you are talking about how often a bill is sent out.
yearly = happening/produced once a year - the bill is made up and sent out not 12 times a year (monthly) but once a year.