Do you have escrows in your current mortgage payment? Some places call these impound accounts rather than escrow accounts, but it is the same thing.
I assume you do from the way your question is phrased. At the closing you paid one year in advance for insurance, this is normal. Each month you pay 1/12 of the insurance amount due into the escrow account and there is about a two month cushion in your escrow at a minimum. Approx two months before your new annual insurance payment is due, your lender will send the payment, from your escrow account, to pay next year's insurance premium.
It will not drop from your payment unless your LTV is lower than 80% AND you specifically request it to be removed from the payment. You have to meet certain conditions in order to drop it from the payment. Your servicer can give you the terms of those conditions.
