When the subscription is cancelled by a user or the credit card declines, the module will receive a silent postback message and the role the user was added to when they registered will be removed. The only case where the expiration date on a role is set is when the subscription interval is NOT set to "9999". 9999 indicates infinite recurring billing.
When you cancel a subscription using either the My Accounts module or in the admin page, the module will post a message to the payment gateway to cancel the subscription. Anytime Authorize.Net receives messages to cancel or change ARB subscriptions they will post a message back to the registration module to return status of the transaction.
Authorize.Net usually send messages postback messages in batch around 2 am. The smith registration module will receive the silent postback from authorize.net and remove the user from the role on their anniversary date of their subscription ending.