Numerous agents have told me about the pro-rated refund. After 37 days, they divide the annual fee by 12. Whenever you call within the 12 month period, they will give you a pro-rated refund of your fee, based on the date you cancel or change products. (Data Point; I did have an agent offer to give a full refund on an old AmEx card many months after the fee posted, YMMV).

UPDATE: According to a representative I spoke with on DEC 28, 2018, the prorated fee refund is a thing of the past. You now have 60 days from the day the fee posted to cancel or convert in order to receive the full fee back. After 60 days, there is no proration.