You're doing that wrong. Good luck fighting with Amex trying to get them to reimburse you on that extended warranty. It takes weeks, if you even get anything. Amex covers almost nothing as shown in the their warranty terms.
See below:
The sleep/wake button got stuck on my wife's iPhone 4s. I put in a call to Amex to file a claim and they reimbursed us $299 plus tax in the next billing cycle, all without even requiring paperwork or asking for the iPhone. I did the out of warranty replacement for $199 plus tax so I actually got my wife a replacement iPhone plus some $$$.
My iPhone 5 needed an out of warranty repair and I paid for it out of pocket. I filed a claim, submitted the receipt and they within days credited my AMEX account. It couldn't have been easier. And they credited me the entire original cost of the phone.
I filed a claim for my iPad 3 because the home button was flakey. Apple wanted 279 for refurbished replacement. I filed a claim with Amex. Next day they credited my account 499 full purchase price on the iPad with no questioned asked. I could've just made it up I guess.