You are correct- annual exclusion ($15,000) does apply in your case.
As to the gift - it is on annual basis... you can give $15K to a person during the year... it can be done in increments, as long they don't exceed the annual amount. Theoretically, if you send 20K to a friend and that friend does send back $15K, then yes - you only gave $5k. But practically, be sure to annotate each transfer so that you can easily prove to the IRS ( if needed) which transaction is gift and which isn't.
