why does my new 16GB iPhone 5S only have 12.9GB of capacity?

It has to do with the difference as to how drive makers and software computes capacity and the fact that iOS 7 and the installed apps (especially those that are 64 bit) take up more space.

In order for the phone to operate correctly, 3.1 GB has to be reserved for the iphone to run on.

Note: A computer does the same thing

the storage is not exact and the manufatures and software makers measure them differnlty also some of the storage is used by the OS (operating system)

See this article for a full explanation:

http://support.apple.com/kb/TS2419