Well, first thing, you cannot have bought a truly new iPhone 5s since that model has been out of production for about 2 years now (the 64GB model for closer to 3 years now). So it is entirely possible the battery is simply worn out, even if it is old unsold stock since it may have been sitting on a shelf for the past 2 years or so.
If you take it into an Apple store, they can check the battery for you for free. If it is just a worn out battery, they can also replace that for you.