Not sure of what kind of arrangement you've got, but will assume that it is a plan through Apple itself or through AT&T, with NEXT, or the one with Verizon (don't know what that one is called). In any event, with such a plan, once you have paid all the monthly payments, the phone is yours to do with whatever you like. The payments may be scheduled for 24 months or they may be longer, but once they're paid off, that's the end of your obligation.
The part which I think is confusing you, the whole yearly upgrade thing, comes into play only if you want to get a new iPhone 12 months after your current one, and in that case, you trade in your old iPhone, get the new one and start a new payment plan. Yes, you are credited for the iPhone you are turning in and start an entirely new payment plan for the new phone. For those who want a new iPhone at each new release, this works out fine, but it is certainly not a requirement that anyone trade in his or her current iPhone in order to get the new one. There is the option to pay off the current iPhone. For instance, right now I am at the halfway mark with my NEXT plan, so that I have the choice of either simply handing in my current iPhone 7 Plus and receiving credit for that while also starting a new plan with an iPhone 8, 8 Plus or X.....OR paying the difference (i.e., balance of my existing NEXT contract) and keeping my current iPhone 7 Plus while also getting a new iPhone and starting a new plan.
I hope this clarifies things for you!