Nope. The round ends just after the creature with the lowest initiative count takes action. This means that if you delay until "just before" the creature with the highest count (other than yours), you act first in the next round. If you delay until "just after" the creature with the lowest count, you act in the current round. Even though, in both cases, you actually act at the same time relative to other creatures.
That said, effects with a duration measured in rounds typically last until the same point in the round in question as when they began. So, if you cast a spell with a duration of 1 round, it ends at the beginning(?) of your next turn, or (if you are no longer acting on the same initiative count), that initiative count. If something else has that initiative count, then I'm not sure how the rules handle it. Were it me, I'd assign the effect an "effective initiative count" equal to your own initiative count at the time you cast it, to be used as a tiebreaker of sorts. Thus, anything that went before you if your initiatives were tied would take its turn before the effect expired. Hell of a corner case, and this paragraph got progressively less and less grounded in real rules with every sentence.
However, actions != durations. Think of actions as costs - you aren't waiting for them to finish, you have to spend them to make an effect happen. I'm not exactly sure what you're asking, to be honest, but I think that's relevant.