What's periodic rent? Here in the US we have 12 months lease or 6 months lease or pay month by month ( which is typically more pricey), never heard or any other options. At the end of your lease you either renew for next term or pay monthly or move out. Is it different in NZ?

If agreement was verbal then there is really no agreement. Verbal agreements don't have any legal value.

Our lease agreement spells everything out. What's in your lease agreement?

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