The cost (living expenses, travel, and insurance) for your outbound student exchange will depend on your destination. For an idea of living expenses see the relevant institution on the exchange partners website:
Standard course fees (as if you were in Dunedin) are paid to the University of Otago when you go on exchange. You can normally continue to receive StudyLink loans and allowances when you are studying overseas on exchange.
There are a number of grants and scholarships which you can apply for as an Otago student going on exchange: