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 fact sheet for the relevant institution on the exchange partners website:

Otago Global Student Exchange Partner Institutions
Visa and insurance information for outbound exchange students

Standard course fees (as if you were on the University of Otago campus) are paid to the University of Otago when you go on exchange. You can continue to receive StudyLink loans and allowances while 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:

Otago Global Student Exchange — Financial Assistance