Institutions often hand out awards that include trophies that are either to be returned after the ceremony, or after one year when a new candidate is awarded the same trophy and their name is engraved on it, alongside its previous recipients'. These trophies are often named after the first recipient or the person who initiated its existence, and are thus rarely referred to by the type of trophy they are. My best guess is 'exchange trophy,' but I'd like to know if there is a proper term for this type of trophy.
Perpetual trophy, such as the auld mug.
Perpetual trophies are held by the winner until the next event, when the winner must compete again in order to keep the trophy. In some competitions winners in three consecutive or non-consecutive events receive the trophy or its copy in permanent ownership.