The presumed drowning has now been referred to the coroner.
Mr Duncan's Hawke's Bay family said in a statement they were "heartbroken" after the boating accident, which happened when deteriorating weather conditions overwhelmed him on the lake.
"He was a most dearly loved son of Barb and Rob, brother of Mark, Ben and William. He will also be hugely missed by his grandparents, aunties, uncles, cousins, and many, many friends."
The family said they appreciated the "care and love" they had received from Mr Duncan's Defence Force mates, as well as the army.
"We will be eternally grateful for their support," the statement said.
"We are also grateful to the military, police, and the navy in their search for Hamish at Lake Coleridge, one of the most beautiful places in the world.
"Hamish was loving his time in the military and basic training was a challenge that he was proud to have achieved. He had commenced his training as a medic within the NZ Army, and the camaraderie he was enjoying with his friends on the course was all a young man could wish for. We have been fortunate to meet many of his classmates who are a credit to their families and the services."
Tributes were flooding in for Mr Duncan on Facebook last night.
James Ireland said: "I couldn't have wished for a better friend than Hamish.
"Mum and I are still finding it hard to come to terms with what has happened as we always enjoyed spending time with Hamish. Hamish will be irreplaceable as a friend in my heart and I hope he will be found soon so that we may all have some closure."
Anne-Marie Reid said: "We got to know Hamish quite well over the Lindisfarne years as he visited up here often when he was able. Such a lovely young man whose company we enjoyed. We will miss his smile and his humour greatly."
Police said Mr Duncan went to Lake Coleridge on his own on Saturday and was not participating in any military exercise or training.
A Defence Force spokesperson also spoke of the sadness of not being able to find Mr Duncan.
"It is with deep sadness that the personnel at Defence Health School and the wider NZ Army will farewell Private Hamish Duncan after the search has failed to find him at Lake Coleridge.
"His peers at Defence Health School will remember him as an outgoing, talented, smart and kind young man.
"The Defence Force will continue to support Hamish's colleagues and his family throughout this difficult time."
Canterbury police are reviewing and evaluating the search for Mr Duncan in consultation with the coroner and members of the police national dive squad.
