Christian is the eldest of Crown Prince Frederik and Crown Princess Mary’s four children; his three younger siblings are Princess Isabella, 12, and twins Prince Vincent and Princess Josephine, both 8.
Fans were quick to send well wishes to celebrate the teenager's birthday.
"Congratulations on your birthday. I wish you a good day with your family and friends," wrote one royal fan.
"Big and heartfelt congratulations to the great young man," wrote a second.
According to an article on Danish media site Avisen under the headline "moving abroad", the pair revealed the news while on a three-day official visit to France.
The children will begin their studies in January 2020.
However, Mary, who hails from Tasmania, told French First Lady Brigitte Macron that it would only be for a short while.
The Royal Head of Communications Lene Balleby confirmed their children's move but she would not say which Swiss school the children would be attending or how long the foreign adventure would last.
Mary said that they see it as a "gift" to their children.
"One where they can be together about it, and a gift they can have with them for the rest of their lives," she said.
This will not be a new adventure for the family. Almost every winter, the family ski the slopes of Verbier during winter holidays.
The Swiss school system, which is multilingual, is known throughout the world for their academia and the stay will also help the children's language skills.