In a medium-large saucepan, combine cider, sugar, water, & cinnamon stick. Place over medium heat and bring to a simmer, stirring to dissolve sugar.
Add cranberries. Simmer and cook over low heat, stirring occasionally, until most of the cranberries have burst and the liquid starts to thicken - anywhere from 10 - 30 minutes (use your own judgement but be careful not to overcook - the sauce should still be liquid)
Remove from heat and allow to cool to room temperature. Transfer to a bowl, cover, and refrigerate for at least 24 hours to develop flavors.
Remove cinnamon stick and serve at room temperature