The quality of their food plays an important part, too, esp when they're younger [I apologize if this was already mentioned], as does the water quality.
I don't have koi, so no experience there, but I did take a summer to grow out some goldfish fry -- and boy, that was the LONGEST summer of my life, or so it seemed, LOL! Anyway, it was a chance to get some practical experience and make my own observations. When I could, I fed the fry live foods, and fed them frequently. They almost seemed to GROW overnight! And each winter I would keep 2 of those fry in my indoor tank so I could be a "mommy" over the winter
. Those fish were always larger than their siblings when released in the spring (for clarification, I only "raised" fry one summer, but the summer prior to that I did find 2 fry in the pond, and both of those came inside that winter).