Why are you thinking "13"?
Do you think at 13 they are
1. Not Shufaha or immature, childish.
2. Balaghul Nikaha or the age of marriage which is synonymous in classical Arabic to maturity.
3. Sound judgement.
4. Mature enough to inherit their wealth and manage their finances.
Another reference to maturity is full physical growth as in Quran 40:67 where Ashaddakum from the root Shadad means full physical growth. So you must know when your child is full grown.
Well its just I've seen some very huge looking physically mature looking 13 year old males and females occasionally, and others (maybe not the same ones, but possibly the same) who are pretty smart or good enough with money (as good, if not better than, some much older people who just never are really). So I thought maybe in an age where people are dying in their 20s from the slightest illness or injury, that they might have to sort of get things going as early as they possibly can for the survival of the human race.