I know that when i personally lead the caravan, i can push the profit through trade via rumor but how can i influence the profit when a companion leads the caravan?
Trade,. Riding speed, Scouting, and city the caravan starts in seem to be the most important factors. Trade obviously affects the prices but since it is % based a caravan can still make money with low trade and the companion trade skill will improve over time though I wouldn't put any companion with 1 main skill to learn trade even if you put focus point into trade it will take forever. Riding speed helps caravan get to more cities quickly and avoid bandits while scouting mostly avoids bandits.
The starting city prosperity affects what products a caravan starts with and how quickly profits accumuilate as the caravan runs it circuit.
In my testing even 'bad' companions with near zero trade skill can earn 300-500 per day starting in a prosperous city. Good traders earn between 900-2000 if they have +50 in trade and the other skills and start in a prosperous city. Not sure what the upper limit is but I've never seen more than 2400 per caravan and in current release version the only 'good' companions are spouses, none of the wanderers start with good caravan skills and only a couple can be trained in the first month or so to be average caravan traders.
After a couple months if you are capturing castles and cities the caravan income is really only to trade your companions and slightly help income but you'll find your caravans attacked and compansion taken prison frequently so training is often better to put companions in their own parties or as governors. Best income is still from battles- vs larger parties 10,000+ plus loot and 50,000+ from selling equipment is normal and you can average 4-5 large battles per war so each war is giving around 200,000+ so after 8-12 wars you can afford to start your own kingdom.