Before revision:  We give several explicit controls of the third term that allows one to outperform a diverse volatile market in the long run . Named energy-entropy portfolios, these strategies work in both discrete and continuous time, and require essentially no probabilistic or structural assumptions. They are well-suited to analyze a hierarchical portfolio of portfolios and attribute relative risk and reward to different levels of the hierarchy. We also consider functionally generated portfolios (introduced by Fernholz) in the case of two assets and the binary tree model and give a novel explanation of their efficacy .
After revision:  We give several explicit controls of the third term that allows one to outperform a diverse volatile market in the long run , a market that is diverse and sufficiently volatile in the sense of stochastic portfolio theory. The framework is illustrated with several empirical examples .