We introduce a framework to analyze the relative performance of a portfolio with respect to a benchmark market index. We show that this relative performance has three components: a term that can be interpreted as energy coming from the market fluctuations, a relative entropy term that measures "distance" between the portfolio holdings and the market capital distribution, and another entropy term that can be controlled by the trader by choosing a suitable strategy. The first aids growth in the portfolio value, and the second poses as relative risk of being too far from the market. 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.
