This paper presents a structural framework for analyzing financial returns, based on the decomposition of logarithmic returns into positive and negative components. This decomposition enables the market to be described across five key dimensions: probability, volatility, risk, bias, and edge. The framework relies on the additivity property of log returns, which allows total return to be represented as the sum of periodic returns. However, while total return is invariant to time partitioning, the decomposition into positive and negative components depends on sampling frequency. Therefore, standardization of the timeframe is required for consistent comparison. The paper distinguishes between volatility, defined as total absolute movement, and risk, defined as the negative component only. This distinction enables separation between beneficial and harmful volatility and allows return to be analyzed as the result of the interaction between movement magnitude and directional bias. In addition, a closed system of metrics is introduced, including PNP, LRB, LRE, and E, which represent the same underlying structure across three different scales: probabilistic, ratio-based, and symmetric. The paper shows that even a small but persistent bias, when operating over large accumulated volatility, can lead to substantial long-term returns. The proposed framework enables a transition from a statistical description of returns to a structural description of the market and provides tools for strategy analysis, asset comparison, and market regime identification.
Motty Shai (Wed,) studied this question.