Bitcoin valuation through power law analysis: Evidence for long-term mean reversion and short-term momentum

Purpose- This study examines the application of power law relationships to Bitcoin valuation and investigates whether deviations from this relationship provide predictive information for future returns across different time horizons. Methodology- Using daily Bitcoin–US dollar (BTCUSD) price data spanning from July 2010 to July 2025, the study estimates a power law relationship between Bitcoin price and time since the Genesis Block. The robustness of the model is evaluated using goodness-of-fit measures. Deviations from the power law-implied fair value are calculated and classified into deciles to analyze their ability to predict Bitcoin returns over short-term (weekly), medium-term (monthly), and long-term (annual) horizons. Risk-adjusted performance is assessed using Sharpe ratios. Findings- The analysis establishes a highly robust power law relationship between Bitcoin price and time, with an R² of 0.9589, indicating exceptional stability over the 15-year period. The estimated relationship, Price = 2.86e-17 × Time_Since_Genesis^5.71, remains consistent throughout the sample. Deviations from the power law-derived fair value exhibit strong predictive power for future returns, with distinct patterns across time horizons. Short-term returns display momentum effects, as the most overvalued decile generates the highest risk-adjusted returns (Sharpe ratio = 1.81). Medium-term returns peak under extreme valuation conditions, particularly in deeply undervalued and highly overvalued states. In contrast, long-term returns demonstrate clear mean reversion, with moderately valued positions yielding the highest absolute annual returns. Conclusion- These findings provide strong evidence that Bitcoin pricing deviates from the assumptions of strict market efficiency. The results offer quantitative support for valuation-based, time-horizon-dependent trading strategies and highlight the relevance of power law frameworks for understanding long-term Bitcoin price dynamics. Keywords: Bitcoin, power law, cryptocurrency valuation, market efficiency, return predictability