How do we calculate the Fair Value of Bitcoin?

1. Introduction

The concept of determining Bitcoin’s fair value has been a subject of extensive research and analysis in financial and quantitative modeling communities. Unlike traditional assets, Bitcoin lacks intrinsic value derived from cash flows, making its valuation challenging. However, empirical studies have demonstrated that Bitcoin’s price follows a power-law growth pattern over time. This methodology details the derivation, historical basis, and accuracy of this approach.

2. Historical Background

The power-law model applied to Bitcoin’s valuation was first popularized by Harold Christopher Burger in 2019. Burger observed that Bitcoin’s price, when plotted on a logarithmic scale against time, exhibited a near-linear relationship. This pattern indicated that Bitcoin’s price followed a power-law function of time rather than a traditional logarithmic or exponential growth model. The power-law model has since been widely discussed in academic circles and has been validated with historical price data spanning over a decade.

3. Mathematical Formulation

The fair value of Bitcoin is computed using the power-law equation:

Pf=a×(T)bP_f = a \times (T)^b

where:

• PfP_f is the fair price of Bitcoin at time TT
• aa is a scaling constant determined through regression analysis
• TT is the number of days since Bitcoin’s genesis block (January 3, 2009)
• bb is the exponent governing the rate of price growth

Empirical analysis suggests the best-fit parameters as: a=1.0117×10−17,b=5.82a = 1.0117 \times 10^{-17}, \quad b = 5.82

Thus, the fair value at any given time TT is calculated as: Pf=1.0117e−17×T5.82P_f = 1.0117e-17 \times T^{5.82}

This model captures Bitcoin’s long-term appreciation while smoothing out short-term volatility.

4. Model Derivation and Empirical Validation

To derive the parameters aa and bb, a regression analysis was performed on historical Bitcoin price data spanning from 2009 to the present. A least-squares regression on the logarithmic transformation of price data yielded an optimal power-law relationship.

Validation steps:

1. Historical Fit: The model was applied retrospectively to Bitcoin’s historical prices. The computed fair value consistently aligned with Bitcoin’s long-term growth trends, confirming the model’s robustness.
2. Out-of-Sample Testing: Data from recent years, not used in model training, was used to predict fair values. The model’s predictions closely matched observed price trends.
3. Comparative Analysis: The power-law model was compared with alternative valuation models such as stock-to-flow (S2F) and Metcalfe’s Law-based approaches. While S2F relies on supply constraints and Metcalfe’s Law considers network effects, both models exhibit periodic deviations from real-world prices. The power-law model, by contrast, has shown superior long-term predictive stability.

5. Interpretation and Accuracy

The power-law model provides a long-term trend estimation rather than a precise short-term price target. Historical testing has shown that Bitcoin’s market price often fluctuates around the computed fair value, sometimes exceeding it during bull markets and falling below it during bear markets.

Key observations:

• Accuracy Over Long Time Horizons: Over multi-year periods, Bitcoin’s actual price tends to revert to the power-law predicted fair value.
• Deviation Ranges: Short-term price movements can deviate by ±30% or more from the fair value, particularly during extreme market conditions.
• Logarithmic Regression Strength: The coefficient of determination (R2R^2) for the power-law fit typically exceeds 0.95, indicating a strong correlation between time and Bitcoin’s historical price.

6. Limitations and Considerations

While the power-law model provides a strong foundation for estimating Bitcoin’s fair value, several limitations must be acknowledged:

• External Market Factors: Regulatory changes, macroeconomic conditions, and institutional adoption can cause significant short-term deviations.
• Potential Model Breakdown: If Bitcoin’s adoption reaches saturation, the power-law trend may weaken over time.
• Market Cyclicality: The model does not account for market cycles, halving events, or speculative bubbles that may temporarily distort Bitcoin’s price.

7. Conclusion

The power-law model remains one of the most robust long-term valuation frameworks for Bitcoin. By leveraging historical trends and empirical data, it provides a strong estimate of fair value while recognizing inherent market volatility. Although it should not be used as a trading signal for short-term decisions, it serves as a valuable tool for understanding Bitcoin’s long-term price trajectory.