The Truth Behind Bitcoin’s Power Law: Rigor Versus Hype

Introduction

Many Bitcoin enthusiasts have likely encountered charts suggesting that Bitcoin prices follow a neat “power law” progression—a visually striking diagonal line on a double-logarithmic scale, or a set of bounded, arcing curves on a log-linear chart. On the surface, such visuals appear to validate price projections as though a mathematical inevitability drives Bitcoin’s growth.

A closer look, however, reveals that power laws do not inherently predict rising prices. By definition, they describe the distribution of extreme values—that is, how often very high (or very low) prices occur relative to more moderate ones. What’s especially appealing is that a heavy-tailed (or “fat-tailed”) distribution implies a higher probability of dramatic price moves than a standard bell-curve model would suggest. This matches Bitcoin’s history of extreme volatility; yet it says nothing definitive about the overall price trajectory.

Researchers—including the team at BlackRock—have noted strong positive skewness in Bitcoin trading, with extreme returns clustered in a small fraction of days (roughly 4% in their study; see my summary at murrayrudd.pro/bitcoin-and-portfolio-allocation-insights-from-blackrocks-research). While such findings excite those hoping for outsized gains, they also underscore how unpredictably these jumps occur.

In this post, I take a deep dive into power law analysis for Bitcoin. I discuss why the idea of a single neat slope extending from pennies to six figures is more myth than math—and why rigorous, tail-focused methods are necessary to truly understand Bitcoin’s extreme moves.

What a Power Law Is—and Why People Embrace It for Bitcoin

The Core Idea of Heavy/Fat Tails

In many natural and financial processes, extreme outcomes—such as earthquakes, market crashes, or even the concentration of city populations—occur more frequently than a standard (normal) distribution would predict. A power law mathematically states that the tail of a distribution follows a relation of the form:

In plain language, this means that as the value of x increases, the probability of observing a value above x decreases. However, it decreases more gently than it would in a normal or exponential distribution. For example, if you double x, the probability might be reduced by a factor of roughly 2^α (where α is the exponent), rather than plummeting to near zero as in a bell-curve.

• Power Law Example: Imagine you set a threshold at $100,000 for Bitcoin’s price. Under a power law, if you double the threshold to $200,000, $400,000, and so on, the chance of the price exceeding that new level falls by a predictable multiplier (roughly “divide by 2^α”) rather than dropping off dramatically.
• Normal/Exponential Comparison: In a bell-curve scenario, doubling the threshold would cause the probability of an extreme outcome to drop very sharply—so much so that such events like a 4X in price become virtually impossible. This is why power law distributions are said to have “fat” or “heavy” tails: they keep the possibility of extreme outcomes (like a massive Bitcoin omega candle) alive at a higher rate than a normal distribution would.

Think of the midwit meme: the tails of the distribution are skinny for the rare low- and high-IQ characters, with only 0.1% odds. With a power law distribution, you get fat distribution tails – maybe the odds for low- and high-IQ characters are 3 to 5% instead of near 0%. With a power law distribution, more people on the fringes of the distribution start to look very smart.

To bring the idea home: when I taught statistics to non-math undergraduates, I often used lottery odds. In a normal distribution, winning the lottery would be an almost unreachable, multi-standard-deviation event (say, 1 in 10 million). But in a power law world, the chance might be 1 in 10,000—a difference that, while still small, is dramatically more “realistic” and could keep ticket vendors busy!

The Appeal in Bitcoin World

• Grand Narrative:
  A neat power law slope on a log-log chart can be interpreted as “Bitcoin is following a powerful force of nature,” bolstering a bullish, almost mythic narrative.
• Skewed Events:
  With positive skewness, historical data show that rare but massive upward moves can dramatically influence overall returns. This supports the idea that “the next surge up could be even bigger.”
• Network Effects:
  Many Bitcoiners refer to reflexivity—where rising prices attract more participants, potentially driving further increases. When this feedback loop appears self-similar across different eras, it aligns with a power law dynamic.

However, there’s a critical difference between identifying heavy tails in price levels (or returns) and asserting that one universal exponent can forecast future price paths. That’s where rigorous methods—such as threshold selection, maximum likelihood estimation (MLE), and goodness-of-fit testing—come into play.

A Step-by-Step Look at Fitting a Power Law

Gathering and Cleaning Data

• Quality matters. Bitcoin data from early exchanges like Mt. Gox or lesser-known platforms may have incomplete records, price anomalies, or thin liquidity. For the analysis below, I used data from CoinCodex, which, despite its limitations, is one of the few freely available full datasets dating back to 2010 (and after threshold selection, only data from 2021 onward were used).

Model Choice

• Determining Best Fit: Even if a log-log plot appears straight, it might be mimicked by other distributions (e.g., lognormal or a Pareto with a cutoff). Determining the best fit requires rigorous statistical tests rather than a simple visual check. For power law modeling, much of the methodology is well explained in: Clauset A, Shalizi CR, Newman MEJ: Power-law distributions in empirical data. SIAM Review 2009, 51: 661-703 (https://doi.org/10.1137/070710111).
• Comparing price-fitting curves goes beyond power law specifics, so I will deal with this topic in a future blog post.

Threshold (x_min) Selection

• Determining the Threshold: A key part of proper power law analysis is identifying a threshold (x_min) above which the power law behavior holds. In Bitcoin’s case, I found that prices below approximately $55,000 do not follow the same slope and are therefore omitted from the tail analysis. This threshold selection is crucial: including all data (when only the top 5–10% truly follows the power law) can “water down” the results.

Estimating the Exponent (α)

• Maximum Likelihood Estimation (MLE): Using MLE, we calculate the exponent 'alpha hat’ (the hat means it is an estimated value of α):

• For instance, if α hat is about 2,  it means that every time x doubles, the probability of exceeding x shrinks by roughly a factor of 2^2.5 (about 5.7).
• Interpretation: An α hat less than 2 suggests a heavy tail, meaning giant outliers aren’t just possible, they’re relatively frequent. Larger values of α hat (>3) means outliers are still more common than in a normal distribution but less so than in very heavy tails.

Goodness-of-Fit Tests

• Kolmogorov–Smirnov (K-S) & Parametric Bootstrap: These tests compare your model’s predicted cumulative distribution with the observed data in the tail.
• p-Value: If we calculate a moderately high p-value from the K-S test, this would indicate we cannot reject the hypothesis that this section of the data (above our threshold) follows a power law with α=2.5. In other words, there’s no strong statistical evidence that the data in this upper region aren’t power-law-distributed at that slope. It doesn’t prove the fit is perfect or exclusive—only that the power law model isn’t contradicted by the observed tail.

My Analysis: Thresholds, Tail Data, and Surprises

Using the poweRlaw package in R—which implements the Clauset–Shalizi–Newman methodology—I analyzed Bitcoin’s daily closing prices (5,297 observations from August 1, 2010 to January 30, 2025):

• Threshold (x_min): The algorithmically determined cutoff was $55,653. Only prices at or above this level were included in the tail analysis.
• Dataset Reduction: More than 91% of historical daily price points fell below the cutoff, leaving 435 observations in the tail set.
• Parameter values: MLE produced an estimated α=5.89, suggesting that while the distribution is heavy-tailed compared to a normal distribution, the tail decays relatively quickly.
• Kolmogorov–Smirnov Distance: The K–S distance for this fit was approximately 0.0909, indicating that the largest difference between the empirical tail cumulative density function (CDF) and the fitted power law’s CDF was around 9% on the [0,1] scale.
• Goodness-of-Fit: The K–S distance of 0.0909, combined with bootstrap p-values of 0.80 and 0.78 for 200 and 1000 sample bootstraps, respectively, indicates no statistically significant deviation from the power law model in the upper tail.

At first glance, these results might seem to confirm a “power law for Bitcoin.” However, note that nearly the entire price history (from near zero) was discarded—only the high-price segment was modeled. This does not prove that Bitcoin has always followed this slope; it merely indicates that the most extreme prices, during a recent period, are consistent with a heavy-tailed behavior.

Also, notice that a higher exponent (around 5.9) means the probability of extremely large outcomes drops off more sharply than it would for lower exponents (like 2 or 3). While extreme events remain more plausible than under a normal distribution, they are not as prevalent as some simplistic analyses might suggest.

A chart (created in R) comparing the empirical tail CDF with the fitted power law line shows that the faint red line on the right, representing the power law, fits only a small proportion of the overall observations. There is even a sharp jag rightwards (e.g., between $66,000 and $74,000) that illustrates how little time Bitcoin spends in certain price ranges—another reminder of the complexity behind the numbers.

Contrast with Lower-Price Eras

Analyses that try to include early data—from Bitcoin’s days at $10, $100, or even $50,000—often fail when forcing a single exponent. This is evidence of multiple market “regimes,” each with its own dynamics (liquidity, participants, external conditions). As Bitcoin evolves, its tail behavior and even the appropriate power law model may change.

Why a High p-Value Doesn’t Guarantee a Good Price Forecast

Distribution vs. Trajectory

The key message for non-math readers: power laws describe the relative frequency of extreme values, not a time-based price trajectory. A heavy-tailed distribution indicates that huge price spikes (or drops) are more likely than a normal distribution would predict, but it does not ensure that future spikes will occur. It merely tells us that if extreme prices do occur, the drop-off in probability is gentler than it would be under a bell-curve.

Market Evolution and Regime Shifts

Bitcoin is known for its changing market regimes. What worked statistically in one phase may not hold in another (e.g., our 2025 institutional inflows and looming government stacking). Liquidity, regulatory shifts, market sentiment, and macroeconomic forces can all alter the underlying distribution.

Comparisons to Other Heavy-Tailed Assets

In traditional finance, similar heavy-tail properties are observed in commodities, emerging-market currencies, and tech stocks. The bottom line is that heavy tails mean extreme events are more common than a normal distribution would predict—but they do not guarantee a continuously rising price.

The Problem with “All Data from 2010”

Visually Compelling but Potentially Misleading

Many double-log charts that span Bitcoin’s entire history look impressive, suggesting a single, unbreakable trajectory. However, these charts often hide crucial breaks between market cycles. By compressing decades of data onto one plot, multiple “bends” or shifts become subtle—even if they are statistically significant.

The Threshold Principle

If a rigorous tail analysis sets a cutoff at $55k, it indicates that data below that point do not follow the same power law. Including all data in one curve can mask these differences and create an oversimplified impression of continuity.

Implications for Forecasting Bitcoin’s Future

Not a Crystal Ball

A well-fitted tail analysis can show how probable extreme price outcomes have been under specific historical conditions. However, it cannot definitively project future behavior if new participants, macro forces, or regulations alter market dynamics.

Understanding “Skewness” vs. “Inevitable Upside”

Research (like BlackRock’s) demonstrates that positive skewness means large upward moves can heavily influence returns. This is one reason many Bitcoiners HODL rather than trying to time the market. However, the data do not guarantee that such rallies will continue at the same frequency or magnitude.

Conversely, changes in supply and demand (i.e., from supply shocks) might one day lead to a heavier tail, with more frequent extreme events.

Balancing Power Laws with Broader Economic Models

For robust forecasting, analysts probably need to combine:

• Power Law Insights: Recognizing that extreme moves are more common than predicted by a normal model.
• Supply–Demand Fundamentals: Factoring in Bitcoin halving events, changes in exchange listings, and institutional adoption. This is what I currently focus on - see our recent Satoshi Action journal article.
• Regulatory and Macroeconomic Considerations: Accounting for large capital inflows, policy changes, and broader economic shifts. This is what I want to build in to our current supply-demand model in the future.

Key Takeaways and Conclusion

What Power Laws Do Say

• Extreme Values: Bitcoin’s tail behavior suggests that both meteoric rallies and steep crashes occur more often than simpler models would predict.
• Threshold Dependence: Discarding lower-priced data in tail analysis is standard—it doesn’t mean early price stages didn’t occur; it only means they follow different statistical patterns.

What Power Laws Don’t Promise

• Guaranteed Growth: A power law does not imply that Bitcoin will perpetually increase in value; it merely characterizes how extreme outcomes scale in probability.
• Single Slope Forever: Different market regimes may exhibit different exponents or thresholds. A universal slope from pennies to six figures is extremely unlikely.
• Safe Predictive Power: Even if current data fits a power law well, shifts in macro conditions or liquidity can change the underlying distribution.

In short, while power law models underscore that extreme events (the so-called “god candles” or “omega candles”) are part of Bitcoin’s inherent volatility, they are not a crystal ball for predicting future prices. They offer insight into the statistical structure of past extremes—but forecasting future price paths requires other approaches.

“Power Law” vs. Log-Scale Curve Fitting

Some Bitcoin commentators claim power law price models by fitting all historical data to a single log-scale line or band. In reality, these approaches generally:

• Use Full History: They include data from near zero upward, without discarding sub-threshold values.
• Perform Simple Log-Scale Regressions: They often regress log(price) on time, forcing one slope across all market cycles.
• Rely on Visual Fits and R²: High R² in a transformed (log-linear or log-log) plot does not prove a nonlinear power law exists. Think about it: you need to take a nonlinear relationship, compress it using a log transformation; run a goodness-of-fit test designed for linear relationships and find, unsurprisingly that you show a significant linear relationship; and then uncompress the data back to nonlinear form, claiming your linear test suggests a significant relationship in the nonlinear dataset. It is flat-out wrong to claim, as we often see in the Bitcoin space, that a high R² ‘proves’ the power law relationship.
• Lack Rigorous Testing: They rarely conduct threshold selection, maximum likelihood estimation, or goodness-of-fit tests (e.g., K-S tests, bootstrap analyses) that are essential to confirm genuine heavy-tailed behavior.

A neat “power law corridor” across Bitcoin’s entire price history might be visually appealing, but without rigorous threshold analysis and statistical testing, it is little more than a curve-fitting exercise rather than true heavy-tail modeling. I once had a perceptive, but disgruntled, student who referred to that kind of curve-fitting as ‘coloring for credit.’

What is the Modeling Goal?

If your goal is to understand why outsized moves occur more frequently than traditional models predict, power law and related heavy-tailed distributions provide a valuable framework. However, if your aim is to predict Bitcoin’s price trajectory over the next few years, you’ll need a broader toolkit.

But Murray, Aren’t You Bullish?

After all this technical discussion, you might wonder if I’m casting doubt on Bitcoin’s future.

On the contrary, if you’ve followed my research (or seen my interview with Robin Seyr on YouTube), you know my view is that Bitcoin could reach very high prices—especially if institutional demand increases and a true supply shock occurs. I use a theory-based demand curve and an inelastic supply curve in my own modeling (see DOI: 10.3390/jrfm18020066). My forecasts, while sharing some price targets with power law advocates, are derived from entirely different premises.

No model is perfect, and I believe that price forecasts should be framed in terms of expected values (EV), with probabilities assigned to a range of possible outcomes.

My modeling suggests that Bitcoin prices of $2- to $5-million (or more) are feasible but say nothing – for now – about the probability of reaching those levels, nor the probabilities of lower Bitcoin prices in the future.

If pressed, I would probably ballpark 1% odds of $5-million and 2% odds of $2-million, which would justify a $90,000 contribution to Bitcoin's EV (0.01[5,000,000] + 0.02[2,000,0000) even if there was a 97% chance of Bitcoin going to zero (0.97[0]), which I certainly do not believe.

Alteratively, if we believed there was a 97% likelihood of Bitcoin staying flat at $100,000 (0.97[100,000]), the EV would be $97,000 + $90,000 = $187,000. Holding a belief that there are non-zero, positive probabilities of reaching extreme prices quickly drives up the EV (something that Greg Foss has been banging the drum on for a number of years).

If a person truly holds beliefs consistent with a $187,000 Bitcoin EV and the market price is $100,000, that implies that person is likely doing some mad stacking!

EV is different for everyone because everyone will put different weights on different kinds of evidence. Every Bitcoiner should critically assess the range of possible price targets they believe are possible and the strength of the evidence informing those beliefs.

Don’t let dubious assumptions or weak statistical analyses mislead you.