The power trend
I first published a chart observing how Bitcoin’s price follows a trendline according to a power law in 2018. My thinking was inspired by famous posts on BitcoinTalk authored by Trolololo from 2014. In his early model, he observed Bitcoin’s price fitting to a logarithmic regression. As the years had gone by, however, it had become evident that Bitcoin had naturally tended toward a power regression with a very good fit.
The inputs
I often discuss four basic types of statistical regressions: linear, logarithmic, exponential, and power. The power law relationship is reflected in the following formula:
In our case, y is Bitcoin’s price, or the dependent variable that we are solving or regressing for: the trendline. Variable a is the intercept coefficient, and b is the slope coefficient
The ten thousandth day of Bitcoin will fall on 21 May, 2036. This model projects out to the end of that month.
The percentile bands shown around the power trendline are similar to typical standard deviations (σ) used in statistics. However, this analysis does not assume a bell curve. Finance is not a bell curve. The bands are calculated as percentiles; that is, a percentage of time, at which price hangs inside certain multiples below or above the regression line. Between the 2.5ᵗʰ and 97.5ᵗʰ percentile red bands represents 95% of observations (similar to 2-sigma). Between the 16.5ᵗʰ and 83.5ᵗʰ percentile blue bands represents 67% of observations (similar to 1-sigma). Again, finance is not a bell curve. Therefore, the 1-sigma and 2-sigma inferred ranges are not equidistant from the trendline nor each other.
The goodness of fit, or the R-squared, on the above relationship is very strong above 95%.
Note: From January 2025, I adjusted the percentile bands toward a more conservative approach. Instead of measuring the historical price multiples above / under the all-time power regression line (and then running percentiles), I am now measuring the historical price multiples above / under the power regression line according to its coefficients derived at that time. As the all-time power trendline has shifted down over time, the prior approach tended to over-estimate the upper multiple percentiles.
Log-log chart
There is an easy way to tell if something in nature follows a power law. When plotted, if both the dependent variable (price) is displayed on log scale (y-axis) and the independent variable (time) is displayed on log scale (x-axis), then the trendline itself becomes a straight line.
This is the exact same chart as above, except plotted log-log, and defining the x-axis as days passed since the Genesis block:
As to why Bitcoin follows a power law, that is a fascinating question. A plausible answer seems to lie in the fact that the adoption of Bitcoin follows a power curve, and thus the price action where supply and demand continually converge on does as well, as the planet discovers this secure, digital base money.
Another way to look at it
Remember, the x-axis is not measured across dates, it’s measured across days. In this case, I’m measuring days from the genesis block on 3 January, 2009 (day 0). This is showing us that the price (adoption) of bitcoin has a relationship to how many days it’s been in existence. The relationship it exhibits is a power trend, which is non-linear.
This table shows how this Bitcoin price-time relationship manifests itself in a proportionate way. Each row in this table shows how long it takes for the trendline to double. Now, if we then measure the increase in days that is reflected in each row, we find a proportional increase in time. Last column.
Stated another way: For every ~13% increase in days in the life of Bitcoin, the price (adoption!) of the trend doubles. It doesn’t matter where we start the increment. This is how power curves work. This is a very exciting, live illustration of how a network grows.
You’re saying this predicts the future?
No. Models are used everywhere in our lives for guidance and baselines, and it is no different here. The interesting thing is that financial markets have traditionally moved in exponential fashion due to the underlying nature of credit and compounding interest. Bitcoin is different. Bitcoin is being adopted.
One other point on models and ‘predictions.’ In my work, when looking into the future with any model, I have always caveated that the word ‘prediction’ is simply a statistical one. Though regression models ‘predict’ a trend, even if a strong goodness of fit exists, there is not much more to say than that. The trend could hold as a helpful guide, or it could break. The rest is probabilities.
How does it evolve?
The one (independent) variable regression analysis is also helpful, not only because it is simple, but it will adjust itself with every new piece of data. In our case, this is every day. If the latest midnight UTC price is below the trendline, it will pull the trendline down. If the latest price is above the trendline, it will pull it up.
So, how has this trendline evolved over the years? Has it actually been helpful?
For now, yes. I have discussed this often on my channel: the all-time power trendline for Bitcoin has been holding at a relatively stable level since 2016.
To understand this last statement, review the last column in this table. Is that price somewhat close to today’s price?
Stay tuned for more. Never financial advice.
Related research
A few months before I published my conclusion in 2018, Dr. Giovanni Santostasi published a chart plotting a power law regression of Bitcoin’s price as a function of days since the Genesis block. In 2014 he published power law relationships of the Bitcoin system’s address and price growth. He continues to develop this theory across an array of factors.