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:

\begin{aligned}
y &= ax ^{b}
\end{aligned}

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 scale coefficient, and b is the rate of change coefficient. These two coefficients can be derived by any statistical computer program upon inputting Bitcoin’s pricing data, and not everyone will get exactly the same answer. My latest coefficients are shown beneath the chart. The independent variable x is time; in our case, days since the genesis block on 3 January 2009.

The chart

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 reflect typical standard deviations (σ) used in statistics. Prices that fall within the 1-sigma blue bands denote 67% of the observations. Prices that fall within the 2-sigma red bands denote 95% of the observations. Note that finance is not a bell curve. Therefore, the 1-sigma and 2-sigma 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 at 95%.

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.

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:

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.