Boring Bitcoin

Bitcoin has been listed as an official Scrabble word

Bitcoin’s price action is subject of tons of blogs, articles and opinions and it is easy to get excited over it. However, bitcoin is intended to become a store of value, and eventually to become ‘good money’. To achieve that, its price needs to gradually stabilize over time and become as boring as possible. And that is what it seems to do, if we zoom out and look through the crazes, the speculations, the manipulations and the hypes.

Allow me to explain.

Bitcoin’s historic price record is not exponential. Exponential means that it follows a straight line up in a log (price) vs. time chart. Such growth is unsustainable and doesn’t hold for anything on this planet. Instead, bitcoin’s price increases nicely with time, but at a slowly decreasing rate. Mathematically, such growth is called logarithmic. Dave the Wave demonstrated this in Bitcoin & the Logarithmic Growth Curve. Another nice paper on logarithmic growth by Awe&Wonder used to be on Medium, but this author removed his work and his account some weeks ago unfortunately. A reference with copies of the original charts was shared by Willy Woo, who also publishes several alternative experimental and established price models.

When studying these articles, lines are drawn all over looking a lot like logarithmic growth curves. The mathematical equations and methods that were actually applied are not specified though, preventing me from reproducing any of these. Hence I started from scratch with some basic models myself. Here are the results, with explicit equations and parameter values.

Logarithmic growth models have 1, 2 or 3 parameters:

Model 1: log (p) = a * log (t): log(p) proportional to log(t)

Model 2: log (p) = a * log (t) + b: straight line between log (p) and log(t)

Model 3: log (p) = a * log (t + s) + b: curved line between log(p) and log(t)

p is bitcoin’s price in USD and t is relative time t in days since the first day where bitcoin got a price. The function log() is the logarithm base 10 and a, b and s are the model parameters.

“All models are wrong, but some are useful“ — George Box (1987)

Data was obtained from Bitcoinity. Based on the period from bitcoin inception (July 17th 2010, t=1) until today (June 7th 2019, t=3248 ), models 1 and 2 were unable to deliver a proper fit. Model 3 with optimal values of a=4.777, b=-13.137 and s=298.127 delivers

Equation 1: log (p) = 4.777 * log (t + 298.127) -13.137

or equivalent: p = 10 ^ (4.777 * log (t + 298.127) -13.137)

Method: Levenberg-Marquardt minimization of sum of squares between actual and fitted y-values (log (p)).

In figure 1 (top), the white line is the bitcoin daily price and the red line represents the fitted model (equation 1). Results were plotted on a linear time scale (x-axis) and a logarithmic price scale (y-axis). The bottom chart shows the derivative of the red line in the top chart and represents bitcoin’s growth rate as a percentage per year.

Figure 1: The logarithmic growth model log (p) = a * log (t + s) + b applied to historic bitcoin price. Top white: actual bitcoin price in USD. Top red: Model bitcoin price in USD given by p = 10^(4.777*log(t+298.127)-13.137). Bottom: Annual growth rate of bitcoin price as a percentage, currently 63.6%.

Bitcoin’s price increases with time at a decreasing rate. Bitcoin’s model growth rate is currently 63.6% annualized, fallen from over 1,000% in 2011. The cursors in green are positioned at today’s values. Everything at the right side of these cursors is extrapolated data. Today (June 7th 2019), bitcoin’s actual price is $7688, while bitcoin’s model price is $6609. Bitcoin’s actual price is thus 16.3% higher than its current model price.

Now that we know the model price as a function of date, we also know when bitcoin’s actual price was below (‘undervalued’) or above (‘overvalued’) its model price. This splits the full price-time dataset into two separate datasets: one where the price was ‘high’ and one where the price was ‘low’. The same logarithmic growth (model 3) was fitted to these two datasets. The results of these fits are:

Equation 2: log (p) = 4.152 * log (t + 179.535) -10.625

Equation 3: log (p) = 5.349 * log (t + 441.591) -15.478

In figure 2, equations 1, 2 and 3 are plotted in the same chart. The red line is equation 1 and the same as in figure 1. The blue line (equation 2) runs nicely through the points where bitcoin was overvalued and the yellow line (equation 3) runs through the points where bitcoin was undervalued.

Figure 2: The red line is the same as in figure 1 (top). The blue line was fitted to only those prices above the red line (bitcoin ~ overvalued), the yellow line was fitted to only those prices below the red line (bitcoin ~ undervalued). The three lines converge, indicating that the future price of bitcoin might slowly stabilize towards the model price.

Figure 3 is perhaps the weirdest format in which you have ever seen the history of bitcoin’s price. Not only the y-scale (price) is logarithmic but the x-scale (relative time since 17 July 2010 plus 298) as well. This shows the linear relationship between x and y, as defined by equation 1.

Figure 3: The same chart as in Figure 2, but the x-scale is now the number of days since 17 July 2010 plus 298 and plotted on a logarithmic scale. The red line is now a straight line and shows the linear relationship between log (price) and log (time+298). The convergence of the price channel is clearly visible again.

Extrapolated to the future the three lines gradually converge to each other, and this is a confirmation of what Dave the Wave also published based on his channel of curves. This was not forced in any way into the model, each of the three lines was fitted independently, and could therefore also have run in parallel or in divergence. The convergence seems to emerge because the peaks and troughs of the actual bitcoin price relative to the model price gradually get less pronounced with time (see figure 4). These results demonstrate one of the most important properties of bitcoin: the volatility of bitcoin diminishes with time and peaks and bottoms get shallower. This trend is likely to continue in the future and is a necessity for bitcoin to take on its intended role as sound money.

Fugure 4: Actual bitcoin price relative to Model bitcoin price as a ratio. There is a tendency for the peaks to be less high with every market cycle, and for the troughs to be less deep. It may take a while but this is price discovery in action.

This modeling work helps to guide our expectations:

Bitcoin’s model price reaches 10k USD around May 2020
Bitcoin’s model price reaches 100k USD around November 2026
Bitcoin’s model price reaches 1M USD around July 2037

Note that these model predictions are not written in stone, and will change a bit with every new daily price added to the data series. But it is hard to imagine that bitcoin wil suddenly break away from this model, and the future parameter values and model price expectations are likely to operate in narrow ranges from here.

Note also that these price milestones happen quite a bit later than the totally unrealistic prognoses of all the time-travelers in the crypto space who keep shouting numbers and dates which are seemingly not covered by a specified mathematical or other model.

That doesn’t mean that the actual price will not surprise on the shorter term, and bitcoin’s high volatility might push its actual price to these 10k, 100k and 1M levels quite a bit sooner. Later is also not impossible but given the high speed of bitcoin’s cycles in the past, combined with bitcoin’s very fast May 2019 price rise, that is unlikely. Such surprises occur because the actual price has shown relatively short-lived, hard-to-time overshoots of a factor of 5-ish and longer-lasting undershoots of a factor of 2-ish from the model price. This pattern will almost certainly repeat, albeit likely with gradually lower amplitudes. Such asymmetric price window provides great trading opportunities for those with the patience to trade these ups and downs properly.