Curvy Mayhem, Stock-to-Flow, and a Critique of Pure SimplicityDisclaimer: This is not financial advice. I am not a statistician. I am not a trading/investing expert. I am a wildlife biologist. This is just a regurgitation of my research, thoughts, and opinions, along with my attempt at having fun with numbers to create an incredibly speculative model for Bitcoin’s future price action. Hang in there folks, this is a long one.
Since I entered the crypto realm in 2017 (I know, such a newbie), I have been obsessed with Bitcoin’s historical logarithmic price chart. Something about the way it smoothly sweeps across the orders of magnitude separating its former obscurity from its financial relevance has drawn me into a fantasy of elegant mathematics, an illusion of design, and a tempting allure for fate. The hindsight is heavy, and it all seems so simple, but it rarely ever is. I often see BTC log charts with curves that march atop the market cycle peaks or support the lengthy slumber of the prices below. I’ve fallen into this habit myself, but these curves are all equally vapid. You can fit infinite curves to any three points after all. (Which of the twelve curves above is the correct one? I personally like light green.) When we create models, we mustn’t be arbitrary for the sake of beauty. What feels right is usually not what ends up being right. Any experienced day-trader will tell you this. We need objectivity.
Financial models are hard to create. For centuries, humans have struggled to keep up with the emergent complexity of the markets they formed. The intricacies of our systems tend to outpace us, and some things forever elude our understanding. However, we desire simple answers to complex questions. We see patterns in everything; it’s just an evolutionary heuristic that our prehistoric ancestors utilized for hunting, gathering, and not dying. But in our hyper-complex modern world, this feature of pattern recognition is usually used to a fault. In the following paragraphs, I outline some issues with models created by others and myself. On the surface, these models appear elegant and well-fit, but when we delve into the assumptions behind such models we often find that simple answers are woefully insufficient to predict the future of a complex and turbulent world.
BITCOIN STOCK TO FLOW MODEL
While the controversial Stock-to-Flow (S2F) model introduced in 2019 by Plan B has proven to be a good fit for Bitcoin’s early price growth thus far, there are several fundamental problems with the model, like failure to account for demand as an influence of price and the lack of a relationship between price and S2F in other scarce stores of value including cryptocurrencies. But perhaps worst of all, this model fails to address the growth-resistant factors that Bitcoin will soon face. Linear regression models on a log-log plot predict infinite growth when extrapolated. Whether limitations arise from resource depletion, social and political behaviour through competition and regulation, or even the laws of physics, nothing can grow indefinitely.
So what will ultimately limit Bitcoin? Let’s start with the energy consumption problem. Bitcoin already consumes about 0.5% of the world’s energy supply, more than most individual countries on the planet, and this percentage is increasing rapidly. The issue lies with Bitcoin’s proof-of-work architecture, an algorithm used in the Bitcoin blockchain that incentivizes miners to expend computational energy to cryptographically secure others’ transactions. As speculation drives the price of Bitcoin higher and the available minable supply decreases, miners face greater competition and expend more energy. Eventually, and probably sooner than later, Bitcoin’s price will rise to such a level that the hash rate, and subsequent mining cost, will no longer be able to keep up. Even putting human behaviour aside, Bitcoin’s energy consumption would exceed the entire energy supply on Earth by the 2030’s given the unfettered growth predicted by the S2F model. This may be the gravest threat to Bitcoin’s development into an economic juggernaut, though some solutions like proof-of-stake have been proposed to address this crisis.
Two more restrictive factors on Bitcoin’s price are governmental regulation and financial pressure. For the most part, Bitcoin has been allowed to grow naturally without too much interference. However, as it becomes a more significant market force, powerful governmental and financial forces will inevitably attempt to influence, control, or even destroy it. Perhaps the latter is unlikely to happen, if not impossible to do, but market adoption can absolutely be decelerated, leading to a suppression of demand and price.
Finally, assuming relatively tame fiat inflation rates, there’s not even enough money on Earth to support the level of growth predicted by the S2F model for even a couple more decades. Eventually, the market will become saturated, demand will diminish, and the price will stabilize. The only way this model works and gives us bitcoins worth $1 trillion in 2050 is if USD inflation goes nuclear and sends the global economy into abject chaos. Even Plan B has admitted as much. By then, your crypto gains would probably be the last thing on your mind.
I think it’s clear that any models attempting to predict the future price of Bitcoin need to include a factor that limits growth over time or extrapolates from existing decelerating price patterns. So I decided to create two alternative models based solely on Bitcoin’s price history. For simplicity’s sake, I chose the more speculative route of creating a model based on the peaks of each of Bitcoin’s bubbles. (Note: Data used in statistical analysis was monthly high bitcoin prices collected from barchart.com and yahoo finance.)
FOUR-PARAMETER LOGISTIC REGRESSION MODEL
Even a brief glance at the logarithmic chart shows a pattern of price bursts steadily decreasing in intensity, revealing a long-term trend of logistic growth. This is not surprising, considering it gets prohibitively harder to 10x a market cap the second, third, or eighth time around. The best-fitting model for four points following a logistic pattern is, of course, the four-parameter logistic model. This provides a moving target for an end to this bull run. (Note: I made this chart before INDEX:BTCUSD was released, so pre-August 2011 prices were drawn in)
Despite giving a tamer near-term outlook, this model still overestimates long-term prices and runs into many of the same problems as S2F, leveling out at a price of 10^230 USD long after our planet is gone and stars stop forming… but at least it levels out. I would also argue that this model is heavily overfitted, using four parameters given only four data points. Furthermore, it places too much emphasis on the starting price of Bitcoin, which may have had little or no influence on its future price.
MARKET CYCLE RATE-OF-INCREASE POWER REGRESSION MODEL
Instead, I looked to a different measure to predict Bitcoin’s bubble behaviour: price increase over time within each market cycle, extrapolated with a power regression model. I defined market cycles as the time between peaks and calculated the percentage price increase over time (in months) from peak to peak. During the first cycle, when Bitcoin jumped from its first-traded value of $0.09 to about $30, the rate of increase over time was astronomical. The percentage rise of each subsequent bubble has decreased since then while market cycles have lengthened. This gives us three complete market cycles ending in June 2011, November 2013, and December 2017, and three data points describing, as an average monthly percentage, the constant rates of increase in price from one peak to the next. Extrapolated with a power regression (y = 2758x^-4.119; R^2 = 0.994), we are left with a shallower rate of increase between the 2017 peak and the approaching peak. This again provides a linear moving target for an end to the run. On a logarithmic chart, the straight lines between peaks look a little different.
This model proves much more flexible than many others. Instead of a specific date or price level, Bitcoin is free to trade however it wishes until the moving target is hit, whereupon the bubble will deflate and we enter a new cycle with a new sloped upper bound. The slope of this bound is determined by the previous market-cycle peak price and the next rate-of-increase value provided by the power regression. These slopes constantly increase, but by less and less each cycle until the price of Bitcoin plateaus. The price level of this ceiling would be determined by the frequency/length of market cycles. Time itself acts as (or at least tracks) the decelerating force.
So, it’s a fun model, and quite pretty on a logarithmic chart, but how good is it actually? Well…
Problems with this model:
It fails to properly define peaks. One can gain an intuitive sense of when each bubble ended, but without an objective definition of this point, the very parameters on which this model relies can be interpreted differently by others. How are we to know if this current run has ended? Was the spike in April 2013 a peak? (Probably not, but you get the point). This one is easy enough to remedy, but I can’t be bothered.
We have only three data points, hardly enough to make a reliable trend, let alone one we can extrapolate (Counterpoint: The power regression extrapolation of only the first two points predicts the third with a surprisingly reasonable margin of error for these scales – about 0.2 orders of magnitude, suggesting this model may already have some predictive power. In other words, if you had followed this dubious two-point model in 2017, you’d have sold at about $12,000.). Additionally, extrapolation leaves us with a much greater margin of error than interpolation, especially when we’re working with such a small sample size. At this point, we risk falling into the trap of moving the goalposts by adjusting our model to match new data as it comes in, not unlike what has been done with the S2F model. This ad hoc method constantly maintains the fit of a model but proves that the initial version had somewhat poor long-term predictive power to begin with.
This model also places too much emphasis on Bitcoin’s starting price in July 2010. I find it unlikely that this asset’s long-term growth dynamics were heavily influenced by this initial value.
It relies on the assumption that the declining rate-of-increase of market-cycle price peaks can be extrapolated into the future. It might be possible to justify this, but I can’t be bothered. This write-up is already nearing 2,000 words.
The use of a power regression forces the assumption that long-term growth will never be negative; instead, Bitcoin will approach a plateau at some point. While there are any number of black swan events that could deflate Bitcoin’s price, no simple price extrapolation model can predict and incorporate these possibilities with any reliability.
If this model somehow plays out perfectly, I’d be elated. But I wouldn’t have been right. I’d have been lucky. The possibilities for Bitcoin’s behaviour during this cycle and the next are innumerable. All you need is 3 data points and you can make anything happen. Perhaps you remember that colorful, curvy chart a bit further up. However, that doesn’t mean it’s not fun to try. Probing the long-term price action of a novel market with statistical fervor has proven to be a rather entertaining and educational experience. It also shows the difficulty, and perhaps the futility, of finding simple solutions to incredibly complex systems.
CONCLUSION
I recently watched a youtube video posted by an astrophysicist. He discussed whether we should rely on beauty and simplicity when creating models to accurately describe the intricate and incredibly complex details of our physical universe. Take the theory of gravity and planetary motion, for example. As physicists, theoreticians, and thinkers studied the skies for millennia and searched for simple answers, the theories progressed from that of circular orbits, to more complex ellipses, to a law for gravitational attraction, to requiring special and general relativity – a dramatic increase in complexity and certainly a less beautiful solution, even if more accurate. I have noticed the same trend in my own field. The theories describing ecosystem equilibrium and the interactions between species have grown more complex as ecologists learn more about the biosphere at various resolutions. I believe these same principles can be applied to most aspects of reality. Simplicity has its place, but we often take it for granted. As tempting as simplicity and beauty are, we mustn’t fail to respect and embrace the complexity of our world, however we interact with it.
