Modeling Value Based on Scarcity
A brief history of Bitcoin Stock-to-Flow models
A little more than one year ago, a Dutch investor that chose to publish his Bitcoin-related work under the pseudonym PlanB published an article in which he introduced the Bitoin Stock-to-Flow (S2F) model. Since then, the model became very popular and a lot has happened. After several independent reviews that provided additional evidence, the relationship between Bitcoin’s value and scarcity was considered provably non-spurious. Recently, those conclusions were shown to be flawed, and a new model that overcomes these flaws was introduced. This article attempts to provide an overview of these developments and breaks down the complex econometric nuances in an easy to understand fashion.
Scarcity
When you ask someone what makes Bitcoin valuable, “there will never be more than 21 million bitcoin” or “you can’t print more of them” are common answers — particularly in times where central bank’s are printing unlimited money. According to certain Austrian Economic theories, scarcity is one of the monetary properties (along with divisibility, durability, portability and recognizability) that gives money value. As Robert Breedlove argues in “The Number Zero and Bitcoin”, Bitcoin even achieves absolute scarcity, a property that is only feasible in the digital domain.
Although the idea of scarcity being a key aspect in Bitcoin’s value proposition has been there since the whitepaper was published, finding an appropriate quantifiable proxy to measure scarcity was less obvious. Inspired by a segment of the book “The Bitcoin Standard”, where author Saifedean Ammous described the scarcity of gold in terms of Stock-to-Flow (S2F) ratio, Plan B decided to explore if a S2F ratio of Bitcoin could be used to model its price.
Stock-to-Flow (S2F) ratio
The S2F ratio is calculated by dividing the stock (the total supply) by the flow (the new production) of an asset. In Plan B’s article, he described gold’s stock to be 185.000 tons, and its flow to be 3.000 tons per year. Hence, gold’s S2F ratio at that time was 185.000 / 3.000 = 61.67, or 62 when rounded up.
However, the S2F ratio of gold fluctuates over time (figure 1). When the gold price is relatively high, gold mining is more viable, incentivizing miners to do so. As a result, the flow increases and S2F ratio decreases. When the gold price is low, mining is less viable, especially in less efficient mines with a high production cost. If these close down or reduce their production, gold’s flow decreases, increasing its S2F ratio again.
Unforgeable costliness
The idea that an asset like gold is difficult to obtain or forge is known as unforgeable costliness, a term that became related to Bitcoin thanks to Nick Szabo, the creator of Bit Gold (a Bitcoin-predecessor). Besides gold (S2F ~62) and silver (S2F ~22), there are few monetary assets that can reliably be expressed in terms of S2F ratio and are considered to be unforgeably costly.
Other metals like palladium (S2F 1.1) and platinum (S2F 0.4) are also relatively rare and difficult to obtain, but are mostly used in industrial production. Their worldwide supply is relatively low in comparison to their yearly production, which means that its producers can have a large influence on the market price by in- or decreasing production, making these assets less optimal to use as a monetary asset.
Bitcoin’s predictable supply issuance
In Bitcoin, the threshold to start mining Bitcoin is very low. Anyone with spare computational power and a power-plug can join the rat-race to be next in line to create a new block and be rewarded in newly minted coins and transaction fees. Due to the competition that has built up over the years it is very difficult to do so profitably, but in essence the network is open for anyone to join.
However, if anyone can just create their own printing-press and start mining Bitcoin, why isn’t its S2F ratio blown to smithereens?
We need to thank Adam Back for this. In 1997, Back introduced the concept of Proof-of-Work (PoW) with Hashcash, a system designed to limit e-mail spam and denial-of-service attacks. Due to a built-in mechanism called ‘difficulty adjustment’, a PoW system periodically adjusts the difficulty of the random number that miners need to guess by adding or removing one or more digits.
In Bitcoin, this difficulty adjustment happens every 2016 blocks, which is about 2 weeks (assuming 10 minute block-intervals). When too much computational power is added to the network and new blocks are found faster than intended (1 block per 10 minutes), the difficulty increases. Miners then need to spend more resources to earn the same reward, incentivizing less-efficient miners to leave the network. Conversely, when miners leave the network and blocks are created slower than expected, the difficulty decreases, giving miners leeway to resume their activities. Thanks to this nifty difficulty adjustment system, Bitcoin’s stock and flow are quite predictable over time.
Bitcoin’s predictable stock and flow
When Bitcoin was launched on January 3rd, 2009, miners received 50 bitcoin mining reward (also called ‘coinbase’; not to be confused with the eponymous exchange) per created block. Every 210.000 blocks (~4 year, assuming 10 minute block-intervals), this reward halves. After the first halving (November 28th, 2012) miners received 25 bitcoin, after the second halving (July 9th, 2016) 12.5 and since the last halving (May 11th, 2020) they receive 6.25.
While we don’t know exactly when blocks are mined, Bitcoin’s stock and flow are completely predictable on a per-block basis (figure 2).
As a logical result, Bitcoin’s S2F ratio can be calculated at any point in time. According to Clark Moody’s dashboard, Bitcoin currently has a S2F ratio of 55, making it almost as scarce as gold. After the 2024 halving, it will surpass that of gold, making it the most scarce monetary asset in the world in S2F ratio terms.
If you have understood this explanation so far, you’ve grasped Bitcoin’s most prominent value proposition (censorship-resistance being another) — with or without the S2F model.
Nonetheless, Plan B tried to take it one step further by attempting to prove that the first-principles based hypothesis that Bitcoin’s price increase can be attributed to it’s ever-increasing relative scarcity is correct by using mathematical models — and thus predict its future price.
The Bitcoin Stock-to-Flow (S2F) model
On March 22nd, 2019, PlanB published “Modeling Bitcoin Value with Scarcity”. To visually assess if bitcoin scarcity, measured in S2F ratio, is indeed related to price, Plan B plotted both on a logarithmic scale. On a logarithmic scale, the distance between 1 and 10 is the same as between 10 and 100, between 100 and 1000, etcetera, making it useful to determine relative price changes.
When S2F increases, so does its market value, as all dots line up in a diagonal line (left graph in figure 3). This is called a ‘linear relationship’ and can be tested using statistical techniques (e.g., based on ‘ordinary least squares’, or OLS). Like the graph suggested, the relationship between Bitcoin’s S2F ratio and market value was indeed significant. According to this model, 94.7% of the historical bitcoin price can be explained by its S2F ratio. Plan B used the S2F ratio and market value of silver (grey dot) and gold (yellow dot) to cross validate the model. The fact that both lined up well with the modeled price was an early sign that this relationship might apply across assets as well.
Since the S2F ratio of Bitcoin can be estimated in in the future, the bitcoin S2F ratio and price can be plotted on a time chart (right graph in figure 3). Despite Plan B rounding down the parameters of the model, it predicted a $55.000 per bitcoin price after the 2020 halving. When Plan B published his article, the bitcoin price was $4.000 and was just recovering from a big price drop.
Over the next few months, several other versions of the S2F became available. These models used slightly different data (e.g., daily instead of monthly, or a different time window) and thus predicted different future prices. The version of the S2F model that became widely popular predicted a bitcoin price of around $100.000 after the May 2020 halving (figure 4).
Although many Bitcoin proponents were ecstatic about the model’s optimistic price predictions, there also were critiques.
It is ‘priced in’
One of the recurring critiques about the S2F model is that since Bitcoin’s supply schedule has been publicly known since its launch, it must be ‘priced in’, as the Efficient Market Hypothesis suggests. According to PlanB, markets are indeed fairly efficient because easy arbitrage opportunities are no longer available. Nonetheless, he thinks markets are structurally overestimating risk, leaving room for the S2F model to be useful as a valuation tool in investing.
Demand is missing
Another recurring comment is that price is a function of supply and demand — and that demand is missing from the S2F model. While this statement is technically correct, it misses the point that statistical models are by definition a simplification of reality and are never 100% accurate, but can still be useful if they are accurate enough. As statistician George Box once put it:
“All models are wrong, but some are useful.” — George Box
Despite demand not being included in the S2F model, the fact that it accounts for almost 95% of the variance in the bitcoin price suggests that it is accurate enough to be useful…. or is it?
Spurious correlations
High correlations like the one we saw in the S2F model are more common than you might expect. Particularly in time series that are both trending in the same direction, finding a high correlation between two variables that have absolutely nothing to do with each other can happen (e.g., see figure 5).
The possibility of the S2F model’s results possibly being spurious was also addressed in a July 2019 review article by a Dutch econometrician called Marcel Burger. In the article, Burger replicated the S2F model and tested whether the model met the needed statistical requirements needed to use those techniques. Burger found flaws related to the model’s underlying assumptions, and suggested that the model should be improved upon.
The rise of cointegration
In an August 11th 2019 publication, phraudsta, an Australian statistician, picked up where Burger left off. phraudsta’s work improved upon the original S2F model by applying a different statistical technique (a Vector Error Correction Model) to overcome the statistical limitations that were identified by Burger. More importantly, phraudsta found that Bitcoin’s S2F ratio and price are ‘cointegrated’, which means that the identified long-term relationship between the two is actually not spurious.
To explain what cointegration entails, phraudsta used an analogy about a drunk walking his dog. Imagine them strolling around, both occasionally going in different directions but remaining in close proximity due to the leash that connects them. Here, the drunk and his dog are ‘cointegrated’; they are connected and will both end up in the same place — wherever that may be.
The opposite would be true if a drunk is on his way home, and a stray dog crosses his path. Both stroll around together for a bit, but this relationship proves to be meaningless if a car drives by and scares the dog away.
In his conclusion, phraudsta suggests that this analogy needs to be changed to apply to the Bitcoin S2F model. Since the S2F ratio variable is actually rather constant, unlike the drunk or his dog, it would be more appropriate to consider Bitcoin’s price to be the drunk and S2F ratio to be the road home.
Shortly after, in September 2019, Marcel Burger replicated phraudsta’s findings. Later that month, a German senior analyst at BayernLB, Manuel Andersch, did the same. After these confirmations, the S2F model was broadly considered to be statistically valid and became even more popular.
Structural breaks
In March 2020, Bitcoin Elf suggested phraudsta to explore if the Bitcoin halvings should be seen as ‘structural breaks’ in the S2F ratio time series. Around the same time, Marcel Burger published an article in which he referred to an academic publication that also covered this topic.
According to that article, a structural break “is a sudden jump or fall in an economic time series which occurs due to the change in regime, policy direction, and external shocks, among others”. Figure 6 illustrates some examples of structural breaks. If these images didn’t already remind you of the increases in S2F ratio after the Bitcoin halvings, they should.
In the article “Stock-to-Flow Influences on Bitcoin Price”, phraudsta applied statistical tests to conclude that the halving events should indeed be seen as structural breaks and need to be accounted for. However, when the effect of the halving events is removed, the S2F variable loses much of its trend. Temporary fluctuations in coin issuance that are corrected for on a two-weekly basis via the difficulty adjustments are then the only remaining source of variance in the S2F variable (figure 7).
phraudsta continued by testing if the S2F variable is ‘stationary’ (without trend) or ‘non-stationary’ (with trend). After removing the effect of the halving events from the S2F variable, it no longer has a long-term trend and becomes ‘stationary’, unlike the bitcoin price that is clearly ‘non-stationary’. In a stationary process, the values can go up and down over time, but stay around a mean (figure 8, top graph). In a non-stationary process, the values also go up and down, but don’t revert back to the mean (figure 8, bottom graph).
While this may seem like a small and overly detailed statistical discussion, its domino effect is rather large: the finding that Bitcoin’s S2F ratio is stationary and price is not means that the cointegration test should not have been applied. Subsequently, this means that it is no longer proven that the relationship between S2F ratio is not spurious. While this doesn’t statistically invalidate the S2F model itself and also doesn’t mean that the relationship between S2F ratio and price is spurious, it re-introduces uncertainty. After all, if the relationship possibly is spurious, it means that there’s no reason why the bitcoin price couldn’t deviate from the S2F ratio trend at any time.
After phraudsta’s article, there was a lot of discussion on this topic. Bitcoin’s S2F ratio was used in the model as a measure of scarcity and the halving events clearly were designed to be the heart and soul of Bitcoin’s long-term scarcity. If you remove the S2F variable’s most important scarcity-component, it seems like you might be missing the plot if you use its remains to test if ‘scarcity drives price’. Is this really necessary?
The bitcoin price as a random walk
A presentation at the Value of Bitcoin conference on May 12th by Sebastian Kripfganz, an assistant professor at the University of Exeter that is an expert in econometric time series analysis, threw fuel on the fire.
In his presentation, Kripfganz described that the effect of the halving events on the S2F ratio’s time series indeed needs to be accounted for, but with a different explanation: because it is deterministic. Kripfganz didn’t explain this with much detail. For him it seemed a fact of life; you simply cannot use a deterministic variable in these time series analyses. The implications are the same as we saw in phraudsta’s analysis: after accounting for the halvings, Bitcoin’s S2F ratio is stationary, making cointegration analysis impossible.
Kripfganz continued by using a different statistical technique (an Autoregressive Distributed Lag or ARDL model) to test if the long-term bitcoin price can be modeled nonetheless. Kripfganz concluded that neither Bitcoin’s S2F ratio or the halving effects explained the long-term bitcoin price, and that it could be best described as a ‘random walk with drift’ from a statistical perspective. This means that while Bitcoin’s price trends upwards so far, it is essentially a ‘random walk’, which means that it could go anywhere.
While Kripfganz’s analysis was highly respected, the necessity of needing to remove the effect of the halving events in the S2F ratio variable because it is deterministic wasn’t immediately well understood. Doesn’t this again take out the essence of what made S2F ratio an interesting proxy for scarcity in the first place, causing us to throw out the baby with the bath water?
The fall of cointegration
An article published on May 20th by Marcel Burger provided clarity on the ‘determinism-debate’ started by Kripfganz. Burger dove deep into academic literature on time series analysis dating back to 1938 and concluded that Kripfganz was right. The cointegration analysis that was performed can only be applied on time series without a deterministic component.
Why you cannot use a time series with a deterministic component is an even deeper and more complex rabbit-hole in statistics. The implications are rather simple though: if you play a game, you have to adhere to its rules. In this case, you cannot use a statistical method to prove something that it cannot test.
Like phraudsta did before him, Burger concluded that in hindsight, the methods of his prior cointegration analysis were improperly applied, invalidating his prior conclusion that Bitcoin’s S2F ratio and price are cointegrated. Burger emphasized that this doesn’t mean that the relationship between between Bitcoin’s S2F ratio and price is spurious and that the S2F model is useless, just that we are now less certain that they aren’t.
After his presentation, Kripfganz mentioned that scarcity could still play a role in the upwards trend (the drift) that he identified in his model, but that it would be impossible to prove that it does from a statistical perspective. This suggests that we have reached the limits of what is statistically possible to prove with the time series analysis methods that are available today.
However, phraudsta disagrees that it would be impossible to prove that S2F ratio and market value are related altogether and implies that using cross asset information could be a way to overcome the limitations of time series analysis (figure 9).
The Bitcoin Stock-to-Flow Cross Asset (S2FX) model
On April 27th, a few weeks before the discussion on cointegration peaked, PlanB already introduced the Bitcoin Stock-to-Flow Cross Asset (S2FX) model that phraudsta hinted at. Like the title suggests, the model is based on data from multiple assets by introducing data from silver and gold to the equation. By doing so, the new model is no longer a time series, since the used datapoints are no longer lined up in a time-ordered fashion.
Deterministic or not, Bitcoin’s S2F ratio as originally defined by Plan B clearly increases over time. To create a cross asset model, which time-point do you use as the datapoint for Bitcoin? Could it be that the monetary properties of Bitcoin changed over time, as Bitcoin gradually became adopted?
Phase transitions
Plan B explored this from the viewpoint of phase transitions. A classical example is that of water, which transitions from a solid form to liquid, gas and eventually ionized when its temperature increases. Plan B went on to describe that you can argue that the dollar also underwent phase transitions. The dollar originally was a gold coin, then transitioned into a silver coin, a gold-backed piece of paper and since 1971 a piece of paper backed by nothing.
In July 2018 Nic Carter and Hasu published “Visions of Bitcoin — How major Bitcoin narratives changed over time”, in which they describe how the way Bitcoin is being described has changed over time (figure 10).
According to Plan B, these can be merged in four overarching phases:
- Proof of concept: immediately after the launch of the network.
- Payments: after bitcoin reached USD parity (1 BTC = $1).
- E-Gold: after the first halving, when bitcoin approached gold parity (1 BTC = 1 ounce of gold).
- Financial asset: after the second halving, when bitcoin reached the $1 billion transaction volume per day milestone.
Bitcoin clusters
Based on these four phases, Plan B applied an algorithm to identify four clusters of monthly bitcoin datapoints. The centers of these clusters (the yellow, orange and red dots in figure 11) represent the datapoints that will be used in the statistical modeling. These datapoints are complemented by two more datapoints for silver (the grey dot) and gold (the gold-colored dot).
Using the same method as for the original S2F model, Plan B found that the model explained 99.7% of the variance in the six cross asset datapoints. Compared to the S2F model, the S2FX model has a higher explained variance and is even more optimistic about the future price, as it predicts a price of around $288.000 per bitcoin in the current halving period (2020–2024)
Is six datapoints enough?
The S2FX model was well received, but received critical notes as well. The most-heard discussion was about whether or not creating a model based on just six datapoints would be robust enough. Due to the low number of datapoints, the model’s parameters and thus predictions could change when more datapoints are added.
For Plan B the results based on these 6 datapoints were indeed enough to convince him that there indeed is a relationship between S2F ratio and market value. As a counterargument against the critique, he calculated the probability of finding 99.7% explained variance with just 6 random datapoints, and that chance is indeed very low (figure 12).
Estimating the ‘Phase 5’ bitcoin price
In phraudsta’s May 7th article “S2FX — Phase 5 Estimations”, he reproduces the S2FX model and calculates margins of uncertainty around the predicted price. With his version of the S2FX model, phraudsta finds a predicted price that is a bit higher ($350.000) than Plan B’s predicted price. While S2F ratio is statistically a very significant predictor of price, the margins of uncertainty around the predicted price are rather large due to the small sample size. According to phraudsta’s calculations, the predicted price for Phase 5 could be anywhere between $83.000 and $1.480.000 (figure 13), while the actual price could further deviate from that predicted price as well.
Other critical notes
One can also question if it is indeed appropriate to split the Bitcoin data up into four different assets and assume those to be independent datapoints. After all, the bitcoin clusters are being formed in a time-contingent manner — otherwise, predicting the fifth phase wouldn’t be possible.
Finally, if you do consider this clustering method to be appropriate, you can still wonder if four is indeed the right number of clusters. Due to the clearly upwards trending price in relation to the S2F ratio an adjustment of the latter will likely still lead to the conclusion that there is a significant relation between the two, but the model’s predicted price could change as a result.
Like Plan B noted in his article, the model ideally needs to be expanded by adding more assets. The theory would be strengthened if it can be proven that there is a relationship between S2F ratio and market value of monetary assets without using Bitcoin data to create the model, and only use Bitcoin as a benchmark. While this sounds nice in theory, it is much harder to apply it in practice, as appropriate assets to use are actually quite scarce.
Does the S2FX model also work for the housing market?
On May 2nd Peter Harrigan, CEO of Grey Swan Digital and former trader at CME, did a first attempt at to extend the cross asset model. In his article “Bitcoin Stock-to-Flow Cross Asset Model Works Well on Housing”, he explores adding another asset class (housing) to the S2FX model. Like the title of his article suggests, this addition appears to rhyme well with the S2FX model.
Based on a detailed calculation, Harrigan determined the S2F ratio and market value of the American housing market in the context of ‘square footage’ and ‘value added’. These two new datapoints appear to align well with the predicted market value estimated by the S2FX model (Figure 14).
Plan B is currently looking into doing a similar analysis to add diamonds and European housing market data to the S2FX model, and has already shared an early indication that at least the latter seems to agree with the model as well.
Final note
Adding more assets to the S2FX model and validating the accuracy of the used data sources should be a main focus in future research. While doing so would likely strengthen the robustness of the model, it could also cause the model’s predicted valuations to change. It is therefore important to realize that one should be cautious in accepting the exact valuations that are predicted by the discussed models and approach this work more as a growing body of evidence that tests the fundamental value proposition that scarcity drives value.
Special thanks go out to Manuel Andersch, Plan B and Guy Swann, who provided feedback during the writing process.
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Disclaimer: This article was written for entertainment purposes only and should not be taken as investment advice.
