A year ago, crypto analyst PlanB published an article introducing the Bitoin Stock-to-Flow (S2F) model. Since then, this value model has become very popular in the field of encryption. After several market verifications, the model relationship between bitcoin’s value and scarcity seems to have been proven. But recently this conclusion model proved to have **certain defects** , and a new model was proposed to overcome these defects. The distribution room (fenbushiBTC) attempts to outline the development of the model and **resolve** the subtle differences between **complex econometrics** in an easy-to-understand way .

**Scarcity**

In the era of central bank unlimited printing of money, when you ask a person what supports the value of bitcoin, the common answer is “never more than 21 million bitcoins” or “you can not make more bitcoins “.

According to some Austrian economic theories, **scarcity is an attribute of currency** (distribution room note: several other attributes are divisibility, permanence, portability, and identifiability), which give monetary value. As Robert Breedlove described in The Number Zero and Bitcoin, Bitcoin has even achieved **absolute scarcity** , an attribute that is only feasible in the Crypto realm.

Since the release of the Bitcoin white paper, the **idea that scarcity is** a key aspect of **Bitcoin’s value proposition** has been around, but finding a suitable quantitative indicator to measure scarcity is not so easy. In the book “The Bitcoin Standard”, the author Saifedean Ammous used the stock to circulation ratio (S2F) to describe the scarcity of gold. Inspired by this, analyst PlanB plans to decide whether to explore whether the S2F ratio can be used to **model Bitcoin price** .

**Stock to flow (S2F) ratio**

The S2F ratio is calculated by dividing inventory (total supply) by the flow of assets (new production). In PlanB’s article, he defined the stock of gold as 185000T and its circulation of 3000T per year. Therefore, the S2F ratio of gold at that time was 185000/3000 = 61.67, which was **62** after rounding .

Figure 1: Gold S2F ratio

However, the S2F ratio of gold **fluctuates over time** (Figure 1). When the price of gold is relatively high, gold mining is more feasible, which also encourages miners to do so. As a result, the flow rate increases and the S2F ratio decreases. When the price of gold is low, the **feasibility of** mining **decreases** , especially in mines with high production costs and low efficiency. If these factories stop production or reduce production, the flow of gold will decrease, thereby again increasing their S2F ratio.

**Unforgeable expensive assets**

The idea that assets such as gold are difficult to obtain or counterfeit is often defined as ” **unforgeable and costly,** ” a term related to Nick Szabo, the creator of Bit Gold. With the exception of gold (distribution room note: S2F 62) and silver (distribution room note: S2F 22), there are few monetary assets that can be reliably expressed in S2F ratios and are considered expensive assets that cannot be counterfeited.

Other metals such as palladium (distribution chamber note: S2F 1.1) and platinum (distribution chamber note: S2F 0.4) are also relatively rare and difficult to obtain, but are mainly used for industrial production. Compared with their annual output, their global supply is relatively low, which means that their producers can have a significant impact on market prices by increasing or decreasing output, making these assets **less suitable for use as monetary assets** .

**Predictable Bitcoin supply**

In the Bitcoin field, the threshold for mining Bitcoin is very low. Anyone with extra computing power can join this fierce competition, become the next producer, and be rewarded with newly minted coins and transaction fees. But because of the competition formed over the years, it is very difficult to make money. But in essence, the network is open and anyone can join.

However, if anyone can start mining Bitcoin, why is there no anomaly in its S2F ratio?

This is mainly because in 1977, Adam Back introduced the concept of **Proof of** Work (PoW) with Hashcash , a system used to limit spam and denial of service attacks. Due to a **built-in mechanism** called ” **difficulty adjustment”** , the PoW system periodically adjusts the difficulty of random numbers that miners need to guess by adding or deleting one or more numbers.

In Bitcoin, this difficulty adjustment occurs every 2016 blocks, about 2 weeks (at 10 minute block intervals). When the network adds too much computing power and finds new blocks faster than expected, the difficulty increases. Then, miners need to spend more resources to get the same return, and miners with lower efficiency will consider leaving the network. Conversely, when miners leave the network and create blocks that are slower than expected, the difficulty is reduced, giving miners room to recover. It is precisely because of this **difficulty to adjust the system** that Bitcoin’s **stock and flow can be predicted** for **a period of time** .

**Bitcoin’s stock and flow are predictable**

When Bitcoin was launched on January 3, 2009, miners could get 50 Bitcoin mining rewards for each block created (distribution room note: also known as “coinbase”, but not an exchange of the same name). For every 210,000 blocks (about 4 years), the **reward is halved** . After the first halving (November 28, 2012), the miner block reward is 25 bitcoins, and after the second halving (July 9, 2016), it is 12.5 bitcoins, this time halving (May 11, 2020) The block reward is 6.25 bitcoins.

Although the exact time when the block is mined is uncertain, the stock and flow of Bitcoin are completely **predictable** on a per block basis .

Figure 2: Changes in Bitcoin supply (blue) and currency inflation (orange)

Therefore, Bitcoin’s S2F ratio can be calculated at any point in time. According to Clark Moody, Bitcoin’s current S2F ratio is 55, which is **almost as scarce as gold** . After halving in 2024, it will surpass gold and become the **world’s most scarce monetary asset in terms of S2F ratio** .

Nonetheless, PlanB tried to prove the assumption based on the first principle by using a mathematical model that Bitcoin’s price increase can be attributed to its growing relative scarcity is correct, thereby predicting its future price.

**Bitcoin S2F model**

On March 22, 2019, PlanB published the article “Modeling Bitcoin Value with Scarcity”. In order to visually assess whether the scarcity of Bitcoin measured by the S2F ratio is really related to price, PlanB on a logarithmic scale, determined that the distance between 1 and 10 is equal to the distance between 10 and 100, which is equal to 100 and 1000 The distance between them, and so on, determine the **relative price** change.

When S2F increases, its market value will also increase, because all points are arranged diagonally (distribution room Note: the left picture in Figure 3), this is called ” **linear relationship** “, you can use statistical techniques, For example, the test is based on “ordinary least squares” or OLS. As shown in the figure, **the relationship between Bitcoin’s S2F ratio and market value is indeed significant** . According to this model, **94.7% of** Bitcoin’s historical price can be explained by its S2F ratio. PlanB uses silver (gray dots) and gold (yellow dots) S2F ratios and market value **cross-validation models** . It was found that the two are consistent with the model price, and this relationship may also apply to early signs of all assets.

Figure 3: PlanB’s original Bitcoin S2F model

Since the future Bitcoin S2F ratio can be estimated, the Bitcoin S2F ratio and price can be plotted on the time chart. Although PlanB rounded this parameter model, it predicted that the price of each bitcoin after halving in 2020 would be $55,000. Distribution Room Note: When PlanB published an article, the price of Bitcoin was $4,000, which had just recovered from a sharp decline.

In the next few months, **several other versions of S2F** came out. These models use slightly different data, for example, predicting different future prices through daily rather than monthly or different time windows. The widely popular S2F model predicts that after halving in May 2020, the price of Bitcoin will be around **$100,000** (Figure 4).

Figure 4: The widely popular S2F model version

Although many Bitcoin proponents are ecstatic about the model’s optimistic price predictions, others have criticized it as ” **priced** .”

One of the criticisms of the S2F model is that since Bitcoin’s supply schedule has been known to the public since its introduction, it must be “priced” as suggested by the efficient market hypothesis. According to PlanB, the market is indeed quite effective because simple arbitrage opportunities no longer exist. Nevertheless, he believes that the market structurally **overestimates risk** , which leaves room for the S2F model as an investment assessment tool.

**Lack of demand**

Another critical point is that **there is no demand for “price is a function of supply and demand” in the** S2F model . Although this statement is technically correct, it ignores the point that statistical models are by definition a simplification of reality and can never be 100% accurate, but they are still useful if they are accurate enough.

As the statistician George Box once said: “All models are wrong, but some are useful.”

Although the S2F model does not include demand, the fact that it accounts for nearly 95% of the Bitcoin price variance seems to indicate that it is accurate enough.

**False correlation**

In the S2F model, you can see that the correlation is higher than expected, especially in the two time series with the same trend, you can find a high correlation between two variables that are completely unrelated (distribution room Note: see Figure 5).

Figure 5: False but very strong correlation between two time series variables

Dutch econometric economist Marcel Burger also mentioned in a review article published in July 2019 that the results of the S2F model may be false. Burger replicated the S2F model and tested whether the model met the statistical requirements required to use these techniques. Burger found defects related to the model’s basic assumptions and suggested improvements to the model.

**Cointegration**

In a publication on August 11, 2019, Australian statistician Nick Emblow (phraudsta) pointed out Burger’s shortcomings. Emblow’s work improved the original S2F model, using a different statistical technique (vector error correction model) to overcome the statistical limitations discovered by Burger. More importantly, Emblow found **S2F rates and prices Bitcoin is “co-integration” ****(cointegration) ****are** , which means that long-term relationship between the two determined not actually false.

To explain what is cointegration, Emblow used an analogy about a “drunk man walking a dog”: imagine that they walked around and occasionally walked in different directions, but because of the bonds connecting them, they still kept a close distance. Here, the drunkard and his dog are “co-integrated together”; they are interconnected and will eventually end in the same place-no matter where it is.

Conversely, if a drunkard is on his way home and a stray dog passes his way, they all walk together, but if a car drives past and scares the dog away, this relationship is meaningless.

In his conclusion, Emblow believes that this analogy needs to be changed to apply to the Bitcoin S2F model. Because the S2F ratio variable is actually **quite constant** , unlike the drunkard or his dog, it would be more suitable to consider that the price of Bitcoin is the drunkard sum, and the S2F ratio is the way home.

Soon after, in September 2019, Marcel Burger copied Emblow’s findings. Later that month, the German senior analyst Manuel Andersch of the Bavarian Bank (BayernLB) did the same thing. After these confirmations, the S2F model is widely considered to be statistically effective and has become more popular.

**Structural mutation**

In March 2020, Bitcoin Elf suggested that Emblow explore whether Bitcoin halving should be regarded as a “structural mutation” in the S2F ratio time series. At about the same time, Marcel Burger published an article in which he mentioned that an academic journal also covered this topic.

Figure 6: Examples of structural mutations in time series

According to the article, structural mutation is a sudden jump or decline in the economic time series due to changes in **institutions, policy directions, and external shocks** . (Distribution room note: Figure 6 shows some examples of structural disruption)

Emblow (phraudsta) used statistical tests to conclude that the halving event should indeed be considered a structural breakthrough and needs to be considered. However, when the effect of the halving event was removed, the S2F variable lost most of its trend. The temporary fluctuation corrected by the difficulty adjustment every two weeks is the only source of variance in the S2F variable (Figure 7).

Figure 7: Correction of Bitcoin S2F ratio (red line) before (left) and after (right)

Emblow continues to test whether the S2F variable is ” **stationary** ” (with a trend) or ” **non-stationary** ” (without a trend). He found that after removing the influence of the halving event in the S2F variable, it no longer has a long-term trend and became “stable”, unlike Bitcoin’s price which is obviously “non-stationary”. In a smooth process, these values will **fluctuate up and down with the change of time** , but will remain near an **average value** (distribution room Note: Figure 8, upper picture). In a non-stationary process, the value will rise and fall, but will not return to the average value (distribution room Note: Figure 8, bottom panel).

Figure 8: Examples of stationary (no trend) and non-stationary (trend) variables

Although this may seem like a small and overly detailed statistical discussion, its domino effect is quite large: it is found that the S2F ratio of Bitcoin is fixed, but the price is not, which means that the cointegration test should not be applied. Subsequently, this means that it is no longer proven that the relationship between the S2F ratios is not false. Although this does not statistically invalidate the S2F model **itself** , nor does it mean that the relationship between the S2F ratio and price is false, but it re-introduces uncertainty. After all, if this relationship may be false, it means that the bitcoin price cannot deviate from the trend of the S2F ratio at any time.

After Emblow’s article, there is a lot of discussion on this topic. The model uses Bitcoin’s S2F ratio to measure scarcity, and the halving is obviously to become the core and soul of Bitcoin’s long-term scarcity. If you remove the most important scarce component of the S2F variable, using the remainder of it to test whether “scarcity will drive prices” may not be necessary.

**Random walk of Bitcoin price**

On May 12, Sebastian Kripfganz, an assistant professor at the University of Exeter and an expert in econometric time series analysis, gave a speech at the “Bitcoin Value Conference”.

In his speech, Kripfganz described the effect of the time series of the S2F ratio on semi-events, which really needs to be explained, but gave a different explanation: because **it is deterministic** . Kripfganz did not explain this in detail. To him, this seems to be the reality of life; you **cannot use deterministic variables in these time series analyses** . This is the same meaning as seen in Emblow’s analysis: after considering it, it is found that Bitcoin’s S2F ratio is fixed, making “co-integration” analysis impossible.

Kripfganz continues to use another statistical technique (distribution room note: autoregressive distributed lag or ARDL model) to test whether long-term Bitcoin prices can be modeled. Kripfganz concluded that neither Bitcoin’s S2F ratio nor the halving effect **can explain the long-term price of Bitcoin** . From a statistical point of view, it is best to describe it as **“random walk”** . This means that although the price of Bitcoin has been on an upward trend so far, it is essentially a “random walk”, which means that it can go anywhere.

Although Kripfganz’s analysis is highly valued, because the S2F ratio variable is deterministic, it is necessary to eliminate the effect of the halving event in the S2F ratio variable, which was not immediately understood.

**Cointegration decline**

On May 20th, Marcel Burger published an article clarifying the “determinism dispute” initiated by Kripfganz. Marcel Burger delves into the academic literature on time series analysis, which dates back to 1938 and concludes that Kripfganz is right. The **cointegration analysis can only be applied to time series without deterministic components** .

Why not use time series with deterministic components is a deeper and more complex problem in statistics, and its meaning is simple: if you play a game, you must obey its rules. In this case, you cannot use statistical methods to prove what it cannot test.

Just like Emblow before, Burger concluded afterwards that his previous cointegration analysis method was improperly applied, making his previous conclusion that Bitcoin’s S2F ratio and price were cointegration invalid. Burger emphasized that this does not mean that the relationship between Bitcoin’s S2F ratio and price is false. The S2F model is useless, but it is not very sure now.

After his speech, Kripfganz mentioned that scarcity still plays a role in the upward trend identified in the model, but from a statistical point of view, it is impossible to prove. This shows that we have reached the limit that can be proved statistically with the time series analysis methods available now.

Figure 9: Nick Emblow’s Tweet

However, Emblow disagrees that it is impossible to prove that the S2F ratio and the market value are completely related, and suggests that the use of cross-asset information may be a way to overcome the limitations of time series analysis (Figure 9).

**Bitcoin Stock to Circulation Cross Asset (S2FX) Model**

On April 27, a few weeks before the discussion about cointegration reached its peak, PlanB had introduced Emblow’s implied Bitcoin stock to circulating cross-asset (S2FX) model. The model is **based on** data from **multiple assets** , introducing silver and gold data into the equation. By doing this, the new model is no longer a time series because the data points used are no longer chronologically arranged.

Whether it is deterministic or not, Bitcoin’s S2F ratio has increased significantly over time. But to create a **cross-asset model** , you need to determine which point in time to use as Bitcoin’s data point, and as Bitcoin is gradually accepted by people, whether Bitcoin’s currency attributes will change over time.

**Cointegration decline**

PlanB explored this point from the perspective of co-integration decline. A classic example is water, which changes from solid form to liquid and gas, and finally ionizes when the temperature rises. PlanB said it can be said that the US dollar has also undergone a **phased change** . The US dollar was originally a gold coin, and later became a silver coin, a paper currency with gold as its background. Since 1971, it has become a paper currency without any background.

In July 2018, Nic Carter and Hasu published “Bitcoin’s Vision-How Bitcoin’s Main Narrative Changes Over Time”, describing how the way Bitcoin is described changes over time (Figure 10).

Figure 10: Various definitions of Bitcoin evolve over time

According to PlanB, these can be combined into four main stages:

- Proof of concept: immediately after network startup
- Payment: After Bitcoin reaches the parity in USD (1 BTC = 1 USD).
- E-Gold: After the first halving, when Bitcoin is close to gold parity (1 BTC = 1 ounce of gold).
- Financial assets: After the second halving, Bitcoin reached a milestone of USD 1 billion in daily transaction volume.

Figure 11: Bitcoin S2FX model

**Bitcoin cluster**

Based on these four stages, PlanB applied an algorithm to identify **four clusters of monthly Bitcoin data points** . The centers of these clusters (distribution room note: yellow, orange, and red dots in Figure 11) represent data points that will be used in statistical modeling. These data points are supplemented by two other data points of silver (gray dots) and gold (gold dots).

Using the same method as the original S2F model, PlanB found that the model can explain **99.7% of the variance in the 6 cross-asset data points** . Compared with the S2F model, the S2FX model has a higher explanatory variance and is more optimistic about future price predictions. It predicts that the price of each bitcoin will be about $288,000 in the current halving period (2020-2024).

**6 data points**

The S2FX model is very popular, but has also received criticism. The most frequently heard discussion is whether creating a model based on 6 data points is persuasive enough, because due to the **small amount** of **data** , the parameters and predictions of the model may change as the amount of data increases.

For PlanB, the results based on these 6 data points are indeed enough to convince him that there is indeed a relationship between the S2F ratio and the market value. But the critic believes that he can find 99.7% of the explained variance with only 6 random data points with a very low probability (Figure 12).

Figure 12: PlanB Twitter

**Estimate the “fifth stage” bitcoin price**

In Nick Emblow (phraudsta)’s May 7 article “S2FX-Phase 5 Estimation”, he copied the S2FX model and calculated the margin of uncertainty surrounding the predicted price. In his version of the S2FX model, Emblow found that a predicted price was slightly higher than PlanB’s predicted price ($350,000). Although the S2F ratio is a statistically very important price forecast indicator, the margin of uncertainty in forecast prices is large due to the small sample size. According to Emblow’s calculations, **the predicted price of Phase 5 may be between US$83,000 and US$1.48 million** (Figure 13), but the actual price may also deviate further from the predicted price.

Figure 13: Emblow predicts Bitcoin price based on S2FX model

One can also question whether it is really appropriate to split Bitcoin data into four different assets and assume that these assets are independent data points. After all, the formation of a Bitcoin cluster has a time limit, otherwise it is impossible to predict the fifth stage.

Finally, if you think this clustering method is appropriate, you will still doubt whether the 4 clusters are indeed the correct number. Compared with the S2F ratio, the price shows a clear upward trend, so adjusting the S2F ratio may still lead to the conclusion that there is a significant relationship between the two, but the price predicted by the model may change accordingly.

Like the PlanB argument mentioned in his article, the **ideal model needs to be expanded by adding more assets** . If you do not use Bitcoin data to build a model and only use Bitcoin as a benchmark to prove that there is a relationship between the S2F ratio and the market value of monetary assets, then this theory will be strengthened. Although this sounds good in theory, it is much more difficult to apply in practice because the proper assets to use are actually quite difficult.

**S2FX model and housing market**

On May 2, Peter Harrigan, CEO of Gray Swan Digital and former CME trader, first tried to expand the cross-asset model. He published an article “Bitcoin’s cross-asset model from stock to circulation works well on real estate”, exploring the addition of another asset class (housing) to the S2FX model. As the title of his article implies, this addition **seems to closely match the S2FX model** .

Based on detailed calculations, Harrigan determined the S2F ratio of the US housing market and the market value in the context of “square feet” and “value added”. These two new data points seem to be very consistent with the market value predicted by the S2FX model (Figure 14).

Figure 14: Expanding the S2FX model by adding data points to the housing market for “value added” (green dots) and “square feet” (blue dots)

PlanB is currently considering a similar analysis, adding **diamond and European housing market data** to the S2FX model, and has shared preliminary results, indicating that at least the latter seems to be equally applicable to the model.

Adding more assets to the S2FX model and verifying the accuracy of data sources used should be **the main focus** of **future research** . Although this may increase the complexity of the model, it may also lead to changes in the model’s predicted valuation. Therefore, it is important to realize that one should carefully accept the accurate estimates predicted by the models discussed and use this work more as evidence to test the basic value proposition of scarcity-driven value.

**This article is the original of “Distribution Room (ID: fenbushiBTC)”. Unauthorized reproduction is prohibited. For reprint or business cooperation, please add WeChat [fenbushi_D].**

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