One of probably the most fascinating points of bitcoin is its historic meteoric worth rise. Is bitcoin going to proceed on this historic path or is progress going to gradual, and even halt?
The stock-to-flow (S2F) model, put ahead by PlanB, means that bitcoin’s future worth may be forecast fairly exactly and that the value will proceed a gradual and spectacular path upwards, with roughly tenfold returns each 4 years. The S2F mannequin has attracted a whole lot of consideration, and PlanB has amassed a formidable variety of followers (1.7 million on the time of writing).
Perhaps partially due to its recognition, the mannequin has extra lately been met with a whole lot of criticism. An instance of such criticism is a harshly-worded current article printed in Bitcoin Magazine. Also, in July 2020, Eric Wall put together a collection of criticisms.
It seems that most individuals discover themselves in both of two camps: the “pro” S2F and the “con” S2F camps. How ought to we place ourselves?
Before I’m going on: I’ve written negatively in regards to the S2F mannequin since 2019, once I predicted that the S2F mannequin’s predictions would prove too bullish. I’ve additionally exchanged with PlanB each publicly on Twitter (e.g. here), and privately. I’ve coauthored a more mathematical article along with InTheLoop, clarifying why we each assume the S2F mannequin is too bullish. It would possibly subsequently come as no shock that I’m not precisely within the S2F camp. However, I’ve additionally seen that a few of the criticisms in direction of the S2F are invalid. Other criticisms purport to deal a loss of life blow to the S2F mannequin, whereas the truth is, they don’t. I subsequently hope to add some readability. It is necessary to be proper for the suitable causes, as a result of right ideas are our solely probability of being proper sooner or later.
The S2F Model
The S2F mannequin states that the value of bitcoin is pushed by its shortage. As the halvings be certain that bitcoin turns into ever extra scarce, its worth ought to repeatedly enhance. The relation between shortage and worth is mathematically outlined (utilizing two empirically estimated parameters) and roughly forecasts a tenfold enhance in worth each 4 years. This offers us a worth of $100,000 per bitcoin for this halving epoch, $1,000,000 for the following, and so forth.
Source: s2fmultiple
What’s flawed with this mannequin? Let’s have a look at some arguments that are put ahead to discredit the mannequin:
Tautological Specification
In their current Bitcoin Magazine article, Level39 had this to say concerning the S2F mannequin:
“Notice how the function says ‘market value’ equals a function of Stock-to-flow? This is a model misspecification with tautological logic and therefore statistically invalid, for the simple reason that ‘market value’ decomposes to ‘stock / price’ while ‘stock / flow’ is on the other side of the equation. In layman’s terms PlanB is essentially asserting that ‘stock is a function of stock.’ A tautology is a trivial statement that is true under any circumstances. It’s like saying a banana is a kind of banana. Of course stock is a function of stock. This is why the data fits, but is scientifically worthless. Tautologies are true but do not tell us anything useful. Rather, they are true because of the meanings of the terms.”
But is this actually so? Has PlanB actually given us a tautological formulation that doesn’t inform us something helpful, a bit as if Isaac Newton had informed us that F = F? Is inventory actually on each side of the equation?
The S2F mannequin as formulated by PlanB makes an attempt to approximate the market cap of bitcoin utilizing stock-to-flow as an enter variable (the place a better stock-to-flow signifies increased shortage). Two parameters (a and b) have to be empirically estimated in order to get the perfect match. Writing this down, it at first would possibly seem that certainly inventory seems on each side of the equation (see the second and third traces, under). However, by merely rearranging phrases, we see that this is effective: the value of bitcoin is on the left-hand facet of the query, inventory and circulation on the suitable facet.
We have clearly demonstrated that the S2F mannequin is not by a tautology that renders it mathematically invalid. Still, there is another level to make. Level39 goes on to clarify:
“PlanB could avoid the tautology by having price alone on one side of the equation and perhaps build a regression of price on flow or stock to flow, but the fit would be different without changing the parameters.”
In different phrases, if PlanB had tried to mannequin (the log of) the value utilizing a linear perform of stock-to-flow as a substitute of the market cap, the inventory wouldn’t seem on each side of the equation, and therefore the supposed tautology would disappear. In different phrases, so as to get a worth forecast primarily based on stock-to-flow, we may both:
- Model the market cap, and translate the market cap again to costs. This is what PlanB did, and Level39 sees a tautology right here, or:
- Model the value immediately. Level39 sees no tautology right here.
Level39 insinuates that A would produce a a lot better match than B due to the supposed tautology. But is this actually the case? In the under plot I’ve in contrast each fashions:
We see the 2 fashions are extraordinarily related to one another. There is no monumental distinction within the high quality of match between the 2 fashions. Hence, even when there have been a tautology within the unique S2F formulation (there isn’t), the purpose could be fairly trivial, since it could not materially matter. The mannequin may very well be rewritten to approximate worth as a substitute of market cap and the consequence could be nearly similar.
Hence, the entire argument concerning a tautology is clearly moot. No loss of life blow to the S2F mannequin right here.
Autocorrelations
Another argument in opposition to the S2F mannequin I’ve regularly heard is additionally talked about by Level39:
“The other problem is that the model is autocorrelated, where the results of today’s value is a function of yesterday’s value. When you adjust for that, the R-squared (R2) value is zero. Thus, scientifically speaking, stock-to-flow is nonsensical and cannot be used to model price.”
Another manner of stating this is to say that as a substitute of making an attempt to discover a relation between stock-to-flow and worth (or market cap) one ought to as a substitute attempt to discover a relation between modifications in stock-to-flow and modifications in worth (or market cap). The declare is that modifications in stock-to-flow on a day-to-day foundation don’t seem to trigger a change in worth on the identical time scale, and therefore there supposedly can’t be a causal relationship between stock-to-flow and worth, that means that the S2F mannequin have to be incorrect.
But is this actually the case? Large modifications in stock-to-flow occur solely as soon as each 4 years. The variations in stock-to-flow between the halvings are principally small and have a powerful factor of randomness. Must we actually count on that each small and huge modifications in stock-to-flow trigger a change in worth? This would imply that we are assuming that there is a linear response, which needn’t essentially be the case: It may very well be argued that solely giant modifications in stock-to-flow are significant.
Hence, the argument of auto-correlations additionally doesn’t yield a loss of life blow to the S2F mannequin.
Ad Hominems
Another argument in opposition to the S2F mannequin I regularly encounter is PlanB’s conduct on Twitter. Level39 has this to say about it:
“[… ] anyone who points out a flaw, potential problem, has a valid question or even “likes” a legitimate inquiry into the validity of his assertions is blocked [by PlanB] […] If PlanB needs to truthfully declare that his fashions have a scientific R2 worth within the excessive 90s, then he can’t be blocking and censoring legitimate criticism that exhibits in any other case.”
The reply I’ve to this is that PlanB can do no matter he appears like on Twitter. He is not obliged to behave in a selected manner or to reply any explicit questions. His conduct has no impression on whether or not the S2F mannequin is legitimate or not.
In addition to this, my very own expertise with PlanB has been very completely different than the one described by Level39. I’ve brazenly criticized his mannequin on Twitter in 2019 (you’ll be able to witness such a dialogue here), and haven’t been blocked. We have exchanged privately and I can not characterize PlanB’s conduct as something aside from very pleasant.
I’ve heard of occasions when individuals have been blocked by PlanB, however I’m not stunned by this: He has to handle an viewers of 1.7 million individuals, which can’t be simple. In any occasion the advert hominem argument says nothing in regards to the validity of the S2F mannequin and ought to be disregarded.
Lack Of Cointegration
There has been a long debate concerning whether or not a sure property often called cointegration (pronounced co-integration, not coin-tegration) exists between stock-to-flow and the value of bitcoin. Cointegration is supposed to trace at a causal relation between the 2 variables. When it finally got here out that the cointegration property does not exist between stock-to-flow and worth, this was interpreted as that means {that a} change in stock-to-flow can not presumably trigger a change in worth. A loss of life blow to the S2F mannequin! But is that basically the case?
I had by no means heard of cointegration prior to 2019, when finding out the stock-to-flow mannequin. It is an idea that is broadly utilized in econometrics, however not in every other fields (so far as I’m conscious). For instance, in March 2020 Judea Pearl, the de facto inventor of causal statistics and writer of “The Book of Why” had not heard of cointegration either! He gave two clarifying statements that cointegration would possibly give an indication that there is causal relation, however that it by no means implies a causal relation. In 2022, Pearl once more lamented that nobody was in a position to satisfactorily explain the concept of cointegration to him.
The incontrovertible fact that the inventor of causal statistics didn’t know in regards to the idea of cointegration is telling: The significance of cointegration appears overblown. The lack of cointegration would possibly maybe trace at bother for the S2F mannequin, however it shouldn’t be thought of a loss of life blow.
Summary Of Anti-S2F Arguments
The arguments in opposition to the S2F mannequin we’ve seen up to now both haven’t any advantage (supposed tautology, advert hominem assaults), or maybe weaken the credibility of the mannequin however don’t rule it out (lack of cointegration, autocorrelations).
What we must always do is depend on empiricism: Is the S2F mannequin in a position to predict future costs appropriately? This is the litmus take a look at for any worth mannequin.
An Empirical Look At S2F
I’ve created a bitcoin worth mannequin referred to as the power-law corridor of growth which depends on the remark (which I owe to Giovanni Santostasi’s reddit post) that bitcoin’s worth follows a straight line when plotted utilizing an x-axis that is scaled logarithmically.
This merely signifies that bitcoin’s worth progress is slowing down. Whereas it used to take solely a couple of 12 months for the value to admire ten-fold, it now takes a number of years. Returns are diminishing, and I count on this pattern to proceed into the longer term.
Yet, many individuals appear to assume that bitcoin’s worth will behave equally sooner or later because it did previously. In different phrases, they count on worth will increase to occur on the identical tempo as previously. I’ve published the below plot in an article on the finish of 2019. Various individuals have made predictions apparently primarily based on the belief of nondiminishing progress (roughly represented by the inexperienced line). I predicted that these forecasts would show to be too bullish, and that the value would extra carefully comply with the orange line, which is ruled by diminishing returns.
How has my prediction fared? The subsequent plot is the very same because the earlier one, however with the addition of worth information (in pink) which is now accessible and that was not accessible on the time I made the prediction.
My 2019 prediction proves to have been prescient. What does this imply for the S2F mannequin? In the same article I defined that S2F forecasts nondiminishing progress, and that I subsequently additionally count on it to be too bullish, equally to the forecasts made by the people above. Below is the plot that I printed:
The identical plot can now be crammed in with newer worth information:
Again, it could seem that bitcoin’s worth extra carefully follows a trajectory with diminishing returns. I subsequently count on the value to transfer additional and additional away from the S2F forecasts in the long run.
The extra mathematically-inclined reader is perhaps in an article I coauthored with InTheLoop which explains in additional element how the form of the S2F worth curve doesn’t match the precise worth information effectively.
The widespread Twitter deal with s2fmultiple tracks how the value is performing in contrast to the S2F forecasts. The metric is referred to because the S2F a number of. A a number of higher than 0 signifies that the value is increased than the S2F a number of, and vice versa.
The historical past of the S2F a number of up to now seems just like the under plot. There have usually been excessive values earlier than 2015, however not a lot after that. This is a touch that the value is not fairly catching up to the S2F mannequin forecasts (and in addition that the form of the S2F worth curve doesn’t match precise worth information effectively).
Source: s2fmultiple
By evaluating my very own power-law hall of progress forecasts to the S2F mannequin, I’m in a position to compute the trendline of how I count on the S2F a number of to evolve sooner or later:
Conclusion
The S2F mannequin has been closely criticized, usually unfairly. I’m extremely assured that the S2F mannequin will fail to predict bitcoin’s worth adequately, however my fundamental argument is merely that the form of the S2F worth forecasts is incorrect and too bullish. The S2F mannequin forecasts nondiminishing progress, which is not justified by empirical observations, which as a substitute strongly trace at diminishing progress.
This doesn’t imply that we must always really feel dissatisfied. Bright days lie forward for the value of bitcoin. In my original article I’ve forecast a worth of $100,000 per bitcoin no sooner than 2021 and no later than 2028, and $1,000,000 per bitcoin no sooner than 2028 and no later than 2037. I nonetheless count on these forecasts to come true.
This is a visitor put up by Christopher Burger. Opinions expressed are solely their personal and don’t essentially mirror these of BTC Inc or Bitcoin Magazine.
