Valuing a Cryptocurrency Network
Not everyone understands crypto, but everybody has an opinion on it.
The value of all cryptocurrency networks is not zero, despite strident statements to that effect by many prominent economists. At no point over the past ten years has a bitcoin been worth zero, and it won’t be worth zero tomorrow. A bitcoin doesn’t generate any cashflows, and as such it trips up many who view it through the more literal lens of received economic wisdom. Similarly, the internet doesn’t have any cashflows, yet few would claim it has no value; instead, it provides utility and the ability to generate future cashflows. What a bitcoin represents is the ability to participate in a global ledger that allows individuals to transfer value to other network participants, cheaply, quickly, and without the need to ask for permission. That by itself is a form of value and has some associated dollar value in utility for the individual.
It stands to reason that this value inherent in the network ought to be somehow dependent on the willingness of others to exchange a bitcoin for goods and services. Hence the value of bitcoin should depend on the number avenues that exist through which it could be transferred, with a proxy for this utility being the number of network participants. Hence, we could propose that the value of the network is proportional to the number of connections between individual participants in the network.
This network valuation rule of thumb is known as Metcalfe’s Law and has been previously used to value internet companies like Facebook and Tencent, whose primary asset is the reach of their social media networks – something quite intangible. It has been proposed that bitcoin and other cryptocurrencies should exhibit scaling properties proportional to the square of the number of users, and here we present a small selection of results of our research into Metcalfe’s law and briefly discuss some other models en-vogue amongst industry commentators.
Should some simple model for network fundamentals prove useful, this raises the possibility of characterizing strong deviations from network fundamentals and searching for universal properties indicating when the market may be irrationally priced. The history of cryptocurrencies has been one of enormous and dramatic bubbles followed by steep and prolonged collapses. Bubbles, although disconcerting for investors, have been an important part of the history of bitcoin and crypto as a whole. In fact, the entire history of bitcoin could be characterized as a series of ever-expanding bubbles and collapses. And as counterintuitive as it may seem, had this bubbly price action not been a feature of bitcoin then bitcoin would have remained, as it started, as an obscure economic toy with – as many an economist would have forecasted – no utility and therefore no value. Rather than a negative, booms and busts are exactly what bitcoin needed to bootstrap itself out of obscurity and will itself into having value. The sequence goes like no bubbles -> nobody would have heard of bitcoin -> no network -> no utility -> no value. Every bubble brings more people, builds out the network, adds legitimacy, increases valuation.
Given that bubbles are an inherent feature of crypto – then understanding their dynamics should yield interesting results. There are a number of models borrowing techniques from statistical physics which can be applied in an attempt to characterize the various bubbles that have permeated the space since inception. Forecasting exactly when a bubble will burst is probably a hopeless endeavor but understanding their behavior may still yield useful results. This is an active area of research at Firinne Capital.
Metcalfe’s Law
“All models are wrong, but some are useful.“– George Box
We want to see if Metcalfe’s Law holds for bitcoin. First, we need a proxy for the number of active users of any given network. Some previously published articles have used the cumulative total of addresses present on the blockchain, then regressed the market cap onto this metric. Arriving at incredibly high r-squared values, the appropriateness of Metcalfe's Law is affirmed, and the analysis typically ends there.
There are issues with this approach, since the time series in the regression is nonstationary; the number of addresses on the blockchain grows almost constantly. Instead, we follow the approach of using daily active addresses as our proxy for the number of users in the network. This approach is far from perfect as the time series used in the regression still exhibits nonstationarity with heteroskedasticity in variance and steadily increasing number of users over time. This issue is partially mitigated by using rolling windows in the analysis below, however we should remain cognizant of the limits of what can be inferred from the regression especially with asserting that number of users is a predicator of market cap and hence price. An interesting avenue of research is to reverse the question and ask to what extent price appreciation in fact drives user activity rather than assuming the causal link runs in the opposite direction. Clearly, as the price of bitcoin rises, hype and headlines follow and therein lies the origin of a bubble formation.
Next, we don’t presuppose Metcalfe's Law (ML), instead we assume that market cap grows as some to-be-determined power of the number of network participants.
Where β0 is exactly 2 for Metcalfe’s Law. We refer to the modification where β0 is to be determined as the Generalized Metcalfe’s Law (GML).
Figure 1 - Log of market cap vs. log of daily active addresses
Figure 1 plots the log of market cap against log of daily active addresses for a variety of starting dates for the regression through to the present day. Depending on what dates are used for the regression we can find quite a variety of possible powers for the GML.
Running a regression from the start of 2011 to the present day gives a beta of 2 with an r-squared of 0.9, remarkably good, however clearly the bitcoin network will have transitioned through several qualitative shifts in network composition over this time period – from early tech enthusiasts, to early adopters and international remittance to mainstream financial acceptance (or at least acknowledgement) and popular speculation. Therefore, it's difficult to make the case for stationarity of the data and, indeed, standard test for stationarity suggest that it is not.
Figure 2 - Ratio of actual market cap to market cap as predicted by Generalized Metcalfe’s Law
Some points of note. Notice that the slope, hence the power in ML steepens until the most recent bear market where it collapses below one. This would imply that the price appreciated too quickly since 2021 relative to what could be explained by ML and even since the price dropped at the end of 2021 the price has still exceeded that predicted by ML. Figure 2 plots the ratio of actual market cap to that predicted by Metcalfe's Law. It’s clear that historically the ratio was reasonably stable and close to one, however in recent years the realized market cap has greatly exceeded that predicted by Metcalfe. Examining the plots below it seems that ML does provide something of a floor on the market cap, however the model alone is not sufficient to support the current price or the price throughout most of the past 18 months.
Figure 3 and Figure 4 below show the realized market cap against expected market cap as predicted by Metcalfe’s Law over time. A choice of model is also used to smooth the number of daily active users and a market cap projection that uses that curve as an input to Metcalfe’s Law is included on the graph. It is easier to see from the log graph (Figure 4) that Metcalfe’s Law has provided a reasonably good fit to the data over the entire lifetime of bitcoin’s existence.
Figure 3 - True market cap data graphed alongside GML projected market cap using a smoothed curve and a curve using raw user data.
Figure 4 - The same curve as Figure 3 on a log scale on the y-axis.
The conclusion of these graphs is that Metcalfe’s Law provided a good fit to the data up until 2020. After 2020 the number of daily active addresses on chain appears to level out or rise much more slowly. Despite this the market cap continues to grow substantially from 2020 onwards. Therefore, either Metcalfe’s Law no longer applies (at least using daily active users as the input variable for network utility) or bitcoin is currently overvalued by a factor of two. We believe that qualitative changes to how bitcoin is used and stored in recent years have contributed to this divergence. Ideally a better measure of network value than daily active addresses incorporating these shifts in network usage will be found.
Examining Metcalfe’s law in time bounded windows may mitigate some of the issues around time series stationarity. Setting the rolling window of the regression to 500 days we find the graph below. The slope of the regressions appears to oscillate around 2 but has extended periods where it is much larger and even negative.
Figure 5. The slope of the regression using a rolling 500-day window plotted over time
Figure 6. Rolling R-squared of the regression from Figure 5.
Looking at the r-squared, it is clear that there are also extended periods of time where the applicability Metcalfe’s Law cannot be supported from the data time bounded within certain periods of history.
To conclude, Metcalfe’s Law historically has given a good fit to the market cap of bitcoin, with some extended periods where the market cap was significantly below where Metcalfe’s Law would have predicted. Furthermore, during periods of extreme euphoria the market cap greatly exceeded that predicted by Metcalfe’s Law. From 2021 onwards however the relationship, as described by Metcalfe’s Law, between daily active users and market cap seems to have broken down. The market cap rose dramatically over this time period, however the number of daily active users on chain did not. This may be due to alternative methods for bitcoin accumulation, via vehicles like the Grayscale Trust or new users holding their bitcoin in third party custodians which pool their assets in fewer addresses.
Alternative Models
For completeness here we include other models with are commonly cited in the industry as providing a basis for valuation of the bitcoin network. Although commonplace in discussion, these models have even less of a firm footing in logic than Metcalfe’s Law.
Stock to Flow
Most attempts have focused on either the supply or demand side of the equation. One such early attempt tried to ascribe intrinsic value to a bitcoin based on the cost of electricity require to produce it – which made about as much sense as claiming that a hole in the ground has intrinsic value because you had to expend energy to dig it.
Continuing in the vein of supply side focus, another popular model come to the fore in recent years is the Stock to Flow model (S2F) which exclusively looks at the supply side of the equation. The model assumes that the price is proportional to the ratio of the number of bitcoins available on the market to the number of new bitcoins being created each year. Since the supply of bitcoin gets constricted over time according to the “halving” schedule every four years, the ratio forecasts a price appreciation over time. The model itself, as described by its author, makes numerous rudimentary errors and its high explanatory power is almost entirely derived from the misapplication of linear regression. Moreover, since the supply of bitcoin, generated by miners following an algorithm is one of the most predictable series of events in all of finance, it’s difficult to argue that supply shocks have not already been priced in. But like most fatuous models, it looked great, until it didn’t. From 2022 the model has deviated from the trend line, but of course models forecasting that everybody holding a bitcoin will be a millionaire in three years tend to have a remarkable longevity in the public consciousness.
Figure 7 - Stock to Flow
Another valuation projection model is the so-called “Rainbow chart” - it is presented without justification and no claims are made to scientific validity. However, there is some method in the madness, an assumption is made that the network should grow exponentially (or perhaps not at all), and if so then a linear regression on a log of price might be able to project a trend in price that could hold over time. The bitcoin rainbow chart below was calibrated to data up until 2014 – fixing the regression parameters at that date and projecting forward has done a remarkable job of providing price boundaries, of course these boundaries are several orders of magnitude apart, so the utility of these charts is fairly limited! Nevertheless, the price of bitcoin has hardly ever fluctuated outside these bounds and it has managed to forecast market tops and bottoms.
Although the rainbow model has no real basis in any fundamental measure, other than the presumption that the price will grow exponentially over time, it does touch on the important concept that linear growth of these types of networks is probably the least likely assumption. As such we can begin to see the genesis of models like Metcalfe’s Law. This leads to the proposal that the value of these networks may be likely to grow according to a power law of some underlying fundamental metric. And if a bitcoin is imagined as a share in a universal ledger which gives the holder the utility of being able to send it to any other network participant effectively instantly and frictionlessly, then clearly the value of the ability to participate in that network should grow in accordance with some proportionality with the number of users that you could potentially transact with. Hence Metcalfe’s Law.
Future extensions of this work
This work will be extended by examining a model attempting to explore commonalities in previous bubbles in cryptocurrency and extract universal properties that can be used to characterize how frothy the market is at any point in time and potentially forecast when a bubble might burst.
Another promising avenue would be to seek alternatives metrics to daily active addresses as a proxy for network users and hence utility.
References
Metcalfe's law and log-period power laws in the cryptocurrencies market (degruyter.com)