Looking forward, our analysis identifies a substantial but not unprecedented overvaluation in the price of Bitcoin, suggesting many months of volatile sideways Bitcoin prices ahead from the time of writing, March In , pseudonymous Satoshi Nakamoto introduced the digital decentralized cryptocurrency, Bitcoin [ 1 ], and the innovative blockchain technology that underlies its peer-to-peer global payment network.
Fuelled by the rise of Bitcoin, a myriad of other cryptocurrencies have erupted into the mainstream with a range of highly disruptive use-cases foreseen. Cryptocurrencies have become an emerging asset class [ 3 ]. At the end of , the price of Bitcoin peaked at almost 20 USD, and the combined market capitalization of cryptocurrencies reached around billion USD. While many pundits have claimed that Bitcoin is a scam and its value will eventually fall to zero, others believe that further enormous growth and adoption await, often comparing to the market capitalization of monetary assets, or stores of value.
By comparing Bitcoin to gold, an analogy that is based on the digital scarcity that is built into the Bitcoin protocol, some markets analysts predicted Bitcoin prices as a high as 10 million per Bitcoin [ 4 ]. But that upper bound is awfully big. There is an emerging academic literature on cryptocurrency valuations [ 6 — 16 ] and their growth mechanisms [ 17 — 19 ].
Naturally, relationships exist between Bitcoin value, adoption and online activity searches, tweets, etc. Macro-economic variables should also determine the attractiveness of the cryptocurrency e. However, existence of large bubbles and crashes—being radically non-stationary with nonlinear tipping point dynamics—makes modelling these mechanisms difficult and risky with stationary linear models and conventional econometric techniques. However, as has been proposed by former Wall Street analyst Tom Lee [ 4 ], an early academic proposal [ 22 ], by now widely discussed within cryptocurrency communities, is that an alternative valuation of Bitcoin can be based on its network of users.
In the s, Metcalfe proposed that the value of a network is proportional to the square of the number of nodes [ 23 ]. This may also be called the network effect, and has been found to hold for many networked systems. Although it seems relatively obvious that bubbles exist within the cryptocurrency market, in finance and economics, the possibility of financial bubbles is often excluded based on market efficiency rationalization, 4 which assume an unpredictable market price, for instance following a kind of geometrical random walk e.
By sharp contrast, Didier Sornette and co-workers claim that bubbles exist and are ubiquitous. Moreover, they can be accurately described by a nonlinear trend called the Log-Periodic Power Law Singularity LPPLS model, potentially with highly persistent, but ultimately mean-reverting, errors. The LPPLS model combines two well-documented empirical and phenomenological features of bubbles see [ 28 ] for a recent review :.
Such log-periodic fluctuations are ubiquitous in complex systems with a hierarchical structure and also appear spontaneously as a result of the interplay between i inertia, ii nonlinear positive and iii nonlinear negative feedback loops [ 32 ]. The model thus characterizes a process in which, as speculative frenzy intensifies, the bubble matures towards its endogenous critical point, and becomes increasingly unstable, such that any small disturbance can trigger a crash. This has been further formalized in the so-called JLS model where the rate of return accelerates towards a singularity, compensated by the growing crash hazard rate [ 30 , 33 ], providing a generalized return—risk relationship.
We emphasize that one should not focus on the instantaneous and rather unpredictable trigger itself, but monitor the increasingly unstable state of the bubbly market, and prepare for a correction. Finding evidence of positive feedbacks between price and online activity, potential for bubble formation was suggested. However, the model focuses on moderate short-term effects, and integrates to produce a linear price trend—neither producing large bubbles, nor a justified fundamental value.
Notably, in [ 36 ] it was claimed that the fundamental value of Bitcoin is zero. Further, explosive unit-roots have been detected in the Bitcoin value e. These tests may identify bubbles—insofar as bubbles can be explained by a consistent mildly explosive unit-root, while perhaps also allowing for a log-linear trend—but are not specific [ 40 ], and have limited descriptive and predictive power.
When both measures coincide, this provides a convincing indication of a bubble and impending correction. If, in hindsight, such signals are followed by a correction similar to that suggested, they provide compelling evidence that a bubble and crash did indeed take place.
This paper is organized as follows. On this basis, we identify a current substantial but not unprecedented overvaluation in the price of Bitcoin. In the second part of the paper, we unearth a universal super-exponential bubble signature in four Bitcoin bubbles, which corresponds to the LPPLS model with a reasonable range of parameters. The LPPLS model is shown to provide advance warning, in particular with confidence intervals for the critical bursting time based on profile likelihood.
An LPPLS fitting algorithm is presented, allowing for selection of the bubble start time, and offering an interval for the crash time, in a probabilistically sound way. We conclude the paper with a brief discussion. From figure 1 , we indeed see a surprisingly clear log-linear relationship. The points becomes darker as time progresses, and the three latest crashes are indicated by coloured points, and arrows indicating the size of the correction. The generalized Metcalfe regression is given in solid red, and with slope forced to be 2 given by the dashed red line.
A scaled Bitcoin market cap is overlaid with the grey line. The red and dashed yellow lines are the nonlinear regression fits of active users, fitting on different time windows. It should be noted that this regression severely violates the assumption that the errors be independent and identically distributed, as there are persistent deviations from the regression line.
This statement deserves to be made in more salient terms: the residuals are in fact the bubbles and crashes! This is the focus of the second part of this paper. Ignoring this egregious violation of the so-called Gauss—Markov conditions is well known to give the false impression of precise parameter estimates. Further, endogeneity is an issue, as the number of active users may determine market cap in the long term, but large fluctuations in market cap can also plausibly trigger fluctuations in active users on shorter time scales figure 1.
We address this by smoothing active users, 10 assuming that this will average out the effects of short-term feedback of market cap onto active users. A multiplier effect is also a plausible consequence of this endogeneity: a jump in user activity causes an increment in market cap, which triggers a smaller jump in user activity, feeding back into market cap, etc. Therefore, we do not claim to isolate the effect of a single increment in active users on market cap, and do not need it.
Of course, one may add other variables to the regression, which further characterize the network, such as degree of centralization, transaction costs, volume, etc. However, the actual volume value of authentic transactions for instance is not only difficult to know, but, in general financial markets, is known to be highly correlated with volatility, of which bubbles and bursts are the most formidable contributors, and may therefore be too endogenous to soundly indicate a fundamental value.
Looking at figure 1 , a clear and important feature is the shrinking growth rate of active users, which we model by a relatively flexible ecological-type nonlinear regression. As in the case of the generalized Metcalfe regression, here there is clear structure in the residuals, as feedback loops develop between the number of active users and price during speculative bubbles.
We opt to fit the curve 2. More generally, within the sample, the fitted curves are similar, but, beyond the sample, differences explode such that there are 4 orders of magnitude difference between the predicted carrying capacities. Here, model uncertainty dominates uncertainty of estimated parameters.
There is also likely to be some non-stationarity and regime-shifts as the Bitcoin network evolves and matures, contributing another level of uncertainty in the long-term extrapolation of our models. Therefore, precise inference based on a single model—notably omitting any limitation imposed by the physical Bitcoin network—is misleading, and long-term predictions effectively meaningless.
However, smoothing of past values is not problematic, and short-term projections may be reasonable. Also, using smoothed active users, the local endogeneities—where price drives active users—are assumed to be averaged out.
The OLS estimated regression, by definition, fits the conditional mean, as is apparent in figure 2. Therefore, if Bitcoin has evolved based on fundamental user growth with transient overvaluations on top, then the OLS estimate will give an estimate in-between and thus above the fundamental value.
For this reason, support lines are also given, and—although their parameters are chosen visually—they may give a sounder indication of fundamental value. In any case, the predicted values for the market cap indicate a current overvaluation of at least four times. In particular, the OLS fit with parameters 1.
Further, assuming continued user growth in line with the regression of active users starting in , the end of Metcalfe predictions for the market cap are 77, 39 and 64 billion USD, respectively, 14 which is still less than half of the current market cap. These results are found to be robust with regards to the chosen fitting window.
Comparing Bitcoin market cap black line with predicted market cap based on various generalized Metcalfe regressions of active users. The remaining lines plug smoothed active users non-parametric up to and the nonlinear regression starting in to project beyond into the generalized Metcalfe formula with different parameters: the smooth green line for the estimated coefficients 1. The lower inset plot with grey line is the price per bitcoin in USD. On this basis alone, the current market looks similar to that of early , which was followed by a year of sideways and downward movement.
Some separate fundamental development would need to exist to justify such high valuation, which we are unaware of. Using the generalized Metcalfe regression onto smoothed active users as well as its support lines, one can identify in figure 2 four main bubbles corresponding to the largest upward deviations of the market cap from this estimated fundamental value.
These four bubbles in market cap are highlighted in figure 3 , and detailed in table 1 —in some cases exhibiting a fold increase in less than six months! In all cases, the burst of the bubble is attributed to fundamental events, listed below, in particular for the first three bubbles, which corrected rapidly at the time of the clearly relevant news. The fourth and very recent bubble was much longer, and it is plausible that the main news there was really the 20 USD value of Bitcoin, i. Upper triangle: market cap of Bitcoin with four major bubbles indicated by bold coloured lines, numbered, and with bursting date given.
Lower triangle: the four bubbles scaled to have the same log-height and length, with the same colour coding as the upper, and with pure hyperbolic power law and LPPLS models fitted to the average of the four scaled bubbles, given in dashed and solid black, respectively. Bubble statistics. The bubbles correspond to the numbering in figure 3.
Bubble 5 corresponds to approximately the last six months of the fourth bubble, and will be used in the next section. The price data for Bitcoin is from Bitstamp, in USD, hourly from 1 January to 8 January ; the Bitcoin circulating supply comes from blockchain. Of particular interest here is that, although the height and length of the bubbles vary considerably, when scaled to have the same log-height and length, a near-universal super-exponential growth is evident, as diagnosed by the overall upward curvature in this linear-logarithmic plot lower figure 3.
And in this sense, like a sandpile, once the scaled bubble becomes steep enough angle of repose , it will avalanche, while the precise triggering nudge is essentially irrelevant. Gox closes. T 1 is the starting time, and t c the stopping or the so-called critical time by which the bubble must burst. The window T 1 , T 2 needs to be specified, with selection of the start of the bubble T 1 often being less obvious.
In this case, generalized least squares GLS provides a conventional solution, which has been used with LPPLS [ 45 — 47 ] and, if well specified, has optimal properties. Here, we opt for a simple specification of the error model, being auto-regressive of order 1, Here, we focus on t c , the critical time at which the bubble is most likely to burst. The sample is taken at equidistant points. Given our proposed fundamental value of Bitcoin based on the generalized Metcalfe regressions presented above, we define the Market-to-Metcalfe value MMV ratio.
In particular, the parameters of the hyperbolic power-law and LPPLS models fitted on the MMV ratio data, for the full bubble lengths, are given in table 4. For the different bubbles, the key nonlinear parameters fall within similar ranges, and calibration of t c is accurate. The apparent bubbles, which radically depart from the fundamental level 1, are coloured and given in table 1 as bubbles 1—3 and 5.
The three thin vertical black lines gives the rightmost edge of the 95, A lower bound for m of 0. First, we provide a simple indication of this potential with two additional sets of fits for each bubble: fitting with bubble data up to 95 and That is, point estimates and confidence intervals are consistent with the true bursting time, noting that t c is in theory both the most probable and latest time for the burst of the bubble [ 30 , 33 , 42 ], as the market is increasingly susceptible as it approaches t c , and can therefore be toppled by bad news.
The three columns are for fits on data up to T 2 , being 95, Next, a more extensive demonstration of the predictability is done for the case of the recent large bubble, summarized in figure 5. At each T 2 from 1 year before the turning point, here specified as 17 December , to two weeks beyond a prediction of the critical time is made, and confidence interval calculated. Further, for each T 2 , a range of bubble starting times, T 1 from to days before the turning point , are considered, and estimates for the critical time combined.
Furthermore, the uncertainty reduces and leads to a strong alarm about two weeks before the eventual turning point. In the upper panel are the aggregated The intervals are shifted such that the origin is the time of the realized tipping point, and the diagonal line defines the lower bound for the prediction i. In doing so, we were able to diagnose four distinct bubbles, being periods of high overvaluation and LPPLS-like trajectories, which were followed by crashes or strong corrections.
Although the height and length of the bubbles vary substantially, when scaled to the same log-height and length, a near-universal super-exponential growth is documented. This is in radical contrast to the view that crypto- markets follow a random walk and are essentially unpredictable. Further, in addition to being able to identify bubbles in hindsight, given the consistent LPPLS bubble characteristics and demonstrated advance warning potential, the LPPLS can be used to provide ex ante predictions.
For instance, a reasonable confidence interval for the critical time indicates a high hazard for correction in that neighbourhood, as any minor event could topple the unstable market. The success of such an approach was shown for the large bubble. Of course, false positives and misses are possible, but somewhat ambiguous in view of the limited number of large bubbles—and also dependent upon the specific decision rule being used potentially including human judgement.
Further, massive exogenous shocks, although rare, could occur at any time, and the LPPLS model can provide no warning there. Further, our Metcalfe-based analysis indicates current support levels for the Bitcoin market in the range of 22—44 billion USD, at least a factor of four less than the current level. On this basis alone, the current market resembles that of early , which was followed by a year of sideways and downward movement.
Given the high correlation of cryptocurrencies, the short-term movements of other cryptocurrencies are likely to be affected by corrections in Bitcoin and vice versa , regardless of their own relative valuations. The likelihood ratio test p -values of bubbles 1—3 and 5, with the pure hyperbolic power-law fit as the null, for the first sub-table are 0.
About years ago, Aristotle was probably the first to argue that money needed a high cost of production in order to make it valuable. In other words, according to Aristotle, the larger the effort to create new money, the more valuable it is. This was later elaborated into the labour theory of value, starting with Adam Smith, David Ricardo, and becoming the central thesis of Marxian economics.
Nowadays, this concept is archaic and it is well understood that money is credit e. It is thus puzzling that cryptocurrencies with proof-of-work designs, which aim at revolutionizing money and exchanges between individuals, use a very old and obsolete concept that has been mostly abandoned in economics. Thus, market crashes result from novel very negative information that gets incorporated into prices [ 26 ].
Limitations: it is difficult to know the true number of active users, in particular because a single user can have multiple addresses that, to an outsider, cannot be distinguished from addresses belonging to multiple users. Moreover, bitcoin. Depending on to what extent this advice is followed, this measure is thus an unclear mix between the number of daily users and the number of daily transactions their activity.
In this case, the causal link between active users and market cap is assumed. On the other hand, it could provide an underestimate of the number of active users if the typical user does not transact daily. This does not seem realistic. The time span was also transformed to 0,1. Predicted values transformed back to original scale for the first day of each year from to in millions of active users, and percentage standard error are 0.
Finally, the estimated carrying capacity is 2. And a massive carrying capacity of 9. For instance, it was revealed on 2 March that Nobuaki Kobayashi, bankruptcy trustee for Mt. Given the regression based de-trending, truly long memory in the errors is not expected, and the auto-correlation of residuals is seen to decay clearly faster than a power law. Further, Dickey—Fuller tests tend to reject that the residuals are unit-root, strongly when significant log periodic oscillations are fit.
These fits are thus not for prediction purpose but for assessing the quality of the hyperbolic power-law versus LPPLS models. With multiple of these curves one for each T 1 , the aggregation is done by simply averaging the curves and taking as the confidence interval the regions where the aggregate curve is above zero.
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|00076 btc to usd||Finance 25— MacDonell A. There is also likely to be some non-stationarity and regime-shifts as the Bitcoin network evolves and matures, contributing another level of uncertainty in the long-term extrapolation of our models. Of particular interest here is that, although the height and length of the bubbles vary considerably, when scaled to have the same log-height and length, a near-universal super-exponential growth is evident, as diagnosed by the overall upward curvature in this linear-logarithmic plot lower figure 3. Read article Calculator.|
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