A Non-Divergent, Long-term, Residual Supply-driven Bitcoin Forecast

Stephen Perrenod

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Jan 26, 2020 · 12 min read

Abstract

I introduce a two-parameter, convergent, residual supply-driven Bitcoin long-term forecasting model that is an alternative to PlanB’s popular but mathematically divergent two-parameter scarcity-driven Stock-to-Flow model. Over a 50-year time span, the models differ by 12 orders of magnitude in their Bitcoin price forecast. The Future Supply model is a linear correlation of the log of market capitalization with the remaining Bitcoin supply yet to be produced (future supply) at any epoch.

The mathematical form of the model is inherently convergent, whereas the Stock-to-Flow model is a power law of an exponential (since stock-to-flow is exponential in block height), and thus is inherently divergent.

The Future Supply model exhibits R² = 0.92 with historical data on quarterly intervals of 13,125 blocks (1/4 of a Block Year). The residuals are well-behaved but skewed positively. The standard deviation of the residuals is 0.374 in base 10 logarithm terms, reflecting Bitcoin’s high volatility.

Using the best fit parameters, the model converges toward an asymptotic value for market cap of $1.63 trillion (in 2020 dollars) and an asymptotic price of $77,500. However, these values have high uncertainty due to the limited price history and high volatility of Bitcoin.

One should be able to easily distinguish whether the Future Supply model or the Stock-to-Flow model is more accurate by the mid-2020s.

PlanB’s Stock-to-Flow model has a large following

Scarcity and security are the fundamental drivers of value for Bitcoin.

PlanB’s Stock-to-Flow (S2F) model is the most popular scarcity model for projecting Bitcoin’s future value. It relates price to S2F as approximately a cubic power law of the latter. S2F increases inexorably with each block (a) because the stock is growing relative to the current block reward, and (b) especially, because every 210,000 blocks the block reward for Bitcoin miners is cut in half. These Halvings occur at nearly four-year intervals.

S2F has the advantages of being a leading indicator and of providing a relatively long-term view. It is fundamentally motivated by increasing scarcity driving value and has a successful analog in the precious metals market (gold has a stock-to-flow of about 60 and Bitcoin will approximately match this after the May 2020 Halving). S2F manifests a good correlation historically and has passed attempts to falsify the model for co-integration of the stock-to-flow relationship with price.

The major disadvantage is that it is intrinsically divergent because price is a (high) power law of S2F, and S2F heads inexorably toward infinity (~2³⁶ at the last halving, followed by infinity) in an exponential fashion. It will already be in uncharted territory by 2024, as S2F exceeds 100. In the limit of later Block Eras, the forecasted market cap increases a factor of 8 after each additional Halving.

In a recent article (https://medium.com/the-capital/modeling-bitcoin-value-three-methods-fb4093b2de71) we compared a difficulty based model(as a security proxy) to PlanB’s scarcity model; the difficulty model is less successful than the stock-to-flow model. We also evaluated a power law of the block number model. This has a correlation similar to S2F but does not explicitly consider either scarcity or security, so lacks fundamental drivers and seems backward looking only, although implicitly growing scarcity and security would be reflected in this model as well.

Modeling scarcity with a convergent formula based on residual supply

Here we present an alternative scarcity model that is a function of only the remaining supply. It is designed to be convergent in the long-term, with a negative correlation between remaining supply (of not yet minted Bitcoins) and the market cap.

We have used quarterly data since Block 105000, that is, subsequent to the first two elapsed Block Years. Before that time, price data was quite sparse.

A Block Year consists of 52,500 blocks and in practice averages about two weeks less than a calendar year. A Block Quarter is 13,125 blocks in length. Currently, almost 12 Block Years have elapsed and the next Halving in mid-May of 2020 marks the completion of 12 Block Years and the end of Block Era 3 and start of Block Era 4. The Satoshi calendar system is described more fully here: https://medium.com/the-capital/living-on-satoshi-time-what-block-is-it-ca30fe307ff6

We looked at both power law and exponential relationships between market cap and future supply. The power law, or log (market cap) vs. log (future supply) model is divergent, since the latter term goes to negative infinity as the remaining supply goes to zero. While such a model has a reasonable R² for historical data, it predicts a market cap of $299 trillion when remaining supply drops to one million. This is equal to all global wealth, and is thus untenable.

However, an exponential relationship of log market cap proportional to future coin supply converges, and our fit has an anti-correlation of -0.96 and an R² of 0.92.

The relationship used is log10 (MC) = b + a*f where f is the future supply, measured in millions of coins. The best fit parameters for a linear regression log10(MC) vs. f on the quarterly dataset are: intercept b = 12.212 and a = -0.3488 for the slope. The parameter b is the asymptotic value for log10 (market cap) as f goes to zero, and corresponds to a market cap of $1.628 Trillion. The slope a is the rate of convergence toward that value.

The relationship can also be written as MC = B*10^(a*f) where B = 10^b, or as MC = B*exp(c*f), where c = 2.303*a. Therefore for the best fit parameters, MC = $1.628 trillion * exp (-0.803*f).

The observed slope parameter a indicates that for every one million reductions in future Bitcoin supply (increase in stock by one million Bitcoins), the market cap increases by 10^(0.349) = exp(0.803) = 2.23 times.

Figure 1 shows the market cap forecast in light blue out to Block Year 17 (2027) and observed quarterly market caps in dark blue since Block Year 2 until the present. The forecast approaches $1 trillion in 2027.

And here is the fundamental ‘physics’ — a second derivative — since the reduction of future supply happens at a slower and slower rate with time, the increase of the market cap slows down as well. In Block Era 2 the future supply was being reduced by over a million Bitcoin per year. Currently, reducing the supply by one million coins requires about 8 years or two Bitcoin eras. By 2031 it will take over 100 years to reduce the future supply by only 1/3 of a million Bitcoin. It takes longer and longer for the market cap to rise in percentage terms in this model.

Another way of saying this is that once the inflation rate drops below 1% (S2F > 100) with 2024 Halving the market cap won’t be very sensitive to the ever lowering inflation rate. The market won’t care as much whether Bitcoin inflation is 0.4% or 0.2% or 0.1%. The differential relative to the dollar inflation supply is what should matter, and a difference of 0.1% or 0.2% will be small in comparison to USD M2 supply inflation of 5% or 6%. And that suggests we should think of this forecast as being in constant 2020 dollars.

The asymptotic market cap for the Future Supply model at roughly $1.6 trillion is about one-fifth of the total gold market cap.

The market cap forecast has been converted to a price forecast using the existing stock of Bitcoins at each point in Figure 2. The asymptotic price for this market cap model is $77,538.

A histogram of the residuals, for actual market cap minus the model prediction, is shown in Figure 3. They are well behaved, with 19 positive and 20 negative values. The standard deviation is 0.374 in log 10 space, corresponding to a factor of 2.37 in either direction. The number of points in the sample of 39 that are greater than one standard deviation is 14, against an expected ~ 12.5. The distribution is skewed positive with excess high observed market caps, as expected with the several strong price run ups over Bitcoin’s history.

However, we have a modest price history, for an unfolding process, with 39 quarterly data points. If we look at subsets of the data there is significant variability in the fitted a, b parameters. If we exclude the first third of the data points, the asymptotic price projection is lowered to $48,321 and R² is just 0.69. In particular, the second Block Era from Block Year 4 until Block Year 8 has a shallower slope and much smaller projected asymptotic price of only $8766 for that portion of the data only, while the third Block Era (not yet completed) has an asymptotic projected price of $416,056.

As usual with models, more data is needed, and time will tell how useful the model remains in the future. We expect to look at monthly data over the available interval as well to see if that lowers uncertainty, but the second Block Era is clearly much flatter on average by visual inspection.

Comparison of Future Supply and S2F models

Let’s look at a comparison between the mathematically convergent Future Supply model and the mathematically divergent Stock-to-Flow model.

Table 1 (a,b) shows the forecast for the long-term convergent Future Supply model. In Table 1a above we show for the start of each block era, through the next half-century, the block reward, the current supply at a given Block Era, and the future supply for Bitcoin. This is somewhat analogous to existing gold reserves. The difference is that new gold supply is always being discovered or developed, but the future supply of Bitcoin is fully pre-determined.

In Table 1b we show the market cap forecast and price forecast for the Future Supply model (price = market cap/stock or existing supply). Note how the forecast grows rapidly during the first several Halvings, reaching nearly a $1 trillion by the late 2027 or early 2028 Halving. But after that, it requires three more Halvings to reach $1.5 trillion and then flattens out toward an asymptotic value of $1.628 trillion.

The predicted long term price for the Future Supply model is around that of the S2F model in the early 2020s but never exceeds $100,000 for the regression on the full data set. However the price volatility is a factor of greater than two (one standard deviation), so $100,000 could be readily breached from time to time.

In Table 1b we also show the stock-to-flow for each era. The forward looking S2F is given by the identity 4*(2^E-2), where E is the block era shown in column 2.

And we show the forecast price with the S2F model using PlanB’s original March 2019 Medium article with a price model of 0.4 * S2F³. He also has published another variant of the model based on yearly data with a 3.3 power law index, that one would rise even more steeply. The market cap is nearly a straight line in the log price vs. Block Era; in the limit for larger Block Eras, the price and market cap rise a factor of 8 as each new era is reached.

What we see is that the S2F model becomes unrealistic during the current decade, as price exceeds $10 million before the end of the decade and thus market cap would exceed $200 trillion, or the value of all real estate on the planet.

In Figure 4 we show the price forecasts for the S2F model and the Future Supply model over the next half-century (Block Era 17, year 2070). The models differ by 12 orders of magnitude by Block Era 17. The S2F model price forecast exceeds $1 billion prior to Block Era 9 and $1 trillion prior to Block Era 12. In practice, by the middle of the 2020s, one should be able to readily determine which of the two models is closer to reality.

Model limitations

As discussed above, the parameter fits for the Future Supply model have substantial uncertainty.

The Future Supply model does not correct for inflation in the US dollar supply. While consumer inflation runs at 2% approximately, the increase in M2 supply is around 5% to 6% per annum. So the forecasting market caps should be considered as in 2020 constant dollars. For example, if M2 grows at 5% over the next 15 years, the USD money supply would double, and the 2035 forecast would be nearly $3 trillion in 2035 dollar terms.

The model does not explicitly consider breakthrough adoption, although it does implicitly reflect the historically growing population of Bitcoin users and HODLers. If central banks start to add Bitcoin to their reserves, or if gold loses favor to Bitcoin as the default asset-based store of value, then the market cap could be driven to something comparable to the total value of gold ($8 trillion) or even some fraction of global monetary reserves ($20 trillion).

The idea of matching total global wealth of $300 trillion seems completely unreasonable to me, both given a convergent model, and considering that one does not need the money supply to reflect all wealth. The money supply has to support the GDP turnover and indeed global money supply (M2-like) and global GDP are roughly comparable at around $90 trillion. Reserve money is a small fraction of total money supply.

Summary

The currently successful stock-to-flow model from PlanB predicts a single Bitcoin would be worth over $1 trillion by 2050 and over $1 quadrillion by 2070.

Clearly one needs a more realistic model for the long term, and we submit the Future Supply model for your consideration. Its advantages include that it is also scarcity based, that it has historical performance comparable to the stock-to-flow model, and that it converges even at the end of this century. It is also very simple, with only two parameters, the ultimate market cap and the rate of convergence with shrinking supply.

Such a flattening out of Bitcoin’s rate of price increase and decreasing volatility seems fully consistent with Satoshi’s vision of a stable global money, one that can act as the decentralized reserve for a future scalable payments system. Second-layer and side chains enhancing the main blockchain will address the scalability issue.

A more sophisticated Future Supply model would incorporate inflation in the dollar supply and perhaps also a parameterization of the adoption rate for Bitcoin. But it is not the velocity of usage that imparts value (high velocity depletes value), it is the willingness to hold in reserve as a store of value.

Apologies to those hoping for $1 million Bitcoin in the next decade, but the forecast returns are still exceptionally healthy out to the start of block era 8, in the middle of the decade of the 2030s. So keep stacking sats. Table 2 shows expected returns in percentage terms per Block Era and per Block Year.

It’s still the hardest money ever created. Bitcoin is a store of value, a money alternative, so it should tend toward stability. If the dollar remains a global reserve currency (perhaps one of several) then the dollar value of Bitcoin should be reflective of relative inflation rates and relative interest rates (as Bitcoin also develops a lending economy).

We expect by the mid-2020s to see whether this model or the Stock-to-Flow model is performing in a more realistic fashion.

Appendix: Quarterly Data

These tables contain the data behind the Future Supply model. The final column in Tables 3a and 3b below shows the differential of the observed market cap and the model predicted market cap. (Log10 of predicted market cap = log10 of observed market cap less the differential.)