# Modeling Bitcoin Value: Three Methods

*Price-Time, Price-Difficulty, Price-Stock2Flow*

**Bitcoin’s Double Feedback Loop**

The beauty of Bitcoin is that its security and scarcity work together in a self-reinforcing pair of cooperating feedback loops as shown in the figure below.

Between Halvings, Bitcoin becomes scarcer since each block’s reward is a decreasing fraction of the outstanding stock (currently around 18 million bitcoins).

But it is even scarcer due to its regularly scheduled Halvings, which occur each 210,000 blocks, almost four years apart. These cut the block reward in half and have the long term outcome of driving the inflation rate in Bitcoin to zero.

In the upper loop in the diagram, we indicate both the effect of the Halvings, and the continual effect of each new bitcoin being a smaller and smaller fraction of the outstanding supply. This increased scarcity drives prices higher.

In the lower loop, we are indicating that higher prices encourage more mining power, more hashing power, and that increases security. Increased security drives prices higher.

And thus scarcity increases security.

Increases in both security and scarcity work to enhance the price over the longer term.

**Basis: Block Time**

I suggest that it is much more natural, appropriate, and accurate to do price, market cap, hashrate, difficulty, transaction value, stock-to-flow, and other studies on a Bitcoin Blockchain calendar basis, when examining correlations and co-integrations of these quantities against a time-related variable.

These results can then be readily converted to regular Gregorian calendar time after the analysis for presentation purposes and for further analysis in calendar time terms.

In the Bitcoin calendar system, a Block Year is 52,500 blocks and a Block Era of four years is 210,000 blocks. (See the article Living on Satoshi Time for more detail).

Bitcoin’s value derives from its security and its scarcity. We take a look at a model of each.

**Security via Difficulty**

For security, one could use hashrate, but it turns out that difficulty is a smoother relationship, since it is only adjusted each 2016 blocks. Hashrate is noiser, but generally trends along with difficulty.

In fact, both hashrate and difficulty have risen with approximately the 12th power of Block time over Bitcoin’s lifetime.

In the article CryptoSupers Smash Moore’s Law we looked at the growth in hashrate over the past 9.5 Block years and found it has been growing with the 12th power of the Block time (or, equally, the block height).

The underlying algorithmic driver for hashrate is Bitcoin’s difficulty adjustment. This happens every two Block weeks (each 2016 blocks). A graph of difficulty is much smoother than for hashrate or even the hashrate weekly averages. Difficulty for the next two weeks is increased (decreased) if blocks are being produced faster (slower) than 10 minutes on average. The average time in seconds for a miner to find a block first is, time = difficulty * 2³² /hashrate.

*Figure 2: log 10 Difficulty vs. Block Year (regression against log Block Year)*

Let us look at a regression for Difficulty with Block time.

Figure 2 shows the log base 10 of the difficulty versus time in Block years, at quarterly intervals. There is a steep rise in years 4 and 5 due to the rapid adoption of ASICs that replaced mining via GPUs, quickly pushing difficulty up by several orders of magnitude.

Table 1 below shows the half-yearly data, however, quarterly data were used in the regression analysis.

A linear regression of all data in log — log space (log difficulty vs. log — Block year) indicates a power law relationship with index 12.38. However if we restrict ourselves to Block year 6 and above (post the ASIC invasion), the power law index is 10.51 (R² = 0.975). Difficulty seems to be very well correlated with the block height or Block year.

Price and difficulty are in a positive feedback loop long term. A higher price attracts more miners, that pushes hashrate up, which causes the difficulty to adjust upwards in order to keep the block time near to 10 minutes. Technological improvements improve miners’ margins and also push hashrate up, which causes difficulty to rise.

So how does price correlate with difficulty?

Woobull likes to look at a Bitcoin Difficulty ribbon composed of multiple moving averages of difficulty. Visual inspection of this chart suggests that strong price pullbacks lead to flattening of the difficulty for some period of time, perhaps as long as a year. These can be buying opportunities because weak miners have been forced out, restoring stability to the market.

This may be the closest we have to a supply — demand relationship since Bitcoin supply does not respond to price except on a very short term basis. The rate of supply emission is essentially fixed with the block reward rate, since difficulty auto-corrects block times toward 10 minutes’ duration.

If we regress log price, starting at Block year 6, against log difficulty, using quarterly data, we find a slope of 0.646, with an R² = 0.919. We can use the difficulty vs. block year relation (power law with index of 10.51) and the price vs. difficulty relation (power law of index 0.646) to arrive at this forecast of price based on the security element, with difficulty as the proxy for security.

The fit to log price for the interval from Block year 6 to Block year 11.5 has a standard deviation in the error terms of 0.289, for a one sigma price variation of a factor of 1.95. Thus, if the difficulty to Block year and price-to-difficulty relations hold going forward, one would expect to see price in the interval [8,30.3] x $1000 for 68% of the time near Block year 12 and [36.4, 138.1] x $1000 for 68% of the time around Block year 15.

**Regression and Co-integration**

One has to be careful with large R² values from regressions between processes that are not normally distributed; spurious correlations may be seen. Difficulty and price are both increasing strongly with time over the price history. In order to have a valid power law regression between price and difficulty, it is important to look at the order of the processes being compared. How many times must the variable be differentiated to yield a stable, normally distributed series?

Fortunately, both log difficulty and log price appear to be second order processes. For log difficulty, the first (second) order deltas have a mean of 0.2306 (-0.0129) and 5.1% (57.9%) of the values are negative. The first order differences do not look close to normal but the second order differences do.

For log price the respective first (second) order means are 0.114 (-0.01) and percent negative values are 33.3% (47.4%) for first (second) order differences. The first order differentiation seems not quite normal. The second order differences appear normal. Thus co-integration between difficulty and price looks feasible, since both look to be second order processes.

**Scarcity via Stock-to-Flow (Plan B model)**

The best known analytical model for Bitcoin value is Plan B’s stock-to-flow model. The idea is simple, scarcity imparts value. Plan B notes that this works for precious metals including gold, silver, and platinum; even diamonds fit onto the same common power-law curve with index ~ 2.2 (note that PlanB subsequent to his article has corrected the stock-to-flow for silver to a much lower value and added platinum and diamonds, and all still fall on essentially the same curve).

He models Bitcoin’s evolution relative to its ever increasing stock-to-flow (log price to log stock-to-flow) and finds that it follows a power law even steeper than that of precious metals, around ~ 3.3.

Perhaps increasing difficulty (security) is a reason why the power law for Bitcoin has a 50% steeper index than for gold and precious metals. Bitcoin is becoming both scarcer and more secure at the same time, while gold’s scarcity and security are essentially static.

Stock-to-flow is the inverse of inflation, it measures the outstanding stock divided by a years’ production including recycling and any shrinkage of stock. In the case of gold, the stock-to-flow ratio (s2f) is 55, or about a 1.8% inflation rate. This is a steady number in recent years.

Bitcoin has a completely predictable stock-to-flow when measured in Block time. There is some variation relative to calendar time, but Block years are quite close to calendar years, a couple of weeks shorter at present.

The article Bitcoin’s Pre-determined Disinflationary Monetary Policy lays out precisely how the s2f grows with Block time. The s2f at each Halving = 4 x (2^E -2) where E is the Bitcoin era. Currently, we are in Era 3, which began with an s2f of 24. Between Halvings, s2f increases gradually, and then it quantum jumps at Halvings.

At the next Halving, in May 2020, it will be the fourth Era for Bitcoin (post third Halving) and the s2f will jump to 56. This is a large impulse for scarcity.

At that time, the s2f for Bitcoin will match gold’s for the first time in its history, and the inflation rate will drop to 1.8%. But Bitcoin becomes scarcer than gold. In 2024, the s2f will jump to 120 and the inflation rate will drop to 0.83%.

Never in history has there been money this hard, this absolutely scarce.

Using quarterly data in the Block calendar system, I find a power law relationship price ~ s2f ^ 3.25, similar to Plan B’s results. For this analysis, I used the mid-point stock-to-flow (looking back half a year, and forward half a year) rather than the fully forward-looking or the only backward-looking options to measure flow. The R² for this power law regression is 0.926.

More detailed studies of co-integration for the two non-stationary processes, price, and stock-to-flow, have shown that the relationship is valid. The analogy often used is that stock-to-flow is the dog leading its master home (toward higher value) in a deliberate fashion, while the drunk master, tethered to the dog, moves to either side in a constrained random walk.

Actually the concern about a spurious correlation seems overwrought with stock-to-flow. After all, stock-to-flow in Block time is not a random process at all but a completely predetermined calculation. Forward stock-to-flow is given by the Halving formula above at each halving and then increases by 1.0 each Block year until the next Halving (since one year’s constant flow is added to stock). As such, stock-to-flow qualifies as a basis vector for measurement of other more derivative processes such as price, difficulty, and hashrate.

The uncertainty in the price forecast is large, with the standard deviation of the errors around the stock-to-flow model prediction of 0.325 in log price, or a factor of 2.11 in either direction.

You might ask, what about forks from the main Bitcoin chain? Bitcoin Cash and Bitcoin SV are the largest hard forks from the original Bitcoin and nominally have the same scarcity attributes, but because they have substantially less security and are not truly decentralized, the marketplace doubts that they will strictly adhere to the Bitcoin core supply algorithm indefinitely. Clearly they have much less security, with Bitcoin Cash having a 36 times lower difficulty as of this writing, and Bitcoin SV even less.

**Price with Block Time**

Some, like HC Burger, even like to model Bitcoin’s price directly against regular calendar time. With Block time one can find a pretty reasonable power law fit. Price ~ Byr ^5.42 where Byr is the number of Block years (one could equally use the block height). The R² is 0.916, somewhat lower than for the stock-to-flow model. The standard deviation is 0.350 in log price, or a factor of 2.24 times.

In this article by Burgercrypto (n.b. HC Burger and Burgercrytpo are different Burgers) he suggests a problem with using log time. He worries about this: “for two time series to be possibly co-integrated, those time series have to be integrated of the same order”.

But time is not a time series! It’s a fully pre-determined basis vector on which to map other data. The only question is what zero points to use if you use Gregorian calendar time. The most obvious is January 9, 2009 (or January 3, 2009) but sometimes people use other arbitrary starting points, and this seems suspect.

In fact, Burgercrypto states time is a relative concept, and discusses the choice of various starting times. This is not an issue if you use Block time, for which there is an absolutely defined starting point in that case. At least for Block time as well as for stock-to-flow, we do have basis vectors that are *completely determined* *in advance*. And using log of Block time is just a transformation of a completely pre-defined basis vector.

**Comparing Models**

We summarize the three forecasts in Table 5, and chart these in Figure 3. For each, the one standard deviation is around a factor of 2 in price on either side from the forecast.

The stock-to-flow model, while being the longest-term and most forward-looking model, also shows the quickest price reaction, as expected from an impulse or shock to the flow rate. We can call this “shock-to-flow” and unlike the strong shock in physics that leads to a factor of four density increase in the shocked fluid, we have a factor of about 10 increase in price with the supply “shock-to-flow”.

We can view the stock-to-flow model as a leading indicator, and the price-based model as a lagging, or backward looking indicator. The difficulty model is a coincident model that, like stock-to-flow, has an identifiable driver.

The s2f model has a clear price driver, namely the impulse or quantum leap that is provided as s2f is pushed to new highs inexorably each four Block years. And it is a steep model, with price as a power law of s2f, and s2f, in turn, an exponential of Block time. Thus it is not surprising that it has very aggressive future price projections. It is also clearly the most forward looking of the models, since the Halving effects are all known in advance, from now until the year 2140.

With calendar time or block time, the only direct driver is persistence, or some sort of Lindy effect, whereby the Bitcoin network is seen to have more staying power and growth potential because its lifetime has been extended another Block year. Of course, difficulty and stock-to-flow increases enter implicitly as time passes, with this model.

Security (difficulty) and scarcity (stock-to-flow) provide fundamental drivers of value. Security makes things more valuable, or at the very least protects the value. People want what is held in vaults. Scarcity certainly makes things more valuable, diamonds are worth more than coal, although both are carbon.

All of these are models, and they work until they don’t. Useful, but not the final truth. We are still learning how this highly dynamical Bitcoin network is evolving, and it has great complexity.

The difficulty model arises from an observation that price and difficulty are correlated with a modest power law while the difficulty itself has been growing very steeply with time.

In the case of difficulty, one wonders about whether it can grow at such a rapid rate given constraints on electricity pricing and availability. In Bitcoin Electricity Usage: Is it worth it? , I noted that electricity consumption of Bitcoin mining has more than doubled each year the past several years. This could lead to rationing or price-rationing for miners.

In the next year or two, we will get some sense as to how much value is derived from security (difficulty) and how much from scarcity (stock-to-flow) as we monitor price behavior. Perhaps someone will develop a compound model taking both factors into account. It is interesting how, over the next 3.5 Block years, the difficulty and stock-to-flow models end up in the same place around $70,000.

The difficulty lags, since it does not have the ‘quantum impulse’ effect from the Halving forcing function. Since miners know what is coming in advance, they take steps to eliminate old equipment and upgrade to new equipment to be best positioned for the post-Halving environment.

As we look at these model results and the large standard deviations, whether using a forecast based on difficulty, stock-to-flow, or block time, it reminds us that one really shouldn’t panic when the price moves the wrong way by $1000 or $3000. These are small excursions relative to the typical volatility of Bitcoin.

Writing this article has allowed me to introduce an alternative model, the difficulty based price model, but has deepened my shared conviction around Plan B’s stock-to-flow model. I believe it is worth tracking both models going forward. Difficulty analysis may also provide some explanatory power as to why forks such as Bitcash and Bitcoin SV, with similar stock-to-flow, have such low values relative to Bitcoin.

I also encourage the use of Block time as the basis for regression and co-integration analysis; it is Bitcoin’s natural rhythm.

**Satoshi’s Monetary Genius**

It appears to have been a genius move by Satoshi to implement Halvings in the block reward algorithm. He could have just proposed, as one example, emitting 1/2 million Bitcoins per year for a 42 year period.

His choice of four Block year Halving cycles providing a “shock to flow” indicates an advanced awareness that technology cycles and perhaps monetary and business cycles would impact the economics of mining and of Bitcoin generally.

Mining hardware was destined to improve even more quickly than Moore’s law. And if Bitcoin succeeded, a rising price would draw more miners into the economy.

For s2f, while it has a strong underlying rationale, the model will be going into uncharted territory by 2024 when the s2f will equal 120 (inflation under 1%). This will be a level of scarcity in a monetary asset that we have never seen before. At some point, the power law will break down, but will it be at the level when Bitcoin’s market cap is equal to that of all gold ($8 trillion), or to the global M2 money supply ($90 trillion) or somewhere lower or higher?

[Two days after this article was published, Swissreg published this very interesting capped stock-to-flow model, that quantifies the relationship between Bitcoin’s future market cap and the money supply, and implies slower ramp in Bitcoin price going forward: https://swissrexag.ch/wp-content/uploads/1910_Newsletter-8_stock-to-flow-model-proof-of-non-linearity_EN.pdf]

Remember it is not just the money supply but its velocity that matters. Currently, Bitcoin has a much higher velocity than the US dollar for example. A higher velocity supports a larger economy, but one that is less stable. As Bitcoin grows more valuable and becomes a more stable asset, the velocity would be expected to drop.

Once inflation is less than 1% for Bitcoin after 2024, will it matter much whether it is 0.4% or 0.1%? All but the last 100 Bitcoins will be mined by 2080. Well before that point, the stock-to-flow model could break to a less steep power law. Perhaps security increases via growing difficulty have been supporting the currently steep power law; one also expects that the youthful, dynamic nature of Bitcoin, with its evolving stock-to-flow, has been a reason.

Precious metals have a power law index in stock-to-flow of around 2.2, so that could be an initial transition, as Bitcoin could move onto the same curve with gold and silver during the next two, three, or four Halving shocks to new supply.

Total global wealth is of order $300 trillion, but one does not need a base money, which is a unit of account, in the same total amount. Two-thirds of that global wealth is in real estate, and it can remain in the same hands mostly and just be re-priced in Bitcoin terms, in the event Bitcoin became the basis for a future monetary system. A future de-central bank of Bitcoin.

Long before we would get to that stage we would expect to see central banks adding Bitcoin to their reserve balances, as a way to defend their institutions and their banking system monopolies. Bitcoin seriously challenges the whole notion of fractional reserve banking with fiat currencies, so many other changes, difficult to calculate, could occur.