What is the relationship between Bitcoin price and time, mining difficulty, reserves-production?

Stephen Perrenod recently published a long article, using three models to analyze the value of Bitcoin, and found that...

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1 Security and Scarcity: Bitcoin's Double Feedback Loop

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As shown below:

Figure 1: Bitcoin’s scarcity and security enhance its value, creating a double feedback loop

Between halvings, bitcoins become more scarce as the rewards for each block decrease in proportion to the amount of coins mined (currently about 18 million bitcoins).

The scarcity of Bitcoin is increasingly evident amid the regular halving of production. The halving occurs every 210,000 blocks, with a period of about four years, which cuts the block reward in half and brings Bitcoin's long-term inflation rate close to zero.

Note: The design of halving the mining reward is a long-term mechanism specially designed to curb the inflation of encrypted digital currency and prolong the mining ecology.

The cycle graph at the top right of the chart above shows that each new bitcoin represents a smaller and smaller percentage of the circulation and has a smaller and smaller impact on the market. this kindGrowing scarcity drives up prices.

The cycle diagram at the lower left of the above figure shows that——The increase in price has encouraged the enthusiasm for mining.

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2 Time Basis: The Bitcoin Calendar System

When doing research on price, market cap, hash rate, difficulty, transaction value, stock, and other variables versus time, I recommend using the Bitcoin blockchain calendar as a basis to examine the correlation and relationship between these factors and time variables. Relationships will be more natural, appropriate and accurate.

After the analysis is complete, these results can be easily converted to regular Gregorian time for presentation and further analysis.

In the Bitcoin calendar system, the "Block Year" (Block Year) is 52,500 blocks, while the four-year "Block Era" (Block Era) is 210,000 blocks。(See "Living in Satoshi Time" for more details).

Bitcoin's value stems from its security and scarcity.

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3 Model 1: Mining Difficulty Obtains Security

The existence of hash rate (computing power) guarantees the security of Bitcoin to a certain extent, but it turns out that the curve of the hash rate is a relatively smooth curve, because the hash rate is adjusted every 2016 blocks.

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3.1 Model establishment

In fact, both hash rate and difficulty increase with block times to the power of 12 over the lifetime of Bitcoin.

In the article "Cryptography Breaks Moore's Law", we looked at the growth of the hash rate over the past 9.5 "block years" and found that it has been increasing by the 12th power of the block time (or equivalently, the block height) increase.

The fundamental algorithmic driver of the hash rate is the adjusted difficulty of Bitcoin.image description

Figure 2: Difficulty log versus block year (regressed against block year log)

Let's take a look at the regression plot of block time and mining difficulty.

Figure 2 shows the logarithm of blockchain years and difficulty (base 10) at quarterly intervals. In the 4th and 5th years, due to the promotion and application of ASICS mining machines, the form of GPU mining was replaced, making the difficulty of mining sharply increased.

The table below cites half-yearly data, but the regression analysis uses quarterly data.

A linear regression analysis of all the data in the logarithmic range (difficulty log versus block year log) reveals a power-law relationship with an exponent of 12.38. However, if we restrict the data to year 6 and beyond (after the advent of ASICS miners), the power law exponent becomes 10.51 (R² = 0.975). Mining difficulty seems to correlate very well with block height or block year.

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3.2 The relationship between price and difficulty

So what is the relationship between price and difficulty?

Woobull likes to study a bitcoin difficulty ribbon that consists of multiple moving averages of difficulty. This chart shows intuitively - a strong price pullback leads to a flattening of difficulty for the index for a period of time (possibly up to a year). This could be a buying opportunity as weaker miners have been forced out, restoring stability to the market.

This is also probably the closest we'll get to a supply and demand balance point, becauseBitcoin's supply hardly responds to price very much, except for extremely short periods of time - the hash rate automatically corrects the block time to 10 minutes, so the release rate of supply is basically fixed.

If we regress the log difficulty on the log price starting from block year 6 using the quarterly data, we find a slope of 0.646, R² = 0.919. Using difficulty as a safe proxy, we can use difficulty versus block year (power law with exponent 10.51) and price-difficulty (power law with exponent 0.646) to obtain the following price predictions:

therefore

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3.3 Regression and cointegration

Care must be taken with R² values ​​resulting from regressions between processes that are not normally distributed, as spurious correlations may emerge.

In historical price data, both difficulty and price have grown strongly over time. 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 obtain a stable normal distribution?

Fortunately, both logdifficulty and logprice appear to be second order processes. For logarithmic difficulty, the average of the first (second) order deltas is 0.2306 (-0.0129), with 5.1% (57.9%) of the values ​​being negative. The first order differences don't look close to normal, but the second order differences look normal.

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4 Model 2: Scarcity and the stock-to-production (S2F) model

The most famous Bitcoin value analysis model is Plan B's reserve-production model (scarcity model).

Note: Plan B is a well-known crypto asset analyst on Twitter.

The expression of this model is simply that scarcity confers value. Plan B states that this applies to precious metals including gold, silver, and platinum; even diamonds follow the same general power law curve with an exponent of 2.2 (note that later in the article, Plan B corrects the stock-to-flow for silver to be lower , and added Platinum and Diamond, and basically both have the same curve).

He modeled the evolution of Bitcoin relative to its ever-increasing reserves-production (recording reserves versus production as a logarithmic price) and found that Bitcoin follows an even steeper power law than precious metals, around 3.3.

Note: The Stock-to-Flow (S2F) ratio model (i.e. the stock-production model) refers to the amount of available assets or reserve assets divided by the amount produced each year. The Stock-to-Flow ratio is an important metric because a higher metric value in S2F reflects a lower incidence of annual inflation for an asset.

Perhaps one of the reasons why the Bitcoin power law index is 50% steeper than the gold and precious metals index is the increasing security/difficulty of Bitcoin. While Bitcoin has become more scarce and more secure, the scarcity and safety of gold have been largely static.

Reserves-production scales inversely with inflation and is measured by the amount in circulation divided by a year's production including recycling and any reduction in reserves. In the case of gold, the reserves-to-production ratio (S2F) is 55, which is about 1.8% inflation. This is a relatively stable figure in recent years.

Bitcoin has a completely predictable inventory if measured by block times. There are some changes with some discrepancies relative to calendar time, but the block year is very close to the calendar year and is currently shortened by a few weeks.

The post “Bitcoin’s Prerequisite Non-Inflationary Monetary Policy” articulates precisely how S2F grows with block time. S2F at each halving = 4 x (2^E -2), where E is the block epoch. Currently, we are in Era 3, which started when the S2F ratio was 24. During the two halvings, the value of S2F gradually increased, and then a huge increase occurred at the time of the halving.

The next Bitcoin halving is expected to be in May 2020, when it will enter the fourth era of Bitcoin (after the third halving), and the estimated S2F value will jump to 56. Its scarcity will also be further amplified.

At that point, Bitcoin's S2F value will match gold's for the first time in history, and the inflation rate will drop to 1.8%. Bitcoin will become scarcer than gold. By 2024, S2F will reach 120 and the inflation rate will drop to 0.83%.

Never in history has there been such a strong, absolutely scarce currency.

Using quarterly data from the block calendar system, I found a power law relation price ~S2F^3.25, similar to Plan B results. In this analysis, I am using the midpoint stock-to-flow (half year back and half year forward) rather than the full forward or only backward options to measure flow. The R² for this power law regression is 0.926.

A more detailed study of the cointegration of two non-stationary processes (price and stock-production) shows that this relationship is valid. A metaphor that is often used is,Stock-yield is the dog that leads the owner home in a deliberate fashion (towards a higher price), while the drunk on the dog, moves either side in a constrained random walk .

In fact, concerns about spurious correlations appear to be overly correlated with excess inventories. After all, the reserve production within a block time is not a random process, but a fully predetermined calculation. At each halving, the forward stock is given by the halving formula above, and then increases by 1.0 units every block year until the next halving (because a year's constant flow is added to the stock), until Until the next halving (because a year of constant flow is added to the inventory). Therefore, reserves can be used as the basis vector for measuring other more derived processes such as price, difficulty and hashrate.

Uncertainty in price forecasts is large, with a standard deviation of 0.325 for the stock-production model forecast error in the logarithm of price, or a coefficient of 2.11 in either direction.

You may ask, what about forks of the main Bitcoin chain?？first level title

5 Model Three: Price and Block Time

Some, like HC Burger, even like to simulate the price of Bitcoin directly based on regular calendar time. Using block times, one can find a reasonably plausible power law. Price~ Byr ^5.42, where Byr is the number of years of the block (block height can also be used). The obtained R² value is 0.916, slightly lower than the stock-flow model. The standard deviation of the logarithm of the price is 0.350, or 2.24 times.

In an article by Burgercrypto (Burger and Burgercrytpo are two different platforms) he asks a question using the logarithm of time. His concern is: "If two time series can be cointegrated, then these time series must be integrated in the same order".

But time is not a time series! It is a completely predetermined basis vector that is used to map other data. The only question that remains is what to use as the zero point if you use Gregorian time. The most obvious is Jan 9, 2009 (or Jan 3, 2009), but sometimes people use other arbitrary starting points, which seems suspicious.

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6 Comparison of three models

We summarize these three predictions in Table 5 and plot them in Figure 3. For each prediction, the standard deviation is approximately 2 times the predicted price on both sides of the price.

The reserves-production model is the longest-term and most forward-looking model and also shows the fastest price response, as would be expected for shocks or shock flows.We might call it "impact flow". Unlike a strong shock wave in physics that causes a 4x increase in shock wave density, a supply "shock flow" increases the price by about a factor of ten.

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

The S2F model has a clear price driver, the impulse or big change provided when S2F is inevitably pushed to new highs every four block years. This is a steep model where price is a power law of S2F which in turn is exponential in block time. So it’s no surprise that its predictions for future prices are quite aggressive. Obviously, this is also the most forward-looking model, as the halving effect is known between now and 2140.

For calendar time or block time, the only direct driver is persistence, or something similar to the Lindy effect (Note: The Lindy effect refers to something that does not die naturally, such as a technology, an idea , their life expectancy is proportional to how long they have currently been around.) things, it is believed that the Bitcoin network has stronger durability and growth potential because its life cycle is extended by another block year. certainly,image description

Figure 3: Year 15 (March 2023) price forecast for block time, inventory, and inventory outflow. The next halving will take place in the 12th block year (May 2020)

installSecurity (difficulty) and scarcity (reserves-production) provide the fundamental drivers of value. Security makes things more valuable, or at least protects value.People want what's in the vault. Scarcity of course makes things more valuable, diamonds are worth more than coal even though they are all carbon.

All of these are models, and they will have reference value until they expire. Useful, but not the ultimate truth. We are still learning how the highly dynamic Bitcoin network has evolved, and it has a lot of complexity.

The difficulty model arises from the observation that price and difficulty are related by a modest power law, while difficulty itself has grown dramatically over time.

In terms of difficulty, one wonders if it can grow so fast given the constraints on electricity prices and availability. In "Bitcoin Power Drain: Is It Worth It?" In the article, I found that the electricity consumption of Bitcoin mining has more than doubled every year for the past few years. This could lead to rationing or price rationing by miners.

Over the next year or two, as we track price action, we'll gain some insight into how much of Bitcoin's value comes from security (difficulty) and how much comes from scarcity (stock-to-flow). Maybe someone will develop a composite model that takes both factors into account. Interestingly, both difficulty and stock-to-flow models end up predicting a price of around $70,000 for the next 3.5 block years.

Difficulty is lagging due to the absence of the "quantum pulse" effect produced by the halving forcing function. Since miners knew what was going to happen in advance, they took steps to retire old equipment and upgrade to new equipment to adapt to the post-halving environment.

When looking at the results of these models and the high standard deviation, whether using predictions based on difficulty, reserves-to-production, or block times, it reminds us that when Bitcoin price moves $1,000 or $3,000 in the opposite direction, People really shouldn't panic. This is a small deviation from Bitcoin’s typical volatility.

Writing this article allowed me to introduce another model, the difficulty-based price model, but deepened my shared belief in Plan B's reserves-production model. I think both models are worth keeping track of. The difficulty analysis may provide some explanations for why forks such as BCH and BSV have similar reserves-production as Bitcoin, but their value is so low compared to Bitcoin.

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7 Genius idea of ​​Satoshi Nakamoto

Implementing halving in the block reward algorithm seems to be a stroke of genius by Satoshi Nakamoto. As an example, he could have proposed releasing 500,000 bitcoins per year for 42 years.

His choice of halving over 4 block years as a cycle provides a "shock flow", showing that he has an advanced awareness that technology cycles, and perhaps currency and business cycles, will affect mining and the entire bitcoin industry as a whole. currency economy.

Mining hardware is destined to advance faster than Moore's Law. If Bitcoin succeeds, the rising price will attract more miners into the industry.

For S2F, while it has a strong rationale, the model enters uncharted territory by 2024 when S2F will equal 120 (inflation below 1%). This will be a level of scarcity of monetary assets that we have never seen before. At some point, the power law will break, but will it be when the market cap of Bitcoin equals the market cap of all gold ($8 trillion), or equals the global M2 money supply ($90 trillion), or whatever ?

The reserve-to-yield model quantifies the relationship between bitcoin's future market value and money supply, and suggests a slowdown in bitcoin prices.

Remember, it's not just the money supply that matters, it's also the velocity of money. Currently, bitcoin circulates much faster than the dollar. Higher velocities support larger economies, but are less stable. As Bitcoin becomes more valuable and a more stable asset, the velocity of Bitcoin is expected to decrease.

After 2024, once Bitcoin’s inflation rate is less than 1%, what does it matter if it’s 0.4% or 0.1%? By 2080, all but the last 100 Bitcoins will be mined. Until then, the reserves-production model may break out of a less steep power law. Perhaps the increased security due to increasing difficulty also supports the current steep power law; one also hopes that Bitcoin's young, dynamic nature, and its ever-changing stock, is one of the reasons.

Precious metals have a power law exponent of about 2.2, so this could be a transitional phase as Bitcoin could be on the same supply curve as gold and silver with two, three, four new supply shock halvings in the future.

The total amount of global wealth is about 300 trillion US dollars, but people do not need the base currency of the same total amount, that is, the unit of account. Two-thirds of global wealth is in real estate, which can be revalued in Bitcoin if it becomes the base currency of the future monetary system. Then there may be bitcoin decentralized banks in the future.

Before we get to this stage, we are likely to see central banks include Bitcoin in their reserve balances as a way to defend the monopoly of their institutions and banking systems. Bitcoin seriously challenges the idea of ​​a fiat-based reserve bank, so many changes are possible, but unpredictable.