Computing power valuation is one of the oldest and most esoteric topics in the cryptocurrency field. Several previous academic papers and industry research reports have discussed the economic and game theory aspects of proof-of-work PoW, but most of them oversimplify the operation of the computing power market in practice, or make unrealistic Hypothesis.
In this article, we explained that computing power operation is essentially equivalent to managing investment portfolios, and computing power pricing includes all aspects of the difficulty of portfolio pricing. We will introduce the mechanism of popular pricing and its flaws. Parameterize the computing power portfolio and further explain the impact of various assumptions in the test on the results. The importance of the valuation framework is not only to practice theory, but to lay the foundation for the development of professional risk management practices in the computing power industry.
Fair value of computing power
Since you can buy coins from the open market, why should you operate your own mining?
This is the most common reaction when people first hear about cryptocurrency mining. As we all know, the huge financial returns stimulated people’s initial interest in mining and matured the industry into a billion-dollar behemoth. A successful miner can produce bitcoin at a price lower than the spot price, so it can open a position at a significant discount compared to buying coins on the open market.
However, low production costs are by no means permanent. Competition in the mining industry has been intensifying over the years, and the market cycle has become increasingly unpredictable. The “discount” so beloved by miners may deteriorate into painful losses at any time. In today’s market, can mining still obtain greater profits than buying from the open market? Considering that this question involves many variables, trying to give a once and for all generalization is futile. But we can divide the market cycle into several prototype stages and observe how the profitability of common mining and trading strategies evolve in each stage.
Let’s start with 2018. In most miners’ memory, it was an extremely bleak year. In the previous article of this series, “Understanding the “Alchemy of Hash Rate”: The Reflexivity of the Cryptocurrency Market and the Four Seasons”, we classified the first three quarters of 2018 as inventory dumping in the mining cycle Inventory-flush, at that time, the price of currency fell, but the growth rate of hash rate was still strong.
Assuming that a miner purchased a Bitmain Ant S9 mining machine with 10 Ph/s computing power in early 2018, and the unit price of each mining machine at that time was about US$2,675, then the total expenditure on 690 mining machines was US$1.85 million. Assuming that the hardware linear depreciation rate is 24 months, and the miner’s electricity cost is $0.0507 per kWh (about 0.33 RMB, data provided by GMR), we can back-test the performance of three common strategies:
Neutral strategy: The tokens sold by the miners are enough to cover the daily electricity bill ($941.38) and the daily depreciation of the miner ($2,563.54). If the mining income for the day is less than the total cost ($3,709.01), only the tokens with sufficient electricity costs can be sold. All the remaining BTC will continue to be held.
Tuning strategy: The miner’s sales income is sufficient to pay the daily electricity bill (941.38 yuan), and all remaining BTC will continue to be held.
Arbitrage strategy: The miner immediately sold all the mined BTC to legal currency. The only goal is to use the difference between the spot price and the cost of production for arbitrage transactions. It is worth mentioning that this strategy will cause tax burdens and will be subject to liquidity-constrained transaction friction. In order to simplify the modeling, these factors are not included in the calculation.
Next, compare the performance of the above mining strategy with the two common strategies of buying coins through the open market:
One-time purchase: The mining evaluation period begins on the same day (January 1, 2018). Based on the BTC spot price (13,465 USD) on that day, the nominal value of the purchase is equal to the total mining capital expenditure + annual operating expenditure (1,845,750 USD + 941.38 *365 USD) tokens, continue to hold tokens until the end of the evaluation period. For the sake of simplicity, we do not consider the transaction friction caused by buying this amount of Bitcoin.
Regular fixed-amount purchase: The nominal value of the purchased tokens is still equal to the mining capital expenditure + annual operating expenditure, but the specific strategy is to purchase a fixed value token (2.26 million USD/365) every day during the entire evaluation period.
In the following year, the daily production cost of each token ($3,709.01 spent divided by the number of tokens mined on the day) exceeded the market price around July and continued to rise in the second half of the year, causing the miner Unprofitable for a long time. It can be seen from the results that after a year of bear market, the “arbitrage strategy” suffered the least losses, while the “tun currency strategy” suffered the most.
This is because the “tun currency strategy” is the only strategy that has no unrealized profit or loss. All other strategies hold long positions to varying degrees. In the dumping phase of mining machine inventory when mining revenue continues to decrease, unrealized positions are likely to result in losses at the end of the valuation period.
In practice, wise miners should shut down their miners after mining at a loss for a long time. If this miner ceases operations at the end of June, their losses will be much smaller. If the miner has been using the “arbitrage strategy” before, the miner can even make a profit.
The neutral strategy and the currency exchange strategy will still be unprofitable, although the losses will be smaller than those of buying coins from the open market. The loss is mainly due to the capital expenditure for purchasing mining machines. The miner had previously purchased these mining machines for a total price of US$1.84 million, but they could only be resold at a price of US$738,000 (excluding transaction friction, freight and taxes). The income from the BTC mined cannot make up for the depreciation of the hardware.
In these two examples, the arbitrage strategy seems to be the safest strategy. But in the opposite phase of the market cycle, what happens when overall mining revenue continues to increase?
After the torment of the shuffle-out phase at the end of 2018, the miners ushered in a bumper harvest in the first half of 2019. Doing the same analysis from January 1, 2019 to June 30, 2019, we can see that the arbitrage strategy has the least profit, and the high-risk aggressive strategy (the currency strategy and one-time purchase) has a higher income than the low-risk strategy. The defense strategy is more than 50% higher.
The life cycle of the mining machine was adjusted to 12 months at the beginning of 2019
Through the post-mortem analysis, you can easily see which strategies have performed Changhong. But when the overall market is in a long-term downturn, to adopt a radical strategy requires firm belief and a deep understanding of macro conditions. This is especially true for the one-time purchase gambling, where the time of opening a position is everything to make or break.
The middle of 2019 is the iterative period of hardware. Miners sell old mining machines and purchase new, more efficient models. In this example, given that the price of BTC has skyrocketed and the price of mining machines has risen, miners can sell mining machines at a premium that is higher than the initial purchase price.
Suppose the miner sold 690 ant mining machines and used the income of 271,000 US dollars to purchase the new Shenma M20 mining machine in the second half of the year. For the whole year of 2019, this miner can earn. If the miner did not replace the mining machine and continued to use the Ant S9 mining machine for mining throughout the year, the income of the miner who updated the mining machine was significantly less.
In fact, during the entire mining cycle, miners do not stick to a fixed strategy. Whenever they think that market trends are changing, they can change their strategy flexibly. In addition, they can use trading strategies to make up for mining costs, or lend tokens to increase the profit of inventory. For example, miners can sell tokens in days when the mining profit exceeds the production cost for profit, and the mining profit is low Purchase tokens from the open market in days of production cost. Adopting the right tactics at different stages of the mining cycle can have a significant impact on performance results. The purpose of these cases is not to generalize a universal money-making strategy, nor is it to prove that mining is definitely better than buying tokens, but to show that managing mining business is essentially managing investment portfolios.
These strategies are the simplest and most common representatives of strategies. Compared with miners who actively adopt multiple strategies, the value that lazy miners who adopt only one simple strategy in the entire market cycle can earn is different. There are endless ways to manage computing power, but no matter what strategy the buyer adopts, the price of the mining machine manufacturer is the same for everyone. Although price is an absolute data point, value is relative. Ideally, the price of a mining machine should represent the average value of all available strategies, but this is impossible. So how should the computing power industry be priced? What does the price of computing power represent? More importantly, how should miners value their computing power in a way that best suits their situation?
Intuitive inference method of computing power pricing
In today’s market, the price of computing power is mainly controlled by hardware manufacturers such as Bitmain, Bit Micro and Jianan Zhizhi. They occupy most of the market share of new mining machines and completely control the initial issuance of mining computing power. The manufacturer’s primary task is to ensure that production can pay back. This has little to do with the cryptocurrency market, but has a greater relationship with supply chain management. The selling price of their products can be adjusted according to market demand, but a certain production profit must be guaranteed. Sometimes manufacturers will artificially lower prices, making them more expensive than competitors. In short, the manufacturer’s pricing does not represent the theoretical fair value of computing power. It is mixed with external factors that reflect the state of mining machine manufacturing enterprises.
In the first article “Understanding Hash Rate “Alchemy”: The Characteristics and Challenges of Bitcoin Hash Power Assets”, we discussed that the most popular indicator for evaluating computing power is the static number of days to balance the profit and loss ( Link Wen Note: China is generally called static days-to-breakeven). This indicator considers the current real-time price of BTC, mining difficulty, fees, and full operating expenditures, and can measure how many days it takes for the purchased mining machine to reach balance. Each miner has different payback days on the same mining machine, because each miner operates in a different way. Miners calculate their payback days based on the electricity bills they use and the electricity cost of the installed capacity. However, mining machine manufacturers cannot take the costs of all miners into account. Therefore, the starting point for the calculation of return days is the average electricity cost of the entire market.
Since collecting data on electricity charges for various miners is extremely challenging, the average cost compiled can only be a rough estimate, and the cost of electricity charges also changes with the seasons. Miners with different cost structures are like the rise and fall of the big waves. The mining machine manufacturer makes this calculation under the best guess of the electricity cost, and determines the price of the mining machine based on a reasonable range of return days.
But what is the industry-wide electricity cost used by mining machine manufacturers as input parameters? We can use the historical price data of the mining machine to reverse the calculation.
We can use the discounted cash flow method to back-test the historical price of the mining machine to find out the underlying assumptions when the manufacturer sets the price for the mining machine. For example, the retail price of Antminer S9 in January 2018 was US$2,675.
Assuming that the life cycle of the Antminer S9 mining machine is 24 months, we can calculate the historical income of a mining machine. Next, we will reverse the electricity cost to make the sum of all the present values of daily free cash flow equal to the total purchase price. Assuming that the annual weighted average cost of capital (WACC) is 12.5%, we conclude that to guarantee an average daily expenditure of no more than $1.57, the electricity cost of the S9 mining machine needs to be $1.57/24/1.365=$0.048/kWh. This means that unless miners can obtain electricity that does not exceed US$0.048 (about 0.33 RMB) per kilowatt-hour, the mining machine will be expensive. The above results are calculated based on strategy 3 (arbitrage strategy). Using other strategies for this analysis, the electricity cost per kilowatt-hour required for the neutral strategy of Strategy 1 is 0.017 US dollars, and the electricity cost per kilowatt-hour required for the strategy 2 Tuncoin strategy is 0.01 US dollars (about 0.07 RMB). This means that in practice, the actual “break-even” electricity cost is in the range of $0.01-0.048 per kWh.
The actual electricity bill paid by most miners in early 2018 is far beyond this range. But this level of premium is not unreasonable. The price of BTC has just hit a record high, the network difficulty has not yet begun to catch up, and the supply of Ant S9 in the market is in short supply. The final determinant of price is still the relationship between supply and demand.
Applying the same method to the pricing of mining machines at other points in time, the table below shows the corresponding “break-even” electricity costs for miners. The electricity cost here is the average of the costs derived from the three strategies.
From another perspective, if the electricity cost of the entire industry is $0.0507 per kWh, what is the fair value of these mining machines at that time? The fair value here is still the average of the fair values of the three strategies.
Please note that since it is difficult to accurately estimate industry-wide average costs and WACC, this analysis does not include industry-wide average electricity costs or WACC changes.
The purpose of this analysis is not to calculate an absolutely objective fair value. Due to different operating expenses and different strategies, the fair value relative to each miner is different. But even assuming the industry-wide average cost, we can still observe the inefficiency of mining machine pricing. During the bull market, mining machine manufacturers greatly increased the price of mining machines, while in the BTC decline, manufacturers were forced to clear their positions at a discounted price below the cost. This is consistent with the historical evidence we have observed in the mining market. When the price of BTC rises rapidly, the price of the mining machine sometimes rises faster than the price of the token.
Theoretically, the price increase also means that the network difficulty growth rate will accelerate in the future, so the rising trend of mining machine prices should be slower than the currency price changes. In fact, market pricing often deviates from the theoretical mechanism under such conditions. In the final analysis, the prices of these mining machines are driven by supply and demand, and the market for mining machines is extremely liquid.
By backtesting the historical pricing of mining machines, we can see that the intuitive inference method of pricing based on static return days is not sufficient to capture the volatility of mining profitability. In order to evaluate the current fair value of mining machines, we need to carry out forward-looking modeling of mining profits, so that our tools or theoretical framework can describe the violent fluctuations of variables.
A more advanced method is to treat computing power as a call option. The principle of this method starts with treating the mining revenue of the mining machine as the underlying asset. Mining income is divided into three elements: price, mining difficulty and cost. Bitcoin price call options are profound enough, but the derivatives that encapsulate these three elements are much more complicated. Using the Black-Scholes Black-Scholes model to describe options based on multiple underlying assets is very simple: additional considerations are related random walks and the corresponding multi-element version of Ito’s Lemma. However, establishing a correlation matrix between the three variables is a difficult task.
As discussed in the “Alchemy” series of computing power, price and computing power are correlated but have a constantly changing lag. Due to the response delay, when checking the relationship between hash power and price in a short time window, the correlation is minimal. Therefore, it is easy to simplify the modelling of the hash rate path into a completely price-independent process. From the perspective of financial theory, computing power is a derivative of Bitcoin, and within a long enough time frame, the two time series are positively correlated.
On the other hand, dynamic modeling of transaction costs is more difficult. Although to some extent, transaction fees are related to price and network hash rate (reversely), it is mainly driven by on-chain activity, which is an external factor. This is why the correlation matrix does not produce meaningful results.
But once an assumption is made about the distribution of the underlying assets, pricing the hash power during the N period is equivalent to pricing a series of zero-execution European call options that expire daily. In other words, as long as the mining machine is turned on, the computing power is a contract that is executed every day and converted into the underlying asset, that is, mining revenue. The cost of the contract is the depreciation of the hardware plus operating expenses. The option premium of the entire asset package should theoretically be the miner price plus the present value of all operating expenses incurred during the N period.
This method has a serious flaw. The contracts on day I and day i-1 of this formula are independently evaluated. In reality, the income on the i-1 day should set the initial conditions for the contract that expires the next day. Any computing power evaluation method based on option pricing and simply summarizing all experiments during the period will face this path dependence problem. Each trial is an unrelated evaluation.
Numerical method valuation
For numerical methods, the problem of path dependence does not exist. Rather than evaluating 10,000 trials per day, use the same 10,000 trials in all trials. Monte Carlo simulation can help complex dynamic modeling by generating random numbers. Use sampling procedures to calculate expected income in a risk-neutral world. Then discount it at a risk-free interest rate. With the help of Monte Carlo simulation, we can simulate the mining profitability of the latest generation of mining machines in the next two years and compare its fair value with the prices in today’s market.
In the first step, we need to make some assumptions about price movements. Many studies believe that the jump diffusion model is most suitable to describe the BTC price distribution. We use the jump diffusion model to simulate 10,000 possible price movements in the next two years. In stochastic simulation, each trend takes a different path.
The jumping diffusion model has two basic parts: diffusion (geometric Brownian motion) and jumping (usually Poisson distribution). To simplify the modeling, we assume that there is a threshold probability of jumping. When the jump is triggered, the amplitude follows a normal distribution.
To calibrate based on historical price data, we use the following as the parameters of the model:
Constant drift: 0.10%
Drift standard deviation: 2.50%
Jump probability: 5.00%
Average jump: 0.10%
Jump standard deviation: 5.00%
In addition to the token price, we also need to predict the hash power of the network to calculate the mining revenue. Modeling hash power is more complicated than price trajectory, because each unit of hash power is different. Although every miner on the network is calculating the hash for the same algorithm, the power consumed varies from miner to miner. The current simplified model of network hash rate abstracts several hardware efficiency categories, which behave differently as the market develops. The classification of models by energy efficiency levels can show us the composition of mining machines on the market, so we can roughly predict how they will evolve in the future.
Unlike price data, the collection of mining information is extremely challenging. The only way to solve this problem is to interview as many miners, distributors and manufacturers as possible. GMR conducted a survey of China’s major mining machine manufacturers and distributors, and obtained an estimate of the market composition as of November 1, 2020.
This structural composition diagram is used as the basis for the initial conditions of the prediction model. Using the estimated industry-wide average total electricity price, we can calculate the break-even threshold for each tier and roughly understand how many mining machines may drop in price if the BTC price falls below the break-even. Using $0.0507 per kWh as the industry-wide average total electricity price estimate, we can draw four possible scenarios based on different price levels.
Please note that this only gives a baseline for hash rate prediction. If the price of BTC rises sharply, miners may put low-priced old mining machines purchased through the second-hand market into production, and mining machine manufacturers may speed up production.
Based on the above situation, we can find a linear function y=4,544x+6e07 to describe the relationship between price and network hash power. For simplicity, we assume that the growth of hash power in the next six months follows a function of the 14-day average BTC price, and the drift term is dW. The drift parameter is set to an average value of 2.5% and a standard deviation of 5%. In addition, based on our estimates of the manufacturer’s mining machine sales, it is assumed that the hash rate will increase by 200 Ph/s per day for the next six months. We simulate the hardware response delay by adding a constant 20-day response delay. This means that the hash rate only reacts to price action that occurred at least 20 days ago.
In reality, the relationship between computing power and price is a chaotic and complicated entanglement. Using a linear function to describe it is like projecting a chaotic system onto a low-dimensional subspace. This function can fail for many reasons. This is the same as the correlation matrix problem described in the option pricing method. However, this architecture allows us to easily add lag time, so this is a significant improvement over assuming that hash power and price are two completely independent distributions. This makes forecasts easier to manage.
In order to further improve our estimation, we can use the Markov Chain Monte Carlo algorithm model. Unlike the Monte Carlo algorithm that extracts independent samples from the distribution, the Markov chain-Monte Carlo algorithm extracts samples when the next sample depends on the existing samples. This solves multidimensional problems better than general Monte Carlo simulation. The exact structure of the algorithm will be discussed in the next article.
Once we have made a prediction for the BTC price and hash rate in the next two years, we can calculate the profitability of mining like a backtest of historical hash power prices. Two years ago, there were very few lending activities backed by crypto assets, but today the crypto lending market has developed into a huge industry. Mortgage loans are one of the most common services that miners often rely on. Assessing the current WACC, this value should be significantly improved. We can reduce it to 10% instead of 12.5% in the 2018 analysis
Using electricity costs at $0.0507 per kWh and assuming a risk-free interest rate of 10%, we can generate a distribution of fair value. The final result is the average of all 10,000 trials. In addition, we assume that after two years, the Antminer S19 Pro and Shenma M30s still retain 20% of the residual value.
Needless to say, this should not be the final verdict on whether the relevant mining machines are overpriced or underpriced. The mean and standard deviation of the price distribution, the function between computing power and price, lag time, electricity cost, discount rate and residual value are all factors that can seriously affect the results of this evaluation. For example, run the simulation with electricity costs of $0.07 per kilowatt hour and $0.03 per kilowatt hour.
We can see that when the electricity cost is higher (left picture), the more efficient mining machines (Ant S19 and Shenma M30s) are closer to the fair value of the lower tier mining machines. When the electricity cost is lower (right), the price of less efficient mining machines (Ant S17 and Shenma M20s) will be more favorable. This proves that if the cost of electricity is sufficiently competitive, miners can benefit from less efficient mining machines.
In our model, we built a switch that will shut down the mining machine if the mining revenue is consistently lower than the expenditure within 14 days. In the real world, miners do not often turn on and off mining machines based on short-term profitability. Most of the time, miners reach an agreement with the colocation data center, agreeing on the minimum amount of electricity they need to consume each month. Even if the profit margin drops below zero, most miners still tend to wait for confirmation of the currency price decline before taking action.
Due to the labor-intensive nature of data center operations and the lack of liquidity in the mining machine market, miners are forced to observe longer-term currency price trends rather than short-term price trends. In recent years, the increase in the number of lending service providers has also increased the affordability of miners in the winter. Instead of selling a large number of tokens, miners can mortgage their tokens or mining machines to borrow fiat currency to pay fees. Nevertheless, this is the lower theoretical limit of mining losses. Miners’ losses cannot exceed capital expenditures plus accumulated operating expenditures.
Like call options, the greater the volatility of the underlying index, the higher the theoretical value of the financial instrument. We can see that the results change as the parameters of the jump diffusion model change. When the volatility is suppressed, the theoretical valuation of the mining machine drops sharply. When the volatility is high, the theoretical value will increase rapidly.
This analysis is based on Strategy 3 Daily Selling Strategy. The same as the backtest analysis, the fair value that can be “unlocked” by running the hash power is within the fair value range (Strategy 1, Strategy 2, Strategy 3). Given that Monte Carlo simulated 10,000 paths, each with a distinct path, running one strategy alone is sufficient to cover each type of market stage.
The future of zero block rewards
Another variable that has a significant impact on mining revenue is transaction fees. Assuming that the cost increases linearly by 5% and 10% each year, the fair value of the mining machine will increase significantly.
In reality, the transaction cost trend is very irregular, and the connection with other endogenous variables is not obvious. Modeling cost trends requires completely independent distributions. In addition, there are many ways to improve its accuracy:
As discussed in the article, the Markov chain-Monte Carlo algorithm model is used to reduce dimensional interference
The dynamic lag is introduced based on the four prototype market cycles, and the Poisson process is used to model the jump.
Use hash rate weighted average electricity cost instead of the median cost of the entire industry.
Calibration parameters using statistical methods
Use mining machine learning tools to describe the relationship between hash power and price.
Incorporate transaction cost forecasts into mining revenue calculations
Agent-based simulation is used for the behavior of miners. Agent-based modeling is a technique used to model complex systems to gain a deeper understanding of system behavior. It is widely used in high-frequency trading or smart contract risk analysis. Under this framework, each miner is a “user” with different strategies and different cost bases. Then we can define some simple reaction types (buy more miners, sell miners, buy more miners but wait for 30 days to arrive, etc.) and build a “user behavior” library. This will allow us to simulate more complex interactions in the hash power market. For more background information, read Conway’s Game of Life.
Nobel Prize winner Myron Scholes said: “All models have flaws, but that doesn’t mean you can’t use them as decision-making tools.”
Like the Black-Scholes Black-Scholes model, the simulation model is a mechanism that tries to reflect the real world through a short description, thereby simplifying its complexity. This reduction makes the model useful, but at the same time limits its practicality. It is important to understand the specific limitations, and the simulation represents only possibility rather than certainty.
But for users who have already formed opinions on the market, this model is the benchmark. Like any predictive model, this simulation is only as good as the assumptions made by the user. One uses modeling tools that transform those views of the future into today’s appropriate prices, in order to explore the problems that will be exposed when the future version appears.
Why is this important? What is the point of developing asset pricing theory in a market that is clearly driven by supply and demand?
Valuation is not just an exercise in theory. For Bitcoin, once the mining industry is completely dependent on transaction fee income, competition will only generate minimal profits, and there is no predictable element in the calculation of mining income. How can we ensure that miners continue to generate computing power? The answer is to maintain the stability of continued investment in mining hardware to increase the security budget of the network. This is critical, because if there is not enough hashing power, the entire system is vulnerable to attacks, and the settlement guarantee of Bitcoin will become worthless.
A rigorous valuation framework is the first step in testing various assumptions and market behavior, and planning accordingly. Given that some mining institutions are too big to fail, assessment is the basis for proper risk management. The purpose of this exercise is to open a dialogue in this general direction. In the next few years, we will continue to work hard to further improve our framework.
