A Quantitative Return Model for Bitcoin

The model described in this article was designed to help a client time the selling down of his bitcoin position.  Bitcoin is extremely volatile, highly speculative, and impossible to value in a traditional way, since it does not provide any cash flows (dividend or interest) or have any intrinsic value.  However, there are a few factors that seem to have some predictive content over a one-month holding period. 

Background Info on Bitcoin

Bitcoin is a “cryptocurrency” intended to provide an innovative medium of exchange that does not rely on any central authority, such as a central bank.  Like paper money, it has no intrinsic value.  Its value is a function of its acceptability by others as payment.  However, whereas U.S. paper money has the full backing of the U.S. government and its taxing power, bitcoin does not have any backing from anyone. 

It is a stretch to call bitcoin a currency.  Bitcoin has limited usefulness as a medium of exchange, one of the primary functions of a currency.  Its value relative to other currencies, such as the U.S. dollar, is extremely volatile.  Someone accepting bitcoin as payment based upon its current dollar value may find that its value has changed substantially by the time the bitcoin is exchanged for dollars. 

The vast majority of bitcoin transactions are probably related to its speculative value as a “crypto-commodity” rather than for buying pizzas.  Another function of a currency is as a “store of value.”   Historically, precious metals, especially gold and silver, were the primary store of value.  Gold still fulfills this function to some extent.  Bitcoin is currently being used primarily as a speculative store of value. 

I hesitate to call bitcoin an “investment,” because exchanging dollars for bitcoins does not contribute to capital investment.  Stocks and bonds provide capital for businesses to grow.  Bitcoin does not.   That is not its intended function.

However, bitcoin does provide a way for those who would like to avoid government scrutiny to store and exchange value.  Although such parties would include drug dealers, cyber blackmailers, and terrorists, it also includes people who live under governments that control the export of capital.  Also, bitcoin could eventually provide some banking functions for poor people who are outside of the traditional banking system.   

Critical to bitcoin is the “blockchain,” or use of transaction records that are linked using cryptography and verified by a large number of independent members of a network, called a “distributed ledger.”  Bitcoin network members are rewarded for providing this verification function for each incremental “block” of transactions by being awarded bitcoin as a byproduct of their effort.  However, not every network member verifying bitcoin transactions is awarded bitcoin for verifying a new block.  Instead, only one network member is awarded the bitcoin in a kind of lottery system.  The lottery is won by solving a very complex mathematical problem (the “hash”).  The first one to solve it wins.  The reward is not only a certain number of bitcoins but also the transaction fees paid by the parties to the transactions in the block.  This activity is known as “mining.” 

The origin of bitcoin was a 2008 paper by Satoshi Nakamoto, a pseudonym for the person or group that developed the logic and code behind bitcoin.  Some important parameters were established in that original paper.  For one thing, only 21 million bitcoins will ever be created.  They are created when a miner solves the “hash,” but the rate of bitcoin creation is set to decline steadily over time.  The number of bitcoins awarded to successful miners was originally 50, but the number is cut in half every 210,000 blocks.  This “halving” is expected to occur about once every four years.  The first halving occurred on November 28, 2012, cutting the rate to 25 bitcoins.  The second was July 7, 2016, when it was cut to 12.5 bitcoins.  The next halving is expected in May 2020, when the new rate will be 6.25 bitcoins. 

Price History

There is no central marketplace for trading bitcoin, so no single market price is universally recognized.  The bitcoin data in this paper comes from Blockchain.com, a widely-recognized source that averages the price of bitcoin over several exchanges.  Its bitcoin price database starts on August 17, 2010, with a price of $.08.  The price broke $1 for the first time on February 10, 2011.  Other pricing milestones were $100 on April 3, 2013, and $1000 on November 29, 2013.  After that, bitcoin fell to a low of $176.50 on January 14, 2015 (a loss of 83%).  It ascended fairly steadily after that, piercing through $1000 again on January 3, 2017.   From there, its rise was meteoric, peaking at $19,289.79 on December 17, 2017.  Its fall was nearly as steep, bottoming on December 15, 2018 at $3,225.30 (another loss of 83%).  It has since partially recovered. 

The price history of bitcoin falls into two regimes:  before and after December 31, 2013.  The first regime is signified by extreme price volatility.  Gandal et. al. examined trading activity in bitcoin during 2013 and found it highly suspicious.  The SEC has cited the potential for price manipulation prominently among its reasons for rejecting applications for a bitcoin ETF.

This paper will focus exclusively on the second regime, starting December 31, 2013.   

Literature Survey

Published papers on bitcoin are sparse, and none that I could find was recent enough to cover bitcoin’s extraordinary price rise in 2017, its crash in 2018, and its subsequent partial recovery thus far in 2019.  Most were written to try to explain and justify bitcoin’s parabolic price rise in 2017.  None can explain the extreme movements over the past couple of years.    

However, there are two papers that are widely cited by others and posit credible theories for the long-term valuation of bitcoin. 

The first paper proffers the theory that bitcoin is supported by its cost of production.  Hayes estimated the cost of production and found that bitcoin’s price converged to the cost of production at the end of the time period studied (April 2016).  The “cost of production” in this cast is the cost of mining bitcoin.  There is some initial capital investment in buying a computer that is optimized for mining, but the primary cost is the ongoing cost of electricity.   

I was unable to replicate his findings, which require data on a number of factors, including computer hash rates (the raw power to solve the “hash”), computer electricity consumption rates, and electricity costs.  I could not find databases for these inputs to the model, and even Hayes made some heroic assumptions, such as holding the cost of electricity constant (at a relatively high) rate of USD $0.135 per kWh. 

Even if I could have replicated his model through April 2016, I very much doubt that the cost of production (which is fairly slow-moving) would explain much of the price gyrations that have taken place since his article was published.  In my opinion, the basic problem with his approach is that instead of the cost of production influencing the price, the relationship is really the other way around.  The Bitcoin code itself provides an ongoing governor on the profitability of mining:  the “difficulty.”  This is a measure of how difficult it is to solve the block algorithm (the “hash”) and win the bitcoins.  The code provides that a solution to the mining algorithm should be found every 10 minutes on average.  The actual discovery rate is measured every 2016 blocks and the difficulty is reset to drive the discovery rate to every 10 minutes.  When the average rises above this level (because the computers have become faster or more miners have joined the network, for example) the difficulty level is increased, driving up the cost of production.  Conversely, if the price of bitcoin falls low enough to discourage miners from continuing to mine bitcoin (as happened in the fourth quarter of 2018), the difficulty level will be reset at a lower level, which will lower the cost of production and encourage more mining.  It’s not the cost of production that causes the price.  The price leads to automatic adjustments in the cost of production.  

To illustrate the point, the graph below demonstrates that price leads difficulty and not the other way around.

In the second paper, Peterson asserts that bitcoin’s price is related to Metcalfe’s Law:  that its value is a squared function of the number of bitcoin market participants.  This idea has some validity on the surface.  “Network effects” are a well-recognized factor in the value of blockbuster technology companies such as Amazon, Google, Apple, Facebook, Twitter, etc.  However, it is not clear that the function is necessarily n-squared (n being the number of market participants).  I could find only one indication of the number of market participants.   Blockchain.com has a database for the number of unique bitcoin addresses.  As shown below, both the price of bitcoin and the number of unique addresses for bitcoin peaked at the end of 2017.  However, both reflect the same underlying phenomena:  enthusiasm (some might say mania) for bitcoin.  Neither caused the other, and the relationship between them is not tight.

A Four Factor Model

One of the major issues with most of the published research on bitcoin is that most papers attempt to explain the price rather than the return.  A basic tenet of financial research is that the dependent variable (the thing to be explained) is always return and not price.  Prices have extreme statistical problems, not the least of which is serial correlation—today’s price is a very good forecast of tomorrow’s price.  Any generally rising time-series will appear visually to have a strong relationship with a generally rising price.  (One example is the so-called “stock to flow model” which compares the total bitcoins mined to date to the number of incremental bitcoins mined monthly.)

This paper will describe a model that forecasts monthly return.  The number of months since December 31, 2013 is only 66.   Trying to use a longer return period (say, three months) would have reduced the number of independent observations and created a problem with overlapping returns.  In practice we expect to update the model and make trading decisions monthly anyway, so monthly periodicity fits our intended use of the model well.    

Having established that the desired dependent variable is monthly return, the next task is to identify the independent variables used to forecast return.  There are four. 

1. Supply Growth

I collected data to the extent that it was available to test out some of the variables mentioned in published papers.  The only one that survived close statistical scrutiny was monthly growth in the number of bitcoins created.  This is an extremely smooth function most of the time, since the Bitcoin code enforces a rate of hash solutions that is very close to every 10 minutes.  That means that the supply growth is very gradually declining as the total number of bitcoins mined to date gradually increases.  The exceptions are the “halvings,” which introduce a jarring and dramatic decline in the supply growth when they occur. 

2. Short-Term Rate Change

The monthly change in the three-month T-bill yield has had an effect on the price of bitcoin.  As an asset with no yield, bitcoin becomes less attractive when interest rates are increasing and the opportunity cost of holding bitcoin goes up. 

3. Return Momentum

Most commodities exhibit some trend-following behavior, and bitcoin seems to follow the pattern as well.  The variable used in the model is an average of exponentially-weighted moving average returns over three trailing time periods (with no lag):  1) three months, 2) six months, and 3) twelve months. 

4. Long-Term Mean Reversion

I struggled to identify some sort of “value” function for bitcoin that had any predictive value at all.  The best factor I could come up with compares the current price to the five-year log linear forecast using the trailing 60-month observations as inputs.  The embedded assumption is that bitcoin’s price should not depart too far from the price predicted by its historical rate of increase.  (If the price were to fall over a five-year time period, it would become a rate of decrease.)  Only statistically meaningful departures from the linear forecast are meaningful.  The variable is the average of the square and the cube of the difference between the forecast and the actual price.  The graph below illustrates the actual and forecast cumulative bitcoin returns since December 31, 2013.

Four Factor Model Construction

Having settled on our four factors, the next step is to use the signals provided by these factors to construct an overall bitcoin return model.  The simplest method would be to equal weight the signals provided by the four factors.  However, not all of them have equal forecasting power.  Also, they each work better at some times than at other times. 

To determine how much weight to put on each factor each month, I use two types of regression analysis.  One type is simple regression analysis which relates the signals of one factor to bitcoin returns in the following month.  For this I typically use up to 120 trailing monthly observations, if they are available.  (We are only using returns since December 31, 2013, so we do not yet have 120 observations.)  The second is multiple regression that relates the signals of all four factors to bitcoin returns simultaneously.  For this I use the trailing 36 monthly observations.  Both regressions are exponentially weighted so that more recent experience is more heavily weighted. 

Using a proprietary process to combine these two methods results in a series of monthly payoff forecasts for each factor.  The overall bitcoin forecast is simply the sum of the four components. 

The model forecasts begin in January 2018 because of the data history needed to specify the model.  Therefore, only 18 monthly forecasts have been generated to date.  The graph below illustrates these monthly forecasts.

The monthly forecasts must be translated into portfolio weightings to be useful for investment purposes.  Typically, I “standardize” model forecasts by comparing the current month to the trailing 120-month history of the forecasts.  Specifically, I subtract the 120-month average forecast from the current forecast and divide the result by the 120-month standard deviation of the forecasts.  This results in a “standardized” indicator (often called a “z score”) that measures how many standard deviations the current forecast is above or below the 120-month moving average. 

However, we only have 18 forecasts.  Consequently, we can only standardize the current forecasts relative to the observations available up to that point.  The graph below shows how the 18 forecasts (blue bars, left hand scale) could be translated into portfolio weightings (orange line, right hand scale).  The most negative forecast results in a -100% position in bitcoin, and the most positive into a +100% position in bitcoin.

Model Results

The graph below shows the monthly returns from the model bitcoin allocations (in blue) and a 100% bitcoin allocation (in orange).  The model returns are much less volatile, as expected. 

The cumulative return of the model has been 54.3%, whereas bitcoin has had a cumulative return of just 15.2%. 

Current Bitcoin Forecast 

The graph below breaks out the current forecasts of the four components of the bitcoin model.  The overall bitcoin return forecast of 20.8% is the highest it has been in the short history of the model.   Supply growth continues to be a very slight negative.  This will be cut in half at the next “halving” expected in May 2020.  The T-bill yield has been falling over the past month, reflecting an expectation that the Fed will lower the fed funds rate by .25% at their upcoming meeting.   Declining rates helps bitcoin because it lowers the opportunity cost of owning a non-yielding asset.  Bitcoin has strong return momentum, which is the main driver of the positive return forecast.  Relative to the its 5-year return trendline, bitcoin is about fairly valued.    

Summary and Conclusion

• Bitcoin returns before December 31, 2013 were wildly volatile and probably manipulated.
• No pricing model can explain bitcoin’s parabolic ascent in 2017, crash in 2018, and partial recovery in 2019.
• Rather than a long-term pricing model, our objective is a monthly return model.
• Four factors are significant: 1) supply growth, 2) short-term rate change, 3) return momentum, and 4) long-term mean reversion.
• The model provided downside protection in 2018 but upside participation in 2019.
• Currently, the model indicates that bitcoin is attractive, mainly because of its strong momentum and fair valuation relative to its 5-year trendline.