Bitcoin & Ethereum — Finding statistical relationships in returns
Comparing the combined distribution of returns of Bitcoin & Ethereum
Ethereum and Bitcoin currently dominate the crypto market, at the moment of writing this piece they have a combined market capitalization of approximately $1.05 Trillion. They are also the oldest cryptocurrencies that still are around and play a significant part in the market. Keeping this in mind, I thought it would be interesting to look at their combined distribution of returns. In simple words how much does the percentage change in bitcoin’s price affect the percentage change in Ethereum’s price?
Ethereum went live on 30th July 2015, about 7 years ago. Initially developed to facilitate decentralized applications. Bitcoin on the hand is the original cryptocurrency created in 2010 by an anonymous creator Satoshi Nakamoto. Both currencies had experienced an enormous surge in price, especially after the pandemic of 2020. The question that seems to plague my mind at least is just how closely are the two related in terms of returns. Are there any patterns to be found in their joint price movements?
This piece seeks to answer these questions by first looking at the distribution of returns concurrently and then using statistical models to find the relationship that exists between their price movements.
Disclaimer:
- The purpose of this piece is educational, this is not financial advice! Please seek financial advice from legally certified professionals.
- The models and data used are all historical. In finance past trends do not necessarily predict future trends.
Data Description & Analysis Methodology
All the information on how the data was sourced and the description of the methodology used:
- Source: The source of the data is cryptocompare. They provide accurate data about cryptocurrency price movements in an easy to analyze format. The accuracy of the data was checked by looking at daily prices for bitcoin and ethereum provided by cryptocompare and exchanges such as Binance. (Note: The prices will not match one-to-one with exchange prices since cryptocompare uses a volume-weighted average price of different exchanges.)
- Granularity: Each data point corresponds to a day, with the day beginning on GMT + 5hrs.
- Timeframe: The timeframe starts from 7th August 2015 to 24th April 2022. This makes the total size of the data equal to 2452 days.
- Methodology: The analysis will explore the return distribution of both BTC and ETH, try to see how they correlate, and build statistical models that will answer how the price movements in one security impact the other.
Comparing Return Distributions
Looking at the distribution of returns of ETH (Ethereum) and BTC (Bitcoin) we can see that in the period August 2015 to April 2022 BTC returned an average of 0.2% per day and ETH returned an average of 0.6%. Bitcoin’s median is closer to its mean indicating a lower skew in the distribution but since the median is lower than the mean in both cases, the distributions are left-skewed. Left skewness tells that it is more likely to get a lower than the mean returns on a day-to-day basis. The spread between the 25th percentile and 75th percentile is larger in the case of ETH, indicative of higher variation. The standard deviation in the case of ETH is 8.15%, as opposed to 3.99% in the case of BTC, which tells that during this period Ethereum has been more “risky” than bitcoin (quotation marks because there is debate in finance whether standard deviation or volatility is the appropriate measure of risk). However, it must be noted that the higher variation in ETH might be due to more uncertainty regarding the asset in its early days!
It comes as no surprise that ETH has a higher 99th percentile and lower 1st percentile returns, a feature of higher variation.
Correlation Between BTC & ETH
Out of 2452 days tracked, bitcoin produced a positive return on 1337 days. While ETH produced a positive return on 1246 days. When bitcoin moved in a positive direction, ETH moved in a positive direction 71.05% of the time. Similarly, when bitcoin had a negative return, ETH also had a negative return 73.45% of the time. In aggregate ETH moved in the same direction as BTC on 1769 days (819+950), which translates to a probability of 72.14%.
Looking more closely at the magnitude of those movements we get:
For the above plot, I have divided the day-to-day returns of BTC & ETH into bands. Please find the band-wise description below:
- +(0–1%) & -(0–1%) indicate days when the cryptocurrency moved between positive 0–1% & negative 0–1%.
- +(1–5%) & -(1–5%) indicate days when the cryptocurrency moved between positive 1–5% & negative 1–5%.
- +(5–10%) & -(5–10%) indicate days when the cryptocurrency moved between positive 5–10% & negative 5–10%.
- +(10+%) & -(10+%) indicate days when the cryptocurrency moved above positive 10% & below negative 10%.
BTC generated a return between 0–1% on 18.60% of days. On those days ETH moved in the same band 21.27% of the time. It generated a return one band higher +(1–5%) 25.22% of the time and one band below -(0–1%) 17.54% of the time. Overall, it remained in the same band or one band above or one band below a total of 64.04% of the time. Indicative of a high correlation in terms of the magnitude of movements. It must also be noted that extreme positive & negative movements (10+%) remained almost equally likely.
BTC closed negative between 0–1% 427 out of 2452 days. ETH moved in the same band 90 days out of 427 total days. It moved one band higher +(0–1%) 14.75% of the time, and one band lower -(1–5%) 33.02% of the time. It must be noted that ETH moved to the +(1–5%) more frequently than the -(0–1%) band showing but the difference is relatively small. Similarly, ETH moved to one band lower -(1–5%) more frequently than staying in the same band -(0–1%) as bitcoin. Indicating that smaller negative movements in BTC correlate with slightly more extreme movements in ETH.
BTC generated a return of +1–5% on 685 of 2452 days. On those 685 days, the band-wise movements of ETH remained in the same or one band above or one band below 69.78% of the days. It must be noted that in this case, extreme positive movements (positive 10+%) were 4.6x times more likely than extremely negative movements (negative 10+%). This shows that when BTC moves in a positive direction ETH is very likely to move in the same direction with a similar magnitude.
BTC is more likely to move in the +(1–5%) band than the -(1–0%) band, about 7% more likely. When BTC moves in the -(1–5%) band, ETH is more likely to follow compared with +(1–5) BTC movement to +(1–5%) ETH movement. BTC -(1–5%) band movement corresponds with a 53.85% ETH -(1–5%) movement. Also, when BTC moves to the -(1–5%) band ETH is 77.51% likely to move in the same band or one band higher or one-band lower. Overall, a -(1–5%) movement in BTC correlates highly with a negative movement in ETH.
As in the previous sections, it is the case that BTC movements correlate with similar directional movements in ETH. The surprising thing to note is that in both BTC+(5–10%) & BTC-(5–10%) cases ETH is more likely to move in the same direction, with one band higher or one band lower than smaller movements. About 83.22% in the positive case and 91.84% in the negative case. This means that high magnitude movements in BTC almost always correspond with high magnitude movements in ETH.
Although the sample size for these movements is low (intuitive since these are extreme movements in either direction), extreme movements in BTC correlate tremendously with extreme movements in ETH. This means that a BTC & ETH portfolio is improperly hedged, mixing these two assets offers very low diversification.
BTC & ETH returns have a high correlation which suggests that there is a strong statistical relationship between these two currencies. In the next section, I will explore this relationship.
Statistical Modelling
Before building any statistical model, it is pertinent to look at the observed relationship between our variables.
Looking at the above plot with a naked eye, it is possible to infer that BTC & ETH undoubtedly have a positive relationship.
Model Description
For discovering our desired relationship I will explore linear models, namely regression. The reasons for this choice are simple I want to make inferences about the model I make and secondly intuitively understand the relationship between BTC returns and ETH returns.
Blackbox techniques (Neural networks, XGboost, etc) perform better on accuracy but it is very hard to make inferences or establish causal relationships using them. Lastly, adding complexity is never an issue, if simple models fail it is easy to build more complex models but the complexity comes at a cost of understanding.
The two sets of models used are:
- Linear Regression (LR) — Chosen because it is simple, intuitive, and serves the task.
- Quantile Regression (QR)— Quantile regression is very similar to Linear Regression with the key difference being that instead of estimating the conditional mean function as is the case for Linear Regression, quantile regression estimates the conditional quantile function at a particular percentile q. You can read more about quantile regression in this research paper from the University of Illinois Urbana-Champaign. It was chosen because it works even if some of the assumptions of Linear Regression are violated and also QR can help in understanding the upper and lower bounds on the relationship between BTC returns and ETH returns.
After regressing ETH Returns with BTC Returns, the following results are obtained.
Looking at the scatter plot above we can see how many returns in ETH one can expect given a change in BTC. The QR Models with q=5 & q=95 give a range where 90% of the time the returns would lie. Similarly, the q=25 & q=75 returns give us bounds where 50% of our returns would lie. The q=50 model tells the separation line where 50% of the time our returns would be smaller and 50% of the time they would be larger. Using these ranges one can infer if one observed an extremely unlikely event, any point below the q=5 line is very unlikely and similarly, a point above the q=95 line is also very unlikely. This analysis holds, as long as this pattern observed between BTC & ETH returns continues in the future.
All the beta coefficients for all the models are statistically significant. The next part compares all the coefficients and their respective 5% and 95% confidence intervals.
Intercept corresponds to when the return of BTC is zero. Looking at the quantile models, we can see that when BTC returns are zero, ETH returns will be between -6.22% & 8.86% 90% of the time. The same concept can be extended when comparing the q=25 & q=75 models, but this time ETH returns would be in those ranges 50% of the time. Similarly, looking at the q=50 model we can see that 50% of the time ETH will have a return above -0.27% when BTC is zero and 50% of the time ETH will have a return below -0.27%. The LR model gives the average case for when BTC has a zero return, of 0.35%.
Looking at the results, we can see according to LR Model, a 1 unit increase in returns for BTC results in a 0.87 unit increase in returns for ETH. For QR when q=50 or the median, a 1 unit incremental return in BTC corresponds to a 0.94 unit incremental return in ETH. The particularly interesting thing is that the QR model at q = 95 has a lower beta coefficient than all our models. The reasoning for this is that it has a larger intercept term. You can think of the predictions of the q=5 & q=95 model giving us a maximum upper and minimum lower bound on how much a change in BTC returns corresponds to a change in ETH returns. Only 10% of the time should an observed event be greater than the q=95 line or smaller than the q=5 line (provided our estimates reflect the true relationship between BTC returns and ETH returns). These extreme events could be indicative of mispricing of ETH relative to BTC. But such an inference requires further investigation, one I would likely do and write about in posts yet to come.
Stay tuned! Please follow me and please clap for the article if you found the content educational and insightful!
Interested in cryptocurrency or statistical analysis in general, please consider having a look at these posts:
- Bitcoin A Visual History — Should you buy at the ‘top’?: https://blog.cryptostars.is/bitcoin-a-visual-history-should-you-buy-at-the-top-20067ae48a14
- Lies, Big Lies & Data Science? : https://medium.com/mlearning-ai/lies-big-lies-and-data-science-6147e81fb9fc
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