Statistical analysis of cryptocurrencies using actual math
Analysis based on data available at 2017–12–26 02:00:00
When reading about the cryptocurrencies use of the word “mathematics” stands out in many places, at times where it should be used and others where it should not. Certain coins attempt use the word “mathematics” to give some sort of legitimacy and affirm trust to their investors that might not understand their risks entirely. In this article I will use math to show how different cryptocurrencies are related and hopefully that would give readers an idea how to better manage your investments.
To understand the risk we need to look at the portfolio distribution. For that we will use daily price data and we will estimate return distribution one day ahead.
Historical data
First of all, let us simply look at the historical daily log-return distribution. We can observe that some cryptocurrencies have higher variance. That is useful to know when constructing a more stable portfolio.
The historical means of the data are the following:
BTC DCR ETC ETH FCT FTC GNO GNT LTC STR
0.01 0.006 0.003 0.005 0.003 0.009 -0.003 0.003 0.011 0.005
VTC WNG XMR XRP ZEC
0.013 0.006 0.012 0.008 0.003The historical deviation of data is the following (lower values means less extreme price movements and vice versa):
BTC DCR ETC ETH FCT FTC GNO GNT LTC STR VTC WNG
0.055 0.09 0.082 0.064 0.1 0.122 0.1 0.095 0.082 0.157 0.126 0.101
XMR XRP ZEC
0.081 0.091 0.077Secondly, we have fitted an ARMA-GARCH model to individual cryptocurrencies and extracted independent and identically distributed errors. This allows us to extract any trends in the data, and further analyse them by observing their dependency structure.
Without unnecessarily over-complicating this demonstration for the reader, I will display the result here and explain what it signifies to the reader.
Dependence structure
The tree graph below shows the strongest dependencies between cryptocurrencies based on Kendall’s τ. It is a more reliable measure of correlation than linear correlation parameter Pearson’s ρ because it does not depend directly on the values and thus is invariant under strictly monotone transformations.
The graph is based on the strongest dependencies. We can therefore visualize and understand which cryptocurrencies are closely related.
Probably the most important take from is article is the dependency structure plotted below :
The contour plot shows the dependency structure between the most dependent pairs. Without explaining in too much detail (of course the reader is very welcome to familiarize him/herself with the copulas and pair copula constructions) the main result is that most cryptocurrencies are dependent when they perform badly, but relatively independent when they do well. In other words they are likely to “crash” at the same time, on the other hand they tend to increase independently. Their dependency structure is asymmetric and it causes the so called fat left tail of the portfolio distribution.
Prediction one day ahead
Below we have constructed a hypothetical portfolio of equal parts of all cryptocurrencies mentioned in this article. Based on the data available at 2017–12–26 02:00:00, our prediction one day ahead is the following:
Value-at-Risk (VaR) at 5% level is -0.141866, in the event of a 5% worst case scenario this portfolio could fall below exp(-0.141866)*100%=86.77 % of its value.
Due to the asymmetry of the dependency structure, the portfolio value is more likely to fall by 20% than to rise by 20% and the expected value of portfolio is negative (-0.009573077).
Concluding remarks
In this article, I have shown that the dependence structure of cryptocurrencies is asymmetric and that should be taken into account because it has a very obvious negative effect on the portfolio distribution. The reader must understand that even with a diversified portfolio there is a risk of losing a relatively high proportion of your investment even during a time period as short as one day.
