Current research has led to a rejection of the hypothesis of a normal distribution of financial assets returns. Under these conditions, portfolio variance cannot serve as a good risk measure. In this paper analyzed the daily returns of the most common cryptocurrencies: Bitcoin, Ethereum, XRP, USDT, Bitcoin Cash, Litecoin. It is shown that the asset returns are not normally distributed, but with good precision follow the Cauchy distribution and Laplace distribution. The analytical expressions for risk measure were obtained using the distribution function and the VaR technique. However, the risk assessment of the return obtained on the basis of the Cauchy distribution is twice as high as the risk assessment obtained on the basis of the Laplace distribution. Therefore, the question arises: what distribution law to use to measurement the cryptocurrency risk? The paper shows that the Laplace distribution is the most adequate basis for measuring of cryptocurrencies risk.
The first complete cryptographic currency appeared in 2008 thanks to the efforts of
Satoshi Nakamoto. It was named Bitcoin. New varieties of digital currency appear each
year due to the information technology active development and the globalization
processes spread. The main advantages of cryptography are that the user controls them
without any regulatory rules in the transaction. Third party costs on a transaction can
be greatly reduced. This has been the main reason for the rapid development of the
market for virtual currencies (crypto-currency) over the past 10 years. More than 2000
varieties of digital money have appeared on the market since the birth of Bitcoin for 5
years. Bitcoin (BTC) remains the most widespread cryptocurrency: there is the largest
market capitalization among other digital currencies (about $220 billion) [
But investments in cryptocurrency can be quite risky as their price is very volatile
[5; 12; 18; 19; 20]. Thus, during the period from July 2018 to July 2019 there were
significant changes in the exchange rate. Initially, the cost of one Bitcoin was $6,600
(July 2018). There was a significant dropping in mid-December 2018 in the price – to
$3,200. Then there was a sharp increasing at the end of June 2019 – to $13,000. The
price of Bitcoin Cash fluctuated from $869 per unit (July 2018) to $77 per unit
(midDecember 2018) to $400 per unit in June 2019. The price of the unit XRP demonstrated
a sharp jump from $0.26 to $0.58 during three weeks in September 2018. Then it began
to fall with slight fluctuations. The course of the ordinary currency (dollars, euros, etc.)
strongly depends on inflation, politic factors and other economic conditions. Thus, its
calculations can be performed fairly accurately, taking into account the influence
factors changing. Instead, fluctuations in the price of cryptocurrency are very difficult
to forecast. Therefore, making the correct decisions in investing and trading
cryptocurrency in order to get the most return is a rather difficult task. The interaction
between supply and demand, the attractiveness for investors, macroeconomic
conditions and financial events are important factors in the formation of the
cryptocurrency price [
Diversification is an important risk reduction tool. Creating a portfolio of financial
assets is one of its instruments. In this paper, the formation of cryptocurrency
investment portfolio based on the Markowitz model is investigated [
It has been shown that the distribution of financial assets contains so-called “heavy
tails”. It indicates a high likelihood of realization of very large and very small return
values. The task of this work is investigating the distribution of the return of virtual
currencies and using it to minimize the risk of working with portfolios of
cryptocurrencies. The results of the study [
Building a portfolio solely on the basis of cryptocurrencies [
The authors of [4] emphasize the importance of modeling nonlinearity and taking
into account the behavior of tail distribution in analyzing the causal relationships
between Bitcoin revenues and trading volume. For analysis the Bitcoin behavior in the
study [
Thus, as shown by the analysis of literary sources, in present-day conditions, not only currencies and valuable metals are used for investment, but also cryptocurrency assets are added to the portfolio. Our analysis was done on the basis of historical data on prices of 6 cryptocurrency (Bitcoin, Bitcoin Cash, Litecoin, XRP, Ethereum, Tether) for the period from January 1, 2018 to June 30, 2020. This data are freely available from the www.coinmarketcap.com site – CoinMarketCap Analytical Services contains historical and actual data about cryptocurrency. The data set is divided into 6 parts, each of which refers to a specific quarter of the study period. The volume of quarterly data is 90–92 records, the total amount of data – 912 records. For comparison, we included in our analysis a study of the stock prices of such leading companies as Amazon and Google. In this case quarterly data volume is 61–64 records, the total amount of data – 628 records.
For further processing, the calculation of the corresponding normalized cryptocurrency return is performed according to following equation = ( / − 1) ⋅ 100%, where xn is the daily return of the n-th asset, Cn is the daily closing price of the n-th asset, i is the observation number.
The dynamics of cryptocurrency Bitcoin return is presented in fig. 1. The main characteristics of the investigated cryptocurrency return for the observed period are given in table 1. As well, for comparison, in table 1, we introduced the statistical characteristics of the two successful companies’ stocks. The analysis of statistical characteristics, given in table 1, showed that the daily stock return of the represented companies is higher than the similar investigated cryptocurrencies return. At the same time, their risk (if we consider the risk as a standard deviation) is much lower (except for the cryptocurrency USDT). From the correlation matrix (table 2) it can be seen that the return of the cryptocurrency is sufficiently correlated with each other (except USDT).
Let’s introduce the concept of the risk zone frontier [
Consequently, the value of the asset risk, in accordance with the above definition, can be estimated by the ratio For statistical research, we divided the data set into 10 time intervals, each of which corresponds to one quarter. As a result of research of the cryptocurrencies Bitcoin, Bitcoin Cash, Litecoin, XRP, Ethereum, Tether using the Pearson, KolmogorovSmirnov, and Shapiro-Wilk tests, in most cases the hypothesis of return normal distribution was rejected (fig. 2). Computer experiments showed that the return of the investigated cryptocurrency with good accuracy is described by both Cauchy distribution and Laplace distribution (fig. 3).
= −
To test the hypothesis of the Cauchy (Laplace) distribution of cryptocurrency returns, we used Pearson’s chi-squared test ( ). To apply this criterion, it is necessary to calculate Pearson statistics using the formula = ∑ ( ) , and compare it with tabular values ( , − 3). Here k is the number of intervals, mi – the theoretical number of the random variable values in the i-th interval, ni – the actual number of the random variable values in the i-th interval, = 0.05 – the level of significance of the test. In our case (0.05,10 − 3) = 14.07. If ≤ the hypothesis of Cauchy (Laplace) distribution is accepted, otherwise it is rejected. The results of test of hypothesis for the cryptocurrency return distribution are shown in the tables 3, 4. It is seen that for most cases the hypothesis of the corresponding distribution is accepted at the level = 0.05. The tables 3, 4 also show the results of testing the hypothesis of the return distribution for Google stocks and Amazon stocks. Comparing table 3 and table 4, we can conclude that the Laplace distribution more accurately describes the distribution of cryptocurrency return compared to the Cauchy distribution.
The Cauchy distribution function has the form ( ) = + .
Here is the mathematical expectation (median) of return, is the coefficient of distribution function chosen by us for each case in accordance with the least squares method.
The Laplace distribution function F(x) has the form ( ) = ( ),
≤ (5) Here x is the return on financial assets, is the mathematical expectation (median) of return, is the coefficient of distribution function chosen by us for each case in accordance with the least squares method.
To determine the coefficient an interval distribution table was constructed. The role of the minimized value was the sum of the squares of the differences between the theoretical and actual values of the frequency at different intervals (equation 3). The parameter (median) for the various cryptocurrencies and periods are shown in table 1.
Using the form of the Cauchy distribution function (4), we can find an analytic
expression for the frontier of risk zone at a given confidence level [
. = + ( ).
(6) (7) Similarly, for the Laplace distribution, from relation (4) we determine an analytic expression for the frontier of risk zone Using (2), (6), (7) we calculated the risk value V at the level of 5% for each cryptocurrency at the appropriate period of time (quarter). However, the risk value calculated on the basis of the Cauchy distribution (riskC) is twice the value of the risk calculated on the basis of the Laplace distribution (riskL). For example, for Bitcoin in the 1st quarter of 2018 the value risk Cauchy = 25.89%, the value risk Laplace = 12.60%. A similar situation is observed for other cryptocurrencies and periods (table 5). For comparison, the Table 5 also shows statistics for Google stocks of and Amazon stocks. The standard deviation, which is a measure of risk in the normal distribution, almost halves the risk compared to the estimate obtained from the Laplace distribution (table 5). In this regard, the question arises: which of the two distributions described above most adequately describes the risks of cryptocurrencies: the Cauchy distribution or the Laplace distribution?
Mediane 0.38 StDev 1.99 RiskC 6.87 RiskL 3.97 StDev/RiskL 0.5 RiskC/RiskL 1.73 Mediane StDev RiskC RiskL StDev/RiskL RiskC/RiskL
Analysis of relations (4) and (5) showed that the Cauchy distribution has very long and heavy tails (fig. 4). In this regard, the risk zone frontier determined on the basis of the Cauchy distribution will be significantly smaller than the risk zone frontier determined on the basis of the Laplace distribution. In this case, the number of return cases that fall into the risk zone determined on the Cauchy distribution basis will be significantly less than the number of cases that fall into the risk zone determined on the Laplace distribution basis. After counting the number of cases that fall into the risk zone, we can conclude which of the two distribution laws more adequately describes the distribution of cryptocurrency returns in the negative return zone. The results of counting the number of critical cases (the case where the return falls into the risk zone) at the confidence level 5% are shown in table 6 and table 7.
20 15 y c n eu10 q e r f 5
2018 2019 2020 1 2 3 4 1 2 3 4 1 2 All cases All days Frequency, %
As can be seen from Table 6, in the case where the risk area is determined based on the Cauchy distribution at the confidence level 5%, the average probability for falling of cryptocurrency return into the risk area is 0.6% – 0.7% (except Litecoin). For the stocks return the average probability for falling into the risk zone is 0.5% – 0.6%. That is, the actual frequency of critical cases is 10 times less than theoretically predicted. Hence the conclusion about the inadequate description of the distribution of cryptocurrency (stock) returns in the negative return zone using the Cauchy distribution. 1 2 3 4 1 2 3 4 210220 All cases All days Frequency, %
2018 2019
In the case where the risk area is determined based on the Laplace distribution at the confidence level 5% (Table 7), the average probability for falling of cryptocurrency return (and stock return) into the risk area is 4% – 5%. Thus, the actual frequency of critical cases is close to the theoretically predicted. Thus, the Laplace distribution is an adequate basis for the risk measure of negative returns of cryptocurrency’s and stock returns. 5
When forming a cryptocurrencies portfolio, first of all it is necessary to take into account their return. Figure 5 shows the average return of cryptocurrencies for the two quarters of 2020. The best cryptocurrencies in terms of profitability are BTC, ETH, XRP and BCH. As can be seen from fig. 5, the average quarterly return on stocks significantly exceeds the average quarterly cryptocurrency return. This means that stocks are more attractive for long-term investments. Cryptocurrencies are a tool for speculative transactions.
Another important aspect of portfolio formation is taking into account the risks of cryptocurrencies and taking into account the correlations of their profitability. The minimum risk is typical for cryptocurrency USDT.
From the correlation matrix (table 8) it can be seen that the return of the cryptocurrency is sufficiently correlated with each other (except USDT). It is clear that the cryptocurrency USDT is the most important component of the portfolio, which will reduce its risk (fig. 6).
For the building the cryptocurrencies portfolio, let’s used the technique, described
in previous research [
So, using the obtained above cryptocurrency risk estimates RiskL (table 5, column 20_Q2), we constructed the set of optimal portfolios (the efficient frontier). Each such portfolio gives maximum return at the established risk level. The table 9 presents the portfolio structure for each, obtained by us, optimal solution. The analysis of the table confirms the well-known statement that a higher return level always requires a higher risk degree. As you can see, the main role in the formation of the portfolio is played by cryptocurrencies ETH and USDT. The first provides high profitability, the second guarantees low risk. Other cryptocurrencies play the role of extras and do not participate in the formation of the portfolio.
To increase profitability, the portfolio can include shares of well-known companies. We will introduce Amazon stocks into the previous portfolio instead of the low-yield cryptocurrency LTC. Similar to the above, we obtained the set of optimal portfolios presented in table 10. The main role in the formation of the portfolio is played by stocks Amazon and cryptocurrency USDT. The stocks provide high profitability, the cryptocurrency guarantees low risk. The introduction of Amazon’s stock to the portfolio halved the portfolio’s risk and doubled its profitability. Thus, we conclude that optimal portfolios should be built by combining cryptocurrencies and stocks of
BTC 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 highly profitable stable companies. The sets of optimal portfolios presented in tables 9 and 10 are illustrated in fig. 7 and fig. 8. Due to its volatility, cryptocurrencies are an attractive tool for short-term investments. However, high volatility is a source of great risk. For assessing of cryptocurrencies risk, it is necessary to identify the return distribution. Numerous studies show that cryptocurrencies return and stocks return are not subject to normal distribution. The aim of our research is to compare the application of the Cauchy distribution and the Laplace distribution to the description of the actual distribution of cryptocurrency yields. A comparison of the actual return frequency in the critically low zone with its theoretical value was used as an evaluation criterion. Calculations performed for six cryptocurrencies over a 30-month period showed that the Cauchy distribution describes well the return distribution in the central part, but greatly overestimates the probability of marginal values of return. In our opinion, using the Laplace distribution is the most adequate approach to measuring the risk of cryptocurrencies (stocks). 0.80 0.70 0.60 0.50 % , rn0.40 u t e R0.30 0.20 0.10 0.00
A comparison of cryptocurrencies returns with the stocks return of leading companies showed that the average quarterly return of cryptocurrencies is low. Thus, it can be concluded that stocks are more attractive for long-term investments. Cryptocurrencies are a tool for speculative trans-actions. Inclusion of stocks of high-yield companies in the cryptocurrency’s portfolio allows in-creasing portfolio profitability and reducing portfolio risk. We have shown that inclusion AMZN stocks into the cryptocurrency portfolio can double the portfolio’s yield and halve its risk.