“The correlation strength of the most important cryptocurrencies in the bull and bear market”

The article explores the correlation strength of the ten most important cryptocurren- cies, emphasizing the examination of differences during the periods of rising and falling prices. The daily and weekly returns of selected cryptocurrencies are taken as the basis for calculating and determining the correlation strength using the Pearson correlation coefficient. The survey covers the period from the beginning of 2017 to Bitcoin’s last local bottom in mid-March 2020. Research findings are as follows: 1) the most important cryptocurrencies are mostly moderately positively correlated with each other over time; 2) correlation strength decreases slightly during the bull period, but mostly remain in the range of moderate correlation; 3) correlation strength increases signifi- cantly during the bear period, with most cryptocurrencies strongly correlated with each other. The results do not change significantly if the daily or weekly cryptocur- rency returns are used as the basis. A strong correlation in the period of falling prices prevents the effective diversification of the cryptocurrency portfolio, which must be considered when investing funds in the cryptocurrency market.


INTRODUCTION
A cryptocurrency is defined as a digital or virtual currency that uses cryptographic algorithms to ensure security. Most cryptocurrencies are based on blockchain technology. This allows for developing distributed systems for conducting currency transactions and maintaining ledgers (Investopedia, 2019). The origin of the first cryptocurrency, Bitcoin, dates back to early 2009, when an unknown programmer, nicknamed Satoshi Nakamoto, produced the prototype of a distributed computer system for processing cryptocurrencies (Nakamoto, 2008). The great success of Bitcoin soon spurred the creation of many new cryptocurrencies, so that now, in mid-2020, thousands of cryptocurrencies could be traded with a total capitalization of more than USD 240 billion (Coinmarketcap, 2020).
The cryptocurrency market slowly picked up in 2019 after the big drop was witnessed the year before. The first half of 2020 also led to quite a few new insights, raising the question of whether investments in cryptocurrencies, particularly in Bitcoin, can, in the long run, become an alternative to investments in traditional investment assets, such as stocks and especially precious metals, the most important of which is gold. It became apparent that cryptocurrencies are not independent of what is happening in the stock markets, as their prices directly followed the fall of price indices on the most important global stock exchanges with the emergence of the COVID-19 virus pandemic. Today, Bitcoin strongly dictates the cryptocurrency market's general mood, accounting for more than 70% of the total market capitalization (TradingView, 2020). This gives Bitcoin an undisputed leading role, and it seems other cryptocurrencies are merely following it, not being able to make their own, independent paths by themselves.
The daily and weekly returns of selected cryptocurrencies are taken as the basis for calculating and determining the correlation strength using the Pearson correlation coefficient. Several analyses can be found on the World Wide Web that attempts to assess the correlation strength between individual pairs of cryptocurrencies, taking into consideration various time periods (mostly from one month to one year). As opposed to those analyses, which are more short-term, the study in this article focuses on a longer period of time (more than three years), with the emphasis on the examination of differences of correlation strength during the periods of rising prices (bull market) and falling prices (bear market).

LITERATURE REVIEW
Although blockchain technology is relatively new and cryptocurrencies have been traded extensively only for the past five years, several articles on this subject in the scientific literature address its individual aspects, both economic and technological, in more detail. Corbet et al. (2019) provide a systematic review of scientific literature, which has considered cryptocurrencies as new financial assets since 2019. Most studies are on the subject of efficiency of cryptocurrencies, finding that cryptocurrencies' efficiency has been relatively low in the past, but is slowly increasing over the years (Tran & Leirvik, 2020; Caporale, Gil-Alana, & Plastun, 2018; Kristoufek & Vosvrda, 2019). The research also shows that the importance of cryptocurrencies to the global economy will increase in the future. This prediction is contingent by several factors, the most important of which are the use of cryptocurrencies as a substitute for traditional currencies, the establishment of appropriate legislation, and the continued development of blockchain technology (Seetharaman, Saravanan, Patwa, & Meht, 2017). The popularity of cryptocurrencies, especially Bitcoin, is largely influenced by the increase of its value over the past decade, driven by growing demand (Ciaian, Rajcaniova, & Kancs, 2015). Demand is contingent on different triggers (De la Horra, de la Fuente, & Perote, 2019) -in the short run, Bitcoin behaves as a speculative asset, while speculations are not expected to have a major impact in the long run. Thus, demand is largely based on expectations about Bitcoin's future usability as a medium of exchange or preservation of financial value. In general, investors' sentiment has a high predictive power on the future of Bitcoin, and the influence of sentiment is great-er in low sentiment regimes than in high sentiment regimes (Burggraf, Huynh, Rudolf, & Wang, 2020). The role of social media (on-line forums, portals, social networks, etc.), whose climate is an important predictor in determining the value of Bitcoin and other cryptocurrencies, should not be neglected (Mai, Shan, Bai, Wang, & Chiang, 2018).
Interestingly, macroeconomic data have no impact on Bitcoin's profitability (Corbet, Larkin, Lucey, Meegan, & Yarovaya, 2020). Opinions are divided on the relationship between traditional markets and cryptocurrencies. Thus, some studies suggest a relatively strong impact on Bitcoin by traditional stock market indices, such as S&P 500 and Dow Jones (Wang, Chen, & Zhao, 2020), while others (Corbet, Meegan, Larkin, Lucey, & Yarovaya, 2018), based on analyses of dependence between cryptocurrencies and various other financial assets, such as gold and bonds, find that cryptocurrencies are quite isolated from other assets. However, the general finding is that the volatility of cryptocurrencies significantly exceeds traditional financial assets' volatility.
For this research, the authors are mainly interested in existing works dealing with the association between cryptocurrency pairs. Giudici and Polinesi (2019) thus find that Bitcoin prices on different cryptocurrency exchanges are strongly associated, with exchanges acting together as an effective single cryptocurrency market. The largest and most stable exchanges (such as Bitstamp) are the most important price-setters. According to their findings, Bitcoin's price is not associated with indices on traditional stock exchanges; however, volatility on traditional stock exchanges has a negative and time-delayed effect on Bitcoin volatility. Sifat, Mohamad, and Mohamed Sharif (2019) note that when considering the association between Bitcoin and Ethereum, based on daily and hourly data within one year, a two-way causality is indicated. The studies addressing multiple cryptocurrency pairs also confirm a strong and positive association between profitability (Katsiampa, Corbet, & Lucey, 2019; Aslanidis, Bariviera, & Martínez, 2019) and the interdependence between the Bitcoin and alternative cryptocurrency markets (Ciaian & Rajcaniova, 2018). In examining different time frames, the finding that stands out is that the price ratio between Bitcoin and alternative cryptocurrencies is significantly stronger in the short term than in the long term. However, cryptocurrencies' association strength is highly variable (Antonakakis, Chatziantoniou, & Gabauer, 2019) and ranges between 25% and 75%. It depends primarily on high or low market uncertainty, with high uncertainty corresponding to the strong association and low uncertainty corresponding to the weak association.
A quick review of basic price graphs of the strongest cryptocurrencies in terms of capital gives the impression that they are interdependent or follow the most important cryptocurrency -Bitcoin. Several analyses can be found on-line, which, using the statistical method of the correlation coefficient, scientifically confirm this impression to a high degree. For example, the Hackernoon website (Hackernoon, 2018) points out that the degree of correlation between Bitcoin and other cryptocurrencies has grown since the beginning of 2017 and stabilized in mid-2018, with the high growth of correlations also confirmed by a study of Cermak (2019), highlighting a weaker association between cryptocurrencies that use the so-called Proof of Work mechanism to validate transactions, compared to cryptocurrencies that use other consensus-building mechanisms. Similar conclusions are reached by the study published on cryptodigestnews.com (Cryptodigestnews, 2018), which takes as a basis the data on the values of ten most important cryptocurrencies for each year (from 2016 to 2018), and clearly shows an increase in the degree of cross-correlation. The analysis conducted by Binance Research (Binance Research, 2020) states that even in 2019, there is still a relatively high degree of the positive correlation between the 20 most important cryptocurrencies, with most ratios exceeding a coefficient of 0.5. The study further focuses on analyzing the relationships between degrees of correlation in individual quarters of the previous year. The largest change in correlations occurred in the third quarter of 2019, suggesting that cryptocurrencies could be more strongly correlated with unfavorable market price movements and less correlated with upward or sideways movements.
Interesting insights can also be gained by using graphs on the coinmetrics.io website (Coinmetrics, 2020). This allows us to obtain a graphical representation of the correlation strength between individual pairs of cryptocurrencies for the selected time period, correlation coefficient (Spearman, Pearson), and the period included in calculating the coefficient (90, 180, or 360 days). The graphs' analysis confirms previous findings, while a slight decline in the association strength can be detected in 2019. Another interesting website is cryptowat. ch (Cryptowat, 2020), which also provides an interactive overview of correlations between selected cryptocurrencies, with the data provided in tabular form.

AIMS
This article aims to investigate the correlations between cryptocurrencies with the largest market capitalization in the first half of 2020. The data from the last three years (from the beginning of 2017 to March 2020) were included in the study, with some cryptocurrencies included in the study only appearing in the second half of 2017. This article aims to explore the correlation strength of the ten most important cryptocurrencies with the largest market capitalization in the first half of 2020.

METHODS
The calculation of the correlation strength between individual pairs of cryptocurrencies is based on the statistical method of the Pearson correlation coefficient. This is the most commonly used measure of linear association of two numerical variables, requiring the use of at least the interval type of both analyzed variables and their linear association. The coefficient ranges between -1 and 1 and is calculated using both variables' covariance and standard deviations. The Pearson correlation coefficient answers whether a linear association between variables exists at all, and if so, how strong is this association. There are two possible types of association: • a positive association exists when the values of both the first (x) variable and the second (y) variable are high or low. In such cases, the coefficient is positive and close to 1; • a negative association exists when the first (x) variable values are high, and the values of the second (y) variable are low, or vice versa. In such cases, the coefficient is negative and close to -1.
The basis for interpreting the results of the Pearson correlation coefficient was defined by Cohen (Cohen, 1988) and upgraded by Rosenthal (Rosenthal, 1996): a coefficient of 0.1 is interpreted as a weak correlation, 0.3 as moderate, 0.5 as strong, and 0.7 as very strong. The basic scale was subsequently expanded and reworked into an interval form; the following intervals are used to interpret the results in the study: The data on the values of the ten most important cryptocurrencies were taken from the Coinmarketcap website (Coinmarketcap, 2020), which provides an archive of average prices for all cryptocurrencies traded in cryptocurrency exchanges. The archive includes daily values of prices at the beginning and end of each day, the highest and lowest achieving daily value, the volume of transactions, and market capitalization. Cryptocurrencies are traded 24 hours a day, throughout the year, which means that there is no opening and closing trading value, as in the case of trading on traditional exchanges; these two val-ues thus only represent the value of cryptocurrency in a certain time interval (the closing price of the previous day and the opening price of the next day are the same). The closing prices of cryptocurrencies are used with two time intervals -day and week -as the basis for calculations.
The correlation strength between cryptocurrencies can be most easily calculated by directly using the previously mentioned closing prices (daily or weekly) for the variables' values. The approach using levels often overestimates the strength of the correlation, which can lead to unreliable estimates. From this point of view, an approach using the mathematical concept of percentage change, which gives the rate of change over time and is often used in financial analysis, is more recommended. The percentage change is the basis for calculating the daily or weekly returns of an individual cryptocurrency, using the following formula: where t represents the unit of time -day or week.
The obtained daily or weekly returns recorded in percentages and rounded to two decimal places were the input data for calculating correlation coefficients between individual pairs of cryptocurrencies in selected periods (the research question dictated the testing of correlation coefficients throughout the studied period, the period of increasing and decreasing cryptocurrency prices).
The study of the correlation strengths cryptocurrencies includes the following periods: • the entire period from the beginning of 2017 to the last local date, i.e., from 1 January 2017 to 12 March 2020. Currencies Bitcoin Cash, Eos, Binance Coin, Cardano, and Tron did not appear on cryptocurrency exchanges until later in 2017, so the calculations take into account the values from the date of registration; • two periods of growth, specifically from 1 January 2017 to 16  The following cryptocurrencies were included in the study: Bitcoin, Ethereum, Ripple, Bitcoin Cash, Litecoin, Eos, Binance Coin, Stellar, Cardano, and Tron. Tether and Bitcoin SV, which otherwise ranked fourth and sixth, were excluded from the study, while Cardano and Tron, which rank eleventh and twelfth, respectively, were included in the study. Tether (Investopedia, 2019) is a special type of cryptocurrency called a stablecoin. Its basic purpose is to maintain the price stability of cryptocurrencies, which is in stark contrast to classic cryptocurrencies' characteristics, which are characterized by high volatility. Thus, any correlation between Tether and other cryptocurrencies could not be expected, as its value hardly changes. However, the reason for excluding Bitcoin SV is its short existence, as it was created as a hard fork of Bitcoin Cash only in November 2018 (Investopedia, 2020); from this point of view, it differs significantly from other cryptocurrencies included in the study. The insufficiently long existence prevents the appropriate statistical analysis for the article, as its primary goal is to address the correlation strengths over a longer period of time. Table 1 shows correlation coefficients between cryptocurrency pairs for the entire period in question, based on their daily return. It shows that there is a positive, mostly moderate correlation between all currencies. Assuming that Bitcoin, due to its market capitalization, has the greatest impact on the cryptocurrency market, the results are first analyzed in terms of the correlation between Bitcoin and other cryptocurrencies. The analysis results for the entire period show a strong or at least moderate association between Bitcoin and other cryptocurrencies. Ethereum and Litecoin have the strongest association with Bitcoin. These are cryptocurrencies that have been present on the market for a long time and have been included in this research for the entire period (since the beginning of 2017). There are two main reasons for the strong association: these cryptocurrencies have a similar set of supporters and investors who invest in less risky investments (relative to others), i.e., in established cryptocurrencies, and at the same time, react more thoughtfully in periods of high volatility (they are more experienced, informed); and second, a longer period of time was included in the calculation compared to other currencies.

RESULTS
Here the year 2017 should be pointed out, which was marked by the constant growth of all cryptocurrencies, without major deviations. In 2018 and 2019, markets were more turbulent, as cryptocurrencies reacted differently to related events (regulation of the market, ban on trading in several countries, theft of coins, etc. Coin with other cryptocurrencies are even weaker, as it has a strong association only with Bitcoin and moderate associations with other cryptocurrencies, and even here, quite a few correlations are bordering on weak. As it turns out, the average value of correlation strength is lowest in Binance Coin among all cryptocurrencies included in the study, at 0.395. Other cryptocurrencies are generally more strongly associated with the strongest cryptocurrencies in terms of capital, and less so to each other. The exception that proves the rule is the pair of Cardano and Stellar, where a strong correlation is found. The results change to some extent if only the period of growth of cryptocurrency prices (Table 2) is included in the calculation. It is generally accepted that the rate of correlation between Bitcoin and other cryptocurrencies falls during periods of price growth. Thus, Bitcoin has a strong associa- In a period of decreasing prices (Table 3), the strength of Bitcoin's correlation with other cryptocurrencies increases significantly, and in most cases, even achieves a very strong association. In the case of Litecoin, Ethereum, Bitcoin Cash, and Binance Coin, the value is very close to 1, which represents a perfect correlation. The strong correlation of these five cryptocurrencies indicates that their movement in the bear period is very similar, so the portfolio's diversification does not result in any advantage. Similar is true for Eos and Cardano, whereas Tron and Ripple deviate from this slightly (although the strength of their correlation with Bitcoin is still very strong or strong). Of the cryptocurrencies examined, Stellar proves to be an exception, where the strength of the correlation with Bitcoin slightly decreases during the period of falling prices, but remains within the range of moderate correlation. During this period, Stellar deviates from the cryptocurrencies included in the study, as it has a strong correlation only with Ripple and Cardano, moderate correlation with other cryptocurrencies, and a weak correlation with Binance Coin. Summarizing the authors' findings, it can be concluded, that during the bear period, cryptocurrencies are generally (very) strongly positively correlated with each other, with individual exceptions. These exceptions give us room for maneuver and the necessary information to decide how to allocate the investments during periods that are not the most profitable for investing in the cryptocurrency market.
Further is the analysis of the differences in the correlation strength coefficients between price growth periods and drop for an individual currency pair ( The results presented earlier are based on the calculation of daily return for an individual cryptocurrency. The correlation strengths between cryptocurrencies, using their weekly returns as the basis, are examined in Table 5, which shows the correlation coefficients between pairs of cryptocurrencies for the entire period weekly. Table  6 shows the differences in the value of correlation coefficients for the entire period, using daily and weekly returns. It turns out that there are certain differences between the two calculations, which means that returns on a daily and weekly basis between individual pairs of cryptocurrencies are not fully aligned. If the average difference in the case of the two strongest cryptocurrencies (Bitcoin and Ethereum) is less than 0.1, it increases to 1.5 for most other cryptocurrencies, and even to over 2 for Tron.
By studying the correlation strength of cryptocurrencies based on weekly returns for the entire period, the findings regarding the correlation strength of cryptocurrencies based on daily returns can be fully confirmed, despite the mentioned differenc-    es. The Table 5 shows a positive, mostly moderate correlation between all cryptocurrencies. The strength of correlation between Bitcoin and other cryptocurrencies is mostly moderate or strong, with Eos surprising weekly, as it is even more strongly correlated with Bitcoin than Litecoin and Ethereum. On the other hand, the cryptocurrencies Ripple and Stellar are on the line between moderate and weak association. When examining Ethereum, a higher degree of correlation is ob-served with other cryptocurrencies weekly than Bitcoin and is most strongly correlated with Eos, while the degree of correlation weekly is slightly reduced in general. Ripple is also interesting, as its correlation strength with quite a few cryptocurrencies increases greatly weekly and even reaches a very strong level with Cardano and Tron. In the case of Bitcoin Cash, there are no major changes, while the correlation strength of Litecoin and Eos with other cryptocurrencies mostly increases significantly weekly, which means that the two cryptocurrencies achieve very similar results as others. Binance Coin is characterized by a decrease in strength of correlation with most other cryptocurrencies, especially with the strongest ones in terms of capital, while not exceeding the weak association with as many as four others. With Cardano and Tron, the situation is reversed -the strength of correlation is mostly increased when using weekly returns, whereas no clear conclusions can be drawn for Stellar.
The analysis of differences in the value of correlation coefficients between the periods of increasing and decreasing prices for an individual cryptocurrency pair, using weekly returns, is shown in Table 7. It also shows that correlation coefficients increase for most cryptocurrency pairs, with a decrease occurring only in individual cases. Similar to the examination daily, Binance Coin (0.55) has the largest average increase weekly, while the order of other cryptocurrencies changes. It is directly followed by Tron and Litecoin, with most cryptocurrencies having a change rate between 0.35

DISCUSSION
Comparing the results of this study to data found on cryptowat.ch and coinmetrics.io (showing the strength of correlations between pairs of cryptocurrencies for a period of at least one day to a maximum of one year in tabular or graphical form) shows that Bitcoin is more weakly associated with other cryptocurrencies than stated by these websites. The research in several scientific papers also suggests a mostly strong association between both Bitcoin and alternative cryptocurrencies ( A detailed examination of the correlation graphs on the coinmetrics.io website for a multi-year period suggests that over time the strength of correlations between individual pairs of cryptocurrencies varies greatly (for example, in 2018, the strength of correlations between Bitcoin and Litecoin varied between 0.8 and 0.48). Similar results for 2019 are also provided by the study conducted by Binance. com (Binance Research, 2020), which examined changes in correlation coefficients by individual quarters. A longer research timeframe thus reduces the extreme values (extremely strong or insignificant correlation) that can be observed in shorter periods (e.g., 1 month), which ultimately leads to more balanced results (smaller differences between pairs of cryptocurrencies). However, the authors were interested in how the strength of correlations between cryptocurrencies differs between periods of rising and falling prices. The results, both with daily and weekly returns, are undoubtedly interesting as they suggest large differences in the strength of correlations between the two periods. If cryptocurrencies are mostly only moderately correlated during the bull market period, the strength of correlation between them increases significantly during the bear market period. The differences are significant: on average, the strength of correlation increased by 0.35, and the largest increase reached a value of 0.8. The increase is present, with rare exceptions, in all cryptocurrency pairs, which means that it is a common phenomenon that should be given full attention when planning the initial portfolio of cryptocurrencies and in its management over time.
Generally, any portfolio diversification reduces financial risk for investors, ensuring better results if investment assets are as uncorrelated as possible. Our study shows that there is always a certain degree of the positive correlation between cryptocurrencies, with at least moderate association even in the long run. Investors who invest all their as-sets only in the cryptocurrency market are thus exposed to extremely high risks, which were also demonstrated in practice in 2018 when the prices of most cryptocurrencies fell by more than 80%. To some extent, the risk can be reduced by a diversification of the portfolio by including weakly associated pairs of cryptocurrencies; however, this approach does not prove effective in a period of falling prices, which is usually the most critical. Thus, this research findings confirm the generally accepted belief that investors should allocate their assets to various types of investments (real estate, stocks, precious metals, etc.), with cryptocurrencies representing only a part of the portfolio. How much should be invested in cryptocurrencies depends on the type of investor, their willingness to take risks, the long-term nature of the investment, and, last but not least, their confidence in the future of blockchain technology.

CONCLUSION
In this article, the results of a study that aims to determine the strength of correlation between the ten most important cryptocurrencies in the last three years or since their inception are presented. The article contributes a new piece to the mosaic of existing scientific and expert contributions examining the issue of association strength between cryptocurrencies themselves and between cryptocurrencies and other investment assets. The key finding is that the strength of correlations between cryptocurrencies is significantly higher during a period of falling prices than during a period of rising prices. Exceptions are quite rare (e.g., EOS), so the cryptocurrency investment portfolio's effective diversification is very difficult to achieve. However, generally, the study results confirm previous findings, which indicate at least a moderate, if not already strong, positive correlation between cryptocurrencies. The differences in the results are primarily due to a longer period included in the calculation than other studies. Thus, the pos-itive correlation is characteristic of all cryptocurrency pairs examined, regardless of whether daily or weekly returns are included in the calculation.
In the future, the research will be expanded to a larger number of cryptocurrencies (50 or 100), while the authors will continue to strive to include the longest possible research period. It would be interesting to know whether the results can also be confirmed in the case of correlation between weaker cryptocurrencies in terms of capital, which have generally been present on the market only for the last two to three years. Potentially different results (weaker association in a period of falling prices) would allow us to change how the investments in cryptocurrencies are viewed in the long run, which could result in the identification of alternative investment strategies. However, until then, the following unwritten rule still applies: Invest in cryptocurrencies only that part of your portfolio that you are also willing to lose.