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Common risk factors in the returns on cryptocurrencies
Journal Pre-proof Common risk factors in the returns on cryptocurrencies Weiyi Liu, Xuan Liang, Guowei Cui PII:
S0264-9993(19)31020-X
DOI:
https://doi.org/10.1016/j.econmod.2019.09.035
Reference:
ECMODE 5002
To appear in:
Economic Modelling
Received Date: 8 July 2019 Revised Date:
22 September 2019
Accepted Date: 25 September 2019
Please cite this article as: Liu, W., Liang, X., Cui, G., Common risk factors in the returns on cryptocurrencies, Economic Modelling (2019), doi: https://doi.org/10.1016/j.econmod.2019.09.035. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Common Risk Factors in the Returns on Cryptocurrencies* Weiyi Liu1, Xuan Liang2†, Guowei Cui3 1
School of Finance, Capital University of Economics and Business, Beijing 100070, China
2
Research School of Finance, Actuarial Studies and Statistics, The Australian National University,
Acton ACT 2601, Australia 3
School of Economics, Huazhong University of Science and Technology, Wuhan 430074, China
ABSTRACT This paper identifies three common risk factors in the returns on cryptocurrencies, which are related to cryptocurrency market return, market capitalization (size) and momentum of cryptocurrencies. Investigating a collection of 78 cryptocurrencies, we find that there are anomalous returns that decrease with size and increase with return momentum, and the momentum effect is more significant in small cryptocurrencies. Moreover, Fama-Macbeth regressions show the size and momentum combine to capture the cross-sectional variation in average cryptocurrency returns. In the tests of the three-factor model, we find most cryptocurrencies and their portfolios have significant exposures to the proposed three factors with insignificant intercepts, demonstrating that the three factors explain average cryptocurrency returns very well. Keywords: Cryptocurrency; Market return; Size; Momentum; Factor model JEL classification: G12
*
This work is supported by grants from the National Natural Science Foundation of China (No. 71601132) and Foundation for the Excellent Talents of Beijing (No. 2016000020124G083). We have no competing interests. † Corresponding Author: Xuan Liang, Research School of Finance, Actuarial Studies and Statistics, The Australian National University, Acton ACT 2601, Australia. Email addresses: [email protected] (Weiyi Liu), [email protected] (Xuan Liang), [email protected] (Guowei Cui).
Common Risk Factors in the Returns on Cryptocurrencies
ABSTRACT This paper identifies three common risk factors in the returns on cryptocurrencies, which are related to cryptocurrency market return, market capitalization (size) and momentum of cryptocurrencies. Investigating a collection of 78 cryptocurrencies, we find that there are anomalous returns that decrease with size and increase with return momentum, and the momentum effect is more significant in small cryptocurrencies. Moreover, Fama-Macbeth regressions show the size and momentum combine to capture the cross-sectional variation in average cryptocurrency returns. In the tests of the three-factor model, we find most cryptocurrencies and their portfolios have significant exposures to the proposed three factors with insignificant intercepts, demonstrating that the three factors explain average cryptocurrency returns very well. Keywords: Cryptocurrency; Market return; Size; Momentum; Factor model JEL classification: G12
1. Introduction Since its inception in 2009 as an open-source digital currency, Bitcoin has inspired and provoked the release of a large number of cryptocurrencies based on blockchain technology. The cryptocurrencies are shown to be extremely volatile with large average returns (Brauneis and Mestel, 2018) in contrast to the traditional currencies. Thus, they are mostly regarded as a new class of assets. In consideration of the huge and rapid growth of the cryptocurrency market, developing an appropriate asset pricing model on cryptocurrencies is an important and emergent complement to modern financial economics and investment theory. Finding common risk factors is always an open challenge in applying the arbitrage pricing theory, in which the expected return of a financial asset can be modeled as a linear function of various factors or theoretical market indices. There has been a great amount of literature focusing on looking for effective factors for stock returns. For example, the market excess return is the first factor that forms the CAPM model (Sharpe, 1964); the size and value factors are proposed in the Fama and French (1993) three-factor model; the momentum factor is further added in the Carhart (1997) four-factor model; the profitability and investment factors instead of the momentum factor are constructed in the Fama and French (2015) five-factor model. The above six factors are the most ones widely adopted in real applications, although there are still many other works aiming to add more contribution to the explanation of average returns. Harvey et al. (2016) catalogue 316 anomalies and consider them to be potential factors in asset pricing models. Nevertheless, Fama and 1
French (2018) argued these various anomalies may produce similar results to the combinations of the six factors mentioned above. Compared with the abundant researches in stock markets, the study on asset pricing for cryptocurrencies is still very limited and not well developed. Although there are several similarities between the stocks and cryptocurrencies on the empirical facts, such as leptokurtosis (Chan et al., 2017), heteroscedasticity (Gkillas and Katsiampa, 2018) and long-memory (Phillip et al., 2019), cryptocurrency assets are fundamentally different from stocks, as the effective stock factors mentioned above are mostly based on the traditional financial theories that suggest stock prices are determined by the present discounted value of fundamentals (Miller and Modigliani, 1961; Campbell and Shiller, 1988), while these fundamentals are not directly related to cryptocurrencies. Gregoriou (2019) regresses the ten most heavily traded cryptocurrencies to the Fama French stock factors and obtains significant positive alphas, indicating that the abnormal returns on cryptocurrencies cannot be explained by stock markets. Liu and Tsyvinski (2018) further show cryptocurrencies have no exposure to most common stock markets and macroeconomic factors or the returns of currencies and commodities. Bhambhwani et al. (2019) use the data of the five most prominent cryptocurrencies and also show there is little relationship between the cryptocurrency returns and stock factors, while they posit that cryptocurrency prices are related to the computing power and the adoption of their respective blockchains. In addition, Corbet et al. (2018) also find evidence that the dynamics of cryptocurrencies are relatively isolated to a variety of other financial 2
assets, indicating that it is practically impossible to construct effective cryptocurrency factors based on the external information from other types of financial markets. In the light of the research above, this paper focuses on the factors that are specific to cryptocurrency markets. We note Sovbetov (2018) find some factors, such as market beta, trading volume, and volatility, are significantly related to the prices of the most common five cryptocurrencies by Autoregressive Distributed Lag cointegration framework. Unlike their approach, the method we use is inspired by Fama and French (1993, 2012, 2015) to construct the cryptocurrency factors, which is quite different from to use the Fama-French stock factors that can be directly obtained from Kenneth French’s website. Since there is no financial data supporting the intrinsic value of cryptocurrencies, we finally obtain three factors, market return, SMB (small minus big), and WML (winner minus loser), that can be considered to analogically construct cryptocurrency factors, based on a collection of all the well-recorded cryptocurrencies (78 complete time series described in Section 2), instead of only several prominent ones, so that the proposed factors can be well defined and the split-sample portfolio analysis become possible. Anomalies and factors are treated different in this paper, where the anomalies are variables which differ from different cryptocurrencies, such as market beta, size and momentum, while the corresponding factors (market return, SML, and WML) are common for cryptocurrencies and their loadings determine cryptocurrency return attributions. The potential economic intuitions behind the three factors can be similar with those in stock markets. The market factor can be defined via the value-weighted 3
cryptocurrency portfolio returns, which is based on the consideration that the cryptocurrencies may have some co-movements that formulate systematic risk across cryptocurrencies, similar to those described in the CAPM model; The SML factor suggests that small cryptocurrencies have higher average returns mainly because they are less liquid and the small coin holders require higher returns for accepting liquidity risk (Amihud and Mendelson, 1986; Liu, 2006). The WML factor assumes buying past winners and selling past losers realize abnormal returns, which might be caused by the delayed price reactions to cryptomarket-related information, as those also discussed in Jegadeesh and Titman (1993), Carhart (1997), and Grobys and Sapkota (2019). As anomaly variables are known to cause problems for the factor models, it is reasonable to analyze the anomalies before the factors. We use two steps to identify anomalies. First, we sort the cryptocurrency returns on anomaly variables, i.e. market beta, size, and momentum. When sorting on market beta, the resulting portfolio return differences are not statistically significant. However, when sorting on size and momentum, it shows significant anomalous returns that decrease with size and increase with return momentum, separately. The long-short portfolio produces an average weekly return of 35.87 with a t-statistic of 3.83 for the lowest level of size to the highest level of size produces and 36.26 percent with a t-statistic of 3.89 for the highest level of momentum to the lowest level of momentum. We also confirm the significant negative relation between size and future returns as well as the significant positive relation between momentum and future returns by using Fama-MacBeth 4
regression. The above results have some interactions with the findings in Liu et al. (2019). Their paper also considered some other cryptomarket-related factors such as volatility and trading volume, while they are not so significant or generally explained by the size and momentum factors. Unlike their factors that are defined directly by anomalies or via the one-dimensional sorting results, our factors are defined different from anomalies and are based on the double sorting results, following the methods by Fama and French (2012, 2015), so that the factor correlation can be reduced and additional results are further investigated. We then use the time-series regression approach to study how cryptocurrencies and their portfolios are priced through the proposed three-factor model. From the double sorting results, we further find the momentum effect is more significant for small cryptocurrencies, which makes a gap of 50.64 percent per week with a t-statistic of 4.42 for buying a small winner portfolio and selling a small loser portfolio. The SML and WML factors in this paper are constructed through the double sorting results, and then their correlation can be reduced to 0.66. Implementing the time-series regressions to the 78 cryptocurrencies on the three factors, it is shown from the significance of the intercepts that 93.99% cryptocurrencies in our sample period can be well explained through our three-factor model. When we turn to the size-momentum portfolios, they generally have significant exposures to the three factors with insignificant intercepts, again demonstrating our three-factor model performs well in explaining the common variation in cryptocurrency returns. 5
The rest of paper is organized as follows. Section 2 describes the data used in our analysis and the anomaly variables we use to predict returns. Section 3 studies the average returns from sorts and cross-sectional regressions on the anomaly variables. In Section 4 our three-factor pricing model is defined and the results of a set of time-series regression tests are discussed. Section 5 discusses the robustness check. Section 6 concludes the paper.
2. Data and variables We obtain daily cryptocurrency prices, dollar volume, and market capitalization from https://coinmarketcap.com/. The sample period is from 07-Aug-15 to 31-Dec-18 with total 1243 trading days, where 07-Aug-15 is the starting date of Ethereum trading. Similar starting date is also used in Liu (2019). We use cryptocurrencies with a complete time series at the end of 2018, leaving a data set consisting of 78 cryptocurrencies. The risk-free rate is defined by the one-month U.S. Treasury bill rate from the Center of Research in Security Prices. We examine the patterns of average returns in a weekly horizon. The weekly returns and corresponding anomaly variables are converted from the daily data of each cryptocurrency (from Tuesday close price to Tuesday close price in next week), which finally leads to 178-week observations. We also carried out the robustness check of daily data in section 5 and the main findings remain the same. Our asset pricing tests are based on two different methods. The first method is conducted by sorting the anomaly variables and the cross-sectional regression of 6
Fama and MacBeth (1973). The second method uses the time-series regression approach on the factor models as Fama and French (1993, 2012, 2015), where the common risk factors are usually formed by excess returns that are related to the anomaly variables. In this section, we introduce the anomaly variables of cryptocurrencies, while we leave the factor definitions and factor models to be discussed in Section 4. We measure the anomaly variables used to forecast returns in a rolling-window scheme. In other words, we use available information at time t to forecast the returns in t to t+1, where the time t+1 means one week after time t in this paper. Here we introduce three variables that could be potential anomalies. The first one is market beta. We define the cryptocurrency market beta as the slope in the regression of a cryptocurrency’s return on the market return, where the market return is the return on the value-weighted cryptocurrency market portfolio. We use 52 week data (nearly 1 year) preceding time t to estimate the betas in a rolling-window. Then, the estimated betas can be treated as proxies for the real ones at time t, although might suffer from estimation errors, to forecast the returns in t to t+1 in the sorting and cross-sectional regression analysis, as we will see these results in Section 3. The second one is market capitalization (size). The definition of the size variable is much easier than that of market betas, since it can be precisely measured for each individual cryptocurrency. We define the market capitalization as the natural log of price times circulating supply at time t to be another anomaly variable. 7
The third one is momentum. We use one-year momentum in cryptocurrency returns, i.e., the cumulative return from t-52 to t for each cryptocurrency. The reason for choosing one-year horizon to define momentum can be attributed to the study of Jegadeesh and Titman (1993), Carhart (1997), and Grobys and Sapkota (2019).
3. Cross-section of expected returns In this section, we first report the characteristics of the portfolios formed by sorting cryptocurrencies into groups based on the anomaly variables introduced in Section 2. Then the interaction between any two of the anomaly variables using double sorts are studied. Finally, we carry out Fama-MacBeth regressions to analyze the cross-section of cryptocurrency returns on the anomaly variables. 3.1 One-dimensional sorts Each Tuesday, 78 cryptocurrencies are ranked into 6 groups with equal group size of 13 cryptocurrencies according to their market value, size, and momentum, respectively. Then, we construct equal-weighted and value-weighted portfolios by the returns in the following week. Notice that the anomaly variables should fall behind the average returns with lag 1 to make sure the information we use is known. Table 1 reports the equal-weighted and value-weighted weekly returns for portfolios formed on market beta, size and momentum, in Panel A, B and C, respectively. The two kinds of weighted average returns both show a decreasing pattern on size and an increasing pattern on momentum. The return for cryptocurrencies with the lowest level of size is 38.67 percent for equal-weighted 8
portfolios and 24.38 percent for the value-weighted portfolios, while the return for cryptocurrencies with the highest level of size is 2.80 percent for equal-weighted and 2.12 percent for value-weighted portfolios. Similarly, the return for cryptocurrencies with the highest level of momentum is 40.68 percent for equal-weighted portfolios and 7.88 percent for the value-weighted portfolios, while the return for cryptocurrencies with the lowest level of momentum is 4.42 percent for value-weighted and 3.01 percent for value-weighted portfolios. However, there is no obvious relationship between market beta and average return. It seems a U shape for the equal-weighted case, and relatively flat for the case of value-weighted case. The insignificant t-statistic values for most of market beta portfolios in the value-weighted case indicate that it lacks strong explanatory power for subsequent returns. [Insert Table 1 here] We also assess the empirical relation between cryptocurrency returns and anomaly variables by adjusting for standard measures of risk from Table 1. The alphas presented in table are the intercepts from regressions of the portfolio returns on the cryptocurrency market return. Note that the alphas are large and statistically significant for the lowest level of size as well as the highest level of momentum, while there are no significant alphas for any of the market beta portfolios in the value-weighted case. Hence, we may still conclude that the size and momentum are reliable indicators of subsequent one-week cryptocurrency returns in cross-section aspect.
9
3.2 Double sorts We further analyze the interaction between size and momentum, size and beta, as well as momentum and beta, respectively. Taking the pair of size and momentum as an example, the double sort is conducted as follows: Cryptocurrencies are first ranked into 3 groups on size, namely small, neutral, and big portfolios. Then, each group is further divided into 3 subgroups (winner, neutral and loser) on momentum. Finally, we can obtain the 3×3 equal-weighted and value-weighted portfolios from two-pass sorts of cryptocurrencies on size and momentum. Panel A of Table 2 shows weighted average returns for portfolios formed on size and momentum. We can observe that, in each column of Panel A, average returns largely decrease from small cryptocurrencies to big cryptocurrencies, showing a significant size effect. In each row of Panel A, the momentum effect of cryptocurrencies is also quite clear for small and neutral groups of cryptocurrencies, while the big cryptocurrencies is the only exception. According to the results of Panel A, the equal-weighted premium of 56.84 percent and value-weighted premium of 35.62 percent can be earned by buying small winner cryptocurrencies and selling big loser cryptocurrencies. [Insert Table 2 here] Panel B and Panel C illustrate similar information. In each column of Panel B, average returns fall from small cryptocurrencies to big cryptocurrencies, similar to Panel A. In each row column of Panel C, the momentum effect also exhibits in different level of market beta. Comparing the results of Panel B and Panel C, we can 10
find that the size effect is more consistent than momentum effect in the double sorting portfolios. In each column of Panel B and Panel C, we can find the structure of market beta is not obvious for any of size-beta portfolios or momentum-beta portfolios, which confirms the findings in Table 1 that the market beta does not have explanatory power for average returns in cross-section aspect.
3.3 Fama-MacBeth Regressions To further evaluate the relation between future returns and anomaly variables, we carry out various cross-sectional regressions using the method proposed in Fama and MacBech (1973). Each week t, we compute the anomaly variables for cryptocurrency i and estimate the following cross-sectional regression:
ri ,t +1 = γ 0,t + γ 1,t βi ,t + γ 2,t log( sizeit ) + γ 3,t momentumi ,t + ε i ,t +1 ,
(1)
where ri ,t +1 is the weekly return (in percent) of the i-th cryptocurrency for week t + 1 ,
β i ,t is the market beta of the i-th cryptocurrency estimated via 52 week data (nearly 1 year) preceding time t, log(sizeit ) is the log of market cap of the i-th cryptocurrency at time t, and momentumi ,t is the lagged return in the past one year of the i-th cryptocurrency. [Insert Table 3 here] Table 3 reports the time series average of the coefficients for seven cross-sectional regression models, where the last column represents the full model (1). The first column presents the results of the regression of cryptocurrency return on market beta. The coefficient associated with market beta is -0.05 with a t-statistic of 11
-0.02. This confirms that there does not seem to be a significant relation between cryptocurrency returns and market betas. The second and third columns confirm the relation between the cryptocurrency return and size and momentum, respectively. In column 2, the coefficient associated with log size is -4.01 with a t-statistic of -3.51. Similarly, in column 3, the coefficient on momentum is 0.017 with a t-statistic of 3.46. In column 4-6, we report regression results to any two of the anomaly variables. We can observe that the coefficients of beta and log size have some changes after adding the momentum term, but the corresponding statistical significance remains unchanged. In the last column, we use all the anomaly variables simultaneously. The coefficients on log size and momentum are again significantly negative and positive, respectively. The results displayed in Table 3 are consistent with the results in Table 1 and Table 2.
4. The three-factor model In this section, we first introduce the definitions of the factors and the design of the factor model. Then the proposed factors are used to construct three-factor model to explain the average returns of cryptocurrencies. Finally, we test the performance of the factor model on the cryptocurrency portfolios formed on size and momentum. 4.1 Factor definitions This paper focuses on three factors that are related to the anomaly variables mentioned in Section 3. The first factor is the market excess return, which has been described before. Here we introduce the construction process of size and momentum factors. 12
As we have stated in Section 3, we use two-pass sorts instead of independent sorts to construct the 3×3 size and momentum portfolios. That is mainly because we have in total 78 cryptocurrencies, and the two-pass sorts can make sure each subgroup has nearly the same number of cryptocurrencies, while the independent sorts define the portfolios by intersections which might lead to imbalance on the number of each subgroup. Moreover, we use 3×3 instead of 4×4 or 5×5 so that each subgroup has a certain number of cryptocurrencies. Notice that the two-pass sort results may be different if we change the order of the first sort and the second sort, i.e., small winner (SW) is usually distinct from winner small (WS), same for the others. Therefore, we define our SMB and WML as follows:
SMBW = SW − BW , SMBN = SN − BN , SMBL = SL − BL ,
(2)
WMLS = WS − LS , WMLN = WN − LN , WMLB = WB − LB ,
(3)
SMB = (SMBW + SMBN + SMBL ) / 3 ,
(4)
WML = (WMLS + WMLN + WMLB ) / 3 ,
(5)
where the return of each subgroup is value-weighted so that the defined factors can be more robust to the outliers. Finally, our factor asset-pricing model for cryptocurrencies can be written as
Ri (t ) − RF (t ) = ai + bi [RM (t ) − RF (t )] + si SMB(t ) + wWML (t ) + ei (t) . i
(6)
[Insert Table 4 here] Table 4 shows summary statistics for factor returns. From the results in Panel A, we can observe the three factors all have significant positive average returns, where the SMB enjoys the highest return that is 16.61 percent per week with a t-statistic of 13
8.57. WML is also very significant at 10.94 percent per week with a t-statistic of 5.47. The market excess return is only 2.12 percent per week with a t-statistic of 1.96. Panel B shows the correlation matrix of the three factors. We can find that SMB and WML are nearly uncorrelated to the market excess return with correlation 0.0027 and 0.0074 respectively, while there is a certain degree of correlation between SMB and WML at 0.6606. Panel C and Panel D exhibit three versions of SMB and WML defined in (2) and (3). The results presented in Panel C and Panel D are consistent with the double sorting results in Table 2.
4.2 Explain average returns of cryptocurrencies We now use the returns of 78 cryptocurrencies to test the explanatory ability of the three-factor model. If an asset pricing model completely captures expected returns, the intercept is indistinguishable from zero. To save space, we only exhibit the cases of three biggest, three medium, and three smallest cryptocurrencies in Table 5. Panel A of Table 5 reports the estimate results for three biggest cryptocurrencies: Bitcoin, Ethereum, and XRP. We can see the intercepts of these three cryptocurrencies are all insignificant, demonstrating that the three-factor model explains the three cryptocurrencies returns very well. The exposures of these big cryptocurrencies are mainly on the market excess return factor, while Bitcoin has significant exposures to all of the three factors with adjusted R2 at 0.89, which indicates that the three-factor model fits the returns of Bitcoin very well. Panel B of Table 5 shows the results of three medium cryptocurrencies: 14
BlackCoin, Potcoin, and MonetaryUnit. From the t-statistics of the three cryptocurrencies we can conclude that the three-factor model also explains their average returns. The BlackCoin and Potcoin have significant exposures to the market factor, while the MonetaryUnit has significant exposure to the size and momentum factor. Unlike the biggest cryptocurrencies, the exposures of the medium cryptocurrencies are diversified. Panel C of Table 5 provides the results for three smallest cryptocurrencies: Bata, TEKcoin, and AnarchistsPrime. We can find that Bata and AnarchistsPrime are well priced by the three factor model, and the exposures of these small cryptocurrencies tend to be located on the size and momentum factors. However, we note that the intercepts of TEKcoin is significantly bounded away from zero and there are no significant exposures to any of the three factors, which means the three-factor model fails to explain the returns of TEKcoin. [Insert Table 5 here] Beyond the above reported 9 cryptocurrencies, we have implemented the regressions to all the 78 cryptocurrencies and obtained the corresponding intercepts and t-statistics. Among the 78 cryptocurrencies, there are only 5 of them rejecting the null hypothesis of intercept being zero at 0.05 significant level. In other words, 93.99% cryptocurrencies in our sample period can be well explained through our three-factor model.
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4.3 Asset pricing tests for size-momentum portfolios To access the model performance, we examine the regressions on the portfolios from 3×3 sorts on size and momentum. The regression details are reported in Table 6, including the intercepts and pertinent slopes of the three-factor model. [Insert Table 6 here] Panel A of Table 6 shows intercepts from the three-factor regressions for the 3×3 size-momentum portfolios. There is no obvious pattern of the positive or negative value of the intercepts for different portfolios. Nevertheless, the t-statistics demonstrate the intercepts are all economically and statistically close to zero. The highest one is the small winner portfolio, whose intercept is 3.15 with a t-statistic of 1.44, still far less than the rejection threshold. In short, the three-factor model works well for the size-momentum portfolios. Panel B to Panel D of Table 6 show the slopes and t-statistics for the three factors. It is observed that the market factor slopes are always close to 1 and are all strongly significant. The SMB slopes are strongly positive for small cryptocurrencies and slightly negative for big cryptocurrencies. The WML slopes are significantly positive for winner cryptocurrencies and also significantly negative for loser cryptocurrencies. From the t-statistics of the three slopes, we observe that all the 9 portfolios have significant exposures to the three factors, so that these three factors play an very important role in pricing cryptocurrencies. [Insert Table 7 here] Table 7 shows matrices of the intercepts and their t-statistics for nested models. 16
Panel A is the CAPM type model for cryptocurrencies. The small winner portfolio’s intercept is still very large at 37.04 with a t-statistic of 7.56. Obviously, it lacks of power to explain the average returns for either small cryptocurrencies or winner cryptocurrencies. Panel B and Panel C add the SMB and WML factors to the model of Panel A separately. While both of them have improved the explanatory power, the t-statistics reduced much more for adding SMB factor. As a consequence, only small winner cryptocurrencies cannot be explained by the model of Panel B, while most of the small cryptocurrencies cannot be explained by the model of Panel C. Panel D is the full three-factor model, which is exactly the same results we presented in Panel A of Table 6. Thus, we can conclude the three-factor model performs well in explaining the common variation in cryptocurrency returns.
5. Robustness analysis In this section, we examine the results under a daily scheme as a robustness check kindly suggested by a referee. The raw daily data is from 07-Aug-15 to 31-Dec-18 with total 1243 trading days. While monthly or weekly data are more frequently adopted for analyzing the long-short strategies in real applications, using daily frequency data would significantly enlarge the sample size, as the development of cryptocurrency markets is still in its infancy. [Insert Table 8 here] Panel A of Table 8 shows the average daily percent excess returns for cryptocurrency portfolios via one-dimensional sorts on market beta, size, and 17
momentum, respectively. Similar to the results in Table 1, the average returns show a decreasing pattern on size and an increasing pattern on momentum, while there is no obvious pattern on market beta. It can be observed that the returns of the long-short zero-investment strategies are significant for size and momentum, but not significant for market beta. Panel B of Table 8 shows the average slopes from day-by-day cross-sectional regressions under the Fama-MacBeth method presented in equation (1). It can be found that no matter the whole model or the nested models, they all demonstrate the significance of size and momentum and the insignificance of market beta, which are consistent with the findings in Table 3. [Insert Table 9 here] Panel A of Table 9 presents average daily excess returns for 3×3 portfolios formed on two-pass double sorts on size and momentum. From each column of Panel A, we can find obvious size effects in winner, neutral, and loser groups. On the other hand, for the rows of Panel A, it reveals significant momentum effect especially for small cryptocurrencies. These findings are quite similar to that in Table 2. Panel B of Table 9 reports the estimation results by regressing the 3×3 portfolios on the three factors, respectively. From the t-statistics presented in Table 9, we can find that the portfolios generally have significant exposures to the proposed three factors, while the regression intercepts are all insignificant, which again shows the three-factor model works well for the size-momentum portfolios.
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6. Conclusion In this paper, three common factors specific to cryptocurrency market, i.e. the market factor and factors related to size and momentum, are proposed to study the average returns of cryptocurrencies. To evaluate the factor performance, we use both the methods of cross-sectional analyses for the anomaly variables and time-series regressions for the risk factors. The results show the size effect and momentum effect are strong in cryptocurrency markets, and the proposed three-factor model has satisfied explanation ability on cryptocurrency returns. On the other hand, the complexities of the cryptocurrency markets are far from fully explored. Although our model has well explained a large amount of cryptocurrencies’ returns, many new patterns beyond those from stock markets could be discovered in future and become potential candidates in factor models for cryptocurrencies. Our analysis could serve as a benchmark for identifying the additional factors.
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Table 1 Average weekly percent excess returns for cryptocurrency portfolios formed on one-dimensional sorts on beta, size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks. At the end of each Tuesday, cryptocurrencies are allocated to six groups based on quintiles (each group has 13 cryptocurrencies), using beta, size and momentum, respectively. In the sort time t, beta means the market beta estimated by the data from t-52 to t; size is the market cap at time t; momentum is the cumulative return from t-52 to t. Then, the equal-weighted and value-weighted returns are calculated from t to t+1. Alpha is the intercept obtained from regression of the portfolio return on market return. Equal-weighted
Value-weighted
Panel A: Sorted on beta Low Return Alpha
2
3
4
5
High
Low
2
3
4
5
High
15.75
5.82
7.18
6.42
9.43
26.95
2.36
4.65
3.23
3.35
2.75
3.68
(2.61)
(3.38)
(3.51)
(3.67)
(4.40)
(3.55)
(2.18)
(2.35)
(2.03)
(1.82)
(1.71)
(1.84)
11.86
3.37
4.49
3.95
6.63
25.54.
0.91
2.69
0.95
0.95
0.70
0.82
(2.04)
(2.78)
(2.90)
(3.19)
(4.07)
(3.32)
(1.13)
(1.45)
(0.86)
(0.68)
(0.57)
(0.59)
Panel B: Sorted on size Low Return Alpha
2
3
4
5
High
Low
2
3
4
5
High
38.67
15.22
6.72
4.87
3.27
2.80
24.38
14.48
6.61
4.84
2.55
2.12
(3.94)
(6.23)
(3.46)
(2.91)
(2.26)
(1.76)
(6.57)
(6.48)
(3.43)
(2.86)
(1.95)
(1.63)
35.40
12.82
3.98
2.49
0.89
0.28
21.43
12.10
3.89
2.46
0.06
-0.01
(3.59)
(5.95)
(2.89)
(2.12)
(1.12)
(0.29)
(6.21)
(6.32)
(2.86)
(2.05)
(0.07)
(-0.72)
Panel C: Sorted on momentum Low Return Alpha
2
3
4
5
High
Low
2
3
4
5
High
4.42
4.43
6.20
6.96
8.86
40.68
3.01
2.99
3.10
3.95
2.89
7.88
(2.92)
(2.98)
(3.64)
(3.48)
(3.76)
(4.19)
(1.86)
(2.05)
(1.72)
(2.05)
(1.61)
(3.24)
2.02
2.12
3.61
4.29
5.98
37.84
0.95
0.58
0.70
1.32
0.35
5.91
(2.18)
(2.38)
(3.30)
(2.87)
(3.18)
(3.87)
(0.76)
(0.72)
(0.52)
(0.94)
(0.28)
(2.62)
22
Table 2 Average weekly percent excess returns for cryptocurrency portfolios formed on two-pass sorts on size and momentum, size and beta, momentum and beta: 07-Aug-15 to 31-Dec-18, 178 weeks. At the end of each Tuesday, cryptocurrencies are allocated to three groups based on tertiles of the first variable in the first step, and then each group is further divided into three subgroups based on tertiles of the second variable. The definitions of beta, size and momentum are the same as Table 1. Equal-weighted
Value-weighted
Panel A: Size-Momentum portfolios Loser Small Neutral Big
Neutral
Winner
9.17
15.51
59.81
(4.88)
(5.04)
(3.91)
4.30
5.75
7.53
(2.77)
(3.01)
(3.51)
2.97
3.18
2.80
(2.05)
(2.12)
Loser Small
Neutral
Winner
8.86
11.24
39.07
(4.57)
(4.35)
(7.94)
4.01
5.55
6.89
(2.72)
(2.96)
(3.14)
3.45
2.71
3.16
(1.62)
(2.15)
(1.60)
(1.73)
Low
Neutral
High
9.27
14.68
25.54
(4.55)
(5.88)
(6.69)
Neutral Big
Panel B: Size-Beta portfolios Small Neutral Big
Low
Neutral
High
22.56
16.03
44.16
(2.61)
(6.07)
(3.58)
6.08
4.63
6.77
(3.60)
(2.65)
(3.03)
2.37
3.06
3.76
(1.81)
(1.98)
Small Neutral
6.06
4.03
6.16
(3.60)
(2.41)
(2.74)
2.54
3.86
1.82
(2.04)
(2.13)
(2.59)
(1.29)
Low
Neutral
High
2.63
4.89
8.53
(1.54)
(2.39)
(3.43)
3.08
2.72
4.04
(1.93)
(1.33)
(2.00)
Big
Panel C: Momentum-Beta portfolios Winner Neutral Loser
Low
Neutral
High
21.86
13.13
41.14
(2.54)
(5.35)
(3.35)
6.00
7.45
6.24
(3.69)
(3.32)
(3.30)
4.19
4.48
4.61
(2.89)
(2.89)
(2.92)
Winner Neutral Loser
23
4.51
2.47
2.56
(2.36)
(1.82)
(1.67)
Table 3 Average coefficients from week-by-week cross-sectional regressions of cryptocurrency returns on beta, size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks. The average slope is the time-series average of the weekly regression slopes, and the t-statistic is the average slope divided by its time-series standard error. The variables are the same as Table 1.
Intercept Market beta
(1)
(2)
(3)
(4)
(5)
(6)
(7)
10.94
76.13
-0.81
66.11
-3.82
26.53
24.08
(3.96)
(3.63)
(-0.25)
(3.90)
(-1.05)
(1.87)
(2.02)
-0.14
1.46
(-0.07)
(0.75)
-0.05 (-0.02)
Log(size)
Adjusted R
2
(0.70)
-4.01
-3.48
-1.64
-1.66
(-3.51)
(-3.50)
(-2.05)
(-2.17)
Momentum R2
1.38
0.017
0.020
0.015
0.017
(3.46)
(4.86)
(3.36)
(5.39)
0.044
0.035
0.109
0.077
0.193
0.132
0.214
0.031
0.022
0.097
0.052
0.171
0.108
0.182
24
Table 4 Summary statistics for weekly factor percent returns: 07-Aug-15 to 31-Dec-18, 178 weeks. At the end of each Tuesday, cryptocurrencies are assigned to 3×3 subgroups through a two-pass sorts described in Table 2. In each subgroup, the return of the portfolio is based on value-weighted. SMBW is the average of return on the portfolios of small winner cryptocurrencies minus the average return on the portfolios of big winner cryptocurrencies, SMBN and SMBL are the same but for portfolios of neutral and loser cryptocurrencies. WMLS, WMLN, and WMLB are defined in the same way. SMB is the average of SMBW, SMBN, and SMBL. WML is the average of WMLS, WMLN, and WMLB. Panel A, Panel C and Panel D show average weekly returns, the standard deviations and t-statistics of the average returns. Panel B reports the correlation matrix of the three factors. Panel A
Panel B
RM -RF
SMB
WML
RM -RF
SMB
WML
Mean
2.12
16.61
10.94
RM -RF
1.0000
Std dev.
12.27
21.93
22.62
SMB
0.0027
1.0000
0.6606
t-statistic
1.96
8.57
5.47
WML
0.0074
0.6606
1.0000
Panel C
Panel D
SMBW
SMBN
SMBL
WMLS
WMLN
WMLB
Mean
35.91
8.52
5.41
Mean
30.21
2.88
-0.29
Std dev.
54.72
25.50
20.07
Std dev.
56.02
18.22
22.84
t-statistic
7.42
3.78
3.05
t-statistic
6.10
1.79
-0.14
‘
25
Table 5 Using three factors in regressions to explain average returns on three biggest, three medium, and three smallest cryptocurrencies: 07-Aug-15 to 31-Dec-18, 178 weeks. RM -RF is the value-weighted return on the cryptocurrency market portfolio; SMB (small minus big) is the size factor of cryptocurrencies; WML (winner minus loser) is the momentum factor of cryptocurrencies. The detailed factor definitions are described in Table 4. RM -RF
SMB
WML
R2
Adj. R2
-0.03
0.91
0.05
-0.06
0.90
0.89
(-0.07)
(32.79)
(2.25)
(-3.16)
0.26
1.13
-0.03
0.11
0.56
0.55
(0.19)
(12.40)
(-0.50)
(1.62)
5.44
1.32
-0.10
-0.06
0.25
0.23
(1.66)
(6.26)
(-0.63)
(-0.36)
-0.84
1.06
0.13
-0.04
0.37
0.36
(-0.43)
(8.47)
(1.41)
(-0.48)
2.49
1.04
0.26
-0.33
0.18
0.16
(0.73)
(4.73)
(1.60)
(-2.05) 0.05
0.03
0.18
0.16
0.07
0.05
0.07
0.05
Int Panel A: 3 biggest cryptocurrencies Bitcoin Ethereum XRP
Panel B: 3 medium cryptocurrencies BlackCoin Potcoin MonetaryUnit
7.51
0.94
0.71
-0.65
(0.93)
(1.81)
(1.82)
(-1.71)
-0.15
1.34
0.76
-0.32
(-0.03)
(4.04)
(3.09)
(-1.33)
27.82
0.82
0.22
0.66
(3.13)
(1.43)
(0.52)
(1.60)
-27.82
2.29
21.29
-11.90
(-0.19)
(0.24)
(2.98)
(-1.72)
Panel C: 3 smallest cryptocurrencies Bata TEKcoin AnarchistsPrime
26
Table 6 Tests of three-factor model for weekly value-weighted returns on cryptocurrency portfolios from 3×3 sorts on size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks. The regressions use the three-factor model to explain the excess return cryptocurrency portfolios formed on size and momentum. Panel A shows three-factor intercepts and their t-statistics. Panel B, Panel C, and Panel D report the slopes for RM -RF, SMB, and WML, and t-statistics for these coefficients.
R(t)-RF(t)=a+b[RM(t)- RF(t)] +sSMB(t)+wWML(t)+e(t) Size\Momentum
Loser
Panel A:
Neutral
Winner
Loser
a
Neutral
Winner
t(a)
Small
1.68
-1.24
3.15
1.10
-0.55
1.44
Neutral
-0.87
-1.83
-2.49
-0.79
-1.25
-1.23
Big
1.44
0.55
1.59
1.01
0.40
1.04
Panel B:
b
t(b)
Small
1.12
1.21
0.93
11.42
8.28
6.59
Neutral
1.05
1.27
1.15
14.75
13.40
8.86
Big
0.99
1.19
1.08
10.65
13.40
10.95
Panel C:
s
t(s)
Small
0.62
0.93
1.16
8.46
8.59
11.04
Neutral
0.27
0.33
0.28
5.04
4.66
2.85
Big
0.18
-0.10
-0.37
2.60
-1.48
-5.03
Panel D:
w
t(w)
Small
-0.50
-0.51
1.34
-7.09
-4.83
13.16
Neutral
-0.16
-0.07
0.21
-3.18
-1.05
2.27
Big
-0.28
0.11
0.50
-4.20
1.78
6.96
27
Table 7 Interprets from nested factor models in regressions to explain weekly value-weighted returns on cryptocurrency portfolios formed on size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks Panel A shows one-factor intercepts and their t-statistics produced by the cryptocurrency market return. Panel B and Panel C show two-factor intercepts and their t-statistics, with adding SMB and WML to the model of Panel A, respectively. Panel D provides the intercepts and their t-statistics of the proposed three-factor model. Size\Momentum
Loser
Neutral
Winner
Loser
Neutral
Winner
Panel A: R(t)-RF(t)=a+b[RM(t)- RF(t)]+e(t) a
t(a)
Small
6.47
8.68
37.04
4.21
3.83
7.56
Neutral
1.78
2.85
4.43
1.85
2.20
2.42
Big
1.35
0.17
0.86
1.11
0.15
0.60
Panel B: R(t)-RF(t)=a+b[RM(t)- RF(t)]+sSMB(t)+e(t) a
t(a)
Small
1.88
-1.03
3.61
1.10
-0.42
1.77
Neutral
-0.80
-1.80
-2.57
-0.70
-1.23
-1.26
Big
1.56
0.55
1.39
1.02
0.39
0.77
Panel C: R(t)-RF(t)=a+b[RM(t)- RF(t)]+wWML(t)+e(t) a
t(a)
Small
7.64
7.72
14.30
4.52
3.07
5.26
Neutral
1.70
1.33
0.17
1.59
0.95
0.09
Big
3.17
-0.39
-1.96
2.43
-0.32
-1.33
Panel D: R(t)-RF(t)=a+b[RM(t)- RF(t)] +sSMB(t)+wWML(t)+e(t) a
t(a)
Small
1.68
-1.24
3.15
1.04
-0.55
1.44
Neutral
-0.87
-1.83
-2.49
-0.79
-1.25
-1.23
Big
1.44
0.51
1.59
1.01
0.36
1.04
28
Table 8 One-dimensional sorts and Fama-MacBeth regressions using daily data: 07-Aug-15 to 31-Dec-18, 1243 days. Pane A reports the average daily percent excess returns for cryptocurrency portfolios formed on beta, size, and momentum, respectively. Panel B shows the average slopes from day-by-day cross-sectional regressions of cryptocurrency returns on beta, size and momentum. Low
2
Panel A: One-dimensional Sorts Beta 3.22 1.02 (3.49) (4.88) Size 6.38 `1.88 (4.47) (6.75) Momentum 0.76 0.63 (3.31) (3.39) (1)
(2)
Panel B: Fama-MacBeth Regressions 2.85 12.73 Intercept (2.27) (3.64) -1.25 Market beta (-0.93) -0.69 Log(size) (-3.52)
4
5
High
High-Low
1.94 (1.79) 0.68 (3.09) 0.89 (4.18)
1.16 (4.70) 0.57 (2.81) 1.18 (4.52)
1.08 (4.69) 0.45 (2.67) 1.15 (4.26)
1.94 (5.90) 0.41 (1.99) 5.75 (4.94)
-1.26 (-1.32) -5.97 (-4.18) 4.98 (3.51)
(3)
(4)
(5)
(6)
(7)
-0.08 (-0.15)
12.67 (3.50) -1.76 (-1.28) -0.59 (-3.28)
1.10 (0.70) -1.21 (-0.72)
6.08 (2.46)
0.0025 (3.24)
-0.36 (-2.59) 0.0018 (2.85)
7.59 (2.46) -1.35 (-0.81) -0.38 (-2.86) 0.019 (3.42)
0.115 0.091
0.091 0.066
0.130 0.094
0.0023 (3.11)
Momentum R2 Adjusted R2
3
0.042 0.029
0.018 0.005
0.075 0.063
29
0.059 0.034
Table 9 Double sorts and factor model regressions using daily data: 07-Aug-15 to 31-Dec-18, 1243 days. Panel A reports the average daily percent excess returns for cryptocurrency portfolios formed on two-pass sorts on size and momentum. Panel B provides the estimation results of the three-factor model in regressions for daily value-weighted returns on cryptocurrency portfolios formed on size and momentum. Size\Momentum
Loser
Neutral
Winner
Loser
Neutral
Winner
Panel A: Double sorts t(µ)
µ Small 1.26 2.62 Neutral 0.66 0.93 Big 0.46 0.47 Panel B: Factor model regressions a
9.05 0.25 0.36
3.82 3.10 2.55
5.63 3.54 2.04
6.32 2.03 1.55
Small Neutral Big
0.12 0.19 0.96
0.59 0.24 0.51 b
0.15 -0.07 0.30
0.81 1.53 0.81
1.43 1.32 1.41 t(b)
0.75 -0.35 1.85
Small Neutral Big
1.09 1.01 0.90
1.03 0.99 1.12 s
0.92 1.07 1.02
31.55 34.75 50.91
18.32 23.44 23.22 t(s)
20.11 22.07 27.31
Small Neutral Big
0.45 0.12 0.09
0.71 0.15 -0.32 w
1.15 -0.01 -0.47
17.88 5.72 7.04
17.11 4.87 -8.94 t(w)
34.20 -0.28 -17.20
Small Neutral Big
-0.38 -0.14 -0.13
-0.41 -0.01 0.24
1.48 0.29 0.58
-14.53 -6.30 -9.56
-9.58 -0.28 6.37
42.21 7.71 20.18
t(a)
30
Common Risk Factors in the Returns on Cryptocurrencies
HIGHLIGHTS
•
We construct three common risk factors which are specific to cryptocurrency market.
•
Size and momentum are strong in sorts, and small cryptocurrencies have more significant momentum effect.
•
The combined effect of size and momentum can largely explain the cross-sectional variation in average cryptocurrency returns.
•
Most cryptocurrencies and corresponding portfolios have significant exposures to our proposed three factors with insignificant intercepts.
S0264-9993(19)31020-X
DOI:
https://doi.org/10.1016/j.econmod.2019.09.035
Reference:
ECMODE 5002
To appear in:
Economic Modelling
Received Date: 8 July 2019 Revised Date:
22 September 2019
Accepted Date: 25 September 2019
Please cite this article as: Liu, W., Liang, X., Cui, G., Common risk factors in the returns on cryptocurrencies, Economic Modelling (2019), doi: https://doi.org/10.1016/j.econmod.2019.09.035. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Common Risk Factors in the Returns on Cryptocurrencies* Weiyi Liu1, Xuan Liang2†, Guowei Cui3 1
School of Finance, Capital University of Economics and Business, Beijing 100070, China
2
Research School of Finance, Actuarial Studies and Statistics, The Australian National University,
Acton ACT 2601, Australia 3
School of Economics, Huazhong University of Science and Technology, Wuhan 430074, China
ABSTRACT This paper identifies three common risk factors in the returns on cryptocurrencies, which are related to cryptocurrency market return, market capitalization (size) and momentum of cryptocurrencies. Investigating a collection of 78 cryptocurrencies, we find that there are anomalous returns that decrease with size and increase with return momentum, and the momentum effect is more significant in small cryptocurrencies. Moreover, Fama-Macbeth regressions show the size and momentum combine to capture the cross-sectional variation in average cryptocurrency returns. In the tests of the three-factor model, we find most cryptocurrencies and their portfolios have significant exposures to the proposed three factors with insignificant intercepts, demonstrating that the three factors explain average cryptocurrency returns very well. Keywords: Cryptocurrency; Market return; Size; Momentum; Factor model JEL classification: G12
*
This work is supported by grants from the National Natural Science Foundation of China (No. 71601132) and Foundation for the Excellent Talents of Beijing (No. 2016000020124G083). We have no competing interests. † Corresponding Author: Xuan Liang, Research School of Finance, Actuarial Studies and Statistics, The Australian National University, Acton ACT 2601, Australia. Email addresses: [email protected] (Weiyi Liu), [email protected] (Xuan Liang), [email protected] (Guowei Cui).
Common Risk Factors in the Returns on Cryptocurrencies
ABSTRACT This paper identifies three common risk factors in the returns on cryptocurrencies, which are related to cryptocurrency market return, market capitalization (size) and momentum of cryptocurrencies. Investigating a collection of 78 cryptocurrencies, we find that there are anomalous returns that decrease with size and increase with return momentum, and the momentum effect is more significant in small cryptocurrencies. Moreover, Fama-Macbeth regressions show the size and momentum combine to capture the cross-sectional variation in average cryptocurrency returns. In the tests of the three-factor model, we find most cryptocurrencies and their portfolios have significant exposures to the proposed three factors with insignificant intercepts, demonstrating that the three factors explain average cryptocurrency returns very well. Keywords: Cryptocurrency; Market return; Size; Momentum; Factor model JEL classification: G12
1. Introduction Since its inception in 2009 as an open-source digital currency, Bitcoin has inspired and provoked the release of a large number of cryptocurrencies based on blockchain technology. The cryptocurrencies are shown to be extremely volatile with large average returns (Brauneis and Mestel, 2018) in contrast to the traditional currencies. Thus, they are mostly regarded as a new class of assets. In consideration of the huge and rapid growth of the cryptocurrency market, developing an appropriate asset pricing model on cryptocurrencies is an important and emergent complement to modern financial economics and investment theory. Finding common risk factors is always an open challenge in applying the arbitrage pricing theory, in which the expected return of a financial asset can be modeled as a linear function of various factors or theoretical market indices. There has been a great amount of literature focusing on looking for effective factors for stock returns. For example, the market excess return is the first factor that forms the CAPM model (Sharpe, 1964); the size and value factors are proposed in the Fama and French (1993) three-factor model; the momentum factor is further added in the Carhart (1997) four-factor model; the profitability and investment factors instead of the momentum factor are constructed in the Fama and French (2015) five-factor model. The above six factors are the most ones widely adopted in real applications, although there are still many other works aiming to add more contribution to the explanation of average returns. Harvey et al. (2016) catalogue 316 anomalies and consider them to be potential factors in asset pricing models. Nevertheless, Fama and 1
French (2018) argued these various anomalies may produce similar results to the combinations of the six factors mentioned above. Compared with the abundant researches in stock markets, the study on asset pricing for cryptocurrencies is still very limited and not well developed. Although there are several similarities between the stocks and cryptocurrencies on the empirical facts, such as leptokurtosis (Chan et al., 2017), heteroscedasticity (Gkillas and Katsiampa, 2018) and long-memory (Phillip et al., 2019), cryptocurrency assets are fundamentally different from stocks, as the effective stock factors mentioned above are mostly based on the traditional financial theories that suggest stock prices are determined by the present discounted value of fundamentals (Miller and Modigliani, 1961; Campbell and Shiller, 1988), while these fundamentals are not directly related to cryptocurrencies. Gregoriou (2019) regresses the ten most heavily traded cryptocurrencies to the Fama French stock factors and obtains significant positive alphas, indicating that the abnormal returns on cryptocurrencies cannot be explained by stock markets. Liu and Tsyvinski (2018) further show cryptocurrencies have no exposure to most common stock markets and macroeconomic factors or the returns of currencies and commodities. Bhambhwani et al. (2019) use the data of the five most prominent cryptocurrencies and also show there is little relationship between the cryptocurrency returns and stock factors, while they posit that cryptocurrency prices are related to the computing power and the adoption of their respective blockchains. In addition, Corbet et al. (2018) also find evidence that the dynamics of cryptocurrencies are relatively isolated to a variety of other financial 2
assets, indicating that it is practically impossible to construct effective cryptocurrency factors based on the external information from other types of financial markets. In the light of the research above, this paper focuses on the factors that are specific to cryptocurrency markets. We note Sovbetov (2018) find some factors, such as market beta, trading volume, and volatility, are significantly related to the prices of the most common five cryptocurrencies by Autoregressive Distributed Lag cointegration framework. Unlike their approach, the method we use is inspired by Fama and French (1993, 2012, 2015) to construct the cryptocurrency factors, which is quite different from to use the Fama-French stock factors that can be directly obtained from Kenneth French’s website. Since there is no financial data supporting the intrinsic value of cryptocurrencies, we finally obtain three factors, market return, SMB (small minus big), and WML (winner minus loser), that can be considered to analogically construct cryptocurrency factors, based on a collection of all the well-recorded cryptocurrencies (78 complete time series described in Section 2), instead of only several prominent ones, so that the proposed factors can be well defined and the split-sample portfolio analysis become possible. Anomalies and factors are treated different in this paper, where the anomalies are variables which differ from different cryptocurrencies, such as market beta, size and momentum, while the corresponding factors (market return, SML, and WML) are common for cryptocurrencies and their loadings determine cryptocurrency return attributions. The potential economic intuitions behind the three factors can be similar with those in stock markets. The market factor can be defined via the value-weighted 3
cryptocurrency portfolio returns, which is based on the consideration that the cryptocurrencies may have some co-movements that formulate systematic risk across cryptocurrencies, similar to those described in the CAPM model; The SML factor suggests that small cryptocurrencies have higher average returns mainly because they are less liquid and the small coin holders require higher returns for accepting liquidity risk (Amihud and Mendelson, 1986; Liu, 2006). The WML factor assumes buying past winners and selling past losers realize abnormal returns, which might be caused by the delayed price reactions to cryptomarket-related information, as those also discussed in Jegadeesh and Titman (1993), Carhart (1997), and Grobys and Sapkota (2019). As anomaly variables are known to cause problems for the factor models, it is reasonable to analyze the anomalies before the factors. We use two steps to identify anomalies. First, we sort the cryptocurrency returns on anomaly variables, i.e. market beta, size, and momentum. When sorting on market beta, the resulting portfolio return differences are not statistically significant. However, when sorting on size and momentum, it shows significant anomalous returns that decrease with size and increase with return momentum, separately. The long-short portfolio produces an average weekly return of 35.87 with a t-statistic of 3.83 for the lowest level of size to the highest level of size produces and 36.26 percent with a t-statistic of 3.89 for the highest level of momentum to the lowest level of momentum. We also confirm the significant negative relation between size and future returns as well as the significant positive relation between momentum and future returns by using Fama-MacBeth 4
regression. The above results have some interactions with the findings in Liu et al. (2019). Their paper also considered some other cryptomarket-related factors such as volatility and trading volume, while they are not so significant or generally explained by the size and momentum factors. Unlike their factors that are defined directly by anomalies or via the one-dimensional sorting results, our factors are defined different from anomalies and are based on the double sorting results, following the methods by Fama and French (2012, 2015), so that the factor correlation can be reduced and additional results are further investigated. We then use the time-series regression approach to study how cryptocurrencies and their portfolios are priced through the proposed three-factor model. From the double sorting results, we further find the momentum effect is more significant for small cryptocurrencies, which makes a gap of 50.64 percent per week with a t-statistic of 4.42 for buying a small winner portfolio and selling a small loser portfolio. The SML and WML factors in this paper are constructed through the double sorting results, and then their correlation can be reduced to 0.66. Implementing the time-series regressions to the 78 cryptocurrencies on the three factors, it is shown from the significance of the intercepts that 93.99% cryptocurrencies in our sample period can be well explained through our three-factor model. When we turn to the size-momentum portfolios, they generally have significant exposures to the three factors with insignificant intercepts, again demonstrating our three-factor model performs well in explaining the common variation in cryptocurrency returns. 5
The rest of paper is organized as follows. Section 2 describes the data used in our analysis and the anomaly variables we use to predict returns. Section 3 studies the average returns from sorts and cross-sectional regressions on the anomaly variables. In Section 4 our three-factor pricing model is defined and the results of a set of time-series regression tests are discussed. Section 5 discusses the robustness check. Section 6 concludes the paper.
2. Data and variables We obtain daily cryptocurrency prices, dollar volume, and market capitalization from https://coinmarketcap.com/. The sample period is from 07-Aug-15 to 31-Dec-18 with total 1243 trading days, where 07-Aug-15 is the starting date of Ethereum trading. Similar starting date is also used in Liu (2019). We use cryptocurrencies with a complete time series at the end of 2018, leaving a data set consisting of 78 cryptocurrencies. The risk-free rate is defined by the one-month U.S. Treasury bill rate from the Center of Research in Security Prices. We examine the patterns of average returns in a weekly horizon. The weekly returns and corresponding anomaly variables are converted from the daily data of each cryptocurrency (from Tuesday close price to Tuesday close price in next week), which finally leads to 178-week observations. We also carried out the robustness check of daily data in section 5 and the main findings remain the same. Our asset pricing tests are based on two different methods. The first method is conducted by sorting the anomaly variables and the cross-sectional regression of 6
Fama and MacBeth (1973). The second method uses the time-series regression approach on the factor models as Fama and French (1993, 2012, 2015), where the common risk factors are usually formed by excess returns that are related to the anomaly variables. In this section, we introduce the anomaly variables of cryptocurrencies, while we leave the factor definitions and factor models to be discussed in Section 4. We measure the anomaly variables used to forecast returns in a rolling-window scheme. In other words, we use available information at time t to forecast the returns in t to t+1, where the time t+1 means one week after time t in this paper. Here we introduce three variables that could be potential anomalies. The first one is market beta. We define the cryptocurrency market beta as the slope in the regression of a cryptocurrency’s return on the market return, where the market return is the return on the value-weighted cryptocurrency market portfolio. We use 52 week data (nearly 1 year) preceding time t to estimate the betas in a rolling-window. Then, the estimated betas can be treated as proxies for the real ones at time t, although might suffer from estimation errors, to forecast the returns in t to t+1 in the sorting and cross-sectional regression analysis, as we will see these results in Section 3. The second one is market capitalization (size). The definition of the size variable is much easier than that of market betas, since it can be precisely measured for each individual cryptocurrency. We define the market capitalization as the natural log of price times circulating supply at time t to be another anomaly variable. 7
The third one is momentum. We use one-year momentum in cryptocurrency returns, i.e., the cumulative return from t-52 to t for each cryptocurrency. The reason for choosing one-year horizon to define momentum can be attributed to the study of Jegadeesh and Titman (1993), Carhart (1997), and Grobys and Sapkota (2019).
3. Cross-section of expected returns In this section, we first report the characteristics of the portfolios formed by sorting cryptocurrencies into groups based on the anomaly variables introduced in Section 2. Then the interaction between any two of the anomaly variables using double sorts are studied. Finally, we carry out Fama-MacBeth regressions to analyze the cross-section of cryptocurrency returns on the anomaly variables. 3.1 One-dimensional sorts Each Tuesday, 78 cryptocurrencies are ranked into 6 groups with equal group size of 13 cryptocurrencies according to their market value, size, and momentum, respectively. Then, we construct equal-weighted and value-weighted portfolios by the returns in the following week. Notice that the anomaly variables should fall behind the average returns with lag 1 to make sure the information we use is known. Table 1 reports the equal-weighted and value-weighted weekly returns for portfolios formed on market beta, size and momentum, in Panel A, B and C, respectively. The two kinds of weighted average returns both show a decreasing pattern on size and an increasing pattern on momentum. The return for cryptocurrencies with the lowest level of size is 38.67 percent for equal-weighted 8
portfolios and 24.38 percent for the value-weighted portfolios, while the return for cryptocurrencies with the highest level of size is 2.80 percent for equal-weighted and 2.12 percent for value-weighted portfolios. Similarly, the return for cryptocurrencies with the highest level of momentum is 40.68 percent for equal-weighted portfolios and 7.88 percent for the value-weighted portfolios, while the return for cryptocurrencies with the lowest level of momentum is 4.42 percent for value-weighted and 3.01 percent for value-weighted portfolios. However, there is no obvious relationship between market beta and average return. It seems a U shape for the equal-weighted case, and relatively flat for the case of value-weighted case. The insignificant t-statistic values for most of market beta portfolios in the value-weighted case indicate that it lacks strong explanatory power for subsequent returns. [Insert Table 1 here] We also assess the empirical relation between cryptocurrency returns and anomaly variables by adjusting for standard measures of risk from Table 1. The alphas presented in table are the intercepts from regressions of the portfolio returns on the cryptocurrency market return. Note that the alphas are large and statistically significant for the lowest level of size as well as the highest level of momentum, while there are no significant alphas for any of the market beta portfolios in the value-weighted case. Hence, we may still conclude that the size and momentum are reliable indicators of subsequent one-week cryptocurrency returns in cross-section aspect.
9
3.2 Double sorts We further analyze the interaction between size and momentum, size and beta, as well as momentum and beta, respectively. Taking the pair of size and momentum as an example, the double sort is conducted as follows: Cryptocurrencies are first ranked into 3 groups on size, namely small, neutral, and big portfolios. Then, each group is further divided into 3 subgroups (winner, neutral and loser) on momentum. Finally, we can obtain the 3×3 equal-weighted and value-weighted portfolios from two-pass sorts of cryptocurrencies on size and momentum. Panel A of Table 2 shows weighted average returns for portfolios formed on size and momentum. We can observe that, in each column of Panel A, average returns largely decrease from small cryptocurrencies to big cryptocurrencies, showing a significant size effect. In each row of Panel A, the momentum effect of cryptocurrencies is also quite clear for small and neutral groups of cryptocurrencies, while the big cryptocurrencies is the only exception. According to the results of Panel A, the equal-weighted premium of 56.84 percent and value-weighted premium of 35.62 percent can be earned by buying small winner cryptocurrencies and selling big loser cryptocurrencies. [Insert Table 2 here] Panel B and Panel C illustrate similar information. In each column of Panel B, average returns fall from small cryptocurrencies to big cryptocurrencies, similar to Panel A. In each row column of Panel C, the momentum effect also exhibits in different level of market beta. Comparing the results of Panel B and Panel C, we can 10
find that the size effect is more consistent than momentum effect in the double sorting portfolios. In each column of Panel B and Panel C, we can find the structure of market beta is not obvious for any of size-beta portfolios or momentum-beta portfolios, which confirms the findings in Table 1 that the market beta does not have explanatory power for average returns in cross-section aspect.
3.3 Fama-MacBeth Regressions To further evaluate the relation between future returns and anomaly variables, we carry out various cross-sectional regressions using the method proposed in Fama and MacBech (1973). Each week t, we compute the anomaly variables for cryptocurrency i and estimate the following cross-sectional regression:
ri ,t +1 = γ 0,t + γ 1,t βi ,t + γ 2,t log( sizeit ) + γ 3,t momentumi ,t + ε i ,t +1 ,
(1)
where ri ,t +1 is the weekly return (in percent) of the i-th cryptocurrency for week t + 1 ,
β i ,t is the market beta of the i-th cryptocurrency estimated via 52 week data (nearly 1 year) preceding time t, log(sizeit ) is the log of market cap of the i-th cryptocurrency at time t, and momentumi ,t is the lagged return in the past one year of the i-th cryptocurrency. [Insert Table 3 here] Table 3 reports the time series average of the coefficients for seven cross-sectional regression models, where the last column represents the full model (1). The first column presents the results of the regression of cryptocurrency return on market beta. The coefficient associated with market beta is -0.05 with a t-statistic of 11
-0.02. This confirms that there does not seem to be a significant relation between cryptocurrency returns and market betas. The second and third columns confirm the relation between the cryptocurrency return and size and momentum, respectively. In column 2, the coefficient associated with log size is -4.01 with a t-statistic of -3.51. Similarly, in column 3, the coefficient on momentum is 0.017 with a t-statistic of 3.46. In column 4-6, we report regression results to any two of the anomaly variables. We can observe that the coefficients of beta and log size have some changes after adding the momentum term, but the corresponding statistical significance remains unchanged. In the last column, we use all the anomaly variables simultaneously. The coefficients on log size and momentum are again significantly negative and positive, respectively. The results displayed in Table 3 are consistent with the results in Table 1 and Table 2.
4. The three-factor model In this section, we first introduce the definitions of the factors and the design of the factor model. Then the proposed factors are used to construct three-factor model to explain the average returns of cryptocurrencies. Finally, we test the performance of the factor model on the cryptocurrency portfolios formed on size and momentum. 4.1 Factor definitions This paper focuses on three factors that are related to the anomaly variables mentioned in Section 3. The first factor is the market excess return, which has been described before. Here we introduce the construction process of size and momentum factors. 12
As we have stated in Section 3, we use two-pass sorts instead of independent sorts to construct the 3×3 size and momentum portfolios. That is mainly because we have in total 78 cryptocurrencies, and the two-pass sorts can make sure each subgroup has nearly the same number of cryptocurrencies, while the independent sorts define the portfolios by intersections which might lead to imbalance on the number of each subgroup. Moreover, we use 3×3 instead of 4×4 or 5×5 so that each subgroup has a certain number of cryptocurrencies. Notice that the two-pass sort results may be different if we change the order of the first sort and the second sort, i.e., small winner (SW) is usually distinct from winner small (WS), same for the others. Therefore, we define our SMB and WML as follows:
SMBW = SW − BW , SMBN = SN − BN , SMBL = SL − BL ,
(2)
WMLS = WS − LS , WMLN = WN − LN , WMLB = WB − LB ,
(3)
SMB = (SMBW + SMBN + SMBL ) / 3 ,
(4)
WML = (WMLS + WMLN + WMLB ) / 3 ,
(5)
where the return of each subgroup is value-weighted so that the defined factors can be more robust to the outliers. Finally, our factor asset-pricing model for cryptocurrencies can be written as
Ri (t ) − RF (t ) = ai + bi [RM (t ) − RF (t )] + si SMB(t ) + wWML (t ) + ei (t) . i
(6)
[Insert Table 4 here] Table 4 shows summary statistics for factor returns. From the results in Panel A, we can observe the three factors all have significant positive average returns, where the SMB enjoys the highest return that is 16.61 percent per week with a t-statistic of 13
8.57. WML is also very significant at 10.94 percent per week with a t-statistic of 5.47. The market excess return is only 2.12 percent per week with a t-statistic of 1.96. Panel B shows the correlation matrix of the three factors. We can find that SMB and WML are nearly uncorrelated to the market excess return with correlation 0.0027 and 0.0074 respectively, while there is a certain degree of correlation between SMB and WML at 0.6606. Panel C and Panel D exhibit three versions of SMB and WML defined in (2) and (3). The results presented in Panel C and Panel D are consistent with the double sorting results in Table 2.
4.2 Explain average returns of cryptocurrencies We now use the returns of 78 cryptocurrencies to test the explanatory ability of the three-factor model. If an asset pricing model completely captures expected returns, the intercept is indistinguishable from zero. To save space, we only exhibit the cases of three biggest, three medium, and three smallest cryptocurrencies in Table 5. Panel A of Table 5 reports the estimate results for three biggest cryptocurrencies: Bitcoin, Ethereum, and XRP. We can see the intercepts of these three cryptocurrencies are all insignificant, demonstrating that the three-factor model explains the three cryptocurrencies returns very well. The exposures of these big cryptocurrencies are mainly on the market excess return factor, while Bitcoin has significant exposures to all of the three factors with adjusted R2 at 0.89, which indicates that the three-factor model fits the returns of Bitcoin very well. Panel B of Table 5 shows the results of three medium cryptocurrencies: 14
BlackCoin, Potcoin, and MonetaryUnit. From the t-statistics of the three cryptocurrencies we can conclude that the three-factor model also explains their average returns. The BlackCoin and Potcoin have significant exposures to the market factor, while the MonetaryUnit has significant exposure to the size and momentum factor. Unlike the biggest cryptocurrencies, the exposures of the medium cryptocurrencies are diversified. Panel C of Table 5 provides the results for three smallest cryptocurrencies: Bata, TEKcoin, and AnarchistsPrime. We can find that Bata and AnarchistsPrime are well priced by the three factor model, and the exposures of these small cryptocurrencies tend to be located on the size and momentum factors. However, we note that the intercepts of TEKcoin is significantly bounded away from zero and there are no significant exposures to any of the three factors, which means the three-factor model fails to explain the returns of TEKcoin. [Insert Table 5 here] Beyond the above reported 9 cryptocurrencies, we have implemented the regressions to all the 78 cryptocurrencies and obtained the corresponding intercepts and t-statistics. Among the 78 cryptocurrencies, there are only 5 of them rejecting the null hypothesis of intercept being zero at 0.05 significant level. In other words, 93.99% cryptocurrencies in our sample period can be well explained through our three-factor model.
15
4.3 Asset pricing tests for size-momentum portfolios To access the model performance, we examine the regressions on the portfolios from 3×3 sorts on size and momentum. The regression details are reported in Table 6, including the intercepts and pertinent slopes of the three-factor model. [Insert Table 6 here] Panel A of Table 6 shows intercepts from the three-factor regressions for the 3×3 size-momentum portfolios. There is no obvious pattern of the positive or negative value of the intercepts for different portfolios. Nevertheless, the t-statistics demonstrate the intercepts are all economically and statistically close to zero. The highest one is the small winner portfolio, whose intercept is 3.15 with a t-statistic of 1.44, still far less than the rejection threshold. In short, the three-factor model works well for the size-momentum portfolios. Panel B to Panel D of Table 6 show the slopes and t-statistics for the three factors. It is observed that the market factor slopes are always close to 1 and are all strongly significant. The SMB slopes are strongly positive for small cryptocurrencies and slightly negative for big cryptocurrencies. The WML slopes are significantly positive for winner cryptocurrencies and also significantly negative for loser cryptocurrencies. From the t-statistics of the three slopes, we observe that all the 9 portfolios have significant exposures to the three factors, so that these three factors play an very important role in pricing cryptocurrencies. [Insert Table 7 here] Table 7 shows matrices of the intercepts and their t-statistics for nested models. 16
Panel A is the CAPM type model for cryptocurrencies. The small winner portfolio’s intercept is still very large at 37.04 with a t-statistic of 7.56. Obviously, it lacks of power to explain the average returns for either small cryptocurrencies or winner cryptocurrencies. Panel B and Panel C add the SMB and WML factors to the model of Panel A separately. While both of them have improved the explanatory power, the t-statistics reduced much more for adding SMB factor. As a consequence, only small winner cryptocurrencies cannot be explained by the model of Panel B, while most of the small cryptocurrencies cannot be explained by the model of Panel C. Panel D is the full three-factor model, which is exactly the same results we presented in Panel A of Table 6. Thus, we can conclude the three-factor model performs well in explaining the common variation in cryptocurrency returns.
5. Robustness analysis In this section, we examine the results under a daily scheme as a robustness check kindly suggested by a referee. The raw daily data is from 07-Aug-15 to 31-Dec-18 with total 1243 trading days. While monthly or weekly data are more frequently adopted for analyzing the long-short strategies in real applications, using daily frequency data would significantly enlarge the sample size, as the development of cryptocurrency markets is still in its infancy. [Insert Table 8 here] Panel A of Table 8 shows the average daily percent excess returns for cryptocurrency portfolios via one-dimensional sorts on market beta, size, and 17
momentum, respectively. Similar to the results in Table 1, the average returns show a decreasing pattern on size and an increasing pattern on momentum, while there is no obvious pattern on market beta. It can be observed that the returns of the long-short zero-investment strategies are significant for size and momentum, but not significant for market beta. Panel B of Table 8 shows the average slopes from day-by-day cross-sectional regressions under the Fama-MacBeth method presented in equation (1). It can be found that no matter the whole model or the nested models, they all demonstrate the significance of size and momentum and the insignificance of market beta, which are consistent with the findings in Table 3. [Insert Table 9 here] Panel A of Table 9 presents average daily excess returns for 3×3 portfolios formed on two-pass double sorts on size and momentum. From each column of Panel A, we can find obvious size effects in winner, neutral, and loser groups. On the other hand, for the rows of Panel A, it reveals significant momentum effect especially for small cryptocurrencies. These findings are quite similar to that in Table 2. Panel B of Table 9 reports the estimation results by regressing the 3×3 portfolios on the three factors, respectively. From the t-statistics presented in Table 9, we can find that the portfolios generally have significant exposures to the proposed three factors, while the regression intercepts are all insignificant, which again shows the three-factor model works well for the size-momentum portfolios.
18
6. Conclusion In this paper, three common factors specific to cryptocurrency market, i.e. the market factor and factors related to size and momentum, are proposed to study the average returns of cryptocurrencies. To evaluate the factor performance, we use both the methods of cross-sectional analyses for the anomaly variables and time-series regressions for the risk factors. The results show the size effect and momentum effect are strong in cryptocurrency markets, and the proposed three-factor model has satisfied explanation ability on cryptocurrency returns. On the other hand, the complexities of the cryptocurrency markets are far from fully explored. Although our model has well explained a large amount of cryptocurrencies’ returns, many new patterns beyond those from stock markets could be discovered in future and become potential candidates in factor models for cryptocurrencies. Our analysis could serve as a benchmark for identifying the additional factors.
References Amihud, Y., Mendelson, H., 1986. Asset pricing and the bid-ask spread. Journal of Financial Economics 17(2), 223-249. Bhambhwani, S., Delikouras, S., Korniotis, G., 2019. Do fundamentals drive cryptocurrency prices? SSRN Working Paper NO. 3342842. Brauneis, A., Mestel, R., 2018. Price discovery of cryptocurrencies: Bitcoin and beyond. Economics Letters 165, 58-61. Campbell, J. Y. and Shiller, R. J., 1988. Stock prices, earnings, and expected dividends. Journal of Finance 43 (3), 661-767.
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Carhart, M. M., 1997. On persistence in mutual fund performance. Journal of Finance 52 (1), 57-82. Chan, S., Chu, J., Nadarajah, S., Osterrieder, J., 2017. A statistical analysis of cryptocurrencies. Risk and Financial Management 10 (12), 1-23. Corbet, S., Meegan, A., Larkin, C., Lucey, B., Yarovaya, L., 2018. Exploring the dynamic relationships between cryptocurrencies and other financial assets. Economics Letters 165, 28-34. Fama, E. F., MacBeth, J., 1973. Risk, return and equilibrium: Empirical tests. Journal of Political Economy 81, 607-636. Fama, E. F., French, K. R., 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33(1), 3-56. Fama, E. F., French, K. R., 2012. Size, value, and momentum in international stock returns. Journal of Financial Economics 105 (3), 457-472. Fama, E. F., French, K. R., 2015. A five-factor asset pricing model. Journal of Financial Economics 116(1), 1-22. Fama, E. F., French, K. R., 2018. Choosing factors. Journal of Financial Economics 128 (2), 234-252. Gregoriou, A., 2019. Cryptocurrencies and asset pricing. Applied Economics Letters 26 (12), 995-998. Gkillas, K., Katsiampa, P., 2018. An application of extreme value theory to cryptocurrencies. Economics Letters 164, 109-111. Grobys, K., Sapkota, N., 2019. Cryptocurrencies and momentum. Economics Letters, forthcoming. Harvey, C., Liu, Y., 2016. … and the cross-section of expected returns. Review of Financial Studies 29, 5-68. Jegadeesh, N., Titman, S., 1993. Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance 48, 35-91. Liu, W., 2006. A liquidity-augmented capital asset pricing model. Journal of Financial Economics 82(3), 631-671. Liu, W., 2019. Portfolio diversification across cryptocurrencies. Finance Research 20
Letters 29, 200-205. Liu, Y., Tsyvinski, A., 2018. Risks and returns of cryptocurrency. NBER Working Paper NO. 24877. Liu, Y., Tsyvinski, A., Wu, X., 2019. Common risk factors in cryptocurrencies. NBER Working Paper NO. 25882. Miller, M., Modigliani, F., 1961. Dividend policy, growth, and the valuation of shares. Journal of Business 34, 411-433. Phillip, A., Chan, J., Peris, S., 2018. On long memory effects in the volatility measure of cryptocurrencies. Finance Research Letters 28, 95-100. Sharpe, W. F., 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19 (3): 425-442. Sovbetov, Y., 2018. Factors influencing cryptocurrency prices: Evidence from Bitcoin, Ethereum, Dash, Litcoin, and Monero. Journal of Economics and Financial Analysis 2(2): 1-27.
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Table 1 Average weekly percent excess returns for cryptocurrency portfolios formed on one-dimensional sorts on beta, size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks. At the end of each Tuesday, cryptocurrencies are allocated to six groups based on quintiles (each group has 13 cryptocurrencies), using beta, size and momentum, respectively. In the sort time t, beta means the market beta estimated by the data from t-52 to t; size is the market cap at time t; momentum is the cumulative return from t-52 to t. Then, the equal-weighted and value-weighted returns are calculated from t to t+1. Alpha is the intercept obtained from regression of the portfolio return on market return. Equal-weighted
Value-weighted
Panel A: Sorted on beta Low Return Alpha
2
3
4
5
High
Low
2
3
4
5
High
15.75
5.82
7.18
6.42
9.43
26.95
2.36
4.65
3.23
3.35
2.75
3.68
(2.61)
(3.38)
(3.51)
(3.67)
(4.40)
(3.55)
(2.18)
(2.35)
(2.03)
(1.82)
(1.71)
(1.84)
11.86
3.37
4.49
3.95
6.63
25.54.
0.91
2.69
0.95
0.95
0.70
0.82
(2.04)
(2.78)
(2.90)
(3.19)
(4.07)
(3.32)
(1.13)
(1.45)
(0.86)
(0.68)
(0.57)
(0.59)
Panel B: Sorted on size Low Return Alpha
2
3
4
5
High
Low
2
3
4
5
High
38.67
15.22
6.72
4.87
3.27
2.80
24.38
14.48
6.61
4.84
2.55
2.12
(3.94)
(6.23)
(3.46)
(2.91)
(2.26)
(1.76)
(6.57)
(6.48)
(3.43)
(2.86)
(1.95)
(1.63)
35.40
12.82
3.98
2.49
0.89
0.28
21.43
12.10
3.89
2.46
0.06
-0.01
(3.59)
(5.95)
(2.89)
(2.12)
(1.12)
(0.29)
(6.21)
(6.32)
(2.86)
(2.05)
(0.07)
(-0.72)
Panel C: Sorted on momentum Low Return Alpha
2
3
4
5
High
Low
2
3
4
5
High
4.42
4.43
6.20
6.96
8.86
40.68
3.01
2.99
3.10
3.95
2.89
7.88
(2.92)
(2.98)
(3.64)
(3.48)
(3.76)
(4.19)
(1.86)
(2.05)
(1.72)
(2.05)
(1.61)
(3.24)
2.02
2.12
3.61
4.29
5.98
37.84
0.95
0.58
0.70
1.32
0.35
5.91
(2.18)
(2.38)
(3.30)
(2.87)
(3.18)
(3.87)
(0.76)
(0.72)
(0.52)
(0.94)
(0.28)
(2.62)
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Table 2 Average weekly percent excess returns for cryptocurrency portfolios formed on two-pass sorts on size and momentum, size and beta, momentum and beta: 07-Aug-15 to 31-Dec-18, 178 weeks. At the end of each Tuesday, cryptocurrencies are allocated to three groups based on tertiles of the first variable in the first step, and then each group is further divided into three subgroups based on tertiles of the second variable. The definitions of beta, size and momentum are the same as Table 1. Equal-weighted
Value-weighted
Panel A: Size-Momentum portfolios Loser Small Neutral Big
Neutral
Winner
9.17
15.51
59.81
(4.88)
(5.04)
(3.91)
4.30
5.75
7.53
(2.77)
(3.01)
(3.51)
2.97
3.18
2.80
(2.05)
(2.12)
Loser Small
Neutral
Winner
8.86
11.24
39.07
(4.57)
(4.35)
(7.94)
4.01
5.55
6.89
(2.72)
(2.96)
(3.14)
3.45
2.71
3.16
(1.62)
(2.15)
(1.60)
(1.73)
Low
Neutral
High
9.27
14.68
25.54
(4.55)
(5.88)
(6.69)
Neutral Big
Panel B: Size-Beta portfolios Small Neutral Big
Low
Neutral
High
22.56
16.03
44.16
(2.61)
(6.07)
(3.58)
6.08
4.63
6.77
(3.60)
(2.65)
(3.03)
2.37
3.06
3.76
(1.81)
(1.98)
Small Neutral
6.06
4.03
6.16
(3.60)
(2.41)
(2.74)
2.54
3.86
1.82
(2.04)
(2.13)
(2.59)
(1.29)
Low
Neutral
High
2.63
4.89
8.53
(1.54)
(2.39)
(3.43)
3.08
2.72
4.04
(1.93)
(1.33)
(2.00)
Big
Panel C: Momentum-Beta portfolios Winner Neutral Loser
Low
Neutral
High
21.86
13.13
41.14
(2.54)
(5.35)
(3.35)
6.00
7.45
6.24
(3.69)
(3.32)
(3.30)
4.19
4.48
4.61
(2.89)
(2.89)
(2.92)
Winner Neutral Loser
23
4.51
2.47
2.56
(2.36)
(1.82)
(1.67)
Table 3 Average coefficients from week-by-week cross-sectional regressions of cryptocurrency returns on beta, size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks. The average slope is the time-series average of the weekly regression slopes, and the t-statistic is the average slope divided by its time-series standard error. The variables are the same as Table 1.
Intercept Market beta
(1)
(2)
(3)
(4)
(5)
(6)
(7)
10.94
76.13
-0.81
66.11
-3.82
26.53
24.08
(3.96)
(3.63)
(-0.25)
(3.90)
(-1.05)
(1.87)
(2.02)
-0.14
1.46
(-0.07)
(0.75)
-0.05 (-0.02)
Log(size)
Adjusted R
2
(0.70)
-4.01
-3.48
-1.64
-1.66
(-3.51)
(-3.50)
(-2.05)
(-2.17)
Momentum R2
1.38
0.017
0.020
0.015
0.017
(3.46)
(4.86)
(3.36)
(5.39)
0.044
0.035
0.109
0.077
0.193
0.132
0.214
0.031
0.022
0.097
0.052
0.171
0.108
0.182
24
Table 4 Summary statistics for weekly factor percent returns: 07-Aug-15 to 31-Dec-18, 178 weeks. At the end of each Tuesday, cryptocurrencies are assigned to 3×3 subgroups through a two-pass sorts described in Table 2. In each subgroup, the return of the portfolio is based on value-weighted. SMBW is the average of return on the portfolios of small winner cryptocurrencies minus the average return on the portfolios of big winner cryptocurrencies, SMBN and SMBL are the same but for portfolios of neutral and loser cryptocurrencies. WMLS, WMLN, and WMLB are defined in the same way. SMB is the average of SMBW, SMBN, and SMBL. WML is the average of WMLS, WMLN, and WMLB. Panel A, Panel C and Panel D show average weekly returns, the standard deviations and t-statistics of the average returns. Panel B reports the correlation matrix of the three factors. Panel A
Panel B
RM -RF
SMB
WML
RM -RF
SMB
WML
Mean
2.12
16.61
10.94
RM -RF
1.0000
Std dev.
12.27
21.93
22.62
SMB
0.0027
1.0000
0.6606
t-statistic
1.96
8.57
5.47
WML
0.0074
0.6606
1.0000
Panel C
Panel D
SMBW
SMBN
SMBL
WMLS
WMLN
WMLB
Mean
35.91
8.52
5.41
Mean
30.21
2.88
-0.29
Std dev.
54.72
25.50
20.07
Std dev.
56.02
18.22
22.84
t-statistic
7.42
3.78
3.05
t-statistic
6.10
1.79
-0.14
‘
25
Table 5 Using three factors in regressions to explain average returns on three biggest, three medium, and three smallest cryptocurrencies: 07-Aug-15 to 31-Dec-18, 178 weeks. RM -RF is the value-weighted return on the cryptocurrency market portfolio; SMB (small minus big) is the size factor of cryptocurrencies; WML (winner minus loser) is the momentum factor of cryptocurrencies. The detailed factor definitions are described in Table 4. RM -RF
SMB
WML
R2
Adj. R2
-0.03
0.91
0.05
-0.06
0.90
0.89
(-0.07)
(32.79)
(2.25)
(-3.16)
0.26
1.13
-0.03
0.11
0.56
0.55
(0.19)
(12.40)
(-0.50)
(1.62)
5.44
1.32
-0.10
-0.06
0.25
0.23
(1.66)
(6.26)
(-0.63)
(-0.36)
-0.84
1.06
0.13
-0.04
0.37
0.36
(-0.43)
(8.47)
(1.41)
(-0.48)
2.49
1.04
0.26
-0.33
0.18
0.16
(0.73)
(4.73)
(1.60)
(-2.05) 0.05
0.03
0.18
0.16
0.07
0.05
0.07
0.05
Int Panel A: 3 biggest cryptocurrencies Bitcoin Ethereum XRP
Panel B: 3 medium cryptocurrencies BlackCoin Potcoin MonetaryUnit
7.51
0.94
0.71
-0.65
(0.93)
(1.81)
(1.82)
(-1.71)
-0.15
1.34
0.76
-0.32
(-0.03)
(4.04)
(3.09)
(-1.33)
27.82
0.82
0.22
0.66
(3.13)
(1.43)
(0.52)
(1.60)
-27.82
2.29
21.29
-11.90
(-0.19)
(0.24)
(2.98)
(-1.72)
Panel C: 3 smallest cryptocurrencies Bata TEKcoin AnarchistsPrime
26
Table 6 Tests of three-factor model for weekly value-weighted returns on cryptocurrency portfolios from 3×3 sorts on size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks. The regressions use the three-factor model to explain the excess return cryptocurrency portfolios formed on size and momentum. Panel A shows three-factor intercepts and their t-statistics. Panel B, Panel C, and Panel D report the slopes for RM -RF, SMB, and WML, and t-statistics for these coefficients.
R(t)-RF(t)=a+b[RM(t)- RF(t)] +sSMB(t)+wWML(t)+e(t) Size\Momentum
Loser
Panel A:
Neutral
Winner
Loser
a
Neutral
Winner
t(a)
Small
1.68
-1.24
3.15
1.10
-0.55
1.44
Neutral
-0.87
-1.83
-2.49
-0.79
-1.25
-1.23
Big
1.44
0.55
1.59
1.01
0.40
1.04
Panel B:
b
t(b)
Small
1.12
1.21
0.93
11.42
8.28
6.59
Neutral
1.05
1.27
1.15
14.75
13.40
8.86
Big
0.99
1.19
1.08
10.65
13.40
10.95
Panel C:
s
t(s)
Small
0.62
0.93
1.16
8.46
8.59
11.04
Neutral
0.27
0.33
0.28
5.04
4.66
2.85
Big
0.18
-0.10
-0.37
2.60
-1.48
-5.03
Panel D:
w
t(w)
Small
-0.50
-0.51
1.34
-7.09
-4.83
13.16
Neutral
-0.16
-0.07
0.21
-3.18
-1.05
2.27
Big
-0.28
0.11
0.50
-4.20
1.78
6.96
27
Table 7 Interprets from nested factor models in regressions to explain weekly value-weighted returns on cryptocurrency portfolios formed on size and momentum: 07-Aug-15 to 31-Dec-18, 178 weeks Panel A shows one-factor intercepts and their t-statistics produced by the cryptocurrency market return. Panel B and Panel C show two-factor intercepts and their t-statistics, with adding SMB and WML to the model of Panel A, respectively. Panel D provides the intercepts and their t-statistics of the proposed three-factor model. Size\Momentum
Loser
Neutral
Winner
Loser
Neutral
Winner
Panel A: R(t)-RF(t)=a+b[RM(t)- RF(t)]+e(t) a
t(a)
Small
6.47
8.68
37.04
4.21
3.83
7.56
Neutral
1.78
2.85
4.43
1.85
2.20
2.42
Big
1.35
0.17
0.86
1.11
0.15
0.60
Panel B: R(t)-RF(t)=a+b[RM(t)- RF(t)]+sSMB(t)+e(t) a
t(a)
Small
1.88
-1.03
3.61
1.10
-0.42
1.77
Neutral
-0.80
-1.80
-2.57
-0.70
-1.23
-1.26
Big
1.56
0.55
1.39
1.02
0.39
0.77
Panel C: R(t)-RF(t)=a+b[RM(t)- RF(t)]+wWML(t)+e(t) a
t(a)
Small
7.64
7.72
14.30
4.52
3.07
5.26
Neutral
1.70
1.33
0.17
1.59
0.95
0.09
Big
3.17
-0.39
-1.96
2.43
-0.32
-1.33
Panel D: R(t)-RF(t)=a+b[RM(t)- RF(t)] +sSMB(t)+wWML(t)+e(t) a
t(a)
Small
1.68
-1.24
3.15
1.04
-0.55
1.44
Neutral
-0.87
-1.83
-2.49
-0.79
-1.25
-1.23
Big
1.44
0.51
1.59
1.01
0.36
1.04
28
Table 8 One-dimensional sorts and Fama-MacBeth regressions using daily data: 07-Aug-15 to 31-Dec-18, 1243 days. Pane A reports the average daily percent excess returns for cryptocurrency portfolios formed on beta, size, and momentum, respectively. Panel B shows the average slopes from day-by-day cross-sectional regressions of cryptocurrency returns on beta, size and momentum. Low
2
Panel A: One-dimensional Sorts Beta 3.22 1.02 (3.49) (4.88) Size 6.38 `1.88 (4.47) (6.75) Momentum 0.76 0.63 (3.31) (3.39) (1)
(2)
Panel B: Fama-MacBeth Regressions 2.85 12.73 Intercept (2.27) (3.64) -1.25 Market beta (-0.93) -0.69 Log(size) (-3.52)
4
5
High
High-Low
1.94 (1.79) 0.68 (3.09) 0.89 (4.18)
1.16 (4.70) 0.57 (2.81) 1.18 (4.52)
1.08 (4.69) 0.45 (2.67) 1.15 (4.26)
1.94 (5.90) 0.41 (1.99) 5.75 (4.94)
-1.26 (-1.32) -5.97 (-4.18) 4.98 (3.51)
(3)
(4)
(5)
(6)
(7)
-0.08 (-0.15)
12.67 (3.50) -1.76 (-1.28) -0.59 (-3.28)
1.10 (0.70) -1.21 (-0.72)
6.08 (2.46)
0.0025 (3.24)
-0.36 (-2.59) 0.0018 (2.85)
7.59 (2.46) -1.35 (-0.81) -0.38 (-2.86) 0.019 (3.42)
0.115 0.091
0.091 0.066
0.130 0.094
0.0023 (3.11)
Momentum R2 Adjusted R2
3
0.042 0.029
0.018 0.005
0.075 0.063
29
0.059 0.034
Table 9 Double sorts and factor model regressions using daily data: 07-Aug-15 to 31-Dec-18, 1243 days. Panel A reports the average daily percent excess returns for cryptocurrency portfolios formed on two-pass sorts on size and momentum. Panel B provides the estimation results of the three-factor model in regressions for daily value-weighted returns on cryptocurrency portfolios formed on size and momentum. Size\Momentum
Loser
Neutral
Winner
Loser
Neutral
Winner
Panel A: Double sorts t(µ)
µ Small 1.26 2.62 Neutral 0.66 0.93 Big 0.46 0.47 Panel B: Factor model regressions a
9.05 0.25 0.36
3.82 3.10 2.55
5.63 3.54 2.04
6.32 2.03 1.55
Small Neutral Big
0.12 0.19 0.96
0.59 0.24 0.51 b
0.15 -0.07 0.30
0.81 1.53 0.81
1.43 1.32 1.41 t(b)
0.75 -0.35 1.85
Small Neutral Big
1.09 1.01 0.90
1.03 0.99 1.12 s
0.92 1.07 1.02
31.55 34.75 50.91
18.32 23.44 23.22 t(s)
20.11 22.07 27.31
Small Neutral Big
0.45 0.12 0.09
0.71 0.15 -0.32 w
1.15 -0.01 -0.47
17.88 5.72 7.04
17.11 4.87 -8.94 t(w)
34.20 -0.28 -17.20
Small Neutral Big
-0.38 -0.14 -0.13
-0.41 -0.01 0.24
1.48 0.29 0.58
-14.53 -6.30 -9.56
-9.58 -0.28 6.37
42.21 7.71 20.18
t(a)
30
Common Risk Factors in the Returns on Cryptocurrencies
HIGHLIGHTS
•
We construct three common risk factors which are specific to cryptocurrency market.
•
Size and momentum are strong in sorts, and small cryptocurrencies have more significant momentum effect.
•
The combined effect of size and momentum can largely explain the cross-sectional variation in average cryptocurrency returns.
•
Most cryptocurrencies and corresponding portfolios have significant exposures to our proposed three factors with insignificant intercepts.