The risks of trading on cryptocurrencies: A regime-switching approach based on volatility jumps and co-jumping behaviours

Figures & data

Table 1. Basic statistics of cryptocurrencies.

	BTC	DASH	LTC	XRP
Mean	0.133	0.218	0.065	0.118
Median	0.156	−0.145	0.000	−0.232
S.D.	3.893	7.414	5.634	6.393
Q1	−1.205	−2.602	−1.961	−2.014
Q3	1.620	2.556	1.892	1.842
Skewness	−0.907	2.870	0.352	2.285
Kurtosis	15.979	48.216	16.632	44.816

This study employs the four most liquid cryptocurrencies as the sample, including Bitcoin (BTC), Dash (DASH), Litecoin (LTC) and Ripple (XRP). The testing period is between 14 February 2014 to 19 November 2020 for 2,471 daily observations. The data are obtained with https://coinmarketcap.com/.

Table 2. Correlation coefficients of cryptocurrencies.

	BTC	DASH	LTC	XRP
BTC	1.000
DASH	0.442***	1.000
LTC	0.658***	0.388***	1.000
XRP	0.375***	0.217***	0.394***	1.000

The correlation coefficients of the four representative cryptocurrencies (BTC, DASH, LTC and XRP) are listed in this table. *** denotes significance at the 1% level. See Table 1 for the data source.

Table 3. Parameter estimates of the bivariate GARCH-CCC model.

$r_t^{BTC}={\mu }^{BTC}+{\phi }^{BTC}\cdot r_{t-1}^{BTC}+e_t^{BTC}$ $r_t^{OTH}={\mu }^{OTH}+{\phi }^{OTH}\cdot r_{t-1}^{OTH}+e_t^{OTH}$ $h_t^{BTC}={\omega }^{BTC}+{\alpha }^{BTC}\cdot (e_{t-1}^{BTC}{)}^2+{\beta }^{BTC}\cdot h_{t-1}^{BTC}$ $h_t^{OTH}={\omega }^{OTH}+{\alpha }^{OTH}\cdot (e_{t-1}^{OTH}{)}^2+{\beta }^{OTH}\cdot h_{t-1}^{OTH}$ $h_t^{BTC,OTH}=\rho \times {\left(h_t^{BTC}\cdot h_t^{OTH}\right)}^{1/2}$
	BTC-DASH	BTC-LTC	BTC-XRP
BTC equation
µ^BTC	0.0177 (0.0109)	−0.0006 (0.0030)	0.0365 (0.0632)
φ^BTC	0.0193 (0.0195)	−0.0214 (0.0208)	0.0284 (0.0185)
ω^BTC	1.2322 *** (0.1458)	1.0191*** (0.1129)	1.2355*** (0.1457)
α^BTC	0.1507*** (0.0160)	0.1614*** (0.0153)	0.1715*** (0.0179)
β^BTC	0.7819*** (0.0180)	0.7931*** (0.0150)	0.7682*** (0.0180)
Other cryptocurrency equation
µ^OTH	−0.2242*** (0.0788)	−0.2031*** (0.0692)	−0.2765*** (0.0920)
φ^OTH	−0.0284 (0.0224)	−0.0336* (0.0178)	0.0581*** (0.0231)
ω^OTH	1.8562*** (0.3046)	2.3005*** (0.2758)	3.0698*** (0.4152)
α^OTH	0.1977*** (0.0201)	0.1011*** (0.0109)	0.3094*** (0.0449)
β^OTH	0.7807*** (0.0195)	0.8290*** (0.0162)	0.6577*** (0.0370)
Correlation
ρ	0.5209*** (0.0147)	0.6943*** (0.0104)	0.4735*** (0.0157)
Log-likelihood	−14030.8747	−13356.6072	−13763.1882

The value in the parenthesis denotes the standard deviation of the estimate. * denotes significance at the 10% level, ** denotes significance at the 5% level, and *** denotes significance at the 1% level. For sample descriptions and data sources, please refer to Table 1.

Table 4. Parameter estimates of the bivariate GARCH-DCC model.

$q_t=\tau +\pi \cdot q_{t-1}+\lambda \cdot e_{t-1}^{BTC}\cdot e_{t-1}^{OTH}/\sqrt{h_{t-1}^{BTC}\cdot h_{t-1}^{OTH}}$ ${\rho }_t=q_t/\sqrt{1+q_t^2}$ $h_t^{BTC,OTH}={\rho }_t\times {\left(h_t^{BTC}\cdot h_t^{OTH}\right)}^{\frac{1}{2}}$
	BTC-DASH	BTC-LTC	BTC-XRP
BTC equation
µ^BTC	0.1471**	0.1507***	0.1494**
	(0.0729)	(0.0037)	(0.0663)
φ^BTC	−0.0153	−0.0086	0.0212*
	(0.0207)	(0.0201)	(0.0112)
ω^BTC	0.5745***	0.6128***	0.8633***
	(0.0787)	(0.0659)	(0.1148)
α^BTC	0.1496***	0.1601***	0.1779***
	(0.0143)	(0.0144)	(0.0178)
β^BTC	0.8383***	0.8334***	0.7954***
	(0.0124)	(0.0117)	(0.0166)
Other cryptocurrency equation
µ^OTH	−0.1058 (0.0943)	−0.0498 (0.0802)	−0.1607 (0.1271)
φ^OTH	−0.0294 (0.0210)	−0.0330*** (0.0005)	0.0557*** (0.0194)
ω^OTH	0.9388*** (0.1548)	1.5775*** (0.1658)	2.7511*** (0.3054)
α^OTH	0.1974*** (0.0159)	0.1164*** (0.0107)	0.2952*** (0.0342)
β^OTH	0.8143*** (0.0120)	0.8471*** (0.0117)	0.6778*** (0.0272)
Time-varying correlations
τ	0.0084*** (0.0028)	0.0045* (0.0029)	0.0179*** (0.0035)
π	0.9583*** (0.0064)	0.9687*** (0.0044)	0.9282*** (0.0069)
λ	0.0410*** (0.0054)	0.0463*** (0.0051)	0.0597*** (0.0059)
Log-likelihood	−13883.9747	−13204.9135	−13618.5079

The value in the parenthesis denotes the standard deviation of the estimate. * denotes significance at the 10% level, ** denotes significance at the 5% level, and *** denotes significance at the 1% level. For sample descriptions and data sources, please refer to Table 1.

Table 5. Parameter estimates of the bivariate SWARCH model with state-dependent correlations.

$h_t^{BTC}=\hspace{0.17em}{g}_{s_t^{BTC}}^{BTC}\times [{\omega }^{BTC}+{\alpha }^{BTC}\cdot (e_{t-1}^{BTC}{)}^2/g_{s_{t-1}^{BTC}}^{BTC}]\hspace{0.17em}$ $h_t^{OTH}=g_{s_t^{OTH}}^{OTH}\times [{\omega }^{OTH}+{\alpha }^{OTH}\cdot (e_{t-1}^{OTH}{)}^2/g_{s_{t-1}^{OTH}}^{OTH}$] $h_t^{BTC,OTH}={\rho }_{s_t^{BTC},s_t^{OTH}}\times {\left(h_t^{BTC}\cdot h_t^{OTH}\right)}^{1/2}$ $P\left(s_t^{BTC}=1|s_{t-1}^{BTC}=1\right)=p_{11}^{BTC},P\left(s_t^{BTC}=2|s_{t-1}^{BTC}=2\right)=p_{22}^{BTC}$ $P\left(s_t^{OTH}=1|s_{t-1}^{OTH}=1\right)=p_{11}^{OTH},P\left(s_t^{OTH}=2|s_{t-1}^{OTH}=2\right)=p_{22}^{OTH}$
	BTC-DASH	BTC-LTC	BTC-XRP
BTC equation
p11^BTC	0.9138***	0.9311***	0.8979***
	(0.0126)	(0.0116)	(0.0165)
p22^BTC	0.7606***	0.8131***	0.8066***
	(0.0363)	(0.0342)	(0.0356)
µ^BTC	0.1256**	0.0383	0.1275*
	(0.0545)	(0.0309)	(0.0695)
φ^BTC	−0.0347**	−0.0475***	−0.0575**
	(0.0162)	(0.0179)	(0.0252)
ω^BTC	3.6642***	4.0457***	2.9639***
	(0.2334)	(0.2701)	(0.2306)
α^BTC	0.1943***	0.1566***	0.1324***
	(0.0319)	(0.0309)	(0.0348)
g2^BTC	10.9830#	9.1008#	11.4481#
	(0.8844)	(0.8032)	(1.0746)
Other cryptocurrency equation
p11^OTH	0.9106***	0.9023***	0.8396***
	(0.0130)	(0.0131)	(0.0212)
p22^OTH	0.8499***	0.7593***	0.6958***
	(0.0240)	(0.0331)	(0.0418)
µ^OTH	−0.2509***	−0.2195***	−0.2743***
	(0.0794)	(0.0534)	(0.0553)
φ^OTH	−0.0572***	−0.0851***	−0.0950***
	(0.0177)	(0.0194)	(0.0321)
ω^OTH	7.8505***	6.3044***	3.0709***
	(0.5190)	(0.5051)	(0.5372)
α^OTH	0.2660***	0.1956***	0.5578***
	(0.0341)	(0.0331)	(0.1481)
g2^OTH	8.8048#	10.1064#	13.8775#
	(0.6448)	(0.7528)	(1.0339)
State-varying correlations
ρ1,1	0.7751***	0.8634***	0.7344***
	(0.0169)	(0.0093)	(0.0214)
ρ2,1	0.3775***	0.6417***	0.3359***
	(0.0679)	(0.0457)	(0.1416)
ρ1,2	0.1543***	0.4511***	0.0000
	(0.0433)	(0.0407)	(0.0377)
ρ2,2	0.8193***	0.8280***	0.7188***
	(0.0290)	(0.0276)	(0.1150)
LR statistic for ρ1,1=ρ2,1=ρ1,2=ρ2,2	117.2884***	134.7098***	93.0692***
Log-likelihood	−13458.5686	−12501.1180	−13029.4481

This study develops the bivariate SWARCH model with four-state variances and correlations to control volatility regimes and correlation dynamics in the digital currency markets. To test significance of the four-state correlations, we implement an identical correlation, i.e. ρ_1,1=ρ_2,1=ρ_1,2=ρ_2,2=ρ into the model and calculate the value of the log-likelihood function of the restricted model. Then we use the difference between the two models (one with four-state correlations vs. one with an identical correlation) to develop the likelihood ratio (LR) statistic. The LR statistic follows a Chi-square distribution with 3 ( = 4–1) degrees of freedom.

The value in the parenthesis denotes the standard deviation of the estimate. * denotes significance at the 10% level, ** denotes significance at the 5% level, and *** denotes significance at the 1% level. # represents that the estimate significantly deviates from the value of one at the 1% level. For sample descriptions and data sources, please refer to Table 1.

Figure 1. Volatility of returns on cryptocurrencies.

Figure 2. Probabilities of volatility state combinations: an example of BTC-DASH portfolio.

Table 6. Comparative analysis of various volatility states.

	BTC = LV and Other = LV	BTC = HV and Other = LV	BTC = LV and Other = HV	BTC = HV and Other = HV
Panel A. Observation percentages of various volatility combination states
BTC-DASH	57.48%	9.65%	21.69%	11.19%
BTC-LIT	65.14%	11.27%	13.01%	10.58%
BTC-XRP	56.91%	16.94%	14.80%	11.35%
Panel B. Risk reduction effectiveness of diversification in various volatility states
BTC-DASH	6.50%	17.49%	13.25%	2.39%
BTC-LIT	15.63%	23.96%	13.94%	6.67%
BTC-XRP	7.26%	16.47%	12.66%	6.63%

On key feature of the bivariate SWARCH model is to provide the estimated probabilities of a specific state for each point in time (the BTC-DASH example is shown in Figure 4). We use these estimated probabilities and a maximum value criterion to define the volatility state. For example, if the estimated probability of the ‘HV-HV’ state is higher than that of the other three states, we identify this point in time as an ‘HV-HV’ state.

In panel B, we show the risk reduction percentage of a cryptocurrency portfolio with equal weight on each cryptocurrency. More specifically, the risk reduction percentage of a cryptocurrency portfolio is presented as follows.

Risk reduction % = −100*[σ_p - (0.5 × σ^BTC +0.5 × σ^OTH)]/(0.5 × σ^BTC +0.5 × σ^OTH)

where σ^BTC and σ^OTH denote the standard deviation of Bitcoin and another cryptocurrency in the portfolio, respectively. The σ_p is the standard deviation of the cryptocurrency portfolio with equal weight on Bitcoin and another cryptocurrency. The figure in bold style represents the minimum value in the row.

Table 7. Cryptocurrency portfolio volatility forecasting.

	BTC-DASH	BTC-LTC	BTC-XRP
Panel A: MAE (Mean Absolute Error)
Bivariate GARCH-CCC model	5.6735	5.5252	5.3308
Bivariate GARCH-DCC model	5.8192	5.6514	5.3778
Bivariate SWARCH model	4.9442 (−5.4657)***	4.4683 (−8.9052)***	4.2007 (−8.7868)***
Panel B: MSE (Mean Square Error)
Bivariate GARCH-CCC model	57.5218	49.7362	52.3870
Bivariate GARCH-DCC model	61.8947	52.6608	53.8310
Bivariate SWARCH model	43.0255 (−2.8237)***	35.4926 (−3.0311)***	34.4661 (−3.1250)***

This study adopts MSE and MAE, the two commonly-used forecasting performance criteria, to evaluate the performance of various models in cryptocurrency portfolio volatility forecasting. The MSE and MAE are defined as follows.

$MSE=T^{-1}\sum _{t=1}^T{\left(r_{p,t}^2-{\sigma }_{p,t}^2\right)}^2$, $MAE=T^{-1}\sum _{t=1}^T\left|r_{p,t}^2-{\sigma }_{p,t}^2\right|$.

where r_p,t is the portfolio return with equal weight on each digital currency and σ_p,t is the estimated standard deviation of the digital currency portfolio. The figure in bold style represents the minimum value in the column. As shown in the table, the bivariate SWARCH model associates with the smallest MSE and MAE.

We adopt the bivariate GARCH-CCC model as a benchmark and calculate the statistic for the difference in MSE and MAE (see the figure in parenthesis). *** represents significance at the 1% level.

Table 8. Portfolio construction performance: Volatility of cryptocurrency portfolio.

	BTC-DASH	BTC-LTC	BTC-XRP
Bivariate GARCH-CCC model	3.8397	3.9954	3.7064
Bivariate GARCH-DCC model	3.8159 (−1.4785)	4.0017 (0.5044)	3.7102 (0.1925)
Bivariate SWARCH model	3.4359 (−10.1508)***	3.4736 (−4.8288)***	3.1974 (−6.7017)***

This table lists the performance of portfolio construction between various models. As our focus is risk of cryptocurrency investment, we adopt a minimum variance portfolio strategy to test the issue and present the volatility (standard deviation) of the cryptocurrency portfolios in this table (e.g. French and Poterba 1991; Tesar and Werner 1992; Ramchand and Susmel 1998).

The figure in bold style represents the minimum value in the column. We use the bivariate GARCH-CCC model as a benchmark to calculate the statistic for the difference in the portfolio volatility (see the figure in the parenthesis). *** represents significance at the 1% level.

French, K. R., and J. M. Poterba. 1991. “Investor Diversification and International Equity Markets.” The American Economic Review 81: 222–226.

 Web of Science ®Google Scholar

Tesar, L. L., and I. Werner 1992 Home Bias and the Globalization of Securities Markets, NBER Working Paper, No. 4218, National Bureau of Economic Research, Cambridge.

 Google Scholar

Ramchand, L., and R. Susmel. 1998. “Volatility and Cross Correlation Across Major Stock Markets.” Journal of Empirical Finance 5 (4): 397–416. doi:10.1016/S0927-5398(98)00003-6.

 Google Scholar

Data availability statement

https://coinmarketcap.com/