# A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020

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## Abstract

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## 1. Introduction

## 2. Basic Statistical Analyses

## 3. Assessing Crypto-Assets’ Risks

#### 3.1. Unconditional Risk

#### 3.2. Conditional Risk

## 4. A Look at Dependence

#### 4.1. Dependence (by Pair-Copulas)

#### 4.2. Dependence (by Cointegration)

## 5. Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The first and second rows show the Bitcoin GPD’s graphical diagnosis for the left and right tails. In the third row the return level is shown in days with the corresponding $\alpha $-VaR, for the left tail (blue) and right tail (black) of Bitcoin.

**Figure 4.**Scatter plots of the standardized residuals from the GARCH fits on the upper-left panel (above diagonal). The corresponding Kendall’s $\tau $ coefficients on the bottom-right panel.

Bitcoin | Ethereum | Ripple | Litecoin | Stellar | Monero | Euro | |
---|---|---|---|---|---|---|---|

Mean | 0.21 | 0.34 | 0.20 | 0.17 | 0.19 | 0.29 | 0.00 |

0.99%[LCL, UCL] | [−0.03,0.47] | [−0.06,0.74] | [−0.25,0.66] | [−0.17,0.52] | [−0.90,1.30] | [−0.14,0.71] | [−0.04,0.04] |

Median | 0.23 | −0.06 | −0.30 | −0.05 | −0.36 | −0.07 | −0.01 |

Standard deviation | 3.96 | 6.33 | 7.18 | 5.49 | 17.37 | 6.60 | 0.48 |

Maximum | 25.49 | 39.94 | 101.97 | 47.97 | 269.14 | 56.47 | 2.47 |

Minimum | −23.97 | −33.35 | −63.15 | −40.60 | −244.65 | −29.18 | −2.88 |

Skewness | −0.21 * | 0.43 * | 2.60 * | 1.02 * | 0.06 | 0.93 * | 0.12 |

Kurtosis | 5.55 | 5.22 | 36.75 | 11.41 | 83.51 | 7.46 | 3.43 |

**Table 2.**GPD estimates and long-run risk measures: shape parameter (standard error); percentage of excesses ${p}^{*}$; 1% and 5% VaR and expected loss (EL) estimates.

Estimates | Bitcoin | Ethereum | Ripple | Litecoin | Stellar | Monero | Euro |
---|---|---|---|---|---|---|---|

Left tail estimates | |||||||

Shape(st. er); ${p}^{*}$ | 0.02(0.07); 14% | 0.04(0.08); 13% | 0.25(0.10); 10% | 0.12(0.08); 12% | 0.59(0.10); 15% | 0.01(0.07); 17% | 0.20(0.11); 9% |

VaR: 1% & 5% | −12.08 & −6.29 | −17.56 & −9.50 | −16.91 & −8.29 | −14.29 & −7.64 | −39.13 & −13.40 | −16.67 & −9.63 | −1.18 & −0.71 |

EL: 1% & 5% | 15.24 & 10.03 | 23.04 & 14.75 | 24.48 & 14.15 | 20.70 & 12.14 | 89.38 & 35.66 | 20.41 & 14.10 | 1.75 & 1.00 |

Right tail estimates | |||||||

Shape(st. er); ${p}^{*}$ | 0.15(0.09); 12% | 0.00(0.06); 13% | 0.33(0.12); 9% | 0.21(0.08); 14% | 0.57(0.12); 11% | 0.15(0.07); 16% | 0.24(0.13); 8% |

VaR: 5% & 1% | 6.23 & 11.41 | 11.15 & 20.23 | 10.06 & 24.61 | 8.63 & 18.00 | 15.68 & 46.98 | 10.83 & 20.60 | 0.76 & 1.25 |

EL: 5% & 1% | 9.78 & 14.81 | 16.84 & 27.93 | 19.81 & 39.91 | 15.43 & 27.75 | 38.74 & 86.72 | 16.91 & 30.10 | 1.07 & 1.83 |

**Table 3.**Summary of the results from the ARFIMA$(p,d,q)$-FIGARCH$(m,D,s)$ fits for all series. All parameters estimates are 5% statistically significant. Notations: LEV: leverage parameter; d.f.: degrees of freedom of the F distribution; skew: skewness estimate of the skew-t distribution.

Crypto-Coin | $({\mathit{\varphi}}_{0},{\mathit{\varphi}}_{1})$ | d | ${\mathit{\theta}}_{1}$ | $\mathit{\omega}$ | ${\mathit{\alpha}}_{1}$ | D | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | LEV | d.f. | Skew | AIC | 1%-VaR |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Bitcoin-LM | (0.201,–) | 0.00 | — | 0.144 | 0.258 | 0.689 | 0.780 | — | — | 3.15 | — | 5.0577 | (−6.80,7.20) |

Bitcoin | (0.207,–) | 0.00 | — | 0.212 | 0.225 | — | 0.401 | 0.418 | −0.089 | 3.11 | — | 5.0584 | (−7.23,7.65) |

Ethereum-LM | (0.128,–) | 0.00 | — | 3.470 | 0.138 | 0.646 | 0.515 | — | — | 3.18 | 1.064 | 6.1096 | (−10.59,12.05) |

Ethereum | (0.128,–) | 0.00 | — | 2.726 | 0.261 | — | 0.738 | — | — | 3.08 | 1.063 | 6.1116 | (−11.28,12.82) |

Ripple-LM | (−0.156,–) | 0.00 | −0.154 | 0.623 | 0.678 | 0.451 | 0.744 | — | — | 2.81 | 1.065 | 5.7761 | (−7.58,8.56) |

Ripple | (−0.160,–) | 0.00 | −0.154 | 2.059 | 0.263 | — | 0.736 | — | — | 2.72 | 1.056 | 5.7967 | (−8.99,10.01) |

Litecoin-LM | (–,–) | 0.00 | — | 0.001 | 0.361 | 0.619 | 0.753 | — | — | 3.11 | 1.048 | 5.5577 | (−11.79,12.80) |

Litecoin | (–,–) | 0.00 | — | 0.092 | 0.107 | — | 0.892 | — | — | 3.10 | 1.045 | 5.5676 | (−13.68,14.77) |

Stellar-LM | (–,–) | 0.00 | −0.097 | 0.815 | 0.278 | 0.374 | 0.375 | — | — | 3.47 | 1.143 | 6.2334 | (−9.61,11.65) |

Stellar | (–,–) | 0.00 | −0.093 | 1.801 | 0.212 | — | 0.787 | — | — | 3.17 | 1.133 | 6.2541 | (−9.75,11.86) |

Monero-LM | (–,–) | 0.0 | −0.110 | 3.256 | 0.115 | 0.652 | 0.582 | — | — | 3.30 | 1.056 | 6.2455 | (−12.40,14.33) |

Monero | (–,–) | 0.00 | −0.110 | 2.321 | 0.206 | — | 0.793 | — | — | 3.18 | 1.053 | 6.2472 | (−13.14,15.09) |

Euro-LM | (–,–) | 0.00 | — | 0.002 | 0.043 | 0.503 | 0.274 | 0.484 | — | 4.82 | — | 1.4120 | (−0.78,0.78) |

Euro | (–,–) | 0.00 | — | 0.001 | 0.019 | — | 0.975 | — | — | 6.73 | 1.088 | 1.2403 | (−0.79,0.88) |

**Table 4.**Pair-copulas’ robust fits: copula family and estimates, dependence measures, the losses (gains) BB7-copula-based joint probability associated 1%-VaR in the upper row, and the $\pi $-copula-based risk under independence (bottom row).

Pairs in | Copula Family | Linear Coef $\mathit{\rho}$ | Kendall’s $\mathit{\tau}$ | Lower TDC ${\mathit{\lambda}}_{\mathit{L}}$ | Upper TDC ${\mathit{\lambda}}_{\mathit{U}}$ | Losses: BB7 Risk | Gains: BB7 Risk |
---|---|---|---|---|---|---|---|

Tree 1 | (Parameters) | (Residuals Based) | (Copula Based) | (Copula Based) | (Copula Based) | Losses: ($\mathbf{\Pi}$ Risk) | Gains: ($\mathbf{\Pi}$ Risk) |

(XMR,BTC) | BB7 | 0.523 | 0.601 | 0.746 | 0.566 | 0.745% | 0.556% |

(1.923,2.361) | (0.01%) | (0.01%) | |||||

(BTC,LTC) | BB7 | 0.630 | 0.674 | 0.829 | 0.627 | 0.833% | 0.635% |

(2.186,3.691) | (0.01%) | (0.01%) | |||||

(LTC,ETH) | BB7 | 0.462 | 0.616 | 0.751 | 0.565 | 0.783% | 0.584% |

(1.918,2.632) | (0.01%) | (0.01%) | |||||

(ETH,XRP) | BB7 | 0.311 | 0.588 | 0.750 | 0.486 | 0.741% | 0.492% |

(1.672,2.422) | (0.01%) | (0.01%) | |||||

(XRP,XLM) | BB7 | 0.236 | 0.608 | 0.740 | 0.605 | 0.746% | 0.599% |

(2.083,2.301) | (0.01%) | (0.01%) |

**Table 5.**Estimates of the speed of adjustment parameters $\alpha $ for all pairs. All estimates are 5% statistically significant.

Pairs | (BTC,ETH) | (BTC,RIP) | (BTC,LTC) | (BTC,XLM) | (BTC,XMR) |
---|---|---|---|---|---|

$\alpha $ | (0.00502,0.00063) | (0.00453,0.000002) | (0.00528,0.00029) | (0.00477,0.000004) | (0.00573,0.00033) |

**Table 6.**Estimates of the speed of adjustment parameters $\alpha $ for the six crypto-asset system. All estimates are 1% statistically significant.

BTC | ETH | RIP | LTC | XLM | XMR | |
---|---|---|---|---|---|---|

1st cointegrating vector | −0.13611 | −0.03555 | −0.00010 | −0.00191 | −0.00003 | −0.00891 |

2nd cointegrating vector | −0.00251 | −0.00037 | −0.000002 | −0.00003 | −0.0000004 | −0.00015 |

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**MDPI and ACS Style**

Vaz de Melo Mendes, B.; Fluminense Carneiro, A. A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020. *J. Risk Financial Manag.* **2020**, *13*, 192.
https://doi.org/10.3390/jrfm13090192

**AMA Style**

Vaz de Melo Mendes B, Fluminense Carneiro A. A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020. *Journal of Risk and Financial Management*. 2020; 13(9):192.
https://doi.org/10.3390/jrfm13090192

**Chicago/Turabian Style**

Vaz de Melo Mendes, Beatriz, and André Fluminense Carneiro. 2020. "A Comprehensive Statistical Analysis of the Six Major Crypto-Currencies from August 2015 through June 2020" *Journal of Risk and Financial Management* 13, no. 9: 192.
https://doi.org/10.3390/jrfm13090192