# Shannon Entropy: An Econophysical Approach to Cryptocurrency Portfolios

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## Abstract

**:**

## 1. Introduction

## 2. Methods, Data and Analysis

#### 2.1. Discrete Entropy Function

#### 2.2. Continuous Entropy Function

#### 2.3. Entropy as a Measure of Uncertainty

- H must be continuous on ${p}_{X}\left({x}_{i}\right)$, with $i=1,\dots ,n$.
- If ${p}_{X}\left({x}_{i}\right)=\frac{1}{n}$, H must be monotone increasing as a function of n.
- If an option is split into two successive options, the original H must be the weighted sum of the individual values of H.

- $H\left(X\right)=0$ if and only if all but one of ${p}_{X}\left({x}_{i}\right)$ are zero.
- When ${p}_{X}\left({x}_{i}\right)=\frac{1}{n}$, i.e., when the discrete probability distribution is constant, $H\left(X\right)$ is maximum and equal to $log\left(n\right)$.
- $H(X,Y)\le H\left(X\right)+H\left(Y\right)$, where the equality holds if and only if X and Y are statistically independent, i.e., $p({x}_{i},{y}_{j})=p\left({x}_{i}\right)p\left({y}_{j}\right)$.
- Any change towards the equalization of the probabilities ${p}_{X}\left({x}_{i}\right)$ increases H.
- $H(X,Y)=H\left(X\right)+H\left(Y\right|X)$; thus, the uncertainty of the joint event $\left(Y\right|X)$ is the uncertainty of X plus the uncertainty of Y when X is known.
- $H\left(Y\right)\ge H\left(Y\right|X)$, which implies that the uncertainty of Y is never increased by knowledge of X, and decreases unless X and Y are independent, in which case it does not change.

#### 2.4. Comparison between Entropy and Variance

#### 2.5. Investment Portfolios

## 3. Results

#### 3.1. Normal Distribution and Q-Q Plots

#### 3.2. Heavy-Tailed Distributions

#### 3.3. Portfolio Entropy

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Probability Density Function Estimation (PDF)

#### Appendix A.1.1. Histogram

#### Appendix A.1.2. Kernel Density Estimation

#### Appendix A.1.3. Parametric Density Estimation

#### Appendix A.1.4. Maximum Likelihood Estimate

#### Appendix A.2. Parametric Distributions Used

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**Figure 1.**(

**a**) Histogram of the probability distribution of the return of Bitcoin. The normal curve is shown in blue, associated with the dataset, with the curve estimated using the Kernel method in red. (

**b**) The log–log plot of the same data is depicted in the inset. Notice that the normal distribution tail decays faster in comparison with the kernel, as expected.

**Figure 2.**Normal Q-Q chart of daily Bitcoin performance. The theoretical normal quantiles are plotted against the observed quantiles.

**Figure 3.**(

**a**) Histogram of the probability distribution of the return of Bitcoin with the adjustments of a normal distribution and Student’s t. (

**b**) The log–log plot of the same data is depicted in the inset. Notice that the normal distribution tail decays faster in comparison with the Student’s t-distribution, as expected.

**Figure 4.**(

**a**,

**c**,

**e**,

**g**) correspond to the adjustments of the probability distribution of the total daily returns with $n\in \{3,8,13,18\}$ assets in the portfolio, respectively, with the kernel, parametric, and histogram methods and the normal curve associated with the data. In (

**b**,

**d**,

**f**,

**h**), the same fits are shown in log–log scale.

**Figure 5.**Normal and empirical entropy against the number of assets in the portfolio. The decrease in entropy as the number of assets in the portfolio increases verifies the diversification effect.

Cryptocurrency | Normal | t-Student |
---|---|---|

BAT | 1531 | 1667 |

BCH | 1555 | 1815 |

BTC | 2111 | 2250 |

DAI | 4395 | 4871 |

EDO | 1239 | 1518 |

EOS | 1583 | 1794 |

ETC | 1608 | 1861 |

ETH | 1831 | 1915 |

ETP | 1393 | 1594 |

LTC | 2111 | 2250 |

NEO | 1631 | 1740 |

OMG | 1401 | 1570 |

TRX | 1694 | 1829 |

XLM | 1590 | 1808 |

XMR | 1814 | 1947 |

XRP | 1595 | 1884 |

XVG | 1305 | 1396 |

ZEC | 1637 | 1722 |

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**MDPI and ACS Style**

Rodriguez-Rodriguez, N.; Miramontes, O.
Shannon Entropy: An Econophysical Approach to Cryptocurrency Portfolios. *Entropy* **2022**, *24*, 1583.
https://doi.org/10.3390/e24111583

**AMA Style**

Rodriguez-Rodriguez N, Miramontes O.
Shannon Entropy: An Econophysical Approach to Cryptocurrency Portfolios. *Entropy*. 2022; 24(11):1583.
https://doi.org/10.3390/e24111583

**Chicago/Turabian Style**

Rodriguez-Rodriguez, Noé, and Octavio Miramontes.
2022. "Shannon Entropy: An Econophysical Approach to Cryptocurrency Portfolios" *Entropy* 24, no. 11: 1583.
https://doi.org/10.3390/e24111583