# An Entropy-Based Approach to Portfolio Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Modern Portfolio Theory

#### 2.1. Markowitz Mean-Variance Portfolio Optimization (MVPO)

#### 2.2. Practical Difficulties with MVPO

#### 2.3. Literature Review

## 3. Entropy as a Risk Measure

#### 3.1. Shannon Entropy (Information Theory)

#### 3.2. Portfolio Optimization Based on Entropy

#### 3.3. Probability Generating Functions

#### 3.4. Portfolio Entropy Objective Function

#### 3.5. Return-Entropy Portfolio Optimization (REPO)

## 4. A Portfolio Selection Example Using REPO

#### 4.1. Data

#### 4.2. Efficient Frontier and Portfolio Selection

#### 4.3. Comparison to MVPO

#### 4.4. Addressing the Five Main Issues with MVPO

## 5. Conclusions

## 6. Materials and Methods

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`R version 3.5.1`) used for the portfolio selection example demonstrated in this paper can be accessed from the following DropBox sharing links:

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Bps | Basis points |

CAPM | Capital asset pricing model |

DEA | Data envelopment analysis |

MED | Maximum entropy diversification |

MQE | Mean-quadratic entropy |

MSSS | Mean-semivariance-skewness-semikurtosis |

MVPO | Mean-variance portfolio optimization |

MVSE | Mean-variance-skewness-entropy |

Nats | Natural units |

PMPT | Post-modern portfolio theory |

REPO | Return-entropy portfolio optimization |

## Appendix A

**Proof.**

**Proof.**

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**Figure 3.**Optimal portfolio actual returns: return-entropy portfolio optimization (REPO) vs. mean-variance portfolio optimization (MVPO).

**Table 1.**The ten randomly selected securities from S&P/TSX 60 and the sample means, variances, and entropies of their mean weekly returns over the ten-year period.

Company Name | Ticker Symbol | Mean (bps) | Variance (bps${}^{2}$) | Entropy (nats) |
---|---|---|---|---|

Loblaw Companies Ltd. | L | 0.006391 | 8.078711 | 2.381352 |

First Quantum Minerals Ltd. | FM | 1.003592 | 61.97863 | 3.277249 |

Thomson Reuters Corp | TRI | −0.019931 | 11.65211 | 2.534170 |

Alimentation Couche-Tard Inc. | ATD.B | 0.495919 | 17.89425 | 2.798943 |

Bank of Nova Scotia | BNS | 0.242633 | 11.32819 | 2.466258 |

Teck Resources Ltd. | TECK.B | 0.729174 | 60.76170 | 3.259236 |

Canadian Tire Corp Ltd. | CTC.A | 0.284006 | 12.18994 | 2.605140 |

Inter Pipeline Ltd. | IPL | 0.211462 | 7.847551 | 2.339923 |

Manulife Financial Corp | MFC | 0.095557 | 24.68777 | 2.746475 |

Suncor Energy Inc. | SU | 0.424803 | 27.36700 | 2.907254 |

**Table 2.**Minimum objective and optimal solutions for mean-variance portfolio optimization (MVPO) and return-entropy portfolio optimization (REPO) methods.

Method | Minimum Objective | Expected Return | Optimal Solution |
---|---|---|---|

MVPO | 3.3993 bps${}^{2}$ | 0.1394 bps | (0.3,0.0,0.2,0.1,0.0,0.0,0.1,0.3,0.0,0.0) |

REPO | 1.9355 nats | 0.1630 bps | (0.2,0.0,0.2,0.1,0.1,0.0,0.1,0.3,0.0,0.0) |

Method | Expected Return | Optimal Solution |
---|---|---|

MVPO | 0.37 bps | (0.0,0.1,0.0,0.4,0.0,0.4,0.0,0.0,0.1,0.0) |

REPO | 0.37 bps | (0.0,0.4,0.3,0.0,0.0,0.0,0.0,0.2,0.1,0.0) |

**Table 4.**Comparison of REPO vs. MVPO portfolios over 20 weeks in 2011: number of portfolios that achieved greater returns.

REPO | MVPO | Total | % REPO > MVPO | |
---|---|---|---|---|

After 2 weeks | 2377 | 1792 | 4169 | 57% |

After 4 weeks | 3115 | 1054 | 4169 | 75% |

After 8 weeks | 2537 | 1632 | 4169 | 61% |

After 13 weeks | 2345 | 1824 | 4169 | 56% |

After 20 weeks | 1699 | 2470 | 4169 | 41% |

Risk Tolerance | Portfolio Entropy | Expected Return | Optimal Solution |
---|---|---|---|

$\alpha =1.0$ | 1.9551 nats | 0.2311 bps | (0.1,0.0,0.1,0.1,0.2,0.0,0.1,0.3,0.0,0.1) |

$\alpha =1.4$ | 2.1317 nats | 0.3588 bps | (0.1,0.1,0.0,0.2,0.1,0.0,0.1,0.3,0.0,0.1) |

$\alpha =1.7$ | 2.1419 nats | 0.3660 bps | (0.1,0.1,0.0,0.2,0.1,0.0,0.2,0.2,0.0,0.1) |

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Mercurio, P.J.; Wu, Y.; Xie, H. An Entropy-Based Approach to Portfolio Optimization. *Entropy* **2020**, *22*, 332.
https://doi.org/10.3390/e22030332

**AMA Style**

Mercurio PJ, Wu Y, Xie H. An Entropy-Based Approach to Portfolio Optimization. *Entropy*. 2020; 22(3):332.
https://doi.org/10.3390/e22030332

**Chicago/Turabian Style**

Mercurio, Peter Joseph, Yuehua Wu, and Hong Xie. 2020. "An Entropy-Based Approach to Portfolio Optimization" *Entropy* 22, no. 3: 332.
https://doi.org/10.3390/e22030332