# Portfolio Optimization with a Mean–Absolute Deviation–Entropy Multi-Objective Model

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Research Development

#### 2.2. Mean-Absolute Deviation Model

#### 2.3. Proposed Mean-Absolute Deviation-Entropy Model

#### 2.4. Model Performance

#### 2.5. Bootstrap Simulation

## 3. Results and Discussion

#### 3.1. Descriptive Statistics of the Returns of Stocks

#### 3.2. Optimal Portfolio Composition

#### 3.3. Model Performance

#### 3.4. Simulation Analyses

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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Symbol | Description |
---|---|

${R}_{j}$ | return of asset j |

${x}_{j}$ | weight of asset j |

$\rho $ | minimum return set by the investor |

$R\left(x\right)$ | portfolio mean return |

$w\left(x\right)$ | portfolio absolute deviation |

$H\left(x\right)$ | portfolio entropy |

${R}^{*}$ | optimal value for portfolio mean return |

${W}^{*}$ | optimal value for portfolio absolute deviation |

${d}_{1}$ | deviation of portfolio mean return from the optimal value |

${d}_{2}$ | deviation of portfolio absolute deviation from the optimal value |

${d}_{3}$ | deviation of portfolio entropy from the optimal value |

Stocks | Mean | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|

MMM | 0.0015 | 0.0322 | −0.7917 | 2.1866 |

AXP | 0.0037 | 0.0474 | 0.1840 | 10.4748 |

AMGN | 0.0022 | 0.0349 | −0.0553 | 0.5722 |

AAPL | 0.0063 | 0.0393 | −0.2796 | 2.8098 |

BA | 0.0048 | 0.0775 | 2.1565 | 31.7449 |

CAT | 0.0053 | 0.0418 | −0.2800 | 1.4749 |

CVX | 0.0014 | 0.0405 | −1.1459 | 11.2163 |

CSCO | 0.0030 | 0.0338 | −0.2415 | 1.9929 |

KO | 0.0012 | 0.0293 | −1.5923 | 12.5853 |

GS | 0.0031 | 0.0449 | 0.4427 | 5.7500 |

HD | 0.0038 | 0.0381 | −0.3786 | 14.4991 |

HON | 0.0035 | 0.0353 | −0.8228 | 11.6518 |

IBM | 0.0006 | 0.0350 | −0.2287 | 3.6550 |

INTC | 0.0031 | 0.0411 | −0.3729 | 2.6244 |

JNJ | 0.0020 | 0.0257 | −0.4792 | 2.5670 |

JPM | 0.0039 | 0.0396 | 0.1057 | 6.3517 |

MCD | 0.0028 | 0.0290 | −0.7580 | 7.1262 |

MRK | 0.0018 | 0.0284 | −0.0812 | 0.9399 |

MSFT | 0.0057 | 0.0301 | −0.4318 | 2.5895 |

NKE | 0.0034 | 0.0380 | 0.6587 | 5.3541 |

PG | 0.0022 | 0.0249 | −0.1018 | 3.4584 |

CRM | 0.0046 | 0.0447 | 1.1580 | 8.4621 |

TRV | 0.0017 | 0.0354 | 0.1498 | 6.1128 |

UNH | 0.0050 | 0.0400 | −0.5416 | 7.4587 |

VZ | 0.0011 | 0.0254 | 0.1536 | 1.0952 |

V | 0.0042 | 0.0310 | −0.6160 | 5.6441 |

WBA | −0.0010 | 0.0408 | −0.0805 | 2.0924 |

WMT | 0.0033 | 0.0280 | 0.1636 | 2.5789 |

DIS | 0.0027 | 0.0354 | 0.0463 | 4.4153 |

**Table 3.**Composition and ranking of stocks in the optimal portfolio (MAD model) for the D1 period and D2 period.

Stocks | Weights (%) [D1 Period] | Ranking [D1 Period] | Weights (%) [D2 Period] | Ranking [D2 Period] |
---|---|---|---|---|

AXP | 0.7904 | 12 | - | - |

AAPL | - | - | 2.9198 | 6 |

BA | 2.3983 | 10 | - | - |

CAT | - | - | 10.1117 | 4 |

CVX | 11.2762 | 4 | - | - |

KO | 9.5720 | 5 | - | - |

JNJ | - | - | 1.4497 | 7 |

JPM | 6.0708 | 7 | - | - |

MCD | 15.8696 | 3 | 7.6486 | 5 |

MRK | 4.5878 | 8 | - | - |

MSFT | 3.7366 | 9 | 23.4267 | 2 |

PG | 18.5039 | 1 | - | - |

CRM | 1.2556 | 11 | - | - |

UNH | 9.2145 | 6 | - | - |

VZ | 0.0759 | 13 | 31.1050 | 1 |

WMT | 16.6482 | 2 | 23.3385 | 3 |

**Table 4.**Composition and ranking of stocks in the optimal portfolio (Proposed model) for the D1 period and D2 period.

Stocks | Weights (%) [D1 Period] | Ranking [D1 Period] | Weights (%) [D2 Period] | Ranking [D2 Period] |
---|---|---|---|---|

MMM | 0.0287 | 27 | 4.0459 | 9 |

AXP | 0.9634 | 20 | 0.0246 | 26 |

AMGN | 0.2127 | 25 | 1.4396 | 15 |

AAPL | 7.7781 | 6 | 4.0520 | 8 |

BA | 7.2796 | 7 | 0.77584 × 10 ^{−7} | 29 |

CAT | 1.3159 | 14 | 8.6001 | 4 |

CVX | 1.0525 | 17 | 0.0140 | 28 |

CSCO | 0.3132 | 22 | 1.8066 | 14 |

KO | 1.3468 | 13 | 0.1865 | 22 |

GS | 0.0261 | 28 | 0.8782 | 17 |

HD | 1.2611 | 15 | 3.0300 | 10 |

HON | 0.9761 | 19 | 0.0896 | 24 |

IBM | 0.0316 | 26 | 0.4680 | 19 |

INTC | 0.3000 | 23 | 0.4319 | 20 |

JNJ | 1.0515 | 18 | 4.3079 | 7 |

JPM | 3.1691 | 11 | 0.6681 | 18 |

MCD | 6.6089 | 8 | 4.9987 | 5 |

MRK | 4.3922 | 9 | 1.3984 | 16 |

MSFT | 15.1745 | 1 | 25.6371 | 1 |

NKE | 1.1007 | 16 | 2.3731 | 12 |

PG | 8.1654 | 5 | 1.9720 | 13 |

CRM | 3.8759 | 10 | 2.4811 | 11 |

TRV | 0.2590 | 24 | 0.0879 | 25 |

UNH | 10.6100 | 3 | 0.2193 | 21 |

VZ | 1.7637 | 12 | 13.2411 | 2 |

V | 9.5210 | 4 | 0.1586 | 23 |

WBA | 0.0015 | 29 | 0.0175 | 27 |

WMT | 11.0766 | 2 | 12.5844 | 3 |

DIS | 0.3443 | 21 | 4.7879 | 6 |

**Table 5.**Performance comparison of the optimal portfolios among the MAD model, the proposed model and the naive diversification strategy for the D1 period.

Optimal Portfolio | MAD Model | Proposed Model | Naive Diversification Strategy |
---|---|---|---|

Portfolio mean return (%) | 0.3000 | 0.4000 | 0.2425 |

Performance ratio | 0.2970 | 0.3254 | 0.2245 |

**Table 6.**Performance comparison of the optimal portfolios among the MAD model, the proposed model and the naive diversification strategy for the D2 period.

Optimal Portfolio | MAD Model | Proposed Model | Naive Diversification Strategy |
---|---|---|---|

Portfolio mean return (%) | 0.3400 | 0.4400 | 0.3185 |

Performance ratio | 0.1762 | 0.2041 | 0.1313 |

**Table 7.**Bootstrap simulation analysis on the portfolio return of the MAD model, proposed model and the naive diversification strategy for the D1 period.

Optimal Portfolio | MAD Model | Proposed Model | Naive Diversification Strategy |
---|---|---|---|

Portfolio mean return (%) | 0.2984 | 0.4011 | 0.2445 |

95% Confidence Interval | 0.2955–0.3012 | 0.3979–0.4043 | 0.2416–0.2474 |

Performance ratio | 0.2909 | 0.3798 | 0.2340 |

**Table 8.**Bootstrap simulation analysis on the portfolio return of the MAD model, proposed model and the naive diversification strategy for the D2 period.

Optimal Portfolio | MAD Model | Proposed Model | Naive Diversification Strategy |
---|---|---|---|

Portfolio mean return (%) | 0.3507 | 0.4341 | 0.3073 |

95% Confidence Interval | 0.3417–0.3597 | 0.4234–0.4447 | 0.2947–0.3199 |

Performance ratio | 0.1927 | 0.2242 | 0.1635 |

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## Share and Cite

**MDPI and ACS Style**

Lam, W.S.; Lam, W.H.; Jaaman, S.H.
Portfolio Optimization with a Mean–Absolute Deviation–Entropy Multi-Objective Model. *Entropy* **2021**, *23*, 1266.
https://doi.org/10.3390/e23101266

**AMA Style**

Lam WS, Lam WH, Jaaman SH.
Portfolio Optimization with a Mean–Absolute Deviation–Entropy Multi-Objective Model. *Entropy*. 2021; 23(10):1266.
https://doi.org/10.3390/e23101266

**Chicago/Turabian Style**

Lam, Weng Siew, Weng Hoe Lam, and Saiful Hafizah Jaaman.
2021. "Portfolio Optimization with a Mean–Absolute Deviation–Entropy Multi-Objective Model" *Entropy* 23, no. 10: 1266.
https://doi.org/10.3390/e23101266