# A Mean-Variance Hybrid-Entropy Model for Portfolio Selection with Fuzzy Returns

^{*}

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## Abstract

**:**

## 1. Introduction

## 2. Mean-Variance Hybrid-Entropy Portfolio Optimization Model

#### 2.1. Fuzzy Returns Predicted by the Markov Method

#### 2.1.1. The Expected Value and Variance of the Triangular Fuzzy Returns

#### 2.1.2. Prediction of Stock Returns

_{i}denote the mathematical expectation of the random variable of ${\tilde{r}}_{i}$; b

_{i}and a

_{i}are the ceiling return and the floor return, respectively. Hence, the membership function of ${\tilde{r}}_{i}$ is defined by the following functions:

- Step 1. Collect the historical trading data in a sample period (in this paper, it is one year or three years), including the opening price R
_{t}_{1}, closing price R_{t}_{2}, ceiling price R_{t}_{3}and floor price R_{t}_{4}, t = 1,2, …, N, where N is the number of sub-intervals. Then we calculate the possible average rates of returns ${r}_{te}=\frac{{R}_{t2}-{R}_{t1}}{{R}_{t1}}$, the highest possible rates of returns ${r}_{th}=\frac{{R}_{t2}-{R}_{t1}}{{R}_{t1}}$, and the lowest possible rates of returns ${r}_{tl}=\frac{{R}_{t4}-{R}_{t1}}{{R}_{t1}}$. - Step 2. Use the classic K-Means cluster analysis method to get the step transition matrix. We divide the range of rate of return into M intervals called state spaces and get mid-points d
_{i}(i = 1,2, …, M) and probability p_{ij}(i, j = 1,2, …, M) that the return is in space j if it was in space i the last state. Then form one step transition matrix by these probabilities:

- Step 3. Develop the state transition equation. The probability of stock return in state space i can be calculated by:$$\mathbf{x}=\mathit{Px}$$
_{1}, x_{2}, …, x_{M})^{T}, x_{1}, x_{2}, …, x_{M}≥ 0, ${\sum}_{i=1}^{M}{x}_{i}=1$.The unique solution of the equation is**x**= (p_{1}, p_{2}, …, p_{M})^{T}. Therefore the probabilities of the stock return in state space i after a long enough time are p_{1}, p_{2}, …, p_{M}. - Step 4. Compute the prediction of the stock return by function $r={\displaystyle {\sum}_{i=1}^{M}{p}_{i}{d}_{i}}$. The highest possible rate of return r
_{th}and lowest possible rate of return r_{t}_{l}can be calculated in the same way. That is how we get the value of the ceiling return b_{i}and the floor return a_{i}. Hence, the prediction of our triangular fuzzy returns turns out to be ${\tilde{r}}_{i}=({a}_{i},{r}_{i},{b}_{i})$.

#### 2.2. Hybrid Entropy

_{h}should meet the following requirements:

- H
_{h}will reach its biggest value if and only if μ_{i}= 0.5 and p_{i}= 1/n (i = 1,2,…,n); - H
_{h}will reach its smallest value 0 if and only if μ_{i}= 0 or 1 (i = 1,2, …, n), p_{i}= 1 and p_{j}= 0(j ≠ i,i,j = 1,2,…,n); - When randomness (ambiguity) disappears, hybrid entropy should be reduced to a normal probability entropy (fuzzy entropy).

#### 2.3. Portfolio Optimization Model

#### 2.3.1. MVM and MEM

_{1}, r

_{2}, …, r

_{N})

^{T}; the wealth fraction invested in the securities X = (x

_{1}, x

_{2}, …, x

_{N})

^{T}; C is the covariance matrix and c represents the given expected return.

_{1}, r

_{2}, …, r

_{N})

^{T}the wealth fraction invested in the securities X = (x

_{1}, x

_{2}, …, x

_{N})

^{T}, and the risk vector of N securities H = (h

_{1}, h

_{2}, …, h

_{N})

^{T}.

#### 2.3.2. MVHEM

## 3. Empirical Comparisons

#### 3.1. Sample Data

#### 3.2. The Empirical Comparisons among MVM, MEM and MVHEM

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Sample Period
| One-Year Period | Three-Year Period |
---|---|---|

Stock Code | ||

600100 | (−0.0314, 0.0070, 0.0494) | (−0.0398, −0.0042, 0.0391) |

600270 | (−0.0411, 0.0095, 0.0583) | (−0.0365, 0.0026, 0.4238) |

600109 | (−0.0433, 0.0009, 0.0496) | (−0.0441, 0.0027, 0.0464) |

600664 | (−0.0382, 0.0008, 0.0349) | (−0.0458, −0.0078, 0.0275) |

600060 | (−0.0445, 0.0045, 0.0527) | (−0.0418, 0.0018, 0.0450) |

600714 | (−0.0392, −0.0065, 0.0399) | (−0.0511, −0.0008, 0.0512) |

600886 | (−0.0351, −0.0043, 0.0353) | (−0.0321, −0.0025, 0.0277) |

600638 | (−0.0357, 0.0057, 0.0420) | (−0.0347, 0.0030, 0.0374) |

600778 | (−0.0354, −0.0031, 0.0362) | (−0.0414, −0.0009, 0.0389) |

600081 | (−0.0409, 0.0058, 0.0527) | (−0.0008, −0.0008, 0.0462) |

Sample Period
| One-Year Period | Three-Year Period |
---|---|---|

Stock Code | ||

002186 | (−0.0380, −0.0042, 0.0428) | (−0.0312,−0.0015,0.0313) |

000791 | (−0.0364, 0.0017, 0.0400) | (−0.0403, 0.0012, 0.0446) |

002032 | (−0.0329, 0.0031, 0.0444) | (−0.0382, −0.0026, 0.0351) |

000002 | (−0.0416, −0.0045, 0.0387) | (−0.0346, 0.0004, 0.0338) |

000768 | (−0.0357, 0.0045, 0.0513) | (−0.0347, −0.0005, 0.0414) |

002226 | (−0.0331, 0.0037, 0.0414) | (−0.0416, −0.0024, 0.0373) |

300027 | (−0.0598, 0.0284, 0.1017) | (−0.0466, 0.0055, 0.0579) |

000088 | (−0.0386, 0.0114, 0.0569) | (−0.0357, 0.0017, 0.0361) |

300005 | (−0.0531, 0.0025, 0.0549) | (−0.0488, −0.0020, 0.0464) |

000001 | (−0.0501, 0.0000, 0.0573) | (−0.0338, 0.0001, 0.0343) |

Sample Period
| One-Year Period
| Three-Year Period
| ||
---|---|---|---|---|

Stock Code | Expected Value | Hybrid Entropy | Expected Value | Hybrid Entropy |

600100 | 0.0079 | 0.6303 | −0.0023 | 0.6237 |

600270 | 0.0091 | 0.7088 | 0.0028 | 0.5525 |

600109 | 0.0020 | 0.5320 | 0.0019 | 0.5852 |

600664 | −0.0004 | 1.0954 | −0.0085 | 1.0520 |

600060 | 0.0043 | 0.6288 | 0.0017 | 0.4766 |

600714 | −0.0031 | 0.6598 | −0.0004 | 0.6944 |

600886 | −0.0021 | 0.6981 | −0.0023 | 0.6944 |

600638 | 0.0044 | 1.1528 | 0.0022 | 0.6767 |

600778 | −0.0014 | 0.6217 | −0.0011 | 0.5536 |

600081 | 0.0058 | 0.6796 | 0.0109 | 0.3789 |

Sample Period
| One-Year Period
| Three-Year Period
| ||
---|---|---|---|---|

Stock Code | Expected Value | Hybrid Entropy | Expected Value | Hybrid Entropy |

002186 | −0.0009 | 1.1940 | −0.0007 | 1.0360 |

000791 | 0.0017 | 0.6555 | 0.0017 | 0.6981 |

002032 | 0.0044 | 1.1211 | −0.0020 | 0.9846 |

000002 | −0.0029 | 0.7656 | 0.0000 | 0.5124 |

000768 | 0.0061 | 0.5451 | 0.0014 | 0.3668 |

002226 | 0.0039 | 0.5890 | −0.0023 | 1.1131 |

300027 | 0.0247 | 0.6167 | 0.0056 | 0.7336 |

000088 | 0.0103 | 1.1977 | 0.0009 | 0.7219 |

300005 | 0.0017 | 0.6466 | −0.0016 | 0.9808 |

000001 | 0.0018 | 0.7208 | 0.0002 | 0.5996 |

**Table 5.**The proportion of one-year period sample stocks in different portfolio selection models in SHSE.

Stock Code
| 600100 | 600270 | 600109 | 600664 | 600060 | 600714 | 600886 | 600638 | 600778 | 600081 |
---|---|---|---|---|---|---|---|---|---|---|

Model | ||||||||||

MVHEM-I | 0.2402 | 0.1574 | 0.1222 | 0.0359 | 0.1590 | 0.0378 | 0.0529 | 0.0416 | 0.0997 | 0.0535 |

MVHEM-II | 0.0551 | 0.1067 | 0.0452 | 0.1804 | 0.2603 | 0.0544 | 0.0654 | 0.1907 | 0.0193 | 0.0220 |

MVM | 0.1500 | 0.7054 | 0.0189 | 0.0012 | 0.0246 | 0.0053 | 0.0004 | 0.0459 | 0.0209 | 0.0281 |

MEM | 0.9153 | 0.0019 | 0.0779 | 0.0002 | 0.0002 | 0.0005 | 0.0017 | 0.0002 | 0.0003 | 0.0008 |

**Table 6.**The proportion of three-year period sample stocks in different portfolio selection models in SHSE.

Stock Code
| 600100 | 600270 | 600109 | 600664 | 600060 | 600714 | 600886 | 600638 | 600778 | 600081 |
---|---|---|---|---|---|---|---|---|---|---|

Model | ||||||||||

MVHEM-I | 0.0102 | 0.0916 | 0.0651 | 0.0231 | 0.0457 | 0.0741 | 0.0920 | 0.0690 | 0.0626 | 0.4650 |

MVHEM-II | 0.0022 | 0.0795 | 0.0450 | 0.0149 | 0.0372 | 0.0500 | 0.4470 | 0.0582 | 0.0553 | 0.2090 |

MVM | 0.0015 | 0.0294 | 0.0844 | 0.0000 | 0.1065 | 0.0003 | 0.0001 | 0.0272 | 0.0002 | 0.7490 |

MEM | 0.0069 | 0.0565 | 0.0146 | 0.0062 | 0.0412 | 0.0147 | 0.1096 | 0.0514 | 0.0306 | 0.6672 |

**Table 7.**The proportion of one-year period sample stocks in different portfolio selection models in SZSE.

Stock Code
| 002186 | 000791 | 002032 | 000002 | 000768 | 002226 | 300027 | 000088 | 300005 | 000001 |
---|---|---|---|---|---|---|---|---|---|---|

Model | ||||||||||

MVHEM-I | 0.0218 | 0.0485 | 0.0369 | 0.0248 | 0.2721 | 0.2360 | 0.1991 | 0.0205 | 0.0697 | 0.0698 |

MVHEM-II | 0.0551 | 0.1067 | 0.0452 | 0.1804 | 0.2603 | 0.0544 | 0.0654 | 0.1907 | 0.0193 | 0.0220 |

MVM | 0.1500 | 0.7054 | 0.0189 | 0.0012 | 0.0246 | 0.0053 | 0.0004 | 0.0459 | 0.0209 | 0.0281 |

MEM | 0.9153 | 0.0019 | 0.0779 | 0.0002 | 0.0002 | 0.0005 | 0.0017 | 0.0002 | 0.0003 | 0.0008 |

**Table 8.**The proportion of three-year period sample stocks in different portfolio selection models in SZSE.

Stock Code
| 002186 | 000791 | 002032 | 000002 | 000768 | 002226 | 300027 | 000088 | 300005 | 000001 |
---|---|---|---|---|---|---|---|---|---|---|

Model | ||||||||||

MVHEM-I | 0.0002 | 0.0544 | 0.0182 | 0.0687 | 0.6542 | 0.0016 | 0.0948 | 0.0239 | 0.0231 | 0.0456 |

MVHEM-II | 0.1518 | 0.0463 | 0.0116 | 0.0533 | 0.4074 | 0.0147 | 0.2351 | 0.0196 | 0.0203 | 0.0398 |

MVM | 0.0153 | 0.0533 | 0.0037 | 0.0384 | 0.4796 | 0.0102 | 0.2885 | 0.0161 | 0.0183 | 0.0772 |

MEM | 0.2690 | 0.0125 | 0.0302 | 0.0050 | 0.0138 | 0.0394 | 0.5278 | 0.0689 | 0.0008 | 0.0329 |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, R.; Zhan, Y.; Cai, R.; Tong, G.
A Mean-Variance Hybrid-Entropy Model for Portfolio Selection with Fuzzy Returns. *Entropy* **2015**, *17*, 3319-3331.
https://doi.org/10.3390/e17053319

**AMA Style**

Zhou R, Zhan Y, Cai R, Tong G.
A Mean-Variance Hybrid-Entropy Model for Portfolio Selection with Fuzzy Returns. *Entropy*. 2015; 17(5):3319-3331.
https://doi.org/10.3390/e17053319

**Chicago/Turabian Style**

Zhou, Rongxi, Yu Zhan, Ru Cai, and Guanqun Tong.
2015. "A Mean-Variance Hybrid-Entropy Model for Portfolio Selection with Fuzzy Returns" *Entropy* 17, no. 5: 3319-3331.
https://doi.org/10.3390/e17053319