# Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation

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## Abstract

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## 1. Introduction

- High concentration: MV-inspired portfolios are highly concentrated on a few securities with the "best: features [13,14]. Assets with either high expected returns or low expected variance will be overweighted, in this way losing the power of diversification that the theory is supposed to ensure [15].
- Instability: MV portfolios tend to drastically re-allocate resources when the asset features change slightly, regardless of transaction costs or data inaccuracy [16,17]. This mainly occurs because MV portfolios do not take estimation inaccuracy into account and concentrate on assets with "good" features.

- Maximize the expected return for a specified risk: the first possible formulation includes the maximization of the portfolio mean in the objective function of the problem and the maximum level of risk that an investor is able to assume as a constraint of the problem.
- Minimize the risk for a pre-determined expected return: the second alternative tries to minimize the risk and introduces the minimum level of the mean return as a constraint of the optimization.
- Minimize the risk and maximize the expected return combining both of the objectives through a user-defined risk aversion parameter.

## 2. The Proposed Method

#### 2.1. Mathematical Formulation of the Model

#### 2.2. Main Foundations of the Model

#### 2.3. Mixed-Integer Linear Programming Reformulation

## 3. Experimental Framework

#### 3.1. Out-Of-Sample Empirical Validation and Portfolio Problems Selected

#### 3.2. Strategies Implemented

#### 3.3. Performance Measures

- The out-of-sample mean returns (MR):$$\mathrm{MR}=(1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}Q)\sum _{q=1}^{Q}{r}_{q}.$$
- The out-of-sample Sharpe ratio (SR), defined as the sample mean of out-of-sample excess returns, MR, divided by their corresponding sample standard deviation:$$\mathrm{SR}=\frac{\mathrm{MR}}{\sqrt{(1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}Q){\sum}_{q=1}^{Q}{({r}_{q}-\mathrm{MR})}^{2}}}.$$

#### 3.4. Hyper-Parameter Optimization

#### 3.5. Statistical Hypothesis Testing

## 4. Results

- to compare the out-of-sample performance of the MSV portfolio with the performance provided by state-of-the-art MV-based strategies (Section 4.1); and,
- to analyse the diversification levels produced by the proposed MSV portfolio and the MV portfolio in problems with different dimensions (Section 4.2).

#### 4.1. Performance Analysis

#### 4.2. Diversification Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

BM | Book-to-market |

GMR | Global maximum return |

GMV | Global minimum variance |

I | Investment |

MILP | Mixed-integer linear programming |

MR | Mean return |

MSV | Mean squared variance |

MV | Mean-variance |

OP | Operating profitability |

QP | Quadratic programming |

SR | Sharpe ratio |

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**Figure 1.**Cumulative monthly returns plot of the four portfolios implemented from 1 September 2017–1 August 2020.

**Figure 2.**The boxplots of diversification for the MV and MSV strategies for $N\in \{7,8,9,10\}$ and $\lambda \in \{0.2,0.4,0.6,0.8\}$.

ID. | Dataset | M | T | N |
---|---|---|---|---|

Industry Portfolios | ||||

$\{1,2,3\}$ | 5 Industry Portfolios | $\{60,120,240\}$ | $\{96,156,276\}$ | 5 |

$\{4,5,6\}$ | 10 Industry Portfolios | $\{60,120,240\}$ | $\{96,156,276\}$ | 10 |

Emerging Market Factors | ||||

$\{7,8,9\}$ | 6 Emerging Market Portfolios Formed on BM and OP | $\{60,120,240\}$ | $\{96,156,276\}$ | 6 |

$\{10,11,12\}$ | 6 Emerging Market Portfolios Formed on Size and BM | $\{60,120,240\}$ | $\{96,156,276\}$ | 6 |

$\{13,14,15\}$ | 6 Emerging Market Portfolios Formed on Size and OP | $\{60,120,240\}$ | $\{96,156,276\}$ | 6 |

Bivariate sorts on Size, BM and I | ||||

$\{16,17,18\}$ | 6 Portfolios Formed on Size and BM | $\{60,120,240\}$ | $\{96,156,276\}$ | 6 |

$\{19,20,21\}$ | 6 Portfolios Formed on Size and I | $\{60,120,240\}$ | $\{96,156,276\}$ | 6 |

Bivariate sorts on Size, BM and I | ||||

$\{22,23,24\}$ | 25 European Portfolios Formed on Size and BM | $\{60,120,240\}$ | $\{96,156,276\}$ | 25 |

**Table 2.**Statistical results of the strategies implemented: the mean returns (MR) and Sharpe ratio (SR) out-of-sample performance of each strategy and dataset (MR and SR), mean MR ($\overline{MR}$), mean MR ranking (${\overline{R}}_{MR}$), mean SR ($\overline{SR}$) and mean SR ranking (${\overline{R}}_{SR}$). The datasets are denoted by their IDs. The best results are in bold face and second-best results in italics.

MR Metric | SR Metric | ||||||||
---|---|---|---|---|---|---|---|---|---|

ID | GMV | GMR | MV | MSV | GMV | GMR | MV | MSV | ID |

1 | 0.9521 | $\mathit{1.6197}$ | ${1.1717}_{0.0065}$ | ${\mathbf{1.8161}}_{\mathbf{0.0000}}$ | 0.1330 | $\mathit{0.2243}$ | ${0.1646}_{0.0015}$ | ${\mathbf{0.2339}}_{\mathbf{0.0000}}$ | 1 |

2 | 1.4004 | $\mathbf{1.8981}$ | ${\mathit{1.8977}}_{\mathit{0.0003}}$ | ${\mathbf{1.8981}}_{\mathbf{0.0000}}$ | 0.2825 | $\mathbf{0.3408}$ | ${\mathit{0.3391}}_{\mathit{0.0008}}$ | ${\mathbf{0.3408}}_{\mathbf{0.0000}}$ | 2 |

3 | 1.3098 | 0.9897 | ${\mathit{1.3146}}_{\mathit{0.0013}}$ | ${\mathbf{1.3478}}_{\mathbf{0.0034}}$ | 0.2698 | 0.1748 | ${\mathit{0.2705}}_{\mathit{0.0008}}$ | ${\mathbf{0.2775}}_{\mathbf{0.0021}}$ | 3 |

4 | 0.6058 | $\mathbf{2.1011}$ | ${1.4175}_{0.0008}$ | ${\mathit{1.2533}}_{\mathit{0.0338}}$ | 0.1411 | $\mathbf{0.3493}$ | ${0.2426}_{0.0005}$ | ${\mathit{0.2965}}_{\mathit{0.0098}}$ | 4 |

5 | 0.6324 | $\mathbf{1.9033}$ | ${\mathit{1.2299}}_{\mathit{0.0012}}$ | ${1.1355}_{0.0004}$ | 0.1541 | $\mathbf{0.3077}$ | ${\mathit{0.2174}}_{\mathit{0.0013}}$ | ${0.2062}_{0.0003}$ | 5 |

6 | 0.7234 | $\mathbf{1.2039}$ | ${0.8252}_{0.0021}$ | ${\mathit{1.0470}}_{\mathit{0.0866}}$ | 0.1752 | 0.2079 | ${\mathbf{0.2093}}_{\mathbf{0.0007}}$ | ${\mathit{0.2084}}_{\mathit{0.0124}}$ | 6 |

7 | 0.5404 | 0.5456 | ${\mathit{0.5477}}_{\mathit{0.0008}}$ | ${\mathbf{0.5577}}_{\mathbf{0.0013}}$ | $\mathit{0.1016}$ | 0.0960 | ${\mathit{0.1016}}_{\mathit{0.0003}}$ | ${\mathbf{0.1028}}_{\mathbf{0.0005}}$ | 7 |

8 | $\mathbf{0.5473}$ | 0.4494 | ${\mathit{0.5342}}_{\mathit{0.0008}}$ | ${0.4494}_{0.0000}$ | $\mathbf{0.1027}$ | 0.0791 | ${\mathit{0.0959}}_{\mathit{0.0002}}$ | ${0.0791}_{0.0000}$ | 8 |

9 | $\mathbf{0.5118}$ | −0.1094 | $-{0.1042}_{0.0118}$ | ${\mathit{0.0021}}_{\mathit{0.0089}}$ | $\mathbf{0.0960}$ | −0.0175 | $-{0.0166}_{0.0019}$ | ${\mathit{0.0021}}_{\mathit{0.0017}}$ | 9 |

10 | 0.1898 | 0.6411 | ${\mathit{0.6524}}_{\mathit{0.0419}}$ | ${\mathbf{0.6848}}_{\mathbf{0.0321}}$ | 0.0324 | 0.1277 | ${\mathit{0.1295}}_{\mathit{0.0072}}$ | ${\mathbf{0.1351}}_{\mathbf{0.0057}}$ | 10 |

11 | 0.4162 | $\mathbf{0.6039}$ | ${0.5962}_{0.0051}$ | ${\mathit{0.6005}}_{\mathit{0.0041}}$ | 0.0742 | $\mathit{0.1268}$ | ${0.1249}_{0.0010}$ | ${\mathbf{0.1270}}_{\mathbf{0.0007}}$ | 11 |

12 | 0.3150 | $\mathit{0.1772}$ | ${\mathit{0.1772}}_{\mathit{0.0000}}$ | ${\mathbf{0.1775}}_{\mathbf{0.0012}}$ | $\mathit{0.0561}$ | 0.0306 | ${0.0306}_{0.0000}$ | ${\mathit{0.0309}}_{\mathit{0.0002}}$ | 12 |

13 | 0.1505 | $\mathbf{0.4075}$ | ${0.1259}_{0.0031}$ | ${\mathit{0.2530}}_{\mathit{0.0810}}$ | 0.0262 | $\mathbf{0.0747}$ | ${0.0220}_{0.0010}$ | ${\mathit{0.0456}}_{\mathit{0.0266}}$ | 13 |

14 | 0.2519 | $\mathbf{0.3078}$ | ${0.2679}_{0.0014}$ | ${\mathit{0.2756}}_{\mathit{0.0294}}$ | 0.0444 | $\mathbf{0.0614}$ | ${0.0474}_{0.0008}$ | ${\mathit{0.0515}}_{\mathit{0.0091}}$ | 14 |

15 | 0.1601 | $\mathbf{0.3181}$ | ${0.1640}_{0.0321}$ | ${\mathit{0.1919}}_{\mathit{0.0705}}$ | 0.0283 | $\mathbf{0.0561}$ | ${0.0291}_{0.0087}$ | ${\mathit{0.0339}}_{\mathit{0.0124}}$ | 15 |

16 | 1.1048 | $\mathit{1.7043}$ | ${1.5803}_{0.1945}$ | ${\mathbf{1.7876}}_{\mathbf{0.0000}}$ | 0.2017 | $\mathit{0.3252}$ | ${0.2992}_{0.0413}$ | ${\mathbf{0.3288}}_{\mathbf{0.0000}}$ | 16 |

17 | 1.4131 | 1.4092 | ${\mathit{1.4394}}_{\mathit{0.0187}}$ | ${\mathbf{1.4656}}_{\mathbf{0.0021}}$ | 0.2534 | 0.2600 | ${\mathbf{0.2682}}_{\mathbf{0.0022}}$ | ${\mathit{0.2639}}_{\mathit{0.0007}}$ | 17 |

18 | 1.0582 | 0.1845 | ${\mathit{1.1500}}_{\mathit{0.0018}}$ | ${\mathbf{1.2829}}_{\mathbf{0.3904}}$ | $\mathit{0.2032}$ | 0.0276 | ${0.2013}_{0.0004}$ | ${\mathbf{0.2280}}_{\mathbf{0.0085}}$ | 18 |

19 | 1.0294 | 1.4941 | ${\mathit{1.4982}}_{\mathit{0.1002}}$ | ${\mathbf{1.5056}}_{\mathbf{0.0196}}$ | 0.1948 | 0.2271 | ${\mathit{0.2445}}_{\mathit{0.0106}}$ | ${\mathbf{0.2621}}_{\mathbf{0.0023}}$ | 19 |

20 | 1.0063 | $\mathit{1.0665}$ | ${\mathbf{1.0790}}_{\mathbf{0.0008}}$ | ${1.0067}_{0.0009}$ | 0.1908 | 0.1721 | ${\mathbf{0.1930}}_{\mathbf{0.0001}}$ | ${\mathit{0.1912}}_{\mathit{0.0003}}$ | 20 |

21 | $\mathit{0.7662}$ | 0.0917 | ${0.7658}_{0.0009}$ | ${\mathbf{0.7664}}_{\mathbf{0.0007}}$ | 0.1488 | 0.0134 | ${\mathbf{0.1497}}_{\mathbf{0.0002}}$ | ${\mathit{0.1490}}_{\mathit{0.0002}}$ | 21 |

22 | 0.4391 | 0.5806 | ${\mathbf{0.5843}}_{\mathbf{0.0091}}$ | ${\mathit{0.5823}}_{\mathit{0.0064}}$ | 0.1011 | $\mathit{0.1019}$ | ${0.0468}_{0.0015}$ | ${\mathbf{0.1048}}_{\mathbf{0.0023}}$ | 22 |

23 | 0.7158 | 0.8431 | ${\mathit{0.8613}}_{\mathit{0.0148}}$ | ${\mathbf{0.8641}}_{\mathbf{0.0217}}$ | $\mathbf{0.1801}$ | 0.1530 | ${0.1548}_{0.0027}$ | ${\mathit{0.1577}}_{\mathit{0.0031}}$ | 23 |

24 | $\mathbf{0.4016}$ | −0.2092 | $-{0.1046}_{0.0113}$ | ${\mathit{0.0032}}_{\mathit{0.0283}}$ | $\mathbf{0.0953}$ | −0.0347 | $-{0.0324}_{0.0023}$ | ${\mathit{0.0008}}_{\mathit{0.0009}}$ | 24 |

$\overline{MR}$ | 0.6934 | $\mathit{0.8426}$ | 0.8196 | $\mathbf{0.8731}$ | 0.1369 | 0.1452 | $\mathit{0.1472}$ | $\mathbf{0.1607}$ | $\overline{SR}$ |

${\overline{R}}_{MR}$ | 3.2500 | 2.5208 | $\mathit{2.4792}$ | $\mathbf{1.7500}$ | 3.0208 | 2.7292 | $\mathit{2.5417}$ | $\mathbf{1.7083}$ | ${\overline{R}}_{SR}$ |

**Table 3.**Statistical results for the Holm test for $\alpha =0.10$ and $\alpha =0.05$ using the mean squared variance (MSV) strategy as the control method: mean MR and SR ranking of the strategies implemented (${\overline{R}}_{MR}$ and ${\overline{R}}_{SR}$), z-statistics and p-values of the Holm tests for the MR and SR analysis and adjusted $\alpha $ Holm values (${\alpha}_{0.10}$ and ${\alpha}_{0.05}$). The best results are in bold face and second-best results in italics.

MR Analysis | |||||
---|---|---|---|---|---|

Method | ${\overline{R}}_{MR}$ | z-statistic | p-value | ${\alpha}_{0.10}$ | ${\alpha}_{0.05}$ |

${\mathrm{GMV}}_{\u2022,\circ}$ | 3.2500 | 4.0249 | 1 × 10${}^{-4}$ | 0.0333 | 0.0167 |

${\mathrm{GMR}}_{\u2022}$ | 2.5208 | 2.0683 | 0.0386 | 0.0500 | 0.0250 |

${\mathrm{MV}}_{\u2022}$ | $\mathit{2.4792}$ | 1.9566 | 0.0504 | 0.1000 | 0.0500 |

MSV | $\mathbf{1.7500}$ | - | - | - | - |

SR Analysis | |||||

Method | ${\overline{R}}_{SR}$ | z-statistic | p-value | ${\alpha}_{0.10}$ | ${\alpha}_{0.05}$ |

${\mathrm{GMV}}_{\u2022,\circ}$ | 3.0208 | 3.5218 | 4 × 10${}^{-4}$ | 0.0333 | 0.0167 |

${\mathrm{GMR}}_{\u2022,\circ}$ | 2.7292 | 2.7394 | 0.0062 | 0.0500 | 0.0250 |

${\mathrm{MV}}_{\u2022,\circ}$ | $\mathit{2.5417}$ | 2.2362 | 0.0253 | 0.1000 | 0.0500 |

MSV | $\mathbf{1.7083}$ | - | - | - | - |

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**MDPI and ACS Style**

Fernández-Navarro, F.; Martínez-Nieto, L.; Carbonero-Ruz, M.; Montero-Romero, T.
Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation. *Mathematics* **2021**, *9*, 223.
https://doi.org/10.3390/math9030223

**AMA Style**

Fernández-Navarro F, Martínez-Nieto L, Carbonero-Ruz M, Montero-Romero T.
Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation. *Mathematics*. 2021; 9(3):223.
https://doi.org/10.3390/math9030223

**Chicago/Turabian Style**

Fernández-Navarro, Francisco, Luisa Martínez-Nieto, Mariano Carbonero-Ruz, and Teresa Montero-Romero.
2021. "Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation" *Mathematics* 9, no. 3: 223.
https://doi.org/10.3390/math9030223