# Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds

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## Abstract

**:**

## 1. Introduction

- virtual weight restrictions with constraints on all decision-making units (DMUs) (see Section 3.1 and Section 4.1);
- virtual weight restrictions with constraints only on the target DMU (see Section 3.1 and Section 4.2);
- weight restrictions with assurance regions (see Section 3.2 and Section 5).

## 2. An Output Oriented DEA Model for Mutual Funds

**Phase I**

**Phase II**

## 3. Restrictions on the Weights of a DEA Model

#### 3.1. Virtual Weight Restrictions

#### 3.2. Restrictions with Assurance Regions

## 4. Matrix Representation of Virtual Weights Restrictions

#### 4.1. Case 1: Constraints on All DMUs

#### 4.2. Case 2: Constraints Only on the Target DMU

## 5. Matrix Representation of Assurance Region Weight Restrictions

## 6. DEA Efficiency Evaluation with Virtual Weights Restrictions and Assurance Regions

**Definition**

**1.**

**Definition**

**2.**

## 7. An Experimental Application to European Mutual Funds

#### 7.1. Design of the Analysis

#### 7.2. Applying Virtual Weights Restrictions with Constraints on All DMUs

#### 7.3. Applying Virtual Weights Restrictions with Constraints on the Target DMU

#### 7.4. Applying Assurance Region Weight Restrictions

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the dynamics of the mean performance scores of virtual weights (VW) restrictions in the cases with constraints on all decision-making units (DMUs) and only on the target DMU as d increases, for $T=1$ and symmetric bounds.

**Figure 2.**Comparison of the dynamics of the mean performance scores of VW restrictions in the cases with constraints on all DMUs and only on the target DMU as d increases, for $T=7$ and symmetric bounds.

**Figure 3.**Dynamics of the mean performance scores of AR restrictions as d increases, for $T=1$ and symmetric bounds.

**Figure 4.**Dynamics of the mean performance scores of AR restrictions as d increases, for $T=7$ and symmetric bounds.

**Table 1.**Lower and upper bounds considered for the virtual weight restrictions applied in the performance analysis of mutual funds; $d\in [0,0.25]$.

Input Variables | ${\mathit{L}}_{\mathit{i}}^{\mathit{X}}$ | ${\mathit{U}}_{\mathit{i}}^{\mathit{X}}$ |
---|---|---|

Symmetric bounds | ||

Initial payout K | d | $1-2d$ |

$\beta $-coefficient | d | $1-2d$ |

Downside risk $DR$ | d | $1-2d$ |

Asymmetric bounds—one-year holding period | ||

Initial payout K | $2d$ | $1-2d$ |

$\beta $-coefficient | d | $1-3d$ |

Downside risk $DR$ | d | $1-3d$ |

Asymmetric bounds—seven-year holding period | ||

Initial payout K | $0.5d$ | $1-2d$ |

$\beta $-coefficient | d | $1-1.5d$ |

Downside risk $DR$ | d | $1-1.5d$ |

**Table 2.**Main results for the virtual weights (VW) restrictions with constraints on all decision-making units (DMUs) in the case of symmetric bounds and $T=1$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.7690 | 0.9233 | 0.9193 | 0.0327 | 6 | – |

0.05 | 0.7598 | 0.9132 | 0.9117 | 0.0325 | 5 | 10.1795 |

0.10 | 0.7593 | 0.9125 | 0.9110 | 0.0325 | 5 | 10.5240 |

0.15 | 0.7582 | 0.9120 | 0.9103 | 0.0325 | 4 | 10.7885 |

0.20 | 0.7542 | 0.9014 | 0.9002 | 0.0293 | 1 | 11.6891 |

0.25 | 0.7542 | 0.9014 | 0.9002 | 0.0293 | 1 | 11.6891 |

**Table 3.**Main results for the VW restrictions with constraints on all DMUs in the case of symmetric bounds and $T=7$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.1583 | 0.5948 | 0.5635 | 0.1401 | 6 | – |

0.05 | 0.1461 | 0.5502 | 0.5277 | 0.1302 | 4 | 11.1442 |

0.10 | 0.1453 | 0.5476 | 0.5256 | 0.1296 | 4 | 11.5272 |

0.15 | 0.1439 | 0.5457 | 0.5240 | 0.1293 | 4 | 11.7564 |

0.20 | 0.1388 | 0.5022 | 0.4873 | 0.1081 | 1 | 12.7340 |

0.25 | 0.1388 | 0.5022 | 0.4873 | 0.1081 | 1 | 12.7340 |

**Table 4.**Main results for the VW restrictions with constraints on all DMUs in the case of asymmetric bounds and $T=1$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.7690 | 0.9233 | 0.9193 | 0.0327 | 6 | – |

0.05 | 0.7598 | 0.9131 | 0.9117 | 0.0325 | 5 | 10.1827 |

0.10 | 0.7586 | 0.9123 | 0.9106 | 0.0325 | 4 | 10.6042 |

0.15 | 0.7542 | 0.9014 | 0.9002 | 0.0293 | 1 | 11.6891 |

0.20 | 0.7542 | 0.9014 | 0.9002 | 0.0293 | 1 | 11.6891 |

**Table 5.**Main results for the VW restrictions with constraints on all DMUs in the case of asymmetric bounds and $T=7$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.1583 | 0.5948 | 0.5635 | 0.1401 | 6 | – |

0.05 | 0.1461 | 0.5502 | 0.5278 | 0.1303 | 4 | 11.1554 |

0.10 | 0.1456 | 0.5480 | 0.5260 | 0.1297 | 4 | 11.5176 |

0.15 | 0.1449 | 0.5467 | 0.5251 | 0.1295 | 4 | 11.7083 |

0.20 | 0.1438 | 0.5454 | 0.5239 | 0.1293 | 4 | 11.8093 |

0.25 | 0.1388 | 0.5022 | 0.4873 | 0.1081 | 1 | 12.7340 |

**Table 6.**Main results for the VW restrictions with constraints only on the target DMU in the case of symmetric bounds and $T=1$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.7690 | 0.9233 | 0.9193 | 0.0327 | 6 | – |

0.05 | 0.7619 | 0.9168 | 0.9142 | 0.0327 | 5 | 6.2997 |

0.10 | 0.7600 | 0.9145 | 0.9121 | 0.0326 | 5 | 8.8013 |

0.15 | 0.7595 | 0.9135 | 0.9118 | 0.0326 | 5 | 9.6747 |

0.20 | 0.7592 | 0.9130 | 0.9114 | 0.0325 | 5 | 10.0994 |

0.25 | 0.7588 | 0.9126 | 0.9108 | 0.0325 | 4 | 10.3157 |

**Table 7.**Main results for the VW restrictions with constraints only on the target DMU in the case of symmetric bounds and $T=7$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.1583 | 0.5948 | 0.5635 | 0.1401 | 6 | – |

0.05 | 0.1488 | 0.5663 | 0.5404 | 0.1341 | 5 | 6.7821 |

0.10 | 0.1463 | 0.5559 | 0.5319 | 0.1315 | 4 | 9.5176 |

0.15 | 0.1456 | 0.5517 | 0.5274 | 0.1304 | 4 | 10.5112 |

0.20 | 0.1452 | 0.5495 | 0.5268 | 0.1299 | 4 | 10.9984 |

0.25 | 0.1447 | 0.5479 | 0.5261 | 0.1296 | 4 | 11.2708 |

**Table 8.**Main results for the VW restrictions with constraints only on the target DMU in the case of asymmetric bounds and $T=1$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.7690 | 0.9233 | 0.9193 | 0.0327 | 6 | – |

0.05 | 0.7619 | 0.9168 | 0.9142 | 0.0327 | 5 | 6.2965 |

0.10 | 0.7600 | 0.9145 | 0.9121 | 0.0326 | 5 | 8.7821 |

0.15 | 0.7594 | 0.9135 | 0.9117 | 0.0326 | 5 | 9.6651 |

0.20 | 0.7590 | 0.9129 | 0.9108 | 0.0325 | 4 | 10.0962 |

**Table 9.**Main results for the VW restrictions with constraints only on the target DMU in the case of asymmetric bounds and $T=7$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.1583 | 0.5948 | 0.5635 | 0.1401 | 6 | – |

0.05 | 0.1488 | 0.5663 | 0.5404 | 0.1341 | 5 | 6.7853 |

0.10 | 0.1463 | 0.5559 | 0.5320 | 0.1315 | 4 | 9.5240 |

0.15 | 0.1456 | 0.5518 | 0.5274 | 0.1304 | 4 | 10.5272 |

0.20 | 0.1452 | 0.5496 | 0.5270 | 0.1299 | 4 | 11.0144 |

0.25 | 0.1449 | 0.5482 | 0.5263 | 0.1297 | 4 | 11.3077 |

**Table 10.**Lower bounds considered for the assurance regions (AR) restrictions in the performance analysis of mutual funds for $d\in [0,0.9]$. The lower bounds are calculated dividing the values reported in the table by the ratio between the mean values of the variables.

Input Variables | K | $\mathit{\beta}$ | $\mathbf{DR}$ |
---|---|---|---|

Symmetric bounds | |||

Initial payout K | 1 | d | d |

$\beta $-coefficient | d | 1 | d |

Downside risk $DR$ | d | d | 1 |

Asymmetric bounds—one-year holding period | |||

Initial payout K | 1 | $0.5d$ | $0.5d$ |

$\beta $-coefficient | $2d$ | 1 | d |

Downside risk $DR$ | $2d$ | d | 1 |

Asymmetric bounds—seven-year holding period | |||

Initial payout K | 1 | $2d$ | $2d$ |

$\beta $-coefficient | $0.5d$ | 1 | d |

Downside risk $DR$ | $0.5d$ | d | 1 |

**Table 11.**Main results for AR restrictions in the case of symmetric bounds and $T=1$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.7690 | 0.9233 | 0.9193 | 0.0327 | 6 | – |

0.10 | 0.7601 | 0.9150 | 0.9126 | 0.0330 | 5 | 8.4119 |

0.20 | 0.7597 | 0.9136 | 0.9114 | 0.0327 | 5 | 9.7500 |

0.30 | 0.7593 | 0.9131 | 0.9111 | 0.0326 | 5 | 10.1122 |

0.40 | 0.7590 | 0.9127 | 0.9109 | 0.0326 | 5 | 10.3429 |

0.50 | 0.7587 | 0.9125 | 0.9107 | 0.0326 | 4 | 10.4679 |

0.60 | 0.7585 | 0.9124 | 0.9105 | 0.0326 | 4 | 10.5417 |

0.70 | 0.7582 | 0.9122 | 0.9103 | 0.0326 | 4 | 10.6298 |

0.80 | 0.7580 | 0.9121 | 0.9102 | 0.0326 | 4 | 10.6955 |

0.90 | 0.7579 | 0.9120 | 0.9100 | 0.0326 | 4 | 10.7372 |

**Table 12.**Main results for AR restrictions in the case of symmetric bounds and $T=7$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.1583 | 0.5948 | 0.5635 | 0.1401 | 6 | – |

0.10 | 0.1464 | 0.5589 | 0.5325 | 0.1341 | 5 | 9.0929 |

0.20 | 0.1458 | 0.5522 | 0.5276 | 0.1310 | 4 | 10.5657 |

0.30 | 0.1453 | 0.5499 | 0.5271 | 0.1304 | 4 | 11.0208 |

0.40 | 0.1449 | 0.5486 | 0.5264 | 0.1301 | 4 | 11.2708 |

0.50 | 0.1446 | 0.5477 | 0.5256 | 0.1299 | 4 | 11.3766 |

0.60 | 0.1442 | 0.5471 | 0.5249 | 0.1297 | 4 | 11.4631 |

0.70 | 0.1439 | 0.5465 | 0.5245 | 0.1296 | 4 | 11.5737 |

0.80 | 0.1437 | 0.5461 | 0.5242 | 0.1295 | 4 | 11.6362 |

0.90 | 0.1435 | 0.5458 | 0.5239 | 0.1294 | 4 | 11.6779 |

**Table 13.**Main results for AR restrictions in the case of asymmetric bounds and $T=1$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.7690 | 0.9233 | 0.9193 | 0.0327 | 6 | – |

0.10 | 0.7614 | 0.9170 | 0.9143 | 0.0330 | 6 | 6.3045 |

0.20 | 0.7600 | 0.9150 | 0.9120 | 0.0330 | 5 | 8.4183 |

0.30 | 0.7596 | 0.9140 | 0.9114 | 0.0328 | 5 | 9.3077 |

0.40 | 0.7593 | 0.9134 | 0.9110 | 0.0327 | 5 | 9.7468 |

0.50 | 0.7590 | 0.9131 | 0.9108 | 0.0327 | 5 | 9.9840 |

0.60 | 0.7588 | 0.9129 | 0.9106 | 0.0326 | 5 | 10.1378 |

0.70 | 0.7586 | 0.9127 | 0.9105 | 0.0326 | 5 | 10.2708 |

0.80 | 0.7583 | 0.9125 | 0.9104 | 0.0326 | 4 | 10.3670 |

0.90 | 0.7582 | 0.9124 | 0.9104 | 0.0326 | 4 | 10.4551 |

**Table 14.**Main results for AR restrictions in the case of asymmetric bounds and $T=7$. Summary statistics on the performance scores (minimum, mean and median values, standard deviation), number of efficient funds and mean absolute variation (divided by 2) of the ranking positions.

d | Minimum Score | Mean Score | Median Score | Std Dev of Score | Number of Efficient Funds | Mean Abs. Variation of Ranking |
---|---|---|---|---|---|---|

0.00 | 0.1583 | 0.5948 | 0.5635 | 0.1401 | 6 | – |

0.10 | 0.1461 | 0.5525 | 0.5281 | 0.1311 | 4 | 10.5978 |

0.20 | 0.1456 | 0.5493 | 0.5271 | 0.1301 | 4 | 11.2772 |

0.30 | 0.1451 | 0.5480 | 0.5255 | 0.1299 | 4 | 11.4744 |

0.40 | 0.1447 | 0.5471 | 0.5248 | 0.1297 | 4 | 11.5689 |

0.50 | 0.1443 | 0.5466 | 0.5244 | 0.1296 | 4 | 11.6490 |

0.60 | 0.1440 | 0.5461 | 0.5241 | 0.1295 | 4 | 11.7324 |

0.70 | 0.1437 | 0.5457 | 0.5237 | 0.1295 | 4 | 11.7869 |

0.80 | 0.1435 | 0.5454 | 0.5234 | 0.1294 | 4 | 11.8093 |

0.90 | 0.1433 | 0.5451 | 0.5232 | 0.1294 | 4 | 11.8189 |

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**MDPI and ACS Style**

Basso, A.; Funari, S. Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds. *Mathematics* **2018**, *6*, 164.
https://doi.org/10.3390/math6090164

**AMA Style**

Basso A, Funari S. Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds. *Mathematics*. 2018; 6(9):164.
https://doi.org/10.3390/math6090164

**Chicago/Turabian Style**

Basso, Antonella, and Stefania Funari. 2018. "Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds" *Mathematics* 6, no. 9: 164.
https://doi.org/10.3390/math6090164