# The Effect of Exit Time and Entropy on Asset Performance Evaluation

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Definitions and Formulas

#### 2.1. Risk Measures (VaR and CVaR)

#### 2.2. SPP-CVaR Risk Measure

**Theorem**

**1.**

#### 2.3. Shannon Entropy

## 3. DEA-Based Evaluation Model of Assets’ Performance

#### 3.1. Evaluation of Assets’ Performance over the Entire Time Horizon Using Traditional Risk Measures

#### 3.2. SPP-CVaR-Based Evaluation of Portfolios’ Performance

## 4. Empirical Application

#### 4.1. The Data Analysis

#### 4.2. Portfolio Selection

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of Coca-Cola (m = 6%).

**Figure 6.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of Amazon (m = 2%).

**Figure 7.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of Tesla (m = 8%).

**Figure 8.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of Pfizer (m = 10%).

**Figure 9.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of oil (m = 12%).

**Figure 10.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of Gold (m = 4%).

**Figure 11.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of Meta (m = 6%).

**Figure 12.**SPP density (

**a**) and real density (

**b**) of the distribution of the exit time of Bitcoin (m = 12%).

Asset | CVaR | Mean Return | Efficiency Score |
---|---|---|---|

Coca-Cola | 0.03745 | 0.00018 | 0.79 |

Amazon | 0.05365 | 0.00018 | 0.68 |

Pfizer | 0.03969 | 0.0000865 | 0.75 |

Tesla | 0.09969 | 0.00229 | 1 |

Oil | 0.11004 | 0.00038 | 0.47 |

Gold | 0.02453 | 0.00035 | 1 |

Meta | 0.06882 | 0.00007469 | 0.58 |

Bitcoin | 0.11175 | 0.00178 | 0.7 |

Asset | Stop-Profit Point | Exit Time | SPP-CVaR | Mean Return | Efficiency Score with Exit Time | Efficiency Score with Exit Time and Entropy | ||
---|---|---|---|---|---|---|---|---|

θ = 0.75 | θ = 1.5 | θ = 2.2 | ||||||

Coca-Cola | 2% | 2 | 0.000230 | 0.00494 | 0.04 | 0.21 | 0.81 | 1 |

Coca-Cola | 4% | 8 | 0.0007 | 0.0031 | 0.02 | 0.14 | 0.36 | 0.76 |

Amazon | 2% | 8 | 0.000916 | −0.0002 | 0.01 | 0.11 | 0.29 | 0.63 |

Amazon | 6% | 9 | 0.00096 | 0.0053 | 0.03 | 0.15 | 0.4 | 0.88 |

Pfizer | 2% | 1 | 0.00016 | 0.00461 | 1 | 1 | 1 | 1 |

Pfizer | 4% | 2 | 0.0001615 | 0.0189 | 1 | 1 | 1 | 1 |

Oil | 2% | 2 | 0.00064 | −0.0229 | 0.01 | 0.06 | 0.18 | 0.63 |

Oil | 4% | 3 | 0.0012 | 0.01119 | 0.05 | 0.26 | 0.71 | 1 |

Meta | 2% | 8 | 0.00088 | −0.0003 | 0.01 | 0.11 | 0.29 | 0.62 |

Meta | 6% | 7 | 0.000297 | 0.00453 | 0.03 | 0.19 | 0.54 | 1 |

Asset | Stop-Profit Point | Exit Time | SPP-CVaR | Mean Return | Efficiency Score with Exit Time | Efficiency Score with Exit Time and Entropy | ||
---|---|---|---|---|---|---|---|---|

θ = 0.75 | θ = 1.5 | θ = 1.7 | ||||||

Coca-Cola | 6% | 16 | 0.001613 | 0.002759 | 0.05 | 0.09 | 0.3 | 0.47 |

Amazon | 8% | 10 | 0.0010479 | 0.00777 | 1 | 1 | 1 | 1 |

Amazon | 10% | 12 | 0.002868 | 0.00758 | 0.35 | 0.54 | 0.65 | 0.65 |

Meta | 8% | 14 | 0.001187 | 0.003991 | 0.27 | 0.28 | 0.5 | 0.68 |

Bitcoin | 2% | 21 | 0.0000577 | 0.00068 | 1 | 1 | 1 | 1 |

Bitcoin | 6% | 21 | 0.0000622 | 0.00267 | 1 | 1 | 1 | 1 |

Asset | Stop-Profit Point | Exit Time | SPP-CVaR | Mean Return | Efficiency Score with Exit Time | Efficiency Score with Exit Time and Entropy | ||
---|---|---|---|---|---|---|---|---|

θ = 0.75 | θ = 1.2 | θ = 1.5 | ||||||

Coca-Cola | 8% | 22 | 0.003711 | 0.002831 | 0.78 | 0.8 | 1 | 1 |

Coca-Cola | 10% | 24 | 0.003532 | 0.003301 | 1 | 1 | 1 | 1 |

Tesla | 2% | 26 | 0.001274 | −0.0022 | 0.38 | 0.4 | 0.43 | 0.64 |

Tesla | 6% | 27 | 0.001165 | 0.0008237 | 0.82 | 0.84 | 0.9 | 1 |

Gold | 2% | 28 | 0.0008493 | 0.0005553 | 1 | 1 | 1 | 1 |

Gold | 6% | 29 | 0.000928 | 0.001130 | 1 | 1 | 1 | 1 |

Asset | Stop-Profit Point | Exit Time | SPP-CVaR | Mean Return | Efficiency Score with Exit Time | Efficiency Score with Exit Time and Entropy | ||
---|---|---|---|---|---|---|---|---|

θ = 0.75 | θ = 1.2 | θ = 2.01 | ||||||

Amazon | 12% | 96 | 0.01394 | 0.00114 | 0.19 | 0.21 | 0.48 | 0.94 |

Pfizer | 6% | 74 | 0.01018 | 0.000794 | 0.21 | 0.22 | 0.36 | 0.68 |

Pfizer | 8% | 79 | 0.01262 | 0.000767 | 0.18 | 0.19 | 0.34 | 0.66 |

Tesla | 10% | 74 | 0.00333 | −0.000459 | 0.22 | 0.24 | 0.51 | 1 |

Tesla | 12% | 76 | 0.006018 | 0.00120 | 0.36 | 0.37 | 0.58 | 1 |

Gold | 8% | 43 | 0.001068 | 0.001377 | 1 | 1 | 1 | 1 |

Gold | 10 | 98 | 0.0010646 | 0.000916 | 1 | 1 | 1 | 1 |

Meta | 12% | 43 | 0.004394 | 0.002012 | 1 | 1 | 1 | 1 |

Asset | Stop-Profit Point | Exit Time | SPP-CVaR | Mean Return | Efficiency Score with Exit Time | Efficiency Score with Exit Time and Entropy | |
---|---|---|---|---|---|---|---|

θ = 0.75 | θ = 1.5 | ||||||

Pfizer | 10% | 140 | 0.02169 | 0.000705 | 1 | 1 | 1 |

Pfizer | 12% | 142 | 0.02233 | 0.00068 | 0.82 | 0.85 | 0.92 |

Oil | 8% | 218 | 0.02490 | 0.000151 | 0.03 | 0.03 | 0.12 |

Oil | 10% | 219 | 0.01765 | 0.000428 | 0.05 | 0.05 | 0.2 |

Oil | 12% | 222 | 0.01308 | 0.000329 | 0.07 | 0.07 | 0.22 |

Gold | 12% | 363 | 0.000915 | 0.000324 | 0.94 | 0.94 | 1 |

Bitcoin | 10% | 177 | 0.000845 | 0.000205 | 1 | 1 | 1 |

Bitcoin | 12% | 179 | 0.000885 | 0.000542 | 1 | 1 | 1 |

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**MDPI and ACS Style**

Ghasemi Doudkanlou, M.; Chandro, P.; Banihashemi, S.
The Effect of Exit Time and Entropy on Asset Performance Evaluation. *Entropy* **2023**, *25*, 1252.
https://doi.org/10.3390/e25091252

**AMA Style**

Ghasemi Doudkanlou M, Chandro P, Banihashemi S.
The Effect of Exit Time and Entropy on Asset Performance Evaluation. *Entropy*. 2023; 25(9):1252.
https://doi.org/10.3390/e25091252

**Chicago/Turabian Style**

Ghasemi Doudkanlou, Mohammad, Prokash Chandro, and Shokoofeh Banihashemi.
2023. "The Effect of Exit Time and Entropy on Asset Performance Evaluation" *Entropy* 25, no. 9: 1252.
https://doi.org/10.3390/e25091252