# An Improved MV Method for Stock Allocation Based on the State-Space Measure of Cognitive Bias with a Hybrid Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Material Preparation

#### 2.2. Methods

#### 2.2.1. State-Space Model

**Definition**

**1.**

**Definition**

**2.**

**Hypothesis**

**1.**

**Hypothesis**

**2.**

#### 2.2.2. Markowitz Mean-Variance Method (MV Method)

**Step 1.**Calculate expected return (single/ portfolio).

**Step 2.**Measure the investment risk (single/portfolio).

**Step 3.**Construct the investment model.

## 3. Results

#### 3.1. The Proposed Stock Allocation Method

**Step 1.**Build a state-space model.

**Step 2.**Kalman filtering.

**Step 3.**EM algorithm.

**Step 4.**Multiple iteration until convergence.

**Step 5.**Calculate EMACB (exponential moving average of cognitive bias).

**Step 6.**Make the EMACB-variance model (EV Model) for risk seekers to obtain the optimal weights of each stock.

**Step 7.**Make the EMACB-variance model (EV model) for risk averters to obtain optimal weights of each stock.

#### 3.2. Illustrative Example for Stock Allocation

**Step 1.**Before measuring, the ADF test needs to be used to recognize whether each time series participated in construction is a stationary sequence, including the series of the volume ratio (VR) and the logarithmic forms of ${I}_{1}-{I}_{3}$, which are defined in Table 1 and Table 2. the results are shown in Table 5.

**Step 2.**to avoid false regression, these variables need to be tested by the Johansen co-integration test, which concludes the trace test and maximum characteristic value test. The results are shown in Table 6.

**Step 3.**Construct the state-space model based on collected data.

**Step 4.**Estimate cognitive bias $xi$ using Kalman filtering.

**Step 5.**Modify the coefficients’ estimation generated in step 4 by the EM algorithm.

**Step 6.**Put the revised estimation into Kalman Filtering to continue the next round of iteration, until convergence. Based on the results, some features of cognitive biases are shown in Figure 1, Figure 2 and Figure 3. Additionally, the estimation results are shown as follows:

**Step 7.**Calculate the 5-day and 12-day EMACB based on the previous measurement outcomes.

**Step 8.**Construct the EV model for the risk seeker and calculate the objective weights of each stock.

**Step 9.**Construct the EV model for the risk averter and obtain the weights allocation. The results of step 8 and 9 are expressed in Table 7.

## 4. Discussion

#### 4.1. General Validation of the Example

#### 4.2. Performance Comparison of Different Methods

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Basis | Meaning | Stimulus Factor |
---|---|---|

Representativeness | A person following representativeness prefers to evaluate uncertain events through the features in the similar events. | Industry Sector Index |

Availability | Availability refers to the situation in which people assess events by the ease with the accessible influence. | PE (Price/Earning) |

Anchoring | Anchoring refers to the situation in which people consider the initial feature as the key factor of decision. | EMA (5) |

Symbol | Variable | Meaning |
---|---|---|

VR | Volume Ratio | VR = trading volume in current period/5-day weighted average trading volume. |

I_{1} | Industry Sector Index | This index reflects the composite stock price changes in corresponding industry sectors. |

I_{2} | PE | PE = Price per shar/Earnings per share. |

I_{3} | EMA (5) | EMA (5) refers to a 5-day weighted average price of exponential decays. |

S1 | S2 | S3 | S4 | S5 | |
---|---|---|---|---|---|

observation | 1706 | 1658 | 1700 | 1687 | 1711 |

mean | 87.4734 | 101.3718 | 428.1459 | 419.2882 | 22.7761 |

median | 87.6000 | 103.1450 | 417.8950 | 411.8350 | 21.0800 |

std. dev. | 69.4728 | 7.7212 | 6.9736 | 5.9473 | 6.5866 |

min | 13.1300 | 94.4700 | 377.7600 | 383.4900 | 18.3300 |

max | 157.4400 | 137.0000 | 445.9900 | 426.1600 | 35.8000 |

S1 | S2 | S3 | S4 | S5 | |
---|---|---|---|---|---|

S1 | 1 | ||||

S2 | 0.0574 | 1 | |||

S3 | 0.1257 | 0.0302 | 1 | ||

S4 | 0.0844 | 0.0717 | 0.2047 | 1 | |

S5 | 0.1024 | 0.2092 | 0.1615 | 0.2286 | 1 |

Variable | ADF Test | ||
---|---|---|---|

T-Statistics | Prob. | Stability | |

dVR | −9.866843 | 0.0000 | stable *** |

dLI1 | −6.634674 | 0.0000 | stable *** |

dLI2 | −7.906096 | 0.0000 | stable *** |

dLI3 | −3.473008 | 0.0007 | stable *** |

Hypothesized No. of CE(s) | Eigenvalue | Trace Test | Max-Eigen | ||
---|---|---|---|---|---|

Statistic | Prob. ** | Statistic | Prob. ** | ||

None * | 0.577532 | 203.5052 | 0.0012 | 55.02875 | 0.0009 |

At most 1 * | 0.480721 | 104.7045 | 0.0000 | 26.11564 | 0.0140 |

At most 2 * | 0.421976 | 37.17026 | 0.0000 | 40.2293 | 0.0007 |

At most 3 * | 0.320903 | 63.28152 | 0.0001 | 19.94836 | 0.0109 |

At most 4 * | 0.151025 | 17.22287 | 0.0000 | 17.22289 | 0.0013 |

^{1}“*” represents rejection of the null hypothesis at the 95% significance level; “**” indicates the p-value in MacKinnon and Huang and Michelis (1999). At the significance level of 5%, there is a co-integration relationship between the variables, which means a long-term equilibrium relationship exists between them.

Type | Weights Allocation | Stock Investment Ratio | ||||
---|---|---|---|---|---|---|

S1 | S2 | S3 | S4 | S5 | ||

Investor_{SK} | 0.477 | 0.146 | 0.195 | 0.093 | 0.088 | 70% |

Investor_{AV} | 0.295 | 0.204 | 0.231 | 0.143 | 0.127 | 30% |

Client Type | Investor-sk | Investor-av | Investor-sk | Investor-av | |||||
---|---|---|---|---|---|---|---|---|---|

Time Span | Return | Variance | Return | Variance | Turnover | Sharp Ratio | Turnover | Sharp Ratio | |

EV: 1 month | 0.05814 | 0.005494 | 0.04646 | 0.003990 | 0.3698 | 0.4452 | 0.4514 | 0.4398 | |

2 months | 0.07849 | 0.012097 | 0.06361 | 0.008504 | 0.5167 | 0.3257 | 0.5759 | 0.3261 | |

3 months | 0.08956 | 0.010745 | 0.08874 | 0.009003 | 0.5384 | 0.3364 | 0.5578 | 0.3349 | |

5 months | 0.10430 | 0.024130 | 0.07963 | 0.006521 | 0.6272 | 0.2681 | 0.6379 | 0.2705 | |

7 months | 0.11309 | 0.016732 | 0.09588 | 0.008136 | 0.6425 | 0.4342 | 0.7100 | 0.4317 | |

MV: 1 month | 0.05575 | 0.005508 | 0.04313 | 0.009940 | 0.4658 | 0.3089 | 0.4776 | 0.3073 | |

2 months | 0.06953 | 0.011986 | 0.05968 | 0.010730 | 0.5783 | 0.2670 | 0.5841 | 0.2660 | |

3 months | 0.07377 | 0.010800 | 0.09040 | 0.010340 | 0.6420 | 0.2683 | 0.6469 | 0.2676 | |

5 months | 0.08841 | 0.034759 | 0.08006 | 0.011406 | 0.6617 | 0.1875 | 0.6672 | 0.1884 | |

7 months | 0.08350 | 0.021804 | 0.07833 | 0.014035 | 0.7125 | 0.2966 | 0.7148 | 0.2948 | |

MVSRB: 1 month | 0.05640 | 0.005507 | 0.04450 | 0.008128 | 0.4463 | 0.4286 | 0.4508 | 0.4185 | |

2 months | 0.07788 | 0.012189 | 0.06117 | 0.009635 | 0.5551 | 0.2968 | 0.5575 | 0.3011 | |

3 months | 0.08007 | 0.010796 | 0.08817 | 0.010131 | 0.5447 | 0.3204 | 0.5485 | 0.3192 | |

5 months | 0.09892 | 0.025488 | 0.08001 | 0.009809 | 0.6314 | 0.2555 | 0.6356 | 0.2560 | |

7 months | 0.09733 | 0.017329 | 0.08727 | 0.012523 | 0.7071 | 0.3987 | 0.7132 | 0.3976 | |

MVSRE: 1 month | 0.05603 | 0.005504 | 0.04374 | 0.003984 | 0.3012 | 0.3839 | 0.4670 | 0.3801 | |

2 months | 0.07840 | 0.011995 | 0.06159 | 0.008498 | 0.4573 | 0.2789 | 0.5820 | 0.2794 | |

3 months | 0.07994 | 0.010787 | 0.08723 | 0.009003 | 0.5218 | 0.3029 | 0.6224 | 0.3018 | |

5 months | 0.10050 | 0.033625 | 0.07977 | 0.006511 | 0.5344 | 0.2417 | 0.6472 | 0.2435 | |

7 months | 0.09853 | 0.016923 | 0.08589 | 0.008127 | 0.5961 | 0.3213 | 0.7076 | 0.7092 | |

Return Comparison: | EV_{SK} ≻ MVSRE_{SK} ≻ MVSRB_{SK} ≻ MV_{SK}; EV_{AV} ≻ MVSRB_{AV} ≻ MVSRE_{AV} ≻ MV_{AV} | ||||||||

Risk Comparison: | MV_{SK} ≻ MVSRB_{SK} ≻ MVSRE_{SK} ≻ EV_{SK}; MV_{AV} ≻ MVSRB_{AV} ≻ EV_{AV} ≻ MVSRE_{AV} | ||||||||

Turnover Comparison: | MV_{SK} ≻ MVSRB_{SK} ≻ EV_{SK} ≻ MVSRE_{SK}; MV_{AV} ≻ MVSRE_{AV} ≻ EV_{AV} ≻ MVSRB_{AV} | ||||||||

SR Comparison: | EV_{SK} ≻ MVSRB_{SK} ≻ MASRE_{SK} ≻ MV_{SK}; EV_{AV} ≻ MVSRB_{AV} ≻ MASRE_{AV} ≻ MV_{AV} |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Wu, H.; Li, G.; Lu, W.
An Improved MV Method for Stock Allocation Based on the State-Space Measure of Cognitive Bias with a Hybrid Algorithm. *Symmetry* **2020**, *12*, 1036.
https://doi.org/10.3390/sym12061036

**AMA Style**

Wang L, Wu H, Li G, Lu W.
An Improved MV Method for Stock Allocation Based on the State-Space Measure of Cognitive Bias with a Hybrid Algorithm. *Symmetry*. 2020; 12(6):1036.
https://doi.org/10.3390/sym12061036

**Chicago/Turabian Style**

Wang, Liwen, Hecheng Wu, Gang Li, and Weixue Lu.
2020. "An Improved MV Method for Stock Allocation Based on the State-Space Measure of Cognitive Bias with a Hybrid Algorithm" *Symmetry* 12, no. 6: 1036.
https://doi.org/10.3390/sym12061036