# Portfolio Optimization Considering Behavioral Stocks with Return Scenario Generation

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## Abstract

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## 1. Introduction

^{B}s) and time duration (T-days) between the cause and effect are also tested and known. Because we can identify the cause-effect pattern of B-stocks, it makes B-stocks predictable and can be considered a good investment pool in portfolio selection. In general, a B-stock contains the information of a cause-effect-P

^{B}-T pattern. From the operational definitions (ODs) of irrational behavior available in related literature, we can identify the cause-and-effect relationship between a stock’s price movement and the effect of the tested behavior. For example, a large price change can represent the cause of the behavior, and a follow-up price reversal can be the ensuing effect. It is also essential to know when the effect will occur after a cause is spotted and the probability that the effect will occur after the said time duration. This is done by looking at the historical data and counting the occurrence of the expected effect after possible causes. Then, using a statistical test, we can verify whether the proportion of occurrence with the possible causes is more than a threshold probability. Those stocks that show a significant cause-and-effect pattern with the corresponding T-days and ${P}^{B}$ are then considered into the B-stock big pool. Depending on the investment strategy, this big pool can be further screened into smaller pools. Since it is known when the effect will occur, one strategy is to invest 1 day before the effect day or during (T − 1)th day. Another strategy is to invest immediately after spotting the cause and exit on the effect day or T-day. For the 1st strategy, the investment pool (small pool 1) are those B-stocks in the big pool that are already on their (T − 1)th on a given trading day. Similarly, for the 2nd strategy, the investment pool (small pool 2) are those B-stocks in the big pool that have their causes spotted on a given trading day. The unique information embedded within these B-stocks is then exploited to have superior portfolios as done by these 5 studies [35,36,37,38,39].

^{B}-T patterns. They exploit the over-reaction and under-reaction B-stocks by considering the small pool 1 investment pool and long position (buy-hold-sell) investment strategy. Regarding return estimation, historical returns were used as return estimates and scenarios. With the assignment of probability weights, SP/A (security, potential, and aspiration) theory is applied to give corresponding probabilities to the return scenarios. The study also proposed the initial concept of a two-dimensional probability weighting which considers the scenarios weights and also the P

^{B}weights of individual B-stocks. This 2-D weighting mechanism was embedded into the generic safety-first scenario-based portfolio selection model to generate the optimal portfolio. The model ensures that scenario weights and the likelihood of each B-stock realizing the expected effect or P

^{B}are consistent with their respective probabilities such that the optimal portfolio will have the highest combined P

^{B}. Then through back-testing, their results show that this framework can outperform benchmarks (e.g., mean-variance portfolio) and the market. The study only considers small pool 1, so the question now can small pool 2 also generate superior portfolios?

^{B}s of B-stocks are now considered for screening further the investment pool (small pool 2) to ensure a ${P}^{B}-T$ efficient portfolio. A ${P}^{B}-T$ efficient portfolio only considers a B-stock in the portfolio if and only if its ${P}^{B}$ is at least higher than the minimum probability (set by the investor) at a given T-day. As for the selection model, a modified scenario-based safety-first model was applied to generate the optimal portfolios. Now, the model’s objective is to have the highest cumulative return over the respective T-days of each B-stock considered in the portfolio. Back-test results show that the proposed behavioral stock portfolio selection framework can also outperform benchmarks (e.g mean-variance portfolio) and the market. The first two studies consider a long position as the investment strategy, so the question is, can B-stocks be profitable using a short position investment strategy?

^{B}s of each B-stock considered. They also considered the different risk attitudes of investors and changed the scenarios’ respective weights through SP/A theory to represent all types of investors during the back-test. Their work provides evidence that these B-stocks can be exploited to generate superior portfolios which can outperform benchmarks and possibly be a viable alternative investment option for investors. In terms of return estimation, the study, similar to the previous 4 studies [35,36,37,38] on B-stocks, only considered historical returns as an estimate for future performances. Thus, the question now is, “What if we can estimate also estimate the returns on the Tth day?”.

## 2. Methodology

#### 2.1. Operational Definitions and Identifying B-Stocks

#### 2.1.1. Disposition Effect OD

#### 2.1.2. Over-Reaction OD

#### 2.1.3. Under-Reaction OD

#### 2.1.4. Ostrich Effect OD

#### 2.1.5. Herding OD

#### 2.1.6. ODs Summary

- Disposition Effect B-stock—a high positive (negative) geometric price change with high (low) abnormal trading volume followed by a high negative (positive) cumulative abnormal return.
- Over-Reaction B-stock—a high positive (negative) price change followed by a high negative (positive) cumulative abnormal return.
- Under-Reaction B-stock—a high positive (negative) price change followed by a high positive (negative) cumulative abnormal return.
- Ostrich Effect B-stock—a high positive (negative) geometric price change with high (low) abnormal trading volume followed by a high negative (positive) cumulative abnormal return with low (high) abnormal trading volume
- Herding B-stock—an abnormally high trading volume followed by a positive or negative cumulative abnormal return.

#### 2.1.7. Identification of B-Stocks

#### 2.2. Estimating Returns of B-Stocks

#### 2.3. Portfolio Selection Model

## 3. Empirical Results

#### 3.1. Data Description

#### 3.2. Portfolio Performance

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Exchange-Traded Funds | Mutual Funds | |||
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Legend | ETF Code | ETF Name | Legend | MF Name |

ETF1 | 0050 | Taiwan Top 50 ETF | MF1 | Yuanta Mainstream Equity Fund |

ETF2 | 0051 | Taiwan Mid-Cap 100 ETF | MF2 | Yuanta Duo Fu Equity Fund |

ETF3 | 0052 | Fubon Taiwan Technology ETF FUND | MF3 | UPAMC Quality Growth Fund |

ETF4 | 0053 | Yuanta Taiwan Electronics Tech ETF | MF4 | Jih Sun Jih Sun Fund |

ETF5 | 0054 | Yuanta S&P Custom China Play 50 ETF | MF5 | Jih Sun Top Five Fund |

ETF6 | 0055 | Yuanta Taiwan Financial Fund ETF | MF6 | Franklin Templeton SinoAm First Fund |

ETF7 | 0056 | Yuanta Taiwan Dividend Plus ETF | ||

ETF8 | 0057 | Fubon MSCI Taiwan ETF | ||

ETF9 | 0058 | Fubon Taiwan Eight Industries ETF | ||

ETF10 | 0059 | Taiwan Finance and Insurance Index | ||

ETF11 | 006201 | Taiwan GreTai 50 ETF | ||

ETF12 | 006203 | MSCI Taiwan ETF | ||

ETF13 | 006204 | Sinopac Taiwan TAIEX Index ETF | ||

ETF14 | 006208 | Fubon Taiwan 50 Index ETF |

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Behavior | Type | Cause | Effect |
---|---|---|---|

Disposition Effect | B-stocks | Disposition effect loser (combination of $-{R}_{G}$ and low $\overline{AV}$) | +CAR |

Short-sell B-stocks | Disposition effect winner (combination of +${R}_{G}$ and high $\overline{AV}$) | −CAR | |

Over-reaction | B-stocks | high − price change (return rate) | +CAR |

Short-sell B-stocks | high + price change (return rate) | −CAR | |

Under-reaction | B-stocks | high + price change (return rate) | +CAR |

Short-sell B-stocks | high − price change (return rate) | −CAR | |

Ostrich Effect | B-stocks | ostrich effect loser (combination of $-{R}_{G}$ and low $\overline{AV}$) | +CAR and high $\overline{AV}$ |

Short-sell B-stocks | ostrich effect winner (combination of +${R}_{G}$ and high $\overline{AV}$) | −CAR and low $\overline{AV}$ | |

Herding | B-stocks | high $\overline{AV}$ (abnormal average volume) | +CAR |

Short-sell B-stocks | high $\overline{AV}$ (abnormal average volume) | −CAR |

B-Stock Type | Cause | Effect |
---|---|---|

disposition-effect | ${R}_{G}\le -10\%$ & p-value < 0.1 | $CAR\ge +1\%$ |

over-reaction | $R\le -5\%$ | $CAR\ge +1\%$ |

under-reaction | $R\ge 5\%$ | $CAR\ge +1\%$ |

ostrich-effect | ${R}_{G}\le -10\%$ & p-value < 0.1 | $CAR\ge +1\%$ & p-value < 0.1 |

herding | p-value < 0.1 | $CAR\ge +1\%$ |

_{G}, CAR, and p-value respectively denotes return, geometric return, cumulative abnormal return, and p-value of abnormal volume.

Portfolio Name | Code | Model | Investment Pool |
---|---|---|---|

MSCI and B-stocks (−5% & 2%) | MB52 | SF (${R}_{L}=-5\%$ & $\gamma $ = 2%) | B-stocks and MSCI Stocks |

MSCI and B-stocks (−2% & 2%) | MB22 | SF (${R}_{L}=-2\%$ & $\gamma $ = 2%) | B-stocks and MSCI Stocks |

MSCI and B-stocks (−5% & 5%) | MB55 | SF (${R}_{L}=-5\%$ & $\gamma $ = 5%) | B-stocks and MSCI Stocks |

MSCI and B-stocks (−2% & 5%) | MB25 | SF (${R}_{L}=-2\%$ & $\gamma $ = 5%) | B-stocks and MSCI Stocks |

MSCI Only (−5% & 2%) | M52 | SF (${R}_{L}=-5\%$ & $\gamma $ = 2%) | MSCI Stocks |

MSCI Only (−2% & 2%) | M22 | SF (${R}_{L}=-2\%$ & $\gamma $ = 2%) | MSCI Stocks |

MSCI Only (−5% & 5%) | M55 | SF (${R}_{L}=-5\%$ & $\gamma $ = 5%) | MSCI Stocks |

MSCI Only (−2% & 5%) | M25 | SF (${R}_{L}=-2\%$ & $\gamma $ = 5%) | MSCI Stocks |

MSCI Taiwan Index | MSCI | Index Return | MSCI Index |

Taiwan Stock Exchange | Market | Index Return | Market Index |

Yuanta Taiwan Financial Fund | ETF | Index Return | ETF Index |

UPAMC Quality Growth Fund | MF | Index Return | MF Index |

362 Trading Days Statistics | MB52 | M52 | MSCI Index | Market | ETF | MF |
---|---|---|---|---|---|---|

Mean Return | 0.002897 | 0.000806 | 0.000059 | 0.000064 | 0.000443 | 0.000057 |

Standard Deviation | 0.0216 | 0.0116 | 0.0103 | 0.0091 | 0.0071 | 0.0118 |

Number of Positive Returns | 216 | 227 | 200 | 196 | 219 | 206 |

Number of Negative Returns | 146 | 135 | 162 | 166 | 143 | 156 |

Ending Cumulative Return | 1.6240 | 0.3064 | 0.0022 | 0.0082 | 0.1634 | −0.0048 |

Number of Postive Cumulative Returns | 361 | 362 | 197 | 204 | 358 | 154 |

Number of Negative Cumulative Returns | 1 | 0 | 165 | 158 | 4 | 208 |

362 Trading Days Statistics | MB22 | M22 | MSCI Index | Market | ETF | MF |
---|---|---|---|---|---|---|

Mean Return | 0.001955 | 0.000829 | 0.000059 | 0.000064 | 0.000443 | 0.000057 |

Standard Deviation | 0.0137 | 0.0100 | 0.0103 | 0.0091 | 0.0071 | 0.0118 |

Number of Positive Returns | 228 | 235 | 200 | 196 | 219 | 206 |

Number of Negative Returns | 134 | 127 | 162 | 166 | 143 | 156 |

Ending Cumulative Return | 0.9607 | 0.3259 | 0.0022 | 0.0082 | 0.1634 | −0.0048 |

Number of Positive Cumulative Returns | 362 | 362 | 197 | 204 | 358 | 154 |

Number of Negative Cumulative Returns | 0 | 0 | 165 | 158 | 4 | 208 |

362 Trading Days Statistics | MB55 | M55 | MSCI Index | Market | ETF | MF |
---|---|---|---|---|---|---|

Mean Return | 0.003023 | 0.000823 | 0.000059 | 0.000064 | 0.000443 | 0.000057 |

Standard Deviation | 0.0219 | 0.0115 | 0.0103 | 0.0091 | 0.0071 | 0.0118 |

Number of Positive Returns | 218 | 227 | 200 | 196 | 219 | 206 |

Number of Negative Returns | 144 | 135 | 162 | 166 | 143 | 156 |

Ending Cumulative Return | 1.7385 | 0.3151 | 0.0022 | 0.0082 | 0.1634 | −0.0048 |

Number of Positive Cumulative Returns | 362 | 362 | 197 | 204 | 358 | 154 |

Number of Negative Cumulative Returns | 0 | 0 | 165 | 158 | 4 | 208 |

362 Trading Days Statistics | MB25 | M25 | MSCI Index | Market | ETF | MF |
---|---|---|---|---|---|---|

Mean Return | 0.002374 | 0.000729 | 0.000059 | 0.000064 | 0.000443 | 0.000057 |

Standard Deviation | 0.0169 | 0.0110 | 0.0103 | 0.0091 | 0.0071 | 0.0118 |

Number of Positive Returns | 222 | 225 | 200 | 196 | 219 | 206 |

Number of Negative Returns | 140 | 137 | 162 | 166 | 143 | 156 |

Ending Cumulative Return | 1.2413 | 0.2741 | 0.0022 | 0.0082 | 0.1634 | −0.0048 |

Number of Positive Cumulative Returns | 362 | 362 | 197 | 204 | 358 | 154 |

Number of Negative Cumulative Returns | 0 | 0 | 165 | 158 | 4 | 208 |

MSCI ONLY | MSCI Index | Market | ETF | MF | |
---|---|---|---|---|---|

MSCI and B-stocks (−5% and 2%) | 0.200 | 0.024 ** | 0.021 ** | 0.098 * | 0.035 ** |

MSCI ONLY (−5% and 2%) | 0.340 | 0.407 | 0.479 | 0.363 | |

MSCI and B-stocks (−2% and 2%) | 0.155 | 0.011 ** | 0.010 ** | 0.042 ** | 0.011 ** |

MSCI ONLY (−2% and 2%) | 0.218 | 0.185 | 0.476 | 0.259 | |

MSCI and B-stocks (−5% and 5%) | 0.071 * | 0.012 ** | 0.010 ** | 0.066 * | 0.019 ** |

MSCI ONLY (−5% and 5%) | 0.330 | 0.395 | 0.465 | 0.353 | |

MSCI and B-stocks (−2% and 5%) | 0.019 ** | 0.018 ** | 0.021 ** | 0.026 ** | 0.008 *** |

MSCI ONLY (−2% and 5%) | 0.456 | 0.377 | 0.688 | 0.396 | |

MSCI Index | 0.431 | 0.899 | 0.512 | ||

Market | 0.569 | 0.932 | 0.510 | ||

Yuanta Taiwan Financial Fund | 0.101 | 0.068 * | 0.283 | ||

UPAMC Quality Growth Fund | 0.488 | 0.490 | 0.717 |

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## Share and Cite

**MDPI and ACS Style**

Young, M.N.; Chuahay, T.T.N.; Lee, Y.-H.; Diaz, J.F.T.; Prasetyo, Y.T.; Persada, S.F.; Nadilfatin, R. Portfolio Optimization Considering Behavioral Stocks with Return Scenario Generation. *Mathematics* **2022**, *10*, 4269.
https://doi.org/10.3390/math10224269

**AMA Style**

Young MN, Chuahay TTN, Lee Y-H, Diaz JFT, Prasetyo YT, Persada SF, Nadilfatin R. Portfolio Optimization Considering Behavioral Stocks with Return Scenario Generation. *Mathematics*. 2022; 10(22):4269.
https://doi.org/10.3390/math10224269

**Chicago/Turabian Style**

Young, Michael N., TJ Troy N. Chuahay, Yen-Hsien Lee, John Francis T. Diaz, Yogi Tri Prasetyo, Satria Fadil Persada, and Reny Nadilfatin. 2022. "Portfolio Optimization Considering Behavioral Stocks with Return Scenario Generation" *Mathematics* 10, no. 22: 4269.
https://doi.org/10.3390/math10224269