# Optimisation of Time-Varying Asset Pricing Models with Penetration of Value at Risk and Expected Shortfall

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## Abstract

**:**

## 1. Introduction

## 2. Review of Literature

## 3. Materials and Methods

#### 3.1. Size Factor

#### 3.2. Value Factor

#### 3.3. Profitability Factor

#### 3.4. Investment Factor

#### 3.5. Value at Risk and Expected Shortfall

#### 3.6. Time-Series Variation in Stock Returns

_{t}= γ

_{t}+ β

_{1}HLCVaR

_{α}

_{,t}+ ε

_{t}

_{t}= α

_{t}+ β

_{1}(Rm − Rf)

_{t}+ β

_{2}SMB

_{t}+ β

_{3}HML

_{t}+ ε

_{t.}

_{t}= α

_{t}+ β

_{1}HLVaR

_{t}+ β

_{2}SMB

_{t}+ β

_{3}HML

_{t}+ ε

_{t}

_{t}= α

_{t}+ β

_{1}HLCVAR

_{t}+ β

_{2}SMB

_{t}+ β

_{3}HML

_{t}+ ε

_{t}

_{t}= α

_{t}+ β

_{1}(Rm − Rf)

_{t}+ β

_{2}SMB

_{t}+ β

_{3}HML

_{t}+ β

_{4}RMW

_{t}+ β

_{5}CMA

_{t}+ ε

_{t,}

_{t}= α

_{t}+ β

_{1}HLVaRt + β

_{2}SMB

_{t}+ β

_{3}HML

_{t}+ β

_{4}RMW

_{t}+ β

_{5}CMA

_{t}+ ε

_{t,}

_{t}= α

_{t}+ β

_{1}HLCVaR

_{t}+ β

_{2}SMB

_{t}+ β

_{3}HML

_{t}+ β

_{4}RMW

_{t}+ β

_{5}CMA

_{t}+ ε

_{t.}

_{t}= α

_{t}+ β

_{1}(Rm − Rf)

_{t}+ β

_{2}HLVaR

_{t}+ β

_{3}SMB

_{t}+ β

_{4}HML

_{t}+ β

_{5}RMW

_{t}+ β

_{6}CMA

_{t}+ ε

_{t.}

_{t}= α

_{t}+ β

_{1}(Rm − Rf)

_{t}+ β

_{2}HLCVaR

_{t}+ β

_{3}SMB

_{t}+ β

_{4}HML

_{t}+ β

_{5}RMW

_{t}+ β

_{6}CMA

_{t}+ ε

_{t.}

#### 3.7. Time Varying Portfolio Structures

#### 3.8. Asset Princing Models

## 4. Results

## 5. Discussion and Conclusion

#### 5.1. One-Factor Model

#### 5.2. Three-Factor Model

#### 5.3. Five-Factor Model

#### 5.4. Six-Factor Model

#### 5.5. Main Findings

#### 5.6. Research Implications

- The risk-return trade-off complexity
- Optimized asset pricing models
- Negativity bias in the market
- Risk-averse behavior of investors in emerging markets
- The predictable model capacity of outliers

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Figure A2.**Daily Returns of Size (S) (S1 Represent the Small Quantile of Size Factor, S4 Is the Large Quantile of Size Factor)) and Beta (b) (b1 Is the Low Beta Quantile and b4 Is the High Beta Quantile, b2 and b3 Represent Medium Beta Quantiles)) Portfolios.

## Appendix C

**Figure A3.**Daily Portfolios’ Returns of Size (S) and VaR (VaR1 Represent the Lowest Value at Risk Portfolio Quantile and VaR4 Represent the Highest Value at Risk Quantile) Factor (95% Level of Confidence).

## Appendix D

## Appendix E

**Figure A5.**Daily Returns of Size (S) and ES Portfolios (95% Level of Confidence) (ES1 is the Lowest Expected Shortfall Portfolio Quartile ad ES4 Represents the Highest Expected Shortfall Portfolio Quartile).

## Appendix F

## Appendix G

**Figure A7.**Daily Returns of Portfolios of Size (S) and Value (BM (BM1 Is the Low Book to Market Value Quartile portfolio and BM4 is the High Book to Market Value Quartile Portfolio)) Factor Data.

## Appendix H

**Figure A8.**Daily Returns of Portfolios of Size Factor (S) and Investment Factor (I (I1 Is the Highly Conservative Investment Portfolio and I4 Is the Highly Aggressive Investment Portfolio)).

## Appendix I

**Figure A9.**Daily Returns of Portfolios of Size Factor (S) and Profitability Factor (P (P1 Is the Highly Weak Profitable Firms and P4 Represents the Highly Strong Profitable Firms)) Ratios.

## Appendix J

## Appendix K

Variables | Mean | Standard Deviation | Minimum Value | Maximum Value |
---|---|---|---|---|

Ri | 0.001 | 0.009 | −0.047 | 0.211 |

RiRf | −0.001 | 0.009 | −0.049 | 0.211 |

Rm | 0.001 | 0.014 | −0.077 | 0.085 |

Rm-Rf | 0.000 | 0.014 | −0.078 | 0.084 |

smb3 | 0.000 | 0.016 | −0.104 | 0.157 |

smb5 | 0.000 | 0.016 | −0.111 | 0.136 |

HML | 0.000 | 0.017 | −0.125 | 0.137 |

CMA | 0.000 | 0.020 | −0.132 | 0.174 |

RMV | 0.000 | 0.022 | −0.189 | 0.141 |

HLVaR95 | −0.001 | 0.018 | −0.257 | 0.102 |

HLVaR99 | −0.001 | 0.022 | −0.899 | 0.090 |

HLCVaR95 | −0.001 | 0.037 | −0.763 | 0.766 |

HLCVaR99 | −0.002 | 0.025 | −1.120 | 0.074 |

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**Table 1.**Portfolio average returns based on size and beta. The table represents the average returns of 16 portfolios formed oversize and stock market beta. A total of 527 listed firms are analyzed, from which we dropped firms with negative book-to-market equity from the sample. Firms with missing values and no data were dropped. S1 to S4 represent small to large portfolio stocks, β1 to β4 represents low market beta to high market beta. Diff. (low–high) is the difference of small size with beta portfolios from large size with beta portfolios. p value is the significant value of two-sample t-test of average return difference of two said portfolios. It is tested at 1%, 5%, and 10% levels of significance.

Size and Beta | ||||
---|---|---|---|---|

β1 | β2 | β3 | β4 | |

S1 | 0.08% | 0.07% | 0.08% | 0.12% |

S2 | 0.13% | 0.15% | 0.16% | 0.19% |

S3 | 0.22% | 0.19% | 0.20% | 0.22% |

S4 | 0.27% | 0.27% | 0.24% | 0.21% |

Diff. (low–high) | −0.0019 | −0.0020 | −0.0016 | −0.0009 |

p value | 0.0394 | 0.0000 | 0.0008 | 0.2811 |

**Table 2.**Portfolio returns based on size with profitability and investment. The table represents the average returns of 24 portfolios formed oversize and profitability, and 24 portfolios developed oversize and investment. A total of 527 listed firms is analyzed, from which we dropped firms with negative book-to-market equity. Firms with missing values and no data were dropped. S1 to S4 represents small to large portfolio stocks, I1 to I4 represents conservative to aggressive investment stocks, and P1 to P4 represents weak to robust profitability stocks. The difference between small and large portfolios is provided with the p value to measure the significant difference between small and large portfolios.

Panel A. Size and Profitability | ||||
---|---|---|---|---|

P1 | P2 | P3 | P4 | |

S1 | 0.09% | 0.09% | 0.07% | 0.09% |

S2 | 0.16% | 0.16% | 0.15% | 0.15% |

S3 | 0.24% | 0.19% | 0.21% | 0.20% |

S4 | 0.26% | 0.26% | 0.23% | 0.21% |

Diff. (Small–Large) | −0.00171 | −0.00168 | −0.00159 | −0.00125 |

p value | 0.038 | 0.0014 | 0.000 | 0.074 |

Panel B. Size and Investment | ||||

I1 | I2 | I3 | I4 | |

S1 | 0.09% | 0.09% | 0.07% | 0.08% |

S2 | 0.16% | 0.16% | 0.14% | 0.15% |

S3 | 0.23% | 0.25% | 0.23% | 0.20% |

S4 | 0.28% | 0.24% | 0.22% | 0.19% |

Diff. (Small–Large) | −0.00193 | −0.00148 | −0.00146 | −0.00114 |

p value | 0.007 | 0.076 | 0.041 | 0.017 |

**Table 3.**Portfolio returns based on size and book-to-market ratio. The table represents the average returns of 24 portfolios formed oversize and book-to-market equity. A total of 527 listed firms is analyzed, from which we dropped firms with negative book-to-market equity from the sample. Firms with missing values and no data were dropped. S1 to S5 represents small to large portfolio stocks, B1 to B5 represents small book-to-market to large book-to-market ratio.

Size and BM Ratio | ||||
---|---|---|---|---|

BM1 | BM2 | BM3 | BM4 | |

S1 | 0.08% | 0.06% | 0.08% | 0.08% |

S2 | 0.14% | 0.14% | 0.15% | 0.16% |

S3 | 0.16% | 0.22% | 0.21% | 0.23% |

S4 | 0.19% | 0.22% | 0.23% | 0.31% |

diff (small–large) | −0.00113 | −0.00165 | −0.00147 | −0.00234 |

p value | 0.000844 | 0.023175 | 0.00155 | 0.000254 |

**Table 4.**Portfolio returns of size, VaR and ES at 95% level of significance. The table represents the average returns of 24 portfolios formed oversize, and VaR and 24 portfolios created oversize and CVaR at 95% level of significance. A total of 527 listed firms is analyzed, from which we dropped firms with negative book-to-market equity. Firms with missing values and no data were dropped. S1 to S5 represents small to large portfolio stocks, VaR1 to VaR5 represents low VaR to high VaR stocks and CVaR1 to CVaR5 means low to high CVaR stocks.

Panel C: Size and VaR 95 | ||||
---|---|---|---|---|

VaR1 | VaR2 | VaR3 | VaR4 | |

S1 | 0.077% | 0.075% | 0.086% | 0.105% |

S2 | 0.170% | 0.164% | 0.146% | 0.159% |

S3 | 0.239% | 0.234% | 0.177% | 0.205% |

S4 | 0.278% | 0.273% | 0.204% | 0.193% |

Diff. (Small–Large) | −0.0020 | −0.0020 | −0.0012 | −0.0009 |

p-value | 0.0593 | 0.0000 | 0.0619 | 0.0043 |

Panel D: Size and ES 95 | ||||

ES1 | ES2 | ES3 | ES4 | |

S1 | 0.087% | 0.083% | 0.068% | 0.101% |

S2 | 0.176% | 0.156% | 0.151% | 0.139% |

S3 | 0.220% | 0.235% | 0.221% | 0.190% |

S4 | 0.302% | 0.260% | 0.233% | 0.176% |

Diff. (Small–Large) | −0.00215 | −0.00177 | −0.00165 | −0.00076 |

p-value | 0.073866 | 0.000114 | 0.0000 | 0.000266 |

**Table 5.**Portfolio returns of size, VaR and ES at 99% level of significance. The table represents the average returns of 24 portfolios formed oversize and VaR and 24 portfolios assembled oversize and CVaR. A total of 527 listed firms is analyzed, from which we dropped firms with negative book-to-market equity. Firms with missing values and no data were dropped. S1 to S5 represents small to large portfolio stocks, VaR1 to VaR5 represents low VaR to High VaR stocks and ES1 to ES5 represents low to high expected shortfall stocks.

Panel E: Size and VaR 99 | ||||
---|---|---|---|---|

VAR1 | VAR2 | VAR3 | VAR4 | |

S1 | 0.086% | 0.078% | 0.081% | 0.099% |

S2 | 0.185% | 0.145% | 0.152% | 0.139% |

S3 | 0.228% | 0.206% | 0.211% | 0.179% |

S4 | 0.317% | 0.244% | 0.237% | 0.194% |

Diff. (low–high) | −0.0023 | −0.0017 | −0.0016 | −0.0010 |

p-value | 0.0382 | 0.0001 | 0.0129 | 0.0006 |

Panel F: Size and ES 99 | ||||

ES1 | ES2 | ES3 | ES4 | |

S1 | 0.101% | 0.072% | 0.081% | 0.098% |

S2 | 0.164% | 0.156% | 0.163% | 0.139% |

S3 | 0.214% | 0.208% | 0.215% | 0.192% |

S4 | 0.283% | 0.242% | 0.221% | 0.196% |

Diff. (low–high) | −0.0018 | −0.0017 | −0.0014 | −0.0010 |

p-value | 0.1281 | 0.0001 | 0.0000 | 0.0001 |

**Table 6.**Single factor systematic risk controlling models. The table represents five models with the dependent variable of excess returns of manufacturing companies (RiRf) listed on the Pakistan Stock Exchange from 2002 to 2014. Independent variables include all systematic risk factors which the study proposed. Excess market returns (Rm-Rf) are calculated from the subtraction of KSE 100 Index returns from the risk-free rate of returns. HLVaR is the subtraction of average returns of High VaR portfolio returns from Low VaR portfolio returns. VaR is observed at both 95% and 99% level of significance. HLCVaR represents subtraction of high ES return stocks from low ES returns stocks. The probability value is provided below, describing the coefficient effect dependent variable’s importance.

Dependent Variable: RiRf | |||||
---|---|---|---|---|---|

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |

RMRF | 0.370 | ||||

(0.000) | |||||

HLVaR 95 | 0.259 | ||||

(0.000) | |||||

HLVaR 99 | 0.125 | ||||

(0.000) | |||||

HLCVaR 95 | 0.062 | ||||

(0.000) | |||||

HLCVaR 99 | 0.079 | ||||

(0.000) | |||||

C | −0.001 | 0.000 | −0.001 | −0.001 | −0.001 |

(0.000) | (0.005) | (0.000) | (0.000) | (0.001) | |

Adj. R Square | 0.595 | 0.442 | 0.487 | 0.560 | 0.501 |

**Table 7.**Three-factor model using market Beta, VaR, and ES. The table represents five models with the dependent variable of excess returns of manufacturing companies listed in Pakistan Stock Exchange. Independent variables include all the systematic risk factors that the study proposed. Market excess return RmRf calculated from the subtraction of KSE 100 Index returns from the risk-free rate of returns. HLVaR is the subtraction of average returns of High VaR portfolio returns from Low VaR portfolio returns. VaR is observed at both 95% and 99% level of significance. HLCVaR represents subtraction of high ES return stocks from low ES returns stocks. SMB provides a size factor; HML is a value factor. All models give results on the three-factor model with substituting systematic risk factor. Italic probability value is provided below every coefficient value representing the significance of the coefficient effect dependent variable.

Dependent Variable: RIRF | |||||
---|---|---|---|---|---|

Model 6 | Model 7 | Model 8 | Model 9 | Model 10 | |

RMRF | 0.401 | ||||

(0.000) | |||||

HLVaR 95 | 0.264 | ||||

(0.000) | |||||

HLVaR 99 | 0.118 | ||||

(0.000) | |||||

HLCVaR 95 | 0.058 | ||||

(0.000) | |||||

HLCVaR 99 | 0.072 | ||||

(0.000) | |||||

SMB | 0.101 | −0.035 | 0.006 | 0.027 | 0.025 |

(0.000) | (0.001) | (0.582) | (0.018) | (0.032) | |

HML | 0.060 | 0.030 | 0.034 | 0.037 | 0.041 |

(0.000) | (0.001) | (0.001) | (0.000) | (0.000) | |

C | −0.001 | 0.000 | −0.001 | −0.001 | −0.001 |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | |

Adj. R Square | 0.454 | 0.585 | 0.491 | 0.569 | 0.553 |

**Table 8.**Five-factor model using market Beta, VaR, and ES. The table represents five models with the dependent variable of excess returns of manufacturing companies listed on the Pakistan Stock Exchange. Independent variables include all systematic risk factors which the study proposed. Market excess return RmRf is calculated from the subtraction of KSE 100 Index returns from the risk-free rate of returns. HLVaR subtracts average returns of High VaR portfolio returns from Low VaR portfolio returns. VaR is observed at both 95% and 99% level of significance. HLCVaR represents subtraction of high ES return stocks from low ES returns stocks. SMB provides size factor, HML is a value factor, and RMW estimates the investment factor represented by CMA and profitability. All models offer results of the three-factor model with substituting systematic risk factor. Italic probability value is provided below every coefficient value representing the significance of the coefficient effect dependent variable.

Dependent Variable: RiRf | |||||
---|---|---|---|---|---|

Model 11 | Model 12 | Model 13 | Model 14 | Model 15 | |

RMRF | 0.437 | ||||

(0.000) | |||||

HLVaR 95 | 0.280 | ||||

(0.000) | |||||

HLVaR 99 | 0.128 | ||||

(0.000) | |||||

HLCVaR 95 | 0.059 | ||||

(0.000) | |||||

HLCVaR 99 | 0.076 | ||||

(0.000) | |||||

SMB5 | 0.152 | −0.101 | −0.056 | −0.020 | −0.027 |

(0.000) | (0.000) | (0.000) | (0.089) | (0.030) | |

HML | 0.040 | 0.045 | 0.043 | 0.044 | 0.048 |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | |

CMA | 0.002 | 0.012 | 0.021 | 0.023 | 0.022 |

(0.824) | (0.117) | (0.012) | (0.007) | (0.008) | |

RMW | 0.002 | −0.028 | −0.034 | −0.027 | −0.031 |

(0.800) | (0.000) | (0.000) | (0.000) | (0.000) | |

C | −0.001 | 0.000 | −0.001 | −0.001 | −0.001 |

(0.000) | (0.008) | (0.001) | (0.000) | (0.001) | |

Adj. R Square | 0.480 | 0.663 | 0.501 | 0.473 | 0.458 |

**Table 9.**Six-factor model. The table represents four models with the dependent variable of excess returns of manufacturing companies listed on the Pakistan Stock Exchange. Independent variables include all systematic risk factors which the study proposed. Market excess return RmRf calculated from the subtraction of KSE 100 Index returns from a risk-free rate of returns. VaR is considered another risk factor with the systematic risk factor in this table. HLVaR subtracts average returns of High VaR portfolio returns from Low VaR portfolio returns. VaR is observed at both 95% and 99% level of significance. HLCVaR represents subtraction of high ES return stocks from low ES returns stocks. SMB provides size factor, HML is a value factor, RMV estimates investment factor represented by CMA and profitability factor. All models give results on the six-factor model. Italic probability value is provided below every coefficient value representing the significance of the coefficient effect dependent variable.

Dependent Variable: RiRf | ||||
---|---|---|---|---|

Model 16 | Model 17 | Model 18 | Model 19 | |

HLVaR95 | 0.183 | |||

(0.000) | ||||

HLVaR 99 | 0.085 | |||

(0.000) | ||||

HLCVaR 95 | 0.034 | |||

(0.000) | ||||

HLCVaR 99 | 0.049 | |||

(0.000) | ||||

RMRF | 0.355 | 0.414 | 0.421 | 0.427 |

(0.000) | (0.000) | (0.000) | (0.000) | |

SMB5 | 0.056 | 0.105 | 0.133 | 0.130 |

(0.000) | (0.000) | (0.000) | (0.000) | |

HML | 0.038 | 0.035 | 0.036 | 0.038 |

(0.000) | (0.000) | (0.000) | (0.000) | |

CMA | −0.004 | −0.002 | −0.001 | −0.001 |

(0.520) | (0.782) | (0.932) | (0.834) | |

RMV | −0.005 | −0.005 | 0.000 | −0.002 |

(0.380) | (0.378) | (0.941) | (0.695) | |

C | 0.000 | 0.000 | −0.001 | −0.001 |

(0.001) | (0.000) | (0.000) | (0.000) | |

Adj. R Square | 0.553 | 0.609 | 0.499 | 0.667 |

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**MDPI and ACS Style**

Nasir, A.; Khan, K.I.; Mata, M.N.; Mata, P.N.; Martins, J.N. Optimisation of Time-Varying Asset Pricing Models with Penetration of Value at Risk and Expected Shortfall. *Mathematics* **2021**, *9*, 394.
https://doi.org/10.3390/math9040394

**AMA Style**

Nasir A, Khan KI, Mata MN, Mata PN, Martins JN. Optimisation of Time-Varying Asset Pricing Models with Penetration of Value at Risk and Expected Shortfall. *Mathematics*. 2021; 9(4):394.
https://doi.org/10.3390/math9040394

**Chicago/Turabian Style**

Nasir, Adeel, Kanwal Iqbal Khan, Mário Nuno Mata, Pedro Neves Mata, and Jéssica Nunes Martins. 2021. "Optimisation of Time-Varying Asset Pricing Models with Penetration of Value at Risk and Expected Shortfall" *Mathematics* 9, no. 4: 394.
https://doi.org/10.3390/math9040394