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This paper proposes a multilayer network risk factor pricing model to depict the impact of interactions between stocks on excess stock returns by constructing the network risk factor based on the stock multilayer network and introducing it to the traditional three-factor pricing model. According to China’s stock market data, we find that compared with the traditional three-factor model, the multilayer network risk factor pricing model can achieve higher fitting degree. Meanwhile, the multilayer network risk factor has a significant positive impact on the excess stock returns in most cases.

The asset pricing puzzle always attracts the focus of the academic and practical circle. The traditional capital asset pricing model holds the view that the differences between stock returns result from their different market risk states. However, Fama and French [

Stocks are affected by the common macroeconomic environment and fluctuate with the stock index as a whole. In addition, there is also a clear interaction between stocks, especially in the regional and industrial sectors. Fama and French [

According to the above analysis, the residual returns of the traditional asset pricing model may contain the structural factors of stock networks. Therefore, it is necessary to construct network risk factors to test their explanations for the excess stock returns. This paper attempts to answer this question. In this paper, we construct the multilayer network risk factor based on the Fama–French three-factor model and test its explanation for the excess stock returns based on China’s stock market data. The contributions of this paper are the following. Firstly, we propose the method to construct the multilayer network risk factor for stock markets. Recent study in network science has moved from single-layer networks to multiplex networks [

The remainder of this paper is organized as follows. Section

In order to solve the asset pricing puzzle, Fama and French [

However, it should be noted that the above studies do not consider the influences of interaction between stocks on excess stock returns. In recent years, the correlation network method is increasingly used to depict the correlation structure of stock markets by regarding stocks as network nodes and their dependence as edges [

Recent studies on financial networks have moved from single-layer networks to multiplex networks [

On the basis of the traditional capital asset pricing model (CAPM), Fama and French [_{it} is the return of stock _{ft} is the risk-free rate at time _{t} is the excess market return at time _{t} is the size factor at the time _{t} is the book-to-market equity factor at time _{t} is the residual at time

In order to depict the influences of stock interactions on excess stock returns, we develop a multilayer network risk factor pricing model by constructing the multilayer network risk factor and introducing it to the three-factor model. Let us construct the following formula:_{t} states the correlation risk return, in which _{t} and

Step 1: constructing the three-factor model to obtain the residual series of excess stock returns.

Step 2: the Pearson, Kendall, and partial correlation coefficients between residual series are calculated to construct three categories of correlation coefficient matrixes, where specific coefficient calculation methods can be seen in Kendall [

Step 3: filtering the redundant information in three correlation coefficient matrixes. In three commonly used network filtering approaches, the threshold method is subjective in the selection of thresholds, which has crucial impacts on the filtering results. The minimum spanning tree (MST) [

Step 4: based on the three-layer stock correlation network, the entropy of the multiplex degree proposed by Battiston et al. [

where

Step 5: calculating the relative multilayer degree centralities

This paper selects the data of Chinese A-share listed companies during July 2005 to June 2018 as the research sample. The sample data can be obtained from the Ruisi financial database (

In this section, we group the 1050 stocks according to their total market capitalization and the ratio of owners’ equity to total market capitalization and further construct portfolios and calculate the traditional three factors. First, based on the 20%, 40%, 60%, and 80% quantiles of total market capitalization in June of

Similarly, we can further calculate the traditional excess market return, size, and book-to-market equity factors. Among them, the excess market return factor is defined as the difference between the stock market return and the risk-free rate, while the size and book-to-market equity factors are calculated by constructing portfolios. More specifically, first, all the stocks are divided into group

To begin with, we make the descriptive statistics for the excess market return factor (RMRF), the size factor (SMB), and the book-to-market equity factor (HML). The descriptive statistical results are reported in Table

Descriptive statistical results of four factors.

Factor | Mean | Std. | Min. | Max. | ADF |
---|---|---|---|---|---|

RMRF | 0.0109 | 0.0891 | −0.2685 | 0.2950 | −10.417 |

SMB | 0.0036 | 0.0563 | −0.2093 | 0.1653 | −10.393 |

HML | −0.0029 | 0.0428 | −0.2775 | 0.1474 | −15.209 |

0.0169 | 0.1000 | −0.2738 | 0.3412 | −10.539 |

From Table

Fitting effect of different factor models.

Size quintile | Book-to-market equity quintiles | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |

Panel A: three-factor model | Panel B: multilayer network risk factor pricing model | |||||||||

Small | 0.642 | 0.937 | 0.942 | 0.939 | 0.930 | 0.720 | 0.952 | 0.955 | 0.953 | 0.937 |

2 | 0.906 | 0.931 | 0.940 | 0.950 | 0.954 | 0.922 | 0.960 | 0.962 | 0.972 | 0.967 |

3 | 0.874 | 0.895 | 0.925 | 0.944 | 0.959 | 0.909 | 0.932 | 0.955 | 0.971 | 0.970 |

4 | 0.860 | 0.889 | 0.900 | 0.938 | 0.942 | 0.900 | 0.947 | 0.934 | 0.958 | 0.946 |

Big | 0.907 | 0.917 | 0.896 | 0.903 | 0.905 | 0.930 | 0.927 | 0.905 | 0.903 | 0.905 |

Table

Grouped regression results of the multilayer network risk factor pricing model.

Size quintile | Book-to-market equity quintiles | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Low | 2 | 3 | 4 | High | Low | 2 | 3 | 4 | High | |

Small | 4.466 | 0.086 | 0.162 | 0.110 | 0.354 | 8.802 | 0.686 | 1.337 | 0.864 | 2.504 |

2 | 0.123 | −0.187 | −0.012 | 0.005 | 0.207 | 0.789 | −1.656 | −0.117 | 0.0489 | 2.045 |

3 | −0.284 | −0.265 | −0.118 | −0.046 | 0.335 | −1.748 | −1.814 | −1.044 | −0.509 | 3.531 |

4 | −0.287 | −0.507 | −0.174 | 0.164 | 0.585 | −1.808 | −4.313 | −1.221 | 1.490 | 4.704 |

Big | 0.126 | 0.462 | 0.472 | 0.991 | 1.098 | 1.097 | 3.561 | 3.077 | 5.970 | 7.217 |

Small | 2.885 | 0.543 | 0.597 | 0.546 | 0.595 | 9.223 | 6.998 | 7.978 | 6.937 | 6.821 |

2 | 0.297 | 0.234 | 0.223 | 0.281 | 0.354 | 3.099 | 3.367 | 3.406 | 4.745 | 5.672 |

3 | 0.011 | −0.115 | −0.022 | 0.018 | 0.266 | 0.106 | −1.284 | −0.321 | 0.321 | 4.554 |

4 | −0.256 | −0.389 | −0.168 | −0.100 | 0.173 | −2.622 | −5.365 | −1.914 | −1.471 | 2.256 |

Big | −0.769 | −0.667 | −0.629 | −0.343 | −0.207 | −10.89 | −8.342 | −6.653 | −3.347 | −2.204 |

Small | −1.537 | 0.018 | 0.043 | 0.130 | 0.161 | −7.720 | 0.363 | 0.906 | 2.604 | 2.891 |

2 | −0.186 | −0.138 | 0.029 | 0.185 | 0.345 | −3.051 | −3.125 | 0.693 | 4.915 | 8.688 |

3 | −0.407 | −0.175 | 0.027 | 0.127 | 0.343 | −6.390 | −3.060 | 0.616 | 3.620 | 9.217 |

4 | −0.358 | −0.263 | −0.162 | 0.060 | 0.512 | −5.755 | −5.694 | −2.895 | 1.400 | 10.49 |

Big | −0.736 | −0.447 | 0.024 | 0.317 | 0.565 | −16.37 | −8.774 | 0.394 | 4.872 | 9.462 |

Small | −3.284 | 0.865 | 0.785 | 0.881 | 0.610 | −6.565 | 6.971 | 6.559 | 7.005 | 4.369 |

2 | 0.868 | 1.180 | 1.003 | 1.018 | 0.802 | 5.657 | 10.62 | 9.596 | 10.76 | 8.044 |

3 | 1.232 | 1.313 | 1.137 | 1.057 | 0.704 | 7.698 | 9.131 | 10.17 | 11.94 | 7.530 |

4 | 1.241 | 1.496 | 1.245 | 0.911 | 0.451 | 7.938 | 12.92 | 8.879 | 8.406 | 3.679 |

Big | 0.796 | 0.581 | 0.577 | 0.106 | −0.127 | 7.046 | 4.546 | 3.819 | 0.650 | −0.849 |

^{∗∗∗}, ^{∗∗}, and ^{∗} indicate significance at the 1%, 5%, and 10% level, respectively.

From Table

In this paper, we choose the monthly data of Chinese A-share listed companies. First, the Fama–French three-factor model is estimated to obtain stock residual series, which can be utilized to construct the stock multilayer network and further calculate multilayer degree centralities. On that basis, we propose a multilayer network risk factor pricing model by computing the multilayer network risk factor and introducing it to the three-factor model. The empirical results show that compared with the traditional Fama–French three-factor model, the multilayer network risk factor pricing model can achieve higher fitting degree. Meanwhile, the multilayer network risk factor has a significant positive impact on the excess stock returns in most cases.

This paper complements the traditional three-factor model by constructing the multilayer network risk factor and confirms that the interactions between stocks have significant influences on the excess stock returns. It further reveals the stock pricing rules, provides important references for investors to make decisions, and offers new thought for the capital asset pricing research. In addition, from the perspective of excess stock returns and multilayer network, this paper provides a theoretical reference for financial regulatory authorities to consider the linkage and multilayer nature of stock prices and then provides index reference and theoretical basis for formulating regulatory policies to prevent excessive volatility of stock prices.

The sample data can be obtained from the Ruisi financial database (

Yu Liu, Lei Wang, and Kun Yang are the co-first authors.

The authors declare that they have no conflicts of interest.

Yu Liu, Lei Wang, and Kun Yang contributed equally to this work.

This study was supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant no. KYCX19_0130) and the Scientific Research Foundation of Graduate School of Southeast University (Grant no. YBPY1942).