^{1}

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In this paper, we extend the 5-factor model in Fama and French (2015) with the non-Normal errors distribution of SSAEPD (Standardized Standard Asymmetric Exponential Power Distribution) in Zhu and Zinde-Walsh (2009) and the GARCH-type volatility. The focus is on finding out whether our new model can outperform the original Fama-French 5-factor model. We use Fama-French 25 value-weighted portfolios to conduct our research. The MLE is used to estimate the parameters. The LR test and KS test are used for model diagnostics. Models are compared by AIC. Empirical results show that with GARCH-type volatilities and non-normal errors, the Fama-French 5 factors are still alive. Our new model can successfully capture the skewness, fat-tailness and asymmetric kurtosis in the data and has better in-sample fit than the 5-factors model in Fama and French (2015). Our study provides an update to existing asset pricing literature and reference for investors.

The capital asset pricing model of Sharpe and Lintner (1965) marks the birth of asset pricing theory [

However, recent studies have discovered that many other important patterns in average returns are left unexplained by the 3-factor model. Panel A of

Author (Year) | Research Purpose | Model | Estimation Method | Data Country | Model factors | Frequency & Period |
---|---|---|---|---|---|---|

Panel A: Extension of Factor Model | ||||||

Fama et.al. (1993) | CAPM Extension | FF3 | - | USA | Mkt, SMB, HML, WML | M1963:7 - 1991:12 |

Carhart (1997) | FF3 Extention | CAPM, FF3, C4 | OLS | USA | Mkt, SMB, HML, WML | M1962:1 - 1993:12 |

Griffin (2002) | FF3 Extention | World, Domestic or International FF3 | - | Global | Mkt, SMB, HML | M1981:1995:12 |

Bali et.al. (2004) | FF3 Extention | FF3 with VAR | OLS | USA | Mkt, SMB, HML, VAR | M1963:1 - 2001:12 |

Chan et.al. (2005) | FF3 Extension | FF3 with IML | GMM | Australia | Mkt, SMB, HML, IML | M1990:1 - 1998:12 |

Chan et.al. (2007) | FF3 Extension | FF3 with Default factor | GMM | Australia | Mkt, SMB, HML, DEF | M1996 - 2004 |

He (2008) | FF3 Extention | FF3, FF3 with State Switch | OLS | China | Mkt, SMB, HML, State Switch | M1995:6 - 2005:12 |

Xiao et.al. (2007) | FF3 Extention | FF3 with Sustainability Factor | GMM | Global | Mkt, SMB, HML, SUS | M1999 - 2007 |

Fama et.al. (2013) | FF3 Extention | FF4 | - | USA | Mkt, SMB, HML, RMW | M1963:7 - 2012:12 |

Yang (2013) | FF3 Extention | FF3 with SSAEPD, EGARCH | MLE | USA | Mkt, SMB, HML | M1926 - 2011 |

Fama et.al. (2015) | FF4 Extention | FF5 | - | USA | Mkt, SMB, HML, RMW, CMA | M1963:7 - 2013:12 |

Mu (2015) | C Extension | C4 with SSAEPD, EGARCH | MLE | USA | Mkt, SMB, HML, WML | M1927:1 - 2014:12 |

Panel B: Fama-French 5-Factor Model comparison | ||||||

Fama et.al. (2014) | Model Comparison | CAPM, FF3, FF4, FF5, FF5 with WML | - | USA | Mkt, SMB, HML, RMW, CMA, WML | M1963:7 - 2014:12 |

Hou et.al. (2015) | Model Comparison | FF5, C, q-factor | - | USA | Mkt, SMB, HML, RMW, CMA, WML | M1967:1 - 2013:12 |

Harshita et.al. (2015) | Model Comparison | CAPM, FF3, FF5 | - | India | Mkt, SMB, HML, RMW, CMA | M1999:10 - 2014:9 |

Chiah et.al. (2015) | Model Comparison | FF3, FF5 | HAC-adjusted OLS | Australia | Mkt, SMB, HML, RMW, CMA | M1982:12 - 2012:12 |

the development of the factor model in stock market. For example, Carhart (1997) incorporates momentum factor into the Fama-French 3-factor (FF3) model and establishes a Carhart 4-factor (C4) model which documents that stocks performing the best in the short run tend to continue this trend [

In 2015, Fama and French proposed a 5-factor model directed at capturing the size, value, profitability and investment patterns in average stock returns and found it performed better than their 3-factor model [

Different from previous researches, our research tries to extend the 5-factor model in Fama and French (2015). Many asset pricing models in the existing literature just assume that financial time series follow the normal distribution, but more and more researches and studies have observed the unique distributional properties of financial data―more kurtosis and higher peak―contradicting the assumption of normality [

1) With GARCH-type volatilities and SSAEPD errors, are the Fama-French 5 factors still alive?

2) Can our new model beat the 5 factor model in Fama and French (2015)?

To answer these questions, we first run simulation to test whether the MatLab program we write can be used in our analysis. Then, Fama-French 25 value-weighted port- folios are analyzed. Data are downloaded from the French’s Data Library, and the sample period is from Jul. 1963 to Dec. 2013. Method of Maximum Likelihood Estimation (MLE) is used to estimate the parameters. Likelihood Ratio test (LR) and Kolmogorov- Smirnov test (KS) are exploited for model diagnostics. Akaike Information Criterion (AIC) is employed for model comparison.

Simulation results show our MatLab program can be employed for our empirical analysis. According to the empirical results, we find out the 5 factors in Fama and French (2015) are still alive! The new model fits the data well and has better in-sample fit than the 5-factor model in Fama and French (2015).

The paper proceeds as follows. The model and methodology are discussed in Section 2. Simulation analysis is reported in Section 3. Empirical results and the model comparisons are presented in Section 4. Section 5 provides the conclusions and future extensions.

Fama and French (2015) propose a 5-factor model (denoted as FF5) to capture the size, value, profitability, and investment patterns in expected stock returns, and show this model empirically outperforms their 3 factor model. The 5-factor model is:

where θ = (β_{0}, β_{1}, β_{2}, β_{3}, β_{4}, β_{5}, μ, σ) are parameters to be estimated in this model. _{ft} is the risk-free return. R_{mt} is the value-weighted market return. SMB_{t} is the return on small stock portfolio minus the return on big stock portfolio. HMLO_{t} is the high book-to-market ratio minus low book-to-market ratio orthogonalized1. RMW_{t} stands for robust operating profitability portfolios minus weak operating profitability portfolios. CMA_{t} stands for conservative investment portfolios minus aggressive investment portfolios. The error term

Based on the GARCH-type volatility in Bollerslev (1986) and non-Normal error distribution of SSAEPD in Zhu and Zinde-Walsh (2009), we extend Fama-French (2015) five-factor model in this section. The new model is denoted as FF5-SSAEPD-GARCH and its math formula is:

where

・ Standardized Standard AEPD (SSAEPD)

The probability density function (PDF) of the SSAEPD proposed by Zhu and Zinde- Walsh (2009) is

where

And_{1} and p_{2} are, the fatter the tails are. If α is smaller than 0.5, it indicates a left-skewed distribution. If α is larger than 0.5, it indicates a right-skewed distribution When α = 0.5, p_{1} = p_{2} = 2, SSAEPD can be reduced to Normal (0, 1). The mean of z_{t} is zero and its variance is 1.

In this paper, we estimate the FF5-SSAEPD-GARCH model with Maximum Likelihood Estimation (MLE). The likelihood function is

where

In this section, we first generate random number series for_{t}, HMLO_{t}, RMW_{t}, CMA_{t} and

We choose

1) Given α = 0.5, p_{1} = p_{2} = 2, we can generate SSAEPD random number series

2) Set the initial value

3) Generate random number series_{t}, HMLO_{t}, RMW_{t}, CMA_{t} from Uniform (0, 1).

4) Set β_{0} = 0.2, β_{1} = 1, β_{2} = 0.5, β_{3} = 0.5, β_{4} = 0.5, β_{5} = 0.5 and we can get

After getting the simulated data_{t}, HMLO_{t}, RMW_{t}, CMA_{t} and

The data we analyze are the monthly returns from the Fama-French 25 value-weighted portfolios for US stock market, which are the same as data used in Fama and French (2015). The descriptive statistics of sample data are calculated by MatLab and listed in

β_{0} | β_{1} | β_{2} | β_{3} | β_{4} | β_{5} | α | p_{1} | p_{2} | a | b | c | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

T | 0.2 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.1823 | 1.0260 | 0.5096 | 0.4801 | 0.5306 | 0.5096 | 0.4951 | 1.9497 | 2.0179 | 0.2998 | 0.4972 | 0.4028 |

P | 8.84% | 2.60% | 1.92% | 3.97% | 6.12% | 1.93% | 0.97% | 2.52% | 0.89% | 0.07% | 0.57% | 0.70% |

T | 0.2 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.1963 | 1.0303 | 0.5234 | 0.4957 | 0.5381 | 0.4601 | 0.5054 | 2.0098 | 2.0179 | 0.2917 | 0.5027 | 0.3943 |

P | 1.85% | 3.03% | 4.69% | 0.85% | 7.62% | 7.98% | 1.09% | 0.49% | 0.90% | 2.78% | 1.17% | 1.43% |

T | 0.3 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.3123 | 1.0108 | 0.5262 | 0.4729 | 0.4787 | 0.4955 | 0.4990 | 1.9998 | 1.9994 | 0.3019 | 0.4929 | 0.4009 |

P | 4.10% | 1.08% | 5.24% | 5.41% | 4.25% | 0.89% | 0.19% | 0.01% | 0.03% | 0.65% | 1.42% | 0.24% |

T | 0.2 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.1911 | 0.5242 | 0.4724 | 0.4775 | 0.5320 | 0.5356 | 0.5000 | 2.0000 | 2.0000 | 0.2918 | 0.5185 | 0.3926 |

P | 4.43% | 4.85% | 5.52% | 4.51% | 6.40% | 7.12% | 0.00% | 0.00% | 0.00% | 2.73% | 3.71% | 2.05% |

T | 0.2 | 1 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.1871 | 1.0070 | 0.9952 | 0.4981 | 0.5193 | 0.5135 | 0.5000 | 1.9998 | 2.0001 | 0.3281 | 0.5289 | 0.3634 |

P | 6.43% | 0.70% | 0.48% | 0.38% | 3.87% | 2.69% | 0.00% | 0.01% | 0.00% | 9.38% | 5.78% | 9.15% |

T | 0.2 | 1 | 0.5 | 1 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.1884 | 0.9851 | 0.5161 | 1.0033 | 0.5163 | 0.4963 | 0.5000 | 2.0000 | 2.0000 | 0.2745 | 0.4918 | 0.4212 |

P | 5.78% | 1.49% | 3.22% | 0.33% | 3.27% | 0.073% | 0.00% | 0.00% | 0.00% | 8.51% | 1.64% | 5.31% |

T | 0.2 | 1 | 0.5 | 0.5 | 1 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.1922 | 0.9619 | 0.4798 | 0.4775 | 1.0282 | 0.5010 | 0.5004 | 1.9959 | 1.9947 | 0.3059 | 0.4688 | 0.4174 |

P | 3.90% | 3.81% | 4.05% | 4.49% | 2.82% | 0.20% | 0.07% | 0.21% | 0.26% | 1.95% | 6.25% | 4.34% |

T | 0.2 | 1 | 0.5 | 0.5 | 0.5 | 1 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 |

E | 0.1944 | 1.0429 | 0.5075 | 0.5375 | 0.4979 | 0.9589 | 0.5000 | 2.0000 | 2.0000 | 0.3088 | 0.5055 | 0.3911 |

P | 2.81% | 4.29% | 1.51% | 7.50% | 0.42% | 4.11% | 0.00% | 0.00% | 0.00% | 2.94% | 1.09% | 2.22% |

T | 0.2 | 1 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.4 | 0.5 |

E | 0.2042 | 1.0001 | 0.4898 | 0.4987 | 0.4843 | 0.4620 | 0.5000 | 1.9996 | 1.9995 | 0.2905 | 0.3948 | 0.5013 |

P | 2.10% | 0.01% | 2.05% | 0.25% | 3.13% | 7.59% | 0.01% | 0.02% | 0.03% | 3.17% | 1.30% | 0.25% |

Notes: T means the true value of parameters. E means the estimates. P means the error in percentage.

Size | Book-to-market quintiles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

Mean | Median | ||||||||||

Small | 0.68 | 1.22 | 1.26 | 1.42 | 1.56 | 1.06 | 1.43 | 1.36 | 1.54 | 1.57 | |

2 | 0.89 | 1.14 | 1.35 | 1.36 | 1.44 | 1.37 | 1.56 | 1.60 | 1.55 | 1.85 | |

3 | 0.91 | 1.19 | 1.21 | 1.29 | 1.49 | 1.51 | 1.47 | 1.57 | 1.50 | 1.61 | |

4 | 1.01 | 0.99 | 1.12 | 1.26 | 1.27 | 1.09 | 1.24 | 1.45 | 1.54 | 1.60 | |

Big | 0.87 | 0.93 | 0.89 | 0.97 | 1.03 | 0.97 | 1.04 | 1.15 | 1.10 | 1.19 | |

Maximum | Minimum | ||||||||||

Small | 39.85 | 38.56 | 28.13 | 27.78 | 33.27 | −34.24 | −30.94 | −28.69 | −28.88 | −28.88 | |

2 | 27.45 | 26.12 | 26.34 | 27.34 | 30.04 | −32.71 | −31.56 | −27.80 | −26.04 | −28.84 | |

3 | 24.69 | 25.03 | 21.94 | 23.40 | 29.20 | −29.79 | −28.99 | −24.26 | −23.03 | −26.17 | |

4 | 25.91 | 20.44 | 24.01 | 24.32 | 27.90 | −25.94 | −28.83 | −26.33 | −21.02 | −23.84 | |

Big | 22.35 | 16.53 | 19.08 | 19.76 | 17.57 | −21.64 | −22.36 | −21.74 | −19.32 | −19.13 | |

Standard Deviation | Skewness | ||||||||||

Small | 8.01 | 6.90 | 6.02 | 5.68 | 6.11 | −0.02 | 0.00 | −0.17 | −0.22 | −0.26 | |

2 | 7.23 | 6.00 | 5.46 | 5.32 | 6.03 | −0.34 | −0.46 | −0.48 | −0.50 | −0.46 | |

3 | 6.67 | 5.51 | 5.03 | 4.94 | 5.52 | −0.36 | −0.52 | −0.53 | −0.33 | −0.41 | |

4 | 5.94 | 5.21 | 5.10 | 4.84 | 5.53 | −0.22 | −0.60 | −0.51 | −0.28 | −0.30 | |

Big | 4.70 | 4.47 | 4.38 | 4.39 | 5.05 | −0.22 | −0.38 | −0.31 | −0.25 | −0.30 | |

Kurtosis | P-value of Jarque-Bera Test | ||||||||||

Small | 5.26 | 6.12 | 5.63 | 5.90 | 6.34 | 0 | 0 | 0 | 0 | 0 | |

2 | 4.45 | 5.33 | 5.91 | 6.11 | 6.16 | 0 | 0 | 0 | 0 | 0 | |

3 | 4.44 | 5.64 | 5.26 | 5.43 | 6.26 | 0 | 0 | 0 | 0 | 0 | |

4 | 4.72 | 5.90 | 6.35 | 5.06 | 5.52 | 0 | 0 | 0 | 0 | 0 | |

Big | 4.67 | 4.76 | 5.24 | 4.96 | 4.07 | 0 | 0 | 0 | 0 | 0 |

The estimates for our new model are displayed in

Size | Book-to-market quintiles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

β_{0} | β_{1} | ||||||||||

Small | −0.30 | 0.10 | 0.00 | 0.69 | 0.00 | 1.03 | 0.95 | 0.94 | 0.87 | 0.97 | |

2 | −0.09 | −0.07 | −0.13 | 0.03 | −0.09 | 1.07 | 1.02 | 0.99 | 0.98 | 1.08 | |

3 | 0.04 | 0.07 | −0.15 | 0.01 | −0.51 | 1.05 | 1.01 | 1.00 | 1.00 | 1.11 | |

4 | 0.16 | −0.08 | 0.04 | 0.06 | −0.07 | 1.05 | 1.07 | 1.05 | 1.02 | 1.13 | |

Big | −0.15 | 0.38 | −0.05 | −0.01 | −0.30 | 1.02 | 0.98 | 0.98 | 0.98 | 1.09 | |

β_{2} | β_{3} | ||||||||||

Small | 1.26 | 1.20 | 1.06 | 1.00 | 1.11 | −0.48 | −0.07 | 0.11 | 0.30 | 0.48 | |

2 | 0.97 | 0.90 | 0.84 | 0.76 | 0.87 | −0.46 | −0.04 | 0.26 | 0.42 | 0.67 | |

3 | 0.69 | 0.63 | 0.53 | 0.48 | 0.65 | −0.43 | 0.00 | 0.30 | 0.51 | 0.63 | |

4 | 0.34 | 0.28 | 0.25 | 0.24 | 0.30 | −0.41 | 0.01 | 0.27 | 0.52 | 0.77 | |

Big | −0.19 | −0.19 | −0.20 | −0.15 | −0.03 | −0.33 | 0.01 | 0.26 | 0.53 | 0.85 | |

β_{4} | β_{5} | ||||||||||

Small | −0.64 | −0.24 | 0.01 | 0.11 | 0.13 | −0.56 | −0.14 | 0.20 | 0.28 | 0.70 | |

2 | −0.21 | 0.04 | 0.22 | 0.23 | 0.23 | −0.58 | 0.03 | 0.37 | 0.53 | 0.74 | |

3 | −0.21 | 0.10 | 0.16 | 0.19 | 0.24 | −0.68 | 0.02 | 0.42 | 0.58 | 0.98 | |

4 | −0.13 | 0.08 | 0.06 | 0.07 | 0.22 | −0.47 | 0.15 | 0.33 | 0.53 | 0.74 | |

Big | 0.13 | 0.12 | 0.02 | 0.09 | 0.06 | −0.26 | −0.01 | 0.38 | 0.50 | 0.77 | |

α | p_{1} | ||||||||||

Small | 0.44 | 0.79 | 0.66 | 0.45 | 0.35 | 1.60 | 3.21 | 2.35 | 1.73 | 1.18 | |

2 | 0.54 | 0.63 | 0.45 | 0.45 | 0.30 | 1.43 | 1.84 | 1.41 | 1.39 | 1.01 | |

3 | 0.38 | 0.72 | 0.72 | 0.80 | 0.62 | 1.27 | 1.68 | 2.07 | 2.03 | 1.80 | |

4 | 0.70 | 0.35 | 0.66 | 0.47 | 0.45 | 2.04 | 1.05 | 1.67 | 1.46 | 1.57 | |

Big | 0.53 | 0.56 | 0.63 | 0.38 | 0.41 | 1.71 | 1.93 | 1.78 | 1.38 | 1.28 | |

p_{2} | a | ||||||||||

Small | 1.57 | 0.80 | 1.08 | 0.93 | 1.70 | 2.73 | 1.72 | 1.27 | 0.42 | 0.60 | |

2 | 1.27 | 1.07 | 1.70 | 1.70 | 1.98 | 1.88 | 1.16 | 0.13 | 1.40 | 0.37 | |

3 | 2.13 | 0.88 | 1.12 | 0.71 | 1.19 | 1.74 | 1.76 | 0.17 | 1.73 | 0.24 | |

4 | 0.91 | 1.69 | 0.97 | 1.56 | 1.68 | 1.94 | 1.89 | 1.21 | 2.15 | 3.28 | |

Big | 1.66 | 1.45 | 1.22 | 1.96 | 1.72 | 0.04 | 0.12 | 2.00 | 0.15 | 0.25 | |

b | c | ||||||||||

Small | 0.37 | 0.27 | 0.22 | 0.08 | 0.22 | 0.00 | 0.00 | 0.00 | 0.82 | 0.49 | |

2 | 0.12 | 0.31 | 0.07 | 0.10 | 0.15 | 0.00 | 0.00 | 0.85 | 0.00 | 0.66 | |

3 | 0.14 | 0.28 | 0.13 | 0.24 | 0.11 | 0.00 | 0.00 | 0.80 | 0.00 | 0.82 | |

4 | 0.14 | 0.33 | 0.53 | 0.15 | 0.12 | 0.00 | 0.00 | 0.14 | 0.00 | 0.00 | |

Big | 0.12 | 0.11 | 0.25 | 0.12 | 0.17 | 0.85 | 0.84 | 0.00 | 0.81 | 0.78 |

To test the significance of coefficients in our new model, Likelihood Ratio test (LR) is applied, LR formula is from Neyman and Pearson (1993), which is Equation (18).

・ Tests for Parameters in the Mean Equation

The P-values of LR are listed in _{0}: β_{1} = β_{2} = β_{3} = β_{4} = β_{5} = 0. The P-values of the joint significance test for all the 25 portfolios are 0, which means β_{1}, β_{2}, β_{3}, β_{4} and β_{5} are statistically jointly significant under 5% significance level. The individual significance tests show that under 5% significance level the coefficient β_{1} in all 25 portfolios are statistically significant; 24/25 portfolios have a statistically coefficient β_{2} and β_{5}; 23/25 and 19/25 portfolios have a statistically coefficient β_{3} and β_{4}, respectively. As for coefficient β_{0}, 16 out of the 25 portfolios don’t have a statistically significant coefficient β_{0} under 5% significance level. Thus, we can conclude that with non-Normal errors such as SSAEPD and GARCH- type volatilities, the Fama-French 5-factor model is still alive.

・ Tests for Parameters in the GARCH Equation

In this part, some restrictions on the parameters in the GARCH equation are tested with Likelihood Ratio test (LR). And the results are listed in _{0}: b = c = 0. The P-values of the LR are all smaller than the significance level 5%, which means our GARCH-type volatility is quite necessary. As for individual hypotheses, we discover that most P-values of LR are smaller than the significance level 5%. And to be specific, ARCH term (H_{0}: b = 0) is significant in 20 out of 25 portfolios and GARCH term (H0: c = 0) is significant in 18 out of 25 portfolios.

・ Tests for Parameters in SSAEPD

We also run significance tests for the parameters in the SSAEPD and the results of parameter restrictions show strong non-Normality. And the results are listed in _{0}: α = 0.5, p_{1} = p_{2} = 2, 21 out of 25 p-values are smaller than the significance level 5%, which means that Normal error assumption is not supported by most of our data. Besides, Asymmetry is documented (H_{0}: α = 0.5 is rejected by 7 out of 25 portfolios). And non-normality is found (H_{0}: p_{1} = 2 is rejected by 8 out of 25 portfolios and 12 out of 25 portfolios reject the null H_{0}: p_{1} = 2.).

In this subsection, the residuals for previous models are checked with both Kolmogorov-Smirnov test and graphs. Our results show that 20 out of the 25 portfolios have residuals which do follow SSAEPD. That means our new model is adequate for the Fama- French 25 portfolios. But the FF5-Normal model is not adequate for the data, since 21 portfolios have residuals which do not follow the Normal error distribution.

Size | Book-to-market quintiles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

H_{0}:β_{1} = β_{2} = β_{3} = β_{4} = β_{5} = 0 | H_{0}:β_{0} = 0 | ||||||||||

Small | 0* | 0* | 0* | 0* | 0* | 0* | 0.08 | 0.95 | 1.00 | 0.31 | |

2 | 0* | 0* | 0* | 0* | 0* | 0.13 | 0.19 | 0* | 0.58 | 0* | |

3 | 0* | 0* | 0* | 0* | 0* | 0.46 | 0.17 | 0* | 1.00 | 1.00 | |

4 | 0* | 0* | 0* | 0* | 0* | 0.01* | 0.24 | 0.57 | 0.36 | 0* | |

Big | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.44 | 0.65 | 0* | |

H_{0}:β_{1} = 0 | H_{0}:β_{2} = 0 | ||||||||||

Small | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

2 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

3 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

4 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

Big | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0.20 | |

H_{0}:β_{3} = 0 | H_{0}:β_{4} = 0 | ||||||||||

Small | 0* | 0.03* | 0* | 0* | 0* | 0* | 0* | 1.00 | 0* | 0* | |

2 | 0* | 0.57 | 0* | 0* | 0* | 0* | 0.96 | 0* | 0* | 0* | |

3 | 0* | 0* | 0* | 0* | 0* | 0* | 0.01* | 0* | 0.01* | 0* | |

4 | 0* | 0* | 0* | 0* | 0* | 0* | 0.91 | 0* | 0.53 | 0* | |

Big | 0* | 1.00 | 0* | 0* | 0* | 0* | 0* | 0.64 | 0* | 1.00 | |

H_{0}:β_{5} = 0 | H_{0}:b = c = 0 | ||||||||||

Small | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

2 | 0* | 0.08 | 0* | 0* | 0* | 0* | 0* | 0* | 0.04* | 0.01* | |

3 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

4 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

Big | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

H_{0}:a = 0 | H_{0}:b = 0 | ||||||||||

Small | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 1.00 | 0* | |

2 | 0* | 0* | 0* | 0* | 0* | 0.02* | 0* | 0* | 0.05 | 0* | |

3 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 1.00 | |

4 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

Big | 0* | 0* | 0* | 0* | 0* | 1.00 | 1.00 | 0* | 0* | 0* | |

H_{0}:c = 0 | H_{0}:α = 0.5,p_{1} = p_{2} = 2 | ||||||||||

Small | 0* | 1.00 | 0* | 0* | 0* | 0* | 0* | 0* | 0* | 0* | |

2 | 0* | 1.00 | 0* | 1.00 | 0* | 0* | 0* | 0.05 | 0.01* | 0* | |

3 | 0* | 0* | 0* | 1.00 | 0* | 0.02* | 0* | 0* | 0* | 0* | |

4 | 0* | 1.00 | 1.00 | 0.95 | 0* | 0* | 0* | 0* | 0* | 0.04* | |

Big | 0* | 0* | 0* | 0* | 0* | 0.35 | 0.37 | 0* | 0.23 | 0* | |

H_{0}:α = 0.5 | H_{0}:p_{1} = p_{2} = 2 | ||||||||||

Small | 0.57 | 0.01* | 0.17 | 0.62 | 0.08 | 0.02* | 0* | 0.01* | 0* | 0* | |

2 | 0.47 | 0.06 | 0.40 | 0.46 | 0.01* | 0* | 0* | 0.02* | 0* | 0* | |

3 | 0.07 | 0.04* | 0* | 0* | 0.18 | 0.01* | 0* | 0* | 0* | 0* | |

4 | 0.02* | 0.08 | 0.03* | 0.69 | 0.42 | 0* | 0* | 0* | 0* | 0.04* | |

Big | 1.00 | 0.52 | 0.14 | 0.46 | 0.33 | 0.20 | 0.21 | 0* | 0.13 | 0* | |

H_{0}:p_{1} = 2 | H_{0}:p_{2} = 2 | ||||||||||

Small | 0.35 | 0.01* | 0.40 | 1.00 | 0.01* | 0.48 | 0* | 0.02* | 0* | 0.16 | |

2 | 0.01* | 0.52 | 0.15 | 0.06 | 0* | 0* | 0* | 0.13 | 0.28 | 1.00 | |

3 | 0* | 0.07 | 0.80 | 1.00 | 0.52 | 0.68 | 0* | 0* | 0* | 0* | |

4 | 0.86 | 0* | 0.18 | 0.04* | 0.14 | 0* | 0.12 | 0* | 0.37 | 0.35 | |

Big | 0.44 | 0.95 | 0.49 | 0.21 | 0.02* | 0.31 | 0.15 | 0.01* | 0.90 | 0.23 |

Note: * means the null is rejected under 5% significant level.

・ Kolmogorov-Smirnov Test for Residuals

To check the residuals, the Kolmogorov-Smirov test (KS) is employed. The P-value of KS test is displayed in

Then, we apply the KS test for the residuals from the FF5-Normal model. The P-values of the KS test are also listed in

・ PDFs of Residuals

By method of “eye-rolling”, the PDF of residuals is compared with theoretical PDFs. Taking the portfolio with Small Size and Low Book-to-market for example, in

Similarly, the probability density function (PDF) for the estimated residuals

In this subsection, we compare our new model with the 5-factor model of Fama and French (2015). The Akaike Information Criterion (AIC) is used as the model selection criterion.

Size | Book-to-market quintiles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

FF5-SSAEPD-GARCH^{a} | FF5-Normal^{b} | ||||||||||

Small | 0.71 | 0.50 | 0.91 | 0* | 0.28 | 0* | 0* | 0* | 0.01* | 0* | |

2 | 0.98 | 1.00 | 0* | 0.94 | 0.76 | 0* | 0.07 | 0.13 | 0.06 | 0.01* | |

3 | 0.67 | 0.87 | 0.17 | 0.46 | 0* | 0* | 0* | 0* | 0* | 0* | |

4 | 0.81 | 0.93 | 0.19 | 0.96 | 0.92 | 0* | 0* | 0* | 0* | 0* | |

Big | 0* | 0* | 0.82 | 0.74 | 0.17 | 0.79 | 0* | 0* | 0* | 0* |

a. The null hypothesis H_{0} is: FF5-SSAEPD-GARCH residuals are distributed as_{0} is: FF5-Normal residuals are distributed as

Size | Book-to-market quintiles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

FF5-SSAEPD-GARCH | FF5-Normal | ||||||||||

Small | 4.19* | 3.60* | 3.29* | 3.45 | 3.49* | 4.32 | 3.71 | 3.36 | 3.43 | 3.60 | |

2 | 3.58* | 3.26* | 3.24* | 3.29* | 3.44* | 3.63 | 3.38 | 3.27 | 3.32 | 3.54 | |

3 | 3.54* | 3.61* | 3.57* | 3.54* | 3.96* | 3.59 | 3.74 | 3.68 | 3.68 | 4.00 | |

4 | 3.60* | 3.76* | 3.80* | 3.77* | 4.16* | 3.67 | 3.85 | 3.93 | 3.82 | 4.18 | |

Big | 2.98* | 3.53 | 3.77* | 3.51* | 4.27* | 3.00 | 3.48 | 3.89 | 3.62 | 4.47 |

Note: Numbers with * are smaller AIC values.

FF5-SSAEPD-GARCH model are smaller than those of the FF5-Normal model. Hence, we can conclude that our new model (FF5-SSAEPD-GARCH) performs better than the 5-factor model in Fama and French (2015).

In this paper, we extend the 5-factor model in Fama and French (2015) by introducing a non-normal error term and time-varying volatilities. The non-normal error assumption we used is the SSAEPD in Zhu and Zinde-Walsh (2009). And the time-varying volatilities are the GARCH model in Bollerslev (1986). For comparison, monthly US stock returns in Fama and French (2015) (1963:07 - 2013:12) are analyzed. Method of Maximum Likelihood is used. Likelihood Ratio Test (LR) is used to test the hypotheses of parameter restrictions. Kolmogorov-Smirnov test (KS) is used to check residuals. Akaike Information Criterion (AIC) is used to compare models.

Simulation results show our MatLab program for the new model is valid. And empirical results show: 1) this new model can capture the skewness, fat tails and asymmetric kurtosis in the data; 2) With GARCH-type volatilities and non-normal errors, the Fama-French 5 factors are still alive, since the estimates are all significant; and 3) our new model can fit the data much better than 5-factor model in Fama and French (2015). Our study provides an update to existing asset pricing literature and reference for investors.

Future extensions will include but not limited to the following. First, we can exam our results with different data. Second, we can compare our results with those from other models such as ARIMA model. Last but not the least, other factors can be introduced into this model.

We also want to thank participants in the 16th^{ }World Business Research Conference at San Francisco (28 - 29 July, 2016) and the seminars organized by Institute of Statistics and Econometrics, Nankai University. The support of Jiayi Zhu and Qingyu Zhu is gratefully acknowledged. The authors are responsible for all errors.

Zhou, W.T. and Li, L.L. (2016) A New Fama-French 5- Factor Model Based on SSAEPD Error and GARCH-Type Volatility. Journal of Mathematical Finance, 6, 711-727. http://dx.doi.org/10.4236/jmf.2016.65050

To test our MatLab program, we also estimate the FF5-Normal model using the program by setting_{1} = p_{2} = 2. The estimates are listed in

Size | Book-to-market quintiles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

a | t(a) | ||||||||||

Small | −0.29 | 0.11 | 0.01 | 0.12 | 0.12 | −3.28 | 1.60 | 0.14 | 2.03 | 1.95 | |

2 | −0.12 | −0.11 | 0.04 | 0.00 | −0.04 | −1.84 | −1.94 | 0.81 | 0.06 | −0.66 | |

3 | 0.02 | −0.01 | −0.06 | −0.03 | 0.05 | 0.38 | −0.20 | −0.99 | −0.40 | 0.59 | |

4 | 0.17 | −0.23 | −0.15 | 0.05 | −0.10 | 2.59 | −3.22 | −1.98 | 0.65 | −1.18 | |

Big | 0.11 | −0.10 | −0.11 | −0.15 | −0.10 | 2.49 | −1.66 | −1.55 | −2.42 | −0.99 | |

h | t(h) | ||||||||||

Small | −0.42 | −0.14 | 0.11 | 0.28 | 0.52 | −9.92 | −4.39 | 4.07 | 10.41 | 17.90 | |

2 | −0.46 | −0.01 | 0.29 | 0.42 | 0.70 | −15.41 | −0.42 | 11.67 | 16.54 | 24.60 | |

3 | −0.43 | 0.12 | 0.37 | 0.52 | 0.67 | −14.63 | 3.81 | 12.25 | 17.15 | 18.83 | |

4 | −0.46 | 0.09 | 0.38 | 0.52 | 0.80 | −15.22 | 2.63 | 11.16 | 15.78 | 20.57 | |

Big | −0.31 | 0.03 | 0.26 | 0.62 | 0.84 | −14.29 | 1.13 | 7.68 | 20.86 | 18.56 | |

r | t(r) | ||||||||||

Small | −0.56 | −0.33 | 0.01 | 0.11 | 0.11 | −13.00 | −10.32 | 0.34 | 3.90 | 3.74 | |

2 | −0.20 | 0.14 | 0.27 | 0.25 | 0.20 | −6.65 | 5.14 | 10.36 | 9.36 | 6.79 | |

3 | −0.20 | 0.22 | 0.32 | 0.28 | 0.32 | −6.77 | 6.77 | 10.30 | 8.83 | 8.58 | |

4 | −0.18 | 0.26 | 0.27 | 0.14 | 0.25 | −5.88 | 7.67 | 7.69 | 4.00 | 6.10 | |

Big | 0.13 | 0.24 | 0.08 | 0.22 | 0.01 | 5.86 | 8.45 | 2.29 | 7.31 | 0.23 | |

c | t(c) | ||||||||||

Small | −0.57 | −0.11 | 0.19 | 0.39 | 0.61 | −12.36 | −3.23 | 6.70 | 13.17 | 18.98 | |

2 | −0.58 | 0.06 | 0.31 | 0.54 | 0.71 | −17.65 | 2.21 | 11.49 | 19.50 | 22.86 | |

3 | −0.66 | 0.13 | 0.42 | 0.64 | 0.77 | −20.53 | 3.83 | 12.67 | 19.13 | 19.58 | |

4 | −0.49 | 0.31 | 0.51 | 0.60 | 0.78 | −14.82 | 8.56 | 13.50 | 16.73 | 18.19 | |

Big | −0.38 | 0.25 | 0.41 | 0.65 | 0.72 | −16.04 | 8.31 | 11.15 | 20.17 | 14.50 |

Size | Book-to-market quintiles | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

a | t(a) | ||||||||||

Small | −0.29 | 0.11 | 0.01 | 0.12 | 0.12 | −3.31 | 1.61 | 0.17 | 2.12 | 1.99 | |

2 | −0.11 | −0.10 | 0.05 | −0.00 | −0.04 | −1.73 | −1.88 | 0.95 | −0.04 | −0.64 | |

3 | 0.02 | −0.01 | −0.07 | −0.02 | 0.05 | 0.40 | −0.10 | −1.06 | −0.25 | 0.60 | |

4 | 0.18 | −0.23 | −0.13 | 0.05 | −0.09 | 2.73 | −3.29 | −1.81 | 0.73 | −1.09 | |

Big | 0.12 | −0.11 | −0.10 | −0.15 | −0.09 | 2.50 | −1.82 | −1.39 | −2.33 | −0.93 | |

h | t(h) | ||||||||||

Small | −0.43 | −0.14 | 0.10 | 0.27 | 0.52 | −10.11 | −4.38 | 3.90 | 10.12 | 17.55 | |

2 | −0.46 | −0.01 | 0.29 | 0.43 | 0.69 | −15.22 | −0.45 | 11.77 | 16.78 | 24.44 | |

3 | −0.43 | 0.12 | 0.37 | 0.52 | 0.67 | −14.70 | 3.71 | 12.28 | 17.07 | 18.75 | |

4 | −0.46 | 0.09 | 0.38 | 0.52 | 0.80 | −15.18 | 2.76 | 11.03 | 15.88 | 20.26 | |

Big | −0.31 | 0.03 | 0.26 | 0.62 | 0.85 | −14.12 | 1.09 | 7.54 | 21.05 | 18.74 | |

r | t(r) | ||||||||||

Small | −0.58 | −0.34 | 0.01 | 0.11 | 0.12 | −13.26 | −10.56 | 0.31 | 3.89 | 3.95 | |

2 | −0.21 | 0.13 | 0.27 | 0.26 | 0.21 | −6.75 | 4.89 | 10.35 | 9.86 | 7.04 | |

3 | −0.21 | 0.22 | 0.33 | 0.28 | 0.33 | −6.99 | 6.77 | 10.36 | 8.98 | 8.88 | |

4 | −0.19 | 0.27 | 0.28 | 0.14 | 0.25 | −6.06 | 7.75 | 7.99 | 4.16 | 6.14 | |

Big | 0.13 | 0.25 | 0.07 | 0.23 | 0.02 | 5.64 | 8.79 | 2.07 | 7.62 | 0.49 | |

c | t(c) | ||||||||||

Small | −0.57 | −0.12 | 0.19 | 0.39 | 0.62 | −12.27 | −3.46 | 6.59 | 13.15 | 19.10 | |

2 | −0.59 | 0.06 | 0.31 | 0.55 | 0.72 | −17.76 | 1.94 | 11.27 | 19.39 | 22.92 | |

3 | −0.67 | 0.13 | 0.42 | 0.64 | 0.78 | −20.59 | 3.64 | 12.52 | 18.97 | 19.62 | |

4 | −0.51 | 0.31 | 0.51 | 0.60 | 0.79 | −15.11 | 8.33 | 13.35 | 16.41 | 18.03 | |

Big | −0.39 | 0.26 | 0.41 | 0.66 | 0.73 | −16.08 | 8.38 | 10.80 | 19.88 | 14.54 |

Note: This table is quoted from the results in