^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

^{4}

^{*}

In this paper, a new model is proposed to empirically test the Capital Asset Pricing Theory. This model is based on the EGARCH-type volatilities in Nelson (1991) and the non-Normal errors of SSAEPD in Zhu and Zinde-Walsh (2009). Is the CAPM theory in Sharpe (1964), Lintner (1965) and Mossin (1966) still alive? Returns of Fama-French 25 stock portfolios (1926-2011) are analyzed. The Maximum Likelihood Estimation Method is used. Likelihood Ratio test (LR) and Kolmogorov-Smirnov test (KS) are used to do model diagnostics. Akaike Information Criterion (AIC) is used for model comparison. Simulation results show the MatLab program is valid. Empirical results show with non-Normal errors and the EGARCH-type volatilities, the CAPM theory is not alive. This new model can capture the skewness, fat-tailness, asymmetric effects and volatility persistence in the data. This new model has better in-sample fit than others. Portfolios with smaller size have larger Beta value.

Capital Asset Pricing Model (CAPM) is first established by Sharpe (1964), Lintner (1965) and Mossin (1966) [

That means, excess return of portfolio ^{1}. Since then, many theoretical and empirical researches about this model have been done.

However, some limitations of the CAPM theory are pointed out by some researchers such as Lucas (1978) [

To empirically test the CAPM theory, it is traditional to assume Normal error terms. However, Normal distri- bution can not capture the skewness, fat-tailness and asymmetric kurtosis of financial data. Thus, a plenty of researches have been done in order to extend the Normal. For instance, Subbotin (1923) [

Based on the SSAEPD in Zhu and Zinde-Walsh (2009) [

1) With non-Normal error terms such as SSAEPD in Zhu and Zinde-Walsh (2009), and EGARCH-type volatilities in Nelson (1991), is the CAPM theory of Sharpe (1964), Lintner (1965) and Mossin (1966) still alive?

2) Can this new model beat the CAPM-SSAEPD model of Zhuo (2013) [

3) Can we find any new patterns for Fama-French 25 portfolios?

^{1}This equation is from page 301 of Bodie, Kane and Marcus (2006). For more reference about CAPM theory, please refer to Investments written by Bodie, Kane and Marcus (2006). To check the CAPM theory, researchers usually use following CAPM-Normal model to test the significance of parameters:

If CAPM theory is alive, then the coefficient of

To answer these questions, simulation is done first. Then, the empirical data of Fama-French 25 stock port- folios are analyzed. Sample period is from January 1926 to December 2011. Method of Maximum Likelihood Estimation (MLE) is used to estimate parameters. Likelihood Ratio test (LR) is used for testing the significance of parameters. The Kolmogorov-Smirnov test (KS) is used to check the residuals. Akaike Information Criterion (AIC) is used for model comparison.

Simulation results show our MatLab program is valid. Empirical results show with non-Normal error terms and EGARCH-type volatilities, the CAPM theory of Sharpe (1964), Lintner (1965) and Mossin (1966) can not explain the US stock market. The estimates of this new model can capture fat-tailness, asymmetric effects, and volatility persistence in the data. The model with EGARCH-type volatilities and SSAEPD error terms has better in-sample fit than others by Akaike Information Criterion (AIC). A portfolio with a smaller Size may have a larger Beta value, which means that they can be more sensitive to the excess return over market.

Author (Year) | Research Purpose | Model | Method | Data | ||
---|---|---|---|---|---|---|

Country | Variables | Frequency & Period | ||||

Sharpe (1964) | CAPM | - | ||||

Merton (1973) | ICAPM | - | ||||

Black (1976) | Wealth CAPM | - | ||||

Lucas (1978) | CCAPM | - | ||||

Bredeen (1979) | ICAPM | - | ||||

Fama et al. (1993) | FF | |||||

Chen (2003) | Consumption beta | CAPM, CCAPM | OLS | Taiwan | Price indices, dividend payments, | M1991:7-2000:3 |

Market beta | Risk-free rate, CPI | |||||

Fletcher (2004) | Predictability | 3-4 m. CAPM | GMM | UK | Excess returns, SMB, HML, FTA, LAB | M1975:1-2001:12 |

David T. (2005) | International asset pricing | D-I-CAPM,VAR | GMM | G7 | Equity returns, exchange rate, | M1978:7-1998:4 |

US inflation, MSCI, dividend yield | ||||||

G7 average forward premiums | ||||||

Lee (2007) | Supply effect | DCAPM | SUR | US | Price, earnings and dividend per share | Q1981:1-2001:4 |

Grauer (2009) | Wide range of betas | CAPM, FF | GLS | Standard | Excess returns, risk premiums, SMB, HML | M1963:7-2005:12 |

Darrat et al. (2011) | Model comparison | CCAPM, | GMM | 17 MSCI | Consumption, CPI, population | Q1970:2-2007:4 |

Surplus CAPM | Countries | Returns on MSCI index, GDP | ||||

Chen et al. (2000) | Estimate of beta | CAPM, ANOVA | OLS | China | Stock price, SSE index, 3-m deposit rate | DWM1994:1:4-1998:12:31 |

Ma (2001) | Robustness exam | CAPM | OLS | China | Shenzhen component index | W1997:9:30-2000:10:29 |

3-y bond rate, size, PE | ||||||

Sun et al. (2002) | Herd behavior | CAPM | GLS | China | SSE index, returns on stock | D1992:1:2-2000:12:29 |

Zhao (2011) | Robustness exam | CAPM | Dual reg. | China | SSE index, 3-month deposit rate, stock price | W2006:1:1-2008:12:31 |

Jin (2011) | Model comparison | CAPM-AEPD | MLE | China, US | Hushen 300 index, 3-m deposit rate | D2006:1:4-2010:12:31 |

DJI, 10-y Treasure bill rate | D2006:1:3-2010:12:31 | |||||

Dai et al. (2011) | Predictability | 2-3-4 m. CAPM | OLS, WNN | China | SHIBOR rate, stock price, SSE index | D2007:1:4-2011:2:1 |

Li et al. (2012) | Robustness exam | CAPM-AEPD | MLE | China | CAC40 index, stock price | D2006-2010 |

Zhuo (2013) | CAPM-SSAEPD | MLE | US | SP500 | D2002-2011 | |

Yang (2014) | CAPM-SSAEPD | MLE | US | Fama and French (1993) 25 portfolios | D1926-2011 |

Note: This table is a revision from Jin (2011).

The organization of this paper is as follows. The model and methodology are discussed in section 2. Simulation analysis is in section 3. Data and empirical results are reported in section 4. Section 5 is the conclusions and future extensions.

Based on the SSAEPD in Zhu and Zinde-Walsh (2009) and the EGARCH-type volatilities in Nelson (1991), in this paper, a new CAPM model is suggested (i.e., CAPM-SSAEPD-EGARCH). The CAPM-SSAEPD- EGARCH (m,s) model has following forms:

Authors | Distributions and their applications |
---|---|

De Moivre (1738) | Normal distribution |

Gauss (1809) | Normal applied in astronomy |

Subbotin (1923) | EPD |

Aitchison J. and Brown J.A.C. (1957) | Lognormal distribution |

Leone F.C., Nottinghan R.B., Nelson L.S. (1961) | Folded normal distribution |

William H. Rogers and John Tukey (1972) | Slash distribution |

Azzalini (1985, 1986) | Skew-normal distribution |

Azzalini (1986) | SEPD |

Zolotarev V.M. (1986) | Stable distribution |

Fernandez et al. (1995)^{2} | Modified SEPD |

Mudholkar and Hutson (2000) | Epsilon-skew-normal family (ESN) |

Swamee P.K. (2002) | Near lognormal distribution |

Ayebo and Kozubowski (2004) | SEPD in finance |

DiCiccio and Monti (2004) | Properties of MLE of the SEPD |

Zhu and Zinde-Walsh (2009) | AEPD |

Notes: EPD = Exponential Power Distribution; SEPD = Skewed Exponential Power Distribution; AEPD = Asymmetric Exponential Power Distri- bution. This table is a revision from Jin (2011).

where

stock portfolio.

are the coefficient parameters in the regression model.

Zinde-Walsh (2009).

distributions.

If

^{3} of Zhuo(2013). If

which is usually used to test the CAPM theory. Different from the CAPM-SSAEPD-GARCH model of Lin (2013), EGARCH-type volatilities of Nelson (1991) is used to consider the leverage effects. If

In this special case, the GARCH parameter

The probability density function (PDF) of the SSAEPD^{4}, proposed by Zhu and Zinde-Walsh (2009), is

where

^{4}If

And

In this paper, we estimate this new model with Maximum Likelihood Estimation (MLE). For simplicity, we define following notations

where

In this section, we simulate the data and derive the simulation results for the CAPM-SSAEPD-EGARCH (1,1).

The true parameters chosen are

1) Given^{5} series

2) Set initial value

3) Get

4) Generate random number series

get

After we have the simulated data

new model. The simulation results are reported in

T | 0.3 | 0.5 | 0.5 | 2 | 2 | 0.3 | 0.5 | 0.4 | 0.6 |

E | 0.3155 | 0.4883 | 0.5002 | 2.0009 | 2.0021 | 0.3034 | 0.4873 | 0.4061 | 0.6387 |

R | 5.17% | 2.34% | 0.04% | 0.05% | 0.11% | 1.13% | 2.54% | 1.53% | 6.45% |

T | 0.3 | 0.5 | 0.5 | 2 | 2 | 0.4 | 0.6 | 0.3 | 0.4 |

E | 0.2737 | 0.5347 | 0.5000 | 2.0000 | 2.0000 | 0.416 | 0.5861 | 0.304 | 0.3974 |

R | 8.77% | 6.94% | 0.00% | 0.00% | 0.00% | 4.00% | 2.32% | 1.33% | 0.65% |

T | 0.3 | 0.5 | 0.5 | 2 | 2 | 0.4 | 0.5 | 0.5 | 0.7 |

E | 0.2877 | 0.5113 | 0.4999 | 1.9996 | 2.0000 | 0.3964 | 0.5063 | 0.5126 | 0.6871 |

R | 4.10% | 2.26% | 0.02% | 0.02% | 0.00% | 0.90% | 1.26% | 2.52% | 1.84% |

T | 0.3 | 0.5 | 0.5 | 2 | 2 | 0.4 | 0.4 | 0.3 | 0.7 |

E | 0.3061 | 0.4932 | 0.5000 | 2.0000 | 2.0000 | 0.4167 | 0.3743 | 0.2951 | 0.741 |

R | 2.03% | 1.36% | 0.00% | 0.00% | 0.00% | 4.18% | 6.43% | 1.63% | 5.86% |

T | 0.3 | 0.5 | 0.5 | 2 | 1.5 | 0.3 | 0.4 | 0.3 | 0.7 |

E | 0.2807 | 0.5557 | 0.5000 | 2.0004 | 1.5004 | 0.3199 | 0.4132 | 0.2751 | 0.6373 |

R | 6.43% | 11.14% | 0.00% | 0.02% | 0.03% | 6.63% | 3.30% | 8.30% | 8.96% |

T | 0.3 | 0.5 | 0.5 | 1.5 | 2 | 0.3 | 0.4 | 0.3 | 0.6 |

E | 0.3048 | 0.4751 | 0.5 | 1.5008 | 2.0018 | 0.3307 | 0.3759 | 0.2766 | 0.6256 |

R | 1.60% | 4.98% | 0.00% | 0.05% | 0.09% | 10.23% | 6.03% | 7.80% | 4.27% |

T | 0.3 | 0.5 | 0.3 | 2 | 2 | 0.4 | 0.5 | 0.3 | 0.6 |

E | 0.2983 | 0.4931 | 0.3000 | 2.0048 | 1.9963 | 0.4107 | 0.4734 | 0.2925 | 0.6037 |

R | 0.57% | 1.38% | 0.00% | 0.24% | 0.19% | 2.68% | 5.32% | 2.50% | 0.62% |

T | 0.3 | 0.5 | 0.5 | 2 | 2 | 0.4 | 0.6 | 0.4 | 0.5 |

E | 0.2557 | 0.5448 | 0.5000 | 2.0000 | 2.0000 | 0.4062 | 0.5856 | 0.3941 | 0.5097 |

R | 14.77% | 8.96% | 0.00% | 0.00% | 0.00% | 1.55% | 2.40% | 1.48% | 1.94% |

T | 0.3 | 0.5 | 0.5 | 2 | 2 | 1 | 0.4 | 0.5 | 0.7 |

E | 0.304 | 0.4949 | 0.5000 | 1.9999 | 1.9999 | 1.0068 | 0.3923 | 0.4939 | 0.7093 |

R | 1.33% | 1.02% | 0.00% | 0.00% | 0.00% | 0.68% | 1.93% | 1.22% | 1.33% |

Notes: T means the true parameters. E means the estimated parameters. R means the relative errors.

The 25 portfolio returns used in Fama and French(1993) are analyzed. Data are downloaded from the French’s Data Library^{6}. Sample period is from January 1926 to December 2011. Caculated by Eviews, ^{7}. We can see that 23 out of 25 portfolios have positive values for the skewness, and all values of the kurtosis are more than

· Estimates and Significant Tests for Parameter Restrictions

^{6}Thanks Din Yin who provides the well organized Excel files. Thanks Professor French for kindly providing the risk free rate by e-mail.

^{7}Excess returns are got by portfolio returns minus the risk free rate.

^{8}Since all values of

^{9}Since most estimates of

The estimates for the new model are listed in ^{8}. Parameters in non-Normal error such as SSAEPD do not capture the skewness and the asymmetric tails^{9}. In contrast, EGARCH-type volatilities could capture the asymmetric effects in the data. Hence, one can conclude that the EGARCH-type volatilities is more powerful to capture the asymmetric effect than non-Normal error such as SSAEPD.

For comparison, we also estimate the CAPM-EGARCH (1,1) model. The results are listed in

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

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

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

Small | 0.73 | 1.09 | 1.30 | 1.45 | 1.66 | 0.55 | 0.95 | 1.25 | 1.46 | 1.49 | |

2 | 0.87 | 1.23 | 1.32 | 1.36 | 1.48 | 1.18 | 1.49 | 1.56 | 1.50 | 1.65 | |

3 | 0.96 | 1.16 | 1.26 | 1.28 | 1.42 | 1.38 | 1.36 | 1.56 | 1.47 | 1.35 | |

4 | 0.97 | 1.03 | 1.12 | 1.23 | 1.32 | 1.24 | 1.36 | 1.54 | 1.53 | 1.54 | |

Big | 0.88 | 0.89 | 0.94 | 0.98 | 0.03 | 1.07 | 1.05 | 1.15 | 1.08 | 1.22 | |

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

Small | 12.23 | 10.58 | 9.21 | 8.64 | 9.57 | 2.71 | 4.40 | 1.77 | 2.73 | 3.07 | |

2 | 7.98 | 7.88 | 7.34 | 7.61 | 8.75 | 0.35 | 1.87 | 2.06 | 1.68 | 1.75 | |

3 | 7.64 | 6.61 | 6.75 | 6.83 | 8.63 | 1.01 | 0.27 | 1.01 | 1.16 | 1.88 | |

4 | 6.24 | 6.30 | 6.41 | 7.02 | 8.98 | 0.21 | 0.82 | 0.94 | 1.79 | 2.02 | |

Big | 5.48 | 5.24 | 5.75 | 6.90 | 13.23 | -0.02 | -0.09 | 0.81 | 1.84 | 4.85 | |

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

Small | 30.86 | 60.01 | 18.48 | 33.33 | 33.26 | 0 | 0 | 0 | 0 | 0 | |

2 | 7.90 | 24.01 | 24.94 | 20.94 | 20.43 | 0 | 0 | 0 | 0 | 0 | |

3 | 13.40 | 9.46 | 17.17 | 15.91 | 22.39 | 0 | 0 | 0 | 0 | 0 | |

4 | 6.45 | 15.00 | 17.40 | 23.24 | 24.78 | 0 | 0 | 0 | 0 | 0 | |

Big | 8.26 | 8.05 | 17.24 | 26.37 | 39.84 | 0 | 0 | 0 | 0 | 0 |

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

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

Small | −0.69^{*} | −0.28^{*} | −0.05 | −0.09 | 0.17^{*} | 1.36^{*} | 1.26^{*} | 1.13^{*} | 1.05^{*} | 1.06^{*} |

2 | −0.22^{*} | 0.02^{*} | 0.31^{*} | 0.15 | 0.20^{*} | 1.25^{*} | 1.20^{*} | 1.09^{*} | 1.19^{*} | 1.25^{*} |

3 | 0.05 | 0.08 | 0.14 | 0.27^{*} | 0.18 | 1.20^{*} | 1.13^{*} | 1.09^{*} | 1.07^{*} | 1.17^{*} |

4 | −0.02 | −0.01 | 0.07^{*} | 0.12^{*} | 0.15^{*} | 1.10^{*} | 1.09^{*} | 1.06^{*} | 1.04^{*} | 1.11^{*} |

Big | −0.07 | 0.01 | 0.07 | 0.06^{*} | 0.17^{*} | 0.98^{*} | 0.93^{*} | 0.91^{*} | 0.95^{*} | 1.07^{*} |

Small | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 1.41 | 1.50 | 1.50 | 1.00 | 1.50 |

2 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 1.50 | 1.20 | 1.90 | 1.20 | 1.50 |

3 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 1.49 | 1.50 | 1.20 | 1.50 | 1.50 |

4 | 0.50 | 0.50 | 0.49 | 0.50 | 0.50 | 1.50 | 1.50 | 1.47 | 1.50 | 1.50 |

Big | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 1.10 | 1.50 | 1.50 | 1.50 | 1.45 |

Small | 1.21 | 1.50 | 1.50 | 1.20 | 1.50 | 0.06^{*} | 0.05^{*} | 0.01 | 0.16^{*} | 0.04^{*} |

2 | 1.50 | 1.20 | 1.30 | 1.20 | 1.50 | 0.52^{*} | 0.04^{*} | 0.23^{*} | 0.25^{*} | 0.23^{*} |

3 | 1.50 | 1.50 | 1.80 | 1.50 | 1.51 | 0.14^{*} | 0.02^{*} | 0.13^{*} | 0.05^{*} | 0.22^{*} |

4 | 1.50 | 1.50 | 1.43 | 1.50 | 1.20 | 0.08^{*} | 0.04 | 0.04^{*} | 0.02^{*} | 0.02 |

Big | 1.00 | 1.50 | 1.50 | 1.20 | 1.19 | 0.06^{*} | 0.05^{*} | 0.07^{*} | 0.09^{*} | 0.13^{*} |

Small | 0.98^{*} | 0.99^{*} | 1.00^{*} | 0.96^{*} | 1.00^{*} | −0.07 | −0.05^{*} | −0.05^{*} | −0.09^{*} | −0.10^{*} |

2 | 0.82^{*} | 0.99^{*} | 0.88^{*} | 0.91^{*} | 0.93^{*} | −0.09 | −0.04^{*} | −0.04 | −0.05 | −0.02 |

3 | 0.93^{*} | 0.99^{*} | 0.93^{*} | 0.98^{*} | 0.91^{*} | −0.02 | −0.04^{*} | −0.03 | −0.04 | −0.05 |

4 | 0.94^{*} | 0.97^{*} | 0.97^{*} | 0.99^{*} | 0.99^{*} | 0.06^{*} | −0.04 | −0.07^{*} | −0.04^{*} | −0.07^{*} |

Big | 0.94^{*} | 0.95^{*} | 0.94^{*} | 0.96^{*} | 0.95^{*} | −0.01 | −0.01 | −0.05^{*} | −0.07^{*} | 0.09 |

Small | 0.25^{*} | 0.33^{*} | 0.17^{*} | 0.31^{*} | 0.22^{*} | |||||

2 | 0.48^{*} | 0.23^{*} | 0.50^{*} | 0.41^{*} | 0.41^{*} | |||||

3 | 0.43^{*} | 0.11^{*} | 0.28^{*} | 0.23^{*} | 0.35^{*} | |||||

4 | 0.29^{*} | 0.26^{*} | 0.27^{*} | 0.22^{*} | 0.27^{*} | |||||

Big | 0.22^{*} | 0.25^{*} | 0.28^{*} | 0.27^{*} | 0.37^{*} |

Note: ^{*}means the parameter is statistically significant under 5% significant level.

^{10}In

^{11}In

^{12}Likelihood Ratio test (LR) is used. The P-values of the joint significance test for all the 25 portfolios are close to 0, which means the coefficients of

^{13}The null hypothesis is

assumptions1^{0}. However, the values of asymmetric parameter ^{11}.

Joint significance tests show both regression parameters are statistically significant (see Panel A of ^{12}. Individual significance tests show all coefficient ^{13}. And most of them concentrate in higher Book-to-market quintiles

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

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

Small | −0.62 | −0.28 | −0.06 | −0.09 | 0.15 | 1.37 | 1.26 | 1.13 | 1.05 | 1.05 |

2 | −0.22 | 0.00 | 0.32 | 0.16 | 0.19 | 1.26 | 1.20 | 1.08 | 1.19 | 1.26 |

3 | 0.06 | 0.08 | 0.15 | 0.26 | 0.19 | 1.19 | 1.13 | 1.09 | 1.07 | 1.17 |

4 | −0.01 | −0.01 | 0.04 | 0.12 | 0.16 | 1.10 | 1.09 | 1.10 | 1.05 | 1.12 |

Big | −0.05 | 0.00 | 0.09 | 0.05 | 0.24 | 0.98 | 0.93 | 0.91 | 0.95 | 1.20 |

Small | 0.03 | 0.05 | 0.00 | 0.17 | 0.03 | 0.99 | 0.99 | 1.00 | 0.96 | 1.00 |

2 | 0.51 | 0.02 | 0.23 | 0.24 | 0.24 | 0.82 | 0.99 | 0.89 | 0.91 | 0.92 |

3 | 0.19 | 0.02 | 0.14 | 0.05 | 0.26 | 0.92 | 0.99 | 0.93 | 0.98 | 0.92 |

4 | 0.08 | 0.04 | 0.14 | 0.02 | 0.03 | 0.95 | 0.97 | 0.91 | 0.99 | 1.00 |

Big | 0.06 | 0.05 | 0.08 | 0.09 | 0.08 | 0.93 | 0.94 | 0.95 | 0.96 | 0.99 |

Small | −0.01 | −0.05 | −0.05 | −0.10 | −0.10 | −0.01 | 0.33 | 0.17 | 0.31 | 0.20 |

2 | −0.10 | −0.03 | −0.07 | −0.06 | 0.00 | 0.48 | 0.22 | 0.50 | 0.42 | 0.41 |

3 | 0.00 | −0.04 | 0.02 | −0.03 | −0.05 | 0.39 | 0.11 | 0.28 | 0.23 | 0.40 |

4 | 0.06 | −0.04 | −0.04 | −0.05 | −0.08 | 0.29 | 0.26 | 0.35 | 0.22 | 0.27 |

Big | 0.00 | 0.00 | −0.06 | −0.07 | 0.09 | 0.22 | 0.23 | 0.28 | 0.26 | 0.43 |

Notes: _{1} = P_{2} = 2.

CAPM-SSAEPD-EGARCH vs. CAPM-EGARCH | |||||
---|---|---|---|---|---|

= | 15 | 17 | 20 | 9 | 17 |

> | 10 | 3 | 5 | 7 | 4 |

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

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

Panel A. | Panel B. | ||||||||||

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^{*} | |

Panel C. | |||||||||||

Small | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} | ||||||

2 | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} | ||||||

3 | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} | ||||||

4 | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} | ||||||

Big | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} |

Note: ^{*}means the parameter is statistically significant under 5% significant level.

or smaller Size quintiles. In conclusion, with non-Normal error distribution and EGARCH-type volatilities, the CAPM theory is not alive since they can earn Alpha returns.

12 out of 25 portfolios have significant parameter

· Residual Checks

Test results for residuals (see ^{14} and the CAPM-SSAEPD-EGARCH model is adequate for data used in Fama and French(1993). However, the CAPM-EGARCH model is not adequate for the data since most of its residuals do not follow the Normal distribution under 5% significance level^{15}. Also, non-Normality^{16} is documented in Panel B of

^{14}The residuals for models are checked with Kolmogorov-Smirnov test. The null hypothesis of KS test is the residuals do follow some distribution. The P-value of KS test is in

From the test results shown in

^{15}Then, we test the residual of CAPM-EGARCH, and the null hypothesis

Based on the test results shown in

^{16}We test the SSAEPD and EGARCH parameters respectively with Likelihood Ratio test. In Panel A

Same conclusions are also can be drawn from the PDFs of the residuals (i.e. method of “eye-rolling”). Taking one portfolio (Size quintile 2 and BE/ME quintile Low) as an example, we plot the residuals of CAPM- SSAEPD-EGARCH and CAPM-EGARCH in Matlab. They are shown in

The Beta value

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

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

CAPM-SSAEPD-EGARCH | CAPM-EGARCH | |||||||||

Small | 0.58 | 0.20 | 0.13 | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} | 0^{*} |

2 | 0.35 | 0.50 | 0.72 | 0.20 | 0.06 | 0^{*} | 0.17 | 0^{*} | 0^{*} | 0^{*} |

3 | 0.22 | 0.53 | 0^{*} | 0.75 | 0.48 | 0^{*} | 0.12 | 0^{*} | 0.28 | 0^{*} |

4 | 0.98 | 0.55 | 0.24 | 0.59 | 0.25 | 0.33 | 0.15 | 0^{*} | 0.36 | 0^{*} |

Big | 0^{*} | 0.77 | 0.92 | 0.38 | 0.22 | 0.33 | 0.11 | 0.20 | 0^{*} | 0^{*} |

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

portfolio gets bigger. Hence, one can draw a conclusion that a portfolio with a smaller Size may have a larger

Then we compare the Beta values with those results in model CAPM-SSAEPD (see Appendix 1). From

The new model is compared with others by AIC criterion (see

Based on the SSAEPD in Zhu and Zinde-Walsh (2009) and the EGARCH-type volatilities in Nelson (1991), a

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

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

CAPM-SSAEPD-EGARCH | CAPM-SSAEPD | |||||||||

Small | 1.36^{#} | 1.26^{#} | 1.13^{#} | 1.05^{#} | 1.06^{#} | 1.43 | 1.27 | 1.25 | 1.16 | 1.19 |

2 | 1.25^{#} | 1.20^{#} | 1.09^{#} | 1.19^{#} | 1.25 | 1.26 | 1.21 | 1.12 | 1.13 | 1.25 |

3 | 1.20^{#} | 1.13 | 1.09^{#} | 1.07^{#} | 1.17^{#} | 1.23 | 1.13 | 1.12 | 1.09 | 1.20 |

4 | 1.10 | 1.09 | 1.06 | 1.04^{#} | 1.11^{#} | 1.08 | 1.05 | 1.06 | 1.08 | 1.25 |

Big | 0.98 | 0.93 | 0.91^{#} | 0.95^{#} | 1.07 | 0.97 | 0.93 | 0.93 | 0.98 | 1.07 |

Note: ^{#} are marked with

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

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

CAPM-SSAEPD-EGARCH | |||||

Small | 6.33^{#} | 5.86^{#} | 5.63^{#} | 5.50^{#} | 5.75^{#} |

2 | 5.46^{#} | 5.05^{#} | 4.92^{#} | 5.03^{#} | 5.51^{#} |

3 | 4.91^{#} | 4.50^{#} | 4.54^{#} | 4.65^{#} | 5.35^{#} |

4 | 4.16^{#} | 3.91^{#} | 4.11^{#} | 4.58^{#} | 5.30^{#} |

Big | 3.65 | 3.50^{#} | 4.02^{#} | 4.46^{#} | 5.55^{#} |

CAPM-EGARCH | |||||

Small | 6.45 | 5.93 | 5.67 | 5.61 | 5.87 |

2 | 5.51 | 5.13 | 4.97 | 5.09 | 5.59 |

3 | 4.97 | 4.53 | 4.55 | 4.67 | 5.43 |

4 | 4.19 | 3.93 | 4.17 | 4.62 | 5.36 |

Big | 3.64^{#} | 3.53 | 4.05 | 4.53 | 5.74 |

CAPM-SSAEPD | |||||

Small | 6.52 | 6.04 | 5.85 | 5.58 | 5.77 |

2 | 5.57 | 5.14 | 5.00 | 5.11 | 5.61 |

3 | 4.98 | 4.55 | 4.56 | 4.81 | 5.47 |

4 | 4.32 | 4.04 | 4.28 | 4.70 | 5.52 |

Big | 3.66 | 3.60 | 4.20 | 4.73 | 5.74 |

Note: ^{# }marks the smallest AIC values.

new CAPM model is suggested in this paper (denoted as CAPM-SSAEPD-EGARCH). And this new model is used to empirically test the CAPM theory with 25 stock portfolios of Fama and French (1993). The sample period is from January 1926 to December 2011. Maximum Likelihood Estimation method is used. Likelihood Ratio test (LR) is used for testing the significance of the coefficients. The Kolmogorov-Smirnov test (KS) is used to check the residuals. Model is compared by the value of Akaike Information Criterion (AIC).

Our empirical results shows 1) With non-Normal error terms and EGARCH-type volatilities, the CAPM theory of Sharpe (1964), Lintner (1965) and Mossin (1966) can not explain the US stock market well. They can earn Alpha returns; 2) The estimates of SSAEPD-EGARCH parameters can capture fat-tailness, asymmetric effects and volatility persistence in the data. The EGARCH-type volatilities is more powerful to capture asymmetric effects than the parameters in SSAEPD; 3) The new model has better in-sample fit than others by Akaike Information Criterion (AIC); 4) A portfolio with a smaller Size value may have a larger Beta value, which means that they can be more sensitive to the market.

Future extensions will include but not be limited to the followings. First, different data can be analyzed. Second, the new model can be compared with others such as ARIMA, ARCH and SETAR. Third, the EGARCH-type volatilities and SSAEPD errors can be used to extend Fama-French 3-factor model. Last, the new model can also be applied to risk management such as calculating Value-at-Risk.

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

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

Small | −0.50 | −0.04 | 0.21 | 0.41 | 0.60 | 1.43 | 1.27 | 1.25 | 1.16 | 1.19 |

2 | −0.22 | 0.16 | 0.32 | 0.34 | 0.39 | 1.26 | 1.21 | 1.12 | 1.13 | 1.25 |

3 | −0.11 | 0.16 | 0.26 | 0.29 | 0.36 | 1.23 | 1.13 | 1.12 | 1.09 | 1.20 |

4 | −0.01 | 0.07 | 0.16 | 0.25 | 0.24 | 1.08 | 1.05 | 1.06 | 1.08 | 1.25 |

Big | −0.03 | 0.01 | 0.06 | 0.06 | −0.41 | 0.97 | 0.93 | 0.93 | 0.98 | 1.07 |

Small | 7.51 | 5.71 | 5.06 | 4.55 | 5.14 | 0.50 | 0.66 | 0.63 | 0.69 | 0.59 |

2 | 4.15 | 3.46 | 3.21 | 3.46 | 4.41 | 0.60 | 0.62 | 0.68 | 0.66 | 0.61 |

3 | 3.11 | 2.42 | 2.56 | 2.86 | 4.14 | 0.72 | 0.63 | 0.68 | 0.52 | 0.60 |

4 | 2.22 | 1.95 | 2.23 | 2.78 | 4.24 | 0.50 | 0.54 | 0.74 | 0.66 | 0.54 |

Big | 1.53 | 1.51 | 2.12 | 2.97 | 6.72 | 0.55 | 0.44 | 0.52 | 0.49 | 0.17 |

Small | 0.85 | 1.22 | 1.27 | 1.56 | 1.29 | 0.71 | 0.66 | 0.72 | 0.65 | 0.69 |

2 | 1.31 | 1.36 | 1.57 | 1.42 | 1.33 | 0.89 | 0.78 | 0.74 | 0.72 | 0.77 |

3 | 1.55 | 1.58 | 1.52 | 1.18 | 1.34 | 0.74 | 1.00 | 0.76 | 0.95 | 0.77 |

4 | 1.07 | 1.20 | 1.68 | 1.56 | 1.13 | 1.03 | 0.92 | 0.68 | 0.75 | 0.81 |

Big | 1.46 | 1.09 | 1.05 | 0.83 | 0.39 | 1.23 | 1.32 | 0.96 | 0.77 | 1.31 |

We also want to thank participants in the conference organized by Risk Management Institute, National University of Singapore (16-17 July, 2010), the Singapore Economic Review Conference organized by Nanyang Technological University (4-6 August, 2011), the International Conference on Applied Business & Economics at Manhattan, NY, U.S.A. (2-4 October 2013), The 8th International Conference on Asian Financial Markets & Economic Development at Nagasaki University (7-8 December 2013), Japan, the World Finance & Banking Symposium at Beijin, P.R.China (16-17 December 2013), the China Finance Reiview International Conference organized by Shanghai Jiaotong University (26-27 July., 2014), and the Canadian International Conference of Social Science and Education (CISSE, 10th & 11th March, 2014) organized by Ryerson University, Toronto, Ontario, Canada. The support of Xuefeng Li, Pin You, Mengyang Lin, Yimeng Hao and Yanjia Yang is gratefully acknowledged. The authors are responsible for all errors.

A new way to empirically test the CAPM theory using SSAEPD errors is suggestedy b Zhuo (2013) as follows:

where

The estimation results of CAPM-SSAEPD based on 25 portfolio returns used in Fama and French (1993) are listed in