Fama and French (1993) add two more factors such as Size factor and Book-to-market factor into the CAPM model and create a 3-factor model, which is capable to explain the stock returns better than the CAPM. Carhart (1997) finds the momentum factor has great effect on stock returns.

After Carhart (1997), many researches about Carhart 4-factor model have been done. And these researches can be divided into two groups. One group apply this model to different countries and show that this model has powerful explanation (see Panel A of Table 1). For example, this model can explain the stock markets well for Europe in Otten and Bams (2002) [3] , China in Guan (2011) [4] , Netherlands in Lopez (2014) [5] .

Others extend this model by finding new factors (see Panel B of Table 1). Fama and French (2012) [6] introduce local and global factors into model. Connor, Hagmann and Linton (2012) [7] find an own-volatility factor. Chai, Fall and Garghori (2013) [8] add Iliquidity factor. Garyn-Tal and Lauterbach (2015) [9] suggest U.S. or global factors to the local model and create a hybrid model. Bradrania, Peat and Satchell (2015) [10] consider Iliquidity factor and Idiosyncratic

Notes: “-” means that no information is available in this paper. Mkt = Market Premium. SMB = Small Size Minus Big Size. HML = High Book-to-market Minus Low Book-to-market. WML = Past Winner portfolios Minus Past Loser portfolios. FF = Fama Frech 3-factor model. C = Carhart 4 factor model. FIML = Information Maximum Likelihood. Four Regions= North America, Europe, Japan, and Asia Pacific. IML = Illiquid portfolios Minus Liquid portfolios. Hybrid C is adding U.S. or global factors to the local model. IV = Idiosyncratic Volatility.

Notes: “-” means that no information is available in this paper. VaR = Value-at Risk. ES = Expected Shortfall. CMMI = Capability Maturity Model Integration. ROA = Return on Assets.

Volatility (IV) in value-weighted portfolios.

Our research falls into the 2nd group and tries to extend the 4-factor model in Carhart (1997). But different from previous researches, instead of introducing different factors, we use a non-normal error of SSAEPD proposed by Zhu and Zinde-Walsh (2009) [11] , and the EGARCH-type volatility of Nelson (1991) [12] . We denote our new model as C-SSAEPD-EGARCH. SSAEPD is capable to show the skewness, fat tails and asymmetric kurtosis of data. Based on the new Carhart 4-factor model, we try to test following hypotheses:

1) With EGARCH-type volatilities in Nelson (1991) and non-normal errors of SSAEPD in Zhu and Zinde-Walsh (2009), are the Carhart 4 factors still alive in the sector of Software & Computer Services?

2) Can this new 4-factor model beat that of Carhart (1997)?

To answer these questions, we run simulation to test the MatLab program used in this paper. Then, the industry of the software & computer services in US, UK and China are analyzed¹. Data are downloaded from the Investing.com, and the sample period is from Nov. 1st, 2012 to Sept. 30th, 2015. Method of Maximum Likelihood Estimation is used to estimate the parameters. Likelihood Ratio test (LR) and Kolmogorov-Smirnov test (KS) are used for model diagnostics. Akaike Information Criterion (AIC) is used for model comparison.

We find out the Carhart 4 factors are still alive! The EGARCH-type volatility can capture the excess kurtosis. The new model fits the data well and has better in-sample fit than Fama-French (1993)’s 3-factor model and Carhart (1997)’s 4-factor model in most cases. The industry of software & computer services in US, UK and China all cannot earn extra Alpha returns since the constant term in this new model is not statistically significant. This industry in US is similar to the market because the Beta coefficient ( β 1 ) is close to 1.

The organization of this paper is as follows. The model and methodology are discussed in section 2. Empirical results and the model comparisons will be presented in section 3. Section 4 is the conclusions and future extensions.

In this paper, sector of the software and computer services is studied. A new Carhart 4-factor model (denoted as C-SSAEPD-EGARCH) is empirically tested using data in US, UK and China. This new model uses the non-normal error term of SSAEPD of Zhu and Zinde-Walsh (2009) and EGARCH type volatility of Nelson (1991) to extend the 4 factor model of Carhart (1997). Software sector indices from Investing.com are analyzed. Sample period is Nov. 1st, 2012 to Sept. 30th, 2015. Likelihood Ratio test (LR) is used for parameter restriction test, Kolmogorov-Smirnov test (KS) for residual check and AIC for model comparison. Maximum Likelihood Estimation method (MLE) is used to estimate models via MatLab.

Empirical results show: 1) with EGARCH-type volatilities and non-normal errors, the Carhart 4 factors are still alive in US! 2) The new model fits the data well and has better in-sample fit than Fama-French (1993)’s 3-factor and Carhart(1997)’s 4-factor model. 4) Software & computer services sector cannot earn extra Alpha returns in US, UK and China. 5) The Beta coefficients of the US are close to 1.

Future extensions will include but not limited to follow. First, we can construct a new index for software & computer services sector. Secondly, the markets in other different countries can be compared. Lastly, the new model can be compared with other models such as Fama-French (2015) 5-factor.

Li, L.L., Zhu, Q.Y. and Mu, Y. (2017) Analysis of the Sector of Software & Computer Services with a New Carhart 4-Factor Model. Journal of Mathematical Finance, 7, 65-82. https://doi.org/10.4236/jmf.2017.71004

The indices about Software and Computer Services are obtained from the Investing.com.

1) Dow Jones (DJUSSV), 2012:11:01-2015:09:30.

2) FTSE 350 (TRINMX9530), 2012:11:01-2015:09:30, 2003:10-2015:10.

3) FTSE China A 600 (FTXIN49530), 2012:11:01-2015:09:30.

US, UK and China have different festivals. For example, China stock market is closed during China Spring Festival while US and UK stock markets are still trading. Hence, we delete all the missing data and get 674 sets of data.

⁸A convenient reparametrization of Equation (19) is obtained by rescaling, where

f AEPD ( x ) = { 1 σ exp ( − 1 p 1 | x − μ 2 α σ K ( p 1 ) | p 1 ) , if x ≤ μ , 1 σ exp ( − 1 p 2 | x − μ 2 ( 1 − α ) σ K ( p 2 ) | p 2 ) , if x > μ . (18)

θ = ( α , p 1 , p 2 , μ , σ )

⁹In this case, μ and σ are not the notations for the population mean and the population standard variance.

¹⁰A convenient reparametrization of Equation (19) is obtained by rescaling, where

f SAEPD ( x ) = { exp ( − 1 p 1 | x 2 α K ( p 1 ) | p 1 ) , if x ≤ 0 , exp ( − 1 p 2 | x 2 ( 1 − α ) K ( p 2 ) | p 2 ) , if x > 0. (22)

θ = ( α , p 1 , p 2 , 0 , 1 )

According to Zhu and Zinde-Walsh (2009), the AEPD density has following form⁸:

f AEPD ( x ) = { ( α α * ) 1 σ K ( p 1 ) exp ( − 1 p 1 | x − μ 2 α * σ | p 1 ) , if x ≤ μ , ( 1 − α 1 − α * ) 1 σ K ( p 2 ) exp ( − 1 p 2 | x − μ 2 ( 1 − α * ) σ | p 2 ) , if x > μ . (19)

where θ = ( α , p 1 , p 2 , μ , σ ) is the parameter vector. μ ∈ R and σ > 0 represent location and scale, respectively⁹. α ∈ ( 0 , 1 ) is the skewness parameter. p 1 > 0 and p 2 > 0 are the left and the right tail parameters, respectively. K ( p ) and α * are defined as

K ( p ) = 1 2 p 1 / p Γ ( 1 + 1 / p ) , (20)

α * = α K ( p 1 ) α K ( p 1 ) + ( 1 − α ) K ( p 2 ) . (21)

If we set the location parameter μ = 0 and the scale parameter σ = 1 , then we say X is a random variable distributed as Standard AEPD, denote it as X ∼ SAEPD ( α , p 1 , p 2 ,0,1 ) . Its PDF¹⁰, mean and variance are

f SAEPD ( x ) = { ( α α * ) K ( p 1 ) exp ( − 1 p 1 | x 2 α * | p 1 ) , if x ≤ 0 , ( 1 − α 1 − α * ) K ( p 2 ) exp ( − 1 p 2 | x 2 ( 1 − α * ) | p 2 ) , if x > 0 , (23)

E ( X ) = 1 B [ ( 1 − α ) 2 p 2 Γ ( 2 / p 2 ) Γ 2 ( 1 / p 2 ) − α 2 p 1 Γ ( 2 / p 1 ) Γ 2 ( 1 / p 1 ) ] , (24)

Var ( X ) = 1 B 2 { ( 1 − α ) 3 p 2 2 Γ ( 3 / p 2 ) Γ 3 ( 1 / p 2 ) + α 2 p 1 2 Γ ( 3 / p 1 ) Γ 3 ( 1 / p 1 ) ,                                                     − [ ( 1 − α ) p 2 Γ ( 2 / p 2 ) Γ 2 ( 1 / p 2 ) − α 2 p 1 Γ ( 2 / p 1 ) Γ 2 ( 1 / p 1 ) ] 2 } . (25)

Then, if we standardize X with its mean and standard deviation, We can get Z = X − E ( X ) Var ( X ) , which we call Standardized Standard AEPD (SSAEPD). The PDF of Z can be got by transformation.

f SSAEPD ( Z ) = | J | f SAEPD ( E ( X ) + Z Var ( X ) ) , (26)

= δ f SAEPD ( ω + Z δ ) . (27)

where ω = E ( X ) ,   δ = Var ( X ) . We can get the probability density function (PDF) of the SSAEPD

f SSAEPD ( z ) = { δ ( α α * ) K ( p 1 ) exp ( − 1 p 1 | ω + z δ 2 α * | p 1 ) , if   z ≤ − ω δ , δ ( 1 − α 1 − α * ) K ( p 2 ) exp ( − 1 p 2 | ω + z δ 2 ( 1 − α * ) | p 2 ) , if   z > − ω δ . (28)

E ( z ) = 0 , Var ( z ) = 1. With α = 0.5 , p 1 = p 2 = 2 , SSAEPD reduces to Normal (0,1).

We draw PDFs of model residuals and compare them with theoretical PDFs such as SSAEPD and Normal, respectively. For C-SSAEPD-EGARCH model, the comparisons of PDFs between reduals z ^ t and SSAEPD ( α ^ , p ^ 1 , p ^ 2 ) are plotted in Figure 1. These two curves are very close to each others for UK, China and US, respectively. Therefore, by method of “eye-rolling”, we conclude the new model fit data well. Figure 2 and Figure 3 show the comparisons of PDFs between reduals u ^ t and Normal ( μ ^ , σ ^ ) for Carhart 4-factor model and Fama-French 3-factor model respectively. These differences are larger than those in Figure 1, which further show that the new model fit data better.

¹¹This MatLab program is written by Mu (2014).

¹²For the method to generate SSAEPD random variable, one can refer to Li, Tian and Zhen (2011).

In this section, we simulate the data and analyze the results to confirm that the program in MatLab is valid¹¹. The C-SSAEPD-EGARCH (1,1) model is simulated as follows.

R t − R f t = β 0 + β 1 ( R m t − R f t ) + β 2 SMB t + β 3 HML t + β 4 WML t + u t ,   t = 1 , 2 , ⋯ , T , (29)

u t = σ t z t ,   z t ∼ SSAEPD ( α , p 1 , p 2 ) , (30)

ln ( σ t 2 ) = a + g ( z t − 1 ) + b ln ( σ t − 1 2 ) , (31)

g ( z t − 1 ) = c z t − 1 + d [ | z t − 1 | − E ( | z t − 1 | ) ] = ( ( c + d ) z t − 1 − d E ( | z t − 1 | ) , if   z t − 1 ≥ 0 ( c − d ) z t − 1 − d E ( | z t − 1 | ) , else . (32)

We choose β 0 = 0.4 , β 1 = 0.4 , β 2 = 0.6 , β 3 = 0.8 , β 4 = 0.5 , α = 0.5 , p 1 = p 2 = 2 , a = 0.8 , b = 0.2 , c = 0.5 , d = 0.5 as the true values of the parameters. The data generation process (DGP) has following steps.

1) Given α = 0.5 , p 1 = p 2 = 2 , we can generate SSAEPD random number series { z t } t = 1 T ¹².

2) Set the initial value σ 0 2 = 1 , z t = 1 , and given a = 0.8 , b = 0.2 , c = 0.5 , d = 0.5 , we can generate { σ t 2 } t = 1 T and { u t } t = 1 T , with the following formula:

g ( z t − 1 ) = c z t − 1 + d [ | z t − 1 | − E ( | z t − 1 | ) ] = ( ( c + d ) z t − 1 − d E ( | z t − 1 | ) , if   z t − 1 ≥ 0 ( c − d ) z t − 1 − d E ( | z t − 1 | ) , else . (33)

ln ( σ t 2 ) = a + g ( z t − 1 ) + b ln ( σ t − 1 2 ) , (34)

u t = σ t z t . (35)

3) Generate random number series { X 1 t } t = 1 T , { X 2 t } t = 1 T , { X 3 t } t = 1 T , { X 4 t } t = 1 T from Uniform(0,1).

4) Set β 0 = 0.4 , β 1 = 0.4 , β 2 = 0.6 , β 3 = 0.8 , β 4 = 0.5 , and we can get { Y t } t = 1 T , with the following formula:

Y t = β 0 + β 1 X 1 t + β 2 X 2 t + β 3 X 3 t + β 4 X 4 t + u t ,   t = 1 , 2 , ⋯ , T . (36)

After getting the simulated data { X 1 t , X 2 t , X 3 t , X 4 t , Y t } t = 1 T , we estimate the parameters in the C-SSAEPD-EGARCH model. The simulation results are reported in Table 8. The estimates from MatLab program are

β 0 = 0.3726 , β 1 = 0.3988 , β 2 = 0.5841 , β 3 = 0.8607 , β 4 = 0.4932 , α = 0.5000 , p 1 = 2.0004 , p 2 = 2.0003 , a = 0.8178 , b = 0.1894 , c = 0.5224 , d = 0.5334 ,

which are very close to the true values of the parameters. For robustness exam, we also change the true values of the parameters and redo the simulation and estimation. All the simulation and estimation show the estimates are very closed to the true values of the parameters, since all error are equal to or less than −25%.

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

Hence, we conclude the MatLab program can be applied to estimate and analyze empirical data for C-SSAEPD-EGARCH.

In this section, we compare the US 4 factors and the Global 4 factors. Data are downloaded from the French Data Library. Because of data availability, we only analyze the monthly data for UK software & computer services sector. Table 9 shows the descriptive statistics of US 4 factors are close to those of Global 4 factors, which shows the strong global effects of US stock market. Estimates in the

Note: The null hypothesis of JB test is H0: Data are distributed as Normal (0, 1).

Note: C-Normal is the model used in Carhart (1997). FF-Normal is the model used in Fama-French (1993).

Note: The null hypothesis of KS test is H0: Data follow a specified distribution. We set the significance level of all tests at 5%. If the P-value of KS test is bigger than 5%, then we do not reject the null hypothesis. Otherwise, we reject the null hypothesis.

Note: *marks the smallest AIC value for each return.

mean equation (Table 10) are all statistically significant. P-value of KS test (Table 11) results shows all models pass the residual test. Table 12 shows Fama-French (1993)’s 3 factor model is better with the lowest AIC values.