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MSc Finance Thesis

Asset Pricing Models in the Chinese Market

Comparison and Choice among the CAPM, the Fama-French

Three-factor Model and the Carhart’s Four-factor Model

Xurui Wang (S2685426)

Supervisor: Dr. Peter Smid

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Asset Pricing Models in the Chinese Market

Comparison and Choice among the CAPM, the Fama-French Three-factor

Model and the Carhart’s Four-factor Model

Abstract

This paper compares the CAPM (using its empirical variant: the Single-index Model), the Fama-French Three-factor Model and the Carhart’s Four-factor Model using monthly data of the Shanghai A Stocks from January 2007 to December 2013 from two perspectives: explanatory power and predictive power over stock returns in the Chinese market. The main results are: Among the three models considered, the Carhart’s Four-factor Model’s explanatory power is only slightly stronger or equal to that of the Fama-French Three-factor Model. The Single-index Model and the Fama-French Three-factor Model have stronger predictive power compared to the Carhart’s Four-factor Model. There is no statistically significant difference between the predictive power of the Single-index Model and that of the Fama-French Three-factor Model. All considered, the Fama-French Three-factor Model might be the most suitable asset pricing model among these three models considered for the Chinese market at the current stage, with comparatively stronger explanatory and predictive power as well as a simpler model.

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1. Introduction

Based on the earlier work of Markowitz on diversification and modern portfolio theory, Sharpe (1964), Lintner (1965) and Mossin (1966) independently developed the Capital Asset Pricing Model (CAPM). The CAPM describes the relationship between risk and expected return given the assumption of Markowitz’s diversification theory that all the investors in the market hold the market portfolio thus the unsystematic risk is well diversified and only the systematic risk is relevant and compensated for. Beta, which measures an asset’s sensitivity to the systematic risk, determines the expected return of the asset.

The CAPM has been the king of asset pricing for a very long time. However, with the in-depth application, there are also louder voices of skepticism. Due to the unsatisfactory results of studies using the empirical variant of the CAPM, the Single-index Model, Fama and French (1993) developed their three-factor model, and the Carhart’s Four-factor Model was introduced in 1997. These models explain the return of stocks better since they reflect some other factors that have influence on the return of stocks yet are not captured by the CAPM.

In the Chinese market, the CAPM is widely used in asset pricing, performance measurement, risk analysis, and cost of capital estimation. And many empirical tests have been done on the applicability of the CAPM as well as other asset pricing models. However, most of them only focus on one specific model and few studies compare different asset pricing models, especially from the perspective of the predictive power of the models.

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Regression (Fama and Macbeth, 1973) is used to construct the portfolios. Then the Single-index Model, the Fama-French Three-factor Model and the Carhart’s Four-factor Model are empirically tested to compare their explanatory powers of returns. Afterwards, we build forecasting models and use them to predict returns and calculate the SSE (sum of the squared errors) between the predicted returns and the actual ones in each month. Whether the differences between the SSE’s of the three models are statistically significant is later tested using t-test. Finally, conclusions will be driven from the comparison and analysis of the results.

This paper continues as follows. The next section reviews the development of relevant theories and summarizes both foreign and Chinese empirical results. Section three describes the data. Section four explains the methodology. Section five presents the results of the empirical tests. Section six shows the results of robustness tests. The final section concludes.

2. Literature review 2.1 Relevant theories

Markowitz (1952) originally uses the mean and the variance of historical returns to measure the expected return and the risk respectively. Investors are assumed to be risk-averse: if two assets or portfolios have the same expected return, investors prefer the one with lower variance. Portfolio return is the proportion-weighted combination of the constituent assets' returns. Portfolio volatility is a function of the correlations of the component assets, for all asset pairs. Markowitz provides us with a way of measuring return and risk from a mathematical perspective, and gives birth to the quantitative era in Finance.

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4 former one is specific to the firm and diversifiable, and the latter one is market-related and cannot be diversified. Since all the investors in the market would hold the market portfolio based on the assumptions of the CAPM, they are only exposed to and rewarded for bearing systematic risk. Thus, systematic risk is the relevant risk and it is measured by beta. The CAPM is a simple linear model that is expressed in terms of expected returns and expected risk. A stock’s expected excess return equals to the market’s expected excess return times beta.

However, with the deeper application of the theory, there are also louder voices of skepticism. Due to the unsatisfactory results of studies using the empirical variant of the CAPM, the Single-index Model, researchers keep digging into this area and bring up the Fama-French Three-factor Model. Instead of using beta as the only variable to describe the return, Fama and French (1993) add two more variables into the equation based on the following observed phenomena: Small firms and stocks with a high book-to-price ratio (the ratio of book equity to market equity) tend to have better performance. These two variables are called SMB and HML respectively, with SMB standing for "Small (market capitalization) Minus Big" and HML for "High (book equity to market equity ratio) Minus Low". They reflect the excess returns of small caps over big caps and of value stocks over growth stocks.

Afterwards, Carhart brings up his four-factor model with a variable reflecting the momentum effect taken into consideration. Momentum in a stock is described as the tendency for the stock price to continue rising if it is going up and to continue declining if it is going down. This variable is sometimes called UMD, which stands for up (stocks with up-going price trend) minus down (stocks with price trend of going down).

Since the existence of the illiquidity premium, a premium demanded by investors when any given security cannot be easily converted into cash and converted at the fair market value, has been verified by a lot of researchers, some work has been done to include it into the CAPM, see e.g. Pástor and Stambaugh (2003), Acharya and Pederson (2006) and Liu (2006).

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5 Before the bring-up of the Fama-French Three-factor Model, the relevant empirical tests focus on the explanatory power of beta to expected returns, and how to improve beta to reach stronger explanatory power. Sharpe and Cooper (1972) take a sample that consists of 34 mutual funds in the U.S.A from year 1931 to year 1967 and use the method of least squares to see whether there is a linear relationship between risk and return in a long period of time. The results show that beta explains more than 95 percent of the variance of returns. Jensen, Black, and Scholes (1972) use monthly data of all the stocks listed on the NYSE from year 1931 to year 1965 and present how the cross-sectional tests are subject to measurement error bias and provide a solution through grouping procedures. The results show that there is a significant positive linear correlation between beta and stock returns. Fama and MacBeth (1973) test the relationship between average return and risk for NYSE common stocks. They conclude that the hypothesis that the relationship between a security’s portfolio risk and its expected return is linear cannot be rejected and that “there seems to be a positive trade-off between return and risk” (Fama and MacBeth 1973, p 633). Later on, an increasing number of things are taken into consideration, like the situation of negative excess market returns and that betas and market risk premiums can vary over time. Pettengill et al. (1995) finds a consistent and highly significant relationship between beta and cross-sectional portfolio returns after adjusting for the expectations concerning negative market excess returns. Jagannathan and Wang (1996)assume that betas and the market risk premium vary over time and find a significant linear relationship between beta and average returns.Durack et al. (2004) use Australian data to test the model brought up by Jagannathan and Wang (1996) and find similar results.

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6 2.2.2 Studies about other factors

At the same time, whether other factors (besides the beta) have significant influences on expected returns are tested. Banz (1981) examines the empirical relationship between the return and the total market value of NYSE common stocks and finds that smaller firms have had higher risk adjusted returns, on average, than larger firms, which is known as the size effect. Basu (1983) introduces the E/P (earning-to-price ratio) effect. He examines the empirical relationship between earnings' yield, firm size and returns on the common stock of NYSE firms and confirms that the common stock of high E/P (earning-to-price ratio) firms earn, on average, higher risk-adjusted returns than the common stock of low E/P firms and that this effect is clearly significant even with firm size controlled. Kleim (1983) finds a “January effect”, that daily abnormal return distributions in January have larger means relative to the remaining eleven months. Bhandari (1988) tests the leverage effect and the results show that the expected common stock returns are positively related to the ratio of debt to equity, controlling for the beta and firm size and including as well as excluding January, though the relation is much stronger in January. The illiquidity premium is considered by Amihud (2002), which shows that over time, expected market illiquidity positively affects ex ante stock excess return. After the brought-up of the Fama-French Three-factor Model, the two new factors(SMB and HML) have been tested in many markets, including Fama and French (1998), Maroney and Protopapadakis (2002) and Drew and Veeraraghavan (2002). And the results show that the SMB factor and the HML factor are both significant.

2.2.3 Researches about the Chinese market

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7 has also been tested by a lot of researchers. Some results show that the Fama-French Three-factor Model explains the stock returns in the Chinese market pretty well and the coefficients of the SMB factor and the HML factor are both statistically significant (e.g. Zhang, Xie and Zhang Binru 2014). Some other papers find that the coefficient of the SMB factor is significant while that of the HML factor is not (e.g. Geng and Zhang 2014). As to the Carhart’s Four-factor Model, Xu and Xiong (2009) find that there are short-term reverse (one to four weeks) and long-term (24 to 72 weeks) momentum effect in the Chinese market (with firm size and book-to-market equity ratio controlled). Most of the studies find that the coefficient of the momentum factor is not significant (e.g. Cheng and Zhang 2010).

3. Data 3.1 Sample

Since the Split Share Structure Reform of the Chinese stock market was completed at the end of 2006, we take all the A stocks listed on the Shanghai Stock Exchange from 01-01-2007 until 31-12-2013 as my sample. ST stocks are eliminated (ST is the abbreviation of special treatment. The Shanghai Stock Exchange implements price limits on those listed companies whose financial positions are abnormal). Finally it comes to a sample of 809 companies from different industries. We do not include year 2014 since the database we use(Datastream)has not updated the financial fundamentals of many companies for that year. But since data like the price and market index are up-to-date, we include year 2014 when we try to compare the predictive power of the three models.

3.2 The use of monthly data

When it comes to what type of data to use, our principle is to reflect as much information as we can but at the same time to keep it clear and simple. Although daily or weekly data contain more information in general, due to the fact that we also use financial fundamentals which are reported only quarterly or even yearly, using these two types of data may not benefit us a lot. Moreover, given that our sample contains 809 companies for a seven-year time period, using monthly data seems feasible.

3.3 Market portfolio and market return

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8 weights in the proportions that they exist in the market. Generally speaking, the market portfolio should also contain bonds, preferred stocks and real estates. Since my sample contains all the A stocks listed on the Shanghai Stock Exchange, the corresponding market portfolio should be a portfolio having all these stocks in it and weighted according to their market values. We use a comparatively narrow definition of market portfolio with only stocks included and we take the return of the Shanghai Stock Exchange A-share Index as a proxy of the market return since it includes all the A stocks and it is weighted using the size of issue. The return is calculated in the following way:

𝐑𝐦,𝐭=

𝐈𝐦,𝐭+𝟏−𝐈𝐦,𝐭

𝐈𝐦,𝐭 (1)

𝐑𝐦,𝐭 is the market rate of return in month t, and 𝐈𝐦,𝐭 is the value of the index in month t . We use the value of the index on the 21st of each month as the index value for that month.

3.4 Risk-free rate of return

It is common to take an interbank offered rate (e.g. LIBOR, London Interbank Offered Rate) as the risk-free rate of return. However, given the special situation of the commercial bank system in China, the interbank monetary market is not completely competitive, thus an interest rate like SHIBOR (Shanghai Interbank Offered Rate) may not be a good approximation of the risk-free rate of return. we use the one-year deposit rate of the lump-sum term deposit (a savings deposit method where the savings term is agreed upon by the customer and the bank, the principal is deposited in lump sum, and the principal and interest will be withdrawn in lump sum at maturity) as the yearly risk-free rate of return, and then calculate the corresponding monthly risk-free interest rates using the method of compound interest as following:

𝐑𝐟,𝐭= √𝟏 + 𝐃𝐑𝟏𝟐 𝐲𝐞𝐚𝐫𝐥𝐲,𝐭− 𝟏 (2)

𝐑𝐟,𝐭 is the risk-free rate of return in month t, 𝐃𝐑𝐲𝐞𝐚𝐫𝐥𝐲,𝐭 is the one-year deposit rate of the lump-sum term deposit in month t. We use the one-year deposit rate on the 21st of each month as the deposit rate for the month, thus we update the risk-free rate of return every month.

3.5 SMB factor and HML factor

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9 companies in the market according to their sizes, and then calculate the performance difference between the portfolio of small caps and that of the big caps. To simplify the process of sorting, We take the monthly return of the performance index for China A share small caps (RChina A,SC) from MSCI website (https://www.msci.com/) as the proxy of the performance of the portfolio of small caps, and that for China A share large caps (RChina A,LC) as the proxy of the performance of the portfolio of big caps. And we have:

𝑺𝑴𝑩𝒕 = 𝐑𝐂𝐡𝐢𝐧𝐚 𝐀,𝐒𝐂𝐭− 𝐑𝐂𝐡𝐢𝐧𝐚 𝐀,𝐋𝐂𝐭 (3)

Still, we use the value of the index on the 21st of each month as the index value for that month and we calculate the monthly return of the index by firstly subtracting the index value of the last month from that of this month and then dividing the difference by last month’s index value (similar to equation 1). Likewise, We take the return of the performance index for China A share standard caps (large and mid cap) of the value style (RChina A,Value) from MSCI website as the proxy of the performance of the

portfolio of value caps, and that of the growth style (RChina A,Growth) as the proxy of the performance of

the portfolio of growth caps. And we have:

𝑯𝑴𝑳𝒕= 𝐑𝐂𝐡𝐢𝐧𝐚 𝐀,𝐕𝐚𝐥𝐮𝐞𝐭− 𝐑𝐂𝐡𝐢𝐧𝐚 𝐀,𝐆𝐫𝐨𝐰𝐭𝐡𝐭 (4)

3.6 UMD factor

Since the MSCI website doesn’t provide us with proper indices for calculating the momentum factor UMD, we calculate it ourselves in the following way: We firstly sort the stocks according to their performances at the end of last year and take the portfolio consisting of the top 30% stocks as the high momentum portfolio and the one consisting of the bottom 30% as the low momentum portfolio. Then, we calculate the differences between the returns of these two portfolios in each month of this year, which are namely the values of the momentum factor in each month:

𝑼𝑴𝑫𝒕 = 𝐑𝐡𝐢𝐠𝐡 𝐫𝐞𝐭𝐮𝐫𝐧 𝐩𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 𝐭− 𝐑𝐥𝐨𝐰 𝐫𝐞𝐭𝐮𝐫𝐧 𝐩𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨𝐭 (5)

The two portfolios stay the same for the whole year and we adjust them at the end of each year.

3.7 Financial fundamentals

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10 Among them, ME is in millions, BE/ME, E/P and the leverage don’t have a unit. For the earning-to-price ratio, we take the reciprocal of the price-to-earning ratio. We use the value of a ratio on the 21st of each month as the ratio value for that month.

3.8 Descriptive statistics of the data

Table 1 concludes the descriptive statistics of the data, with panel A including all the variables in the form of panel data and panel B including variables in the form of time-series data.

Table 1 Summary Statistics

This table presents summary statistics of all the variables we use. Panel A contains panel data variables and Panel B contains time-series variables.

N Mean Maximum Minimum Variance Std. Skewness Kurtosis

Panel A Ri 67956 0.0204 10.7525 -0.6319 0.0297 0.1723 8.1838 409.9205 D/E 67884 125.8543 42648.03 -3035.21 586715.6 765.9736 39.3283 1944.819 ME 67956 12544.14 2010208 183.17 3.54E+09 59491.34 16.6809 345.1389 BE/ME 67908 370.343 9047.904 -11900.35 113991.4 337.6261 -2.3348 140.9985 E/P 56603 0.0351 2.0000 9.82E-06 0.0029 0.0538 15.2708 372.9634 Panel B Rm 84 0.0019 0.2218 -0.2011 0.0070 0.0823 0.1400 0.4890 Rf 84 0.0024 0.0034 0.0019 0.0000 0.0005 1.2640 0.6790 SMB 84 0.0143 0.1782 -0.1780 0.0040 0.0660 -0.2640 0.6070 HML 79 0.0049 0.0922 -0.0821 0.0010 0.0339 -0.0100 0.0870 UMD 84 0.0004 0.3061 -0.2799 0.0130 0.1124 0.6610 0.8270

Notes: Ri is the stock return. D/E is the debt-to-equity ratio. ME is the market capitalization. BE/ME is the book-to-market ratio. E/P is the earning-to-price ratio. Rm is the market return. Rf is the risk-free rate of return. SMB, HML, UND are the small-minus-big factor, the high-minus-low factor and the up-minus-down factor respectively. N represents the number of observations. Std. reports the standard deviation. It is worth noting that the HML factor only has 79 observations, this is because the relevant index values from June 2009 to October 2009 are missing.

Sources: Datastream and Wind

4. Methodology

4.1 Explanatory power

Step 1: Fama-Macbeth regression

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11 following model:

𝐑𝐢,𝐭 = 𝛂𝟎,𝐭+ 𝛂𝐢,𝐭∗ 𝐅𝐢,𝐭+ 𝐞𝐢,𝐭 (6)

Where 𝐑𝐢,𝐭 is the return of stock i in month t. 𝐅𝐢,𝐭 is value of the financial fundamental for stock i in month t. We calculate the return of stock similarly as before: we take the stock price on the 21st of each month as the stock price for that month. The monthly stock return equals to the difference between the stock price in this month and that in last month divided by the latter one. Considered financial fundamentals include: stock capitalization ME (which reflects the size of a company), the book-to-market ratio BE/ME (which reflects whether a company is a value company or a growth company), the ratio of earnings to price E/P (which reflects a company’s profitability), and the ratio of debt to equity D/E (which reflects a company’s financing structure).

Step 2: Portfolio construction

According to the results of the Fama-Macbeth regressions, portfolios will be constructed according to those financial fundamentals that have significant impacts on stock returns: The companies will firstly be ranked in descending order according to the value of the financial fundamentals of year T-1, after that, they will be equally divided into 5 groups. Since we have 809 companies in our sample, each of the first four groups has 162 companies in it while the last group has 161 companies. The portfolios will stay the same for the whole year and be reconstructed next year. Consequently, we will have five sub-panels according to each of the financial fundamentals that have significant impacts on stock returns.

Step 3: Empirical test

First of all, we test the Carhart’s Four-factor Model, to see whether the momentum factor (UMD) is significant. If it’s not, we will leave it out and test again, which leads to the test of the Fama-French Three-factor Model. If the coefficient of the momentum factor is significant, we will test the Fama-French Three-factor Model as well, in order to see how big the difference between the explanatory power of these two models is by comparing their adjusted R-squares. The Single-index Model will then be tested. The regression models are as the followings.

For the Single-index Model, we have

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12 𝐑𝐢,𝐭− 𝐑𝐟,𝐭 = 𝛂𝐢+ 𝛃𝐢(𝐑𝐦,𝐭− 𝐑𝐟,𝐭) + 𝐚𝐢𝐒𝐌𝐁𝐭+ 𝐛𝐢𝐇𝐌𝐋𝐭+𝐞𝐢,𝐭 (8)

For the Carhart’s Four-factor Model, we have

𝐑𝐢,𝐭− 𝐑𝐟,𝐭= 𝛂𝐢+ 𝛃𝐢(𝐑𝐦,𝐭− 𝐑𝐟,𝐭) + 𝐚𝐢𝐒𝐌𝐁𝐭+ 𝐛𝐢𝐇𝐌𝐋𝐭+ 𝐜𝐢𝐔𝐌𝐃𝐭+𝐞𝐢,𝐭 (9)

𝐑𝐢,𝐭 is the return of stock i at time t. 𝐑𝐦,𝐭 is the return of the market index at time t. 𝐑𝐟,𝐭 is the risk-free rate of return at time t. 𝛂𝐢 is the intercept, 𝛃𝐢, 𝐚𝐢, 𝐛𝐢 and 𝐜𝐢 are the corresponding coefficients and 𝐞𝐢,𝐭 is the error term.

4.2 Predictive power

Step 1: Forecasting model building

For each year, we run three regressions using the three regression models (equation 7, 8, and 9) to get our forecasting models. As mentioned before, the reason that we don’t use the data of the latest fiscal year 2014 in the explanatory power part is that some financial fundamentals have not been updated yet. Since data needed in this predictive power part is already up-to-date, we also do the regressions for year 2013 to forecast the stock returns of year 2014. Thus we come to seven forecasting models for each of the three models being considered (21 forecasting models in total).

Step 2: Forecast

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13 Step 3: The sum of the squared errors

Sequentially, we will subtract the actual excess return from the predicted one in each month, square the errors (there will be 79 instead of 84 numbers for each model since there are five missing data in variable HML so we exclude that five months in this step) and add them all together (i.e. calculating the SSE of each model).

4.3 Comparison and analysis

For the explanatory power part, we mainly focus on comparing the adjusted R-squares and to see whether the models work differently when it comes to companies with different financial fundamentals. For the predictive power part, we test whether the differences between the predictive powers of these three models are statistically significant by testing whether these three models have the same SSE (Since the number of months for each model is the same, this is equal to testing the mean of SSE) and the model with the smallest SSE is regarded to have the strongest predictive power. We use t-test to do this: the null hypotheses are the SSE of the Single-index Model equals to that of the Fama-French Three-factor Model (H0 1), the SSE of the Single-index Model equals to that of the Carhart’s Four-factor Model (H0 2), and the SSE of the Fama-French Three-factor Model equals to that of the Carhart’s Four-factor Model(H0 3) respectively. We have three alternative hypotheses for each null hypothesis: the SSE of a model is smaller than that of another model (H1 1), the SSE of a model is not equal to that of another model (H1 2) the SSE of a model is bigger than that of another model (H1 3).

5. Results

5.1 Explanatory power

5.1.1 Results of the Fama-Macbeth Regression

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Table 2 Results of the Fama-Macbeth Regressions

This table presents the results of the Fama-Macbeth regressions (equation 6) of stock returns on four financial fundamentals for the time period from January 2007 to December 2013.

D/E ME BE/ME E/P

coefficient 7.89E-07 -6.05E+00 -0.0000377 -0.2331882

t-statistic 0.91 5.95 -19.29 -17.33

p value 0.361 (0.001)*** (0.000)*** (0.000)***

Notes: The results are estimated using OLS regressions:Ri,t= α0,t+ αi,t∗ Fi,t+ ei,t. Ri,t is

the stock return, Fi,t is a specific financial fundamental among those four considered

(D/E is the debt-to-equity ratio, ME is the market capitalization, BE/ME is the book-to-market ratio, E/P is the earning-to-price ratio).

* Statistical significance at 10% level. ** Statistical significance at 5% level. *** Statistical significance at 1% level.

5.1.2 Results of the empirical test

Before we test the Carhart’s Four-factor Model, we first take a look at the correlations between the market factor MKT (Rm-Rf), SMB, HML and UMD. We can see from Table 3 that the correlation between any two of them is smaller than 0.3, which indicates that the correlations between these variables are weak and that there is no multicollinearity problem.

Table 3 Correlation Matrix

This table presents the correlation matrix of all the variables in the three models considered. MKT SMB HML UMD MKT 1 0.291 0.016 0.197 SMB 0.291 1 0.057 -0.044 HML 0.016 0.057 1 0.032 UMD 0.197 -0.044 0.032 1

Notes: The MKT is the market factor, which stands for the excess return of the market (Rm-Rf).SMB, HML and UMD stands for the small-minus-big factor, the high-minus-low

factor and the up-minus-down factor respectively.

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Table 4 OLS regressions of the three models: companies sorted according to size

This table reports the results of OLS regressions of the three models with companies sorted according to their sizes.

Carhart's Fama-French Single-index

α β a b c α β a b α β ME 01 coefficient 0.0173 1.0420 0.2930 -0.0905 0.0296 0.0176 1.0433 0.2926 -0.0845 0.0256 1.0628 (biggest) t-statistic 6.91 111.74 17.95 -2.91 3.23 7.06 111.95 17.92 -2.72 11.07 119.14 p value 0.000 0.000 0.000 0.004 0.001 0.000 0.000 0.000 0.007 0.000 0.000 adj R-square 0.5249 0.5245 0.5106 prob>F 0.000 0.000 0.000 ME 02 coefficient 0.0248 1.0741 0.5671 -0.6175 0.0122 0.0250 1.0747 0.5669 -0.5931 0.4142 1.1131 t-statistic 9.41 109.05 32.89 -1.88 1.26 9.47 109.21 32.88 -1.81 16.40 114.16 p value 0.000 0.000 0.000 0.060 0.208 0.000 0.000 0.071 0.821 0.000 0.000 adj R-square 0.5381 0.5381 0.4892 prob>F 0.000 0.000 0.000 ME 03 coefficient 0.0281 1.0813 0.6064 -0.0189 -0.0122 0.0280 1.0808 0.6066 -0.0213 0.0454 1.1199 t-statistic 10.77 110.88 35.52 -0.58 -1.27 10.73 110.92 35.53 -0.66 18.01 115.18 p value 0.000 0.000 0.000 0.561 0.204 0.000 0.000 0.000 0.511 0.000 0.000 adj R-square 0.5501 0.5500 0.4936 prob>F 0.000 0.000 0.000 ME 04 coefficient 0.0106 1.0810 0.6864 0.1135 -0.0376 0.0291 1.0794 0.6870 0.1059 0.0487 1.1215 t-statistic 11.00 107.77 39.09 3.39 -3.81 10.84 107.65 39.10 3.17 18.51 110.39 p value 0.000 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.002 0.000 0.000 adj R-square 0.5461 0.5457 0.4725 prob>F 0.000 0.000 0.000 ME 05 coefficient 0.0379 1.0833 0.7003 0.2572 -0.9073 0.0369 1.0794 0.7018 0.2390 0.0610 1.1339 (smallest) t-statistic 8.62 65.97 24.36 4.70 -5.62 8.38 65.71 24.38 4.37 14.89 71.72 p value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 adj R-square 0.3151 0.3135 0.2755 prob>F 0.000 0.000 0.000

Notes: The results are estimated using OLS regressions. Carhart’s:Ri,t− Rf,t=αi+ βi(Rm,t− Rf,t)+ aiSMBt+ biHMLt+ ciUMDt+ei,t

Fama-French: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ei,t Single-index: Ri,t− Rf,t= αi+ (Rm,t− Rf,t)βi +ei,t

adj R-squre is the adjusted R-square. ME is the market capitalization with the group ME 01 containing the biggest 20% stocks and group ME 05 containing the smallest 20% stocks. The p values in bold are those ones for which corresponding coefficients are not statistically significant at 10% level.

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16 stock returns that much. And we do see that for two out of the five portfolios, the coefficient of the UMD factor is not significant. Besides, we can also see that with the decrease of the ME, the coefficient of the SMB factor (a) becomes bigger (in the Fama-French Three-factor Model as well as the Carhart’s Four-factor Model). Since the SMB factor measures the premium of the size effect, it makes sense that its coefficient becomes bigger when explaining a group containing companies of smaller sizes, indicating the existence of the size effect in the Chinese market. Although all the adjusted R-squares of the regressions are significant, they differ when it comes to company groups with different financial fundamentals and we can see that these three models work not that well for small size firms.

Table 5 OLS regressions of the three models: companies sorted according to the book-to-equity ratio

This table reports the results of OLS regressions of the three models with companies sorted according to their book-to-market ratios.

Carhart's Fama-French Single-index

α β a b c α β a b α β BE/ME 01 coefficient 0.0378 1.0979 0.5045 0.1105 -0.0139 0.0376 1.0973 0.5047 0.1078 0.0526 1.1301 (highest) t-statistic 11.55 89.93 23.6 2.71 -1.16 11.52 89.96 23.61 2.65 17.22 95.86 p value 0.000 0.000 0.000 0.007 0.248 0.000 0.000 0.000 0.008 0.000 0.000 adj R-square 0.4347 0.4347 0.4031 prob>F 0.000 0.000 0.000 BE/ME 02 coefficient 0.0310 1.0855 0.5789 0.0457 -0.0440 0.0305 1.0836 0.5796 0.0369 0.0483 1.1245 t-statistic 11.93 111.84 34.07 1.41 -4.60 11.73 111.65 34.09 1.14 19.20 115.79 p value 0.000 0.000 0.000 0.158 0.000 0.000 0.000 0.000 0.254 0.000 0.000 adj R-square 0.5512 0.5505 0.4963 prob>F 0.000 0.000 0.000 BE/ME 03 coefficient 0.0259 1.0709 0.6116 -0.0048 -0.0124 0.0257 1.0704 0.6118 -0.0073 0.4385 1.1119 t-statistic 9.94 110.21 35.95 -0.15 -1.30 9.89 110.25 35.96 -0.23 17.44 114.57 p value 0.000 0.000 0.000 0.881 0.193 0.000 0.000 0.000 0.821 0.000 0.000 adj R-square 0.5484 0.5484 0.491 prob>F 0.000 0.000 0.000 BE/ME 04 coefficient 0.0238 1.0607 0.5816 -0.0433 -0.0077 0.0237 1.0604 0.5817 -0.4483 0.0407 1.0997 t-statistic 6.98 83.23 26.06 -1.02 -0.61 6.96 83.28 26.07 -1.06 12.81 89.69 p value 0.000 0.000 0.000 0.309 0.541 0.000 0.000 0.000 0.291 0.000 0.000 adj R-square 0.4063 0.4063 0.3715 prob>F 0.000 0.000 0.000 BE/ME 05 coefficient 0.0180 1.0413 0.5714 0.0755 -0.0197 0.0178 1.0405 0.5717 0.0715 0.0353 1.0797 (lowest) t-statistic 5.36 82.98 26.01 1.80 -1.59 5.30 82.98 26.02 1.71 11.23 88.98 p value 0.000 0.000 0.000 0.071 0.111 0.000 0.000 0.000 0.087 0.000 0.000 adj R-square 0.4070 0.4070 0.3692 prob>F 0.000 0.000 0.000

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17

Fama-French: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ei,t Single-index: Ri,t− Rf,t= αi+ (Rm,t− Rf,t)βi +ei,t

adj R-squre is the adjusted R-square. BE/ME is the book equity to market equity ratio with the group BE/ME 01 containing the top 20% value stocks and group BE/ME 05 containing the top 20% growth stocks. The p values in bold are those ones for which corresponding coefficients are not statistically significant at 10% level.

Similarly, we can see from Table 5 that all the regressions are significant and the explanatory power of the Carhart’s Four-factor Model is only slightly stronger than the Fama-French Three-factor Model. Now, only the portfolio BE/ME 02 has a significant coefficient for factor UMD. Moreover, we can see that the higher the BE/ME ratio is, the bigger the coefficient of the HML factor (except for portfolio BE/ME 05). Since the HML factor reflects the premium of the BE/ME ratio, it seems reasonable for the portfolio consisting of value companies (companies with high BE/ME ratio) to have a bigger coefficient of the HML factor (although in portfolio BE/ME 02, BE/ME 03 and BE/ME 04, the coefficient of factor HML is not significant). Based on the adjusted R-squares we can see that these models work not as well for high or low BE/ME ratio companies as for companies of medium BE/ME ratio.

Table 6 OLS regressions of the three models: companies sorted according to the earning-to-price ratio

This table reports the results of OLS regressions of the three models with companies sorted according to their earning-to-price ratios.

Carhart's Fama-French Single-index

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18 E/P 05 coefficient 0.0316 1.0927 0.6351 0.0978 -0.0303 0.0312 1.0913 0.6355 0.0917 0.0502 1.1334 (lowest) t-statistic 11.51 106.68 35.41 2.86 -3.00 11.39 106.61 35.43 2.69 18.80 109.83 p value 0.000 0.000 0.000 0.004 0.003 0.000 0.000 0.000 0.007 0.000 0.000 adj R-square 0.5357 0.5354 0.4714 prob>F 0.000 0.000 0.000

Notes: The results are estimated using OLS regressions. Carhart’s:Ri,t− Rf,t=αi+ βi(Rm,t− Rf,t)+ aiSMBt+ biHMLt+ ciUMDt+ei,t

Fama-French: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ei,t Single-index: Ri,t− Rf,t= αi+ (Rm,t− Rf,t)βi +ei,t

adj R-squre is the adjusted R-square. E/P is the earning to price ratio with the group E/P 01 containing the 20% stocks that rank highest in the E/P and group E/P 05 containing the 20% stocks that rank lowest in the E/P. The p values in bold are those ones for which corresponding coefficients are not statistically significant at 10% level.

Similar conclusions about the significance of the regressions and comparative strength of the explanatory power of the three models can be drawn from the results concluded in Table 6. And we can also see that these models explain returns of companies with a high E/P ratio less well.

Combining Table 4-6, we can infer that: Firstly, many coefficients of the momentum factor UMD are not significant (nine out of 15) and the adjusted R-squares of the Carhart’s Four-factor Model are just slightly higher than or even the same as those of the Fama-French Three-factor Model, which indicates that the momentum effect is not a good explanatory factor of stock returns in the Chinese market. However, it is worth noting that this paper only tests the one-year momentum effect (when we calculate the UMD, the portfolios are adjusted only once a year), so we are not in the position to state whether the momentum factors of a shorter period (quarterly, monthly or even weekly) do have significant impacts on stock returns. Secondly, the adjusted R-squares between the Fama-French Three-factor Model and the Single-index Model differ a lot in many groups, indicating that compared to the latter one, the former one has a stronger explanatory power of stock returns. Finally, beta and a (the coefficient of the size factor SMB), are always significantly positive, indicating a positive trade-off between stock returns and the excess market return and the existence of the size effect in the Chinese market. However, the coefficient of the HML factor is not significant in some portfolios (see Table 4, 5, and 6), which means that the observed phenomenon that value companies tend to have higher returns than growth companies may not be the case in the Chinese market.

5.2 Predictive power

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19

Table 7 Coefficients of the forecasting models

This table shows the coefficients of the three forecasting models built using data in each of the seven years from 2007 to 2013.

year model α β a b c 2007 Single-index 0.0584 1.0996 Fama-French 0.0349 1.0449 0.7604 -0.6293 Carhart's 0.0372 1.0519 0.7501 -0.6148 -0.0674 2008 Single-index 0.0577 1.1046 Fama-French 0.0307 1.0446 0.7531 -0.5005 Carhart's 0.0329 1.0511 0.7432 -0.4853 -0.0641 2009 Single-index 0.0199 1.2323 Fama-French 0.0080 1.7396 0.1984 -1.2073 Carhart's -0.0030 1.6226 0.5171 -0.5791 -0.1811 2010 Single-index 0.0229 0.9435 Fama-French 0.0124 1.0681 0.5565 0.3007 Carhart's 0.0122 1.0764 0.4692 0.2519 0.0436 2011 Single-index 0.0103 1.5146 Fama-French 0.0097 1.4091 0.3727 0.2543 Carhart's 0.0097 1.3907 0.3926 0.2474 -0.0766 2012 Single-index 0.0004 0.8470 Fama-French 0.0036 0.7022 0.4963 -0.1204 Carhart's 0.0036 0.7002 0.5032 -0.1161 0.0067 2013 Single-index 0.0190 1.1996 Fama-French 0.0236 1.1880 -0.2068 -0.1202 Carhart's 0.0224 1.1974 -0.2841 -0.0761 -0.0613

Notes: The results are estimated using OLS regressions based on 809 stocks. Carhart’s: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ ciUMDt+ei,t

Fama-French: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ei,t

Single-index: Ri,t− Rf,t= αi+ (Rm,t− Rf,t)βi +ei,t

Those coefficients in bold are not statistically significant at 10% level.

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20

Table 8 SSE of the three models

This table concludes the SSE (sum of squared errors) of the 79 months of the three models respectively. SSE SSE N (max excluded) Single-index 0.2163 0.1653 32 Fama-French 0.2699 0.1972 23 Carhart's 0.3549 0.2562 24

Notes: SSE=∑(ERft-ERrt)^2 ERft is the forecast average excess return in month t, ERrt is the

actual excess return in month t. N is the number of months in which the model has the smallest SSE.

Table 9 concludes the results of the t-tests. We can see that the SSE of the Single-index Model is not statistically different from that of the Fama-French Three-factor Model, but significantly smaller than that of the Carhart’s Four-factor Model. And the SSE of the Fama-French Three-factor Model is also significantly smaller than that of the Carhart’s Four-factor Model. These indicate that there’s no significant difference between the predictive power of the Single-index Model and the Fama-French Three-factor Model, but the Carhart’s Four-factor Model predicts stock returns of the Chinese market less well.

Table 9 Results of t-tests

This table concludes the results of the t-tests.

H0 1 (SSE1=SSE2) H0 2 (SSE1=SSE3) H0 3 (SSE2=SSE3)

t-statistic -0.9955 -1.8569 -1.6385

p value (H1 1) 0.1613 0.0335 0.0527

p value (H1 2) 0.3226 0.0671 0.1053

p value (H1 3) 0.8387 0.9665 0.9473

Notes: SSE 1 represents the sum of squared errors from the Single-index Model. SSE 2 represents the sum of squared errors from the Fama-French Three-factor Model. SSE 3 represents the sum of squared errors from the Carhart’s Four-factor Model. H0 stands for the null hypothesis. H1 stands for the alternative hypothesis. H1: the SSE of a model is smaller than that of another model (H1 1), the SSE of a model is not equal to that of another model (H1 2) the SSE of a model is bigger than that of another model (H1 3). Numbers in bold are statistically significant at 10% level.

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21 6. Robustness test

6.1 Statistically insignificant coefficients in the forecasting models

Since some of the coefficients of our forecasting models are not significant (see Table 7), we substitute these coefficients with 0 (see Table 10) and redo the predictive power part to see whether we can come to the same conclusions. From Table 11 we can still see that: no matter in sense of value or of number, the Single-index Model performs the best (has the smallest SSE and the largest number of months in which the model has the smallest SSE) in forecasting stock returns among these three models.

The results concluded in Table 12 differ a little bit from those in Table 9, with no statically significant difference between the SSE of the Fama-French Three-factor Model having and that of the Carhart’s Four-factor Model (almost significant though). Thus, we cannot say that the predictive power of the Fama-French Three-factor Model is stronger than that of the Carhart’s Four-factor Model anymore. However, this doesn’t change our final conclusion: as long as the former is not weaker than the latter, taking both explanatory power and predictive power into consideration, the Fama-French Three-factor Model is still the most suitable one for the current Chinese market among the three models considered.

Table 10 Coefficients of the forecasting models (2)

This table shows the coefficients of the three forecasting models built using data in each of the seven years with coefficients that are not significant equal to 0.

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22 2012 Single-index 0 0.8470 Fama-French 0.0036 0.7022 0.4963 -0.1204 Carhart's 0.0036 0.7002 0.5032 -0.1161 0 2013 Single-index 0.0190 1.1996 Fama-French 0.0236 1.1880 -0.2068 -0.1202 Carhart's 0.0224 1.1974 -0.2841 0 -0.0613

Notes: The results are estimated using OLS regressions based on 809 stocks. Carhart’s: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ ciUMDt+ei,t

Fama-French: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ei,t

Single-index: Ri,t− Rf,t= αi+ (Rm,t− Rf,t)βi +ei,t

Now, those statistically insignificant (at 10% level) coefficients in Table 7 are replaced by 0 and in bold.

Table 11 SSE of the three models (2)

This table concludes the SSE (sum of squared errors) of the 79 months of the three models respectively, using forecasting models of which the statistically insignificant coefficients are replaced by 0. SSE SSE N (max excluded) Single-index 0.2165 0.1655 33 Fama-French 0.2706 0.1979 26 Carhart's 0.3484 0.2381 20

Notes: SSE=∑(ERft-ERrt)^2 ERft is the forecast average excess return in month t, ERrt is the

actual excess return in month t. N is the number of months in which the model has the smallest SSE.

Table 12 Results of t-tests (2)

This table concludes the results of t-tests after substituting those statistically insignificant coefficients in the forecasting models by 0.

H0 1 (SSE1=SSE2) H0 2 (SSE1=SSE3) H0 3 (SSE2=SSE3)

t-statistic -1.0096 -1.6658 -1.2577

p value (H1 1) 0.1579 0.0499 0.1061

p value (H1 2) 0.3158 0.0998 0.2122

p value (H1 3) 0.8421 0.9501 0.8939

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23 6.2 The frequency of forecasting model adjustment

We now adjust our forecasting models every month instead of once a year to see whether the frequency of forecasting model adjustment influences our results. We select year 2008 for this robust test: we firstly use data from January 2007 to December 2007 to build our models to forecast the excess average return in January 2008 then we use data from February 2007 to January 2008 to build our models to forecast the excess average return in February 2008, and so forth. The coefficients of the resulting new forecasting models are concluded in Table 13.

Table 13 Coefficients of the forecasting models (3)

This table shows the coefficients of the three forecasting models for each of the 12 months in year 2008 with the forecasting models adjusted every month.

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24 Carhart's 0.0365 1.0571 0.7417 -0.4874 -0.068 11/2007-10/2008 Single-index 0.0617 1.1113 Fama-French 0.0341 1.0502 0.7571 -0.4864 Carhart's 0.0366 1.0576 0.7459 -0.4691 -0.0721 12/2007-11/2008 Single-index 0.0611 1.1149 Fama-French 0.0337 1.0542 0.7521 -0.4795 Carhart's 0.036 1.0611 0.7418 -0.4634 -0.0671

Notes: The results are estimated using OLS regressions based on 809 stocks. Carhart’s: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ ciUMDt+ei,t

Fama-French: Ri,t− Rf,t= αi+ βi(Rm,t− Rf,t) + aiSMBt+ biHMLt+ei,t

Single-index: Ri,t− Rf,t= αi+ (Rm,t− Rf,t)βi +ei,t

Column Time shows the time period of the data used to get the corresponding forecasting models for the very next month after the time period.

All the coefficients are significant at 10% level.

Afterwards, we recalculate the SSE in each of the 12 months in year 2008 of these new models (SSEnew) and compare them with those calculated before. Then, we use t-test to check whether SSEnews are statistically equal to SSEs as well as whether the results are still the same. We can see from Table 14 that now the Fama-French Three-factor Model and the Carhart’s Four-factor Model have stronger predictive power than the Single-index Model but there’s no statistically significant difference between the predictive powers of them. Thus this doesn’t change our final conclusion that the Fama-French Three-factor Model is the most suitable one. Besides, we can see from Table 15 that except for the Carhart’s Four-factor Model, of which the SSEnew is statistically smaller than the SSE at 10% significance level (SSEnew equals to SSE at 5% significance level), the SSEnew and the SSE are statistically the same for the other two models at 1% significance level. Thus we can say that the difference between adjusting the forecasting models once a year and every month is not statistically significant.

Table 14 Results of t-tests (3)

This table concludes the results from the t-tests to compare the SSEnews.

H0 1 (SSEnew1=SSEnew2) H0 2 (SSEnew1=SSEnew3) H0 3 (SSEnew2=SSEnew3)

t-statistic 2.3811 2.6967 0.9633

p value (H1 1) 0.9818 0.9896 0.8220

p value (H1 2) 0.0364 0.0208 0.3561

p value (H1 3) 0.0182 0.0104 0.1780

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25

of a model is smaller than that of another model (H1 1), the SSEnew of a model is not equal to that of another model (H1 2) the SSEnew of a model is bigger than that of another model (H1 3). Numbers in bold are statistically significant at 10% level.

Table 15 Results of t-tests (4)

This table concludes the results from the t-tests to see whether the SSEnews are statistically the same as the SSEs before.

H0 1 (SSEnew1=SSE1) H0 2 (SSEnew2=SSE2) H0 3 (SSEnew3=SSE3)

t-statistic 0.0860 -0.0205 -1.6819

p value (H1 1) 0.5335 0.4920 0.0604

p value (H1 2) 0.9330 0.9840 0.1207

p value (H1 3) 0.4665 0.5080 0.9396

Notes: SSEs are the sums of squared errors for the models before changing the forecasting model adjustment frequency while SSEnews are those after the change. 1 represents the Single-index Model. 2 represents the Fama-French Three-factor Model. 3 represents the Carhart’s Four-factor Model. H0 stands for the null hypothesis. H1 stands for the alternative hypothesis. H1: the SSEnew of a model is smaller than the SSE of the model (H1 1), the SSEnew of a model is not equal to the SSE of the model (H1 2), the SSE of a model is bigger than the SSE of the model (H1 3). Numbers in bold are statistically significant at 10% level.

7. Conclusions

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26 ratio and not-too-high earning-to-price ratio.

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27 References

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Lagged NPL is impaired loans over gross loans at time t-1, lagged reserve ratio is the loan loss reserves over impaired loans at time t-1, Slope EU/US is the yield curve

Table 8: The effect of the four components of Corporate Social Responsibility on Corporate Financial Performance as measured by return on assets for European companies from the

Above all, the disaggregated analysis implies that in subgroups of female and high-trust respondents, the happiness positively affects their holding of risky