The Stability of Beta Coefficient in China ’ s Stock Market

CAPM (Capital asset pricing model) is widely used in asset pricing, project evaluating and investment deciding. Beta coefficient, one of the core tasks of CAPM, its accuracy and stability are of great significance. Weekly China’s stock return data have been used. Firstly, analyzed the differences of mean value, maximum value and minimum value of beta coefficients which regressed by different length of time. Secondly, introduced T statistic to test the mean difference of beta which regressed by different length of time. Thirdly, used dummy variables to test the stability of beta coefficients and found that the optimal length of time for beta estimating was 12 months. In addition, several investigations about the relationship between the stability of bate coefficients and markets, industries, market size have been done finally.

The formula of CAPM is following: where, i r is weekly return of stock ( ) r is the risk free rate.m r is weekly return of market portfolio.i β is the beta coefficient of stock ( ) Beta coefficient, the slope of CAPM, measures the return sensitive between a single stock and the market index which is regarded as a measurement of systematic risk.Beta is 1 for market portfolio.
In researching about the stability of beta coefficients, Fabozzi, F.J. and Francis, Y. P. Ye J.C. (1978) used 700 samples in the NYSE (New York Stock Exchange) and drew a conclusion that the beta is unstable [2].In researching the relationship between the estimate duration and the beta stability, Marshall E. Blume (1971) did the researches and proved the beta was more stable with the increases of the estimation duration and more stable in bigger [3].Rodeny, Gar and Griepentrog (1978) used the sampled in Standard and Poor and found that the optimal duration for beta estimation are 4 years [4].Robert D. Brooks, Robert W. Faff and Mohamed Ariff (1998) used the data from Singapore Exchange and the time phase was from 1986 to 1993 and drew a conclusion that the beta is unstable in Singapore's stock market [5].
For the beta stability between single stock and portfolio, Levitz (1974) found that the beta is stable in portfolio and highly unstable in individual [6].
The researches above are mainly in a quote-driven market and developed market.For order-driven market, Keith S. K. Lam (1999) took investigations on Hong Kong market for the period 1980-1993 and found that the beta is stable in short and median term but unstable in long term [7].
For developing countries, Soumya Guha Deb and Sagarika Misra (2011) used dummy variable model test the beta stability in Indian stock market, time phase was from 1996 to 2010, proved that the beta was unstable in the short-term [8].
The paper explores the beta stability in China's stock market, aims to find the relationship between the beta stability and the estimate duration, enhances the beta stability studies in China.

Sources of Data
The numbers of Sample were 208 and were selected in the Shanghai Stock Exchange and the Shenzhen Stock Exchange which were listed before 2008.The method for sample selection was stratified sampling method according to the market size and industries.Firstly, the proportion of sample is determined by the population in different industry.For example, if the proportion in the manufacturing industry according to the population is 30%, then the sample numbers in it are 62 (calculated by 208 × 30%).Secondly, as the sample numbers in each in different industry has been determined.Population in each industry is ranged from smallest size to biggest size.After that, divided the population into several cells and take the samples from different cells in order to make the samples differentiate in market value.
The Shanghai Composite Index has been selected as a proxy of market portfolio.Time periods for this study last from January 2008 to December 2013.
Weekly China's stock return data have been used and the formula for return calculating is following: where, i r is the weekly return of stock i .t p is the close price for stock i for week t. 1 t p − is the close price for stock i for week 1 t − .Y. P. Ye

Using t Statistic to Test Beta Difference
Based on the weekly yield data of the listed companies, the estimated length of 6 months, 12 months, 18 months, 24 months, 36 months, 48 months, 60 months and 72 months using OLS beta Value, the calculation model for the capital asset pricing model, as follows: ( ) where, i r is weekly return of stock ( ) m r is weekly return of market portfolio.i β is the beta coefficient of stock ( ) Calculate and compare the difference between the mean value, the maximum value and the minimum value of the beta at different length of time and using T statistic model to test the mean difference [8].
where, i x is the mean value of beta in different length of time for time i .j x is the mean value of beta in different length of time for time j .n − .The null hypothesis is there is no difference in beta between time I and time j and the significant level are 5% and 10%.

Using Dummy Variable to Test the Beta Stability
The model used to test the stability of beta coefficient including several dummy variables.[8] ( ) where, i r is weekly return of stock ( ) is the numbers of dummy variables.
When the length of time for estimation is 6 months, k is 11.When the length of time for estimation is 12 months, k is 5.When the length of time for estimation is 18 months, k is 3.When the length of time for estimation is 24 months, k is 2.
When the length of time for estimation is 36 months, k is 1. j D is dummy variable j. ji b is the coefficient of dummy variable j and stock i.i ε is the residual item.
The significant level is 5% and 10%.If the dummy variable is significant means beta is unstable, conversely, if the dummy variable is un-significant means the beta is stable.Otherwise, beta is supposed to increase in the corresponding time phase if the dummy variable is significant and positive, it is sup-Y.P. Ye posed to decrease if the dummy variable is significant and negative.
All the regression mentioned above were made by Excel and Eviews.

Additional Investigations
Additional investigations include testing the difference of beta stability in difference industries, difference markets and the relationship between the beta stability and the market size.

Sample Selection
The numbers of sample are 208 and the industry distribution has been showed in Table 1.Companies listed before 2008 mainly concentrated in manufacturing industry and the proportion is 56.59%.In order to make more researches in different industries this paper lower its proportion to 26.92%.The five highest proportion industries including manufacturing, real estate, wholesale and retail, utilities, transportation.As showed in Table 1.average beta mainly range from 1.1 to 1.2 and the standard deviations are less than 0.1 which indicates that the beta difference is small.As the market beta is 1.0 which means the systematic risk of the listed companies are highly similar with the systematic risk of the market.With the assumption of the stocks' price are equal to their value and there is no excess return for unsystematic risk, there is little difference of return in investing in single stock and market index.

2) Mean difference of beta coefficients
As showed in Table 3, T statistic test result which used to examine the mean difference of beta that estimated by different length of time has been showed in exhibit 3. Group 1 to 8 is the same as Table 3, group 1 means the length of time is 6 months and group 8 means the length of time is 72 months.
When the significant level is 5%, the number of significant in the group is 7, the probability is 25.00%, that is, the probability of the mean betas are different in different estimated length of time is 25.00%.When the significant level is 10%, the number of significant is 10, the probability is 32.14%, that is, the probability of the mean betas are different in different estimated length of time is 32.14%.It can be said that the length of time for betas estimating cause a considerable difference and the selection of different estimation times is very important for beta estimating.

Stability of Beta Coefficients 1) Regression results in different length of time for estimating
As showed in Table 4, the numbers of dummy variable that are significant in the significant level are 5% and 10%.As the time phase for the regression is from 10%, the numbers of significant dummy variable are 514 which mean there are 514 unstable betas.Otherwise, the dummy variables which are significant and positive are more than that are significant and negative in b1, b2 and b7 which mean the betas are supposed to increase, the corresponding time phases in this As shown in Table 5, the significant numbers of dummy variable are 179 when significant level is 5% and 257 when it is 10%.Similar to above, the betas were supposed to increase when the market was in a bull market and decrease when the market was in a bear market.
As shown in Table 6, when the length was 18 months, January 2008 to June 2009 was the base period and b1 is from July 2009 to December 2010, and so on.
As shown in Table 6, the significant numbers are 158 for significant level is 5% and 210 for significant level is 10%.
As shown in Table 7, when the length was 24 months, January 2008 to December 2009 was the based period.The significant numbers are 96 when significant level is 5% and 119 when it is 10%.
As shown in Table 8, when the length was 36 months, January 2008 to De-Y.P. Ye cember 2010 was the based period.The significant numbers are 47 when significant level is 5% and 63 when it is 10%.

2) Length of time for estimating and beta stability
As shown in Table 9, when the significant level is 10%, the proportion of unstable beta is 22.47% for 6 months and is 30.29% for 36 months.When the sig-

Beta Stability in Different Industries
As shown in Table 11, the industry difference in beta stability showed that manufacturing, real estate and wholesale and retail have the highest unstable beta while utilities and transportation have the lowest unstable beta.Manufacturing, real estate and wholesale and retail are all cyclical industries.The proportion of beta unstable in real estate industry is 35.60% when the significant level is 10% and it is the highest among all.The high growth rate of housing market and the high leverage in china's real estate companies could be the reason about the high unstable of their beta coefficients.At the same time both manufacture and wholesale and retail have experienced a high growth rate during the past ten years which could be the main reason.
Utilities and transportation are both noncyclical industries mostly have the characteristics of low growth rate, stable income and cash flow, low correlation with the economic growth and those can be the main reasons that course they have the lowest stable of beta.

Beta Stability and Market Value
This part aimed to make researches about whether bigger companies have more stable beta coefficients.As the bigger companies mostly have various businesses and higher diversified, their business may more resistant to risk and their beta coefficients may more stable.In this part, the samples were divided into 5 groups according their market value which group 1 represents the smallest companies and group 5 represents the biggest companies.As showed in Table 12.

s
is the variance of the beta for time i and 2j x s is the va- riance of the beta for time ( ) j i j ≠ .Degrees of freedom for T statistic are 2 2 three coefficients are July 2008 to December 2008, January 2009 to June 2009, July 2011 to December 2011 when the market in China were mostly bull markets.Otherwise, the numbers of dummy variables which are significant and negative are less than that are significant and positive for the other coefficients when the market were mostly a bear market.Thus, beta is tend to increase in the bull market and decrease in the bear market.When the length of time for estimation is 12 months, the based time phase is January 2008 to December 2008, the corresponding time phase for b1 is January 2009 to December 2009, and so on, b5 is from January 2013 to December 2013.

Table 1 .
Industries distribution of samples.
1) Beta characters in different length of timeAs showed in Table2, all the beta coefficients are range from 0.5 to 2.0, the

Table 2 .
Beta characters in different length of time.

Table 3 .
T statistic test result.

Table 4 .
Regression results for 6 months.

Table 5 .
Regression results for 12 months.

Table 6 .
Regression results for 18 months.

Table 7 .
Regression results for 24 months.

Table 8 .
Regression results for 36 months.

Table 10 .
Beta stability in different markets.

Table 11 .
Beta stability in different industries.