_{1}

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.

CAPM (Capital asset pricing model) was initially proposed by Sharpe (1964) [

where,

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, J.C. (1978) used 700 samples in the NYSE (New York Stock Exchange) and drew a conclusion that the beta is unstable [

For the beta stability between single stock and portfolio, Levitz (1974) found that the beta is stable in portfolio and highly unstable in individual [

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 [

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 [

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.

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,

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,

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 [

where,

The model used to test the stability of beta coefficient including several dummy variables. [

where,

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 supposed to decrease if the dummy variable is significant and negative.

All the regression mentioned above were made by Excel and Eviews.

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.

The numbers of sample are 208 and the industry distribution has been showed in

1) Beta characters in different length of time

As showed in

Industry | Market proportion | NO. of samples | Sample proportion |
---|---|---|---|

Manufacturing | 56.69% | 56 | 26.92% |

Real estate | 8.98% | 50 | 24.04% |

Wholesale and retail | 8.38% | 30 | 14.42% |

Utilities | 4.86% | 25 | 12.02% |

Transportation | 4.19% | 19 | 9.13% |

Others | 16.90% | 28 | 13.47% |

Total | 100% | 208 | 100% |

Group | Length of time | Max | Min | Average | Standard deviation |
---|---|---|---|---|---|

1 | 6 months | 1.9728 | 0.1866 | 1.1305 | 0.0794 |

2 | 12 months | 2.1555 | 0.5288 | 1.1916 | 0.0649 |

3 | 18 months | 2.3744 | 0.6496 | 1.2257 | 0.0627 |

4 | 24 months | 2.0877 | 0.6181 | 1.1930 | 0.0546 |

5 | 36 months | 2.0069 | 0.5443 | 1.1634 | 0.0508 |

6 | 48 months | 1.9497 | 0.5461 | 1.1762 | 0.0464 |

7 | 60 months | 1.9127 | 0.5460 | 1.1716 | 0.0435 |

8 | 72 months | 1.8801 | 0.5059 | 1.1647 | 0.0416 |

Group | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

1 | ― | 2.321** | 3.643** | 2.463** | 1.314 | 1.858* | 1.693* | 1.417 |

2 | 2.321** | ― | 1.377 | 0.059 | 1.197 | 0.666 | 0.875 | 1.190 |

3 | 3.643** | 1.377 | ― | 1.377 | 2.669** | 2.162** | 2.393** | 2.725** |

4 | 2.463** | 0.059 | 1.377 | ― | 1.317 | 0.763 | 0.984 | 1.317 |

5 | 1.314 | 1.197 | 2.669** | 1.317 | ― | 0.593 | 0.388 | -0.062 |

6 | 1.858* | 0.666 | 2.162** | 0.763 | 0.593 | ― | 0.219 | 0.560 |

7 | 1.693* | 0.875 | 2.393** | 0.984 | 0.388 | 0.219 | ― | 0.344 |

8 | 1.417 | 1.190 | 2.725** | 1.317 | -0.062 | 0.560 | 0.344 | ― |

Note: ** indicates the coefficient is significant at 5% level. * indicates the coefficient is significant at 10% level.

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

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.

1) Regression results in different length of time for estimating

As showed in

When the significant level is 5%, the numbers of significant dummy variable are 348 which mean there 348 betas are unstable. When the significant level is

Dummy variables | Significant level 5% | Significant level 10% | ||||
---|---|---|---|---|---|---|

No. of significant | No. of significant and positive | No. of significant and negative | No. of significant | No. of significant and positive | No. of significant and negative | |

b1 | 46 | 32 | 14 | 65 | 45 | 20 |

b2 | 42 | 39 | 3 | 59 | 49 | 10 |

b3 | 39 | 10 | 29 | 54 | 14 | 40 |

b4 | 23 | 13 | 10 | 43 | 20 | 23 |

b5 | 54 | 9 | 45 | 69 | 11 | 58 |

b6 | 9 | 5 | 4 | 22 | 11 | 11 |

b7 | 24 | 17 | 7 | 41 | 29 | 12 |

b8 | 28 | 12 | 16 | 36 | 18 | 18 |

b9 | 27 | 10 | 17 | 45 | 20 | 25 |

b10 | 27 | 13 | 14 | 36 | 14 | 22 |

b11 | 29 | 14 | 15 | 44 | 19 | 25 |

Total | 348 | 174 | 174 | 514 | 250 | 264 |

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 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.

As shown in

As shown in

As shown in

As shown in

2) Length of time for estimating and beta stability

As shown in

Dummy variables | Significant level 5% | Significant level 10% | ||||
---|---|---|---|---|---|---|

No. of significant | No. of significant and positive | No. of significant and negative | No. of significant | No. of significant and positive | No. of significant and negative | |

b1 | 31 | 18 | 13 | 40 | 24 | 16 |

b2 | 50 | 4 | 46 | 67 | 7 | 60 |

b3 | 26 | 16 | 10 | 48 | 27 | 21 |

b4 | 38 | 16 | 22 | 50 | 20 | 30 |

b5 | 34 | 11 | 23 | 52 | 14 | 38 |

Total | 179 | 65 | 114 | 257 | 92 | 165 |

Dummy variables | Significant level 5% | Significant level 10% | ||||
---|---|---|---|---|---|---|

No. of significant | No. of significant and positive | No. of significant and negative | No. of significant | No. of significant and positive | No. of significant and negative | |

b1 | 68 | 5 | 63 | 79 | 6 | 73 |

b2 | 42 | 19 | 23 | 60 | 30 | 30 |

b3 | 48 | 13 | 35 | 71 | 20 | 51 |

Total | 158 | 37 | 121 | 210 | 56 | 154 |

Dummy variables | Significant level 5% | Significant level 10% | ||||
---|---|---|---|---|---|---|

No. of significant | No. of significant and positive | No. of significant and negative | No. of significant | No. of significant and positive | No. of significant and negative | |

b1 | 46 | 11 | 35 | 58 | 13 | 41 |

b2 | 50 | 18 | 32 | 61 | 21 | 40 |

Total | 96 | 29 | 67 | 119 | 34 | 81 |

Dummy variables | Significant level 5% | Significant level 10% | ||||
---|---|---|---|---|---|---|

No. of significant | No. of significant and positive | No. of significant and negative | No. of significant | No. of significant and positive | No. of significant and negative | |

b1 | 47 | 25 | 22 | 63 | 31 | 32 |

Length of time | Proportion of beta unstable | |
---|---|---|

Significant level 5% | Significant level 10% | |

6 months | 15.21% | 22.47% |

12 months | 17.21% | 24.71% |

18 months | 25.32% | 33.65% |

24 months | 23.08% | 28.61% |

36 months | 22.60% | 30.29% |

Note: the proportion calculated by the number of significant dummy variables divided by the total dummy variables. For the length of time is 6 months and significant level is 10%, the proportion 22.47% is calculate by 514/(208 × 11), 514 are the numbers of the significant dummy variables, 208 are the sample numbers, 11 are the dummy variables for each sample’s regression. For the length of time is 36 months and significant level is 10%, the proportion 30.29% is calculate by 63/(208 × 1),

nificant level is 5%, the proportion of unstable beta is 15.21% for 6 months and is 22.60% for 36 months. Mostly, the proportion was increasing with the length of time increased which means the beta is likely to become less stable as the estimation duration increases in china’s stock market. This conclusion is completely adverse with the investigations that made in the developed market which the beta tends to be more stable with the increased of the estimation duration.

Since the stability of the beta decreases with the increase of the estimated duration, the most stable for beta estimation is 6 month. In spite of that, the proportion of 12 months is slightly higher than that in 6 months. As weekly data has been used in the regression, there are nearly 32 data if 6 months are selected and there are nearly 52 data can be used if 12 months are selected. In thinks about this, the optimal estimation time is 12 months.

In china, the public stock market has been divided into four part which are main-board Market of Shanghai (SH), Main-board Market of Shenzhen (SZ), Small and Medium Enterprise Board that for the small and median size companies, Growth Enterprise Market that for the companies have a high growth rate. This paper only make researches on the first three markets as the Growth Enterprise Market was set up after 2009.

As showed in

Markets | Proportion of beta unstable (length of time: 12 months) | |
---|---|---|

Significant level 5% | Significant level 10% | |

Main board market of SZ | 16.63% | 25.12% |

Main board market of SH | 16.53% | 23.32% |

Main board market | 16.57% | 24.04% |

Small and medium enterprise board | 20.00% | 30.00% |

Total | 17.21% | 24.71% |

Industries | Proportion of beta unstable (length of time: 12 months) | |
---|---|---|

Significant level 5% | Significant level 10% | |

Manufacturing | 18.57% | 27.14% |

Real estate | 24.40% | 35.60% |

Wholesale and retail | 23.33% | 32.00% |

Utilities | 12.80% | 16.00% |

Transportation | 9.47% | 15.79% |

total | 17.21% | 24.71% |

As shown in

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.

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

Group | Proportion of beta unstable (Significant level 10%) | ||||
---|---|---|---|---|---|

6 months | 12 months | 18 months | 24 months | 36 months | |

1 | 24.68% | 26.67% | 33.33% | 29.76% | 26.19% |

2 | 19.48% | 24.29% | 30.95% | 35.71% | 28.57% |

3 | 21.00% | 27.14% | 35.71% | 35.71% | 28.57% |

4 | 23.59% | 30.95% | 40.48% | 35.71% | 30.95% |

5 | 31.36% | 23.50% | 34.17% | 23.75% | 32.50% |

When the length of time for estimation is 6 months, proportion of beta unstable group 1 and group 2 are 24.68% and 19.48, it is 23.59% and 31.36% for group 4 and group 5 which are much higher than group 1 and 2. It is similar when the length of time for estimation is 12 months, 18 months, 24 months and 36 months which means that smaller companies mostly have more stable betas and it is completely opposite with the assumption above.

This paper examined the stability of beta in China’s stock market across estimation duration, markets, industries and market size. The t statistic result shows that the estimation duration is really important for beta estimating, different length of time for regression can create a great difference in beta. Completely adverse with the conclusion in the developed market, the beta stability is likely to be more unstable with the increases of the estimation duration and beta tends to increases in the bull market and decreases in the bear market. In china, beta risk is much higher in the Small and Medium Enterprise Board than in the Main-board market, and it is higher in the Main-board market in SZ than in the Main-board market in SH. For industries, beta is less stable in cyclical industries and high growth industries. Also beta is supposed to be more stable for small companies than big companies.

However, more research still should be done further according to the article. For example, one of the conclusion that the beta tends to increase during the bull market and decrease during the bear market. Further investigation can be made to test if there is any arbitrage opportunity for this conclusion? Otherwise, the article hasn’t researched deeply on the reasons about the longer the estimation duration is, the more unstable the beta in China’s stock market which completely adverse with the conclusion in the developed market.

Ye, Y.P. (2017) The Stability of Beta Coefficient in China’s Stock Market. Journal of Service Science and Management, 10, 177-187. https://doi.org/10.4236/jssm.2017.102016