Estimating and Testing the Captial Asset Pricing Model

Introduction

Captial Asset Pricing Model (CAPM) is used to describle the relationship between the return of individual asset and the return of market portfoilo. Given the equation below:
(1)
Where represents the actual return of stock j in day t, is the return of market portfoilo in day t and represents the influences on its returns.

留學生dissertation網In the CAPM model, the parameter is 0 as usual. The parameter is the coefficient which measures the sensitivity of individual return against market return. This model has two assumptions: (1) market operates efficiently and (2) all investors are fully informed.

In this paper, the CAPM model is tested by running linear regression using TSM software. Two stocks chose from different industrials with 5-year montly data will be used to test the CAPM model econometrically. The two stocks are Citicorp (CITCRP) from Bank industrial and Mobil (MOBIL) from Oil industrial with the monthly data from Jan 1983 to Dec 1987.

Tests

1. Plots of the individual return against market return.

Firstly, the return of individual stock against market return can be shown graphically. Here are the time plots and scatter plots for returns of CITCRP and MOBIL against market.

The time plots of both stocks show the same characteristic that individual return moved with the market, even in the stock crisis in 1987. Both individual returns dropped dramatically along with the market in that year and rose back simultaneously with the market later. It predicts that both betas of the two stocks is close to 1.

The scatter plots describles the degree of correlation between the individual and market. Both two graphs illustrate that more spots stay in the two quadrants where means that both stocks have a positive correlation with the market.

2. Confidence intervals and significant test for the parameters
The parameter and can be estimated by ordinary least squares using TSM software. (see the TSM outputs in appendix 1)

The t statistic value is calculated by the equation below while t critical value can be looked up in TSM:
(2)

http://www.400190.com/dissertation_writing/Finance/The confidence intervals at a certain confidence level are calculated by the following equation:
(3)

Then the equation (3) can be rewrite as:
(4)

In this case, the t critical value can be looked up in TSM. With 58 denominators and 2.5% tail probability in t distribution, the t critical value is 2.00172.

The 95% confidence intevals for parameters of CITCRP and MOBIL can be calculated using the equation (4), which are :
CITCRP: ,
MOBIL: ,

#p#分頁標題#e#
 

The rule of the significance test here is that reject the null hypothsis : when t statistic value critical value. The null hypothsis for parameters of the CAPM model are =0 and =1 against >1 respectively.

For CITCRP, t statistic value 0.136 of parameter is less than the t critical value 2.00172. So it cannot reject the null hypothsis =0. The t statistic value 0.352 of parameter is less than the t critical value 1.67155. So it cannot reject the null hypothsis =1. For MOBIL, it also gets non-rejection on both parameters significant test.

The above tests shows that the CAPM model is fitted with both stocks owing to the satisfaction which is =0.

3. The individual risk of the stocks

The individual risk of the stocks can be measured by the standard deviation of . The risk of CITCRP is 0.0585 and the risk of MOBIL is 0.0599. (see the values from appendix 1) is introduced to measure how much percentage of the risk can be explained by the market. Given the equation below:
(5)
The higher the , the higher the porpotion of the risk is contributed to the market. The of CITCRP is 0.5347, which indicates that 53.47% of its risk is contibuted to the market; the of MOBIL is 0.3818, which suggests 38.18% of its risk is contributed to the market.

4. Comparing betas of the stocks

The beta of the CAPM model can also be calculated by another equation:
(6)

The beta of CITCRP and MOBIL can be calculated by equation (6) taking the values from appendix 3, which is approximate 1.43 and 3.32 respectively. Both betas are different from the estimates derived from running the linear regression for the CAPM single index model. The difference of MOBIL is even more than three times.

5. Chow stability test

In TSM, the stable of the model over different periods can be tested automatically. In this case, the first 5-year data can be used as sample and the second 5 years are chose for forecast period. The rule of chow stability test can be explained by the following equation:
(7)
Where represents the whole sample’s sum of squares, and is the sub-sample’s sum of squares.

Given the null hypothsis that the model is stable over the whole period against that the model is not stable over the whole period. Reject when . Alternatively, P-value can be used to determine this hypothsis. Reject when P-value is less than the certain significance level. (see the TSM outputs of this test in appendix 3)

The P-value of chow stability test for CITCRP is 0.004 which is less than 0.05. Reject the null hypothsis that the model are stable over the whole 10-year sample. So the CAPM model for CITCRP is not stable over the whole 10-year period. The reason may be various, such as the changes of the interest rate, financial regulation or money policy. As for MOBIL, it gets non-rejection in this test. So the CAPM model for MOBIL is stable over the whole 10-year period. The reason may be that the retrun of oil industrial is less susceptible than bank industrial.#p#分頁標題#e#

6. Strict CAPM against the Arbitrage Pricing Model

Test the significance of additional macroeconomic variables is to test the strict CAPM against the Arbitrage Pricing Model. In this case, three additional varibles are introduced, which are RINF representing the rate of inflation, GING representing the growth in industrial production and ROIL representing the changes in the real oil price.

The significance of the three additional variables can be considered individually by t statistics. Also it can be considered by testing their joint significance. The null hypothsis of joint significance is that all restricted parameters equal to 0 against that at least one rectricted parameter is not equal to 0. The rule of joint significance test can be explained by following equation:
(8)
Where represents the sum of squares of restricted regression and represents the sum of squares of unrestricted regression.

Reject the null hypothsis that all restricted parameters equal to 0 when > . P-value can also be used here. Reject the null hypothsis when P-value is less than the certain significance level such as 0.05.

For CITCRP, the significance of individual variable can be test by t statistic value which can be calculated by equation (2) . The t statistic value for variable GIND is 1.06 which is less than the t critical value 2.00404. The t statistic value for variable RINF is 1.012 which is less than the critical value 2.00404. The t statistic value for variable ROIL is 0.388 which is also less than the critical value 2.00404. All these varibles get non-rejection in the individual significance test. On the other hand, the result of joint significance test also get non-rejection, because the P-value 0.063 is bigger than the given significance level 0.05. For MOBIL, it also gets both non-rejections on individual and joint significance test. (see the TSM outputs of this test in appendix 4)

The results of this test show that the three addtional variables which are RINF (the rate of inflation), GING (the growth in industrial production) and ROIL (changes in the real oil price) have no effects on both stocks. The CAPM model is strict CAPM.

Conclusion

In this paper, the single index CAPM model is tested in four aspects, which are the significance test of model parameters, the estimate beta comparing with the beta calculated by mean, chow stability test and strict CAPM against Arbitrage Pricing Model. The result of these tests with selected stocks can be concluded that: (1) the single index CAPM model is fitted with both stocks. (2) the beta estimated by the linear regression is different from the beta calculated by expect return. (3) the CAPM model for CITCRP is not stable while the model for MOBIL is stable over time. (4) the model for both CITCRP and MOBIL are strict CAPM.#p#分頁標題#e#

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Appendix

1.
***********************************************************************
Dependent Variable is CITCRP
60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)
used for estimation
with 60 pre-sample observations.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.00099 0.00727 0.136 0.892
MARKET 1.03941 0.11188 9.29 0
R-Squared = 0.5347
Residual SD = 0.0585

***********************************************************************
Dependent Variable is MOBIL
60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)
used for estimation
with 60 pre-sample observations.
留學生dissertation網Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.00912 0.00756 1.207 0.232
MARKET 0.78093 0.14168 5.512 0
R-Squared = 0.3818
Residual SD = 0.0599
2.
*** Summary Statistics for CITCRP ***
Using 60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)
Mean = 0.0100333

*** Summary Statistics for MARKET ***
Using 60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)
Mean = 0.0087

*** Summary Statistics for RKFREE ***
Using 60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)
Mean = 0.005592

*** Summary Statistics for MOBIL ***
Using 60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)
Mean = 0.0159167

3.
***********************************************************************
Dependent Variable is CITCRP
60 observations (25-84, dates 1978, Mth 1 to 1982, Mth 12)
used for estimation
with 60 ex-post forecasts 1983, Mth 1 to 1987, Mth 12.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.00517 0.00879 0.588 0.559
MARKET 0.44137 0.1394 3.166 0.002
Chow Stability Test: ChiSq(2) = 11.2359 {0.004}

***********************************************************************
Dependent Variable is MOBIL
60 observations (25-84, dates 1978, Mth 1 to 1982, Mth 12)
used for estimation
with 60 ex-post forecasts 1983, Mth 1 to 1987, Mth 12.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.00339 0.00786 0.431 0.668
MARKET 0.67838 0.11102 6.11 0
Chow Stability Test: ChiSq(2) = 0.6869 {0.709}

4.
***********************************************************************
Dependent Variable is CITCRP
60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)#p#分頁標題#e#
used for estimation
with 60 pre-sample observations.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.02006 0.01502 1.335 0.187
MARKET 1.03338 0.11308 9.138 0
GIND -0.01212 0.01143 -1.06 0.294
RINF -0.04903 0.04845 -1.012 0.316
ROIL -0.00051 0.00131 -0.388 0.7
Wald Test of Zero Restrictions on:
GIND
RINF
ROIL
F(3,55) = 2.5709 {0.063}

***********************************************************************
Dependent Variable is MOBIL
60 observations (85-144, dates 1983, Mth 1 to 1987, Mth 12)
used for estimation
with 60 pre-sample observations.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.00895 0.0169 0.53 0.598
MARKET 0.76585 0.14731 5.199 0
GIND -0.00226 0.01079 -0.209 0.835
RINF 0.00995 0.04817 0.206 0.837
http://www.400190.com/dissertation_writing/Finance/ROIL 0.00118 0.00151 0.781 0.438
Wald Test of Zero Restrictions on:
GIND
RINF
ROIL
F(3,55) = 0.6193 {0.605}