Investigating The Relationship Between Firm Characteristics And Audit Costs
Hypothesis
The Capital Asset Pricing Model is used to analyze the tradeoff between the risk and the return of the stock. It is considered as one of the most important and valuable contribution to the finance. It is known that to earn higher return the investors have to take higher risk, however it is important to know how much risk is optimal. The CAPM model was first introduced by the Sharpe in 1964 followed by Treynor in 1961. Furthermore, later academicians such as Mossin (1966), Black in 1972 has also contributed to the Capital Asset Pricing Model(Lee et al., 2016; Pennacchi, 2008).
One of the main assumptions of the CAPM model is that the positive tradeoff between risk and return asserts that the expected return on any assets/stock is a positive function of only one variable which is the market beta(Arx and Ziegler, 2008; Tille and Wincoop, 2013).
In the current research the Capital Assets Pricing Model has been tested for 6 different portfolios. Monthly return for the entire portfolio and the return on market have been taken into consideration. The risk free rate has been used to calculate the excess return for each portfolio. The CAPM model has been tested using the regression model whereas the goodness of fit of the model has been tested on the basis of R squared and the F statistics. Since the data is time series following hypothesis has been tested which is the standard time series CAPM model:
Null Hypothesis: The intercept is not significantly equal to zero.
Alternative hypothesis the intercept is significantly equal to zero.
Results from the correlation matrix are shown in the table below and the results show that the return on the market and the returns on the six different portfolios included in the study are positively and significantly correlated. In other words, if one variable increases the other variables also increases. However the correlation do not guarantee the causation .
|
Correlations |
||||||||
|
mrt_rf |
Small med |
Small high |
Big med |
|||||
|
mrt_rf |
Pearson Correlation |
1 |
.868** |
.880** |
.840** |
.973** |
.927** |
.874** |
|
Sig. (2-tailed) |
.000 |
.000 |
.000 |
.000 |
.000 |
.000 |
||
|
N |
492 |
492 |
492 |
492 |
492 |
492 |
492 |
|
|
Small low |
Pearson Correlation |
.868** |
1 |
.933** |
.871** |
.823** |
.722** |
.699** |
|
Sig. (2-tailed) |
.000 |
.000 |
.000 |
.000 |
.000 |
.000 |
||
|
N |
492 |
492 |
492 |
492 |
492 |
492 |
492 |
|
|
Small med |
Pearson Correlation |
.880** |
.933** |
1 |
.968** |
.800** |
.815** |
.814** |
|
Sig. (2-tailed) |
.000 |
.000 |
.000 |
.000 |
.000 |
.000 |
||
|
N |
492 |
492 |
492 |
492 |
492 |
492 |
492 |
|
|
Small high |
Pearson Correlation |
.840** |
.871** |
.968** |
1 |
.739** |
.799** |
.837** |
|
Sig. (2-tailed) |
.000 |
.000 |
.000 |
.000 |
.000 |
.000 |
||
|
N |
492 |
492 |
492 |
492 |
492 |
492 |
492 |
|
|
Big low |
Pearson Correlation |
.973** |
.823** |
.800** |
.739** |
1 |
.866** |
.789** |
|
Sig. (2-tailed) |
.000 |
.000 |
.000 |
.000 |
.000 |
.000 |
||
|
N |
492 |
492 |
492 |
492 |
492 |
492 |
492 |
|
|
Big med |
Pearson Correlation |
.927** |
.722** |
.815** |
.799** |
.866** |
1 |
.899** |
|
Sig. (2-tailed) |
.000 |
.000 |
.000 |
.000 |
.000 |
.000 |
||
|
N |
492 |
492 |
492 |
492 |
492 |
492 |
492 |
|
|
Big high |
Pearson Correlation |
.874** |
.699** |
.814** |
.837** |
.789** |
.899** |
1 |
|
Sig. (2-tailed) |
.000 |
.000 |
.000 |
.000 |
.000 |
.000 |
||
|
N |
492 |
492 |
492 |
492 |
492 |
492 |
492 |
|
|
**. Correlation is significant at the 0.01 level (2-tailed). |
For all six different portfolios six different regression model was performed and the result are discussed below.
- Small low and market return
As the summary output shows the value of R squared is 0.75 which shows that the goodness of fit is good. It shows that 75 % of the variation in the dependent variable is explained by the independent variable and the rest of the variation is due to some other factors.
|
SUMMARY OUTPUT |
|
|
Regression Statistics |
|
|
Multiple R |
0.868463 |
|
R Square |
0.754228 |
|
Adjusted R Square |
0.753726 |
|
Standard Error |
3.306693 |
|
Observations |
492 |
Table 1Summary output from the regression analysis
|
ANOVA |
|||||
|
df |
SS |
MS |
F |
Significance F |
|
|
Regression |
1 |
16441.93 |
16441.93 |
1503.713714 |
1.9872E-151 |
|
Residual |
490 |
5357.766 |
10.93422 |
||
|
Total |
491 |
21799.7 |
Correlation matrix
Table 2 ANOVA table from regression analysis
Similarly the results from the ANOVA table also shows that the F statistic of 1503.71 is statistically significant which as the p value is less than 0.05. So the cumulative impact of the independent variable on the dependent variable is statistically significant.
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
-0.22111 |
0.150452 |
-1.46963 |
0.1423048 |
-0.516719713 |
0.074503 |
-0.51672 |
0.074502596 |
|
X Variable 1 |
1.303058 |
0.033603 |
38.77775 |
1.9872E-151 |
1.237033961 |
1.369082 |
1.237034 |
1.369082421 |
Table 3 Results for regression coefficients
The results shows that the intercept is not 0 and the p value is also more than 0.05 we cannot reject the null hypothesis. So the CAPM do not hold for this portfolio. The beta in this case is more than 1 which shows that the return on portfolio is higher than the risk free rate(Çelik, 2012).
- Small mid and market return
|
Regression Statistics |
|
|
Multiple R |
0.880209 |
|
R Square |
0.774769 |
|
Adjusted R Square |
0.774309 |
|
Standard Error |
2.454692 |
|
Observations |
492 |
Table 4 Summary output from the regression analysis
In case of small mid portfolio also the R square and the adjusted R squared are both 0.77 indicating that the change in the market return explains 77 % change in the return in the portfolio, which is considered to be a very good goodness of fit.
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
1 |
10156.25 |
10156.25 |
1685.541 |
1E-160 |
|
|
Residual |
490 |
2952.501 |
6.025513 |
|||
|
Total |
491 |
13108.75 |
Table 5 ANOVA table from regression analysis
The statistically significant F statistics of 1685 also indicates that the regression model is fit, and it also indicates that the cumulative impact of the independent variable on the dependent variable is significant.
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
0.345717 |
0.111687 |
3.095413 |
0.002078 |
0.126273 |
0.565161 |
0.126273 |
0.565161 |
|
X Variable 1 |
1.024128 |
0.024945 |
41.05535 |
1E-160 |
0.975115 |
1.07314 |
0.975115 |
1.07314 |
Table 6 Results for regression coefficients
Results from the regression coefficients show that that the intercept is 0.34 which is statistically significant at 5 %. So the null hypothesis can be rejected and the alternative hypothesis can be accepted, which indicates that the CAPM model holds for this portfolio. The beta in this case is more than 1 indicating that the return is higher than the risk free rate(ShwetaBajpai and K.Sharma, 2015; Smith and Walsh, 2013).
The third portfolio is the small high portfolio and the results from the regression analysis are shown in the table below.
The summary statistics shows that the 70 % variation in the return for this portfolio is being explained by the variation in the market whereas rest of the variation is due to some other factors.
|
Regression Statistics |
|
|
Multiple R |
0.83951 |
|
R Square |
0.704777 |
|
Adjusted R Square |
0.704174 |
|
Standard Error |
2.914165 |
|
Observations |
492 |
Table 7 Summary output from the regression analysis
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
1 |
9934.037 |
9934.037 |
1169.762 |
6.6E-132 |
|
|
Residual |
490 |
4161.255 |
8.492358 |
|||
|
Total |
491 |
14095.29 |
Table 8 ANOVA table from regression analysis
Similarly the results from the ANOVA table suggests that the F statistics of 1169.762 is statistically significant at 5 % significance level suggesting that the cumulative impact of the independent variable on the dependent variable is statistically signficant.
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
0.444474 |
0.132593 |
3.35218 |
0.000864 |
0.183954 |
0.704994 |
0.183954 |
0.704993981 |
|
X Variable 1 |
1.012862 |
0.029614 |
34.20178 |
6.6E-132 |
0.954675 |
1.071049 |
0.954675 |
1.071048704 |
Table 9 Results for regression coefficients
Furthermore the results from the regression coefficients show that the intercept of 0.44 is statistically significant so the null hypothesis can be rejected and the alternative hypothesis can be accepted. This implies that the CAPM holds for this portfolio. The beta value in this case is more than 1 indicating the return on the portfolio is higher than that of the risk free return.
- Big low and market return
Regression analysis
Similarly the regression result for the big low asset portfolio is shown in the table below. The Adjusted R square fo 0.94 suggests that the variation in the market is able to explain94 % of the variation in this portfolio. In other words the return on these assets moves together with return on the market.
|
Regression Statistics |
|
|
Multiple R |
0.972708 |
|
R Square |
0.946161 |
|
Adjusted R Square |
0.946051 |
|
Standard Error |
1.067202 |
|
Observations |
492 |
Table 10 Summary output from the regression analysis
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
1 |
9807.5 |
9807.5 |
8611.229 |
0 |
|
|
Residual |
490 |
558.0707 |
1.13892 |
|||
|
Total |
491 |
10365.57 |
Table 11 ANOVA table from regression analysis
Results from the ANOVA table suggest that the F value of 8611.29 is statistically significant as the p value is less than 0.05. On the basis of this, it can be said that the cumulative impact of the independent variable on the dependent variable is statistically significant.
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
||
|
Intercept |
-0.03019 |
0.048557 |
-0.62183 |
0.534344 |
-0.1256 |
0.065211 |
-0.1256 |
0.065211 |
|
|
X Variable 1 |
1.006391 |
0.010845 |
92.79671 |
0 |
0.985082 |
1.027699 |
0.985082 |
1.027699 |
Table 12 Results for regression coefficients
Furthermore the results from the regression coefficients suggest that the intercept of -0.03 is not statistically significant as the p value is more than 0.05, so the null hypothesis cannot be rejected. The beta value in this case is also more than 1 suggesting that the return on the portfolio is more than the risk free rate.
- Big medium and market return
The regression results for the big medium portfolio shows that the R squared is 0.85 which suggests that 85 % of the variation in the dependent variable is due to the independent variable and rest is due to some other factors. The R squared value of 0.85 shows better goodness of fit of the regression model.
|
Regression Statistics |
|
|
Multiple R |
0.927107 |
|
R Square |
0.859527 |
|
Adjusted R Square |
0.85924 |
|
Standard Error |
1.628737 |
|
Observations |
492 |
Table 13 Summary output from the regression analysis
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
1 |
7953.622 |
7953.622 |
2998.217 |
5.6E-211 |
|
|
Residual |
490 |
1299.864 |
2.652784 |
|||
|
Total |
491 |
9253.486 |
Table 14 ANOVA table from regression analysis
Similarly the results from the ANOVA table shows that the F statistics is statistically significant so the cumulative impact of the independent variable on the dependent variable is significant.
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
0.114203 |
0.074106 |
1.541071 |
0.123945 |
-0.0314 |
0.259809 |
-0.0314 |
0.25980885 |
|
X Variable 1 |
0.906296 |
0.016552 |
54.75597 |
5.6E-211 |
0.873775 |
0.938816 |
0.873775 |
0.938816368 |
Table 15 Results from the regression analysis
As shown in the table above the regression coefficient of intercept is 0.11 which is not statistically significant so the null hypothesis cannot be rejected. Furthermore the bête coefficient is less than one suggesting that the return on the risk free rate is higher than the return on this portfolio.
- Big high and the market return
The last regression a result is for the big high and the market return and the results shows that 76 % of the variation in the portfolio return is due to change in the market return and other variation is due to some other factors. The adjusted R squared value indicates that the model goodness of fit is more than the minimum requirement of 0.6.
|
Regression Statistics |
|
|
Multiple R |
0.874268 |
|
R Square |
0.764344 |
|
Adjusted R Square |
0.763863 |
|
Standard Error |
2.350438 |
|
Observations |
492 |
Table 16 Summary output from the regression analysis
|
ANOVA |
|||||
|
df |
SS |
MS |
F |
Significance F |
|
|
Regression |
1 |
8780.199 |
8780.199 |
1589.303 |
6.7E-156 |
|
Residual |
490 |
2707.035 |
5.524561 |
||
|
Total |
491 |
11487.23 |
Table 17 ANOVA table from regression analysis
Similarly the significant value of the F statistics also indicate that the model is a good fit and the cumulative impact of the independent variable on the dependent variable is significant.
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
0.14151 |
0.106943 |
1.323223 |
0.186378 |
-0.06861 |
0.351634 |
-0.06861 |
0.351634 |
|
X Variable 1 |
0.952225 |
0.023886 |
39.86606 |
6.7E-156 |
0.905294 |
0.999156 |
0.905294 |
0.999156 |
Table 18 Results for regression coefficients
Results from the regression coefficients shows that the intercept is not statistically significant so the null hypothesis cannot be rejected so the CAPM model does not satisfy. Furthermore the beta coefficient in this case is less than 1 which suggests that the return on this portfolio is less than the risk free rate.
For the Fama –French three factor model regression analysis was conducted where the dependent variable as the excess return of the portfolio and the independent variable includes three factors namely market excess return, size (smb) and book to market (hml). Separate regression was performed for each portfolio and results from regression analysis are shown below:
- Small low portfolio
|
Regression Statistics |
|
|
Multiple R |
0.988219 |
|
R Square |
0.976576 |
|
Adjusted R Square |
0.976432 |
|
Standard Error |
1.022931 |
|
Observations |
492 |
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
3 |
21289.06 |
7096.353 |
6781.758 |
0 |
|
|
Residual |
488 |
510.6375 |
1.046388 |
|||
|
Total |
491 |
21799.7 |
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
||
|
Intercept |
-0.22444 |
0.047178 |
-4.75732 |
2.59E-06 |
-0.31714 |
-0.13174 |
-0.31714 |
-0.13174 |
|
|
mkt-fr |
1.099176 |
0.010969 |
100.2115 |
0 |
1.077625 |
1.120727 |
1.077625 |
1.120727 |
|
|
smb |
0.99952 |
0.016119 |
62.00852 |
1.5E-233 |
0.967849 |
1.031191 |
0.967849 |
1.031191 |
|
|
hml |
-0.26497 |
0.017074 |
-15.5192 |
1.94E-44 |
-0.29851 |
-0.23142 |
-0.29851 |
-0.23142 |
- Small medium portfolio
|
Regression Statistics |
|
|
Multiple R |
0.987547 |
|
R Square |
0.975249 |
|
Adjusted R Square |
0.975097 |
|
Standard Error |
0.815398 |
|
Observations |
492 |
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
3 |
12784.29 |
4261.432 |
6409.383 |
0 |
|
|
Residual |
488 |
324.4585 |
0.664874 |
|||
|
Total |
491 |
13108.75 |
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
0.106088 |
0.037607 |
2.820994 |
0.004983 |
0.032197 |
0.179979 |
0.032197 |
0.179979 |
|
mkt-rf |
0.961442 |
0.008743 |
109.9639 |
0 |
0.944263 |
0.978621 |
0.944263 |
0.978621 |
|
smb |
0.785238 |
0.012849 |
61.11359 |
8.1E-231 |
0.759992 |
0.810484 |
0.759992 |
0.810484 |
|
hml |
0.358109 |
0.01361 |
26.31295 |
1.19E-95 |
0.331369 |
0.38485 |
0.331369 |
0.38485 |
- Small high portfolio
|
Regression Statistics |
|
|
Multiple R |
0.994066 |
|
R Square |
0.988168 |
|
Adjusted R Square |
0.988095 |
|
Standard Error |
0.584592 |
|
Observations |
492 |
|
ANOVA |
||||||
|
df |
SS |
MS |
F |
Significance F |
||
|
Regression |
3 |
13928.52 |
4642.84 |
13585.57 |
0 |
|
|
Residual |
488 |
166.7729 |
0.341748 |
|||
|
Total |
491 |
14095.29 |
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
0.052773 |
0.026962 |
1.957347 |
0.050876 |
-0.0002 |
0.105749 |
-0.0002 |
0.105749 |
|
mkt rf |
0.997806 |
0.006268 |
159.1806 |
0 |
0.98549 |
1.010123 |
0.98549 |
1.010123 |
|
smb |
0.857891 |
0.009212 |
93.12913 |
0 |
0.839791 |
0.875991 |
0.839791 |
0.875991 |
|
hml |
0.701605 |
0.009757 |
71.90563 |
7.8E-262 |
0.682433 |
0.720776 |
0.682433 |
0.720776 |
- Big low portfolio
|
SUMMARY OUTPUT |
|
|
Regression Statistics |
|
|
Multiple R |
0.98948 |
|
R Square |
0.979071 |
|
Adjusted R Square |
0.978943 |
|
Standard Error |
0.666744 |
|
Observations |
492 |
|
ANOVA |
|||||
|
df |
SS |
MS |
F |
Significance F |
|
|
Regression |
3 |
10148.63 |
3382.877 |
7609.705 |
0 |
|
Residual |
488 |
216.9393 |
0.444548 |
||
|
Total |
491 |
10365.57 |
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
0.106894 |
0.030751 |
3.47615 |
0.000554 |
0.046474 |
0.167314 |
0.046474 |
0.167314 |
|
mkt-rf |
0.98431 |
0.007149 |
137.6796 |
0 |
0.970263 |
0.998357 |
0.970263 |
0.998357 |
|
rmb |
-0.16706 |
0.010506 |
-15.9008 |
3.56E-46 |
-0.1877 |
-0.14642 |
-0.1877 |
-0.14642 |
|
hml |
-0.28192 |
0.011128 |
-25.3329 |
5.32E-91 |
-0.30378 |
-0.26005 |
-0.30378 |
-0.26005 |
- Big medium portfolio
|
Regression Statistics |
||
|
Multiple R |
0.960867 |
|
|
R Square |
0.923266 |
|
|
Adjusted R Square |
0.922794 |
|
|
Standard Error |
1.206251 |
|
|
Observations |
492 |
|
ANOVA |
|||||
|
df |
SS |
MS |
F |
Significance F |
|
|
Regression |
3 |
8543.426 |
2847.809 |
1957.2 |
1.5E-271 |
|
Residual |
488 |
710.0605 |
1.455042 |
||
|
Total |
491 |
9253.486 |
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
-0.00297 |
0.055633 |
-0.05331 |
0.957505 |
-0.11228 |
0.106344 |
-0.11228 |
0.106344 |
|
X Variable 1 |
0.988368 |
0.012934 |
76.4148 |
1.1E-273 |
0.962954 |
1.013781 |
0.962954 |
1.013781 |
|
X Variable 2 |
-0.16498 |
0.019008 |
-8.67938 |
5.98E-17 |
-0.20232 |
-0.12763 |
-0.20232 |
-0.12763 |
|
X Variable 3 |
0.324993 |
0.020133 |
16.14206 |
2.8E-47 |
0.285434 |
0.364551 |
0.285434 |
0.364551 |
- Big high portfolio
|
Regression Statistics |
||
|
Multiple R |
0.974915 |
|
|
R Square |
0.950459 |
|
|
Adjusted R Square |
0.950154 |
|
|
Standard Error |
1.079897 |
|
|
Observations |
492 |
|
ANOVA |
|||||
|
df |
SS |
MS |
F |
Significance F |
|
|
Regression |
3 |
10918.14 |
3639.38 |
3120.777 |
0 |
|
Residual |
488 |
569.0946 |
1.166177 |
||
|
Total |
491 |
11487.23 |
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
-0.17013 |
0.049805 |
-3.41592 |
0.000689 |
-0.26799 |
-0.07227 |
-0.26799 |
-0.07227 |
|
mkt-rf |
1.085606 |
0.011579 |
93.75334 |
0 |
1.062854 |
1.108358 |
1.062854 |
1.108358 |
|
rmb |
-0.02531 |
0.017017 |
-1.48733 |
0.137574 |
-0.05874 |
0.008126 |
-0.05874 |
0.008126 |
|
hml |
0.751493 |
0.018024 |
41.69326 |
5.9E-163 |
0.716078 |
0.786908 |
0.716078 |
0.786908 |
Residuals from each regression line are shown in the excel sheet. The residual shows the difference between the actual and the predicted values. Lower the values of residuals better the prediction. The residuals can be used to test whether the prediction made by the regression analysis are in line with the actual values or not. Another important assumption is that the residuals should follow the normal distribution. In case of high residual values more variables can be included in the regression or different market can be used which can explain the movements in the portfolio return more accurately.
References:
Arx, U. Von, Ziegler, A., 2008. The Effect of CSR on Stock Performance?: New Evidence for the USA and Europe Economics Working Paper Series The Effect of CSR on Stock Performance?: New Evidence for the USA and Europe 43.
Çelik, ?., 2012. Theoretical and Empirical Review of Asset Pricing Models: A Structural Synthesis. Int. J. Econ. Financ. Issues 2, 141–178.
Lee, H.-S., Cheng, F.-F., Chong, S.-C., 2016. Markowitz Portfolio Theory and Capital Asset Pricing Model for Kuala Lumpur Stock Exchange: A Case Revisited. Int. J. Econ. Financ. Issues 6, 59–65.
Pennacchi, G., 2008. Theory of Asset Pricing. Pearson/Addison, Boston.
ShwetaBajpai, K.Sharma, A., 2015. An Empirical Testing of Capital Asset Pricing Model in India. Elsevier 189, 259–265.
Smith, T., Walsh, K., 2013. Why the CAPM is half-right and everything else is wrong. Elsevier 49, 73–78.
Tille, C., Wincoop, E. Van, 2013. International Capital Flows under Dispersed Private Information 1. Geneva.