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 low	Small med	Small high	Big low	Big med	Big high
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.

1. 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).

1. 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.

1. 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.

1. 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.

1. 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.