In this particular project the strict model of CAPM is examined and try to understand if the equation manage to fully predict the beta of the stock. Two stocks are being examined and the assignment focus on the betas of the stocks Texaco and Public Service of New Hampshire.
The structure of the report is as follows: First the method of ordinary least squares is used to produce the estimated values of α,β. Second the results obtained are discussed, their significance is tested and their confidence interval calculated. The stability of the model and it's possibility to explain the returns of the stock are evaluate. Comparison is also being made between CAPM and the arbitrage model. Third, the shortcomings of the method are being reported and the conclusion is being drawn based on the reported results of the CAPM and its prediction ability in the stock price.
METHODOLOGY
The data used in the assignment are monthly stock returns of different industries the first one is TEXACO and the second Public Service of New Hampshire, the dates under examination are from January of 1978 up to December 1982. The New York Stock Exchange market is used as a market of reference. All the tests are executed in 5% significance level.
Using the methodology of ordinary least squares and the equation the estimated values of the intercept and the beta of each stock are obtained, the stands for the actual return of company's j stock in month t and is the market's return for the same month. Their confidence intervals are calculated. Moreover the hypothesis of heteroscedasticity is under test.
The explanation power of the regression on the stock returns is evaluated. In addition the indiosygratic risk of the stock is estimated throw the residual sum of squares.
The chow stability and chow forecast test are used to test the stability of the model over the long run and the forecast power of the regression. The chow stability test check for structural breaks in the sample period while the chow forecast test takes into account the previous period and try to forecast the stock prices in the future and then anticipate these results to the true results of the forecasted period.
Comparison is made between the CAPM and the APM model ,using the joint significance test to examine which is the most efficient in evaluating the stock price. The APM model test separately the significance of the macroeconomic variables which are inflation, growth in industrial production, changes in the real oil price. While the CAPM theory imply that that contains all the non asset specific information relevant to individual asset prices at date t.
At the end comparison is made between the outcome of the regression regarded the beta of the stock and the equation of the CAPM model.
The CAPM implies that all un-diversifiable risk is in the beta of the stock which is also the market risk while the diversifiable risk lies in the error term.
RESULTS
3.1 PLOTS
The plots below present the relation and the correlation of the two stocks and the market.
Texaco is not very volatile it moves with the market with the only few exceptions as in 09/1980 and the period between 03/1981 and 06/1981 where seems to move opposite from the market so the beta of the stock is expected to be between 0.5 and 1. Texaco has a pick point around November of 1980 and that maybe be due to the expectation of the investors prior to the accident in Lake Peigneur later the same month, which follows by a great fall of the stocks returns. The pick might also be an indicator of the acquisition of The Neches Butane Products Company from the Texaco. The market reach its down point between the period 13/02/1980 - 27/03/1980 due to the Hunt Silver Crisis and especially the day of the complete collapse was on 27/03/1980.
The Public Service of New Hampshire seems to be uncorrelated with the market, in some periods they seem to be negative correlated, for example between the 06/1980 and 05/1981 and in others positive correlated as in 05/1982 - 12/1982. In the first half of the sample size market is slightly overperfome the stock in average. There are not many pick points for PSNH, the only exception would be around May of 1980. The returns of the particular stock are not seems to be very volatile.
The upward slopping declares that the texaco is possitive correlated with the market, the same is valid for the case of PSNH below.
Significance of the estimated values
3.2.1 TEXACO
Dependent Variable is TEXACO
60 observations (25-84, dates 1978, Mth 1 to 1982, Mth 12) used for estimation.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept -0.00063 0.00883 -0.071 0.943
MARKET 0.64466 0.11336 5.687 0
Diagnostic Tests:
B-P Heterosced. (LM): ChiSq(1) = 0.0852 {0.77}
(appendix 1.1)
The βj in this case is 0.64466 that means that the stock is not relative risky neither its return is relatively high, the beta of the stock is less than one. At 64.5% the stock moves with the market for the specific period. The return of the stock is lower of the return of the market.
In addition the null hypothesis of no existence of heteroscedasticity is not rejected in 5% significance level.
The 95% confidence intervals is calculated for both α,β
For α
a-(1.96 * s.e.(a)) =< α =< a+(1.96 * s.e.(a))
-0.00063-(1.96*0.00883)=< α =< -0.00063+(1.96*0.00883)
-0.0179=< α =< 0.01668
And for β
b-(1.96 * s.e.(b)) =< β =< b+(1.96 * s.e.(b))
0.64466-(1,96*0.11336) =< β =< 0.64466+(1,96*0.11336)
0.42247 =< β =< 0.866846
The null hypothesis of Ho: α=0 is examined against the alternative H1: α=/ 0
It is a two tail test so the p-values are examined.
The p-value of α=0.943> 0.05 so for significance level 5% the null hypothesis is not rejected. The intercept is insignificant for the return of the stock.
In order to compare the return of the Texaco with the market return the null hypothesis Ho: β=1 is examined against the alternative β>1.
We perform an one tail test of linear restrictions.
We will reject the null hypothesis H0: β=1 if (b-β)/s.e.(b)> 1.64485
But (b-β)/s.e.(b)=(0.64466-1)/ 0.11336= -3.13462 <1.64485 so the null hypothesis is not rejected. The stock is has not higher return than the market portfolio return. The stock does not over perform the market.
PSNH
Dependent Variable is PSNH
60 observations (25-84, dates 1978, Mth 1 to 1982, Mth 12) used for estimation.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.00487 0.00597 0.816 0.418
MARKET 0.20971 0.07672 2.733 0.008
B-P Heterosced. (LM): ChiSq(1) = 0.3243 {0.569}
(Appendix 1.2)
The βj = 0.20971 declares an even less risky stock than this of Texaco. Only at a presentence of 20.97% the stocks moves with the market. A lower return is expected in relation with the market return. Neither in this case the null hypothesis of no heteroscedasticity is rejected. That means the variance of the us , which in the CAPM case is the idiosyncratic risk of the stock, is constant in each observation as defined by Dougherty (2002, pp220)
The confidence intervals for a is:
a-(1.96 * s.e.(a))=< α =< a+(1.96 * s.e.(a))
0.00487-(1.96 *0.00597) =< α =< 0.00487+(1.96 *0.00597)
-0.00683=< α=<0.016571
For b
b - (1.96 * s.e.(b))=< β =< b+ (1.96 * s.e.(b))
0.20971-(1.96 * 0.07672)=< β =< 0.20971+(1.96 *0.07672)
0.059339=< β =<0.36008
Τhe null hypothesis α=0, is not rejected, it is a two tail test so for the α the p-value is 0.418> 0.05, this also can be concluded from the confidence interval of α (-0.00683=< α=<0.016571)
The hypothesis test H0: β=1 against H1:β>1 is an one-tail test of linear restrictions. The t-statistics for the test is calculated using the formula [(b-β)/s.e.(b)]=> (0.20971-1)/0.07672=-10.301<1.64485 so the null is not rejected, neither PSNH has higher return than the market portfolio return.
4. R-squared, standard deviation of the residuals
Texaco
Residual SD = 0.0663 declares the difference between the estimated values received from the regression and the true values, in this case defines the risk which is uncorrelated with the market risk and represents the diversifiable risk of the stock.
R-Squared = 0.358 the goodness of fit, is the variance of the stock's price explained by the regression line as stated by Doughtery (2002, pp 66) Only the 35,8% of the movement of the stock is attributed to the volatility of the market.
PSNH
The residual standard deviation is lower (0.044) than the Texaco's . That suggests that PSNH is a less volatile stock since it is inherited less unsystematic risk. A presentence of 4.4% of risk can diversified away. This is also illustrated in the time plot.
The R-Squared is 11.1% which actually means that the movement of the market does not explain the movement of the stock, this can be also shown from the plot since the market and the stock seems to be pretty uncorrelated with each other. Such low r-squared proves that the risk of the stock depends mostly in the company's specific risk as mentioned by Berndt, E. R. (1996 pp 40).
5. Chow stability test
Texaco
Chow Stability Test: p-value=0.803 > 0.05 So for significance level 5% the null hypothesis that the model is stable over the 10 year period is not rejected. The hypothesis of parameter stability is not rejected.
Chow Forecast Test: ChiSq(60) = 69.3671 {0.191}> 0.05 We don't reject the null hypothesis of the variance of forecast error not to be significant different from the variance of fitted model. They have been drawn from the same distribution, with same mean and variance.
(Appendix 2.1)
PSNH
Chow Stability Test: ChiSq(2) = 1.4686 {0.48}
The p-value of chow stability is 0.48> 0.05. so for significance level 5% the null hypothesis that the model is stable over the long run period is not rejected.
(Appendix 2.2)
6. CAPM against Arbitrage model
Another model of same importance as the CAPM is the Arbitrage Pricing Theory The difference between them is that the APM considers , as stated by Cuthbertson(1996, pp 60),two essentials for the stock's return, the expected and the surprise element, which splits even more into general and specific news. The macroeconomic variables which are examined in this assignment consider as general news.
The new equation is:
rjt=aj+bj+b2 RINFt+b3 GINDt+b4 ROILt+ ujt
where: RINF is the rate of inflation, GIND is the growth in industrial production, ROIL is the real oil price. The new variables RINF, GIND and ROIL contain the percentage monthly changes in CPI, FRBIND and POIL/CPI respectively.
The null hypothesis here is H0:b2=b3=b4=0, for the test the whole sample size could be used, since is proved that the sample in both cases is stable over the long run, but here the test is executed for the period of interest.
Dependent Variable is TEXACO
60 observations (25-84, dates 1978, Mth 1 to 1982, Mth 12) used for estimation.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept -0.02099 0.01973 -1.064 0.292
MARKET 0.65739 0.1151 5.711 0
GIND 0.00331 0.00842 0.393 0.696
RINF 0.02815 0.02393 1.176 0.245
ROIL -0.00078 0.00207 -0.375 0.709
Wald Test of Zero Restrictions on:
GIND
RINF
ROIL
F(3,55) = 0.5975 {0.619}
(Appendix 3.1)
Dependent Variable is PSNH
60 observations (25-84, dates 1978, Mth 1 to 1982, Mth 12) used for estimation.
Estimation Method: Ordinary Least Squares
Estimate Std. Err. t Ratio p-Value
Intercept 0.02387 0.01302 1.833 0.072
MARKET 0.19683 0.07595 2.592 0.012
GIND -0.00568 0.00556 -1.021 0.312
RINF -0.02664 0.01579 -1.687 0.097
ROIL 0.00085 0.00137 0.62 0.538
Wald Test of Zero Restrictions on:
GIND
RINF
ROIL
F(3,55) = 1.5834 {0.204}
(Appendix 3.2)
It is apparent that for both Texaco and PSNH these macroeconomic variables are not important for the return of each stock. The joint significance test in both cases doesn't reject the null hypothesis. The CAPM and APM model are not consistent with each other according to Cuthbertson (1996, pp 67). That may reflect the fact that CAPM already has take into account for these macroeconomic variables in the return of the market portfolio.
Looking at the t-ratios of the variables GIND, RINF, ROIL the null hypothesis that they don't have explanatory power is not rejected. Each absolute value of the t-ratios is smaller than 1.96. But also their p-values are greater than 0.05.
7. Informal test on CAPM
Texaco
*** Summary Statistics for MARKET ***
Mean = 0.0192833
*** Summary Statistics for RKFREE ***
Mean = 0.00808517
*** Summary Statistics for TEXACO ***
Mean = 0.0118
(Appendix 4.1)
Bj=(0.0118-0.00808517)/( 0.0192833-0.00808517)= 0.3317366
Value of the beta from original regression 0.64466
PSNH
*** Summary Statistics for MARKET ***
Mean = 0.0192833
*** Summary Statistics for PSNH ***
Mean = 0.00891667
*** Summary Statistics for RKFREE ***
Mean = 0.00808517
