Ex2 Estimating and Testing the Capital Asset Pricing Model
Preamble
In this exercise, I use monthly data and ordinary least squares (OLS) method to estimate the ‘alphas’
‘ betas’ and ‘least square residuals’ of two stocks (equities)—Mobil and Tandy separately. According to
the results from TSM, I can calculate the 95% confidence intervals for β, use the t-test or p-values to test
the significance of α and t-test to test the null hypotheses that β=1 against the alternative hypothesis β>1,
and find the coefficient of determination and the standard deviation of the residuals to measure the
goodness of fit of regression, in the other word in this exercise, the proportion of the risk attributable to
the market and the individual risk of the stocks. To check the stability of the models over the full 10-year
period of the sample, I use the method of forecast adequacy test (Chow stability test 2). The strict form of
the CAPM assumes that markets are efficient. This assumption can be tested by including macro variables
in the regression—the rate of inflation, the growth in industrial production, and changes in the real oil
price (the Arbitrage Pricing Model). I use F-test to test the joint significance of these macro variables.
𝐸(𝑟𝑗𝑡 )−𝑟𝑓𝑡
Finally, the values of 𝐸(𝑟𝑚𝑡 ), 𝐸(𝑟𝑗𝑡 ) and 𝑟𝑓𝑡 is used to calculate the CAPM equation 𝛽𝑗 =
,
𝐸(𝑟𝑚𝑡 )−𝑟𝑓𝑡
which is compared with the OLS estimate of 𝛽𝑗 for the same time period.
Stock 1(Mobil)
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It can be seen from the two plots in the first line that the trend of the stock return followed that of
market return. In the special time October 1987 which witnessed the market crash, the return dropped
sharply. The scatter diagram of returns disregards the time ordering but represents the relationship
between market and stock apparently. The rate of change in stock return was bigger than that in market
return.
Estimate Std. Err. t Ratio p-Value
R-Squared = 0.3653
Intercept
0.00339 0.00914
0.37
0.712
Residual SD = 0.0686
MARKET
0.67838 0.11742
5.777
0
Chow Stability Test: ChiSq(2) = 0.6869{0.709}
Using ordinary least squares in TSM, I get the estimation of 𝛼̂=0.00339 and 𝛽̂ =0.67838 which means that
if the market return is equal to zero, the stock return will be 0.00339 and when the market return grows
every one unit, the stock return will increase by 0.67838 units accordingly.95% confidence intervals of β
is 𝛽̂ -1.96𝜎𝛽̂ ≤β≤𝛽̂ +1.96𝜎𝛽̂ . 𝜎𝛽̂ is in practice unknown, it is replaced by the estimate, S.E.(𝛽̂ )=0.11742 in
this model. According to my calculation, 95% confidence intervals of β is (0.448 , 0.909). To test the null
hypothesis, α=0, I use the t-test |
̂
𝛼
0.00339
|=|
̂)
𝑠.𝑒.(𝛼
|≈0.37 (t Ratio), which is less than the critical value (1.96)
0.00914
of 5% significance level from normal distribution (2-tailed test). I have no reason to reject the null
hypothesis, so α is equal to zero. I can draw the same conclusion from testing p-value of α, which is 0.712
more than 0.05. Another null hypothesis is β=1 against the alternative hypothesis β>1. On the basis of ttest,
̂ −𝛽∗ 0.67838−1
𝛽
̂ )= 0.11742 ≈-2.74<1.64
𝑠.𝑒.(𝛽
(1-tailed test), I cannot reject the null hypothesis that β is equal to one.
𝑅 2 is equal to 36.53% which means that 36.53% of the risk is attributable to the market. The standard
deviation of the OLS residual is 0.0686 which measures the individual risk of the stock. Through running
Chow stability test, I get the result that ChiSq(2)=0.6869{0.709}. P-value is equal to 0.709 which is more
than 0.05. I do not reject the null hypothesis. On the other word, the model is stable over the full 10-year
period of the sample. I change the Capital Asset Pricing Model (CAPM) instead of Arbitrage Pricing Model
(APM) by including macro variables in the regression. I use an F-test to decide whether the null
hypothesis that 𝛽2 =𝛽3 =𝛽4 =0 is reasonable. According to the result that p-value is equal to 0.697 which is
bigger than 0.05, I cannot reject the null hypothesis. So the error term 𝑢𝑗𝑡 depends solely on individual
firm effects, and is not predictable from macroeconomic variables.
Wald Test of Zero Restrictions on:
GIND
RINF
ROIL
F (3,115) = 0.4795 {0.697}
𝐸(𝑟 )−𝑟
The CAMP equation is 𝛽𝑗 =𝐸(𝑟 𝑗𝑡 )−𝑟𝑓𝑡 . Through the TSM, I get the sample means, and
𝑚𝑡
̅̅̅̅̅−𝑟
𝑟𝑗𝑓 ̅̅̅̅̅
𝑓𝑡
𝑓𝑡
0.0164667−0.0080517
calculate 𝛽𝑗 ≈̅̅̅̅̅−𝑟
=
≈0.74848.The OLS estimate of 𝛽̂𝑗 for the same time period is
̅̅̅̅̅ 0.0192833−0.00808517
𝑟
𝑚𝑡
𝑓𝑡
0.67838, which is near to the CAMP estimate of 𝛽𝑗 .
***Summary Statistics for MARKET***
Mean = 0.0192833
***Summary Statistics for MOBIL***
Mean = 0.0164667
***Summary Statistics for RKFREE***
Mean = 0.00808517
Stock 2(Tandy)
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It is apparent from the information given the market return and the stock return changed around the
value of zero. The trend of stock return usually followed that of market return, but the stock return of
Tandy had a larger fluctuation. In October 1987, the market experienced a crash, the market return
dropped down quickly, and so do the stock return of Tandy. The scatter plot shows the relationship
between market return and the stock return.
7
Estimate
Std. Err.
t Ratio
p-Value
R-Squared = 0.3056
Intercept
0.03084
0.01591
1.939
0.057
Residual SD= 0.1194
MARKET
1.03233
0.20433
5.052
0
Chow Stability Test: ChiSq(2) = 4.4871 {0.106}
The estimation of α and β is equal to 0.03084 and 1.03233 separately, which means that if the market
return grows every one unit, the stock return will increase by 1.03233 units, and if the market return is
zero, the stock return of Tandy will be 0.03084. In this model, S.E.(𝛽̂ ) is equal to0.20433, so 95%
confidence intervals of β is (0.632,1.433). On the basis of t-test, |
̂
𝛼
0.03084
|=|
̂)
𝑠.𝑒.(𝛼
0.01591
|≈1.939<1.96 (2-tailed
test), I cannot reject the null hypothesis that α is equal to zero. P-value test can also be used in this
hypothesis test. The value is 0.057, more than 0.05. I get the same conclusion from this result that α is
equal to zero. To test the whether β is equal to one or it is bigger than one, I calculate the t-value of it,
̂ −𝛽∗ 1.03233−1
𝛽
̂ )= 0.20433 ≈0.158<1.64
𝑠.𝑒.(𝛽
(1-tailed test), I have no reason to reject the null hypothesis 𝛽=1. The
goodness of fit of regression (𝑅 2) is 30.56%. The probability of risk attributable to the market is 30.56%.
The individual risk of the stock is measured by the standard deviation of the OLS residual which is 0.1194.
I use the Chow stability test to check whether the model is stable over the full 10-year period of the
sample. P-value is equal to 0.106 bigger than 0.05, therefore the model is stable. To test the assumption
that markets are efficient, macro variables (growth, inflation and oil price) is included. Though the Wald
test of zero restrictions on GIND, RINF and ROIL, I get the p-value (0.397) which is more than 0.05. The
assumption is acceptable.
Wald Test of Zero Restrictions on:
GIND
RINF
ROIL
F (3,115) = 0.9964 {0.397}
̅̅̅̅̅−𝑟
̅̅̅̅̅
𝑟
0.05075−0.0080517
𝑗𝑓
𝑓𝑡
In the model, I calculate the CAMP estimate of 𝛽𝑗 ≈̅̅̅̅̅−𝑟
=
≈3.80910. However the
̅̅̅̅̅ 0.0192833−0.00808517
𝑟
𝑚𝑡
𝑓𝑡
OLS estimate of 𝛽̂𝑗 for the same time period is 1.03233. There is a big difference between these two
values. OLS regression is estimated on the basis of a quadratic weighting approach that conflict the
assumption of risk aversion, and the data do not show a normal distribution, because the probability
distribution of market returns are with “fat” tails. Due to these factors, beta is sensitive to market
fluctuations, which may make OLS inappropriate for beta estimation.
***Summary Statistics for MARKET***
Mean = 0.0192833
***Summary Statistics for RKFREE***
Mean = 0.00808517
***Summary Statistics for TANDY***
Mean = 0.05075
Conclusion
In the Capital Asset Pricing Model (CAPM), 𝑟𝑗𝑡 =𝛼𝑗 +𝛽𝑗 𝑟𝑚𝑡 +𝑢𝑗𝑡 , 𝛼𝑗 and 𝛽𝑗 are constant coefficients. Usually,
𝛼𝑗 is equal to zero. Seeing from the tests done separately in the two stocks (Mobil and Tandy), I both
have no reason to reject the null hypothesis that α is equal to zero. 𝛽𝑗 is the basic indicator of the stock’s
risk and return characteristics. CAPM says that it is impossible for an asset to have a higher than average
return, or 𝛽𝑗 >1, without being relatively risky, and/or relatively highly correlated with the market. In the
CAPM of stock 1 and stock 2, the null hypothesis of 𝛽𝑗 =1 is not refutable. Through the project, I am
familiar with the Capital Asset Pricing Model (CAPM) and ordinary least squares (OLS) method. I know
how to do some estimates and tests, and check the stability of the models. I can analyse the stocks of the
companies with the conclusion drawn from TSM.