Introduction
In this assignment, we will discuss and describe the application of statistics and probability theory to a real world problem. Our task is to evaluate and summarize the performance of a given stock (y). This report consists of the four parts. The first part is descriptive statistics of the given data between the two time periods and between the given stock and the Standard & Poor’s 500 index. Next, a hypothesis test will be performed in order to examine the performance of stock (y). The third part of the report includes Capital Asset Pricing Model estimation by the mean of linear regression analysis. Finally, the conclusions and recommendations are given in a short ending paragraph.
Descriptive Statistics
The descriptive statistics is calculated for the variable “dly”. This statistics includes measures of central tendency and measures of variability. The results include mean, standard deviation, variance, skewness, kurtosis and 1-99% percentiles for two time groups:
. by time_dum, sort : summarize dly, detail
-> time_dum = 0
dly
Percentiles Smallest
1% -.148241 -.1626573
5% -.0604362 -.148241
10% -.0196972 -.1480532 Obs 126
25% -.0080602 -.1341331 Sum of Wgt. 126
50% -.0002813 Mean .0010931
Largest Std. Dev. .0430453
75% .011327 .0974424
90% .0385268 .1364441 Variance .0018529
95% .0671248 .1479988 Skewness -.3312571
99% .1479988 .1529715 Kurtosis 8.913084
-> time_dum = 1
dly
Percentiles Smallest
1% -.0976059 -.1123998
5% -.0641885 -.0976059
10% -.0334775 -.0847702 Obs 130
25% -.0102179 -.0745916 Sum of Wgt. 130
50% -.0011226 Mean -.0004958
Largest Std. Dev. .0333125
75% .0100777 .0745916
90% .0267828 .0988824 Variance .0011097
95% .0616145 .1014411 Skewness .1647289
99% .1014411 .1249423 Kurtosis 6.307856
The descriptive statistics indicated that the average return on stock y was changed from 0.19% in the first half of the observations to -0.004% in the second half of the observations. The standard deviation in the first half is 4.3% compared to 3.33% in the second half. This indicates that the relative risk of the stocks was lowered in the second half. Generally, the stock is quite risky, because the ratio between the mean and standard deviation (coefficient of variation) is very high. The data is approximately symmetric in both time periods, because skewness is relatively close to 0.
Hypothesis Tests
In the previous paragraph, we indicated that the return has been reduced from positive value in the first half of the data to the negative value in the second half. In other words, on average, the share became unprofitable. In this part of the report, we want to examine the significance of this change. Research question: Has the average return of the stock dropped significantly from the first half to the second half of observations?
Null hypothesis: there is no significant difference in the average stock returns between the first and the second halves.
Alternative hypothesis: there is a significant difference in the average stock returns between the first and the second halves.
Since the task is to compare mean values of the two returns of the same stock, Student’s t-test with equal variances has been used.
. ttest dly, by(time_dum)
Two-sample t test with equal variances
Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
0 | 126 .0010931 .0038348 .0430453 -.0064965 .0086826
1 | 130 -.0004958 .0029217 .0333125 -.0062764 .0052849
---------+--------------------------------------------------------------------
combined | 256 .0002862 .0023965 .0383446 -.0044333 .0050058
---------+--------------------------------------------------------------------
diff | .0015888 .0048021 -.0078681 .0110457
diff = mean(0) - mean(1) t = 0.3309
Ho: diff = 0 degrees of freedom = 254
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Pr(T < t) = 0.6295 Pr(|T| > |t|) = 0.7410 Pr(T > t) = 0.3705
CAPM Estimation
The classic CAPM model has the following form (Investopedia, 2003):
ra=rf+βa(rm-rf)
Where rm is expected market return, rf - risk free rate, ra – return of the given security, βa- beta of the security. The CAPM model can be also expressed in terms of expected return (BAUER College of Business, 2016):
ERa=βaERm+rf
This equation has a form of the simple linear regression model. Note, that since we decided that the return on the stock y is approximately equal for the whole time period considered, we should not include the time_dum variable in the regression. Consider the return of stock y as a dependent variable and the return of S&P 500 as independent variable and run the regression procedure:
. regress dly dlsap
Source | SS df MS Number of obs = 256
-------------+------------------------------ F( 1, 254) = 10.06
Model | .014280307 1 .014280307 Prob > F = 0.0017
Residual | .360647987 254 .001419874 R-squared = 0.0381
-------------+------------------------------ Adj R-squared = 0.0343
Total | .374928293 255 .001470307 Root MSE = .03768
dly | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dlsap | -.2405554 .0758528 -3.17 0.002 -.3899359 -.091175
_cons | .0004666 .0023558 0.20 0.843 -.0041727 .0051059
The analysis of variance indicated the joint significance of the coefficients of the model (F=10.06, p=0.0017). The coefficient of determination shows that only 3.38% of variance in stock y is explained by this model. The regression equation has the following form:
dly=-0.2406*dlsap+0.0005
Conclusion and Limitations
In this paper, we have examined the performance of stock y. It is appeared that the average return of the stock is approximately equal for the two time periods considered. The stock is quite risky, because the variance of the return is very large compared to the value of the expected return. It was determined that the Standard and Poor’s 500 index has a significant impact on the y’ return. However, it explains only a little portion of its variance. There might be other factors that significantly impact the return of stock y and these factors were not included in this research.
This analysis is valid only if the assumptions for Student’s t-test and linear regression analysis are met.
References
Investopedia. (2003). Capital Asset Pricing Model (CAPM) Definition | Investopedia. [online] Available at: http://www.investopedia.com/terms/c/capm.asp [Accessed 28 Apr. 2016].
BAUER College of Business. (2016). CAPM as a Regression. [online] Available at: http://www.bauer.uh.edu/rsusmel/phd/lecture%208.pdf [Accessed 28 Apr. 2016].
