Analysis of the performance of stocks is essential

In order to make the right decision in the investment of stocks, analysis of the performance of stocks is essential. The two factors which are essential in stocks are the risk factors and return on stocks. In the present report we analyse the performance of stocks of Boeing and IBM for the period starting 1/12/2010 and ending at 31/05/2016. The closing prices of the stocks have been obtained from yahoo finance. In addition, the prices of S&P 500 index and US TN have also been used for the analysis.

The analysis in the report provides information for the stated period based on which investment decision can be made.

Herein we present the time line charts of prices of stocks

Figure 1: Comparison of prices of Stocks

Figure 3: Closing stock prices of Boeing

Figure 4: Closing stock prices of IBM               

From the above chart it is clear that the prices of S&P is way above the prices of Boeing and IBM during the same period. In addition, it is also seen that the prices of Boeing show a steady growth during the period while the prices of IBM initially rose and then fell during the period. Moreover, the prices of both IBM and Boeing are very close to each other.

Table 1: Statistics for the returns on stock prices of Boeing and IBM

Statistics	BA	IBM
Mean	1.0140	0.0715
Standard Error	0.7427	0.6267
Median	1.1997	-0.0741
Mode	#N/A	#N/A
Standard Deviation	5.9877	5.0523
Sample Variance	35.8520	25.5255
Kurtosis	0.6987	0.6553
Skewness	-0.4699	-0.1368
Range	30.8197	28.8656
Minimum	-18.5328	-14.3826
Maximum	12.2870	14.4829
Sum	65.9092	4.6464
Count	65	65

From the above table it is found that the average returns on the stock prices of Boeing during the period of 1/12/2010 to 31/05/2016 is higher than the returns on IBM. However, it is found that the risk associated with the stock prices of IBM is lower than that of Boeings. The higher average return associated with the stocks of Boeing can be attributed to the higher risks for the prices of the organizations.

Hence it is found that the stock prices of IBM is relatively riskier than that of Boeings.

In order to further analyse the data a basic requirement is the presence of normality.  The requirement of normality of the data checks whether the data follows a normal distribution. The Jarque-Bera test is used to test for normality of the data. Skewness and Kurtosis are two important statistical features which are used to calculate the Jarque-Bera Statistics.

The formula for Jarque-Bera statistics is:

The Jarque-Bera statistics follows a Ch-square distribution with 2 degrees of Freedom.

Organization	Skewness	kurtosis	Count
BA	-0.4699	0.6987	65
IBM	-0.1368	0.6553	65

From the above calculations the Jarque-Bera statistics for Boeing:

Time line charts of prices of stocks

Since the p-value is less than 0.05 hence it can be inferred that the data for IBM follows a normal distribution. 

Thus it is found that the data for both Boeing and IBM follows normal distribution. Hence the returns on the closing stock prices can be used for further calculations.     

To test if the average return of the stock prices of Boeing is at least 3% the one-sample t-test is used. The one-sample t-test is used since the test is used to assess how a sample data compares with a given value. In the present condition we have to compare if the average return of the stock prices of Boing is less than or greater than 3%.

Null hypothesis: The average return of the stock prices of Boeing is less than 3%

Alternate hypothesis: The average return of the stock prices of Boeing is at least 3%

Statistics	Value
count	65
mean	1.0140
standard deviation	5.9877
standard error	0.7427
Hypothesized mean	0.03
a	0.05
Tails	1
df	64
t stat	1.3249
p value	0.0950
t crit	0.0630
sig	No

The t-statistics is calculated as :

Since the t-stat is more than the critical value hence we do not have sufficient evidence to reject the Null Hypothesis. Thus, it is found that the average return of the stock prices of Boeing is less than 3%  

In order to compare the risks associated with both the stocks independent sample t-test assuming unequal variance was used.

Null Hypothesis: The risk on stock prices of both the stocks are similar

Alternate Hypothesis: The risks on stock prices of both the stocks are similar

t-Test: Two-Sample Assuming Unequal Variances
	BA	IBM
Mean	1.0140	0.0715
Variance	35.8520	25.5255
Observations	65	65
Hypothesized Mean Difference	0
df	124
t Stat	0.9699
P(T<=t) one-tail	0.1670
t Critical one-tail	1.6572
P(T<=t) two-tail	0.3340
t Critical two-tail	1.9793

From the above test we find that at 0.05 level of significance p-value is 0.3340. Since the p-value is more than the level of significance hence we do not reject Null Hypothesis.

Thus the risks associated on both the stocks is similar.

In order to compare the risks associated with both the stocks independent sample t-test assuming unequal variance was used.

Null Hypothesis: The average return on stock prices of both the stocks are similar

Alternate Hypothesis: The average return on stock prices of both the stocks are similar

t-Test: Two-Sample Assuming Equal Variances
	BA	IBM
Mean	1.0140	0.0715
Variance	35.8520	25.5255
Observations	65	65
Pooled Variance	30.6887
Hypothesized Mean Difference	0
Df	128
t Stat	0.9699
P(T<=t) one-tail	0.1670
t Critical one-tail	1.6568
P(T<=t) two-tail	0.3339
t Critical two-tail	1.9787

From the above test we find that at 0.05 level of significance p-value is 0.3339. Since the p-value is more than the level of significance hence we do not reject Null Hypothesis.

Thus the risks associated on both the stocks is similar.

However, since the risk of IBM is lower than that of Boeing hence we would choose IBM for further analysis.

The excess return and excess return for IBM is calculated using the stated formula.           

CAPM has been calculated using the regression model.

From the above regression analysis, we find that the coefficient for market return of Boeing is 1.1193 and for IBM is 0.7204. Thus, we can say that the volatility of Boeing is 111.93% as against 72.04% for IBM. Since the market returns for IBM is less volatile hence the stock prices are less risky and thus may be more profitable.

From the above tables we find that R² for Boeing is 0.4048 and for IBM is 0.2380. R2 defines the relation of excess market return of the stocks with excess return. From the values of R² we find that the correlation of the returns for Boeing is 40.48% and for IBM is 23.80%. Thus, the prices of Boeing is more closely associated with the market returns.

Hence it would be wise to invest in the stocks of IBM

At 0.05 level of significance and 63 degrees of freedom the critical value is 1.9983.

For IBM:

Coefficient = 0.7204

Standard Error = 0.1624

Thus the confidence interval:  à 0.3959, 1.0450

Thus the 95% confidence interval for the coefficient of the market return of IBM is 0.3959, 1.0450.

Statistics	Values
Mean	0.0715
Standard Deviation	5.0523
Count	65
Standard Error	0.6267
confidence Level	95%
Lower critical value	-1.9600
Upper critical value	1.9600
Margin of Error +/-	1.2282
Confidence Interval Lower Limit	-1.1567
Confidence Interval Upper Limit	1.2997

 The 95% confidence interval for IBM is -1.1567, 1.2297. Hence, it is reasonable to suggest that the prices of IBM are within the interval at 5% risk. In other words, we can say that when stock prices of another period is taken then there is a 95% chance that the returns would lie between -1.1567 and 1.2297.

	BA	IBM
Skewness	0.0927	-0.3866
Kurtosis	0.8889	1.7733
Jarque-Bera	12.1631	5.6947
χ2(0.05,2)	0.0023	0.0580

The distribution of the error term was analysed using the Jarque-Bera test. From the test results it is found that while the error term of the market return for Boeing is normally distributed, the error term for IBM is not normally distributed.