Analyzing Four Different Stock Closing Prices

Methodology

The prices of stocks keep on changing everyday depending on the market forces due to demand and supply. In this report, we present an analytic view of four different stock closing prices. The stocks prices are related to S & P 500 index, IBM stock price, US TN (10 year) and Boeing stock price. The aim is to choose the best stock out of the given four using certain criteria. The data used was obtained from the Yahoo Finance for the period spanning from 01/12/2010 to 31/05/2016 (a total of 65 months were included for analysis). Table 1 below presents the first 10 observations of the data.

Table 1: Sample portion of the data

Data was collected from the Yahoo Finance where a criteria was used for the range of data to be downloaded. The interest was on the closing prices where for each and every stock we had to download the data and then select the closing prices. Stock returns for each of the stocks was then computed using the following formula;

After computing the return rates for the four stocks, we then computed the excel returns on the stock that had the highest average rate of returns as follows;

The first analysis done was the descriptive statistics where we looked at the measures of central tendency (mean and median) as well as the measures of dispersion (standard deviation and range). The maximum and minimum returns are also presented.

Table 2: Descriptive statistics

 From table 2 (descriptive statistics table) above, we clearly see that Boeing stock returns was the highest (M = 1.014) and it was followed by that of S & P 500 index (M = 0.787). IBM had an average stock return of 0.071 while US TN had the lowest average stock returns (M = -0.906). In terms riskiness of the stocks, US TN is the riskiest stock while S & P 500 index is the least risky stock.

1. Line charts

The following plots shows the time series plots for the four different stocks. The first one is that of S&P 500 index, as can be seen, the plot shows a continuous increase in the closing prices.

For the case of IBM closing prices, there was a continuous increase in the closing prices till around the end of 2011 before which the prices began to fall again 

1. a) Returns computations

The returns for the different stocks were computed using the following formula;

Descriptive Statistics

We show the first returns for each of the stock, the rest of the values are shown in excel;

S & P 500 index; 

IBM Stock; 

US TN (10 year) stock;

Boeing stock; 

1. c) Jarque-Berra test of normality

We used Jarque-Bera test to test for the normality of the rate of returns for the IBM and for the Boeing stocks. Jarque–Bera test is simply a test for goodness-of-fit where it checks whether sample data have the kurtosis and skewness matching a normal distribution. The null hypothesis of the test is that the data is normally distributed. The results of the tests are given below; 

As can be seen, the p-value is 0.2104 (a value greater than α = 0.05), we fail to reject the null hypothesis and conclude that the Boeing stock returns are normally distributed at 5% level of significance.

We also tested the normality for the returns on IBM stock. Again, the returns were found to follow a normal distribution (p-value > 0.05).

In this section, we sought to test whether the average return on Boeing stock is greater than or equal to 3%. The tested hypothesis s; 

Results are presented in table 3 below;

              Table 3: T-test of Boeing Stock price

In table 3, we present the one-sample t-test where the aim was to test the claim that the average return on Boeing stock is greater than or equal to 3% , the p-value for the one-tailed is 0.094; this value is greater than α = 0.05. The null hypothesis cannot be rejected in this case hence we conclude that the mean average returns for Boeing stock is not at least 3% as is claimed.

In this section, the aim was to compare the risk associated with the returns on Boeing and the returns on IBM stocks. We used F-test to make a comparison of the two. The following was tested; 

The P value associated with the test statistics (F-test) is 0.0885. This value is greater than 5% level of significance. This means that we cannot reject the null hypothesis and by not rejecting the null hypothesis we conclude that the risk associated with the two stocks is not significantly different at 5% level of significance.   

Next, we sought to compare the average returns for the Boeing stock and that of the IBM stocks. Are the average stock returns for the two stocks the different? The following hypothesis was used to answer the research question presented;

Time Series Plots

Where,  is the average returns for Boeing stock while  is the average returns for the IBM stock.

Results from the two-sample t-test assuming equal variances revealed that the average returns for the two stocks are not significantly different (p-value > 0.05, two-tailed).  Though the average returns for the Boeing stock were slightly larger (M = 1.01, SD = 5.99) than the average returns for the IBM (M = 0.07, SD = 5.05), the differences were found to be insignificant at 5% level of significance.

Using a simple linear regression, we estimated the Capital Asset Pricing Model (CAPM) in which case the dependent variable was the excess return on Boeing stock and explanatory variable was the excess market return (computed as return on S&P 500 minus the risk fee rate). The results of the regression analysis are presented in the table below;

We first check for the goodness-of-fit of the model. We observe that the p-value of the F-statistics is 0.000 (a value less than 5% level of significance). We reject the null hypothesis and conclude that the model is significantly different from zero hence it is fit and appropriate.

1. b) Interpretation of Coefficients

In terms of the coefficients, we see that the coefficient for the independent variable is 1.0969. This value is positive implying a positive relationship between the dependent variable and the independent variable. The value further shows that a unit increase in the independent variable would result to an increase in the dependent by 1.0969. Likewise, a unit increase in the independent variable would result to a decrease in the dependent variable by 1.0969.

1. c) Interpretation of R²

The value of R-squared is 0.8026; this means that 80.26% of the variation in the dependent variable is explained by the independent variable in the model. A large percentage of the variation is explained within the model, this further points to the appropriateness of the model.

1. d) Interpretation of confidence interval for beta

Results also presented the 95% confidence interval. The results shows that we are 95% confident that the true population mean returns is between 1.00 and 1.26. This shows that the slope is statistically significant at 5% level of significance.  

We tested the neutrality of the stock returns (Boeing stock returns) using 95% confidence interval. In doing this, we aimed at testing the following hypothesis; 

The 95% confidence interval results shows that we are 95% confident that the true mean is between -.3879 and 3.7731. This range incorporates 1 and as such we can say that the hypothesis is insignificant (i.e. different from 1). By failing to reject the null hypothesis we conclude that the beta is not different from 1 hence the returns on the stock is equal to 1 hence the stock is neural.

The last test is regarding the normality of the error term. The hypothesis tested is;

H₀: The error term follows a normal distribution

H_A: The error term does not follow a normal distribution

The error term was stored as e after the regression and then the normality was tested using Shapiro-Wilk test. The p-value is 0.5963 (a value greater than α = 0.05), we fail to reject the null hypothesis and conclude that the error term is normally distributed at 5% level of significance. 

As can be seen, the above chart further shows that the data is normally distributed. 

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