Comparing Monthly Returns of General Electric to the Returns of the S&P500 Index

Cole DavidsonOctober 7th, 2015

This paper will test the hypothesis that the year over end of month returns of General Electric Company from January 2005 through July 2016 equals the returns of the S&P500 Index over the same period of time.

1 DataThe end of the month price data are from Yahoo Finance. The year over year returns are computed as follows:

r= 100*ln(Price[t]/Price[t-12])

seeing Price[t] is either the end of month stock price of General Electric or the S&P500 Index value at time t and Price[t-12] is the price or index value 12 months earlier. With the year over year returns being used, the first year of data is lost so the returns data may may be available from January 2006 through July of 2015 for a total of 115 observations.

The returns are plotted in the following time series graph.

-80

-60

-40

-20

0

20

40

60

80

100

120

140

2006 2008 2010 2012 2014 2016

GenElectricRRSP500RR

Figure 1: Year over year percent returns of General Electric and the S&P500 Index. From January 2006 through July 2015.

The plot shows that the both returns move together and have a positive correlation. The simple correlation between the returns of General Electric and the S&P500 Index is .6713 and the t-value for the two-tailed null hypothesis is that this correlation is zero is t(128)= 10.25, with a two-tailed p-value of 0.0000. It can be confirmed that the returns are indeed quite positively correlated.

The sample means and standard deviations of the returns of data are listed in Table 1.

Table 1: Sample means and standard deviations for the year over year percent returns of General Electric and the S&P500 Index from 1/2005 through 7/2015.

Summary Statistics Mean Standard DeviationRGE 20.689 5.571RSP 1408.400 327.34

2 Hypothesis TestsThe results of the t-tests for the two tailed hypothesis:Ho : RGE = 0 versus Ha : RGE 6(does not)= 0 are: Null hypothesis: population mean = 0Sample size: n = 115Sample mean = 20.69, std. deviation = 5.57Test statistic: t(114) = (20.69 - 0)/0.519405 = 39.834Two-tailed p-value = 9.647e-69(one-tailed = 4.824e-69)

It can shown that the returns of General Electric are very different from zero over this time period. Along with that finding, the returns of S&P 500 Index are easily comparable too. The results for this is as follows: Null hypothesis: population mean = 0Sample size: n = 115Sample mean = 1408.42, std. deviation = 327.344Test statistic: t(114) = (1408.42 - 0)/30.525 = 46.1397Two-tailed p-value = 1.353e-75(one-tailed = 6.766e-76)

Since the both firms have very different standard deviations, the test will be tried assuming different variances for each sample. This assumption can be confirmed if done by the “2 variances test in gretl. Here is the unpooled test given by the Welch-Satterthwaite equation:

d.f. = (s ^2 1/n1 + s ^2 2/n2)^ 2 (s^ 2 1 /n1)^ 2/(n1 − 1) + (s^ 2 2 /n2) ^2/(n2 − 1)

where s squared and n= 1,2, are the sample variances and sample variances and sample sizes of the two samples in the test the results were:

Ho : RGE = RSP versus Ha : RGE 6(does not)= RSP

Null hypothesis: population variance = 6Sample size: n = 130Sample variance = 107154Test statistic: chi-square(129) = 129 * 107154/6 = 2.30382e+06Two-tailed p-value = 0(one-tailed = 0)

Where the standard deviation of RGE- RSP is equal to:

Square root of: 5.57^2/115 + 327.34^2/115= 30.53

3 SummaryNow, I have mad a successful computation of the year over year percentage returns of the end of month stock price for General Electric and the S&P500 Index from 1/2005 to 7/2015 for a totally of 115 monthly observations. The returns do display a strong correlation, yet the S&P500 Index’s returns suggest that they are way less volatile than General Electric’s returns. But this could be because your returns from year over year accrue much less risk than month over month returns. However, it was also discovered that General Electric and the S&P Index were significantly different through this given timeline.

It can be seen that neither firm was normally distributed during this period of time. Unfortunately, this means the t-test are not highly consistent. But with 115 observations, it can easily be drawn to the central limit theorem and to realize that the test statistics are normally distributed.