The formula to calculate the continuous compounded daily returns is as follows:

r^{t} = ln(P^{t} / P^{t}-1)

where:

- r
^{t}is the continuously compounded one period return - P
^{t}is the value of our index at time t - P
^{t}-1 is the value of the index at time t - 1 - ln is the natural log function

Summary statistics give us a quick look at the data. From calculating these numbers, we found that both indexes have a mean and median close to zero. While the SSE appears more volatile than the S&P 500, the latter index had the largest one-day gains and losses. In addition to the summary statistics, we also tested the returns for normality, as the correlation calculation works only with normally distributed variables. We also calculated the correlation coefficient between them.

Summary statistics (returns)

S&P 500 SSE-0.14 -0.13

Min

Mean-0.00031 -0.00007

Median0.00082 0.0014

Max0.11 0.09

Std Dev0.018 0.023

The correlation coefficient between SSE and S&P 500 returns is 0.081. A plot of SSE returns vs. S&P 500 returns confirms the small correlation between them (see “Plotted,” below).

The next step is to test for autocorrelation. Autocorrelation searches for relationships within data that have been shifted through time. Lagging one index and comparing it to itself shows the SSE and the S&P have little or no internal autocorrelation, but a lag of one day of just the S&P returns results in a correlation coefficient of 0.1726, which is significant, and more than twice the size of the standard correlation of 0.081 (see “Time shifts,” below).

These results raise many new questions that deserve being explored:

- If the lag of S&P 500 results in a significant correlation on a time window of over three years, what will the values be of the correlations on windows that are smaller: one year, one month or five days?
- If we find a high correlation, does it hold for at least two days?
- If it holds for at least two days can we take advantage of it?

**Testing the lag**

We calculated, first, the yearly correlation between the returns of S&P 500 and the one-day lagged returns of the SSE. The results were comparable to the three-year period. In 2007, the correlation was 0.164. In 2008, the correlation was 0.161. In 2009, the correlation was 0.183. We also calculated the monthly correlations. The monthly correlations ranged from -0.31 to 0.71 with the mean around 0.166, median around 0.173 and 43 observations.

We can pinpoint three interesting observations: The correlation is not uniformly weak through time -- it goes from weak to strong; the mean is similar in the three-year and one-year values and the values of the correlation tilt toward the positive (see “Skewed,” below).

These results encourage the exploration of whether the five-day correlations will provide further findings.