# S&P 500 monthly returns

The S&P 500 is a value weighted index published since 1957 of the prices of 500 large cap common stocks actively traded in the United States.Many index funds and exchange-traded funds attempt to replicate (before fees and expenses) the performance of the S&P 500 by holding the same stocks as the index, in the same proportions.

Theoretically, market returns are not serially correlated, and, thus, studying the past returns/prices can't help in predicting future returns. In this study, we'll examine this assumption using S&P 500 as a proxy.

### Objective

For our purpose, we try to model the S&P 500 returns using pure econometric techniques.

### Data

Source: Yahoo Finance.

Date Range: January 1980 to March 2009.

### Preliminary Analysis

Examining the graph for S&P 500 levels above, it becomes obvious that index levels follow a non-stationary process. To proceed with our analysis, we convert levels into logarithmic returns using $$r_{t} =\ln(\frac{P_{t}}{P_{t-1}}) = (1-L)\ln(P_{t})$$
Using NumXL and Excel built-in function, we can generate the required time series (i.e. DIFF(ln(G2:G711),1))

Next, we examine the series descriptive statistic and test the hypothesis whether the log returns are not serially correlated (i.e. white noise):

The log returns is not a white noise, and its probabilistic distribution exhibits fat tails than normal distribution.

 ACF PACF

### Conclusion

The preliminary analysis shows negligible serial correlation or ARCH effect. Furthermore, the returns probability distribution exhibits some excess kurtosis, but negligible skewness. In sum, the S&P 500 monthly returns appears as non-Gaussian white noise.