Benford’s laws tests on S&P500 daily closing values and the corresponding daily log-returns both point to huge non-conformity

The so-called Benford’s laws are of frequent use to detect anomalies and regularities in data sets, particularly in election results and financial statements. However, primary financial market indices have not been much studied, if studied at all, within such a perspective.

This paper presents features in the distributions of S&P500 daily closing values and the corresponding daily log-returns over a long time interval, [03/01/1950 - 22/08/2014], amounting to 16265 data points. We address the frequencies of the first, second, and first two significant digits and explore the conformance to Benford’s laws of these distributions at five different (equal size) levels of disaggregation. The log-returns are studied for either positive or negative cases. The results for the S&P500 daily closing values are showing a remarkable lack of conformity, whatever the different levels of disaggregation. The causes of this non-conformity are discussed, pointing to the danger in taking Benford’s laws for granted in large databases, whence drawing “definite conclusions”. The agreements with Benford’s laws are much better for the log-returns. Such a disparity in agreements finds an explanation in the data set itself: the index’s inherent trends. To further validate this, daily returns have been simulated via the Geometric Brownian Motion and calibrating the simulations with the observed data averages and testing against Benford’s laws when the log-returns distribution’s standard deviation changes. One finds that the trend and the standard deviation of the distributions are relevant parameters in concluding about conformity with Benford’s laws.