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.

所谓的本福德定律经常用于检测数据集中的异常和规律性，特别是在选举结果和财务报表中。但是，从这种角度看，如果对基础金融市场指数进行了研究，甚至还没有进行太多研究。

本文介绍了S＆P500每日收盘价的分布特征以及相应的长时间间隔[03/01/1950-22/08/2014]的每日对数回报，总计16265个数据点。我们处理了第一，第二和前两个有效数字的频率，并在五个不同（相等大小）的分解级别上探索了这些分布与本福德定律的一致性。对正数或负数情况下的对数返回都进行了研究。标准普尔500每日收盘价的结果表明，无论不同的分解水平如何，都严重缺乏一致性。讨论了这种不符合的原因，指出了在大型数据库中将本福德定律视为理所当然的危险，从而得出“确定的结论”。与本福德定律达成的协议对于返回原木要好得多。协议之间的这种差异在数据集本身中找到了一种解释：指数的内在趋势。为了进一步验证这一点，已经通过几何布朗运动模拟了日收益率，并使用观察到的数据平均值对模拟进行了校准，并在对数收益率分布的标准偏差发生变化时根据本福德定律进行了测试。人们发现，分布的趋势和标准偏差是结论是否符合本福德定律的相关参数。通过几何布朗运动模拟了日收益，并使用观察到的数据平均值对模拟进行了校准，并在对数收益分布的标准偏差发生变化时根据本福德定律进行了测试。人们发现，分布的趋势和标准偏差是结论是否符合本福德定律的相关参数。通过几何布朗运动模拟了日收益，并使用观察到的数据平均值对模拟进行了校准，并在对数收益分布的标准偏差发生变化时根据本福德定律进行了测试。人们发现，分布的趋势和标准偏差是结论是否符合本福德定律的相关参数。