Checking financial markets via Benford’s law: the S&P 500 case 95digit of such prices/returns should follow the Benford distribution. But, if some ex-
ceptional event affects these stocks in ways similar among them, the corresponding
asset prices/returns could be considered “less random” than that stated in [6], and the
probability distribution of their first significant digit should depart from the Benford
one. In this sense the fitting, or not, to Benford’s law provides an indication of the
ordinary working, or not, of the corresponding financial market.
3 Benford’s law: financial applications
Investigations similar to ours have been sketched in a short paper by Ley (see [8]),
which studied daily returns of the Dow Jones Industrial Average (DJIA) Index from
1900 to 1993 and of the Standard and Poor’s (S&P) Index from 1926 to 1993. The
author found that the distribution of the first significant digit of the returns roughly
follows Benford’s law. Similar results have been obtained for stock prices on single
trading days by Zhipeng et al. (see [12]).
An idea analogous to the one traced in the previous section, namely that the
detection of a shunt from Benford’s law might be a symptom of data manipulation,
has been used in tax-fraud detection by Nigrini (see [10]), and in fraudulent economic
and scientific data by Gunnel et al. and by Diekmann, respectively (see [5] and [4]). ̈
Benford’s law has been used also to discuss tests concerning the presence of
“psychological barriers” and of “resistance levels” in stock markets. In particular
De Ceuster et al. (see [3]) claimed that differences of the distribution of digits from
uniformity are a natural phenomenon; as a consequence they found no support for the
psychological barriers hypothesis.
All these different financial applications support the idea that in financial markets
that are not “altered”, Benford’s law holds.
4 Do the S&P 500’s stocks satisfy Benford’s law?
The data set we consider consists of 3067 daily close prices and 3067 daily close
logarithmic returns for 361 stocks belonging to the S&P 500 market,^4 from August
14, 1995 to October 17, 2007. The analysis we perform proceeds along three steps:
- in the first one we investigate the overall probability distribution of the first sig-
nificant digit both on the whole data set of prices and on the whole data set of
returns; - in the second step we study the day-by-day distribution of the first significant digit
of returns; - finally, in the third step we analyse the sequences of consecutive days in which
the distribution of the first significant digit of returns does not follow Benford’s
law, i.e., the consecutive days in which anomalous behaviours happen.
(^4) In this analysis we take into account only the S&P 500 stocks that are listed for each of the
days belonging to the investigation period.