Mathematical and Statistical Methods for Actuarial Sciences and Finance

Checking financial markets via Benford’s law: the S&P 500 case 97

looking at chi-square as a distance, the empirical probability distributions are closer to
Benford’s law than to the uniform probability distribution. In this sense we agree with
Ley (see [8]) claiming that the distributions of the first significant digit of prices and
returns essentially follow Benford’s law. In other terms, the S&P 500 stock market
behaviour as a whole in the period August 14, 1995 to October 17, 2007 can be
considered as “ordinary”.
Finally, we observe that the empirical probability distribution related to returns is
significantly closer to Benford’s law than the empirical probability distributionrelated
to prices. In particular, the latter is 19.77 times further away from Benford’s law than
the former. This evidence is theoretically coherent with that stated in the paper of
Pietroneroet al.(see [11]), since logarithmic returns are obtained from prices by a
multiplicative process.

4.2 Day-by-day analysis

Here, we address our attention to returns since their empirical probability distribution
is closer to Benford’s law than that of prices. We day-by-day perform the same kind
of analysis considered in the previous subsection, but only with respect to Benford’s
law.
Over the investigated 3067 days, the null is rejected 1371 times, i.e., in about
44.70% of cases. In Figure 2 we represent the values of the day-by-day calculated chi-
square goodness-of-fit tests (the horizontal white line indicates the value ofχ 82 , 0. 05 ).

Day-by-day analysis with respect to returns: 14 August, 1995 – 17 October, 2007

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Computed chi-square

Fig. 2.Day-by-day calculated chi-square