94 M. Corazza, A. Ellero and A. Zorzi
guess that the first significant digit of financial prices/returns follows Benford’s law
in the case of ordinary working of the considered financial markets, and that it does
not follow such a distribution in anomalous situations.
The remainder of this paper is organised as follows. In the next section we provide
a brief introduction to Benford’s law and the intuitions underlying our approach. In
section 3 we present a short review of its main financial applications. In section
4 we detail our methodology of investigation and give the results coming from its
application to the S&P 500 stock market. In the last section we provide some final
remarks and some cues for future researches.
2 Benford’s law: an introduction
Originally, Benford’s law was detected as empirical evidence. In fact, some scientists
noticed that, for extensive collections of heterogeneous numerical data expressed in
decimal form, the frequency of numbers which havedas the first significant digit,
withd=1, 2,...,9,wasnot1/9 as one would expect, but strictly decreases asd
increases; it was about 0.301 ifd=1, about 0.176 ifd=2,..., about 0.051 ifd= 8
and about 0.046 ifd=9. As a consequence, the frequency of numerical data with the
first significant digit equal to 1, 2 or 3 appeared to be about 60%. The first observation
of this phenomenon traces back to Newcomb in1881 (see [9]), but a more precise
description of it was given by Benford in 1938 (see [2]). After the investigation of
a huge quantity of heterogeneous numerical data,^2 Benford guessed the following
general formula for the probability that the first significant digit equalsd:
Pr(first significant digit=d)=log 10(
1 +
1
d)
, d= 1 ,..., 9.This formula is now called Benford’s law.
Only in more recent times the Benford’s law obtained well posed theoretical
foundations. Likely, the two most common explanations for the emergence of prob-
ability distributions which follow Benford’s law are linked to scale invariance and
multiplicative processes (see [11] and [6]).^3 With attention to the latter explanation –
which is of interest for our approach – and without going into technical details, Hill
proved, under fairly general conditions, using random probability measures, thatif
[probability]distributions are selected at random and random samples are taken
from each of these distributions, the significant digits of the combined sample will
converge to Benford distribution(see [6]). This statement offers the basis for the
main intuition underlying our paper. In fact, we consider the stocks of the S&P 500
market as the randomly selected probability distributions, and the prices/returns of
each of these different assets as the generated random samples. The first significant
(^2) For instance, lake surface areas, river lengths, compounds molecular weights, street address
numbers and so on.
(^3) In other studies it has been proved that also powers of [0,1]-uniform probability distribution
asymptotically satisfy Benford’s law (see [1] and [7]).